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M(*o) = V(,'>(*o),
»=0,l,...,n-l.
This lemma is a special case of a theorem which has been proved in [4]. Now we are going to define the notion, which already turned out to be a very useful tool in the theories of ordinary differential and functional inequalities and convex functions. DEFINITION 3 Let hypothesis H2 be fulfilled. Let us fix an arbitrary point We define the function R: D x R n - 1 —► R by the formula
(eJ.
*(*,vV,...,v"-,) = ?(0, where ip € F is the unique function satisfying the conditions V>W(x) = y'
for
i = 0 , 1 , . . . ,n - 1.
Let us notice that the function R is a "first integral" of the family F, because it is constant on every member
(0). Suppose also (i) Given « € C(7), initial value problem ">' = f(x,y,u(x,y),w) be a.c. with (0). Does (1)&(2) have an a.c. solution? 5. The General Goursat Problem The characteristic problem (l)fc(2) is "harmless" compared to problem (1)&(3) with {y) = —y and 0(y) = y, we get the functional equation e(x) = a'(x) + v(o(x)) a'(x) a.e. in Jx which may not be solvable in Ll(Ji). The only result we have is the "Satz" from [4] in case / is independent of (», to), which is sharp in some sense shown by counter-examples given there and reads as follows. Th«r>r»m 9. Let / ; J x JR. -» M be measurable in (x, y) € J, continuous in u € M and such that |/(x, y, u)| < M(1 + |u|) on J x R for some Af > 0. Let <*(•) and /?(•) be increasing with ct(0) = /?(0) = 0 such that the corresponding curves intersect only in (0,0), «(•) is Lipschitz, /?(•) is a.c. and /3(ct(-)) is Lipschitz of constant Jb < 1. Let 0, ^ < ip < TT/2. 4 2> where fC n Af- -. The linear operator M2 in (2.5) belongs to C(D;K) and satisfies M2 : Af -> K n AfL. § (2.5) and thus, due to the continuity of dF/dy, for all "usual" approximations w. Obviously, the constant K in (2.2) is a bound for the (ff_i — ► /f^-norm of L~l, if we choose the usual norm ||u||#» := ||Vu||i, in H^(il). In Section 4, we will show how the computation of K can be reduced to an eigenvalue problem. Suppose finally that some monotonically nondecreasing function G : [0, oo) —► [0, oo) has been computed such that ||f (•,« + u) - F{-,u) - c ■ till,., < GdlVulli,) for all u € J^(ft), + F(; u)ip] dx= fVu-V -VT) + F(;u)ri]dx for all i; e H°(Sl) (3.4) n n which exists due to the Lax-Milgram-Lemma. (3.4) provides, for q := Vw — Vtp, {divq)[r)] = - fq-Vridx= f F(-,w)r) dx for all rj € H°(tt), a n i.e., div q = F(-,w). Furthermore we derive, inserting IJ := (t) - v(t)) = 2e 0 on (?„ since ip = 0 is incompatible with the boundary conditions. Hence u < u*(T). This argument, applied to G, 0 {(t,x) : 0 < t < to}, gives the full assertion. The proof of (c) and (d) can be found in [12]. ■ These statements are only interesting for the increasing problem. In the decreasing case we usually have u'(t) = uo(0) and M = {0}. Theorem 3. Suppose uo(0) > u0(x) on [0,s o ]. Let a'(t) := max{a(r) : 0 < r < t} and a > 0. Then every solution of (P) satisfies u(t,x) <-a"(t) a (u - |v|) < pz^'V 0 on r„ U T,. Now (MP) implies 0 on (?«,. From u > 0 on G, follows o-(t) < s(t) since u > 0 on G,. ■ Remark. Theorem and proof remain true if u and v satisfy the differential inequa lities u r z — u< < 6V on G, and vxx — vt> S\v\p on G„ with 6 > 0. Theorem 9 (Uniqueness). £e< (u,s) and (v,a) be two [in the decreasing case: smooth] solutions o/(P). Then u = v holds. Remark. The definition of a solution implies u > 0 on G. and v > 0 on Ga. Proof. We apply the previous theorem to (u, v) and (v, u) and obtain v < u and o- < s and vice versa u < v and s < c ■
a.e. on J,, w(0) = V'(y)
(14)
has a unique solution, for almost all y 6 /». Then (8)fc(9) has an a.c. solution. Notice that we get the same limit w for almost all y € «/», hence (13) and (8), since we may pass to the limit also inside (11). Concerning this uniqueness assumption it is worth to recall that there are not only continuous g : JR? -» R which are not locally Lipschitz but such that through every point of JR.2 there exists a unique solution of z' = g(x,z), see e.g. §585 in [1], but also continuous g such that there exist infinitely many solutions through every point of JR3, see e.g. Chap II.5 of [9]. Without uniqueness there is just one simple case, covering also the situation of Example 1 ( / independent of u), namely Proposition 2. Let / , tp and V> be as in Proposition 1, but instead of (i) assume (ii) / is increasing in u. Then (8)fc(9) has an a.c. solution. Proof. For simplicity, let / be continuous and bounded by M, and tp = i> = 0. Let tto = Mxy and
*o(x,») = y /((,v,uo(f,v).«i>o(£,v))<%,
Mx
144 we get u>i < Wo. and therefore «>n(x,y) = / f(t,y,Un(t,v),*>„({,y))d{ Jo
and u B+1 (x,y) = / wn(x,rj)dt) Jo
for n > 0
generates a decreasing sequence (un,tD„), the limit « = lim un of which is an a.c. solution fl—»00
such that u_ = lim tD„.
O
n—»oo
Since we don't know whether there is a counter-example, let us state Problem 1. Let / : J x 2R7 -* ]R be continuous (or measurable in (x,y) and continuous in (u,u>)) and satisfy (7), y> and i/> be a.c. (or continuously differentiable) with
(15)
has at most one solution w £ C(7i). (iv) For almost all x € Ji, given u £ C(Jj) and w 6 Ll(J3), initial value problem «' = f(x,V,u(y),v,w(y))
a.e. on J2, v(0) = v?'(x)
(16)
has at most one a.c. solution v. Then (l)fc(2) has an a.c. solution. This is a special case of Theorem 1 in §2 of [2]. The meaning of (15) is: given that the limit exists and w € C(Jj) satisfies the same equation, possibly with the limit for a subsequence of («„), then w = w. Notice that for v„ = v this implies unique solvability of (6), given u and v. Then (15) is good enough for application of an old compactness criterion of M. Frechet for convergence in measure, and this yields relative compactness in C\(J) of the set
145 of these solutions to as (u, t>) varies in an appropriate subset of C(J) x Cj(J). Therefore it is clear that we need no limit in (16) if we consider only such w. It is easy to see that the standard uniqueness conditions w.r. to (v,w), see e.g. §21 in Walter [12], imply (15) and (16). Example 2. Consider the special case «»» = fo(x,y,u,)itt
a.e. in J,
u(x,0) = (p(x) in Jlt
u(0,y) = i>(y) in J ,
(17)
where / 0 : J x JR —» M is continuous and bounded by M 0 , y> and V" are C" (or a.c). So, we have f(x,y, u,v,v>) = fo(x,y,w)v and may assume that / is bounded by M > 0, since we get a priori bounds and may replace v by g(v) with g(p) = p for \p\ < r and g{p) = rsgnp for \p\ > r, for some large r. Since / is Lipschitz in v, we change the roles of v and w in (iii) and (iv). Then, obviously, v
(y) = V>'(x) +
um
/ /o(*,'?,u>„('0) v ( , 7)*/
on
J
i
has a unique solution, and w' = f0(x,y,w)v(x,y)
a.e. on Ji,
w(0) = V'(y)
has a unique solution if / 0 is locally Lipschitz, say, and therefore (l)fc(2) has a unique classical (a.c.) solution in this case. If /o is only continuous we don't know whether there is an a.c. solution. Notice that v(x, y) = exp ( J f0(x,v,
v>(x, r)))
hence (1)&(2) is equivalent to finding a solution w € Ci(J) of
Mx, i/) = V(y) + [ Hi, v, «>(£. v)) <*P (jT Me, v, «(t i») *,) v'(« # which is not what one would call an integral equation of standard type. So, as an extension of Problem 1, the reader is asked to solve Problem 2. Let / : J x ]R3 -> M be continuous (or measurable in (x,y) and continuous in (u,v,tu)) and bounded, and let
146 increasing a(-) and /?(•) such that or(0) = /3(0) = 0, as can be expected by looking at the equivalent system u(z,y) = ¥,(x) + V(»)-V»(0)- f v(fi(r,),r,Mr,)dr,-
f
w(t,a(t))a'(t)d{
+{/ / - / / }m,vM(,v)XZ,n)M(,v))didr, JO Jfifr)
JO J0
v(x,y) =
+ f
f(x,r),n(x,T)),v{x,r]),w(x,r)))dt)
(19)
JO(T)
»(*, y) = V-'(lO - «(/3(»), y)/3'(») + / " / « , V, «(£.»),»«, v), « « , I/)) <*£•
(20)
The trouble starts already in case / = 0: if the edge at (0,0) of the region enclosed by the two curves is to sharp then one may not get a solution, roughly speaking. Notice, for example, that in this case v is independent of y and w does not depend on i , hence, choosing
«(/3(y),y) = tf(v) °«> **,
with p € Lip(Ji) to be determined such that a corresponding ti also satisfies the first part of (3). This leads to «(x, y) = p(x) + ^(y) - p{fHv)) + f
f M, 1,»((, V), «(f, V), «<*, V)) dndi
y«(,) Jo
147 v(x,y) = p'(x) + I Jo
f(x,T],u(x,Ti),v(x,T)),w(x,Ti))dT),
equation (20) for w and the additional equation p(x) = p( 7 (x)) + 9{x) - 0(a(x)) - f
["
/ ( ( , V, u(i,,,),,;({,
V),
w(t, r,)) dr,di
(21)
J-,(z) JO
with 7(1) = /3(a(x)). The given functions in (21) are assumed to be Lipschitz and 7 is Lipschitz of constant k < 1, hence the linear operator T, defined on Lip(7i) by (Tp)(x) = pd(x)), has norm less than one; consider e.g. IIPII = j M 0 ) ! + SUP{\P(X) ~ /K*)!/!* - i | : x, x e Ji and x # x} as norm on Lip(7i). So, (21) is solvable for p, but the outcome of this approach is essen tially the same as the one considered in [4], i.e. the question what one really needs about / w.r. to (v, w) is not simplified this way. Therefore the reader is asked to solve Problem 3. Let / : J x 2R3 — ► M be continuous (or measurable in (x, y) and continuous in (u,v,w)) and satisfy (7). Let the other data given for problem (1)&(3) be as in The orem 2. Under which (additional) conditions about / w.r. to (v,w) does (1)&(3) have an a.c. solution?
References [1] C. Caratheodory: Vorlesungen uber reelle Funktionen. Chelsea Publ. Comp., New York 1968 ( 3 r d ed.) [2] K. Deimling: A Caratheodory theory for systems of integral equations. Ann. Mat.. Pura Appl. 86 (1970), 217-260. [3J K. Deimling: Das Picard-Problem fur ti ly = f(x,y, u,«»,«,) unter CaratheodoryVoraussetzungen. Math. Z. 114 (1970), 303-312. [4] K. Deimling: Das Goursat-Problem fur ux> = f(x,y,u). Aequationes Math. 6 (1971), 206-214. [5] K. Deimling: Sample solutions of stochastic ordinary differential equations. Stoch. Anal. Appl. S (1985), 14-20. [6] K. Deimling:. Multivalued Differential Equations. W. De Grayter, Berlin 1992. [7] J. Diestel and W.M. Ruess and W. Schachermayer: Weak compactness in ^'(/i.A'). Proe. Amer. Math. Soc. 118 (1993), 447-453. [8] P. Garabedian: Partial Differential Equations. J. Wiley, New York 1964. [9] Ph. Hartman: Ordinary Differential Equations ( 2 n d ed.), Birkhauser, Boston 1982. [10] Ph. Hartman and A. Wintner: On hyperbolic partial differential equations. Amer. J. Math. 74 (1952), 834-864.
148 [11] M. Krzanski: Partial Differential Equations of Second Order II. Polish Sci. Publ., War8zawa 1971. [12] W. Walter: Differential and Integral Inequalities. Springer, Berlin 1970. [13] W. Walter: On the non-existence of maximal solutions for hyperbolic differential equa tions. Ann. Polon. Math. 19 (1967), 307-311.
WSSIAA 3 (1994) pp. 149-177 © World Scientific Publishing Company
149
Comparison Theorems for a Bight Disfocal Eigenvalue Problem Jerry Diaz U.S. Air Force Academy Colorado Springs, Colorado, USA and Allan Peterson Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, Nebraska 68588-0323, USA Abstract. Assuming a certain two term nth order differential equation is right disfocal we show under certain assumptions the existence of a smallest positive eigenvalue for a related problem. Under further assumptions we prove the existence ofa corresponding dgenfunction in a certain cone. Finally we prove a comparison theorem for smallest positive eigenvalues for two eigenvalue problems. 1. Introduction Let n > 1, m > 1, k a fixed element of {1,2,---.n-l} and I = [a,b]. We define the linear differential operator L, by Lu = vSn' + r(t)u, where u(t) is an m-column vector function of class Cn[a,b], and r(t) is a continuous scalar function on [a,b]. Also let P(t) = (pj.-O)), Q(t) = (
150
We will make assumptions so that the Green's function G(t,s) for the scalar boundary value problem, (-l)n"kLy = 0 (2) Ty = 0, where Ly and Ty are as above, but defined appropriately for the scalar case. Under certain assumptions we will show the existence of a smallest positive eigenvalue for (1). And with further conditions on P(t), that its corresponding eigenvector is essentially unique with respect to a 'cone'. We also prove comparison results for the eigenvalue problems (1) and (3), (-l) n_k Lu = AQ(t)u (3) Su = 0, where Su = 0 are boundary conditions similar to those above. Our results are new, even in the scalar case. Our technique will be to use the theory of uo-positive operators with respect to a cone in a Banach space. We then will use sign conditions on a Green's function and then appropriately define an integral operator which will map a cone back into itself. The theory of operators acting on a cone, can be found in books by Krasnosel'skil [9], Deimling [1], and Guo and Lakshmikantham [4]. Related papers include those of Eloe and Henderson [2], Gentry and Travis [3], Hanlcerson and Peterson [5,6], Kreln and Rutman [10], Keener and Travis [7], Kreith [11], Schmitt and Smith [13], Tomastik [14,15], and Travis [16]. 2. The Green's Function: In this section we give sufficient conditions for the existence and give an explicit form for the Green's function for our problem (2). Also, we will give certain sign conditions of the Green's function. Some of these sign conditions are new. Theorems 1 and 2 and Lemmas 3, 4 and 5 appear in [12]. We state these results
151
here for easy reference and because we need them to prove some additional facts. We will need the following definition. Definition: The differential equation Ly = 0 is called right disfocal on an interval I if there does not exist a nontrivial solution y of Ly = 0 and points t x < t 2 < ••• < t n 6 I such that yt1-1)(tj) = 0 for i = l,2,---,n. We will also need to introduce some notation. For each fixed s in the interval [a,b], let {y0(t,s),y1(t)8),y1(t,8)1-",yn_1(t,s)} be the set of (linearly independent) solutions of Ly = 0, satisfying the initial conditions: u; y,(j) (M)l t = g = *£, o < j, k
where S-. is the Kronecker-delta function _ JO, for j i k * " l l , for j = k. Theorem 1. Let Ly = 0 be right disfocal on [a,b]. Then, for each fixed s € [a,b], the Green's function for (2) exists and is given by, for a < t < s, 0 n-k
(ii)
G(t,s) = B f
h^M
y k (t,a)
•••
y^Va)
-
yiW(b..)
-
y„_i(*.*)
viV.a) ^0...)
If s < t < b, then we replace the sero in the first row, first column by y - i t M ) with everything else remaining the same. In the aboveformula,D is given by:
152
y^V.a) D =
(i ) yk 2 (b,a)
y^jV.a) (i 2 )
yk+? (b,a)
y n -? ( b > a )
y^va)
^n-k),.(b,a) N yk^n-k^/v 0>>a)x y k+1
To present the next theorem, we need to consider the following partition of n-k-tuples. We will say that (iiii2>"">*n-k)
<
(Ji^'""'^-^ '* t ' iere
ex 8ts
'
ia
integer m such that i)
iii)
if = jjfor k = 1,2,---.m - 1
i. < j , for / = m + 1, m + 2,">,n-^.
We can now give a comparison theorem and sign conditions on the Green's functions for different boundary value problems. Theorem 2. Let Ly = 0 be right disfocal on [a,b], and suppose that
(il'i2'""'in-i) < Ol'h'"'k-i> G:
VVk
where
° > Jl < -b < *' * < Jn-^ < n - 1. If
(t,s) is the Green's function for the boundary value problem Ly = 0 yW(a) = 0, i = 0 , l . . . , k - l y j(b) = 0, j = l . V - . n - k ,
then c[P). ., (t,s) < G J( P ) . . , (t,s) on (a,b)2 for p = 0,1,• • • ^ . 1 n-k l Ti-k We note that the above theorem gives us a sign condition on G(t,s). Since it is well known that GQ 1#--n _ kl (t,g) > 0 on (a,b) we have that if
GiV•••**-*) > C' 1 '*••• n - k - 1 )> then G i . . . i (M) > ° o n (a>b)21
n—k
153
The above two theorems are proved using the following lemmas (see [12]). Lemma 3. Let L* be the adjoint operator defined by L*z = z^n' + (-l)nr(t)z, corresponding to our operator Ly = y*n' + r(t)y. Then Ly = 0 is right disfocal if and only if L*z = 0 is right disfocal. Our next lemma gives a relation between our set of solutions to Ly = 0 and a set of solutions to L*z = 0. Lemma 4. For i = 0,1,"-.n-l, let z.(t,s) be solutions on [a,b] to L*z = 0
4 J) (M)l t=g = fy 0 < j < n - 1. Then, for 0 < i, j < n - 1, we have
ylj)(t,8) = (-Di+jz<>j^(M) for all t, s e [a,b]. This lemma is proved by applying the Lagrange identity to y-(t,r) and z . ,(t,s) and evaluating at t = s and t = r. By using the adjoint relations of Lemma 4, it is easy to show that for any fixed t,
(w)r{y?V.»)} = (-i) r y£M
w
on [a,b] x [a,b] for 0 < i, j < n - 1, 0 < r < j . The next lemma is a crux to all of our results. Lemma 5. If Ly = 0 is right disfocal, and y is a nontrivial solution to (2) except for one of the boundary conditions at b, then y(t) # 0 for all t 6 (a,b). This last lemma is proved by assuming that there exists a t o 6 (a,b) and a nontrivial solution y, to Ly = 0, satisfying the hypotheses of Lemma 5, such that y(t ) = 0. Then using the boundary conditions with a Rolle's Theorem argument,
154
one can contradict Ly = 0 is right disfocal (see [12]). It is important to note that this lemma also holds for t h e adjoint equation. W e can now give a sign condition on certain derivatives of the Green's function for (2) a t t h e end points. T o establish t h e sign condition at t = b , we need a bit more notation.
Consider our boundary conditions at t = b . Suppose that i . > 0,
then we define ko = 0. Now assume i l. = 0 . If i_n—K . = n - k - 1 ,' set ko = n - i . If i n _^ > n - k - 1 pick k o so that i. = j - 1, for j = 1,2...-,k 0 a n d k Q < i k for example, if we have the (n-k)-tuple O i ^ ' " " ' ^ - ^ k
=
r
So,
(0»1»2»^»"""^n_i)> then
o = 3-
Theorem 6. Let Ly = 0 be right disfocal on [a,b] and G(t,s) be the Green's function for (2). Then G(k>(a ,t) > 0 for all s E (a,b). Further, if we define k as k
„ ( k J(b,s)
above, then (-1) G
> 0 for all s E (a,b).
Proof: We will first show that G^k\a,s) > 0, for all s E (a,b). After taking k derivatives and evaluating at t = a, we have that the first row 4j. of G ^ a , s) is &1 - ( 0 j { k W ) 4 + l ^ ) » — ^ £ 1 ( * * ) ) " (O.M,—,0). Now, define f(s) on [a,b] by
yk+J (b,a) (i 2 ) f(s) = <***>
yn_r(b,s)
y k+ ? (b,a) (in-*) yk+P(b,a)
(i 2 )
y n -? 0>>a)
&**+)
»n-k+l So we have that G^(a,s) = ^ - i l p f(s) on [a,b]. Now, we can show that f(s) # 0 for all s € (a,b). We first transform f into its equivalent adjoint form using
155
Lemma 4. Then f satisfies L*z = 0 and the equivalent adjoint boundary conditions. So from the adjoint form of Lemma 5, we have that f(s) £ 0 for all s 6 (a,b). Now, consider any element in the first column of f. By using (4), we have that
(w)j{y£i(b>8)>l8=b = ("1)jyn-i-j(b'b) = 0, for 0 < j < (n-1) - i n _ k and
i 6 ihJi'"'
^n-k^' W 'n-k
=
n
~ * t * len
rJ^(b) = 0 for 0 < j < (n-1) - i n k >
we
define J = ")• ^bis
te s u s
^
*bat
Now, letting j = n - 1 -i _^ and s = b, we
have that
y^jV.a)
^(b.b) n—k 2
^
Vk
f(j)(b) = (-l)j
fj
(b,b)
y k+ ? (b,a)
(b,a)
1
n—k
k+l
(b'a^
j0
yj(b
&>< b,a)
y^jV.a) (i2)
n k+1
= (-l)VD -
y k+ ? (b.a) (i
*k+ j
(b,a)
•••
y
y^V*) (i2) y k + 2 (b«a)
*k+2
y n -? (b>a)
...
&*W> a)
£V
a)
= ("!>
•
(i 2 )
...
#o.
o 4!i'0>,a)
(ii) n-1 (b,a)
...
(b» a)
.a)
(b,a)
n-l
... ...
•••
y^lVa) (i2)
yn_? 0>.») ( y B _1
(b»a)
It it a standard argument to show that the above determinant is positive.
156 This gives us that (-l)J(-l) n "* + 1 f^(b) > 0. we can use a Taylor series expansion on ( - l )
Now, since f^(b) = 0, for 0 < i < j , n
f(s), about b, to get that
(_if-k+lf(J)(b) l£*l± + 0((s-b)J+1).
(_l)^+lf(8) =
J *
This tells us that for a sufficiently small S > 0, if j is even, then ( - l ) n - k + 1 f ( s ) > 0 If j is odd, then (-l) n - k + 1 f(8) < 0 for all s 6 (b-tf,b), or
for all s € (b-tf,b).
(-l)j{(-l) n '* + 1 f(8)} > 0 for all s 6 (b-«,b). we have that ( - l )
n
In either case, for a small enough S,
f(s) > 0 for all s € (b-£,b). n
But, we have already shown
+1
that f is of one sign on (a,b). Thus (-l) "* f(s) > 0 for all s 6 (a,b). This gives n-k+1 us that GM(a,s) = - ^ f(s) > 0 for s 6 (a,b), and so the first part of our theorem is proved. To prove our sign condition at t = b, we suppose that i, = 0 and kQ is defined as before.
We define the function f on [a,b] by y n - i 8)
y^ } (b,a)
y^V.a)
n-k
«•) - <4
y^Va) y n -! ) ( b « a )
<W o>.») yk(i„-k) (b,a)
JW-I
By defining f in this manner, we have that G(
k
„).(b,s) = ^-^4^ - k f(s) for s e [a,b].
Like before, we show that f(s) / 0 for all 8 e (a,b) by first transforming f into its equivalent adjoint form using Lemma 4. Then f satisfies L*z = 0 and the equivalent adjoint boundary conditions.
So from the adjoint form of Lemma 5, we must have
that f(s) / 0 for all s E (a,b). Consider any element in the first column of f.
By using (4), we have that
157 for 0 < j < (n-1) - i n - k and i € {»1»»2'"'>iii-k}-
( If 'n-k =
n -
j = 0). This tells us that f<j)(b) = 0 for 0 < j < (n-1) - in_^.
*
then we
Now, letting
j = (n-1) - i n _j t , we have that (ko).
yi°(b,b) n-k
(kc).
yk"(b,a)
(ii)
P(b) = (-l)J
(ii)
y- * (b,b) Vk
y k * (b,a) *
y{ W tb,b)
^^(b.a)
Vk
0
•••
y^'V.a)
(k D ). y n _j (b,a) yn-!)(b>a)
^V*)
*
y n -i
(b,a)
0
y k * (b,a)
y^iW)
1
y[W(b,a)
T^O...)
-(-I)1
(k„) yk (b,a) n-i y k * (b,a) = (-i)j(-i)
(k0) yk (b,a) y k + { (b,a)
T^V-a) yk!rk)(b»a)
define
ViV.*) VT^CM)
158
(i2)
yk
2
(i 2 )
(*2>
(b,a)
y n -? a)
yk+? (b,a)
(ik )
n
k
(-l)J(-l) -*(-l) °
?k
° (b,a)
y( k o)(b,a)
yk + l° ( b ' a )
yn_J° (b,a)
y&^.a)
y&^b.a)
'n-r yk
0>,a)
yk+1
(b,a)
Now, from our definition of ko, we have the i.
yn-i
< kQ < i. o
(b,a)
(b'a)
,«, so again from o
a standard argument we have that the above determinant is strictly greater that zero. This gives us that (-1)>(-I)n"~k(-1) °f(^(b) > 0.
Now, since fW(b) = 0, for _ . kQ 0 < i < kQ, we can again use a Taylor series expansion on (-1) (-1) f(s), about b, to get
( - i r t - u S w = (_i)»-k(-i)k«»f(j)(b) &pi + o((x-bJ+1)). From this we can again see that for 8 > 0 sufficiently small, that if j is even, then k k ( - l ) n _ k ( - l ) °f(s) > 0 for s e (b-«,b). If j is odd, then ( - l ) n - k ( - l ) °f(s) < 0 for all s 6 (b-5,b), and so (-1)J{(—l)n-k(—1) °f(s)} > 0.
In either case we have that
k
(-l) n ~ (-1) °f(s) > 0 for all s e (b-£,b) for small enough 6. But, we have already shown that f is of one sign on (a,b). Thus (-l) n ~ (-1) °f(s) > 0 for all s 6 (a,b). k (k ) , .vn-i k This gives us that (-1) °G ° (b,s) = ^ — (-1) °f(s) > 0 for all s e (a,b), and so our theorem is proved.
159
4. Existence and Comparison Results: We will now introduce a suitable Banach space for our eigenvalue problem (1). Recall that the boundary conditions Tu = 0, for u an m-column vector, are uW(a) = 0, for i = 0,l,---,k-l, and u
J
(b) = 0, for j = 1,2,---.n - k, where
n-
0 < i, < i, < • • • < • - £ < l - First, let us suppose that i, t 0. When i, ^ 0, we will denote these boundary conditions as T.u = 0. We now introduce the Banach space Bj = {u6Cn([a,b]lrftt)|iiW(a) = 0, i = 0,1,-...k-1} with norm ||u|| = maxj..-. {maxr ,i|u^'(t)|} where | • | is the Euclidean norm. Following ideas from Hankerson and Peterson [5,6], and Tomastik's papers [14,15], we let I, J C {1,2,---.m} be such that I U J = {l,2,---,m} and I fl J = 0. (It is permissible for I = 0 or J = 0). Let K be the 'quadrant' cone in R m defined by K = {x = (x 1 ,---,x m )|x i > 0 if i e I, xj < 0 if i e J}. Although some of our results will hold for any solid cone in Rm. Define &• to be the discrete function 6. = 1 if i 6 I, and 6- = -1 if i e J. We can then equivalently define the cone K to be K = {x 6 Rm|£.x. > 0 for i = 1,2,- ".m}. It follows that the interior of K is K° = {x 6 R ™ ! ^ > 0 for i = 1,2,• --.m}. We now define the reproducing come P. c B. by P, = {u e B.|u(t) e K, t 6 [a,b]}. This gives us the following Lemma concerning the interior of our cone P,. Lemma 7. Let the cone P, in the Banach space B, be defined as above. Then the interior of P, is given by Pj = {a 6 BjMt) 6 K°, t e (a,b] and uM(a) 6 K 0 }, or equivalently P° = {u 6 Bj|^Ujft) > 0, t e (a,b] and JLuf^a) > 0, i = l,2,---,m}.
160 Proof: Let Q = {u e B^uft) 6 K°, t 6 (a,b] and J k )(a) 6 K 0 }. First we will show that Q C P°
Let u be an arbitrary element of Q, so we want to find an
e > 0 so that the ball B(u;£) C P,. For a vector function x(t) on [a,0\ C [a,b] we define the distance function dr a(x(t),dK) to be the distance between the function x(t) on [a,0\ and the boundary of the cone dK. Let e, = *dr . ,(u^ '(a),0K), so we have that e, > 0 since u^ '(a) € K°. Now u' ' is a continuous function, so there exists a 6 > 0 so that u ( k \ t ) 6 B(u(k)(a);e1) c R m , for all t € [a,a+fl. that this gives us that dr
We note
(k)
, «(u (t),aK) > ev
Thus we have that u(t) e K° for all t 6 [a+£,b]. Then, if we now let £
2 = S^ra+tfblM*)'^) w e ^ ^ n a v e t n a t £2 > " 8*nce t n e K^Pk °* u ( t )' w ^ c ' 1 ' 8 compact on [a+£,b], and dK do not intersect. We note that in this case, we have that d
[ a + * . b ] ( n ( t ) , a K ) > e 2Let e = min{e 1 ,e 2 } > 0. Then we have that B(u;c) C P^. To show this, we
let z € B(u;e). Then ||z-u|| < e so in particular we have that | z ( k ) ( a H ( k ) W I < ex = j d k b ] ( n W ( a ) , « ) .
This tells us that iW(a) € K°. Now
||z-u|| < e also tells us that |*' '(tj-n* )(t)| < e for all t € [a,a+£]- This gives us that z( k )(t) 6 K for all t € [a,a+j]. If this were not so, then since z( k )(a) E K° and z* > is continuous, there would exists a t € [a,a+d] so that z^ '(t ) 6 dK. But from the note above we know that dr
, «(u' '(t),5K) > e, > e. This gives us
k
that |zM(t o )-ii( )(t 0 )| > e which is a contradiction. Thus z^ k \t) e K° for all t € [a,a+fl. Now, this last statement tell us that for i = l,2,---,m,tfjsVk)(t) > 0 for all t 6 [a,a+J|. Thus Az> ~ '(t) is a strictly increasing function on [a,a+i] with ^.^""^(a) = 0 for each i. Hence we have that
tfjyk-1\t)
> 0 for all t e (a,a+fl,
for i = 1,2,• ",m. Thus &zA ~ '(t) is strictly increasing on (a,a+fl, with &,Y
'(a) = 0 for each i. Thus 6ff
>(t) > 0 on (a.,&+6\ for each i. Hence for
161 each i = 1,2,• --.m we have that &zA &z.'
'(a) = 0.
'(t) is strictly increasing on (a,a+fl with
Continuing in this manner, we eventually come to the conclusion
that z(t) 6 K for t € [a,a+£]. Also, we have that |z(t)-u(t)| < e < « 2 for all t e [a+*,b]. or else we contradict e„ < c 'r a+ ri,i(u(t),0K).
^ i n c e z ( & + *)
€
Thus, z(t) I dK an<
^°
^
z is
continuous, we must have that z(t) 6 K for all t 6 [a,+£,b]. Thus z(t) € K for all t € [a,b].
But his means that z 6 P., and since z was
an arbitrary element of B(u;e), we have that B(u;e) C P..
But u was an arbitrary
element of Q and we found an e > 0 so that B(u;e) C P..
Thus we have that
QCP;. We now show that P? C Q. Let u be an arbitrary element of P?. Suppose there exists a l o E (a,b] so that u(tQ) € dK. This give us that there exists a component of u, say u- so that u- (t ) = 0. Considering the scalar equation, o
o
& u- (t) > 0, it can be seen that for any e > 0, since & u- (tQ) = 0, we can find O
O
O
0
a function 6z.1 (t)' e B(&1 u-1 ;e)' so that 6z-1 *(t O') < 0. If we let the vector 1 1 x
OO
v
OO
0
0
function z(t) equal u(t) in each component except in the iQ slot, and then in the slot let (z(t))j = Zj (t), then z 6 B(b;«). But z(tQ) t K since ^ Zj (t o ) < 0. Thus 0
z i P..
0
0
0
Now z was based on t > 0. Thus, for any e > 0 we can find a
z € B(u;e) and z f Pj. This contradicts u e Pj. Thus u(t) e K° for all t e (a,b). Now suppose u* '(a) I K°.
So there exists an i so that &u> '(a) < 0. Then
for any e > 0 we can find a z e B(u;t) so that &z> '(a) < 0.
Thus &z> > is
strictly decreasing at a. We have that z> ~ '(a) = 0 so we can find a 6 > 0 so that ^ ^ " ^ ( t ) < 0 for any t 6 (a,a+fl. (a.a+fl and ^^""^(a) = 0.
Thus, Sffk-2) k 2
Hence Sf} ~ \t)
is strictly decreasing on
< 0 for all t e (a,a+«j. Like
before, by continuing in this manner we come to the conclusion that z(t ) i K and
162 so z f P j , which contradicts u G P?. Thus we must have that u^ '(a) G K°. So if u 6 P j we have that u(t) 6 K° for all t 6 (a,b], and also that u^ '(a) G K°.
Thus u G Q, and since u was an arbitrary element of P?, we have
that P? C Q.
Thus our lemma is proved.
Now let us suppose that i. = 0. We will denote these boundary conditions as TQu = 0.
As in the last section, let kQ, 1 < kQ < n - k, be such that i. = j - 1
for j = 1,2,. ..,k 0 and i k
< kQ < i k O
j (if kQ < n-k). 0
We now introduce the Banach space B o = {u G Cn([a,b], R m )|uW(a) = 0, 0 < i < k-1, u W(b) = 0, 0 < i < k Q -l}, with norm ||u|| = max 0< . < {maxr . 11 ^ ' ( t ) |} where | • | is the Euclidean norm. We now define the reproducing cone PQ c BQ by PQ = {u G B o |u(t) G K, t G [a,b]}.
We also have a lemma concerning the interior of this cone P .
r Lemma 8 Let the cone P o in the Banach space Bo be defined as above. Then the interior of P is given by k (k ) P° = { u G B j u ( t ) G K°, t G (a,b), uM(a) G K°, and (-1) °u ° (b) G K°}
or equivalently P
o = {» 6 B<J W * ) > °» * 6 ( a ' b )' *i u i k) ( a ) > °. " d
K (K) (-1)
"AJUJ
° (b) > 0, 1 < i < m}.
Proof: The proof of this lemma is very similar to the proof of Lemma 7. Let Q = {u G B01 ^ ( t ) > 0, t 6 (a,b), 5.^%) 1 < i < m}. First we will show that Q C P°.
> 0, (-lf0^0
(b) > 0,
Let u be an arbitrary element of Q,
so we want to find an e > 0 so that the ball B(u;«) C P,.
Now, from the
argument in Lemma 7, we see that if we let e, = jdr , ,(u^ '(&),dK) > 0, then there exists a S1 > 0 such that uM(t) G B(u( k \a);e 1 ) c R m , for all t G [a,a+£J.
163 k (k ) If we let e 2 = jdt y ( ( - l ) °u ° (b),0K) > 0 then there exists a Sj > 0 such k (k ) k (k ) that (-1) °u ° (t) 6 B((-l) °n ° (b);e2) C R m , for all t e [b-tf^b]. this gives us that dr. r vi((-l)
We note that
ko (ko) u ° ( 0 ) ^ ) > eo-
Since u(t) € K° for all t e [i+S^b-ty
if we let c 3 = Ij&u.g ^_s ,(u(t),3K),
then we have that e, > 0 since the compact graph of u(t) on [a+6pb-&,], and dK do not intersect. Let e = min{£,,e 2 ,£ 3 } > 0. Then we have that B(u;e) C PQ. we let z E B(U;E).
To show this,
Then, similar to the arguments in Lemma 7, we have that z(t) €
K for all t 6 [a,b-<5], for some 5 > 0. Also, since ||z—«|| < e, we have in particular we have that |z ° (b)-u ° (b)| = |(-1) °z ° (b)-(-l) °« ° (b)| < e2.
Then since
k (k ) k (k ) (-1) °u ° (b) e K° and e2 = 5
This gives us
that
k (k ) Thus (-1) °z (t) € K° for all
t 6 [b-<5,b]. This last statement tells us that for i = 1,2,■ --.m, r5j(-l)
k0 (k ) Zj (t) > 0 for all
164 t 6 [b-S,b].
Now, since zj"(b) = 0 for j = 0,1,---,^-1, i = 1,2,---.m, we can use
a Taylor series argument (as in the proof of Theorem 6), to show that tfjZj(t) > 0 for all t E [b-S2,b], i = 1,2,---.m.
Hence z(t) E K for all t E [b-62,b].
Combining our cases we have that z(t) E K for all t E [a,b].
But this means
that z E PQ> and since z was an arbitrary element of B(u;c), we have that B(u;c) c PQ.
But u was an arbitrary element of Q and we found an e > 0 so that
B(u;e) c P r
Thus we have that Q c P°.
We now show that P° C Q.
Let u be an arbitrary element of P°.
Now,
following arguments similar to the ones in Lemma 7, we see that u(t) E K° for all t € (a,b), and that uW(a) E K°. k (k 0 ) Now suppose (-1) °u ° (b) i K°.
So there exists an i so that
4j(-l) °u ° (b) < 0. Then for any e > 0 we can find a z € B(u;e) so that k (k ) JL(-l) °zi ° (b) < 0.
Now, z["(b) = 0 for j = 0,1,- ••,k 0 -l.
So, again using a
Taylor series argument, we can show that S.t-(t) < 0 on (b-£,b) for a sufficiently small 6 > 0.
But then z(t) ( K for t E (b-£,b) which tells us that z t PQ. This k (k ) contradicts u E P°. Hence we must have (-1) °u ° (b) E K°. So if u E P° we have that u(t) E K° for all t 6 (a,b), and also that k (k ) v[\a) e K° and (-1) °u ° (b) E K°. Thus u E Q, and since u was an arbitrary element of P°, we have that P° C Q. Thus our lemma is proved. With our Banach spaces and cones suitable defined, we can now proceed on to our first existence result. Theorem 9.
Let Ly = 0 be right disfocal, and assume that SiS^M
> 0, for
t E [a,b], 1 < i, j < m, and that there is a t 0 E [a,b] such that p. • (t 0 ) > 0. 0 O
Then for the eigenvalue problem (1)
165
(-l) n-1 Lu = AP(t)u TjU = 0, (so (ij > 0), there exists an eigenvector z € P, with corresponding positive eigenvalue A which is a lower bound for the modulus of any other eigenvalues for the corresponding problem. Proof: To solve this problem, we will seek the eigenvalues of the linear integral operator M: B, —» B. defined by b Mu(t) = [ G(t,8)P(8)u(s)d8, a < t < b, a where G(t,s) is the Green's function for (2). Now the eigenvalues of the boundary value problem (1) are reciprocals of the eigenvalues of the operator M. We note that zero is not an eigenvalue of (1) since Ly = 0 is assumed to be right disfocal. Now an argument using the Arzela-Ascoli Theorem shows that M is a compact operator. We now show that M: P. —• P.. Let u be an arbitrary element of P.. If we can show that ^(Muft)); > 0 for all t € [a,b], i = 1,2,--.m, then Mu 6 Pj. m Consider the ith component of P(t)u(t), (P(t)u(t))i1 = E P;i(t)U;(t). Now 6.6. = 1, j = l »J
J
J J
and <5jU-(t) > 0 so we have that for all t € [a,b],
«i(P(t)«(t))i = l ^ / t ^ t ) > o, since 6-6jp..(t) > 0 by hypothesis. From the note following Theorem 2, we have that G(t,s) > 0 on (a,b) « (a,b). Thus
r ^(Mnyt) = J G(t,s) E tyiftypjMM > 0, a *~ for t € [a,b], 1 < i < m and so Mu € P,. Since u was an arbitrary element of P., we have that M is a positive operator, that is M: P. —» P.. In order to apply Theorem 1, [7], we must find a nontrivial uo 6 P,, and an
166 e0 > 0 so that Mu0 ~> 0a 0u . Let u0*(t)' = ^ KY?'i
6-l e- , where 1e- is the unit
l
0
0
0
vector in R m in the iQ direction. We note that u 6 B.. Now the jth component of u0 (t) is u (t) = ' kL! T 6- &■ •■ where &■■ is the Kronecker delta function. Thus ' °y ' »0 »0J iJ S.n .(t) = {6-6. ftr?) } 6- ■ > 0, so u 6 P , . We have that £. u o i (t) = ( f f i 0
0
> 0, on (a,b] and that dj uW(a) = 1 > 0.
*
0
0
We now consider MuQ(t). Since M: P. —• P,, we know that WMuWt) > 0 = 6-u.{t) for 1 < j < m, j # iQ. When j = iQ we have that «i (Mujj (t) = o
J
o
G(t,s) E 6{ 6. Pi .(s)5.u .(sjds i=l
o
J
oJ
J
J
a
* b = fG(t,s)«i 6i P j j (s)6. u o i (s)ds J
0
O
0 O
a b = [G(t,8)Pi . (s) ^ i °° k! a since by Theorems 2 and 6 0 for t > 0, for t € (a,b],
0
0
k
ds is continuous,
since by Theorems 2 and 6 G(t,s) > 0 for t e (a,b], s e (a,b), pj j is continuous, 0 0
Theorem ■"d PiNow. i (*„) > °- 6 tells us that
so we can see
0 0
from above that Hence, we can find an sufficiently Now, Theorem 6 tells us that G^ '(a,s) > 0 for all s € (a,b), so we can see small,above so that from that 6- (MuQ)> '(a) > 0. Hence, we can find an e. > 0, sufficiently 0
O
small, so that 6i (Mu Q )[ k \a) - e^ uW(a) > 0. Now Thus we can find a 0
0
O
O
&j (Mu )[^(a) - e^ u(|)(a) = 0, for j = 0,l,---,k-l. Thus we can find a 6 > 0 so0 that o 0 0 0 so that 6i (Mujj (t) - e^. u o i (t) > 0, for all t e [a,a+d]. 0
0
O
O
Now both 6- (MuY (t) and 6- u • (t) are positive on [a+£,b] so we can let 0
0
O
O
167
62 =
■in[a+«>b]V1Uo)io(t) ■"*M,b]«i»»i ™ L
J
This gives us that ^ (Mu o )j (t) - e^ 0
0
°-
O
u o i (t) > 0, for all t G [a+tf,b]. 0
Finally, letting eQ = min{e,£ 2 }
0
>
0
we nave tnat
$ (Mu0)i (0 ~ 0
for all t 6 [a,b]. This gives us that Mu
0
e
0^
u
oi (4) - ®'
O
0
> e u with respect to the cone P , .
By
applying Theorem 1 in [6] the conclusions of our theorem follow. We have a parallel theorem in the case that i, = 0. Theorem 10. Let Ly = 0 be right disfocal, and assume that &&p..(t) > 0, for t 6 [a,b], 1 < i, j < m, and that there is a tQ € [a,b] such that p. . (t Q ) > 0. oo Then for the eigenvalue problem (1) (-l) n _ 1 Lu = AP(t)u T o u = 0, (so i x = 0), there exists an eigenvector z 6 P
with corresponding positive eigenvalue A which
is a lower bound for the modulus of any other eigenvalue for the corresponding problem. Proof:
Like before, we solve this problem by seeking the eigenvalues of the linear
integral operator M: BQ —• BQ defined by b Mu(t) = [G(t,s)P(s)u(s)d8, a < t < b, a where G(t,s) is the Green's function for (2), with the boundary conditions TQy = 0. Again, the eigenvalues of the boundary value problem (1) are reciprocals of the eigenvalues of the operator M, and we note that zero is not an eigenvalue of (1) since Ly = 0 is assumed to be right disfocal. Now, an argument identical to the one in the proof of Theorem 9, shows that our compact operator M maps P Q into P .
168
In order to apply Theorem 1 in [6] we must find a nontrivial uQ E P , and an e > 0 so that Mn > e u . In this case we let 0
O
O O
O
0
0
where e- is the unit vector in Rm in the i direction. We note that u 6 B . It is 1 0
0
0
easy to see that 6- times the jth component of uQ(t) is nonnegative. Hence u 6 P . We also have that <5- u ; (t) > 0 on (a,b) and that O
1 0 1 v 0 O
O
h *0k)w - (-Dk° ^ 0
0
/
\ * /
> °. »* H ) % M O ) M ■ ^ O
0
> °-
0
We now consider Ma (t). Since M: P —» P , we know that 0* *
O
0'
MMuWt) > 0 = foi -0) for 1 < j < m, j # iQ. When j = iQ we have that k b k k o \ (Muo)j (t) = [G(t,s)Pi j (.)(-!) ° **j*L i £ $ L ds > 0, 0
J
O
0 0
0*
a for t E (a,b) since by Theorems 2 and 6, G(t,s) > 0 for t E (a,b), s E (a,b), and p- • (t 0 ) > 0, p. . continuous. So we have that 6- (MuY (t) > 0 for all 0 0
0 0
O
O
t E (a,b). Now, similar to the proof of Theorem 9, we can find an e. > 0, and a S1 > 0, so that ^ (Mu0)j (t) - e ^ uoi (t) > 0, for all t e [a,a+£]. 0
0
0
0 k
o
(ko)
Also, Theorem 6 tells us that (-1) G
(b,s) > 0 for all s E (a,b), so we can
see from above that (Mu o )[ k °\b) = |(-l) ko G (ko) (b,s) Pi j (s)(-l)k° l * j ^ I t * ^ ds > 0.
(-1)\ 0
0
J
0 0
o'
a k k Thus, there exists an ^ > 0 so that (-1) °4L (MuQ)[k)(b) - e 2 ( -1 ) °*i O
O
tt
i^(b) > °-
0
k k Now (-1) °fct (Mn0)jJ)(b) - e 2 (-l) ^ uW(b) = 0, for j = O,l,--.,ko-l.
0
Thus by
169 using a Taylor series expansion, we can find z 6„ > 0 so that 4 (Muo)j (t) - e2«j u oi (t) > 0, for all t 6 (b-^,bj. 0
0
0
0
Now both 6- (MuV (t) and 6. u . (t) are positive on [&+S,,b-SJ so as in the 0
0
0
0
proof of the last theorem, we can find an e, > 0 so that \ (Mu0)i (t) - e2S{ u o j (t) > 0, for all t 6 [a+^.b-A,]. 0
0
O
O
Finally, letting eQ = min{£,,£2,£3} we have that s
i (
Mu
0
( 0 - *0&i u oi (*) > 0, for all t e [a,b]. This gives us that Muo > eQu0
0 )j
0
0
0
with respect to the cone P . By applying Theorem 1 in [6] the conclusions of our theorem follow. If we have stronger conditions on P(t), we get better results.
Again we will
have parallel theorems. Theorem 11.
Let Ly = 0 be right disfocal on [a,b] and assume &£p--(t) > 0,
1 < i, j < m, for all t 6 [a,b], and p.. equals zero only on a set of measure zero. Then for the eigenvalue problem (1), (-l) n - 1 Lu = AP(t)u T x u = 0, (so i x > 0), there exists an essentially unique eigenvector zg in P,, and its corresponding eigenvalue is simple, positive and smaller then the modulus of any other eigenvalue for this eigenvalue problem. Proof:
As in the last proof, we define the compact linear integral operator M by b Mu(t) = [G(t,s)P(s)u(s)d8, a where G(t,s) is the Green's function for (2). We wish to show that M is a u -positive operator so that we can apply Theorem 2 in [6]. To do this, we will show that M: P,/{0} —• P. which implies [9] that M is a uQ-positive operator.
170 Let u be an arbitrary element in P./{0}.
then, there exists an
i 0 6 {1,2,-••,m} and a tQ € (a,b), so that & Uj (tQ) > 0. o
(By the continuity of Uj
o
o
we can assume, without loss of generality, that tQ € (a,b)).
Since u- is a o
continuous function we have that there exists an interval to the right of tQ on which 6. u- is positive. 0
O
Now for each i = 1,2,-".m, S-S, p« 0
> 0, p.. is continuous and zero only on
0
O
a set of measure zero. Thus, for each i, we can find an interval to the right of tQ, on which each 6.6. p.. is positive. O
Taking the intersection of these m + 1 right
0
intervals, we have an interval (o,b) c [a,b] such that 6-6- p.. (t)<5- u. (t) > 0, for all O O
t 6 (a,0), i = 1,2,---,m.
0
0
thus, since G(t,s) > 0 for all t 6 (a,b], s 6 (a,b) by
Theorem 2 and Theorem 6, and since by hypothesis 6-S- p=j > 0, we have that for 0
O
each i = 1,2,-■ -,m,
r ^(MuJjW = J 0(1,8)^ E pij(8)uj(s)ds J a
m
r
= jG(t,S) E ^ ^ . J ^ l j d i J_ a
f >
0 ( 1 , 8 ) ^ ^ 0P y 0(8)^ 0Uj 0(8)d8 •*
a
o
a
a
a
> 0. Thus we have that ^(Mu^t) > 0 for all t 6 (a,b].
But this gives us that Mu(t) 6
K° for all t e (a,b]. Now we also know by Theorem 6 that G( k )( a,s) > 0 for all s E (a,b). Following the same argument as above, this gives us that (Mu)*- '(a) 6 K°.
Since
Mu(t) 6 K° for all t 6 (a,b] and (Mu)W(a) 6 K° we have by Lemma 7 that
171 Mu E P . .
Now u was an arbitrary nontrivial element of P..
M: Pi/{0} —•P?-
So [9] M is a uo-positive operator.
Thus we have that
Hence we can now apply
Theorem 2 in [6] and the conclusions of our theorem follow. If we have that i, = 0 then we have results similar to Theorem 11. Theorem 12.
Let Ly = 0 be right disfocal on [a,b] and assume that S-Ss>-U) > 0,
1 < i, j < m, for all t 6 [a,b], and p.- equals zero only on a set of measure zero. Then for the eigenvalue problem (1), (-l) n _ 1 Lu = AP(t)u T o u = 0,
(so i : = 0),
there exists an essentially unique eigenvector z in P , and its corresponding eigenvalue is simple, positive and smaller then the modulus of any other eigenvalue for this eigenvalue problem. Proof:
As in the last proof, we define the compact linear integral operator M by b Mu(t) = |G(t,s)P(s)u(s)ds, a where G(t,s) is the Green's function for (2), with boundary conditions TQy = 0. We will again show that M is a u -positive operator and then apply Theorem 2 in [9]. To do this, we again show that M is a u -positive operator by showing that M: P o /{0} -
P^-
Let u be an arbitrary element in P /{0}. Then following arguments identical to those in the last Theorem, we have that there exists an interval (a,/3) C [a,b] and an iQ so that Sfi Pjj (t)S. u. (t) > 0, for all t € (a,/9), i = 1,2,---.m. O
O
0
Then, since
0
G(t,8) > 0 for all t 6 (a,b), s € (a,b) by Theorem 2 and Theorem 6, and since by hypothesis S-S- p-
> 0, we have that for each i = 1,2,• ".m,
172
^(Mn)j(t) = JGCt.s)^ #E p y t i j u ^ d ! a *~~
m
r = jG(t,s)
f
E «i«jpij(s)«juj(8)d8 ■*
a
> G f t . s ) ^ pH (s)tfj nj (s)ds "
0O
O 0
O
O
a > 0. Thus we have that ^(Mu^t) > 0 for all t e (a,b), 1 < i < m.
But this gives us
that Mu(t) 6 K° for all t € (a,b). Now we also know by Theorem 6 that G^ '(a,s) > 0 for all s € (a,b). Following the same argument as above, this gives us that (Mu)*- '(a) e K°. k (k) Theorem 6 also tells us that (-1) G (b,s) > 0 for all s E (a,b). Hence, similar to the argument above, we have that (-l) ko «i(Mu)! ko) (b) > 0, k (k ) and so (-1) °(Mu) ° (b) € K°. Since Mu(t) E K° for all t e (a,b), (Mu)W(a) E K° and k (k ) (-1) °(Mu) ° (b) E K° we have by Lemma 8 that Mu E P°.
Now u was an
arbitrary nontrivial element of PQ. Thus we have that M: P o /{0} —» P° and we have that M is a uQ-positive operator.
Hence we can now apply Theorem 2 in [6]
and the conclusions of this theorem follow. We have now come to our main theorems which give comparison results for eigenvalue problems with different boundary conditions. will consider pertain to the n-tuples (^i>"'An^) ^
The boundary conditions we (Ji>J2>",>Jn_jc)- We let
Ty = 0 denote the boundary conditions y^'(a) = 0 for i = 0,1,• --.k - 1, and y
J
(b) = 0 for j = 1,2,-",n - k.
Also we let Sy = 0 denote the boundary
173
conditions y ^ ( a ) = 0 for i = 0,1,---.k - 1, and y
(b) = 0 for i = 1,2,.--.n - k.
Theorem 13. Let Ly = 0 be right disfocal and assume that the continuous matrix function P(t) and Q(t) have the properties: a)
There is an iQ e (1,2,-..,m} and a t 0 e [a,b] such that p- . (tQ) > 0;
b)
0 < J^jPyW < Vflift), for t e [a,b], 1 < i, j < m;
c)
Each q-• = 0 only on a set of measure zero.
o o
Further assume that ( i ^ r "4 n _^) < (i^fy- "\-^)
mA
tnat
'i > °-
Tnen
there exists smallest positive eigenvalues AQ, A of (1) and (3), respectively, both of which are positive, AQ a lower bound in modulus and A strictly less in modulus then any other eigenvalue for their corresponding problems, and both of their corresponding eigenvectors belong to P.. Further, AQ is a simple eigenvalue and its corresponding eigenvector belongs to P?. Moreover, AQ < AQ and if A = AQ, then P(t) = Q(t) on [a,b]. Proof: Let G(t,s) be the Green's function for Ly = 0, T x y = 0 and H(t,s) be the Green's function for Ly = 0, S,y = 0. We define the integral operators. M, N: B x - . Bl by b b Mu(t) = [G(t,8)P(s)u(s)ds and Nu(t) = JH(t,s)Q(s)u(s)ds. a a o By Theorem 2, we have that 0 < G(t,s) < E(t,s) on (a,b) . So from earlier proofs, we know that M, N: P. —• P.. Now by Theorem 9, M possesses a positive eigenvalue 1/A which is an upper bound, in modulus, for all other eigenvalues of M, and its corresponding eigenvector zg belongs to P.. By Theorem 11, we have that N has a positive, simple eigenvalue 1/A , which is strictly greater, in modulus, than all other eigenvalues of N, and its essentially unique eigenvector vQ belongs to P..
174 To get a comparison between these two eigenvalues we need to show that M < N, with respect to P..
Let u be an arbitrary element in P..
Then for each
fixed i e {1,2,-".m}, we have from the hypothesis, V f l j W > ^ j P i / t ) > 0 for t 6 [a,bl,
1 < j < m.
Since u € Pj, we know that tf-u.(t) > 0 for all t e [a,b], 1 < j < m.
This gives us
that
for t E [a,b], a < j < m. Then from Theorem 2 we have that b b fH(t,s) E &q,ii)u,(i)di > fG(t,s) E &.Pii(s)u.(s)ds > 0 b b m m ^(JH(M) E ^)nfs)ds > ^(jG(t,.)_S Pij(s)uj(s)ds) > 0. J J a a Since i was arbitrary, this tells us that component wise b b «j(JH(t,8)Q(s)u(s)ds)i > &1(JG(t)8)P(8)u(8)ds)i > 0 a a for all t e [a,b], i = 1,2,---.m. Thus b b ([H(t,8)Q(8)u(s)ds) - [G(t,s)P(8)u(s)ds) = (N-M)u(t) 6 K a a for all t e [a,b]. Thus Nu > Mu with respect to the cone P,. Since u was an arbitrary element of P., we have that M < N. Now ( j - ^ 0 ) wid ( T - . V 0 ) »*e eigenpairs of M and N respectively, so we have o o that the inequalities of Theorem 3 in [6] hold. Theorem 8, we have that N is u0-positive.
Also, similar to the proof in
From above we have that M < N, and
so we can apply Theorem 3 in [6] to give as that -r— < -r- or Ao < AQ.
175 Finally, suppose that AQ = AQ = A, then Theorem 3 in [6] tells us that z = kv for some nonzero scalar k. Then 0
O
AP(t)zo = Lzo = kLvo = kAQ(t)v0 = AQ(t)zQ. Thus AP(t)zQ = AQ(t)zQ or (Q(t) P(t))z = 0 since A i 0. Comparing each component i of (Q(t) - P(t))z , gives us that
| foyW - PijMKjW = °. * 6 W>lSo that
Ij^W° " PijW^fo/0 = °' * 6 [a,b]Since zQ 6 Pj we have that S-zo(x) > 0 for all t 6 (a,b). This plus the fact that
VftjW - ^Piffl - ° for * € k b l' * - '• J - m> 8"™ ^ Pi/t) = QijW,
t « W>],
1 < i, j < m.
Finally, by continuity it follows that P(t) = Q(t) on the closed interval [a,b]. Our companion theorem for Theorem 13, requires more of a correlation between the boundary conditions. Theorem 14. Let Ly = 0 be right disfocal and assume that the continuous matrix function P(t) and Q(t) have the properties: a)
There is an iQ E {1,2,---.m} and a t 0 E [a,b] such that p- ■ (tQ) > 0;
b)
0 < ^jPyft) < yfift),
c)
Each q.- = 0 only on a set of measure sero.
o o
few
» € [a,b], 1 < i, j < m;
Further assume that ( i ^ i ' " ^ ^ ) < Gi>V"»Jn-k)> h ~ ° integer k0 defined for ( i ^ i - " ^ ^ ) «
aho
the k 0
and tnat the
defined for ( j ^ , - - J,,^)-
Then there exists smallest positive eigenvalues AQ, AQ of (1) and (3), respectively, both of which are positive, AQ a lower bound in modulus and AQ strictly less in modulus then any other eigenvalue for their corresponding problems, and both of their corresponding eigenvectors belong to P g . Further, A g is a simple eigenvalue
176 and its corresponding eigenvector belongs to P . Moreover, A < A and if A0 = A o , then P(t) = Q(t) on [a,b]. Proof:
The proof for this theorem is virtually identical to the proof of the last
theorem. We let G(t,s) be the Green's function for Ly = 0, TQy = 0 and H(t,s) be the Green's function for Ly = 0, SQy = 0. We define the integral operators M,N : B o -
B o by b b Mu(t) = [G(t,8)P(s)u(s)ds and Nu(t) = [H(t,s)Q(s)u(s)ds. a a 9
By Theorem 2, we have that 0 < G(t,s) < H(t,s) on (a,b) . So from earlier proofs, we know that M, N: PQ —» P . Now by Theorem 10, M possesses a positive eigenvalue 1/AQ which is an upper bound, in modulus, for all other eigenvalues of M, and its corresponding eigenvector ZQ belongs to PQ.
By Theorem 12, we have that
N has a positive, simple eigenvalue 1/A , which is strictly greater, in modulus, than all other eigenvalues of N, and its essentially unique eigenvector v belongs to P°. Now, the argument to show that M < N with respect to the cone P is identical to the argument in Theorem 13. Thus, by applying Theorem 3 in [6} we have that Ao < AQ. If A o = A0 = A, then by following the argument in Theorem 13, we see that P(t) = Q(t) on [a,b]. fypfiffTfllllCftB
1.
K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, 1985.
2.
P. Eloe and J. Henderson, Comparison of eigenvalue problems for a class of multipoint boundary value problems, to appear.
3.
R.D. Gentry and C.C. Travis, Comparison of eigenvalues associated with linear differential equations of arbitrary order, Trans. Amer. Math. Soc. 223(1076), 167-179.
4.
D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones", Academic Press, Inc., 1988.
177 5.
D. Hankerson and A. Peterson, Comparison of eigenvalues for focal point problems for nth order difference equations, Differential and Integral Equations, An International Journal for Theory and Applications 3(1990), 363-380.
6.
, Comparison theorems for eigenvalue problems for nth order differential equations. Proceedings of the American Mathematical Society 104(1988), 1204-1211.
7.
M.S. Keener and C.C. Travis, Positive cones and focal points for a class of nth order differential equations, Trans. Amer. Math. Soc. 237(1978), 331-351.
8.
, Sturmian theory for a class of nonselfadjoint differential systems, Ann. Mat. Pura. Appl. 123(1980), 247-266.
9.
M.A. Krasnosel'skil, "Positive Solutions of Operator Equations," Fizmatgiz, Moscow, 1962; English translation P. Noordhoff Ltd. Groningen, The Netherlands, 1964.
10.
M. Krdn and M. Rutman, Linear operators leaving invariant a cone in a Banach space, Trans. Amer. Math. Soc. 26(1950), 1-128.
11.
K. Kreith, A class of hyperbolic focal point problems, Hiroshima Math. J. 14(1984), 203-210.
12.
A. Peterson and J. Ridenhour, Comparison theorems for Green's functions for right disfocal boundary value problems, Recent trends in Ordinary Differential Equations, Edited by R.P. Agarwal, WSSIAA, 1(1992), 493-^506.
13.
K. Schmitt and H.L. Smith, Positive solutions and conjugate points for systems of differential equations, Nonlinear: Anal. 2(1978), 93-105.
14.
E. Tomastik, Comparison theorem* for second order nonselfadjoint differential systems, SIAM J. Math. Anal. 14(1983), 60-65.
15.
, Comparison theorems for conjugate points of nth order nonselfadjoint differential equations!, Proc. Amer. Math. Soc. 96(1986), 437-442.
16.
C.C. Travis, Comparison of eigenvalues for linear differential equations of order 2n, Trans. Amer. Math. Soc. 177(1973), 363-374.
WSSIAA 3 (1994) pp. 179-185 © World Scientific Publishing Company
179
A L O W E R B O U N D FOR T H E ZEROS OF T H E BESSEL F U N C T I O N S A R P A D ELBERT Mathematical Institute of the Hungarian Academy of Sciences Budapest, P.O.B. 1S7, H-1S64, Hungary and ANDREA LAFORGIA Dipartimento
di Matematica,
Universita dcgli Studi delta Basilicata
Via Nazario Sauro, 85 — 85100 Potenta,
Italy
ABSTRACT. We establish the following lower bound A3 i ivk >v + ak(v + - y - ) * ,
">0,
* = 1,2,...
"k
where j„i denotes the tth positive zero of the Bessel function J*(T) of the first kind, a* — H 2 - 1 ' 3 , A/, = 2otV2ok/3, and n is the Jbth positive zero of the Airy function Ai(x). The result gives stringent numerical results for large values of i/, but also for u near to zero it improves known results. Similar inequality holds for the kth positive zero yvk of the Bessel function Yv(z) of the second kind. Additionally some comparisons of inequalities in question are given.
1. Introduction In the literature there exist many lower bounds for the zeros j„i, (k = 1,2,...) of the Bessel function Jv(x) of the first kind. For example we recall here the following ones rfc > jot +",
v>0
.*>** + " | - j ,
(LI)
0
, ! > « ' + 1.865757f* + 0.5«/-*,
(1.2) v> 1
,i > [ i / 2 - 1 9 + 6(2i/2 + 10i/+17)i] ,
(1.3) «/>-l
(1.4)
3,i > [ f ( " + !)[-(" + 2)(^ + 7)+ +[(«/ +2)(i/ 3 + 19i/2 +131./+ 257)]*]] ,
«/>-!
(1.5)
1980 Uatkemaiics Subject Classification (1085 Revision). Primary 33C10. Key words and phrases. Bessel functions, zeros, asymptotics, inequalities. Work sponsored by Consiglio Nasionale delle Rkercbe of Italy and (partially) by Hungarian National Foundation for Scientific Research Grant No. 6032/6319 .
180
" > -1
irt > >/(" + !)(«' +«). 2 4 ( , / + 1)2
(!■«) 2
& > r - 2{v - 1 ) , y 1 V " l-2
v >-l.
(1.7) '
V
The bound (1.1) given by Laforgia and Muldoon" is the best possible linear bound valid for any v > 0. The bound (1.2) was established by Hethcote4 and it gives good numerical results for large k. Also inequality (1.3) is due to Hethcote5. This inequality is very stringent for large v, but it has the disadvantage to hold only for ifc = 1. Inequalities (1.4) and (1.5) are due to Ahmed and Calogero1 and they are stringent for small values of v. Finally (1.6) and (1.7) have been recently established by Lorch7. The need of these inequalities was evinced by M.S. Ashbaugh and R.D. Benguria2 who have applied them to find bounds on the ratios of eigenvalues of some partial differential equations. The inequality (1.6) was proved by A. Elbert* for — 1 < v < 0, who indicated a proof for 0 < v < 1 and conjectured the validity for v > 1. The purpose of the present paper is to establish a new bound for the positive zeros jVk (Jfc = 1,2,...). The advantage of our result is that it gives good numerical results for small and for large values of u, as well. Our result reads as follows: THEOREM. For v>0 and k = 1,2,... let j„t and xk denote the k-th positive zero of the Bessel function J„(x) and the k-th positive zero of the Airy function Ai(x), respectively. Then
j,*>i' + a*(«'+4h*.
«S'
"£°»
* = 1,2,...
(1-8)
where ||,
Ak = |o*>/E£.
(1.9)
The basic ingredient to our approach is the Olver's asymptotic formula 8 : j»k = v + akvl + — aji/"i + 0 ( i / _ 1 ) ,
i/-»oo.
(1.10)
2. Some comparisons In this section we carry out some comparisons concerning the accuracy of inequalities (1.1-7) for the zero j u \ on (0, oo) and also for our new inequality (1.8). At v = 0 inequalities (1.4-7) give the following lower bounds for j'oi: 2.3955446..., 2.40438046..., 2.23606 2.3576306..., reap. These values suggest that among the inequalities (1.4-7) the sharpest inequality is (1.5), the next one is (1.4), which is followed by (1.7) and the weekest one is (1.6). The fact that (1.7) provides a sharper bound for j„i than does (1.6) was settled by L. Lorch7. In what follows we are going to show that A) (1.4) is sharper than (1.7), and then B) (1.5) is sharper than (1.4). A) Since the fraction in (1.7) can be written in the form \{v + l)[ v /(2i/ -I- 3)(2i/ + 11) + 2v — 1], by using the notation I = I/ + 1 we have to show 2x2 - 5x - 24 + 8\/2i 2 + 6x + 9 > xs/lx1
+ 20i + 9,
i > 0.
(2.1)
181 Clearly, it is sufficient to show that the equation 2x 2 - 5x - 24 + 8 \ / 2 x 2 + 6x + 9 = xs/ix1
+ 20x + 9
(2.2)
has no zero in (0, oo). Eliminating the square roots in two steps we get 64x 4 (7x 2 - 4x - 160) = 0. This equation has the only zero x\ = (2 + \/1124)/7 in (0, oo) which is no proper zero of (2.2) because in (2.2) we have at x = i i for the left-hand side 78.439308... while 74.16195... for the right-hand side, and this contradiction proves (2.1). B) As in case A) we have to show that the equation 2xy/x* + 17x 3 + 112x 2 + 240x + 144 - 18\/2x 2 + 6x + 9 = 2x 3 + l l x 2 + 6x - 54
(2.3)
has no zeros in (0, oo). Eliminating again the square roots we get 81x 4 (97x 4 + 680x 3 + 304x 2 - 2304x + 100) = 0.
(2.4)
By numerical calculation we get two positive zeros of Eq. (2.4): Xi = 0.04367926... and x 2 = 1.47399891... and (2.4) can be written in the form 81x 4 (x - n ) ( x - x 2 )(97x 2 + 827.2147 ...x
+ 1553.200...) = 0.
Hence the only positive zeros could be x i , x 2 . However, a direct calculation gives in (2.3) -53.7053227... >
-53.7053238...
at x = n
-1.3943672... >
-1.8162129...
at x = x 2 ,
which proves that inequality (1-5) is stronger than (1.4). Comparing the inequalities (1.1-3) with the ones (1.4-7) we can make the following observation. Undoubtedly, (1.1) is precise around v = 0 but for large i/'s it gives poor results. E.g. inequality (1.6) yields j v i > v + 3 + O(^) for large v while (1.1) gives j„i > 2.4048 ■ ■ ■ + v. Inequality (1.2) holds only on the finite interval (—1/2,1/2) so its use is rather restricted. Thus we arrive at inequality (1.3) which is comparabile with (1.8) since a\ = 1.855757... according to Tricomi 8 . In (1.8) we get for large i/'s: j„i > v + a\vll* + 0{v~2l3) while in (1.3) the third term is 0.5i/ - 1 / 3 , i.e. (1.3) gives a sharper lower bound for j„i than (1.8) provided v is sufficiently large. By numerical evidence we find that this occours already when v > 16. In the interval (0,16) our formula (1.8) is in any respect better than (1.3) not mentioning the fact that (1.8) holds also for j„ 2 , j„3, . . . . Applying (1.8) to k = 1 and comparing with inequalities (1.4-7) we find that inequality (1.8) is always sharper than the ones (1.6-7) and it is sharper than (1.4) for u > 3 or (1.5) for v > 9, resp.
182 3. Proof of the result Let us define the function H = H{v) in the following way i/arccos V- ,
H = H(u) = y/p -v*-
v >0
(3.1)
where j = j„t. First we wish to prove that H > At where Ak has been defined by (1.9). In the sequel the subscripts k in At and a* will be omitted. In order to prove the inequality H > A we show that lim H{u) = A, H'(u) < 0. (3.2) V
'OO
We recall the well-known Watson formula 9 p.508 j ' = ^ = 2j f°°K0(2j sinhtje- 2 "' dt dv J0 where Ko(x) denotes the Bessel function with imaginary argument. By the decreasing character of K0(x) which follows immediately e.g. from the integral representation 9 p.181 K0(x) = (°° Jo
e-xco*zc
we find j ' <2j I Ko{2it)e-2vt Jo
dt.
With the substitutions u = 2jt,
*=cos0 J
(O<0<^) 2
the inequality above can be written in the following way j'<
Jo
rK0(u)e-"com,du.
The integral on the right-hand side can be found in 9 p.388
H K0{u)c-*™tdu = -?-.
Jo
sin a
thus we have sinfr
Clearly, the substitution K = cos8 in (3.3) is possible because 9 p.485 Uk>v,
u>0,
A: = 1,2,....
(3.3)
183 With the notations (3.3) the function H can be written as H{v)=
j{s\ad-0coa9).
Taking into consideration the asymptotic formula (1.10) we get lim„_oo 0{v) = 0 and lim H(u) = Urn j[sin0 - flcosfl] = lim j • {6* = i lim j sin3 0 = ± lim j ( l - cos2 0)* 3
3
I/--00
X-.00
= i U m [ , ( l - 7 ) ' ] ( l + 7)? = i 2 t a t = ^ . This shows the limit relation in (3.2). For the proof of the second relation, a differenti ation of H{v) gives H\v) =
J
/ ~
V
- (arccos V- -
Now we use the inequality (3.4) and the relation K = cosfl. A simple calculation shows that the inequality H' < 0 is equivalent to 6 sin 0(sin 6 — 1) < 0 which is clearly true. Thus we have H(u) > A as we have claimed. By (3.1) the function H(u) has the following integral representation ax
H(u) = / Jv
and H(v) > A. Therefore if we put K = Kk(w) = v + a(v + £)i,
(3.5)
then in order to show the inequality K < j it is sufficient to prove the inequality
I
-dx < A.
This inequality is equivalent to /
f
V K2 — v2 — v arccos — < A K VK* - i/» - A
v < arccos — . v will imply alsoKthe relation We may remark here that this inequality
VK* -u2-A 0<-
* <-r.
v
2
(3.6)
184 With the substitution v = mA the function K defined by (3.5) can be written as A3 l 9 1 K = mA + a(mA + —)» = A[m + (1 + -m)3] and we have to show in (3.6) that ^[m + O + f n O t y - m ' - l -* < arccos m m Let us introduce the following new notations
+
(i
r
+
.
(3.7) >
|m)i
* = * ( m ) = (l + fm)* p = p(m) = m + X
(3g) 2
2
. . m V'P - m - 1 ft = «(m) = arccos . p m Thus inequality (3.7) is reduced to h(m) > 0. Since lim h(m) = £ - — > 0,
m—0
2
8
lim Wro) = 0
m—oo
v
'
we need to show only that h'(m) < 0. A differentiation gives W^PVP* ~ m2h'(m) = —m(p2 — m2)p' + p(p2 — m 2 ) — py/p2 — m 2 and we must prove that the right-hand side is negative, or equivalently, that (p2-m2)(l
+
fm)2
By means of the notations introduced in (3.8) this inequality is equivalent to P{X) = (1 + \m)X2
+ ( - 1 + i m + \\m2)X
- 2m - 2m2 > 0
where on the left-hand side the function P(X) is a quadratic polynomial of X. We observe that X > 1 and by (3.8) it follows * > l + fm-£m2. Moreover
1 + |m - £m2 > 1
for 0 < m < | .
The polynomial P(X) has the following properties: P ( l ) < 0, P'(l) > 0. Thus when 0 < m < | we have only to check the sign of P(X) at X = 1 -I- | m — g^m2. A direct calculation gives Pfl J. 3 _ ,
8 „ 2 \ _ a S _ 2 , 1S3„,3
17-81 _ 4 , 9-81 „ 5 ^ 1 5 3 ^ 3 / ,
0
\
185 This proves the inequality P(X) > 0 when 0 < m < | . Now let m > | . We find
»
X = (l + fm)i>(l + f . f ) i = 2 i > § . Moreover the polynomial P(X) can be written as a polynomial of m, too: P(X) = (%X-
2)m2 + (fX 2 + \X - 2)m + X2 - X.
Using X > | we can see that every coefficient of this polynomial is positive. Thus P(X) > 0 also for m > | . The proof of the theorem is complete. Remark. Looking over the above proof we can observe that the proof is essentially based on three facts: on the Olver's asymptotic formula, on the Watson formula for dj/dv and on the inequality j„i > u. Concerning the first one we have used actually a weaker result, namely liml,_<x> j ( l - y) 3 ' 2 = a 3 / 2 . Thus our inequality (1.8) could be extended to other zeros c„t of the cylinder functions Cv(x) = cos aJ„(x) — sin ay„(i), (0 < a < *•), subject to c„» > v, where Yr(x) denotes the Bessel function of second kind, provided the corresponding asymptotic formula (1.10) is already known. This is the case concerning the zeros y„i of Y„(x) (where a = 7r/2) and according to 8 Vvk = v + bkf* + 0(y~*)
as v —* oo.
Particularly, we have &i = 0.931577.... Thus we can derive the inequality
(
£3\i
9
v + Jf) , Bk = -bky/2bk which could be of good use because for the zeros y„fc there is not so great variety of inequalities as it is for the zeros of jy*. REFERENCES 1. S. Ahmed and F. C&logero, On the zeros of Bessel functions, Lett. Nuovo Cimmlo, 2 1 (1978), 531-534. 2. M.S. Ashbaugh and R.D. Benguria, A second proof of the Payne- P61ya- Weinberger conjecture, Comm. Math. Pkyt., 147 (1992) 181-190. 3. A. Elbert, Some inequalities concerning Bessel functions of the first kind, Studia Set. Math. Hungar., 6 (1971), 277-283. 4. H.W. Hethcote, Bounds for zeros of some special functions, Proc. Amer. Math. Soc., 26 (1970), 72-74. 5. H.W. Hethcote, Error bounds for asymptotic approximations of zeros of trascendental functions, SIAM J. Math. AnaL, 2 (1) (1970), 147-152. 6. A. Laforgia and M.E. Muldoon, Inequalities and approximations for zeros of Bessel functions of small order, SIAM J. Math. Anal., 14 (1983), 383-388. 7. L. Lorch, Some inequalities for the first positive zeros of Bessel functions, SIAM J. Math. Anal., 24 (3) (1993), 814-823. 8. P.W.J. Olver, The asymptotic expansion of Bessel function of large order, Philos. Trans. Roy. Soc. London, 2 4 7 a (1954), 328-368. 9. G.N. Watson, A Treatise on the Theory of Bessel functions, 2nd ed., Cambridge University Press, 1944.
WSSIAA 3 (1994) pp. 187-196 © World Scientific Publishing Company
187
Comparison of Eigenvalues for a System of Two-Point Boundary Value Problems Paul W. Eloe Department of Mathematics University of Dayton Dayton, Ohio 45469, U.S.A. Johnny Henderson Department of Discrete Mathematics and Statistics Auburn University Auburn, Alabama 36849, U.S.A. Abstract The theory of tio-positive operators with respect to a cone in a Banach space is applied to a class of two-point boundary value problems for a system of linear ordinary differential equations. The existence of a smallest positive eigenvalue is established, and then a comparison theorem for smallest positive eigenvalues is obtained.
1
Introduction
Let n > l , l < f c < n - l , and 0 < j < k be given. We are concerned with comparing eigenvalues for the eigenvalue problems,
(l.i)
y(n) = A . £ * ( * ) y ( 0 , .=0
and (1.2)
y'-^A^OiWyW, 1=0
with eigenvectors in each case satisfying the boundary conditions,
(1.3)
y'')(o) = 0, 0 < t < k - 1, y(0(6) = 0,
j
188 where Pt(x) and Qt(x), 0 < t < j , are continuous mxm
matrix functions on [a, b]. In particular,
when P,(x) and Qi(x) are positive with respect to an appropriate cone in a Banach space and satisfy certain inequality comparisons with respect to that cone, we shall compare the smallest eigenvalues A, and A3 of (1.1), (1.3), and (1.2), (1.3), respectively. Our methods for the comparison of these eigenvalues involve applications of sign properties for a scalar Green's function, followed by applications from the theory of uo-positive operators with respect to a cone in a Banach space. Several authors have successfully applied these techniques in comparing eigenvalues for two-point conjugate or two-point right focal boundary value problems; see, for example, [1, 2, 9, 13, 15, 16, 18, 19, 20, 22]. And in fact, for the scalar case, the cone theory methods were applied in [5] in comparing eigenvalues for (1.1), (1.3) with eigenvalues for (1.2), (1.3). We also mention that similar comparison results have been obtained in [6] and [10] for eigenvalue problems for multipoint conjugate boundary value problems. Fundamental to our approach for the systems (1.1) and (1.2) are the constructions used in [13]. Moreover, we also cite the works in [12] and [14] in which the cone theoretic methods are used for eigenvalue comparisons in the setting of difference equations. Finally, in some related papers, the theory of cones in a Banach space, to characterize solutions of eigenvalue problems that lie in a cone and corresponding extremal points, has been used effectively in [3, 4, 7, 8, 11]. This paper is developed so that, to be self contained, we include in Section 2 preliminary definitions and results from the theory of cones in a Banach space. In Section 3, we apply the results of Section 2 in comparing the smallest positive eigenvalues of (1.1), (1.3) and (1.2), (1.3). Also, in Section 3, sign properties of scalar Green's functions from [5] are stated, which are principal in this application of the theory of u0-positive operators with respect to a cone in a Banach space.
2
Cones and uo-positive operators
In this section, we provide definitions and auxiliary results from cone theory which we will apply in the next section to the eigenvalue problems (1.1), (1.3) and (1.2), (1.3). Most of the discussion of this section involving the theory of cones in a Banach space arises from the results in Krasnosel'skii's book [17]. Let calB be a Banach space over the reals. A closed, nonempty set V C B is said to a cone
189 provided (i) ou + /3« e V, for all u, v G V and all a, f) > 0, and (ii) u, - u G V implies u = 0. A cone, V, is said to be reproducing, if B = V - V. A cone, V, is said to be solid if V / 0, where P 0 is the interior of V. Remark. Krasnosel'skii [17] proved that every solid cone is reproducing. A Banach space B is called a partially ordered Banach space, if there exists a partial ordering, <, on B satisfying (i) u < v, for u, v G B, implies tti < tv, for all I > 0, and tu > tv, for all t < 0, and (ii) ui < vt and Uj < Uj, for «i, Uj, »i, t>j 6 B, imply i»i + Uj < t>i + t>j. Let V C B be a cone and define u < v, if and only if » - u G V. Then < is a partial ordering on B, and we shall say that < is the partial ordering induced by V. Moreover, B is a partially ordered Banach space with respect to <. Let M, N : B -» B be bounded, linear operators. We say that M < N with respect to V, provided Mu < Nu, for all u 6 V. We say that a bounded, linear operator M : B —► B is lie-positive with respect to V, if there exists tio G V, UQ / 0, such that for each nonzero uGV,
there
exist positive real numbers fc,(u),fc2(u) such that iiu 0 < Afu < A;}uo. Of the next three results, the first two can be found in Krasnosel'skii's book [17], and the third result is proved by Keener and Travis [15] as an extension of results from [17]. Lemma 2.1 Let B be a Banach space over the reals and let V C B be a solid cone. If M : B -» B is a linear operator such that M : P\{0} —► V,
then M is u0-positive with respect to V.
Theorem 2.2 Let B be a Banach space over the reals and let V C B be a reproducing cone. Let M : B -* B be a compact, linear operator which is u0-positive with respect to V. Then M has an essentially unique eigenvector in V, and the corresponding eigenvalue is simple, positive, and larger than the absolute value of any other eigenvalue. Theorem 2.3 Let B be a Banach space over the reals and letV C B be a cone. Let M, N : B -► B be bounded, linear operators, and assume that at least one of the operators is uo-positive with respect toV.
If M < N with respect to V, and if there ezist nonzero ult uj G V and positive real number*
A[ and Aj, such that Mut > A[Ui and Nu? < AJUJ, then Ai < Aj. Moreover, if A] = Aj, then ut is a scalar multiple of u2.
190
3
Comparison of eigenvalues for (1.1), (1-3) and (1.2), (1.3).
In this section, we apply the results from Section 2 in comparing smallest positive eigenvalues Ai and Aj of (1.1), (1.3) and (1.2), (1.3), respectively. The techniques we present here are valid for 0 < j < k — 1 and are not valid for the case j = it. However, the case j = k has been resolved in [21]. We assume hereafter that 0 < j < k - 1 is fixed. Our pattern of development involves sign properties of a scalar Green's function. Namely, let G(x,s) denote the Green's function for the scalar equation
yM = 0,
(3.1)
satisfying the boundary conditions (1.3). Eloe and Henderson [5] proved that
(-l) n -*£-rG(x,a) > 0 on (a,b) x (o,6),0 < i < j .
(3.2)
It was also established in [5] that
(-ir-k-^iG{a,s)>0,a<s
(3.3) and (3.4)
£^
J
G(M>0,a<3
We now introduce a suitable Banach space and appropriate cone in order to apply the results on Uo-positivity from Section 2. Let the Banach space B be given by 8 = { t i 6 C ( "-"([a,6],R m ) | u satisfies (1.3)}, with norm WI =
O
0<>Sn
where | ■ | denotes the Euclidean norm on Rm. Let AC be a solid cone in R"1 and then define the cone
VCBby V = {u e B | ( - l ) - ' u ( , ) ( i ) 6 AC,a < x < 6,0 < t < j).
191 We shall also assume hereafter that, for each 0 < i < j , Pt(x) and Qi(x) are continuous
mxm
matrix functions on [0,6], (-1)—'fi(z)AC C AC and (-l)—*0i(z)A: C AC on [a,b], and that, for each nontrivial u € V, there exists an x„ € [a,b] such that both (-1)""*P 0 (x u )u(x u ) n
G AC" and
0
(—l) ~*Q<,(z„)u(zu) € AC . (We will repeat these assumptions in some of our theorems that follow.) Now, for the eigenvalue problems (1.1), (1.3), and (1.2), (1.3), we consider in an equivalent manner eigenvalues of the linear operators M, N : 8 —» B defined by
G(x,s)Y^PM^i)(.3)ds,
(3.5)
Mu(x)=
a<x
(3.6)
7Vu(i)= / G(i,«)^Qi(a)u ( "(s)rfa,
a < x < b.
Remarks, (a) Standard arguments employing properties of G(x,s) and applications of the Arzela-Ascoli Theorem yield that M and N are compact operators. (b) Ai / 0, for all eigenvalues Ai of (1.1), (1.3). So, if u(i) is an eigenvector corresponding to an eigenvalue A] of (1.1), (1.3), then /•»
i
u(x) = / G(x, s)Xl Y, Pi(s)vf-'\s)d3,
a<x
(i.e., — u = Mtt). That is, eigenvalues of (1.1), (1.3) are multiplicative inverses of eigenvalues of Ai
(3.5), and conversely. (Analogous statements hold for eigenvalues of (1.2), (1.3), and (3.6).) Theorem 3.1 The cone V is solid, and hence reproducing, and the operators, M and N, are Uo-positive with respect to the cone V. Proof: We will show that the operator M : 7>\{0} -♦ V,
(duplicate arguments verify the same
for N). We first verify that M : V -* V. Thus, choose u 6 V. From (3.2) and the assumption (-l)"-kPi(x)fC
C AC, o < x < 6, 0 < : < j , it follows that
(-l)-*(tf «)<'>(*) = A-l)-*£jG(*,.) J](-l)-»P<(«)(-ir-*i«
Thus, Mu e V.
192 Next, let 0 j* u e V, and define h by />(x) = M « ( i ) = /
G(*,«)£P<(«)u w («)
It follows again from (3.2), the assumption (-l)°~*P,(x)AC C AC, a < i < 6, 0 < t < j , and the existence of an z . 6 [a, 6] such that (-l)"-tP0(xu)u(xu)
6 AC', that for 0 < « < j ,
(-l) n -'/.('>(z) € K',a < x < b. As pointed out in [13], this does not imply that h €
V.
To see that, in fact, h € V, assume for the sake of contradiction that A £ 7". Bnt /i 6 ? C 8 , and so there exists a sequence {y r } C B\V such that y, -» h in C("_1)-norm. Thus, for each r > 1, there exist xr 6 [a, 6] and 0 < i, < n — 1 such that
(-1)~ V't*,) * AC. By pawing to a subsequence and relabeling, we may assume all ir are equal, say tf = t0, r > 1 (i.e., y^*\xr)
$ Ky r > 1), and also that xr -> z 0 , some x0 € [a,b]. There are 3 cases: x0 € (a,6),
If *o G (a, 6), since y<*»(xr) -> h("(*o), 0 < t < n - 1, and since (-l)"-*Aj<'»>(xo) € AC% then for r sufficiently large, (-l)"~*y£'°'(x,.) G AC°, which is a contradiction. Next, let x0 = a (the argument for i 0 = 6 is similar). Since
*(i)(a) = [ £ G (<M)E*W u(,) w
where £ « , , , . . . , { „ „ • ) = (yiJ +1, (fir),...,»< m ', +1) «m,)) r , a < {., < x r , 1 < i < m, r e N. Thus,
and since y<*'(o) -> A»<*>(q), (-1)-*/><*>(O) 6 AC', and iJ(fir,-.-,{mr) n
o)
0
h ( t + 1 ) (a), as r -
follows that for r sufficiently large, (-l) "*y}' (i r ) 6 AC ; again, a contradiction.
oo, it
193 We thns conclude that h 6 V so that, in turn, M : V\{0} -> V.
In particular, V ± 9 so that P
is solid and hence a reproducing cone. Lemma 2.1 implies M is tio-positive, since M : P\{0} —» 7 " . The proof is complete. Theorem 3.2 Assume that, for0 < « < j , Pi(x) is a continuous mxm k
on [a,b], that (-l)"~ Pi(x)/C that (-1)""''Pa(z a )u{x u )
real-valued matrix defined
C K, and that, for each 0 / « £ P , there exists an xa € [o,6] such
6 K.'. Then, M has an essentially unique eigenvector u € V,
and the
corresponding eigenvector A is positive, simple, and larger than the absolute value of any other eigenvalue. Proof: Theorem 2.2 establishes the existence of such an eigenvalue A and an eigenvector u € V. From the proof of Theorem 3.1, Afu 6 V°. Since Mu = Au, it follows that u e P°. Theorem 3.3 Assume that, for each 0 < < < j , P,(x) and Q.(z) are continuous mxm
real-valued
n
matrix functions defined on [a,b], that (-l)—*P<(x)JC C K and (-l) -*<3.(z)£ C K on [a,b], and that, for each 0 ^ u € V, there exists an xu 6 [a,6] such that both (-l)"-tP0(xu)u(x.)
€ K° and
n
(-l) -*Qo(*uM*o) € K°. Moreover, assume that (-l)"-*P,(z) < (-1)—*
Let Ai and Aj 4e the largest positive eigenvalues of M and N, respectively.
Then A] < A3. Furthermore, if Ai = Aj, then, for each 0 < » < j , Pi(x)ut(x) = QMx),a
< x < b,l = 1,2,
uiAere Ut(x), I = 1,2, is the essentially unique eigenvector in V° corresponding to A<, I = 1,2. Proof: The hypotheses on Pj(i) and Qi(x), 0 < » < j , imply that M < N with respect to V. If U( 6 V, t = 1,2, are the essentially unique eigenvectors given by Theorem 3.2 corresponding to At, I = 1,2, then Theorem 2.3 implies A, < A,. Now, if it is the case that, for some 0
j,
(-lr-'p^oKOro) ? (-lr-'QiMufa), for some zo, then there exists x € [a, 4] such that,
(-iy-*W4(*)-fi(*))m(*) €«:•.
194 One can then argue as in the proof of Theorem 3.1 that (N - M)U] 6 V° (using (<Jj — P() in place of Q, for the argument). But u\ € V, and so for sufficiently small t > 0, (N - Af )t»i > tut. Thus A'ui > Afti! + tui = (A! + e)t»|. Yet Nu2 = AJ«J and so applying Theorem 2.3 to N, we have A! + e < A2, or Ai < A3. Thus, A t = Aj implies P,(*)u,(*) = Q i ^ M * ) ,
o<x<6,
0
Finally, since Theorem 2.3 implies U] and u3 are scalar multiples whenever Ai = A,, we also have that Ai = A] implies P , ( I ) U J ( Z ) = 0,(z)u 2 (z),
a < x < b,
0
The proof is complete. In view that eigenvalues of Af are multiplicative inverses of eigenvalues of (1.1), (1.3), and conversely, and in view of Theorems 3.2 and 3.3, we conclude by stating a comparison theorem for smallest positive eigenvalues At and A2, of (1.1), (1.3), and (1.2), (1.3), respectively. Theorem 3.4 Assume the hypotheses of Theorem S.S. Then, there exist smallest positive eigen values At and A2 of (1.1), (1.3), and (1.2), (1.3), respectively, each of which is simple and less than the absolute value of any other eigenvalue for the corresponding problem, and the eigenvectors, u, and u 2 , corresponding to X\ and Aj, may be chosen to belong to V.
Finally Aj < Ai, with Ai = Aj
implying Pi(x)ut(x)
= Qi(x)ut(x),
a<x
0 < t'j,
*=1,2.
References [1] S. Ahmad and A. Lazer, Positive operators and Sturmian theory of nonselfadjoint second-order systems, Nonlinear Equations in Abstract Spaces, Academic Press, New York, 1978, pp. 25-42. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. [3] P. W. Eloe, D. Hankerson and J. Henderson, Positive solutions and conjugate points for mul tipoint boundary value problems, J. Differential Equations 05 (1992), 20-32.
195 [4] P. W. Eloe, D. Hankerson and J. Henderson, Positive solutions and ./'-focal points for two point boundary value problems, Rocky Mtn. J. Math. 22 (1992), 1283-1293. [5] P. W. Eloe and J. Henderson, Comparison of eigenvalues for a class of two point boundary value problems, Appl. Anal. 34 (1989), 25-34. [6] P. W. Eloe and J. Henderson, Comparison of eigenvalues for a class of multipoint boundary value problems, Recent Trends in Ordinary Differential Equations 1 (1992), 179-188. [7] P. W. Eloe and J. Henderson, Focal points and comparison theorems for a class of two point boundary value problems, J. Differential Equations 103 (1993), 375-386. [8] P. W. Eloe and J. Henderson, Focal point characterizations and comparisons for right focal differential operators, J. Math. Anal. App., (in press). [9] R. D. Gentry and C. C. Travis, Comparison of eigenvalues associated with linear differential equations of arbitrary order, Trans. Amer. Math. Soc. 223 (1976), 167-179. [10] R. D. Gentry and C. C. Travis, The existence and extremal characterization of eigenvalues for an n " order multiple point boundary value problem, Ann. Mat. Pura. Appl. 126 (1981), 223-232. [11] D. Hankerson and J. Henderson, Positive solutions and extremal points for differential equa tions, Appl. Anal. 39 (1990), 193-207. [12] D. Hankerson and J. Henderson, Comparison of eigenvalues for n-point boundary value prob lems for difference equations, Differential Equations: Stability and Control 127 (1990), 203-208. [13] D. Hankerson and A. Peterson, Comparison theorems for eigenvalue problems for n'h order differential equations, Proc. Amer. Math. Soc. 104 (1988), 1204-1211. [14] D. Hankerson and A. Peterson, Comparison of eigenvalues for focal point problems for n1* order difference equations, Diff. and Int. Equations 3 (1990), 363-380. [15] M.S. Keener and C. C. Travis, Positive cones and focal points for a class of n"1 order differential equations, Trans. Amer. Math. Soc. 237 (1978), 331-351.
196 [16] M.S. Keener and C. C. Travis, Sturmian theory for a class of nonselfadjoint differential systems, Ann. Mat. Ptira Appl. 12S (1980), 247-266. [17] M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Fizmatgiz, Moscow, 1962; English translation P. Nordhoff Ltd., Groningen, The Netherlands, 1964. [18] K. Schmitt and H. L. Smith, Positive solutions and conjugate points for systems of differential equations, Nonlin. Anal. 2 (1978), 93-105. [19] E. Tomastick, Comparison theorems for second order nonselfadjoint differential systems, SIAM J. Math. Anal. 14 (1983), 60-65. [20] E. Tomastick, Comparison theorems for conjugate points of n"* order nonselfadjoint differential equations, Proc. Amer. Math. Soc. 96 (1986), 437-442. [21] E. Tomastick, Comparison theorems for focal points of systems of n"1 order nonselfadjoint differential equations, Rocky Mtn. J. Math. 18 (1988), 1-11. [22] C. C. Travis, Comparison of eigenvalues for linear differential equations of order 2n, Trans. Amer. Math. Soc. 177 (1973), 363-374.
WSSIAA 3 (1994) pp. 197-205 © World Scientific Publishing Company
DIFFERENTIAL INEQUALITIES FOR A SINGULAR BOUNDARY VALUE PROBLEM
PAUL W. ELOE AND JOHNNY HENDERSON
ABSTRACT. A boundary value problem for a singular ordinary differential equation, defined on a bounded domain, is studied. A uniquely determined and absolutely integrable Green's function is constructed such that standard fixed point theorems can be applied. In this paper, sign properties of the Green's function are determined in order that monotone methods and the Schauder fixed point theorem can be applied. Several examples are given and in one example, a delay equation is studied.
Let a(z) 6 C'(0,1]. In this paper, we shall employ a Green's function and differential inequalities to study the singular boundary value problem (BVP),
(i)
{<*)v'Y = !(*, y, v'), o<x< 1, y (i) = o,
where we assume a(0) = 0, limx_g+ a '( x ) exists and is +oo. Throughout this paper we shall denote this limit by a'(0). We also assume that a(z) > 0,0 < z < 1. Note then that the linear differential operator associated with (1) has a singular point at x = 0. We shall say that y is a solution of (1) if y satisfies (1) and y 6 0^(0,1] n C 1 [0.1]Applications of differential inequalities to BVPs without singular points are vast. We refer the reader to some of the references provided in [1]. In this paper, we shall obtain a uniquely determined Green's function associated with (1). In addition, we shall show that the Green's function is absolutely integrable on [0,1] and show that standard fixed point theorems can be employed to obtain solutions of the singular BVP, (1). In particular, once the integrability of the Green's function has been addressed, we shall apply differential inequalities, monotone methods, and the Schauder fixed point theorem to obtain solutions of the singular BVP, (1). Existence of solutions of singular BVPs has been of interest to many authors in recent years. See [2] for a partial list of references. The techniques in this paper are most closely related to the techniques employed in [3] where a Green's function for a linear differential operator with a singularity is employed; however, in [3] the authors also allow for singularities in the nonlinear term, / , and so, the intention of this paper is different than that of [3]. 1991 Af«(*em«li
197
198 Define 0(x) = ff l/a(u)du. Note that / 0 (s)ds exists. To see this, integrate /» 4>{*)ds by parts to obtain Jx 4>(s)ds = —x
lim x
If 0(0) is finite, then (2) is clear. If 0(0) is unbounded, (2) follows by L'hospital's rule. It also follows that fg (s/a(s))ds exists since /»'mx_0+(*/<»(*)) exists and is finite. In particular, x/a(x) is a continuous, bounded function on [0,1]. Thus, <j> is absolutely integrable on (0,1), since 0 is of constant sign. Remark. If 0(0) is finite, our assumptions on a relate to condition E in [2] which places restrictions on the singularity with respect to the independent variable. Define (
In what follows, we show that G(x, s) defined by (3) is the Green's function for the BVP, (ay7)' = 0,0 < x < l,y(l) = OJy'l bounded. Note that this BVP is uniquely solvable since 1 and 0(x) are linearly independent solutions of (ay1)' = 0,0 < x. Since 0 is absolutely integrable on (0,1), and since, x/a(x) is bounded and continuous on [0,1], the following lemma is valid and the details of proof are straightforward. We omit the details since similar details will be provided in the proof of Lemma 2. Lemma 1. Let h € C[0,1]. Then y(x) satisfies (ay1)' = h,0 < x < l,y(l) = 0. ly'WI bounded for 0 < x < 1 if, and only if, y(x) = f0 G(x,s)h(s)ds. Remark. If y(x) = /„ G(x,s)h(s)ds, then |y*| bounded follows from the mean value theorem and the fact that x/a(x) is bounded. Let B = Cl[0,1] with \\y\\ = max{||j/||o, |||/||o}i where || • ||0 is the supremum norm. Let / : [0,1] x RJ —» R be continuous. Define an operator, M, on B by (4)
My(x)=
[
G(x,8)f(s,y(s),y'(s))ds.
Jo
We first claim that M maps B into B. Let y 6 B. We argue that My is continuous at x = 0 and {My)' is continuous at x = 0. We do not provide details for x > 0. My(0) = Jo *(»)/(«, VWYM)** w well-defined since 4> is absolutely integrable on (0,1). lim r _ 0 + My(x) = lirn / % ( * ) / ( * , y(«), •(*))<**+ / *(«)/(«, V(«). •(»))<*•• «-o+ Jo Jx Thus, it is sufficient to argue that limz_o+ f0' d>(x)f(s,y(s),y'(s))ds = 0. This follows immediately from the mean value theorem and (2). To argue the continuity of (My)' at x = 0, note that for r > 0, (My)'(x) = f Gt(x, s)f{s, y{s), y'(s))d» = / * ( / ( « , y(s), y'(s))/a(x))ds. Jo Jo
199 Apply the mean value theorem and L'hospital's rule to obtain lim(My)'(r) = /(0,y(0),y'(0))/a'(0). x-»0+
Forx = 0,(My)'(0) = lim (My(fc) - My(0))/h = lim ( /*(*(*) - *(,))/(,, y(s),
AW')/*-
Jo
Since My is continuous at 0, the ratio here is an indeterminate form. Apply L'hospital's rule and then the mean value theorem to obtain that (My)'(0)= lim /
^'(/.)/(*,y(*),y'(*))<<S = /(0,y(0) 1 y'(0))/a'(0).
Thus, (My)' is continuous at 0 and we conclude that M : B —► B. Lemma 2. Let f : [a,b] x R2 -► R, and let M : B —■ B be defined by (4) . Then M is completely continuous. Proof. Let D C B be bounded. We argue that {M(D)} is uniformly bounded and that each of the sets {My{x) : y 6 D) and {(My)'(x) : y 6 D) are equicontinuous for each x 6 [0,1]. Let K = maxa<x
- *(*))/(,, y{s), y>(s))d,\.
In particular, since
K I f (du/a{u))duds = K f f (l/a{u))dsdu = K f (u/o(u))
200 We now address the equicontinuity of {(My)'(x) : y € D) for each z € [0,1]. Let 0 < z < 1,0 < * < 1, and let y € D. Then |(My)'(x) - (My)'(x)\ = I f'f(»Ms),A'))d*Mx)Jo < K\l/a(x)
[' Jo
f(sM»U('))ds/a(z)\
- l/a(z)\ + | j f /(.,y(,),y'(,))d,/a(*)|.
In particular, the equicontinuity of {(Af y)'} at z > 0 follows from the continuity of 1/a at z > 0. For z = 0, let 0 < * < 1, y £ D. Apply the mean value theorem to find c € (0, *) such that |(My)'(0) - (My)'(z)\ = |/(0,^0),i/(0))/a'(0) - /(c,,( C ), j/(c))z/a(*)| <
Kz/a(z).
In particular, the equicontinuity of {(My)'} at z = 0 follows from the property a'(0) = oo. We point out that it is only in the argument to show equicontinuity at x = 0 that the condition a'(0) = oo is required. This completes the proof of Lemma 2. The following theorem is an easy consequence of Lemma 1. Theorem 3. Assume y € B. Then y is a solution of the BVP (1) if, and only if,
y(x)= f Gfx.^.yW.y'OO)^. Jo
Now that Lemma 2 and Theorem 3 have been established, note that standard fixed point theorems can be applied to obtain the existence of solutions of the BVP, (1). In addition, observe that (5)
G(x, a) < 0, (x,«) € (0,1) x (0,1), Gx(x,») > 0, (x, s) 6 (0,1) x (0,1),
where Gz = (d/dx)G(z,$). For the remainder of this paper, we shall exploit Theorem 3 and (5) and employ the method of upper and lower solutions to obtain the existence of solutions of the BVP, (1) . We shall present three examples. For notational simplicity in the presentation let Ly(x) denote the differential operator (a(zy(x))',0<x
201 In particular, if yi < ri.zj < yj, then (6)
/ ( * . Vi, !fe) > / ( * , ^i. *?)■ * € [0,1].
Define a partial order on fl as follows: for y, z 6 B, y < z if, and only if, y(x) < *(x),0 < x < 1 and ^(x) < y'f.xJ.O < x < 1. From (5) and (6) , it is clear that the operator M : B —» B, defined by (4) , is an increasing map with respect to the partial order on B. In particular, if y, z € B, then (7)
V
My<
Mz.
Remark. Throughout the remainder of the paper, if an inequality is employed and no argument is supplied, we mean to employ the partial order on B. If an argument is supplied, then the partial order denotes the usual partial order on R. Theorem 4. Assume f : [0,1] x RJ is continuous and satisfies (6). Assume there exist wi.Vi £ fl satisfying i) xti\
ii)
W l (l)
< 0 < t/,(l),
Hi) Lwi(x) > / ( x , t » i ( x ) , !»;(*))■£«,(*) < /(X,W,(X),U',(X)),0< X < 1.
Then there exists a solution, u, of the BVP, (1) , satisfying u>i < u < V\. Moreover, if one defines inductively the sequences, u;jt+1 = Mwk,vt+i = Mvt,k = 1,2,..., then wt converges in B to a solution, w , of the BVP, (1) , and vk converges in B to a solution, v, of the BVP, (1) , such thai (8)
u>t < u>t+i < w < v < v t + i < vt,t = 1,2,
Finally, w and v are respectively minimal and maximal solutions in the sense that if u is a solution of the BVP, (1), satisfying wi < u < vi, then u satisfies w < u < v. Proof. Define D = {y € B : u>i < y < vi). One first argues that conditions i) ii) and iii) imply that (9)
u>\ < u<j < «j < »i.
First note that by Lemma 1, TDI(X) = u
for(yi,yj),(*i,zj)eR,> (10)
|/(*,yi,y») - / ( * . * i . * ) | < KilVi - r,| + K2\yj - r,|,0 < x < 1.
202 As for assumptions on upper and lower solutions, assume there exist w t ,vi € B satisfying conditions i) and ii) of Theorem 4 and Lwt(x) > f(x,w1(x),xt/1(x)) + A(wltvi)(x),Q
< x < 1,
W x ) < /(*,»i(«), »!(*))- i4(i»i,»i)(x).0 < x < 1, where A(w, v)(x) = Ki\v — w\(x) + Kj\i/ - u/|(x). Define sequences {wk}, {«*} by u>*+i(x) = / (12)
G(x,s)(f(s,wk(s),w'k(s))
+
A(wk,vk)(s))ds,
^ f*+i(x)= / Jo
G(x,s)(f(s,vk(s),v'k(s))-A{wk,vk)(s))ds.
The following theorem follows again by the Schauder fixed point theorem and con sequences of monotone increasing operators. See [5] for complete details. We omit the details since similar details will be provided in the proof of Theorem 7. Theorem 5. Assume f : [0,1] x R3 is continuous and satisfies (10). Assume there exist W\, vi g B satisfying conditions i) and ii) of Theorem 4 and (11), and define sequences inductively by (12). Then there exists a solution u of (I) satisfying (13)
u»i < u < vi.
Moreover, for each k, wk < wk+i < vk+i < vk, and if u is a solution of (1) satisfying (13) , then wk < u < vk,k = 1, Remarks, i) In Theorem 5, one obtains iterative improvement, as in (8) , and so, limiting functions, w and v, exist. However, the limiting functions satisfy the BVP (1) if, and only if, w = v. ii) Theorem 5 remains valid if (10) holds on the set D = {(x.yi.yj): 0 < x < l,u>,(x) < y, < v,(x),t/,(x) < y, < u>i(x)}. Example 3. For our final example, we again employ solutions of differential inequalities and monotone methods to study a delay differential equation related to the differential operator, L. The arguments here are modeled after those in [6] or [7]. Let x 0 € [0,1] and consider the BVP, (14)
Ly(x) = /(x,y(x),y(x - x 0 )),0 < x < l.y(l) = 0,
(15)
y(x) = y ( 0 ) , * o < x < 0 .
We shall say that y is a solution of the BVP, (14) , (15) , if y satisfies (14) , (15) , andyeC^O.lJnCHxo.l]. We shall require the following fundamental result in the proof of Theorem 7.
203 Lemma 6. Let F : [0,1] x R —► R be continuous and bounded. Then there exists a solution of the BVP, Ly(x) = F{x,y(x)),0 < x < l,y(l) = 0. Proof. Suppose \F(x,y)\ < K on [0,1] x R. Let G\ = max t€ [ 0 ,i)/ 0 -G(x,s)ds, and Gi = max r€ [ 0| i] /„ Gx(x,s)ds. G\ exists and is finite by the integrability of
vi(x)>wi(x),z0
(17)
«,(1) = u,,(l) = 0,
(18)
«i(z) = v,(0), u.,(x) = u,,(0), x0 < x < 0,
(19)
Lvi(x) - f(x,vi{x),vi(x Lwi(x)-
- x0)) + A(vuwi)(x)
f(x,wi(x),wi(x
< 0,0 < x < 1,
- x0)) - A(vi,wi)(x) > 0,0 < i < 1.
Define inductively {vm}, {wm} by « m+1 (x) =
(20)
{ ,
«m+i(0),ro<*<0, , /mv(s),vm(sj "" G(x,s)(f{s,v "" i - x0)) - A(vm,wm)(s))ds,0
<x < 1,
and ti) m + i(*) = V>m+l(0),XQ<X<0,
(21)
/ G(x,s)(f(s,wm(s),wm(s . Jo
- x0)) + A(vm,u>m)(s))ds,Q < x < 1.
Theorem 7. Let f : [0,1] x RJ —► R 6e continuous and satisfy (10) . Assume a € C l (0,l],a(0) = 0,o'(0) = +oo,a(i) > 0,0 < * < 1. i4**«m« there exist wuvi satisfying (16) - (19) and define the sequences {vm}and{wm} by (20) and (21) , respectively. Then there exists a solution, u, of the BVP, (14) , (15) , such that for each m , (22)
vm(x) > vm+l(x) > u(x) > w m + ,(x) > u-m(r),0 < x < 1.
Proof. We first show that vm(x) > u;m(x),0 < x < 1 for each m. Since f(x,vm(x),vm(x
- x0)) - f(x,wm{x),wm(x
- xo)) - 2X(wmit«m)(x) < 0,
204 and G satisfies (5) , this first set of inequalities follows. We now show vm(x) > vm+l(x),wm+i(x) > t« m (x),0 < x < 1 and do so, inductively. Note that "i(0), x0 < x < 0, f G(x,s)Lv1(t)da,Q < x < 1. Jo Thus, vi(r) > fj(i),0 < x < 1, now follows from (19) and (5) . Similarly, wj(x) > w i(x),0 < x < 1. Assume inductively that "*(*) > "i+iC*) > w*+i( r ) > «"t(x),0 < x < l,k Then L(vm - vm+l)(x)
<m.
=
/ ( * . » m - l ( * ) . « m - l ( * - *o)) - /(^.WmW.Vm^ - *o))
-A(vm-i,wm-i)(x)
+ A(vm,wm)(x)
=
f{x, » m _ l ( x ) , « m - l ( x - X0)) - / ( x , V m (x), V m (x - X0)) - t f l ( « m - l - «m)(x) ~ # 3 ( « m - l - «m)(x - X 0 ))
+A'i(«;m_i - wm)(x) + Ki(wm-i
- wm)(x - x 0 ) < 0.
wm(x) > « m + 1 (x),0 < x < 1, now follows by (5) . Similarly, wm+i(x) < tum(x),0 < x < 1. The inequalities, independent of u in (22), are thus obtained. It follows by Dini's theorem that each of {v m } or {w m } has a uniform limit, v or w, respectively, on [x 0 ,1]. Also, since
i
v(0),xo<x<0, f G(x, *)(/(*, v(s), v(s - x0)) - A(v, w)(s))ds, 0 < x < 1,
and a'(0) = +oo it follows that t; € C*(0,1] n C[x0,1). Similarly, u> 6 C 2 (0,1] D C[z 0 ,l]. For each y g C[0,1], define z € C[x0,1] by
i
v(x),ify(z)>w(z), y(x),ift>(x)>i/(x)>ti;(x), w(x), ify(x) < w(x),
for 0 < x < 1, and z(x) = z(0) for x0 < x < 0. For y € C[0,1), define F(x, y(x)) = / ( x , z(x), z(x - xo)). By Lemma 6, there exists a solution, u, of the BVP, Ly(z) = F(x,y(x)),y(l) = 0. We now show that w(x) < u(x) < v(x), 0 < x < 1, and the proof of Theorem 7 will be complete. L(v — u)(x) = f(x,v(x),v(x
- xo)) - Ki(v - tu)(x) - K3(v - w)(x - x 0 ) - /(x,*(x), z(x - x 0 )).
Thus, L(v — u)(x) < 0, since / satisfies (10) and by the definition of z. Hence, v(x) > u(x) follows by (5) . Similarly, w(x) < ti(x).
205 REFERENCES [1] R.P. Agarwal, Boundary value problem* for higher order differential eenation*, World Scien tific, Singapore, 1986. [2] P.W. Eloe and J. Henderson, Exiatence of tolution* for gome aingular higher order boundary value problem*, 2. angew. Math. Mech. 73 (1993), 315-323. [3] A.M. Fink, J.A.Gatica, G.E. Hernandez, and P. Waltman, Approximation of lolution* of tingular second-order boundary value problem*, SIAM J. Math. Anal. 22 (1991), 440-462. [4] L. Collate, Functional tntlfti* and numerical mathematict. Academic Preu, New York, 1966. [5] P.W. Eloe and L.J. Grimm, Monotone iteration and Green1* function* for boundary value problem*, Proc. Amer. Math. Soc. 78 (1980), 533-538. [6] P.W. Eloe and L.J. Grimm, Conjugate type boundary value problem* for functional-differential equation*, Rocky Mountain J. Math. 12 (1982), 627-633. [7] L.J. Grimm and K. Schmitt, Boundary value problem* for differential conation* with deviating argument; Aequationet Math. 4 (1970), 176-190. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF DAYTON, DAYTON, OHIO 45469-2316
E-mail addre**: eloeOudavxb.oca.udayton.edu DEPARTMENT OF DISCRETE AND STATISTICAL SCIENCES, AUBURN UNIVERSITY, ALABAMA 36849-
5307 E-mail addre**: hendej20mail.auburn.edu
WSSIAA 3 (1994) pp. 207-217 © World Scientific Publishing Company
EXISTENCE A N D NONEXISTENCE OF POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN A N ANNULUS L.H. ERBE*
Department of Mathematics University of Alberta Edmonton, Alberta Canada T6G 2G1 HAIYAN WANG
Department of Mathematics Michigan State University East Lansing, Michigan 48824
ABSTRACT. We study the equation-Au = A0(|x|)/(u) fit < |x| < R,, i f R " , TV > 1 subject to linear boundary conditions at R\ and Rj. Under assumptions concerning sub- or superlinearity of / , we establish existence, non-existence, and multiplicity results for positive solutions.
'Research supported by NSERC Canada.
207
208 1. Introduction In this paper we consider existence and nonexistence of positive radial solutions of the equation - A u = A ff (|x|)/(u),
^ <M<Jfe
(1.1)
where x 6 IR", N > 1, along with linear boundary conditions at R\
and R? which
include
u = 0 on
| i | = Ri
and
|x| = Rt
u = 0 on |x| = Ri <\ -3- = 0 on |x| = Ri or
and
— =0
on
|i| = R2
(1.2b)
and u = 0
on
| i | = R7
(1.2c)
Here r = | i | and
Or
(1.2a)
4^ denotes differentiation in the radial direction, 0 < R\ < Rj <
oo. Because of the radial symmetry, we seek criteria for existence (or nonexistence) of positive radial solutions of (1.1) which then must satisfy _ u ' ' ( r ) _ ^ z i u ' ( r ) = A S (r)/( U (r)),
R1
(1.3)
Further, via a standard change of variable, (1.3) may be written as an ODE on (0,1). Consequently, we shall consider the following BVP with linear boundary conditions
-u" = Xh(t)f(u),
0
(I)
au(0) - 0ti'(O)= 0 (II) 7u(l) + W ( l ) = 0 where o, f), 7,6
are nonnegative real constants with p = ~f/3 + ay + aS > 0. We shall
also assume that /(u) > 0 for u > 0 and h(t) > 0 for t > 0, with both /
and
h continuous on their respective domains. The BVP (1.1), (1.2) and (I), (II) has been
209 the subject of numerous investigations (cf. [1-17]) under various assumptions. In [7], [8], the BVP (I), (II) (with
A = 1)
was investigated and various criteria for existence of one
(or several) positive solutions were obtained. We introduce the notation
/o =
lim ^ «-»0+
and we shall say that Similarly,
/
/
and
/ « , = lim ^
U
»—>oo
is sub (super) linear at
is sub (super) linear at
co
0
(1.4) U
in case
in case /o = 0
/o = -f-oo
( / » = oo).
(/o = 0).
The main result
of this paper may then be stated as T h e o r e m 1. Let /(u) > 0
for
f,g
be continuous real valued functions
u > 0
and
neighborhood of t = 1/2. a) if
/o = /oo = 0
there exists
fa = /oo = oo
/o = 0
or
d) if
/o = 00
or
e) if
/ ( u ) > cu
for
/(ti) < cu
for
in some
Ai > 0
such that (I), (II) has at least two
distinct
Aj > 0
such that (I), (II) has at least two
distinct
there exists
there exists
A3 > 0
such that (I), (II) has at least one
A4 > 0
such that (I), (II) has at least one
0 < A < A4. u > 0
and some constant
that (I), (II) has no positive solution for f) if
g(t) ^ 0
A > Aj.
/oo = 00
positive solution for
and assume
0 < X < Xt.
/oo = 0
positive solution for
t > 0
with
A > A].
there exists
positive solutions for c) if
for
[0,oo)
Then:
positive solutions for b) if
g(t) > 0
defined on
u > 0
there exists
A5 > 0
such
there exists
A« > 0
such
A > A5.
and some constant
that (I), (II) has no positive solution for
c > 0,
c > 0,
0 < A < A«.
The proof of Theorem 1 is based on the following Fixed Point Theorem of cone expansion/compression type. (See [6] for a proof and additional details.)
210 T h e o r e m 2. Let X open subsets of X
be a Banaek apace, K C X
a cone and assume
Q\,Sli
are
with 0 € fti, Hi C ft2 and let F : K n (G3\Cli) -> K
be a
completely continuous operator suck that either (i)
||.Fu|| < ||u||,
u&KndQi
\\Fu\\ > \\u\\,
and
ueKndili
or (ii)
||fu|| > ||u||,
ueKndClr
and
\\Fu\\<\\u\\, u e K n a n , . Then F
has a fixed point in K D (n 2 \fti).
2. Notation and Proofs The BVP (I), (II) is equivalent to the integral equation
u(t) = A / k(t,s)h(s)f(u(s))ds Jo
= Fu(t)
(2.1)
where « € X : = C(0,1] and k(t, s) is the Green's function for the problem —u" = 0 subject to the boundary conditions (II) and is given explicitly by
{
(ili)
' >~ p\(p + c,t)(
The operator
F : X —► X
is completely continuous and if we define the cone K C X
by K = {u € X : u(<) > 0, min u(f) >
||u|| = sup |u(t)| and a = min { j ^ y , $ & } ,
verify directly that
k(t,3)/k(s,s)
> a for
t 6 J, s € [0,1]. Hence, for
(2.3) then one can u € X
we
211 have
m in m
(Fu)(t) = ™ i | '
k(t,S)h(s)f(u(s))ds
teJ
>A
k(s,s)h(3)f(u(s))ds
> ACT max j
k(t,s)h(s)f(u(s))ds
Jo
=
Consequently, Fu G K
for any u € X.
Proof of Theorem la. For u 6 K, {Fu){\) =
we have \fk{\,S)h{s)f{u{s))ds
Jo
>A /
(2.4) \(±
,,)h(s)f(u(s))ds.
•Zl/4 '1/4
For any p > 0 we define ,, 3 3 // 4 4
m(p) = min { /
*(i ,*)/»(<)/(«(*))<*» :
U
6 AT, ||u|| = p } .
(2.5)
Jl/4
Since />(a) ^ 0 and /(u) > 0 it follows from ? < u(.s) < p on J
that m(p) > 0.
Let 0 < pi < pa be arbitrary and define
Ai=max{ From (2.4) - (2.6) and for
4o-^)}-
A > A, it follows that
(2 6)
-
||Fu|| > ||u|| for
||u|| = pi
and
||u|| = p j . Since /o = 0 for any A > Aj we can choose gj > 0 and f/ > 0 such that 2gi < p! and such that
/(u)
t)X f k(s,s)h(s)ds
where
(2.7)
212 Then for u G K
and ||u|| = qj we have
Jo
<X f Jo
k(s,s)h(3)f(u(s))
<xv\\u\\fk(3,3)h(3)d3<\\u\\. Jo
Thus,
||Fu|| < ||u|| for u e K
and ||u|| = q\.
Also, since
/<» = 0 we may
choose qi > 2p? such that /(ti) < t)u for u > q? where rj satisfies (2.7). Now if /
is bounded, say /(ti) < N for all u € (0, oo), then we may also suppose that
NXf0 k(s,s)h(s)d3 < q?. Then for u 6 K (Fti)(r) = A I Jo
and ||u|| = 92 we have
k(t,s)h(a)f(u(s))ds
and ||u|| = q2.
is unbounded then qi > 2pj is chosen so that /(ti) < f(qj)
0 < u < q2. Then for u € K
for
and ||u|| = q2 we have
(Fu)(t) = X f Jo
k(t,s)h(a)f(u(3))ds [1k(3,3)h(3)d3
<Xf(q2) Jo
< Arjft / Jk(s, s)h(s)d3 < q2 and so again we have \\Fu\\ < \\u\\ for u € K
and ||u|| = q2. If we now set
fix = {ti e X
IHKPI}
fis = {« e x
IMI
213
fi4 = { u e X : |u||
F has a fixed point u t 6 K f"l (JJj\fti)
with qi < ||«i|| < p i
and pj < ||u 2 || < ?j. Since pi < P2,
and ui.uj
are distinct and are both positive for 0 < t < 1. Proof of Theorem lb. For any u e K
we have
(Fu)(t) = A f k{t,s)h{a)J{u(s))ds Jo Let 0 < p i < p i
<\f
k(s,s)h(s)f{u(a))ds.
(2.8)
Jo
be given and let Af; = max {/(u) : 0 < u < pi],
i = 1,2.
Then
we have, from (2.8) (Fu)(t) < (A / k(s,s)h(s)
||u|| = Pi
(Fu)(t) < (A f k{s,s)h(s)dS)Mi, Jo
||u|| = p,.
and
Therefore, we may choose Aj > 0 such that for 0 < A < Aj we have (Fu)(t) < pi and (Fu)(t)
for ||u||=pi
and |MI=PJ. respectively. If we set
« , = { « € * : ||«||
||Fu|| < ||u|| for A < A2 and u 6 K D dil2
and ||Fu|| < ||u|| for A < A,
and u £ ATlSflj. Now since /o = oo, there exists ?i < § Pi such that /(ti) > Aiu for 0 < u < gi where M > 0 satisfies ,3/4
A
(2.9)
214 Then for u £ K
and ||u|| = q\ we have (Fu)(I) = A/ 1 *(i,3)A( J ,)/(u(3))d, Jo >xf
*(±
,s)h(s)f(u(s))ds
Ji/*
>\
we have
||Fu|| > ||u||,
ueKHdili.
Similarly, since /„, = oo, it follows that there is q > 0 such that u > q where M
/(u) > Afu for
satisfies (2.9). If we now put qi — max {2p2, J }, then for u 6 K
and ||u|| = qi we have min u(t) >
(Fu)(i) = A / ^ ( l , j W j ) / ( U ( < ) ) < k Jo ,3/4
>A/
*(1 ,*)*(-)/(«(*))*
Jl/4 ,3/4
>A/A/
i(i,i)^)u(i)i!
Jl/4
>AfA
Therefore, if we define £1*
fi4 = { « € * :
||u||
then ||.Fu|| > ||ti|| for u e K n 3J24. Thus, if we apply the Fixed Point as in the proof of part a), we conclude the existence of ui 6 K n ( J ^ f l i )
and u? g K n (il^fij)
which are positive solutions of (I), (II) with 0 < qi < ||«i|| < pi < pj < \\ui\\ < qj.
215 Proof of Theorem lc and Id. The proofs of parts c) and d) are similar to those for a) and b). If we consider
fti
and
Q? in the proof of a), then it follows that c) is true for
/o = 0. Similarly, c) is true for /,» = 0 by considering CI3 and ft^. Likewise, d) follows from the proof of b). Proof of Theorem le. Suppose
v(t)
is a positive solution of (I), (II). Therefore, since
veK v(11) =
\[lk(\,s)h(3)f{v(s))ds
Jo
>\f3\($,s)h(s)f{v(s))ds Jl/* ,3/4
> Ac / Ji/i >\ca([
Jfc(i s)h(s)v{s)ds k(\
,s)h(s)ds)\\v\\.
Hence, if A is sufficiently large so that ,3/4
Ac
k{±,s)h(s)ds>
1
then we have v(j) > ||v||, which is a contradiction. This proves part e). Proof of Theorem If. Suppose v(t)
is a positive solution of (I), (II). Then we have
„(t) = A / Jo
k(t,s)h{3)f(v(s))ds
< cX I k(s, s)h(s)v(s)ds
k(3,3)h(3)
Jo Thus, if A is sufficiently small so that Thus, if A is sufficiently small so that cA / k(3,s)h(s)ds < 1 Jo
216 then we have v(t) < \\v\\ for 0 < t < 1, which is a contradiction. This proves part f).
REFERENCES 1. D. Arcoya, Positive solution* for temilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217-227. 2. C. Bandle, C.V. Coffman and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differ ential Equations 69 (1987), 322-345. 3. C. Bandle and M.K. Kwong, Semihnear elliptic problems m annular domains, J. Appl. Phyt. (ZAMP) 40 (1989), 245-257. 4. C.V. Coffman and M. Marcus, Existence and uniqueness results for semilinear Dirichlet problems in AnnuH, Arch. Rational Mech. Anal. 108 (1989), 293-307. 5. E.N. Dancer, Global breaking of symmetry of positive solutions in two dimensional equations, Differ ential and Integral Equations 5 (1992), 903-913. 6. K. Deimling, Nonlinear Functional Analysis, Springer, 1985. 7. L.H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. (to appear). 8. L.H. Erbe, S. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. (to appear). 9. Xabier Garaisar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations TO (1987), 69-72. 10. J. Kasdan and F. Warner, Remarks on some quasiUnear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597. 11. S.S. Lin, On non-radialty symmetric 251-279.
bifurcation in the annulus, J. Differential Equations 80 (1989),
12. S.S. Lin, On the existence of positive radial solutions for semilinear elliptic equations in anntiiar domains, J. Differential Equations 81 (1989), 221-233. 13. P L . Lions, On the existence of positive solutions of temilinear elliptic equations, SIAM Review 34 (1982), 441-467. 14. F. Pacard, Radial and non-radial solutions of — Au = A/(u) on an annulus of R*, n > 3, J. Differ ential Equations 101 (1993), 103-138. 15. Jairo Santanilla, Existence and nonexistence of positive radial solutions for some semilinear problems in annular domains. Nonlinear Analysis, TMA 16 (1991), 861-879.
elliptic
217 16. Haiyan Wang, On the existence of positive solutions for semiHnear elliptic equations in the annuhu, J. Differential Equations (to appear). 17. Haiyan Wang, Existence of positive solutions for nonhnear elliptic equations in the annuhu, preprint.
WSSIAA 3 (1994) pp. 219-232 © World Scientific Publishing Company
219
ON SOME PROPERTIES OF THE T-MODULUS
H. ESSER Instxtvt fir Geometric und Prakttichc Mathematik, RWTH Aachen, Templergraben 55, 5t06S Aachen, FRG
N. KIRCHHOFF, G. LUTTGENS, R.J. NESSEL Lehrstuhl A fur Mathematik, RWTH Aachen, Templergraben 55, 5206S Aachen, FRG
ABSTRACT The present paper is devoted to some aspects, concerned with the T-modulus, a measure of smoothness for integrable functions on compact intervals. While at first structural properties such as continuity and saturation are worked out, the paper continues with considerations on an improper extension to unbounded inter vals. Finally an estimation between the T-modulus and the ordinary L°°-modulus is discussed for convex functions, which illustrates some effects, observed in numerical analysis.
1
Introduction
Around 1968 P.P. Korovkin and Bl. Sendov independently introduced a new kind of measure of smoothness, the so-called T-modulus, which turned out to be most appropriate to deal with approximation problems in spaces of Riemann integrable functions (in this respect compare with property (1.4)). In the meantime the theory and wide applicability of the r-modulus were worked out in detail by the Bulgarian school of approximation, the material being summarized in [10]. Nevertheless (and of course), there are still some nice properties left, and it is the purpose of this paper to add four of them which may be useful in connection with further applications. To this end, for a compact interval [a, 6] C R of the real axis R let B = B[a, 6] be the space of real-valued functions, everywhere defined and bounded on [a, 6]. Consider the subspaces C[a, 6] and R[a, b] of functions, continuous and Riemann integrable,
220 respectively. Obviously C C R C B. Let ^[0,6] be the space of (extended) realvalued functions, Lebesgue integrable over [a, b]. For r £ N (set of natural numbers) and / € B the r th local modulus of continuity is defined by (x e [a, 6], 6 > 0) Wr{SJ,x) : = s u p { | A ; / ( y ) | : y,y + rh € Us{x)},
Ai/(»):=EQ(-1),"*/(» + **), Us(x):=[x-6,x
+ S]n[a,b].
(1.1)
In these terms the r th r-modulus may be introduced by (cf. [10]) i T,(*.
/ ) == rr(6, / ; B[a, 6]) := J «,(*, / , *) efa, a
where J denotes the upper Riemann integral (of bounded functions). Recalling the standard definitions of the (L°°-) modulus of continuity ( / 6 B) « , ( * , / ) := u,r(SJ;B[a,b})
:=sup{|A r k /(y)| : y,y + rh e [a,6],|A| < *}
as well as of the (L1-) integral modulus ( / € L1)
7 " K/(y)|dy , iffc>0 w r (6,/;L 1 [a,6]) := sup • IM<*
a
/ |A;/(y)|rfy I «+rW
,
iffc<0,
it is an immediate consequence of the definitions that Wr(SJ;ll[a,b])
< TT(6,f) < (b-a)u>T(6J)
(1.2)
for any / € R, say. On the other hand, one has the following nontrivial connection: If r > 2, then (cf. [10,p.l5]) T,(*,/)<«V*uv_ 1 (*,/';L 1 [a,6])
(1.3)
for every function / , absolutely continuous on [a, 6]. In Section 5 this compaxison of the moduli is then further considered in the special case of convex functions. As mentioned the r-modulus serves as a measure of smoothness, particularly suit able for the approximation of Riemann integrable functions. This is already indicated by the fact that for / € B there holds true the equivalence (cf. [10,p.ll]) /€R
<^
Um_Tl{6,f) = 0,
(1.4)
221 which is completely parallel to the classical assertion that for / 6 B one has feC
<=*
lirn w,(5,/) = 0.
(1.5)
In Section 2 this is extended to the following strengthening of (1.4): For / € B one has / € R if and only if TI(S, f) is a continuous function of S on [0,oo). In this connection let us mention that additional regularity assumptions upon / also imply that the upper Riemann integral in the definition of the r-modulus may be replaced by the ordinary Riemann integral. Indeed, whereas it is a familiar fact that feC
= >
u>r{6, f, x) e C[o, 6] for each 6 > 0,
=►
u>i(6,/, x) e R[a, b] for each 6 > 0.
(1.6)
it was shown in [8] that / £ R
(1.7)
Note that the corresponding assertion for arbitrary r G N is still open. The discussion of further properties of the r-modulus is continued in Section 3 with the following o/t-saturation theorem: If / g R satisfies TT(6,f) = o(ST),6 —► 0+, then there exists an algebraic polynomial pT-\ of degree at most r — 1 such that / ( x ) = p r _i(x) for all x € [<*,&]• Finally, in Section 4 the notion of bounded coarse variation is considered in relation with the existence of an improper extension of the definition of the r-modulus to unbounded intervals.
2
On the Continuity of the First r-Modulus
For / € C[o, 6] property (1.5) and the subadditivity of u)i(6, f) immediately imply that Wi(6,/) is a continuous function of 6 at all 6 > 0 upon setting u)j(0,/) = 0. Concerning regularity properties of T\ (8, / ) as a function of 6, the (obvious) monotonicity already ensures r i ( 6 , / ) € R[0,c] for any c > 0 (and / 6 B[a,6]). On the other hand, though the counterpart to (1.5) is now given by (1.4) for / € R[a, 6] and thus handles the continuity of Ti(S,f) at S = 0, the continuity of Ti(6,f) for 6 > 0 is less obvious since appropriate counterparts to the subadditivity are not known. In fact, the proof of the following theorem is more intricate, it being based on Lebesgue's criterion on Riemann integrability. Theorem 1: Let f g B[a,6]. Setting ri(0, / ) = 0 there are equivalent: (a) /€R[«,6], T (b) i{$yf) is a continuous function of 6 on [0, oo).
222 Proof: (b) => (a): See (1.4). (a) =*• (b): By (1.4) it already follows that Tj(£, / ) is continuous (from the right) at 6 = 0. Moreover, since Ti(6, / ) = Tj(6—a, / ) for all 6 > b—a, continuity also holds true for these values of 6. In view of this suppose that there exists some 6o € (0,6 — a) such that Ti(6,f) is not continuous at 6 = So (note that the following arguments apply to the still missing (left hand) case of 6 = 6 — a, too). Then, without loss of generality, there exist a quantity mo > 0 and a strictly decreasing sequence (£„)JJL, with limn-,00 6n = So such that for all n € N (cf. (1.7)) 6
/ [ w i ( < S „ , / , x ) - W!(«o,/,*)]<& > m0.
(2.1)
a
Let gn(x) := w , ( 6 n , / , x ) - u>i(So,f,x) and Bn := {x € [a,6] : gn(x) > m0/2(b - a)}. Since gn € R[a,6] C L'fa.&j and gn(x) > gn+i(x) for all x € [a,6], the sets i?„ are (Lebesgue) measurable with Bn D B„+i. Now suppose that A(B n ) = o(l), where A(B„) denotes the Lebesgue measure of Bn. Then 6
/ 5„(x) dx= (
+ B„
/
W ( x ) dx
[o,i]\B„
< 4 sup | / ( y ) | • X(Bn) + -pL-\([ayb]\Bn) vefo.b] 2(6-a) a contradiction to (2.1). Thus for B := (XLi Bn Lebesgue measure that
< o(l) + one nas
X(B) = A( lim Bn) = lim \(Bn) n—»oo
^ , 2
by 'he continuity of the
> 0,
n—»oo
in other words, on the set B of positive measure there holds true vi(6n,f,x)-ui{6oJ,x)
>
™° . > 0 forall x € B , n € N .
(2.2)
4(0 — a)
The next step is to show that for each x € B at least one of the points x ± So is a point of discontinuity of / . To this end first suppose that both x ± S0 0 [a, 6]. In view of [x - So, x + S0] \ [a, b] = [x - 60, a) U (6, x + S0] one then has [a, 6] C [x — <, x + <] and therefore wi(«,/,x)= wi(£0,/,x)= w , ( 6 - a , / , x ) for all t sufficiently close to So, a contradiction to (2.2). Assume x ±6o € [a, 6] and that / is continuous at both points X ± 5 Q (otherwise the property under consideration
223 trivially holds; note that if one of the points x±S0 does not belong to [a, b], the relevant part of the argument just cancels). Then for each e > 0 there exists T; > 0 such that (cf. (1.1))
l/(v)-/(*
-M/<£
Since u>i{t,f,x) = M'(t,f,x)
forally6
— m"(t,f,x),
M'(t,f,x):=
U(*-M •
(2 3)
-
where
sup / ( y ) ,
m*(t,/,x):=
y6t/,(i)
inf
/(y) ,
yet/,(l)
one has \w1(t,f,x)-uh(So,f,x)\<\M'(t,f,x)-M"(6o,f,x)\
+
\m'(t,f,x)-m'(6o,f,x)\.
It follows for 6o < t < So + r) that 0 <
M'(t,f,x)-M'(60,f,x)
I
0
<
, \fM'(t,f,x) = M'{S0,f,x) sup f(y)-M'(60,f,x) , \{M-(tJ,x)>M-(S0J,x) yeiM*)\
y6
the latter because of (2.3) and y € Ut{x)\Ut,{x)
<=* y € ( [ i - ( , i - « i i ) U ( i + 4 ) i + l ] ) n [ a , i ] C ([x - £0 - »7. x - S0) U (x + So, x + S0 + r)]) fl [a, b] .
If So — T]
- M'(t,f,x)
< t.
Altogether this leads to I™ (*i/> x ) thus \u)x(t,f,x)-
ux(6o,f,x)\
m
(So,J,x)\
J
< 2e for all \t-S0\
< »?, contrary to x € B (cf. (2.2)).
Since X(B) > 0, the sets {x G f? : x — So € [a, i] and / not continuous at x — 5 0 }, {x 6 B : x + S0 € [a, b] and / not continuous at x + So} cannot both be of Lebesgue measure zero. This implies that the set of discontinuities of / has positive (outer) measure so that by Lebesgue's criterion / € B[a, 6] cannot be Riemann integrable, a contradiction to the assumption (a). Therefore / € R[<J,i] implies the continuity of Ti(6,f) at each S € [0,oo). D
224 3
Saturation
For r £ N suppose that / € R[a, b] is such that lim S-rTr(S,f)
= 0.
(3.1)
4-.0+
In view of (1.2) this also implies lim 6-ru>r(6,f;V[a,b})
S—*o+
= 0.
The well-known saturation property (cf. [2,p.40]) of the L 1 -modulus uT(8, f;Ll[a,b]) then immediately delivers the existence of an algebraic polynomial p r _i of degree at most r — 1 (i.e., p r _i € VT-\) such that f(x) — p r _i(x) for (Lebesgue) almost every x € [a, 6]. But the uncertainty of the Riemann integrable / on a set of Lebesgue measure zero is not natural. In fact one has the following o/i-saturation theorem for the r-modulus. Theorem 2: Let r € N and suppose that f g R[a,6] satisfies (S.l). Then there exists a polynomial p r _i € VT-i such that f(x) = p r _i (x) for all x € [a, b]. Proof: The aforementioned saturation result for ujr(8,f; Ll[a,b]) already estab lishes the existence of some p r _i € VT-i such that f(x) = p r _i(x) a.e. Obviously, AT!j>r-i(x) = 0, hence w r (6,p r _i,x) = 0 for all x € [a,6],8 > 0, and therefore TT(S,pT-i) = 0 for all 6 > 0. Thus, setting g := / — pr-i € R[a, 6], one has #(x) = 0 a.e. and lim 8-TTT(6,g)=0.
(3.2)
Now suppose that there exists xo € [a, b] such that |ff(xo)| = T*O > 0. Without loss of generality one may assume x0 G (a, 6) (the cases x0 = a or x0 = b follow by analogous (onesided) arguments). Choosing 6 > 0 such that [x0 — 26, x 0 + 25] C [a,b], the next aim is to show that u>r(6,g,x) > mo > 0
for all x € Xo
8
8 , Xo + -
r
(3.3)
r
Indeed, since x 0 G [x — 8, x + 8] and [x — 6, x + 8) C [xo — 28, xo + 28] C [o, 6] one has (cf. (1.1)) ur{S,g,x):= sup \&\g(y)\ > sup |A^(x0)| y,V+rhe(/,(x)
sup xo+rfc€[:
z„+rheUt(x)
|<7(xo) + y ^ V - l ) * « 7 ( z o + M)l-
(3.4)
225 Obviously, i 0 + rh € [x — 6, x + 6] if and only if
hel-.=
X —
XQ
—S I—
ZQ + 5
For 1 < Jfc < r consider the sets Nk := {h 6 / : g(x0 +fc/i)= 0}. Since g(x) = 0 a.e. on [a, 6], one has \(Nk) = A(/), i.e., almost every h 6 / is contained in TV*, 1 < k < r. Moreover, A(r^ = 1 Ni,) = X(I) > 0 which in particular implies that there exists h € / such that g(xo + kh) = 0 for each 1 < k < r. In view of (3.4) one then has Ur{6J,x)>\g(x0)\, thus (3.3). On the basis of (3.3) one may now continue to
6~Trf{6,g)>6-r
I
—61-Tj:o{l),
m0dx =
xo-S/r
a contradiction to (3.2). 4
□
On the Existence of r-Moduli on Unbounded Intervals
Let us turn to functions, defined over unbounded intervals. Particularly in connection with the approximation by discrete operators (i.e., by operators, based upon point evaluation functional, e.g., sampling sums), one then is asked to extend the definition of the T-modulus suitably. For simplicity, let us confine the discussion to the interval R + := [0, oo) (and to the case r = 1). Thus, consider real-valued functions / which are well-defined and bounded on [0, oo), i.e., / € B[0, oo). A natural candidate for an extension would certainly be given by
Tl(S,
/ ; R + ) := lim r,(5, / ; B[0,6]) := lim / ^(6, f, x) dx. 6-*oo
(4.1)
6—*oo J 0
But then the question is posed under what conditions the latter limit exists as a finite real number. For example, consider the function /o £ B[0, oo), given by
/.(-)=={; ; xx e€ NR
+
\N.
Obviously, /o is improper Riemann integrable on R + , but for the local modulus of continuity one has
*,/..«) = { J ;
«i(*./o.*)-<
n
elsewhere,
226 and thus (6 < 1/2)
oo *y r,(*,/o;R+) = £
\dx = oo.
*=> kit Therefore, in order to ensure the existence of (4.1), apart from the boundedness some further nontrivial conditions have to be imposed on the functions (indeed, this point of view is still missing in the relevant literature, compare, e.g., with the treatment of (generalized) sampling sums as given in [3]). In this connection it may be of some interest to look upon the problem in the light of recent developments (see [5]), concerned with a strengthening of the notion of improper Riemann integrability. To this end, let e > 0 be arbitrary fixed. The set M C R + is said to be e-separated if |x — y\ > e for all x,y € M with x ^ y. For a function / G B[0,oo) the quantity V.(/):=sup{£|/(*fc)-/(**-i)|} k
is called e-variation of / , where the sup is taken over all e-separated (finite or countably infinite) sets {x 0 < xi < . . . } C R + . Then a function / G B[0, oo) is defined to be of bounded coarse variation (i.e., / G BCV) if / has a finite e-variation for every e > 0. Obviously, if Ve(f) < C < oo, independent of e > 0, then / is of bounded variation over R + . On the other hand, the notion of bounded coarse variation just reduces to usual boundedness of / in case the interval is compact. The following criterion may be found in [9]: For / G B[0, oo) there are equivalent: (i)
/ G BCV
(ii)
J2
sup
\f(y) - f(z)\ < oo,
where ( x t ) t l 0 ls a n v allowable partition of R + , i.e., 0 = x0 < Xi < ... < Xk < x/fc+i < . . . with 0 < inf (xk - xk-i) < sup(x t - xk-i) < oo. *eN
(4.2)
jtgN
In these terms one has Theorem 3 : For f g B[0, oo) there are equivalent: (i) / G BCV, (ii) Ti(6, f; R + ) < oo for every 6 > 0. Proof: (i) => (ii): Let / € BCV and S > 0 be arbitrary. If x € [4kS, (4k + 2)S\, k e N , then [x - 6, x + 6} C [(4ik - 1)6, {4k + Z)S) C [0, oo),
227 which implies «,(*,/,*):=
sup
|/(y)-/(*)|<
sup
V,zeUt(x)
|/(y) -
f(z)\.
»,Z6[(4t-l)«,(4A:+3)«]
If x € [(4k + 2)6, (4k + 4)6], then analogously
«i (*,/,*)<
sup
|/(v)-/WI-
V,t€[(4k+l)6,{4k+!>)6)
Thus for any b > 46
J u,1(6,f,x)dx<J2(
J t=1
44
S
<26J2(
«P
+
)<*{6,f,x)dz
(4t+2)«
^p
V,ie[(4*-l)«,(4*+3)«]
t=1
+ J
4JW
)l/(y)-/WI<<»,
y,ze|(4*+l)«,(4*+5)<]
independent of 6 > 46, the series being convergent in view of (4.2), (i) => (ii). This already establishes (ii). (ii) => (i): Obviously, for x e[k-l,k],k e N, one has [fc-l,Jfc] C [x - l , x + l ] n R + , and thus the assumption for 6 = 1 delivers k OO
oo
£
sup
oo
=: £
|/(y)-/(*)|<£
*
/
r
/[
sup
\f(y)-f(z)\]dx
°°
ul(\,f,x)dx
=J
Hence (i) follows by (4.2), (ii) => (i).
Ul(\,f,x)dx
= r1(l,/;R+) <
oo.
D
Let us mention that the result of Theorem 3 may also be expressed in terms of the so-called simple Riemann integrability as developed in [5]. Indeed, if / is improper Riemann integrable over R + , then / € BCV if and only if / is simple Riemann integrable. Therefore, for a function / , bounded and improper Riemann intergable over R + , the r-modulus T\(8, / ; R + ) exists as a real number for each 8 > 0 if and only if / js in fact simple Riemann integrable. In some of the applications, however, the latter class turned out to be too narrow so that a weighted definition of a r-modulus for unbounded intervals was suggested in [7].
228 5
Convex Functions
In view of the inequality (1.2): Tr(6J)<(b-a)wT(6J), error estimates are certainly improved if the L°°-bound u>r(6,f) can be replaced by TT{8, f). It is the purpose of this section to explore a different behaviour of the moduli (S —» 0+) for convex functions which illustrates an effect, observed in numerical analysis. Indeed, the following considerations originate (see [4]) in the numerical treatment of the boundary value problem (0 < a < 1) y"(t) = \t-l-\°ior0
y(0) = y(l) = 0.
(5.1)
On the basis of usual 0(/i 2 )-discretisations on uniform grids with step size h = l / n , n 6 N , the standard estimates for the approximation error between the solu tion y of (5.1) and the approximate solution t//, of the associated discrete problem lead to the L°°-bound wj(fc,y") + wj(A,y'). For the present problem (5.1), however, this would only deliver the rate 0(ha) whereas numerical experience even suggests the rate 0(ha+1). The true rate 0(ha+1) was in fact established in [4] by some more refined argument in terms of local moduli of continuity. In [1] the matter was then revisited in the light of the theory of r-moduli. Indeed, the relevant approximation error can also be estimated by T2(h, y") + r2(h, y') + u^(3h, y', 0) + wjflA, y', 1), which reproduces the rate 0(ha+l)
in case of the example (5.1).
By means of these results one is led to consider the example fa(x)
■=-(
r^— )
for 0 < a < 1, x e [o, 6]
in some more detail. Obviously, ^(SJa)>\A}fa(a)\
= - ^ ^ S
a
.
On the other hand, for o < x < a + 26 < b one has (cf. (1.1)) <*(6Ja,x)<4
max \fa(y)\ y€Ut(z)
<4
fx + s-ay \ b-a h
229 whereas for a + 26 < x < b
«*(«, /„, x) < P mj*) \f!(t)\ = f - £ (x-S- .)-» 6\ Therefore, since fa is continuous, a+16
b
" R i b «*>•" "*""' + <J^F«* ^ - < l — s r ' l In other words, for the particular function fa one indeed has T2(6,fa) "»(*,/.)
O(f).
(5.3)
Let us extend this improvement to some more general class of functions. To this end, / G B[o, b] is said to be convex on [a, b] if for every x, y € [a, 6]
/( £ y ? )<5l/(x)+ /(»)]•
(5-4)
For the basic facts on convex functions see [6, Chapter III]. Theorem 4: Let f g B[a,6] be convex and absolutely continuous. Then for 6 > 0 t i ( * , / ) < 8 * w , («,/).
(5.5)
Proof: Since / is absolutely continuous, one may proceed via (1.3) and estimate u>i (tf, / ' ; L1 [a, 6]) in terms of LO\ (6, f) for convex functions. Here we prefer the following more direct argument which uses the convexity of / right from the beginning. If x e [a,a + 6} U [6 - 6, b], then u2(6,f,x)<
wj(*,/)< 2«,(*,/).
If x € (a + 6, b — S), thus [x — S, x + S] C [a, b], consider 0 < e < h < 6. Since / is convex (and continuous), one has (cf. [6,p.72]) /(Az + ( l - A ) y ) < A / ( * ) + ( l - A ) / ( y )
230
for every A 6 (0,1) and y, z € [o, b], hence
/(*)-/(v)>}[/(y + A(*-y))-/(y)]. For y € [a+6, b—S\,z = y+h and A = e/h this in particular implies that for 0 < e < h f(y + h)-f(y)>f(y h ~
+
e)-f(y) e
Analogously one also shows that for 0 < e < h f(y)-f(y-h) h
f(y)-f{y-e) e
~
Now the assumptions upon / ensure (cf. [6,p.91]) that the one-sided derivatives /,'.*, //_) of / exist and are nondecreasing on (a, 6). Therefore /(y + t
»- /M >/,V,M. M-to-V
),
and it follows for y, y + 2h € [x - S, x + S) that (cf. (5.4)) 0 < f(y) - 2/(y + h) + /(y + 2ft) < M-/('+)(y) + /('-)(!/ + 2A)] < * [/(_)(* + S) - f'M(x - 6)}, hence for x £ (a + 6, b — 6) ^(SJ,x)<6[fl_)(x
+
6)-f'(+)(x-6)).
In view of the absolute continuity of / the derivative f'{x) exists (Lebesgue) almost everywhere, thus f'(x) = f!_\{x) = f'i+\(x) a.e., so that onefinallyobtains (cf. (1.6)) a+6
6-6
6
MSJ) = ( / + / + a
o+o
J)^(SJ,x)dx
6—5
6-«
< 4 S a,, (5, / ) + 6 f [/'(* + S) - f'(x - 6)} dx a+6
= 46uh(6J) + 6[f(b)-f(a
+ 26)-f(b-26)
< 46ul(6J)+26ui(26,f)
+ f(a)}
< 85wi(5,/).
O
231 Let us point out that Theorem 4 includes the improvement (5.3) for the example fa. To this end, obviously ui(6,fa)
= 77
r—sup{(x — a)° — ( i — 6 — a)a : a < x — 8 < x < b}.
(b — a)a
Since the function ga(x) := (x — a)a — (x — S — a)a is decreasing on a<x one has ga(x) < ga(a + 8) = 6a for a + 6 < x. In other words,
—
S<x
Moreover, for a < x < a + 6 < 6, thus x — 8 < a
<*(*, /«,,«) > |AL./-(a)| = |
^
(* - «)"<
and therefore 9 — 1a
/
2 — 2°
(R'"'''ta(t-#ti)1,t'
a
In view of (5.2) this implies T2(* > /.) = 0 ( * a + 1 ) , but
?o(6a+1),
as well as a +I Thus the estimate (5.5) is sharp in the sense that the factor S on the right hand side cannot be replaced by a quantity with the behaviour 0(8). References 1. B. Buttgenbach, H. Esser, R.J. Nessel, On the comparison of error bounds for finite difference schemes, Numer. Math. 64 (1993), 477-486. 2. Z. Ditzian, V. Totik, Moduli of Smoothness, Springer, New York, 1987. 3. D.P. Dryanov, On the convergence and saturation problem of a sequence of discrete linear operators of exponential type in Lp(—00,00), Ada Math. Hung. 49 (1987), 103-127. 4. H. Esser, Stabilitatsungleichungen fur Diskretisierungen von Randwertaufgaben gewohnlicher Differentialgleichungen, Numer. Math. 28 (1977), 69-100.
232
5. S. Haber, 0. Shisha, Improper integrals, simple integrals, and numerical quadra ture, J. Approx. Theory 11 (1974), 1-15. 6. G. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge Univ. Press, Cam bridge, 1952. 7. N. Kirchhoff, Uber die Approximation Riemann-integrierbarer Funktionen durch diskrete Operatoren auf der reellen Achse, Dissertation, RWTH Aachen, 1993. 8. H. Mevissen, R.J. Nessel, E. van Wickeren, On the Riemann convergence of positive linear operators, Rocky Mountain J. Math. 19 (1989), 271-280. 9. C.F. Osgood, 0. Shisha, On simple integrability and bounded coarse variation, in Approximation Theory II (Proc. Conf. Austin 1976, Eds. G.G. Lorentz et al.), Academic Press, New York, 1992, 491-501. 10. Bl. Sendov, V.A. Popov, The Averaged Moduli of Smoothness, Wiley, New York, 1988.
WSSIAA 3 (1994) pp. 233-239 © World Scientific Publishing Company
233
SOME BEST CONSTANT INEQUALITIES OF L(IogL)°-TYPE
Matts Essen Department of Mathematics, University of Uppsala Box 480, S-751 06 Uppsala, Sweden and D.F. Shea Department of Mathematics, University of Wisconsin Madison, WI 53706-1313, USA and C.S. Stanton Department of Mathematics, California State University San Bernardino, CA 92407, USA Dtiiadti
U PnffOT
Wotff s*f Wtlttr en kit SStk Hrtki*i.
ABSTRACT.
Let F = f + if be analytic in the unit disc U. If a € [1,2] is given, we discuss inequalities of the type ( 2 T 1 j r V | ( l o g + |F|)«- 1 )(e i »)d» < M „ / 2 » ) £\\f\(l<*+
\f\)')('")M
+ *o
where R* is an error term. We prove that the best value of the constant Aa is 2/(*a). In the case or = 1, this is result is known.
0.
INTRODUCTION
Let F = f + if be analytic in the unit disc U with /(0) = 0. If a is a given positive number, we define La(F) = sup(2x)- 1 / " |F(rew)|(log+ |F(re»)|)-<W. r
Jo
We wish to discuss inequalities of the following type. Does there exist a constant Aa and an error term Ra such that (0.1)
La.i(F)
< AaLa{f) + R*
?
234 The case a = 1 was discussed in Pichorides [5] and Essen [2] who proved that the best value of Ai is 2/ff. In the present paper, we discuss the case 1 < or < 2 and prove that the best value of Aa in this range is 2 / ( x o ) . Let us here quote a theorem of Cole dating back to 1970 (cf. Gamelin [4], Theorem 8.3). T h e o r e m C . Let H be a continuous, reaJva/ued function on C. Then the following are equivalent. (i) There exists a subharmonic function h on C such that h < H, while h > 0 on the reai axis. (ii) For all trigonometric polynomials u, it
1
tf(u(e'''),£(*•'*))<#
>0
where u is the harmonic conjugate of u. Clever choices of the subharmonic minorant h in Cole's theorem can lead us to sharp constants in several inequalities. Examples can be found in [4, Ch. 8] and in [1] and [2]. The purpose of the present paper is to give such an argument proving inequality (0.1) with Aa = 2/(xa) in the case 1 < a < 2: it is similar to the proof in [2], which deals with the case a = 1. However, the inequalities in the case 1 < a < 2 are much more complicated than in the case a = 1 (cf. the proof of (1.2)!). In a later paper, we are going to discuss the case a > 2, as well as applications to value distribution theory. 1.
INEQUALITIES FOR SUPERHARMONIC FUNCTIONS
If z = x + iy = re'' and x > 1, we define G„(z) - - # { z ( l o g z ) ° } = j^T" sin a s xT° cos a s , where logz = Te". We assume that argz = 0 and s = arg(logz) vanish on the positive real axis. L e m m a 1. Let 1 < a < 2 and let Aa = 2 / ( x a ) . Ifx > 1, then (1.1)
r ( l o g r ) 0 - 1 - xOogr)*- 1 - Aax(logr)°
If (cos s)"-11 sins | < 2 / x and |«| < i r / 2 a . (1.2)
rilogr^icoss)1-"
<
AaGa(z),
tBen
- Aax(\ogr)a
>
AaGa(z).
L e m m a 2 . Let 1 < a < 2 and Jet c 0 = e x p { i / 2 } . We define Ka(z) Then Ka is superharmonic in C ifc> L e m m a 3 . If a > 1, x > e (1-3)
a_1
= Ga(\x\ + iy + c). CQ.
and y > 0, then
*(logr)° < x(log(* +j/))° < x(\ogx)a
The proofs of these lemmas will be given in Section 3.
+
ay(\ogx)a-\
235 2.
T H E MAIN RESULT
We define £ . - i ( / , / ) = sup(2x)-' / " |/(r e ' # )|(log+ l A r e " ) ! ) - 1 * . r
Jo
T h e o r e m 1. Let F = / + if be analytic in U with f(0) = 0, let c > e x p { x / 2 ) be a given constant, let Fc - | / | + c + if and let fc = \f\ + c.Ifl
L a - i ( ^ ) < AaLa(fc)
+ RF„
where Rr. = i — 1 ( | / | + e) + aA.L.-M,
\f\ + c) - c'(logc') a
with d = | / ( 0 ) | + c. T i e constant Aa = 2 / ( x o ) is best possible. Proof. Let tu = u + iv. From (1.1), we have 11«| + c + te|(log | \u\ + c + Mil)0"-1 <
A„K„(to)+
+A„(\u\ + c)(log I H + e + fill)" + (|«| + c)(log 11«| + e + t«|)—». To prove (2.1), we put w = F, use Lemma 3 and integrate with respect to 0. According to Lemma2, Ka is superharmonic in C. Hence KaoF is superharmonic in U and (2.1) follows from (1.3) and from the superharmonic mean value inequality. To prove that Aa = 2/(xa) is best possible, we recall some estimates from [3]. Let 6 and B be given positive numbers and let D = Z)»,B = {w = u + iv : u > B, \w\ < i u l o g | w | } . Let us say that F € L ( l o g L ) a _ l if L„_i(F) is finite. Let //» be a conformal mapping of U onto Dh,B with / ( 0 ) = IB and / ( 0 ) = 0. Applying techniques from Sections 5 and 6 in [3], we deduce that H\, e I ( l o g L ) 0 - 1 if and only if 6 < 2 / ( * a ) . To construct the desired example, we wish to replace the complex variable z in (1.1) and (1.2) by Hi = ft» + ihi and then integrate with respect to 0. We note that if B is large, |s| = | a r g l o g f f , | < T / ( 2 1 o g B ) . If e > 0 is given, we choose B so large that ( c o s s ) 1 - ° < 1 + e. Since Ga o H\, is harmonic in U, it follows from (1.1) and (1.2) that (2.2)
L«-i(/r») - AaLa(hh)
+ #„> <
(2.3)
(1 + e)L„-i(J/») - AaLa(hh)
>
AaGa{2B), AaGa(2B),
where RfHt = £ a _ i ( / u ) + aAaLa-i(ht,ht). As mentioned above, we have (2.4)
limL„_i(if»)
= 00
** *> T 2 / ( ™ ) = Aa.
Let T = dU be the unit circle. Since (fc»/A»)|r —► 0 as Hi\r —» oo, it is easy to see that RHiILa-i(Hi) tends to 0 as 6 - » j 4 a . From (2.2) and (2.3), it is now clear that La(hh) ->■ oo as b —♦ J4 0 and that limsupL a _ 1 (i/»)/L 0 (At) > Aa, b ] Aa, which proves our assertion.
236 3. PROOFS OF LEMMAS
1-3
We use the following notation. If x = x + iy = re'* with x > 1, we put i = logr and logz = Te". Then we have logr = »'*, \T\ = (ti + 6i)1'3
and t = 0/(tans)
Proof of Lemma 1. A computation shows that (1.1) is equivalent to Q
- ( 1 - cos0)(cos s ) ° - 1 > Aa which is its turn can be rewritten as which is its turn can be rewritten as
{cos(0 + as) - cos 0(cos s)a)
(3.1) A,,-—^—j{sino» + cottf((cos«) a -coso«)} > (sins)(cos«) a-1 1 — COS V from Lemma 4 below and the fact that This relation follows easily This relation follows easily from Lemma 4 below and the fact that /««■.
(3.2)
Os\\kO
W
IT
...
,„
T^r9±r W<*/2-
Lemma 4. For 0 < s < x/2 and 1 < a < 2, we iave (3.3)
(cos*)0 > cos as
(3.4)
sin a* > a(cos s ) " - 1 sin s
It remains to prove Lemma 4. We note first that (3.3) follows from (3.4) since —((coss)" — cos as) = a{sino« — sins(co8») a-1 } > o(o — l)sin«(cos*)<*~1 > 0. as To prove (3.4), we study for s fixed in (0, ir/2) g{a) = log(8in as) — log a — (a — 1) log(cos s) — log(sin 5). Differentiating, we see that since sin as < as for s > 0. g"(a) = -s'isinas)-2
+ a"' < 0.
Hence g is concave on [1,2]. Since j ( l ) = g(2) = 0, g is positive in (1,2) which proves (3.4). We have proved that (1.1) holds. To prove (1.2), we require two lemmas. Lemma 5. If 1 < a < 00, we have (3.5)
sin as < a sins, 0 < * < x/a.
237 Lemma 6. If I < a < 2, we nave (3.6)
sinars < asinscos(a — 1)«, 0 < * < r,
(3.7)
(coe s)a - cos as < a sin2 «(cos s)"'1, 0 < s < x/2a,
(3.8)
(coe ip)-1 - cos
The proof of Lemma 5 is easy and is omitted. To prove (3.6) we define yo(«) = asinscos(a— l)s — sin as. Differentiating, we obtain g'0(s) = a{cosscoe(a — 1)* — (a — l)sinssin(a — 1)* — cos as} > a{coe*coe(a — l)s — sin«sin(a — 1)« — coe a*} = 0. Since jo(0) = 0, we have proved (3.6). To prove (3.8), we introduce 9\{
A(V) > s'i(*/4) = (3/V5) - (2/x) > 0, J < y> < x/2. Since furthermore $i(x/4) = (l/\/2") - 1/2 > 0, we conclude that (3.8) holds. Let us now consider the function j 2 («) = a sin3 «(cos*)~1 + (cos as)(cos s)~a — 1. To prove (3.7) is equivalent to proving that gi is nonnegative on (0, »/2or). Differentiating, we obtain g'2(s) = a(coe s)'1'" (sin s)h(s), where
M.)M«»r,+<«-rl-==fe^.
sin* Using the notation in the proof of (3.6), we see that Using the notation in the proof of (3.6), we see that j > n ( a - 1 ) a = S o(«)(-in*)- 3 > 0, 0 < s < , / a , ds sin s and that h' is negative on (0, */2a) and thus that k is decreasing on this interval. If h(*/2a) is nonnegative, g'7 will be positive on (0, */2a) and we conclude that 52 is positive on this interval since j j (0) = 0. If/i(»/2a) is negative, we note that A(0) = 3—a is positive and thus
238 that g'3 is positive in an interval (0,«o) and negative in an interval (s0,x/2a). If g2(r/2a) is nonnegative, it follows that g? will be positive on (0, x/2a). To prove that j2(x/2a) > 0, we put isjla =
Using the expression Ga{z) = T"{y sin as — x cos as},
and recalling that t = 8/ tan * and that T = (
■vl-nr
= 6(coesY
a
.(COS*) 0 —COS a s
{coe8-
'— sins
Applying (3.5) and (3.7), we obtain Applying (3.5) and (3.7), we obtain
81110 sin OS-
+
:
}.
sins
(3.9) /„(*) < flafcosffsins + sin^coss) 1 - 0 '}. We wish to prove (1.2) which is equivalent to We wish to prove (1.2) which is equivalent to (3.10) 70(z)<^(c«.s)1-<'. From (3.9), it is clear that it suffices to deduce that From (3.9), it is clear that it suffices to deduce that 0{cos08in* + sin0(co8«)1_<*} < ^(coss) 1 "", or equivalently that (cos s)"-1 sin s < ( £ - 8sin8)/(6 cos8). Thus (1.2) will hold if (2/*) < ( f - 0sinff)/(0cos0), 0 < 0 < x/2. It suffices to prove that (3.11)
g3(8) = £ - 0(sin 8 + - cos 9) > 0, 0 < 6 < jr/2. 2
7
Since g'3 is negative and js(ir/2) — 0, (3.11) holds which concludes the proof of (1.2). Proof of Lemma 2. A computation shows that (3.12)
dG -^- = -T* coe as-aT0-2 ox
{t coe as-r 8 sin as).
The function cos as = cos(a arctan(0/i)) is positive if and only if logr = <> 0/tan(x/2a).
239 If 1 < a < 2, the right hand member is at most » / 2 . Thus cos a s is positive if logr > x / 2 , i.e. if r > co. The function Ga is harmonic if Sir > 1. On the line { £ * — c} where c > Co, *jf- is negative. Locally around {tRz = 0}, we have min(G 0 (r + c + iy), Ga(-x
+ c + iy)) = Ga(\x\ + c + iy).
Hence Ka is superharmonic near {9tz = 0} and thus everywhere. We have proved Lemma 2. Proof of Lemma S. If x > e a _ 1 , the function r _ 1 ( l o g x ) a _ 1 is decreasing. If y > 0, there exists by the mean value theorem ( € (x,x + y) such that (log(r + y))a - (log*)" = yaCHlofsO"-1
< yax-^logr)"-1.
This proves the right hand inequality in (1.3). The left hand inequality is trivial. REFERENCES
1. M. Essen, A superharmonic proof of the M. Ritsz conjugate function theorem. Ark. Mat. 22 (1984), 241-249. 2. M. Essen, Some best constant inequalities for conjugate functions. International Series of Numerical Mathematics, vol. 103, ed. W. Walter, Birkhauser Verlag, Basel (1992), 129-140. 3. M. Essen, D.F. Shea and C.S. St an ton, A value-distribution criterion for the class LlogZ, and some related questions, Ann. Inst. Fourier (Grenoble) XXXV (1985), 127-150. 4. T.W. Gamelin, Uniform algebras and Jensen measures, London Math. Soc. Lecture Note Series 32, Cambridge University Press (1978). 5. S.K. Pichorides, On the best value of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov. Studia Math. 44 (1972), 165-179.
WSSIAA 3 (1994) pp. 241-254 © World Scientific Publishing Company
241
MAJORIZATION FOR FUNCTIONS WITH MONOTONE NTH DERIVATIVES.
A. M. FINK
Mathematics Department Iowa State University, Ames, Iowa 50011 U.S.A. This paper u dedicated to Wolfgang Walter on occasion of kit 66th birthday. ABSTRACT. We find connections between Hardy, Littlewood, and Polya's majorization inequalities, Farkas' Lemma, and some interpolation problems. We also consider what "majorization" should look like for functions with higher monoticities.
1. Introduction. A well known theorem of Hardy, Littlewood and Polya [1, Theorem 108] is that N
N
£>(*.■) <E#vO
(i)
holds for all convex <j> if and only if x is majorized by y, written as x -< y. This means, that if {ZJ} and {y,} are arranged in non-decreasing order then N
N
£ x i < £ y . - j = 1,...,N-1; and i=j
i=j
N
N
•=1
i=l
(2)
If instead of convex functions, as in [1, Theorem 107], one asks that (1) is to hold for all increasing <j>, then (2) is replaced by Xi
i = l,...,JV.
(3)
In this paper we do two things. First we will show how these two theorems of Hardy, Littlewood and Polya are connected to linear inequality theorems, and
242 interpolation problems. Secondly, we ask what we think is a natural question: what happens if <j> is required to have monotone derivatives of higher order? Specifically, let Mn = {^|^' n - 1 ' is increasing and convex}
(4)
where we require that n > 1. Note that M 1 is the class of convex functions. It is well-known that if <j> G Mn, then ^ " ' exists a.e. and is an increasing function. We address ourselves to the question of the notion of inequality between {x,} and {j/i} that will give a necessary and sufficient condition for (1) to hold for all <j> 6 M " . See Mitrinovic [3, page 166] for one solution to this problem. Our formulation gives new insights into the problem. 2. Reformulation. To connect the Hardy, Littlewood and Polya results to linear inequalities we now do the reformulation of the problem (1). Let z = {x\,..., x/v, j/i, • • ■, VN}We then write z = {z\,..., *2N} with Z2AT > *2N-t ^ • • • > z\ ■, and consider 2N
!>#*«) > 0
(5)
i=i
where pi = 1 if Zi = yj for some j and pi = —1 if Zi = Xj for some j . This inequality is then equivalent to (1). This embeds (1) into inequalities of the form N
5>*W > 0
(6)
«=1
for arbitrary real numbers {p<} and various classes of functions. We will always assume that the sequence {z,} is written in non-decreasing order. Alternatively, we may consider the inequality N
5>1K > 0
(7)
where {j/j} is always a non-decreasing sequence. In succeeding sections more will be required of the sequence {$ft}. We will attempt to prove "best possible" inequalities in the sense of Fink and Jodeit [3]. We now formulate this. Suppose we are considering the inequality (7). Let Q be a class of sequences {p,} and Y be a class of sequences {j/i}.
243 Definition 1. The classes Q and Y are in duality with respect to the inequality (7) if (i ) the inequality (7) holds for all p € Q if and only if y e Y; and (ii) the inequality (7) holds for all y € Y if and only if p € Q. We find dual classes for the inequality (7) in the next section. For the inequality (6) we must reformulate the duality in a slightly different way. The results of the next section will motivate this reformulation. 3 . The n-monotone sequences. We devote this section to the case when the class Y consists of sequences with higher order monotonicity. Let Yn = {yi\(n + 1)*' order differences are non-negative} where yi < • • • < yjv and the nth order differences are defined by A ^ y , - = y,- — y,_i and A y j r + 1 ' = A ^ y y — A^ r 'y 7 _i so that A ' n - , ' y , is defined for j = n , . . . , N. We always assume N>n. The set of inequalities that define Yn are linear and can be written in the form An y > 0
(8)
where y = ( y i , . . . , y n ) T and -^n is an appropriate matrix. See below, where we modify A„ to be an n x n matrix. It is easy to write a formula for the dual class Q using Farkas' Lemma, [4, pg.7]. Farkas' Lemma. For each matrix A and vector 6 exactly one of the following alternatives has a solution (I ) Ay > 0 and yTb < 0; (II) zTA = bT, z > 0. Here x > y means n > yj. We may reformulate this as a Corollary. Corollary 1. Ay > 0 implies yTb > 0 if and only if there is a z > 0 such that zTA = bT. T h e o r e m 1. Suppose we consider the inequality N
$>y.>o.
(9)
i
Let A be a n x n non-singular matrix, Y = {y\Ay > 0}, and Q = {p\{AT)-'p
> 0}.
244 T i e n the classes Y and Q are in duality with respect to inequality (9). Proof:. According to the Corollary, Ay > 0 implies pTy > 0 if and only if there is a z > 0 so that zTA = pT. This means (AT)~1p = z > 0. Hence condition (i) of Definition 1 is satisfied. To complete the proof of (ii) we need to show that if y is a fixed vector so that pTy > 0 for all p € Q, then y 6 Y. Let p be the ith row of A. Then ( A T ) - 1 p = ( 0 , . . . , 1,0,... , 0 ) T with the 1 in the i'fc position. Hence (AT)~1p > 0 with p € Q. Thus pTy > 0. Since p is an arbitrary row of A we have Ay > 0 and y € K. It may be remarked that (ii) follows from (i) generally if Y is a closed convex cone in a Banach space, see Luenberger [5, page 215]. We have included the proof because we think it is instructive. 4. Application t o Hardy, Littlewood, and Polya. To apply Theorem 1 to the inequality (7) and (8) we must construct An as a square non-singular matrix. Consider first, the case n = 0 for which Y° is the nondecreasing sequences. N
Let J/o = (1, ■ • •, l ) r then ±yo € Y° and inequality (7) then is ^Pi
=
0- We
l
assume this necessary condition. Then without loss of generaUty we take all elements in Y to have yi > 0. Indeed, p • y = p • (y — yiyo). So Ao may be taken to be • 1 -1
1 . 0
. 0 . . -1
1..
It follows that Ajj"1 is rl 1
1 .
. 0
.1
1
1.
1
and Q° defined by (A'[)~ p > 0 is given by N
N
Q° = {P\ £ Pi = °; E P>i ^ °>i = 2> • • • > N ) - ^ i=i
the
interpretation of (1) as in (6),
i=j
these requirements are that for each real number x, (the number of yi 's > x} > (the number of x< 's > x), with equality for x = - c o . This holds if and only if *i is y«" f ° r each t.
245 In a similar way, one can reproduce ordinary majorization (2). This involves first showing that N
N
£ « = £•'ft = 0 1
1
is a necessary condition. These come from taking the specific elements in Y1, ± y 0 , ± y _
where yo = ( 1 , . . . , 1 ) T , and y = ( 1 , 2 , . . . ,N).
N
Without changing EP>'V> we thus l
add an appropriate sequence to make yj = y2 = 0. Then A\, defining Y1 may be taken to be ■1 0 -2 1 0 1 - 2 1 0 0 1 - 2 1
1
-2
1
one may verify that this is A% so that Q1 = {P\(AI)-2P>
0} = {P\ £
£ P» > 0, j = l,...,N;jrPi
i=j n=i
= f > < = o}. 1
1
Remark. It is interesting to observe that Hardy, Littlewood and Polya did not make this connection between what is called majorization, (conditions 2), their Theorems 107 and 108 and Farkas' lemma. They quote a variant of FWkas' lemma, namely a result of Steimke on page 171 where they note its similarity to an integral version of Theorems 107 and 108. This circle of ideas is also connected to the solvability of certain interpolation problems. See section 7. below. 5. The general n-monotone case. One can construct Q" explicitly as above although we prefer to do it in a way that shows the difference between inequalities (6) and (7). Recall a summation by parts formula, N
N
^ h a t = kisi + ^2Sj(kj i=l
j=2
with
»= £ a -
s
»=n
-
kj-i)
(10)
246 We let a< = pi and ki =
N
N
5>**o = #*i)5>+5>Atf*i). <=1
1
(ii)
j=2
We now take
EPIK-WEW + E ^ W W 1
1
(12)
;=2
with
p/»+D = £ > ] » > , i > n
+
l
(13)
7=»
and P/0)=p,-, i = l,...,JV. It follows by induction that
£>y,- = EA ( °y i + 1 P i + 1 + E if+,)A<"+1>y,. 1
i=0
(14)
J=n+2
T h e o r e m 2. Let Q n = {p|PJ ; ) = 0, j = 1 , . . . , n ; P / n + 1 ) > 0 i = n + 2 , . . . , N}. Then Qn is in duality with Yn with respect to inequality (7). Proof. We begin by deriving the necessary conditions, i.e., that p 6 Qn if (7) holds for all y 6 Yn.
N
As before, we take yo = ( 1 , . . . , 1) T to get ^Pi
= 0. In (14) this is
l N
± j^pjyo = i y i P j
= iPi
i hence P{
= 0. By induction, we assume
• l
py' = 0, j = 1 , . . . , Jfe and take for y a sequence such that A^fc+1^yfc+2 = A^ f c + 1 )yt + 3 = . . . = A ( t + 1 ) yAf = 1. Clearly this y has A ( n + 1 ) y , = 0 if Jfc < n. Now (14) becomes
o<E«yi = i«t , ) l
and for —y, the reverse inequality so that P^ + i 1 , . . . , n. Now (14) has become
= 0. Hence P-
X > * = E ^i n+1) A (n+1) y i . 1
j=n+2
= 0 for j =
(15)
247
A sequence {y,} such that A^ n+, ^y ; - = 6jj0,jo ,
n
P>
.
= n + 2,...,N
N
shows that
n
> 0 is necessary for X^PiVi > 0 to hold for all y G y . The conditions l
of Q„ are now obviously sufficient. To complete the proof we need to show that if (7) holds for a particular y and for all p G Qn, then y G Yn. Let y be given and let p be the sequence that computes A^n+1^t/j0, i.e.,
P = ( O , . . . , O , ( - + 1 ) , - ( - + 1 ) , . . . , ( - I ) - + ' ( : + > ) , o,...,or N
where ( - l ) n + I ("+}) is in the j{,h position. If y G Y" then ^ p . y , = A( n+1 >y >0 > 0. _ Consequently, p G Q".
N
We are assuming that ^ f t y ; > 0. Hence A ( n + 1 >y i o = l
N
n
YlPiyt ^ 0. Since jo is arbitrary, y G Y . l
We note from the proof that the conditions P-
= 0 j = 1 , . . . , n may be replaced
by E«'*Pi = 0, * = 0 , . . . , n . •=i
6. Application t o Mo and Mi. We can only partially apply this result to the question of finding the dual of Mn. For example, for n = 1 we find that N
N
1
N
N
i
Q = {p\52pi = J2 p< = °> £ £ p ^ ° 1
1
k=j
J = 3,...,AT}.
i=k
Let Q1 be the dual of M 1 with respect to inequality (6). If <j> is a convex function, then y< = <^(z,) is a convex sequence. Hence Qx C Q 1 . Conversely, if {y,} is a convex sequence, then there is a <j> G Af1 such that
(6).
7. The case n > 2. As noted above, the conditions for a sequence p 6 Qn were rather simple and provided a class in duality with Y". This same class is in duality with M" precisely
248
when there is a one- to-one correspondence between functions in M" and thenrestrictions to {zi} being in Yn. This correspondence is obvious for n = 0,1 and was used above. For n > 2 this correspondence fails, in particular it is not always possible to find a function
(16) (17)
'n+2 — *1
Then a function <j> is in M" when A ( B + , ) # t i , . . . , t „ + 2 ) > 0 for all (ii,...,<„ + 2 ). For a given sequence (z\,..., ZN) in increasing order as before, we consider inequal ities (6) and (7) for all
on the formula that
Borel measure ft, and irn a polynomial of degree n, whenever <j> 6 Mn. The measure ft has compact support. Suppose that for every sequence {yi} with ASn+l'yi > 0 we can find a
Vi =*„(*)+ J(zi-t)Zdn(t).
(18)
—oo
When we compute the (r» +1)*' divided difference of {j/j} we may take the operator under the integral sign. If
A
(n+1)
Vi = I M(t;zi,...,zi+n+1]dn(t).
(19)
—oo
that is, under our assumption, there is a regular Borel measure satisfying (19) whenever {yi} is in Yn. Now as {y,} runs over Yn, the sequence yi runs through the non-negative sequences. So our assumption leads to the solvability (for /i)of OO
J M(t;Zi,...,zi+n+l]dft(t) —oo
for every non-negative sequence {a,}.
= a,
(20)
249 Lemma. For a fixed sequence {a,}, (20) has a solution \i or there is a vector A so
that J2 KM(t-, Zi,..., i + n + 1] > 0 and A • a < 0,
(21)
t
but not
both. oo
Proof. (Sketch) Let 5 = {x\xi = J M[t;zu...,zi+n+l]dfi(t),/i
> 0}. Then 5 is a
—oo
closed cone. One normalizes the measures and uses weak compactness to show S is closed. If (20) has no solution, a ^ S. So find a hyperplane A • x so that A • x > 0 for x 6 S, and A • a < 0. oo
Now A • x =
/ £ \iM[t; Zi,...,
Zi+n+i]dfi(t)
> 0 for all /i > 0 implies that
—oo
J2AjAf[t;Zi,...,Zi+n+i] > 0. It is clear that (20) and (21) cannot both have solu tions. If (20) now has a solution for all {a,} non-negative, then the lemma says that £XiM[t;zh...,Zi+n+1] *
>=► Xi > 0
(22)
is a necessary condition, since we may take a, = 6ij. Although we do not use it, (22) is also a sufficient condition for (20) to be solvable. For n = 1, M[t; Zi, Zj+i, £.+2] is & continuous "roof top" function with support on [zj,«i + 2], piecewise linear and non-negative. Thus if one computes
^2XiM[t;zi,Zi+uzi+2} 1
at t = z,+i, this sum reduces to a single term XiM[zi+i; Zi, zt+i, Zj+2]. So (22) is clearly satisfied, that is, the convex case is well behaved. But if n = 2 the function M[t;Zi,Zi+i,Zj+2)z>+i\ is a continuous piecewise quadratic with support [«i,zx+3] and is strictly positive on (zi,Zi+s). So for example, take Ai = A3 = 2, then M[t; zi,z2,z3,z4] + M[t; z3,z4,z5,zt] >0 for t € [z2>*s]> the support of M[t;z2,Z3,zt,zs]. that
Hence there is a small A2 < 0 so
3
5 3 XiM[t;zi,zi+i,zi+J,zi+3] i=l
and (22) is violated.
>0
250 v
One sees that the choice of yt,Zi are restricted if <j>(zi) = j/j is to hold. We now are in a position to formulate the problem of finding Q„. We invoke a theorem of Sonnevend [7] which says that for given sequences z = ( « i , . . . ,ZN) and y = ( y i , . . . ,J/A/), there is a <j> € Mn such that 4>(zi) = y< if and only if there is a function in M" of the form N
(18)
5(0 = E °i(* - *••)++E *«(* - *o++*«(«) i
/
such that 5(«i) = <j>(zi) = y,-. Here irn is a polynomial of degree n, {si} is an increasing sequence with at most
— entries in each interval (zi,Zi+\);
n
and (t —
n
Zi)+ = (t — Zi) when t > Zi,0 else. Since S € M we see that the jump of 5*"' at Zi or a,- is n!aj or nlbi respectively. We must have a,- > 0 and 6; > 0. Since the inequality (6) is linear in ^, we see that the inequality (6) for <j> G M" is equivalent N
to X^PiS'(zj) > 0 for all 5 of the form (18). Finally, again by linearity and nonl
negativity of a,,bi, this is equivalent to (6) holding for all polynomials nn and for the single inequality N
EP>(Z«-S)+^0
S&R
(19)
-
1
Since for any polynomial wn, —ir„ is also such a polynomial, we have N
E^ 7 r »( z ') = 0
(20)
for all polynomials 7r„ of degree n. Hence we have the necessary conditions N
* = 0,...,n.
EM'=0,
(21)
l
In order to introduce the appropriate divided differences we return to the identity (11). Define
*«> = (*,•-*_,■) £>«-!>,
i>j
(22)
k=i
with P;
= pi as before and (11) becomes
£>«(*) = # * , ) f > + E P i 1)A(1) #*i-i.*>)1
1
j=2
(23)
251 Again repeated use of summation by parts leads to N
n 3 0+l)
P<"+» A<"«>*(z,_„_„...,,,).
+ £ j=n+2
If we take <j>(z) = zk
k = 0 , . . . , n we get from (21) and (24) (by induction) that N
0 = ]►> **) = W^ftt"-
(25)
I
Hence for all
E P * ) = 1
£ PJ-+,)A(-+,)^>_n_1,...,^).
(26)
J=n+2
It is now obvious that if p G Qn, we have $^Pi0(z,) > 0 for all ^ 6 Mn.
This
l
sufficient condition fails to be necessary. 8. Duality for n > 2. In order to formulate duality properly we must introduce the third variable in inequality (6). Definition 2. We will say that pairs (p,z) G Q" are in duality with Mn respect to inequality (6) if and only if (i ) inequality (6) holds for all <j> G Mn if and only if (p, z) G Q"; and (ii) inequality (6) holds for all (p, z) € Q" if and only if
with
As we have seen above, Q° and Q1 are independent of z. We now look at (26) for the special case when <j>{z) = (z — s)+ (see (19)). Then as above A^""1"1^ is the B-spline of Schoenberg and Whitney. We immediately have our theorem. T h e o r e m 3 . Let N
N
Q" = {(P, *)\ £ ikPi = 0, Jfc = 0,..., n; £ 1
for all
s&
R).
;=n+2
PJn+l)M[s, *,-_„_„..., Zj] > 0
252 T i e n Qn is duality with Mn with respect to inequality
(6).
n
Proof. To show that Q is necessary, we have only to combine (21), (25) and (26). The sufficiency also follows from (26), and the preceding remarks. To complete the proof of (ii) of Definition 2, we let ^ be a fixed function such that (6) holds for all (p, z) € Q n . If z\,..., zn+2 is an arbitrary increasing sequence, we let p be the n
sequence for which ^2p~iip(zi) = A^ n + 1 V(' J r i>-••i 2 n+2)- If V1 £ Mn then this is l _t
non-negative, s o p g Qn. Applying this to <j> we get A^ n+1 ^
This is local because M[x,Zj-n-\,...,Zj\ = 0 if a 6 (—00, z,-_„_i] U \ZJ,00). Again it is instructive to look at the low order cases. For Z{ < s < Zj+i> we have M[s; Zi,Zi+\] — 1 and zero else. Thus (27) yields P% > 0 as the correct condition, as in Theorem 2. For n — 1, M[s; Zj-2,Zj-\, Zj] is a "rooftop" function, [ZJ— Zj-2]M[s; Zj-2, %j-\, Zj] for Zj-i < s < Zj and Zj — Zj-i
Zj-2 for Zj-2 < * < zj-\, Zj-i — Zj-2
with the value
zero elsewhere. In this case (27) becomes for z,- < s < z,+i p(2)
S-Zi Zi — Zi-\
1 Zi+i
(2)
— Zi-i
Zi+2
- S
Z{+2 — Zi+l
1
v,
n
(28)
Zi+2 — Zi
This implies immediately that pJ + j and P^'2 aie ^ 0- For n > 2, we get more than 2 terms and the conditions could be simplified if we knew exact conditions for the positivity of a linear combination of B-splines. 9. S o m e Calculations for n = 2. We do some calculations for n = 2 to see what they look like. It may be verified that (**-s)2 (*4 - X3)(xt
1(xi-Xi)M[s;x1,x2,x3,xA]
- X2)
x3 < s < xA;
(X2-3)*
(*4 - x2)(x3
- x2)
=•
(x 3 -x2)(x3 X2 < s < x3;
(*i~')2
(x2 0
-xx){x3
X\ < 3 < X2', *i)
else.
-xv) (29)
253 Long and tedious calculations lead to Theorem 4. We omit the proof. Readers may ask the author for a copy. T h e o r e m 4 . Q2 is given by pairs (p,z) such that
52**P« = °> k = o,...,n; j(3)
Pi'
(*{+! - Zi){zi+2
(41)
(3) > 0, Ptf> > 0;
- Z ^ O P W + ( Z < - 2i-l)(zi+l
(42)
~ Zi-2)P£\
> 0
fort = 3 , . . . , J V - 2 ;
(*» - z 2 ) ( « 3 - *i)(*4 - zi)P 5 < 3 ) + (*« ~ *a)(*2 - *i)(*5 - «2)P 4 ( 3 ) > 0;
(43)
(44)
and if [(zi+2 - *.•_,)*»&> - (* i+1 - Zi-aJPi+a] [(*i+a - * - i ) P ,(3) !+3 -(*i+s " *O)JVM] > 0 for
some t = 3 , . . . , N — 3, then
(45)
10. Concluding remarks. We have written out Q2 explicitly. It does not seem feasible to do higher order explicitly for M n , n > 3. We however note that one can derive similar results for anchored sequences and functions. Suppose, for example, Y0n = {y|A<" +1 > yi > 0
i = n , . . . , N and A ^ y , = 0 j = 0 , . . . , n } ,
then QQ its dual is simply obtained by removing the equalities Pjj)=0, j = 0 , . . . , n . (See (14)). Acknowledgements. The Author would like to thank the Mathematics Department of the University of Queensland and the Australian Research Council for supporting this research, especially Professor L. Bass for arranging the support.
254 References. 1. Hardy, G.H., Littlewood, J.E., and Polya, G. Inequalities, Cambridge, 1952. 2. Mitrinovic, D.S., Analytic inequalities, Springer-Verlag, 1970. 3. Fink, A.M. and Jodeit, M., Jr., On Chebyshev's other inequality, Inequali ties in Statistics and Probability, Lecture Notes, I.M.S., Vol. 5, 1984. 4. Kuhn, H.W. and Tucker, A.W., l i n e a r inequalities and related systems, An nals of Mathematical Studies 38, Princeton, University Press, Princeton. 5. Luenberger, D.G., Optimization Sons, New York, 1969.
by vector space methods, John Wiley and
6. Schoenberg, I.J., Cardinal spline interpolation, Regional Conference Series in Applied Mathematics. No. 12, SIAM, Philadelphia, 1973. 7. Sonnevend, G.Y., Necessary and sufficient conditions for interpolation with functions having monotone r" 1 Derivatives, Analysis Mathematics, 14 (1988), 273-285.
WSSIAA 3 (1994) pp. 255-268 © World Scientific Publishing Company
255
Stability of polynomial mappings controlled by n—convex functionals Roman Ger
Abstract The notion of delta-convexity (yielding a generalization of functions which are representable as a difference of two convex functions) has been extended to the case of higher order convexity. Examples are given and various charac terizations presented. Finally, some stability type results (in the spirit of [3]) are established, including a corollary on supporting polynomial functionals.
1.
Introduction
It is well-known that the functional equation (1)
AJ+V(*) = 0,
where A£ stands for the p—th iterate of the difference operator A^y(x) := tp{x + h) —
(2)
A2+V(*)>0,
where x 6 IR, h 6 (0, oo), are just C"1-1 —functions whose derivatives y>(n_1) are convex (see e.g. M. Kuczma [5, Chapter XV]). Therefore, the solutions
256
(2) are used to be called n—convex functions. For n = 1 inequality (2) states that V
\ ~ ) 2 ' I,yGlR' which is the functional inequality denning Jensen-convex functions. Moti vated by this fact, in what follows, we shall be using the operator
w.) := g(-ir-' (" t >
(d -
^+£?),
instead of AJ +1 . We have *>(*) = A ^ ( x ) . ; thus (f is n—convex (resp. n—concave ) if and only if (3)
x
=> * > ( x ) > 0 ,
(resp. (3')
x < y = * ^ ( i ) < 0 ).
It is not hard to check that, for odd n' s, condition (3) is equivalent to the following inequality (4)
* > ( * ) > 0.
Recently, an interesting and exhaustive study of the class of delta-convex mappings (yielding a generalization of functions which are representable as a difference of two convex functions) has been given by L. Vesely and L. Zajicek [9]. Their definition of delta-convexity reads as follows: Let (X, || • ||) and (Y, || • ||) be two real normed linear spaces and let D be a nonempty open and convex subset of X. A map F : D —► Y is termed delta-convex provided that there exists a continuous convex functional / : D —► 1R such that / + y* o F is continuous and convex for any member y* of the space Ym dual to Y with || ym || = 1. If this is the case then F is called to be controlled by / or F is a delta-convex mapping with a control function /•
257
It turns out that a continuous function F : D —► Y is a delta-convex mapping controlled by a continuous function / : D —► IR if and only if the functional inequality
is satisfied for all x,y G D (see Corollary 1.18 in [9]). In a natural way, this leads to the following Definition. Let (X, || • ||) and (K, || • ||) be two real normed linear spaces and let n G IN. Assume that we are given a proper cone C C X and a nonempty open and convex set D C X. Write x < y whenever y — x G C. A mapping F : D —► Y is termed delta-convex ofn—th order if and only if there exists a (control) functional / : D —► IR such that for all x,y G D one has (6)
x
\\S;F(x)\\ <«?/(*).
In the case where n is odd and the order relation < is linear (or, what amounts the same, C U (—C) = X ) relation (6) is equivalent to
(7)
n w o n <«;/(*),
and the order structure in X is not needed any more; in particular, for n = 1 inequality (7) reduces to (5). In the case where n is even, the restriction x < y in (6) turns out to be essential. Indeed, having just (7) for every i , y 6 D and for an even n G IN we obviously get (4) (with y> = / ) for all x,y G D whence, by interchanging x and y, we obtain
W)
Examples
Now, we are going to present some examples of delta-convex mappings of n—th order. We begin with
258
Proposition 1. In the case where Y = IR a function F : D —► IR is delta-convex of n—th order if and only if F is a difference of two n—convex functions. Proof. Assume / : D —> IR to be a control function for F. Then, for all x,y G D we have x
Conversely, let F = y>i — y?2, where
259 such that
\
,
xeL"(n).
Proof. First we observe that the Nemyckii operators: F and G{x)(i) := yj(t,x(t)), t G ft, x G £"(ft), act (continuously) from L p (ft) into L x (ft) (see M. M. Vajnberg [8] and L. Vesely & L. Zajicek [9]). Now, to check (6), fix arbitrarily x,y G Lp(Cl), x < y , and put Zj := (l ~ d r ) x + ^ forjG{0,l,...,n + l}. Then
IIW*)II =
Jn
f\Wx)){t)\dtk(t) dtk(t)
= / E(- 1 ) n + w n T )v(«.*i«))k*w Jn ^ JO
\
J /
| JO
260
n+l
= B-i)n+w(n;pW) = wo, i=o and the proof is completed.
\ J /
Proposition 4 (n—th order delta-convexity of the Hammerstein operator). Under the assumptions of Proposition 3 if, additionally, K : JR. x fi —► 1R is a Lebesgue measurable function such that for some c > 0
/or tk—almost allteft,
then the Hammerstein operator
H{x):= [ K(;t)
/tf(•,*(•))<%,
xelf{tt).
Proof. We argue like in [9, Proposition 6.9]. It is not hard to check that the linear operator T(z)(s) := / K(s,t)z{t)dlk(t),
z e L 1 ^ ) , s € IR,
Jti
acts continuously from LX(Q,) into L1(1R) and ||T|| < c. Moreover, H = ToF, where F is the Nemyckii operator spoken of in Proposition 3. In view
261 of the (just established) n—th order delta-convexity of F, for arbitrarily fixed i , y € L"(ft), x < y, we get ||£*(«)||
= <
K(ToF)(x)|| = ||T(^(x))|| ||r||KF(x)||
which was to be proved. 3.
Equivalent conditions
The following result establishes necessary and sufficient conditions for a given map to be delta-convex of n—th order. Theorem 1. Under the assumptions of the Definition the following condi tions are pairwise equivalent: (i) F is a delta-convex mapping controlled by f ; (ii) for every y* G Y* the function y* o F — ||y*|| • /
is n—concave ;
(iii) for every y* G Y* the function y* o F + ||y*|| • /
is n—convex ;
(iv) for every y* G Y", \\y*\\ = 1, the function y' o F + f (v) for every y* G Y*, \\y*\\ = 1, the function y" o F — f
is n—convex ; is n—concave ;
(vi) for every choice of rationals 0 = A<j < Ai < • • • < An < A n+ i = 1 and for every pair x,y G D, x < y, one has n+l
(8)
|l53(-l)^wV(Ao,A1,---,Aj.1>A,-+1,...AIl,AB+1) i=o x ^ l - A ^ x + A^H n+l
<E(-1)n+WV(A<"Al'---'A>-1'Vl»--^n,An+1) xfUl-X^x
+ Xjy),
where V stands for the Vandermonde's determinant of the variables considered .
262
//, moreover, the function D 3 x ■—► \\F(x)\\ + \f(x)\ G 1R is upper bounded on a second category Baire subset of D, then each of these conditions is equivalent to (vii) for every choice of real numbers 0 = A0 < Ai < • • • < An < A n+ i = 1 and for every pair x,y G D, x < y, one has (8). Proof, (i) implies (ii). Let F : D —► Y be an n—th order delta-convex mapping with a control functional / : D —► 1R. This means that relation (6) holds true for all x,y G D. Fix arbitrarily a nontrivial continuous linear functional y* : Y —► IR. Obviously, it follows from (6) that
jj£y(W«))<W*). whenever x, y G D, x < y, whence, in view of the linearity of the operator 6^, we infer that WoF-||yl/)(z)<0 provided that x, y G D,x
Replace y* by —y* in (ii).
(iii) implies (iv).
Trivial.
(iv) implies (v).
Replace y* by —y* in (iv).
(v) implies (vi). Fix arbitrarily points x,y G D, x < y, rational numbers 0 = Ao < Ai < • • • < An < A n+i = 1 and a continuous real functional y* G V*, ||y*|| = 1. On account of (v), the function tp := y* o F — f is n—concave , i.e. * > ( * ) < 0. Since the points (9)
XJ := x + \j(y - x) = (1 - A,-)* + A,-y, j G {0,1,..., n + 1} ,
divide rationally the segment [x,y], in virtue of T. Popoviciu's result from [6] (see also: M. Kuczma [5] and R. Ger [1], [2]) we get n+l
52(-l)B+1"iV(Ao,A„-",Aj.„Ai+1,---AB,AB+1)V(aJi)
263 i.e. /n+l
y*(E(-1)n+1"'V(A<"Al'---'Vi,V1,---A„,An+1)F(a;i)j n+l
< E(-ir+1--'V(Ao,A1,...)Ai_1,Ai+1).-.An,An+1)/(xi) , j=o
whence, in view of the arbitrarness of y', we get (vi). (vi) implies (i). n
'
An elementary calculation shows that the number
/n+l\
I
[
\
)
'
i 1 ' ""'
n+l
.
1 '
i 1 ' '"'
n+ln+1
i 1 '
n+l
/
/
is positive and does not depend upon j 6 {0,1, ...,n + 1}. Therefore, having arbitrarily fixed x,y € D,x < y, and putting A^ := j^,j G {0,1,...,n + 1}, in (vi), we get n+l
i=o
D-'r-^^'Xd-^).^,). n+l
<
which gives (i). To prove the last part of the theorem assume (i) and take an arbitrary functional y* 6 Y*, \\y*\\ = 1. By means of (iv), the function y> := y* o F + / is n—convex. Since \
+ \f{x)\,
x€D,
we infer that both / and y> are n—convex functions bounded category Baire subset of D and hence continuous (see R. Ger quently, F is weakly continuous. Since (i) implies (vi), we have choice of rational numbers 0 = Ao < Aj < • • • < An < A n+ i = 1 pair x,y £ D, x < y. Thus
on a second [2]). Conse (8) for every and for every
n+l
(10)
|^(-ir+1-^(A0,A1,...,Ai_1,Ai+1,..-An,An+1)(y*oF)(xJ)|
264 n+l
< E ( - 1 ) n + 1 _ i y ( A o ' A»'• • •' A'-i> A ^> • • •A»' A »+i)/( x i) ' ;'=0
where the i j s are defined by (9). In view of the continuity of / , y* o F and V inequality (10) holds true for all real numbers 0 = A0 < Ai < • • • < An < An+i = 1, and condition (vii) is proved. Since the converse implication is trivial, the proof has been completed. 4.
Stability results
Recently, the following result was obtained in [3]: under some mild regular ity condition upon the control function / , for every solution F of inequality (5) there exists an affine mapping A (i.e. a polynomial function of the first order) and a point x0 such that || F(x) — A(x)\\ < f(x) — f{x0) for all I ' S from the domain of F. In what follows we are going to extend this result to the case of polynomial mappings of higher orders. Theorem 2. Let (X, || • ||) and (Y, || • ||) be two real normed linear spaces and let n be a fixed odd positive integer. Assume that we are given a nonempty open and convex set D C X. If F : D —► Y and f : D —► 1R are two Cn+1—mappings such that inequality
(?)
iiwon <$/(*),
holds true for all x,y 6 D, then for every x0 € D there exist C°°—polynomial functions Q : D —► Y and q : D —► IR of at most n—th order such that F{x0) = Q(x0), f(x„) = q(x0), and \\F(x)-Q(x)\\
x 6 X,
265
(a monomial function ofp—th order) has the following property (see e.g. L. Szekelyhidi [7] or L. M. Kuczma [5]): for all i , h 6 X one has
Akm(x\ - I
kl m(A) ioTk
A„m(*) - \ 0
=P
for k > p
In what follows, Dkg(x) will stand for the Jfc—th Frechet differential of a map g; plainly, Dkg(x) is a k—additive (actually, k—linear) and symmetric mapping. The monomial generated by Dkg(x) will be denoted by dkg(x). Fix arbitrarily an x0 G D. For each y* € V, || y'\\ = 1, the C n+1 -function if := y* o F — f is (unconditionally) n—concave, whence for any x,y £ D, we get 0 > *>(*) = A ^ ^ ( x ) = A g i ( £ ±i*
(^TTji ^ ^
(x +
° *(* " Xo)) {x ~ Xo))
(n+1)! gf
=
< 0.
Consequently, for every x 6 D we have
y* (F(x)) - /(*) =
- x.)
k=oK-
n
I
- E ZTdV(*o)(z - x.)
= »* ( E i / ^ x * - *■)) - E ^*/(x0)(x - Xo). \k=0
'
/
k=0
'
266
Thus, setting
for x £ X, we get two C°°— polynomial functions such that F(x0) — Q(x0), f{x0) = q(x0), and
f(F(x)-Q(x))
for
z£B(x0,e),
then I|I| FF(r\ Mr\\\ ( x )-- Q ( x ) |<| < 7 - f - T | | x - x 0 | r 1
(» + !)'
for all x £ Proof.
B(x0,e).
As a matter of fact, we have proved that II F(x) - Q(x)\\ < (n H +r 1)! n T ^ (*• + '<* " *•)) <x -
x
»)
whence || F(x) - Q(x)\\ < ? - i - _ | | D«»f (n + 1)!
(x0 + 6(x - x 0 )) || || x - x 0 | r + 1
for all x £ B(x0, e). Corollary 2 (on supporting polynomial functionals). Let (X, || • ||) a real normed linear space and let n be a fixed odd positive integer. Assume that we
267
are given a nonempty open and convex set D C X and a C n + 1 —functional f : D —► IR such that inequality
W(*)>0 holds true for all x,y € D. Then, for every x0 £ D, there exists a C°°—poly nomial functional q : D —► IR of at mostn—th order such that f(x0) = q(x0) and for all x 6 D. Proof. Take F := 0 in Theorem 2. R e m a r k 1. As pointed out in the Introduction, to avoid a reduction to polynomial functions in the case of even n' s, the n—convexity is denned by (3). However, in such a case, for even n' s, even the Corollary is no longer valid. To see this, consider the cubic function on IR: / ( x ) = x 3 , x 6 IR. We have 6*f(x) = | ( y — x) 3 > 0 whenever x < y, x,y £ IR, but, obviously, there exists no quadratic polynomial supporting / .
References 1. R. Ger, Convex functions of higher orders in Euclidean spaces. Ann. Polon. Math. 25 (1972), 293-302. 2. R. Ger, n— Convex functions in linear spaces. Aequationes Math. 10 (1974), 172-176. 3. R. Ger, Stability aspects of delta-convexity. In: "Stability of HyersUlam type" (ed. Th. M. Rassias & J. Tabor), Hadronic Press, Inc. (to appear). 4. Ch. 0 . Kiselman, Fonctions delta-convexes, delta-sousharmoniques et delta-plurisousharminiques. Lecture Notes in Mathematics 578, Springer Verlag, 1984, 98-118.
268
5. M. Kuczma, An introduction to the theory of functional equations and inequalities. Polish Scientific Publishers & Silesian University, Warszawa-Krakow-Katowice, 1985. 6. T. Popoviciu, Sur quelques proprietes des fonctions d'une ou de deux variables reelles. Mathematica (Cluj) 8 (1934), 1-85. 7. L. Szekelyhidi, Convolution type functional equations on topological abelian groups. World Scientific, Singapore-New Jersey- London-Hong Kong, 1991. 8. M. M. Vajnberg, Variational methods for the study of nonlinear op erators. [in Russian] Moscow, 1956 (English translation: Holden-Day, San Francisco, 1964). 9. L. Vesely ic L. Zajioek, Delta-convex mappings between Banach spaces and applications. Dissertationes Math. 289, Polish Scientific Publish ers, Warszawa, 1989.
Roman Ger, Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
WSSIAA 3 (1994) pp. 269-284 © World Scientific Publishing Company
269
A MULTIDIMENSIONAL NIKOLSKII-TYPE INEQUALITY IN THE BALL
P. GOETGHZLOCK U n i v e r s i t e de P a r i s - s u d , Departement de Mathematiques, B3t. 425 91405 Orsay Cedex, France
ABSTRACT The
purpose
Nikolskii-type
of
inequality
this
paper
in a generic
is
to
prove
particular
a
multivariate
case
and to show
that the best exponent of the estimate depends on the fact set of zeros of the weight-function
that the
is transversal or tangential to
the boundary of the domain.
0. Notation. T
We denote by
n
the set of algebraic polynomials of a single or
several variables (according to the context) of degree at most
n . The
set of 2ii-periodic trigonometric polynomials of order at most
n
is
denoted by T n
For
E c R
,11.11
is the usual norm is the usual norm
(1 s p a oo) or pseudo-norm (1 s p a oo) or pseudo-norm
p,E
(0 < p 2w-periodic < 1) on E . function i3 considered as a function defined on (the Every Every circle)
2w-periodic function i 3 considered as a function defined on
(the
K - R/2irZ ; K i s provided with t h e m e t r i c
(x,y) - Min ( |x-y+2k7t| ; k e Z ) . For a 2n-periodic dfunction f we define For a 2 n - p e r i o d i c function
f
we define
llfll
: - IIfII p
p, (0,21t)
Let x - (x , x , . . ., x ) € F , we set x' - (x , x , . . ., x and we write x - (x', x ) . The boundary of a set Q c R is denoted
)
270 by flQ and the unit open ball in R
with center at origin by
B :
B = ( x e RN ; dist(x,0) < 1 ) For equal to
x € R , we denote by
[x]
the greatest integer less than or
x .
1. Introduction. Let
P € y
be a polynomial of a single variable and
I - [-1,1] .
n
We have
tlPH
s (n+1)
HXPH
00, I
IIPII
.
(1.1)
00, I
s (n+1)
2
B(l-x)Pll
03, I
.
(1.2)
00, I
The first inequality was proved by Schur [6] in 1919 and the second is an immediate consequence of Taylor's formula and the classical Markov's inequality. These inequalities have been generalized
(see [1] and [5])
in the one-dimensional case: Let
<5 ,5 ,y ,...,y 1
2
1
i
be rpositive constants, a ,...,a 1
laj < 1 (i - l,...,r) , a 4 * a We set
,
p > 0 , a > 0. \7r
w(x) - |l-x| l |l+x| 2 |x-a | y i ...|x-a 1
k = Max ( 2 6 ^ 2« 2 , y , ..., r^
and
satisfying
r
Then there exists a positive constant
C
r
■
such that for any
P e ¥ n
ka
a
IIPII 3 c n llPw ll p,I P,I furthermore the exponent ka i3 sharp.
Inequalities 1.1, 1.2 and 1.3 show that the exponent of on the location of zeros of of
I
w
in
I
(1.3)
n
depends
: zeros located on the boundary
have a double influence on the exponent.
271 In this paper we are concerned with the generalization of these inequalities to the N-dimen3ional ca3e. 1) The problem has been investigated in [2] with (the interval
I
p = oo
and
Q c R
is then replaced by a bounded open set
Under general assumptions on w for any P e 9
IIPII „ s c n d HPwH
,
oo, Q
where the optimal exponent d where the optimal exponent d only on the geometric relations only on the geometric relations vanishes. vanishes.
„ oo, £J
is effectively is effectively between dil between dil
2) In [4] it was proved that for any P € P when the Markov's inequality with exponent w
) .
and (1 it was shown that
n
and for functions
a - 1
computable and depends computable and depends and the set where w and the set where w
IIPII _ s C n r
, r
HPwII _
holds on the set Q C R
satisfying :
For some positive constant there exists
«
s € N
d
and
such that
for any x € fl
Is I s d
(3)
and w
(x) * 0 .
If
there exists s € N such that Is I s d and w an can be locally defined by x - f(x ,..., x
(x) * 0 . ) where f
If
an
)
can be locally defined by
is a lipschitz function and x
x
where
f
is the transversal variable, we know by
(3] that the Markov's inequality holds in then if grad w * 0
- f(x ,..., x
0
with exponent 2
(i.e. d-1) we have
IIPII _ s C n
HPwII
r - 2 ; .
2 2 It is easily shown that when It is easily shown that when if
w(x,y) - 1-x
Q = [-1,1) Q = [-1,1)
the exponent the exponent
IIPII
k = 1 or 2
is optimal is optimal
and is not if w(x,y) = x .
These results suggest that if grad w * 0
where
2 2
then for any P € f
. 5 C nk(X HP|w|a|| _ p,n p,u
according to the fact that the set
is transversal or tangential
to
3Q .
n
{ x ; w(x) - 0 )
272 The purpose of this paper Is to prove this conjecture In the special case when ft Is the unit ball In R
Theorem.
p > 0 , a > 0 ; there exists a positive constant
Let
C - C(ot,k)
and w(x) - x - a .
such that, for any P € "P , n
IPN
C n s<X II | x - a | a
s P,B
where
s - 0 if
P U
|a| > 1
Furthermore/ the exponent
, s-1 sa of n
(1.4)
p,B
N
if
|a| < 1
, s - 2
if
|a| - 1.
is sharp.
2. Soaa preliminary rasults. First, we recall the classical Bernstein inequality : For any
T € T
,
IIT'fl
n
Lemma 1. Let T e T , 0 n
co
a n llTll oo
6 [0, 2 K ] be such that
|T(6„)| - HTH
0
09
0
For any 6 6 [ Q- - , 0 + - ] we have o n o n
|T(0) | t i ITH* . z co
Proof.
Since T'(8 ) - 0 , for any 8 € [ 8 - — , 6 + — ] we have o o n o n T(8) - T(8 ) + -z (8-8 ) 2 T"(8 ) for some 8 between 8 and 8 . 0
2
0
1
Using Bernstein inequality yields I T(8) - T(8) I
1
0 2
*
|T"(8 )| a n 11TB 1
* | (8"e„)2 lT"(8,)|
00
. Then
s i i- n2 HTll* n
whence the lemma follows Immediately.
Laana 2. Let b € [0, n/2] . For any 8 € K we have o 4b I cos 8 - cos b| £ — |8 - 8 | K
for either 8 e [ 8 - b/2, 8 ] or 8 e [ 8 , 8 + b/2 ]
- \ NTH*
273 Proof.
Draw the graph of
slopes of line segments and
Laoma 3 .
Let
9 —» I cos 9 - cos b|
( (0,1-cos
b)
,
(b,0)
and calculate the
)
( (b,0) , <w,1+cos b) ) ) .
b € (0, n/2] , p > 0 , k > 0 , a > 0 ; there exists a
positive constant
C
- C(p,k,a)
such that for any
T e T
we have n
I |sin 9|k T II* s
C b"" Max(l,b_1) n" II |cos 8 - cos b | a |sin e|k T U
p
p
Proof. It is not restrictive to make the proof only for Let
T € T
n
and
9
0
be such that
|T(9 ) I 0
n £ Max( k+1, 2/b )
* HTII oo
1) We have : n | c o s 8 - c o s b | ° * |T(6) | p l a i n 9 | k p d9
f
I Icos 9 - c o s b l 0 * | T ( 9 ) | P I s i n 9 | ( [ k l + l l p d9 '0
i Since
n
(sin 8)
£
*
k+1
T(9)
,
the
order
is less than
2n . Let
I (sin 9 )l k l * 1 T(9 ) I 0 0 By Lemma 2, since
l/(2n) 3 b/2
of
the
trigonometric
polynomial
9
be such that o II (sin 9) " t 1 * 1 T(8) II* . eo
the inequality
I cos 9 - cos b| £ 4n" b 18 - 9 | holds
for either
d e n o t e by f*\co3 o
J
9 e [9 - — , 9 ] or 8 € [ 8 , 9 + ^ - ] . 0 2n 0 0 0 2n t h e i n t e r v a l where t h e i n e q u a l i t y i s v a l i d . We have
9 - c o s b | a p |T(9) | p | s i n e | " k l t l l p d8
J
£
Icos 8 - c o s bl"* | T ( 8 ) | P I s i n e | U k l + 1 , P d9
J 0
We
274 H ( s i n 9 ) | l c ! + 1 T ll* p 2
t
P
f
| c o s 6 - c o s b l " * d9 j
o
Le (we u s e Lemma 1) <Xp
*
ii (sin e ) l k l + 1 T II*P 2"p ( * | ) *
-
P ( , i n e ) l k I + 1 T ll*pC b ^ n - 0 * " 1 a 1
| e - e o l a p de
J,J o
(we U3e Lemma 2)
Summarizing : II | c o s 9 - c o s b | a
| s i n 9 | k T ll*p ( s i n 8 ) I k l + 1 T II*p oo
2) Let
J
- [ b - 1/n , b + 1/n )
J
- J
u J
and
J
1 , 2 1 2
Since
1/n s b/2
J
then |sin 9|k s c [ |T(9) | p |sin 9|kp d9
(2.1)
- K \ J 1,2
J n J
sin | |sin 8|k
.
1
- [ -b - 1/n , -b + 1/n ] ,
3
we have
| sin b Isin 6|k s
,
C b^n"^"1
= 0 s
and,
for any
|sin 6||sin 9|k
b"1 |sin 9||kl+1
,
s
8 € J |sin 01tk)*1
and
j
1,2
s
c 3 b"p
f
s
P
1
|T(0) | p | 8 i n e | , l k l + 1 , p de 1,2
C b" 4
Using i n e q u a l i t y 2 . 1
II ( s i n 8 ) l k l + 1 T ll* p oo
.
yields
|T(9) | p I s i n 9 | k p d8
[ *
n"
T
a
C
b"
(a+1)p
n a p II |cos 9 - cos b| a |sin e|k T II*p .
5
P
Furthermore, it is elementary to prove that for I cos G - cos b | > This last inequality gives |T(6) | p Isin 9|kp de
f 3
b C —. 6 n
9 e J
(2.2)
275 s
b""* n 0 * f
c
Jj
Icos 0 - c o s bl"* |T(8) | P I s i n 8 | k p d8 3
C? b _ 0 t p n a p fn\co3 •'o
s
8 - c o s b l 0 * |T (8) | P I s i n 8 | k p d6
Then |T(8) | p | s i n 8 | k p d8 3
C b" a p n 0 * II Icos 8 - cos b | a | s i n eI^ T II*p .
s
^
(2.3)
p
Inequalities 2.2 and 2.3 together give II Isin 8|k T H*p p
s
b"0* Max(l,b~p) n0* II |cos 8 - cos b | a |sin 0|k T U p
C 8
and
raising
both
p
members
to
the
power
1/p
gives
the
required
inequality.
TiOTim 4. C - C(p,a)
Let
p > 0 , k > 0 , a > 0 ; there exists a positive constant
such that for any k
II Isin 8|
Proof.
T € T n
we have
?tt
*
T II
s
p
C
n
tt
k
II (1 - cos 6)
Isin 8|
*
T II . P
We give only a sketch of the proof which is made in exactly the
same lines as proof of Lemma 3.
* 1) Let
T € T
, n
9
be such that
|T(8 ) | = HTll
0
and
J
0
00
- [8 + 1/n, 8 - 1/n] . 0
0
0
We have
f
|1 - cos 8| ap |T(8)|p |sin 8|kp d8 a
f |1 - cos 8| ap |T(8) |p Isin 8|kp d8
*
II T II *p C oo 1
o n"^- 2 ^- 1 .
276 2) Let
J
-
[ - 1 / n , 1/n ]
J
- J U J 1,2 1 2 P |T(6)| I s i n e | k p d9
f Jj
,
and
J J
s
-
[ w - 1/n , n + 1/n ] ,
- K \ J . W e have 3 1/2 kp X C n' ' * T II*p 2
1.2 s
n20Cp II |1 - c o s B | a I s i n e l " T II*p .
c 3
Furthermore, s i n c e f o r f Jj
|T(8) |
p
Isin 8|
kp
(2.4)
P _2
8 € J
|1 - c o s 81 >
C n
we have
d8
3
C n2<Xp f
s
| 1 - c o s 81°* | T ( 8 ) | P I s i n 8 | k p d8 J
s
c
n
2<Xp
3
II |1 - c o s e | "
I s i n 8 | k T II*p .
5
(2.5)
P
Inequalities 2.4 and 2.5 together give I Isin 8|k T II*p
Lanaa 5 .
Let
C - C(p,a)
p > 0
C n2(Xp II (1 - cos 8 ) " |sin 8|k T ll*p .
a
p
S
and
p
a > 0 ; there e x i s t s
such t h a t f o r any
c £ 0
and any
a positive
constant
P € T n
f C |P(x) | p dx J -c I
Proof.
C
|P(x) | p dx
s
s
C c" 0 * n 0 * f
C
J
-c
C c" 0 * n 20tp |
C
|P(x) | p I x l 0 * dx , |P(x) | p W-lf*
dx .
This is an immediate corollary of Lemmas 3 and 4.
(See also [5 cor. 15 p. 114, cor. 26 p. 126] and [1]).
3. Proof of the theorem (inequality 1.4). In this section, we prove inequality
1.4 of the theorem. The
sharpness of the exponent will be proved in the next section.
277 3.1.
Th« c«s«
|a | < 1 .
We assume that
0 a a < 1
which is not restrictive. Let
P € ¥ n
1. We f i r s t prove t h e
inequality
IIPII „ s C ( a , a , p ) n" II |x - a | a P II _ p,n N p,n N a+1 where ft -= { x € R ; - — s dist (x, 0) 3 1 ) . p , B,
Spherical coordinates
. . .,
(3.1)
are used in the unit ball,
not exactly in the usual way. Correspondance formulas are here for
p 6 [0, 1]
,
9 £ [0, 2w)
,
- p sin 6 sin w
x
r
1
r
r
2
x
... 3in
l
- p sin 8 sin w
x
(i-1,2, ...,N-2) sin
N-3
... sin
N-2
cos
r
r
H-3
*1
,
N-2
— p sin 8 sin op ... cos
3
x
r
l
*N-3
- p cos 6
N N —?
N-1
dx - p
Let
| sin 81
N—3
I sin if \
...
p € [ (a+l)/2, 1 ] ,
fixed.
Then
8
> T(8)
-
|sin qp
(i-1,2, .. .,N-2)
T(p, 8, ip^,
....
is a trigonometric polynomial of order at most We have
I x - a| -
If we define
\p cos 8 - a | -
b e [0, it/2]
|x - a|
-
| dp d8 dip
¥>N_2>
-
...dip
be choosen and Pfx^ ...,
xj
n .
p | cos 8 - — I .
by
cos b - — . W e have P p |cos 8 - cos b| £ —=— I cos 8 - cos b|
where b e [ Arc cos (2a/(a+l)) , n/2 ] . Then, using Lemma 3 yields n |T(8) | P | sin 8|""2 d8
r
*
Ap n t t P f
n Icos 6 - cos bl 0 * |T(9) | p |sin el"" d6
278 where
A
is the is the constant
2a replaced by Arc cos —-r- . Thus
r
n
|T<e> l p Isin e |
C b
Max(l,b )
of Lemma 3 with
b
A does not depend on p . Then
de
o
A"
s
f _ £ _ 1"* n «P [""lp c o s 8 - a |"* |T(8) | P l a i n '0
e|""'d8
and now, multiplicating both sides by p
... | sin
I sin ^ |
and integrating with respect to p to
(f> from
—=— to
0 to it (i - l,2,...,N-3)
2. Given a point E
from
S € B \ fi with
some positive constant
change of variable
S
1
and with respect
there exists a neighbourhood
for any point in V . Then for
C
HPII _ s c II | x - a | a P II . p,D l N p,C
3. Given a point
. . -d*>N.2
we get inequality 3.1.
x * a
D c B and x * a
of S such that
dp dip i
s
C
l
n" II | x
= (s',s ) e B \ £3 with
y' - x'- s' , y - x - a
N
- a | a P II
„ . p,B
(3.2)
s - a . We make the
in order that
S
becomes
the origin. Let c > 0 be such that A := { y ; lyj < c , (i-1,2, . . . ,N) } c B Then, using Lemma 5 gives J
lP
s
c2 n a p j
| P ( y ' , y N ) l p l y j 0 * dyN .
Integrating both sides with respect to y' in
[-c, c]
yields
IIPH * C n" II |y | a P II . 3 C n" II |y | a P H „ 1 p,4 3 "H p,4 3 -"N p,B HPB . s c n a II |x - a| a P II . s c n a II |x - a| a P II . p,j< 3 N p,i< 3 N p,B
and (3.3)
279 Taking
account
of
the compactness
of
the closed
ball
B
,
inequalities 3.1, 3.2 and 3.3 together give the required result.
3.2.
Tha casa
We suppose
|a| - 1 .
that
a - 1 . Let
P € f
.
For any fixed
x'
satisfying
n
llx' II s V5/2 We have
r I
-c
let
c - / 1 - llx' II2
c £ 1/2 and p ap
IP<X\X N >i
u-xj N
, and (1 - B n { x ; llx' II s • 3 / 2 ) .
dx^ N
| P ( X ' , X N ) | p ( l - c ) a p dxN
£
J
*
C cap
*
C 2"°* n" 2ttp fC
-C
n" 2 a p
|P(X',XN)|P
J '
dxN
(we use Lemma 5) r J
that i s
[
c
|P(x'fx)|pdx N
-c
N
* C n2CCp\ 2
r J
| P ( x ' , x ) | p dx
J
c
N
N
|P <x',x ) | p 11-x | a p dx . N '
-c
'
N
Then integrating both sides of this inequality with respect to llx' II s ^3/2 f J
N
x'
with
yields
|P(x)| P dx
s
n 2 °* f
2
n
Furthermore if
C
x € B \ CJ we have
J
|P(x) | p |l-x | a p dx .
(3.4)
N
n
1-x
S 1/2 and N
[ |P(x)| P dx 3 2ttp [ |P(x) | p U-x l°* dx . Inequalities 3.4 and 3.5 together give the result. Inequalities 3.4 and 3.5 together give the result. 3.3.
Tha casa Obvious!
|a| > 1 .
(3.5)
280 4 . Shaxpnaas o f t h a e x p o n e n t .
4 . 1 Son*
l o u a
We need some preliminary estimates on Jacobi polynomials for
notation).
The
Jacobi polynomial
P '
(see [7]
(whose degree
is
n)
n
satisfies ([7pp. 168-169]) : |P(d,0)(cos 9) | s
nd
if
0 s 9 s 1/n
(4.1)
C n"1/2 e"d"1/2
if
1/n a 6 s ir/2
(4.2)
1
if
TI/2
s e a K .
(4.3)
n
Lama 6. L o o m 6.
Let
p > 0 , a £ 0
and
d > a + - ; there exists a positive p
constant C » C(p,a)
s u c h t h a t f o r any
II I x l " P < d ' 0 , ( l - 2 x 2 ) II n p, [-1,1] II 11 i a „ ( d ' 0 ) / > II II | l - x | P (x) II n p,1-1,1] Proof.
C nd"a"1/p ,
(4.4)
r. d-2a-2/p C n .
(4.5)
s „ 3
We have
r1|P(d'°,(l-2x2)|p -l n
J
n
Ixl^dx
-
2
fl|P(d'0,(l-2x2)|px,,"dx J0 "
.11 [ | P < d , 0 , ( c o s 6) | P ( s i n ( 6 / 2 ) l1*" c o s ( 6 / 2 ) d9 . J n o U s i n g e s t i m a t e s 4 . 1 , 4 . 2 and 4 . 3 we g e t r e s p e c t i v e l y -l/n [ | P ( d ' 0 , ( c o s 6) | p ( s i n (6/2) ) * * c o s ( 6 / 2 ) d8
° Jt/2 J
l/n
S
f 1 / n n d p ( 6 / 2 ) » * d6 o
| P ( d ' 0 , ( c o s 6) | P ( s i n "
( 6 / 2 ) ) ^ cos (6/2)
* c f/2 J
l/n
-
Ci n ^ - ^ " 1 , d6
(n"1/2 e" d - 1/2 ) p e"" de
a c n^-^" 1 ,
281 I |P(d'°'(cos 6) | p (sin (0/2))p0t cosO/2) d9 n H/2
s
JI
C . 3
These three estimates together give inequality 4.4. Inequality 4.5 is proved in exactly the same way with the change of variable
Lamma 7. Let C - C(p,d)
p > 0
and
d > 0 ; there exists a positive constant
such that for any
n > 0
we have
r 1/4n |P (d ' C) (l-2x 2 )| p dx J
-l/«n
x - cos 8 .
C n^" 1 ,
*
(4.6).
"
f1 J J
, |P(d'0,(x)|Pdx 2 n
Cndp-? .
*
(4.7)
l-l/(2n )
Proof. Let Q ( x ) - p ( d , 0 > (l-2x 2 ) n
1) Inequality 4.6. We h a v e
IIQll
=■
Q(0)
Pld,0)(l)
-
00, I
and I nd
-
[-1, 1]
.
n
By t h e Markov-Bernstein i n e q u a l i t y , |Q'(x) |
3
for any
x e I ,
(l-x2)'1/2 .
2n IIQD 00, I
Then for for some
|x| 3 l/(4n) c
between
0
,
| oo, I a n d x . Thus
-
|Q(x)-Q(0)|
L. 2n (i-i_)" 1 / 2 ||Q( 4n 16 oo, I , | Q ( x ) | fc - IIQll -
|Q(x)-BQ»
| s oo, I |x| s l/(4n)
and t h e n f o r
lQ(x)-HQN
3
l Q ( x ) | p dx
[
-
|x Q' (c) I
a
£ IIQll 3 oo, I - nd , therefore
oo. I
3
C ndp_1 .
£
•*-l/4n
2) I n e q u a l i t y Let
4.7.
T(6) - P < d , 0 ) ( c o s 6)
. Then
IITII*
n
-
T(0)
-
nd
03
and u s i n g Lemma 1 we g e t
|T(8)|
t
- n
for
8 6 [0, 1/n]
-l/n
We have
\
I p ( d ' ( " (cos 8)
| p s i n 8 d8
t
C n dp " 2
.
282 1
f
which the same as
\ J J
and s i n c e
c o s (1/n)
I P(d,0)(x) n
| p dx
C n dp " 2
£
2
cos(1/n) I"' J
> 1 - 1/(2n ) : | P,d'C1(x) | 2 n
p
dx
C n dp " 2 .
*
4.2 Proof of thm sharpnass. 1)
Case
la| < 1 .
We assume that
0 a a < 1.
By the change of variable becomes
the origin
Q c [-1/ 1]
" I
parallelepiped
y' - x' , y - (x - a)/(a+1) ,
and the ball
B
. There exists
s , s , ...» s
A :- ( y ; |y | s s
(0,0,...,0,a)
is transformed
into
a set
such that the
(i=l,2,...,N) ) c fl .
Let
n be such that 1/n < 4s 0 O N In order to prove the sharpness of the exponent a , we exhibit a In order to prove the sharpness of the exponent a , we exhibit a sequence of polynomials of degree 2n such that sequence (R ) of polynomials of degree 2n such that 2n
n " II |x - a ) " R N
or equivalently a sequence or equivalently a sequence
II
2n
s
C II R
p,B
(S )
II
2n p,B
of polynomials of degree of polynomials of degree
2n 2n
such such
2n tt
a
that n II | y | S II for some positive constant N 2n p,U for some positive constant C . td 0) 2 We set S (y) - P ' (l-2y ) with 2n
f jfl
IS 2
p
ly I
n ap
N
s
C II S
II
2n p , R
d > a + - . We have p
dy
N
a
dp-pOt-l
C n
(we use Lemma 6, Ineq 4.4)
283
5
^""""Ln ' P"0,(1-2^ ' ^
S
Cn""* [ 4 J
| P,d-0,(l-2y^) n N
Cn-"* f 4 J
I P,d'0,(l-2y^) n N
(we u s e Lemma 7, |Pdy
C rf"0' II S 4
2n
II
4.6)
( s i n c e l / n <4s ) 0
*
Ineq.
N
|pdy
p,l2
which completes the proof.
2) Case
|a| - 1.
We assume that
a = 1 .
P
A
n
P(x)
" 'x
; |x
-
J iS 5 ^2Nn (i-l,2,...,N-l) , l - -i_ s |x |N< 1 - J__ ( _ 2 2 2n
Clearly,
A
c B n
r iP(x> ip u-x i ap dx J
s
N v
s
i-l
J N
N
' J-l
Ndp-N-l-2<Xp ,
*
r |P(X) ip ii-x i ap dx
-1 C n C
3n
and we have
n" 2 a p
4
(by Lemma 6 Ineq. 4 . 4 and 4.5)
N-l
n i =l
„ l/4n
J J
lpId'0,a-2<,
-l/4n
I'oxJ '
2
- ,11--11// (3n <3n ))
I Pn
X J
l-1/
(XN)
I
dxN
(2n )
(we use Lemma 7, Ineq. 4.6 and 4.7) |P(x) | p dx
s
-2a j C4 nif'"*P I
| P ( x ) | p dx
which c o m p l e t e s t h e p r o o f .
284 R*f*r*nc*a
1. P. GOETGHELUCK, Polynomial inequalities and Markov's inequality in weighted Lp-spaces,
Acta Math.
Acad.
Sci.
Hungar.
33 (1979), 325-331.
2. P. GOETGHELUCK, Une inegalite polynomiale en plusieurs variables, J. Approx.
Theory
40
(1984), 161-172.
3. P. GOETGHELUCK, Markov's inequality on locally lipschitzian compact subsets of R
in
Lp-spaces, J. Approx.
Theory
49 (1987), 303-310.
4. P. GOETGHELUCK, Polynomial inequalities on general subsets of Coll. 5.
Math. P.
57
NEVAI,
R,
(1989), 127-136. "Orthogonal
polynomials,"
Memoirs
of
the
AMS
213,
Providence RI USA, 1979. 6. I. SCHUR, Uber das maximum des absoluten Betrages eines Polynoms in einen gegebenen
Interval, Math.
Z. 4 (1919), 271-287.
7. G. SZEGO, "Orthogonal polynomials," Amer. Math. Soc. Coll. Pub. Vol. 23, New York,
1959.
WSSIAA 3 (1994) pp. 285-292 © World Scientific Publishing Company
285
COMPUTATIONAL INEQUALITIES FOR F U N C T I O N S
Dedicated to Professor Dr. Wolfgang Walter WERNER HAUSSMANN Department of Mathematics, University of Duisburg 47048 Duisburg, Germany and KARL ZELLER Department of Mathematics, University of Tubingen 72076 Tubingen, Germany ABSTRACT Using Chebyshev expansions and coefficient estimates, we obtain inequalities concerning functions, derivatives and integrals. In particular these inequalities are useful for the treatment of differential equations (cf. Breuer-Everson [1] and Plum [7]). Introduction It is a common task in analysis and numerical mathematics to estimate a func tion and its derivatives (often the function is or describes a remainder). Here we deal with methods based on biorthogonal systems (BOGS), in particular using Cheby shev polynomials and degrees of approximation. We state three general purpose results, while two results refer to papers by Breuer-Everson [1] and Plum [7]. First we treat the derivative of a series (where the coefficients are small enough). Next we estimate a general remainder. Then we consider the maximum modulus of a function (starting from grid points). Breuer-Everson [1] are interested in the sources of error occuring with deriva tives. Our Proposition 4 gives a certain explanation and specification. Plum [7] treats differential operators and estimates an integral over f = g -q where / is fixed and q is variable. We describe another approach (Proposition 5).
286 1. Series Let C[—1,1] denote the vector space of real valued continuous functions on the compact interval [—1,1] provided with the norm ll/IU := H/lloo.I-i.11 :=
sup
|/(*)|
(/eC[-l,l]).
*€[-l,l)
For k 6 No w e denote by T* the Chebyshev polynomial of degree k Tjb(x) = cos(k arccosx)
for x € [—1,1].
For sufficiently smooth functions in C[—1,1] we have 00
where the coefficients at can be estimated with the aid of integration formulas or by approximation procedures. By D we denote the differentiation operator D := — . ax P r o p o s i t i o n 1. oo
For n € N let e :=
* 2 | a t | . Then we have:
^ t=n+l n
\°f(x) ~ Y,akDTk^\ ^
£
for
-1<X<1,
k=0
where equality holds for x = 1 if a t > 0 for k > n. Here we allow e = oo, but in practice we shall usually have t < oo and so small, that the estimate is satisfactory. A modification of this approach is mentioned in Section 6. The proof rests upon (cf. Markov's inequality)
IIDiviu < k2,
rk(i) = k\
In the case e < oo tennwise differentiation is allowed, also leading to equality in the mentioned case a* > 0 (or < 0). Considering x = — 1, alternating a* are relevant. For —1 < x < 1 we have | T'k{x) \ < k2, thus we obtain equality only if a* = 0 for k > n.
287 2. B O G S We introduce continuous linear functionals Lk on C[— 1,1] defined by l
£ * ( / ) := \ j f(x)Tk(x)-j^L=
for / € C [ - l , 1], k = 0 * , 1 , 2, . . . ,
-l
2 1 (where 0* means, that for k = 0 the factor — has to be replaced by —). Then (Tt,X./fc)t6j^ constitutes a biorthogonal system (BOGS). We note that the norms of the functionals are ||£o|| = 1,
||L*|| = \ .
Using the best approximant p j — 1 of degree k — 1 to / , we get the estimate oo
\ak\ = 1 X 4 ( ^ 0 ^ ) 1 = \Lk(f)\
= \Lk(f - pl_x)\
< -Ek.i(f)
for k > 1,
with Em(f) := inf \\f — p||oo, p ranging over all polynomials of degree < m. For more details see [5, 6]. A general result is Proposition 2. For a continuous linear functional R on C[—1,1] and for f € C[— 1,1] we have the estimate \R(f)\
<
£ • £
£*_,(/)-UKTOI,
where B _ , ( / ) := U / I U oo
This follows immediately from the representation / = % , a t^/t, and from the t=o above estimates for the \ak\. In practice often R(Tk) = 0 for small k (e. g. for functionals based on interpo lation formulas), determining a certain degree of exactness. The ensuing remainders might be small, allowing to handle a middle part of the series - while the tail will be estimated globally. In suitable symmetric cases, all remainders with odd (resp. even) index vanish; this can be exploited in the BOGS method. Further we could employ the ak directly, or estimate them in another way. Altogether these consider ations give good insight concerning the structure of the remainder and its sources. Thus we have better chances to develop convenient numerical strategies.
288 3 . Grids In practice functions / defined on the interval [—1,1] can be considered and computed only on & grid G, i.e. afinite subset of [—1,1]. Besides the norm ||. Hoo^-i.i] we shall use the grid semi-norm H/IU.O := m a x | / ( z ) | . x€Cr
In addition, we introduce the grid constant C„ta which is defined to be the smallest value satisfying IIP IIoo.l-l.l] ^
C'n.G- | | p ||oo.G
for any polynomial p of degree not exceeding n. Proposition 3. Let G be a given grid in [—1,1], Then for any f € C[—1,1] the estimate l|/|loo,[-i,i] < C „ , 0 ■ l l / I U o + (1 + C»,c) • £ „ ( / ) holds true. For the proof we consider a proximum p* to / with respect to ||. ||oo,[-i,i]- Then l|/l|oo,[-l,l]
<
ll/-P*lloo,[-l,l] +
< E„(f)
llp'IU.I-1,1]
+ C , c ( | | p * - / | U . o + 11/Hoo.o)
289 4. Derivatives In order to compute the derivative Df of a function / , Breuer-Everson [1] consider oo
n
akT
and
/ = Y, >"
p = Z^ 6 * Tt '
i=0
i=0
where the (exact) coefficients ak are replaced by computed values 6* (using dis cretization, approximation, rounding) for 0 < ik < n, and bk = 0 for ik > n + 1. They find (by computation and consideration) that the maximum error for the first derivative grows as n 2 and write: "The source of the error is found to be the magnification of roundoff error by the recursion equation, which links coefficients of a function to those of its derivative." Our point of view is somewhat different, leading to complementary explanations based on the following result (see also Section 6). Proposition 4. oo
V J k2la*l> ^en
Let again be e :=
'
*=n+l n
II Df - DplU < J^\ak-bk\-k2
+ e,
k=0
where we have equality if ak>h <**>0
for * = 0,...,n, for k>n
+l
.
The proof is based on the fact that ||DTt||oo = DTk(l)
\\Df
- DplU
< f^|a*-64|.||W*||oo t=0
+
\\D[
= k2. We have
f) \*=TI-(-l
akTk)
|U. /
In the case e < oo the derived series is uniformly (absolutely) convergent in [—1,1], thus it can be handled term by term. The assumptions leading to equality are satisfied for certain (nontrivial) series and suitable numerical procedures (cf. Fox-Parker [4, p. 67]).
290 5. P r o d u c t s Plum [7] treated differential operators by a numerical homotopy method. Thereby he evaluates an integral with integrand / = g • q, where g is known and q is a computed polynomial (changing). He uses Simpson's rule and the remainder with D*f. This derivative is estimated using Leibniz' rule and a grid method for the occuring derivatives of q. Other approaches for controlling the error are given by the basic methods described above - for instance the differentiation employing Fourier-Chebyshev series (watching the critical points + 1 and —1). A more direct way is based on the expansion n
oo
/
=
9■9
=
^2 h Tk ■ ^2 c, T,
k=0
n
oo
1=0
=
^2 5Z i=0
bhC
< TkT'
1=0
(assuming convergence). P r o p o s i t i o n 5. For a linear continuous functional R and a function g with oo we have For a linear continuous functional R and a function g with \^ \bk\ < oo we have k=o n
R(g
=
J2Ck<7'>
1=0
where oo
a, := £ 6 t - J l ( T * T , ) , i=0
R(TkT,)
=
\(R(Tk+l)
+
R(Tk_,)).
Proof. By our assumptions we can rearrange the series and apply R termwise. The last equality follows from the functional equation of the cos-function (with T _ m := Tm). In many cases the cr/ can be computed or estimated beforehand (using analytical and numerical methods). For the truncation of the series some principles have been mentioned above. Once we have enough information about the
291 6. Remarks In our context two expansions are useful: oo
oo
p=l
k-0
«'=£«„•T; = 5>.r 4 where the coefficients are related by hck
- 2•
^
p-ap
(CQ := 2, ck := 1 else)
p-Jt=l,3,5,...
(we assume suitable convergence properties). Breuer-Everson start from a finite expansion (ap = 0 for p > N) and rearrange «' (now a polynomial). But it is also possible to begin (theoretically) with an infinite expansion. The influence of the ap with p > N on the bk (with k < N) is the same for all odd k (resp. all even k; observe Co = 2). It leads to an odd (resp. even) basic polynomial, multiplied by a factor (which could be estimated or computed in various ways). Such corrections will improve the approximation. Further Breuer-Everson mention matrix techniques. Thereby the functions u and «' are replaced by vectors v and v' (using N + 1 collocation points). This leads to the relation v' = Dv with a certain matrix D. In computing D and the inner products in Dv an increased (scientific) accuracy is beneficial - it will control the errors effectively. In addition one could employ some sort of interval arithmetic - replacing bounds of other provenance. For the speed of the calculation it is important to have available fast methods (like FFT) or parallel computations (of different kinds). Theory and application of Chebyshev polynomials are described in several books, e.g. Fox-Parker [4] and Rivlin [8]. In particular we mention the Tau method (for differential equations and integral equations). For certain purposes other poly nomials (functions) are better suited. Altogether the BOGS method (with supplements) gives insight into the sources of the error occuring here (calculation and estimation of derivatives). Thus it indi cates methods for fast and stable computation.
292 7. References 1.
K. S. Breuer and R. M. Everson, On the errors incurred calculating derivatives using Chebyshev polynomials, J. Comp. Pkys. 99 (1992), 56-67 .
2.
H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten, Math. Z. 86 (1964), 4 1 ^ 4 .
3.
H. Ehlich and K. Zeller, Numerische Abschatzung von Polynomen, Z. Angew. Math. Mech. 45 (1965), T120-T122.
4.
L. Fox and I. B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.
5.
W. Hauflmann, E. Luik and K. Zeller, Cubature remainder and biorthogonal systems, In: Multivariate Approximation II (W. Schempp and K. Zeller, Eds.), Internat. Ser. Numer. Math. 61,191-200, Birkhauser, Basel-Boston-Stuttgart, 1982.
6.
W. HauBmann and K. Zeller, Quadraturrest, Approximation und ChebyshevPplynome, In: Numerical Integration (G. Hammerlin, Ed.), Internat. Ser. Numer. Math. 57, 128-137, Birkhauser, Basel-Boston-Stuttgart, 1982.
7.
M. Plum, Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method, / . Appl. Math. Phys. 41 (1990), 205-226.
8.
T. J. Rivlin, Chebyshev Polynomials : From Approximation Theory to Algebra and Number Theory, Wiley, New York-Chichester, 1990.
WSSIAA 3 (1994) pp. 293-301 © World Scientific Publishing Company
293
ON FIRST ORDER DIFFERENTIAL EQUATIONS IN ORDERED BANACH SPACES S. HEIKKILA Department
of Mttkemttict,
Untveraity of Omln, 90510 Omlu,
FinUni
and V. LAKSHMIKANTHAM Detriment
of Applied Moikemniict,
Florid* Inttiinie
Melbourne, FL 31901-69SS,
of Technology,
USA
ABSTRACT In this paper we shall derive existence and comparison results for extremal solutions of a first order ordinary differential equation in an ordered Banach space whose order cone is regular and has a nonempty interior. We shall assume that the dependent variable in the equation is decomposed into two parts, one being Lipschits continuous and quasimonotone nondecreasing, while the other is aondecreasing, but not necessarily continuous. No kind of compactness conditions are assumed.
1. Introduction. Given an ordered Banach space E, an interval J = [0, T\, T > 0 and a mapping f: J xExE-r E, consider the IVP x'=/(<,x,x),
x(0) = x..
(1.1)
We shall prove the existence of extremal solutions of the IVP (1.1) if the order cone K of E is regular and has a nonempty interior, and if the following conditions hold. (fO) f(-,x,y(-)) is strongly measurable whenever y: J —* E is continuous and x €E. (fl) There is a Lebesgue integrable mapping p: J —> R+ such that \\f(t,x,y) - f(t,x,y)\\ < p(t)\\x - x\\ for all x, x, y € E and for almost all (a.a.) teJ.
294 (f2) f(t, •, y) is quasimonotone nondecreasing and f(t, y, •) is nondecreasing for a.a. t € J and for all y € E. (f3) ||/(t,0,y)|| < q(t)h(\\y\\) for a.a. t € J and for all y € £ , where 9: J -» R+ is Lebesgue integrable, h: R+ —> (0,00) is nondecreasing and /o°° j4jjy = 00. 2. Preliminaries. Let £ be an ordered Banach space with order cone K. boundary of K, and by K" the dual cone of K, defined by
Denote by dK the
K* = {c € E* I ex > 0 for each x € K). Given V C £ and g: V —► E, we say that y is gu
Given J = [0, T], T > 0, denote by C(J, E) the space of all continuous mappings x: J —> E, equipped with the norm ||x|| a = max{||x(i)|| 11 6 J}. Define a partial ordering in C( J, E) by x < y if and only if x(t) < y(t) for each t € J. If a, 0 € C(J,E), denote [a,£] = {x € C(J, £ ) | a < x < /9}. The space of all absolutely continuous mappings x: J —► E is denoted by AC(J, E). The following result which is proved in [5]. L e m m a 2 . 1 . Given g: J x E —► E, assume there is a Lebesgue integrable p: J —► R+ such that ||y(r,x)-y(t,y)||
mapping
(2.1)
for all x, y € E and for a.a. t £ J, and that g(t, ■) is quasimonotone nondecreasing for a.a. t € J. If the order cone of E has a nonempty interior, then x, y € AC(J,E), x' — g(t,x) < y' — g(t,y) almost everywhere (a.e.) on J and x(0) < y(0) imply x(t) < y(t) for each teJ. Assume now that the order cone K of E is regular and has a nonempty interior, and let e be a fixed vector from the interior of K. It is well-known (cf. [2]) that K is also normal, and that the equation ||x||,. = inf{o > 0 I -at
< x < ae}
295 defines a norm || ■ || e in E which is equivalent to the original norm || • || of E, i.e. there exist positive numbers /J, V such that u 11*11. < ||x|| < v \\x\\t for each x € E. It is easy to see that the hypotheses (fl))-(f3) hold when the norm || • || is replaced by || • He and the functions p and h by the functions pe and he, defined by pe(t) = -p(t)
and he(v) = -h(uv),
t € J, v € R+.
Given x0 € E, denote pi = pt + 9> and let w: J —► R+ be the solution of the IVP w' = P i ( i ) ( M » ) +1»),
u>(0) = ||«.||..
(2.2)
Condition (f3) ensures that w exists and is uniquely determined. Choose a(t) = x„ + \\x0\\te - w(t)e
and 0(t) = x0 - \\xo\\te + w(t)e,
t € J.
(2.3)
L e m m a 2.2. For each choice of y € [or, /9] the IVP x'= f(t,x,y(i)),
x(0)=xo
(2.4)
has a unique solution x, and this solution belongs to [<*,{)]. Proof. Let y € [a, /?] be given. From the definition of the norm || • || e and from (2.3) it follows that ||y(r)|| e < w(t) for each t € J. The hypotheses (ffl) and (fl) imply that the equation g(t,x) = f(t,x,y(t)),
t€J,xeE,
(a)
defines a Caratheodory function g : J x E —► E. From (fl), (f3) and (2.2) it follows that for each x € [<*,&] ||*(*,x(0)||. < pi(t)(he(w(t))
+ w(t)) = »'(«) for a.a. * € J.
(b)
In particular, ||ff(t, x 0 )|| e < w'(t) for a.a. t E J , whence ff(<,z0) is Bochner integrable. Condition (fl) implies that ||*(t,*)-*(M)||.
296 for all x, y € E and for a.a. t 6 J. Thus the IVP (2.4) has a unique solution x (cf. [3, 7]). Moreover, x is obtained as the uniform limit of the iteration sequence (•F n y 0 )£L 0 , where y0(t) = x0 and F is defined by Fz(t) = *o + J
f(s, z(s), y(a)) ds,
t€J.
(c)
If z € (a, 0], it follows from (a), (b) and (c) that \\Fz(t)-x0\\t<£\\g(s,z(s))\\tds < /
w\s)ds
= w(t)-\\x0\\t,
teJ.
Jo
This implies by the definition of || • ||e that (||x.||« - w(t))e < Fz{t) -x0<
(w(t) - \\x0\\c)e,
t € J.
Thus Fz € [a,0] for all z € [a,0]. Since y0 € [a,0], then x = lim„ Fnx0 because [a, 0] is closed.
6 [a,0], D
L e m m a 2.3. Let x = Gy denote the solution of the IVP (2.4) for given y € [<*,/?]• ffyi, V2 6 [<x,0], and Vi < Vt, *nen Gyi < Gy 2 . Proof. Denote Xi = Gyi, i = 1,2, and g(t,x) = / ( t , x , y 2 ( t ) ) , t € J, x e E. Conditions (f0)-(f2) ensure that g satisfies the hypotheses of lemma 2.1. Since / ( t , x , •) is nondecreasing by (f2), then x\ — g(t,xi) < 0 = x'2 — j ( t , x j ) a.e. on J, whence lemma 2.1 implies that xi < x2. □ The following fixed point result which is a special case of [4, Thm. 3.1], is also needed in the proof of our main results. L e m m a 2.4. Let [a, 0] be a nonempty order interval in C(J, E), and let G: [or, 0] —* [a,0] be a nondecreasing mapping. If the sequence ( G y n ) ^ 0 converges uniformly on J to a function of [a, 0] whenever (yn)^L0 ^ * nondecreasing sequence in [a, 0], then G has the least fixed point x , and the greatest fixed point x*. Moreover, x, = min{y e [a,/?] | Gy < y}, and x* = max{y 6[a,0]\y<
Gy}.
(2.5)
297 3 . Main results. By using the results of lemma 2.3 and lemma 2.4 we shall prove. T h e o r e m 3 . 1 . Let E be an ordered Banacfa space whose order cone K is regular and has a nonempty interior. If there is a null setZinJ such that f: JxExE —► E satisfies conditions (f0)-(f3), then for each x0 € E the IVP (1.1) has the extremal solutions. Proof. Given x0 € E and e € int K, let w be the solution of the IVP (2.2) on J. Choosing a, fi by (2.3), let G: [a,0] —» [ot,/3] be the operator which assigns to each y € [a, p] the solution of the IVP (2.4). Lemma 2.3 implies that G is nondecreasing. Moreover, Gy is for each y € [ot,0] the only function in [or,/?] which satisfies the integral equation Gy(t) = x„ + J
f(s, Gy(s), y(s)) ds.
(a)
Prom (fl), (f3) and (2.2) it follows that \\f(t,Gy(t),y(t))\\e
+ q(t)ht(w(t))
< w'(t),
for a.a. t 6 J and for all y € [a,0\, where w is the solution of the IVP (2.2). This and (a) imply that \\Gy(t) - Gy(s)\\e < \w(t) - w(s)\
whenever y € [a,0] and s, t € J.
(b)
Assume now that (y n )^L 0 is a monotone sequence in [a, /?]. Because the sequence (Gyn)%L0 is pointwise order bounded and monotone, and since the order cone K of E is regular, it follows that y(t) = Urn Gyn{t) (c) exists for all t € J■ Since the sequence (Gyn)%L„ is equicontinuous by (b), then the convergence in (c) is uniform, so that the limit function y is continuous. Obviously, y belongs to [a, 0]. The above proof shows that the hypotheses of lemma 2.4 are valid, whence G has the least fixed point x . and the greatest fixed point x". In view of the definition of G we see that x , is the minimal and x* is the maximal solution of the IVP (1.1) in If x is any solution of (1.1), then it satisfies also the integral equation *(*) = *„ + / f(s, x(s), X ( J ) ) ds,
t£j.
(d)
298 Thus
t
IK«)ll.
< IM« + J*Pi(s)(K(\\x(s)\\e) + ||*(*)||.)d» for all f € J. This implies (cf. [3], lemma 1.5.3) that ||x(t)|| e < w{t) for all t e J, where w is the solution of the IVP (2.2). Prom (f3), (2.2) and (d) it then follows that ||*(t)-*.||. <
J*pt(s)\\x(3)\\t+q(s)ht(\\x(s)\\e)ds
< I Pi(s)(ht(w(s))
+
w(s))ds=w(t)-\\x0\\t
Jo
for each t 6 J. In particular, x € [or, /?]. Because x, and x* are the minimal and the maximal solutions of (1.1) in [a,0], then x € [x,,x*]. Since x was an arbitrary solution of (1.1), then x , and x* are the extremal solutions of the IVP (1.1). D As a consequence of theorem 3.1 we obtain. Corollary 3 . 1 . Given / ; : J x E —» E, i — 1,2, assume there is a nuii set Z in J such that (i) / i ( - , x ) is strongly measurable for each x € E and / i ( t , 0) is Bochner integrable. (») fi{', !/(•)) is strongly measurable for all y € C( J, £ ) . (iii) There is a Lebesgue integrable function p: J —► R+ such that ! ! / ! ( < , * ) - / ! ( t , x ) | | < p ( t ) | | x - x | | whenever t 6 J\Z and x, x € E. (>v) / i ( ' i ' ) JS quasimonotone nondecreasing and f-i{t, •) is nondecreasing for all t£j\Z. (v) ||/a(*,y)|| < q(t)h(\\y\\) for all t e J \ Z and y e E, where q: J -> R+ is Lebesgue integrable, h: R+ —► (0, oo) is nondecreasing and /o°° ^ r = co. If the order cone of E is regular and has a nonempty interior, then the IVP x' = / i ( t , x ) + / 2 ( t , x ) , has for each x0 € E the extremal
x(0) = x o
solutions.
Proof. Conditions (i)-(v) imply that the function / : J x E x E —» E, defined by /(*, *, y) = /i(*. *) + M*> y)i
teJ,x,yeE,
has properties (f0)-(f3), whence the assertion follows from theorem 3.1.
O
299 R e m a r k s 3 . 1 . If £ is separable, then conditions (fD) and (ii) hold if / ( - , x , •) and }i are standard functions in the sense defined in [6]. Replacing condition (f3) by the existence of a lower solution a and an upper solution 0 of (3.1) we obtain existence results for extremal solutions of (3.1) in the order interval [a, fi]. Moreover, if / is continuous, then the Lipschitz condition (fl) is not needed (cf. [1]). From [1, Satz 3j it follows that if E = C(fi, R), ordered by K = C(fi, R+), where ft is a compact metric space possessing at least one accumulation point, there is a bounded, continuous and nondecreasing mapping g: E —» E and x0 6 E such that thelVP x' = g(x),
x(0) = xa
(3.5)
does not possess any absolutely continuous solution on any interval J = [0,T], T > 0. A similar counter-example can be constructed in the case when £ is a Banach space (c„) of all real-valued sequences y = (yi)iSi with l i m , - ^ y, = 0 and norm ||y|| = sup< \w\, ordered by the cone A" of all nonnegative-valued sequences of E (cf. [3]). In the former example K is normal but not regular and has a nonempty interior. In the latter case K is regular but has an empty interior. Thus the results of theorem 3.1 and corollary 3.1 don't hold in general if K is not regular or if K has an empty interior. If / is defined in J x V x V where V C E, and if the hypotheses of theorem 3.1 hold, then the solution x(-,x 0 ) of the IVP (1.1) exists on the interval [0, c], where c € (0, T] is so chosen that a(t), 0(t) € V for each t € [0, c] where a, 0 are defined by (2.3). This holds if w(c) - \\x0\\t < d, where d = inf{||y - * 0 ||« | y g V) and w is the solution of the IVP (2.2). 4 . D e p e n d e n c e o n t h e data. In this section we shall consider the dependence of solutions of the IVP (1.1) on the initial value x0 and on the function / . Our main result is. Proposition 4 . 1 . Let f, f: JxExE —► E satisfy conditions (f0)-(f3), and assume that the order cone KofEis regular and has a nonempty interior. Given x0, i0 6 E, let x, denote the minimal solution of (1.1), and let x* be the maximal solution of the IVP x'=/(t,x,x),
x(0) = x o .
(4.1)
Ifx0<x0 and f(t, x, y) < / ( r , x, y) for a.a. t £ J and for all x, y G E, then all the solutions of (1.1) and (4.1) belong to the order interval [x»,x*].
300 Proof. Let x be any solution of the IVP (4.1). Denoting x = Gx, where G is defined as in lemma 2.3, we have x' = /(*, x, x(t)) < f(t, x, x(t)) for a.a. t € J,
x(0) = x . < x„.
(a)
The function g(t, y) = f(t, y, x(t)), t 6 J, y e E, satisfies the Lipschitz condition (2.1). From (4.1) and (a) it follows that x' - g(t, x) < 0 = x' - g(t, x) and x(0) < x(0). This implies by lemma 2.1 that x < x, i.e. Gx < x. Because x . is the least fixed point of G, it follows from (2.5) that x, < x. Noticing also that x , is the minimal solution of the IVP (1.1), then x» < x for each solution x of (1.1) or (4.1). Dual reasoning shows that x < x* for each solution x of (1.1) or (4.1). D A an immediate consequence of proposition 4.1 we have. Corollary 4 . 1 . Under assumptions of theorem 3.1 the minimal and the maximal solutions of the IVP (1.1) are nondecreasing with respect to x0 and f. Denoting u <; v if v — u belongs to the interior of K, we obtain. P r o p o s i t i o n 4 . 2 . Given / , / : J x E x E -* E, and x„, x„ € E, x„ < x0, as sume that the TVP's (1.1) and (4.1) both have solutions on J. Assume also that f(t, x, x) < f(t, x, x) for a.a. t € J and for all x 6 E, and there is a Lebesgue integrable function q: J -* R such that y t-» f(t,y,y) + q(t)y or y >-* f(t,y,y) + q(t)y is nondecreasing for a.a. t € J. If x is any solution of (1.1) and x any solution of (4.1) on J, then x(t) < x(t) for each t € J. Proof. Let x and x be solutions of (1.1) and (4.1), respectively. Denoting Q(t) = J0 q(s) ds, t 6 J , it is easy to see that e^x(t)
= x„+ f\e^'\f(s,
x(s), x(s)) + q(s)x(s)) ds,
teJ
(a)
t 6 J.
(b)
Jo
and tt>Wx(t) = x0-rJ
leQW(f(s,x(s),x(s))
+ q(s)x(s)}d3,
Denote t\ = sup{s € J | x(t) -C x(t) for each t € [0, s]}, and make a counterhypothesis: t\ < T. The continuity of x and x implies that x ( t i ) = x ( t i ) + ifc for
301 some ife € dK. Since x(0) < x(0), then *i > 0. Since x(t) < x(t) for each t € [Q,ti], it follows from the given hypotheses that f(s,x(s),x(s))
+ q(s)x(s)
< f(s,x{s),x(s))
+ q(s)x(s),
0 < < <
(c)
FVom (a), (b) and (c) it follows that x0-x0
+ eQMk
= eQl*l)x(h)
= £
e«"[f(S,x(S),x(s))
- x(0) - ( e 0 ( , , ) * ( * i ) - x(0))
+ q(s)x(s) - f(s,x(s),x(s))
- q(s)x(s))ds
> 0.
But then 0 <. e~^tl\i0 — x„) < k, which would imply that 0 < k, contradicting the fact that ife € dK. Thus x(t) < x(t) for each t e (0,T). This and the above reasoning with t\=T imply that also x(T) < x(T). D R e m a r k 4 . 1 . In proposition 4.2 we assumed an existence of at least one solution to the IVP (1.1). As for conditions which ensure this, see, e.g., [3]. References. 1. A. Chaljub-Simon, R. Lemmert, S. Schmidt and P. Volkmann, Gewohnliche Differentialgleichungcn mit quasimonoton wachsenden rechten Seiten in geordneten Banachraumen, Int. Ser. Num. Math. 103 (1992), 307-320. 2. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Aca demic Press, New York-London, 1988. 3. S. Heikkila and V. Lakshmikantham, Monotone iterative techniques for discon tinuous nonlinear differential equations, Marcel Dekker, New York (to appear). 4. S. Heikkila, V. Lakshmikantham and Y. Sun, Fixed point results in ordered normed spaces with applications to abstract and differential equations, J. Math. Anal. Appl. 163,2 (1992), 422-137. 5. R.M. Redheffer and W. Walter, Remarks on ordinary differential equations in ordered Banach spaces, Monattschefte Math. 102 (1986), 237-249. 6. I. V. Shragin, On the Caratheodory conditions, Russian Math Surveys 3 4 , 3 (1979), 183-189. 7. W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin - HeidelbergNew York, 1970.
WSSIAA 3 (1994) pp. 303-308 © Worid Scientific Publishing Company
303
ON I N F I N I T E SYSTEMS OF L I N E A R B O U N D A R Y VALUE P R O B L E M S
GERD HERZOG Mathematisches Institut I, Universitat
Karlsruhe,
Postfack 6980, 76128 Karlsruhe, Germany Postfach 6980, 76128 Karlsruhe,
Germany
and and
ROLAND LEMMERT Mathematisches Institut LEMMERT I, Uhiversitat Karlsruhe, ROLAND Mathematisches Institut I, Universitat Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany ABSTRACT Boundary value problems of the first kind for infinite systems of second order ordinary differential equations of the form a(n) * n ( 0 + X > n * Z n ( 0 = r„(<) *=1
are discussed with respect to existence and uniqueness of coordinatewise solutions. No a priori quantitative restrictions to the a„k's and the r n 's are imposed.
1. Many problems in applied analysis lead to infinite systems of differential equations, the most simple case of which reads <■(»)
*'«(0 = 5^«n***(0 + M0. *=i
n = l,2,...,
(1)
304 with prescribed initial values *n(0)=*O,n,
n = l,2,...,
(2)
compare [1] and [7], for instance. According to various properties of the data a„* and r n one can look at these problems in different frameworks and prove existence and/or uniqueness in different function spaces like CQ, P or /°°, for example. Each solution in the aforementioned cases is also a solution componentwise, so it seems natural to consider this concept of solution, too. The natural frame for this is w = JRN (or w = C N ) provided with the product topology. Here, no a priori quantitative conditions on the matrix A = (a„t), on the functions r n or the initial values xo,„ are imposed. Combining results from [2] and [4] we have T h e o r e m 1. If in (1), (2) each function r n : [0,1] —► 1R is continuous, then this problem is always solvable. It is uniquely solvable for each xo = (*o,n) */ and only if the spectrum cr(A) of A is at most countable. Here the spectrum
x € w , y€
where the sum has only a finite number of terms. By i ( w ) respectively L(
i) a{A) = a(A) ± 0.
Then
305 ii) cr{A) is at most countable or has at most countable complement in C. iii) o~(A) ist at most countable if and only if span {TAky : k = 0 , 1 , . . . } is finitedimensional for each y € f. (Such a 34 is then called locally algebraic.) iv) o~(A) is at most countable if and only if ar(TA) = o-(A) (o-p(A) the point spec trum). oo
v) If f(z)
= £ afcZ* is an entire function and a(A) is at most countable, then
oo
oo
k=0
Jt=0
2 <*kAkx and J3 ajl*y ore convergent for each x € w, y € y>, and define elements f(A) € L(u), f(TA) € L(
Furthermore f(a(A))
=
2. We fix B € L(
<€[0,l],
y(l) = ih.
(3) (4)
If (3), (4) has a solution y, M = {y(i) : 0 < t < 1} is compact and so dim span M < oo. Since each subspace of ip is closed and because j/'*'(<) € span M, A: = 0 , 1 , . . . , we get Bkn0 = -y<2*>(0), B*r;, = -y (2 *>(l), and so span {Bkn0 : Jfc = 0 , 1 , . . . } and span {Bkr)i : k = 0,1,...} are finite-dimensional. This motivates Theorem 2. The following assertions are equivalent: i) (3), (4) is uniquely solvable for each pair r/0, »/i € f. ii) (7(B) is< a< most countable and k2ir2 £ o(B), k € IN. Proof. The remarks just made show that B is locally algebraic if i) holds, and so o~(B) is at most countable. If k2n2 € o{B) = ap{B) for some k € IN, y(t) = sin (fcirt) • y 0 , yo a corresponding eigenvector, would be a nontrivial solution to (3) with boundary values 0, so (3), (4) was not uniquely solvable. To prove the converse, it will be sufficient to solve (3), (4) for t}\ = 0, n0 ^ 0. (Remark that if y(t) is a solution to (3), then y(l — t) is also a solution.) Since B is locally algebraic, there is a minimal k > 1 such that {n0, Bkt)o} is linearly independent, so that k
Bk+lr,0 = 5 > . flV
Br)0,...,
306 Let E — span {770, Brjo, • • •, BkVo}'< then BE C E, and B\E has a matrix representa tion
/0 * =
0 ••■ 0 AA
; •
-
• *
.
\ o 0 ••■ 1 A / None of the numbers pw2 is an eigenvalue of MB, SO that / z"(t) + MB ■ z(t) = 0, \ z(0) = ( l , 0 , . . . , 0 ) T , *(l) = 0 is uniquely solvable. Then y(t) = £ Zj(t)B'r]0 is the desired solution. J=0
Uniqueness is proved by means of the same ideas. We now look a the inhomogeneous equation y"(t) + By(t) = s(t),
(5)
where s : [0,1] —»
_s
5(2)
i n
^_V(-l)
2 t + 1
,*
- —-§(2TrrT
2
and set h(t) = tg(t B)t], r? to be determined. y(t) = h(t) + y,(t) solves (5), and y(l) = /i(l) + y r (l) = g(B)ri + j/ r (l) = 0 is uniquely solvable, since g(B) is invertible: 0 € o-(g(B)) = g(a(B)) would imply that 17(B) contains a zero of g which are exactly the numbers fc2ir2, k € IN.
3 . We consider the problem x"{t) + Ax(t) = 0,
t € [0,1],
(6)
307 x(0)=&,
x(l) = fi,
(7)
where £<>i£ € u>, >1 6 £(<*>), and prove the analogue to Theorem 2. Theorem 4. 77ie following assertions are equivalent: i) (6), (7) is uniquely solvable for each £oifi € w. ii) o~(A) is at most countable and fc2T2 £ c(;4), A; € IN.
Proof. Let i) be valid, set f0 = 0 and solve (6), (7) for each & € w. Then (i —* x (the solution) is a closed linear map from w to C([0,1], w) and therefore continuous-. So to each ifr € ^ , t € [0,1], there exists an element y(t) € y? such that (x(0,«h> = «i,y(0>We have clearly y(0) = 0, y(l) = t)x, and since 4x(<) solves (6), (7) with boundary values 0 and i4&, we get (Ax(t),m)
= (Atuy(t))
=
(ZuTAy(t)),
(-x"(0,»/,) = -{6,!/"(*)) = {6,-S/"(<)>Since £i is arbitrary, we infer y"(t) +TAy(t) = 0. Theorem 2 gives ii). The proof that ii) implies i) follows the same ideas and is omitted. Concerning the inhomogeneous equation x"(t) + Ax{t) = r(t),
t € [0,1],
(8)
where r : [0,1] —> u> is continuous, we have Theorem 5. Ifcr(A) is at most countable with k7r2 £
References 1. K. Deimling, Ordinary Differential Equations in Banach spaces, Springer, Ber lin - Heidelberg - New York, 1977. 2. G. Herzog, Uber gewohnliche Differentialgleichungen in Frechetraumen, Disser tation, Karlsruhe, 1992.
308 3. K.H. Korber, Das Spektrum zeilenfiniter Matrizen, Math. Ann. 181 (1969), 8-34. 4. R. Lemmert and A. Weckbach, Charakterisierungen zeilenendlicher mit abzahlbarem Spektrum, Math. Z. 188 (1984), 119-124.
Matrizen
5. H.H. Schaefer, Topological Vector Spaces, Springer, Berlin - Heidelberg - New York, 1971. 6. H. Ulm, Elementartheorie unendlicher Matrizen, Math. Ann. 114 (1937), 493505. 7. W. Walter, Ordinary differential inequalities in ordered Banach spaces, J. DHL Eqs. 9 (1971), 253-261. 8. J.H. Williamson, Spectral representation of linear transformations Cambridge Philos. Soc. 47 (1951), 461-472.
in w, Proc.
WSSIAA 3 (1994) pp. 309-324 © World Scientific Publishing Company
O n t h e relative
309
P—Capacity
a n d i t s a p p l i c a t i o n t o d e g e n e r a t e elliptic e q u a t i o n s TOSHIO HORIUCHI
Department of Mathematics, Ibaraki University Mito, Ibaraki, 310, Japan. Dedicated to Professor Wolfgang Walter on his 66th birthday Abstract. The purpose of this paper is to study the weighted relative p—capacity C£(E,F) and to apply it to examine solutions for certain degenerate elliptic equations. First, we shall establish metric properties of the p—capacity in terms of weighted Hausdorff measure using the relativity of E and F. Secondly, we shall apply these to degenerate elliptic equations and describe fundamental theorems for weak solutions.
1. Introduction Let E and F denote compact and open subsets of R n , respectively, E c F. The number (1.1)
CZ(E,F) = mtlf
\Vu\'u>dx:u€CE(F)\
is called the weighetd p-capacity of a compactum E relative to F. Here 1 < p < °°> V u = ( ^ 7 ' ^ 7 ' • • • ' ^ ) u d CE(F) is the class of functions u(x) € C 0 - 1 ^ ) with u(x) > 1 for x € E and compact support contained in F. The weight function u>(x) will be a non-negative, measurable function on R n satisfying the following condition (S). (S): T H E SOBOLEV INEQUALITY.
Given 1 < p < +oo, there exist positive constants C and q> p such that for all balls B with diameter d(B), it holds that:
for any u € C$(B). Here w(B) = fgudx
and C is independent of each u.
In §8, we shall also assume the next property on the weight u:
310 (P): THE POINCARE INEQUALITY.
Given 1 < p < +00, there exist positive constants C and q>p such that for all balls B with diameter d(B), it holds that:
(1.3)
( ^ j g Hx) - ti(B)*|M*)«k)' < < Cd(B) (-Lj JB |V«(x)|M») dxj ',
for any u e CX(B), where either u(B)* = -^hx fBu(x)w(x)dx AT JB u(x) dx, and C is independent of each u.
or u(B)* =
In §3, we shall give sufficient conditions for a weight function u to satisfy these properties (S) and (P). In this paper, the following degenerate elliptic operator Lu fulfils an important role; (1.4)
Luu = - divMaOlVul''-2 Vu).
In fact, the capacitary extremal u to the variational problem (1.1) satisfies, in the weak sense, the Euler-Lagrange equation Luu = 0 in F \ E ( For the definition of the capacitary extremal, see the remark just after Proposition 2.2 in §2.). The purpose of this paper is to study the weighted p-capacity C£(E, F) and to apply it to examine solutions to the equations represented by "Luu = f" under some assumptions. First, we make clear the behavior of C%(E, F) as a set function from the point of view of the relativity of E and F. Then, by making use of this, we shall give simple proofs of metric properties of the p-capacity in terms of weighted Hausdorff measure. Our methods in this paper are based on the effective use of the theory of the Dirichlet problem for non-linear elliptic differntial equations and the Sobolev inequality (1.2). We note that when u> = 1, these topics were already treated in the author's paper [11]. Secondly, we shall apply our results to degenerate elliptic equations in §8, which are essentially of the type Lu = f, and describe fundamental theorems for weak solutions. The results are the unique existence of weak solution, a Haraack inequality for positive solutions, and the interior and boundary regurality for weak solutions ( cf. [5], [6], [7], [8], (14) and [15] ). This paper is organized in the following way: In §2 we prepare notations and collect the basic properties of the weighted p-capacity. In §3 we describe some sufficient conditions for a weight function u> to satisfy the properties (S) and (P). In §4 our main results for the weighted relative p—capacity will be stated, and the proof of Corollary 4.2 is also given there. §5 is devoted to prepare the
311
proposition concerned with the classical Dirichlet problem for quasi-linear elliptic equations of second order. Under these preparations we shall establish Theorem 4.1 in §6. The proof of Proposition 4.4 will be given in §7. Lastly in §8, we shall give the application to the degenerate elliptic equations.
2. Preliminaries In this subsection we prepare notations to be used throughout the paper and present a very brief introduction to the weighted p—capacity C£(E, F) defined by (1.1). We begin with recalling some simple properties of C£(E, F), which are mostly obvious consequences from the definition (cf. [4] and [13]): PROPOSITION 2.1. Let p satisfy p > 1 and let a weight function w satisfy the property (S) in %1. (1) Let E\ and E^ be compact sets C F. The inclusion E\ c £2 implies C?(EltF) <(%(&,F). (2) In the definition ofC£(E,F), the space CE(F) can be replaced by the space DE(F) = {u e CE(.F);Q < u < 1,« = 1 m o neighborhood of E}. (3) For any compact set E C F and e > 0, there exists a bounded open set G such that £ C G C F , 9 G M smooth and
C;(E, F) < C^(G, F) < C%(E, F) + e. It is very useful to know there exist extremal functions where the infimum in the definition of C°(E.F) is achieved. To this end we denote by Wo'p(tl;<j) the closure of Co°(ft) in the space W1'p(il;u>), where the space W1,p(il;u)) is the set of functions on fi c R", whose generalized derivatives dPu of order < 1 satisfy
(2.1)
||u : W*'"(il;u>)\\ = V ) ( [ IWvfudx)
' < +00.
Conventionally, we set (2.2)
Lp(n;w) = W 1 *(n;«)
Then it follows from the Clarkson inequalities (cf. [15]) and the Sobolev inequal ity (1.2) that:
312 PROPOSITION 2.2. Let p satisfy p > 1 and let u satisfy the property (S). Assume that F is a bounded open set whose boundary is smooth. Then
(2.3)
C%(E,F)=intl
f \Vu\*dx;u 6 Wi*{F;u),u
> 1 on E q.e. \.
Here by the term q.e. we mean that u > 1 on E everywhere except possibly on a set of the weighted relative p—capacity zero. Moreover if C%(E,F) < +oo, then there exists u € WQ*(F;U) such that 0 < u < l , « = l everywhere on E and (2.4)
{ fFWu dx =
CZ(E,F),
in F\E
{ Luu = 0,
in the weak sense,
where Lu is defined by (1.4) This distribution u is called " capacitary extremal of E relative to F ", and it is essentially unique up to values on sets of vanishing weighted p-capacity. Here we note that the assumptions on F may be avoided by the use of definition of W 0 lj, (F;u) which does not require u € W{F;u) for u € Wl'*{F\u). For more precise informations, see [15; Chapter 2 ] for example. In the rest of this subsection we prepare more notations. Let us set, for an arbitrary compact set E of R n , (2.5)
dist(x, E) = inf \x - y\ and £ , = { i € R n : dist(x,
E)
y€£
where En is called a tubular neighborhood of E in R n . If dE is smooth, then 8En is also smooth for almost all n > 0 by Sard's lemma. But even if dE,, is not smooth, we can always approximate dEn by compact smooth manifolds. There fore we assume throughout this paper that the family of tubular neighborhoods denned by (2.5) is smooth as well without loss of generality. Lastly we give the definition of the d-dimensional weighted Hausdorff mea sure. Let 5 be a bounded set in R". Consider various coverings of S by balls Bj = Bri{ij) with centers Xj and radii r,, we put (2.6) AJ(S) = vd inf £ rfuirj,
Xj)
and
«(r, x) = y ^ - r y /
^
«(*) dx,
l # r ( Z ) | JBr(x)
where v* is the volume of the unit ball in R d and the infimum is taken over all such coverings. It is of no importance if Bj are assumed open or closed. If we also assume r, < e, we get a corresponding lower bound H%C{S). The limit (2.7)
^(5)=tUmo^,e(5)
clearly exists and is called the d-dimensional weighted Hausdorff measure of S. Here we note that h%(S), H%t(S) are zero simultaneously.
313
3. Some examples of the weight function u In this subsection, we shall give some sufficient conditions for a weight func tion u> to satisfy the properties (S) and (P). First, let us recall the Muckenhoupt's Ap condition. 1. A non-negative measurable function u(x) on R n is an Ap weight for p > 1 ( or simply u € Ap) if EXAMPLE
rt
»"
\P-1
T(l5[/.-*)(l!l/.- *)
< 00,
where the supremum is taken over all balls B in R n . Then, it is well known that u> € Ap is a doubling measure and satisfies the properties (S) and (P) ( For the proof, see [7]). 2. Let p > 1 and let M be a (n — k)— dimensional subspace of R (1 < Jfc < n - 1). We set u(x) = dist(x, M)a. Then u> € Ap, iff -k < a < k(p—l). Moreover, it follows from Example 3 below that u satisfies (S) if -k < a, and u> satisfies (P) if either — k
n
EXAMPLE
3. We recall the property P(s) (0 < s < n) which is defined in
[10]. DEFINITION OF P(S). Let s be a positive number satisfying 0 < s < n and .8* = min(l, s). A closed set M is said to have the property P(s) if \M\ = 0 and there exists positive number C such that (3.2)
|B n (M„ \ M„.)| < Crf—(r, -
rf)"d(B)",
where B is an arbitrary ball with diameter d(B)
{
a> -s
if
0>o>-3
l<«
Then u satisfies the required properties (S) and (P) (For the detailed, see [10] and [12]). We remark that: there are positive numbers C and C such that, for any x € R n , r > 0, m > 0 and any ball B , we have C" 1 (r + dist(x, M))a < u/(r, x) < C(r + dist(x, M))°, so that H£(M r\B)
Hl+^M
n B).
314
In particular, if a > n - m, then H£(M n f l ) < +00 for any ball B. We also remark that a fairly large class of sets M of R n has this property P(s), for example, a set offinitepoints (« = n), an (n—k)—dimensional Lipshitz manifolds of R n (s = k) and a Cantor set (« = 1 — log3 2) and so on. As is easily seen, if M" and M' satisfy P(s) and P(t) respectively, then it follows from the "linearity" of P(s) with respect to M that M* U M* satisfies P(min(s,t)) as well. See §2 in [10], for the examples and counterexamples to this property P(s). In the rest of this subsection, we give an example of capacity and capacitary extremal with a weight u satisfying P(s). EXAMPLE 4. Let us set, for 1 < p # n + a, 1 < / < n, a >
-I,
w(x) = Ix1]", for x = (x1, x") € R' x Rn"', A A R -\x\ . A p-n-a „ «(*) = n i i . for A = £ — , 0 < r < R < +00.
(3.5)
RA - rA
p- 1
Then we have (3.6) C;(BT, BR) = f
\Vu\"udx = ,JAl*
W ^ - 1 " * dx
I
\R* ~ rA\p
JBR\BT
Jr<\x\
A | " - | 5 ' - | | 5 " - ' - | 5 ( ^ , ^ ) \ R -r^l "" ■{ |lAI"!^- !!^ - r'*! "" if either I = 1 or n. 1
1
and
1
1
1
1
A
£1
1
if 2 < I < n - 1.
it
- div(w|V«| p-2 Vu) = 0
in
BR\Br.
Here | 5 n - 1 | is the area of the (n — 1)—dimensional unit sphere.
4. Main results In this subsection we shall state our main results, which are concerned with the weighted relative p-capacity. First we give a theorem on the behavior of the weighted p-capacity as E,, is shrinking away to E, which characterizes in some sense the sets of non-vanishing p-capacity and provides the upper estimates for the p-capacity. Then we state the result on the lower estimate which will be established later, by using the property (S). 4.1. Letp satisfyp > 1. Assume thatw(x) is non-negative, mea surable and has the property (S). Let E be a compactum in R n . IfC£(E, F) > 0 for some open set F c R " , then it holds that THEOREM
(4.1)
lhn(%{E,EJ
= +oo.
315
Moreover, we have
(4.2)
q[(E,FY
\VuJTudxJ ,
for any n € (0,dist(E,dF)). Here u is the extremal function where the infimum in the definition of C£(E, F) is achieved. Here we note that this inequality (4.2) becomes an equality when u = 1, E and F are concentric closed balls. From this theorem we can easily derive the connection between the weighted p—capacity and the weighted Hausdorff measure. A non-negative measurable weight function w is said to be a doubling weight if there is a constant Co such that for all balls B, ui(B') < Co • u>(B) where B' is the ball with the same center as B but with twice the radius - the double of B. The following is a direct consequence, and the proof will be given in the last of this section. COROLLARY 4.2. Let p satisfy 1 < p < n. Assume that u>(x) is a doubling weight and has the property (S). Let E be a compactum in R n . Assume that H"-P(E) < +00. Then it holds that
(4.3)
C£(£,R n ) = 0.
We also have the following ( The proof is omitted, cf. [11]). PROPOSITION 4.3. Let p satisfy 1 < p < n. Assume that w(x) a is nonnegative, measurable function with the property (S). Let E be a compactum in R n . IfC^(E,Kn)= 0, then for any open set F containing E we have
(4.4)
CpJ(E,F) = 0.
Here we note that if p > n, the assertion fails to hold. In fact C*(.E,Rn) = 0 (a; = 1.), for any compactum E. But CP(E, F) is away from 0 in general. Secondly we give a proposition on the sets of vanishing weighted p—capacity which is well known if u> = 1. Let us recall the definition of the class Ap in §3. PROPOSITION 4.4. Let p satisfy 1 < p < n. Assume that u 6 Ap and that C%(E, F) = 0 for some open set FcRn. Then it holds that
(4.5)
HZ_p+e(E) = 0
for an arbitrary e > 0. We note that, for a general weight u>, this property is not completely clear. In the rest of this section we shall establish Corollary 4.2 which is rather elementary if we admit Theorem 4.1.
316 PROOF OF COROLLARY 4.2. Let T? > 0. Since H^_p(E) < +oo, we can
construct a locally finite open cover of E by balls BTj (XJ) with radius rj, center Xj(j = 1,2,...) such that (4.6)
B2ri(^)cEv>
rj<\
where H = vnlpmax(l,2H%_p(E)). tions ipj(j = 1,2,...) so that (4-7)
and
Y.r'F>w{Ti,xi)<
H
>
Let us choose a sequence of smooth func
0 < V , ( x ) < l ,
where C is a positive number depending only on the dimension of the space. Then we immediately get C~(£, E„) < J |V sup
where C" is a positive number depending only on the dimension of the space. Thus we see that (4.1) does not hold, hence we have Cp(E,Kn) = 0.
5. e—Reguralization Throughout this section we assume that E and F denote bounded smooth open subset of R n . We shall explain that the extremal function in Proposition 2.2 in §2 can be approximated by smooth solutions of regularized problems. Since this fact seems to be familiar, one may skip this section at first and return here if necessary when motivated by the use made later on. For any 0 < e < 1, we set (5.1)
Je(ti) = J (|Vu|2 + £2)*0;e dx
and consider the variational problem (5.2)
inf{J £ («):«€C £ (F)},
where {w t (x)} 0 < t
317 to see that uc also satisfies the property (S) if so w does. By C£e(E,F), we denote the minimal value of this problem. Then obviously we have (5.3)
C?aiu'lM(E,F)
< C£ e (£,F) < C(p)(C%(E,F) + ep\F\ + epu(F)),
where u(F) = JF u/(x) dx and C{p) is a positive number depending only on p. This implies C%(E,F) and C%e(E,F) behave in a similar way, provided F is bounded and e is sufficiently small. We shall collect basic properties of this regularized p-capacity C%e(E,F) which are useful in this paper. The following is well-known. For the proof, see [15;Chapter 2] for instance (See also the remark just after Proposition 2.2). LEMMA 5.1. Assume that E and F are smooth and bounded. Then, there exist extremals ue 6 WQ'P(F;u)e) for e > 0 such that
(5.4)
Jc(uc) = q^(E, F)
for e > 0,
(5.5)
IimJe(iie) = q r ( E , n
Bm«, =
tti»W0,*(F;w).
Moreover, from the well known resuls on the existence of the classical so lution of the quasi linear elliptic differential equation, the capacitary extremal functions {ue}o<e satisfy the Euler-Lagrange equation:
(
-dtv(u;e(|V«e|2 + e2)*?Vue) ue = 1
on E, uc = 0 2
u€ e C -"(F \ E)
=0
in F\E,
on dF,
(classical Holder space).
Here, /i € (0,1) may depend upon e. 6. The proof of Theorem 4.1 We shall show the inequality (4.2), assuming that (6.1)
C?(£\F)>0,
for some bounded smooth E and F. It follows from the argument in the previous section that the capacitary exremal can be approximated by a suitable sequence of C2'**—functions. Therefore we assume in the proof, without loss of generality, the regularity of the capac itary extremals. Let U and Vq for a small T? > 0, be solutions of the following Dirichlet problems:
(
-div(u>\VU\p-2VU) = 0 in F \ E, V\E = 1< 2
U\0F = °>
ueC -"(F\E)r\c°'1(F),
318
and ' -dtt/(a/|VV„r 2 Vig = 0 in Ev \ E, (6.3) VV€C2'"(E^E)C\C0'1(E^). Then we have (6.4)
J \VU\'tjdx = CZ(E,F)
and
J \VVv\>udx = C^(E,E„).
We take a family of Lipschitz functions ipp for p € (0,1/2) so that 1 1 - p-x 1>p{x) = < l-2/> 0
(6.5)
0 < x < p. p <x <1— p, 1 - p < x.
Then (6.6) f
V^,(V„). Vtf| VU\*-'LJ dx = /
=
^(V,)|V^r2^wd5
/ iwr'^ds.
Here we denote by S the (n — 1)—dimensional Lebesgue measure, and v is the unit outward normal. Let fl be an arbitrary open set such that dil is smooth mdilcF\E. Then we have / \VU\"-2^-wdS= Jan °v Therefore we have (6.7)
(6.8)
/ div (w|Vtf|"-2Vt/) dx = 0. Jii
\VU\"-2^-uidS=
/ JdE,
OV
f
\VU\r~2?¥-<jdS.
J0E
OV
Here we note that \VU\>u>dx = / JOE JdE Combining this withh (6.6) and (6.8) we have (6.9)
(6.10)
/
IF\E Jf\E
C%(E, F)= f
\VU\'-2^-udS. ov
#(V,)VV, • VU \VU\"-2udx.
JE,\E
So we get (6.11)
CZ(E,F)<maxWp\CZ(E,E„)i(J
\VU\"udx\
' .
Since p is an arbitrary positive number, we have the desired estimate by letting p -► 0.
319
7. The proof of Proposition 4.4 We begin with preparing two propositions, the one is a variant of the theo rem due to L. Carleson [3; Chapter II], and the other is due to D.A. Adams [1; Theorem 7.1]. PROPOSITION 7.1. Let d be a positive number < n. Then there exists a constant C, only depending on the dimension, such that for every compact set E, there exists a nonnegative measure with compact support on R n satisfying
u(Bp(x)) -< r pd-u(p,; ,( r-,-„x-„ ,.,-, \ u(E) > ChV(E).
(71)
for every Bp(x),
Here u/(p,x) and h%(E) are defined by (2.6) in %2. PROPOSITION 7.2. Let 1 < p < q < +00 and p < n. Letui 6 Ap and u be
a nonnegative measure on R n . Then the inequality (7.2)
(^j
W\"duy
IVurwdx)',
for all u € CS°(Rn), holds if and only if (7.3)
K =
c
sup
j- < +00,
*6R-,P>O p-iu)(Bp(x))'
where <J(BP(X)) = fB ,^u(x)dx. C in (7.2).
Moreover K is equivalent to the best constant
PROOF OF PROPOSITION 4.4. Without loss of generality we assume that 0 < e < p < n. By Proposition 7.1 there exist a nonnegative measure u and a ball BQ such that
u(Bp(x)) < ^ n_p+e a;(r, x), (7.4)
u(E) > Ch%_j+e(E), . suppu c Bo
and
for any p and x 6 R n , for some constant C > 0,
E C F c Bo-
Next we modify w in Fc to obtain u* such that UJ*(X) = u>(x) on F, w*(i) = 1 on BQC and u* € Ap. Then, it follows from Proposition 7.2 that the inequality (7.5)
( f
|u|« da J ' < C ( j
I Vu|pw* dx\',
for some q > p,
320
holds for all u € Co°(Rn). Admitting this for a moment we first establish Proposition 4.4. Since C£(E,F) = 0, for any 6 > 0 we can find an element us € C 0 , 1 ^ ) with compact support contained in F such that (7.6)
us>loaE
and
/
\uf\qdn
Then fi(E) < C6. Thus we have (i(E) = 0, and hence /#_,*,(£) = 0, which implies HZ-ri.t(E) = 0. PROOF OF (7.5). Since w* € Ap, there exist positive numbers C and t such that infx€flo um(Bp(x)) > Cp* for any 0 < p < 2d(B0) (cf. [1]). Since u>*{x) = 1 (x € BQC) and p < n, we see that the condition (7.3) holds when we choose q so that; q = p*~t^€, if p < t and 4 = p + l , i f t < p .
8. Applications to degenerate elliptic equations In this section we shall apply our imbedding results to certain degenerate elliptic equations and describe fundamental theorems for weak solutions. Since the structure of the proofs of these results are quite similar to those in the nondegenerate case, which are essentially due to J. Moser [14] and P. Gariepy, W.P. Ziemer [8] and E.W. Stredulinsky [15], we do not give precise proofs here but mention the basic ideas. Let 1 < p < oo and n > 2. In this section we always assume that a given non-negative measurable function u> satisfies the properties (S) and (P) defined in §1. The equations to be treated are of the form (8.1)
-divA(x,Vu)
+ B(x,Vu) = 0
in D,
n
where D is a domain of R ,and the functions (8.2)
A:DxRn-»Rn
and
B:DxRn-^R1
satisfy the four hypotheses listed below: (H-l) x ~* A(x, h), B(x, h)
are measurable for all A € R n ,
h -» A(x, h), B(x, h)
are continuous in a.e. x € £>.
(H-2) wlAI" < A(x, h) ■ h, \B(x, h)\ < bu\h\p,
\A(x, h)\ < wlAl""1, for some positive constant b.
(H-3) Monotonicity: (A{x, h\) — A(x, A2)) • (hi - h2) > 0 if hi jt A2(H-4) Homogeneity: A(x,th) = \t\p~2tA(x,h),
for all t € R 1 .
321
By W^(D; u) we denote the local version of the space W1'P(D; u). DEFINITION OF WEAK SOLUTIONS. A function u e WJ£(D;u) is said to be a weak solution of the equation (8.1) in D if it holds that
(8.3)
/ (A(x, Vu) ■ V
for all ip e Cg°(D).
JD DEFINITION OF A-HARMONIC.
A weak solution denned above is called A-
harmonic if B = 0. Now we state our results. By V we denote the dual space of w£,p(D;u>). Then we have 8.1. Let D be a bounded domain o/R n whose boundary is smooth. Let p satisfy p > 1. 27ien for any f 6 V, there exists a unique weak solution u e WQ'P(D;U) satisfying PROPOSITION
(8.4)
-div A(x, Vu) = f(x)
in D.
Since the operator Lu = —divA(x,Vu) is monotone and coersive, the unique existence follows from the general theory for monotone operators. Fol lowing the arguments by E.B. Fabes, C.E. Kenig, R.P. Serapioni [7], E.W. Stredudinsky [15; Chapter 3] we have: 8.2. Let 8 satisfy 0 < 6 < 1 and let L be a positive number. Let Br(a) C D be a ball with its radius r and center a. Assume that u € Wte(Br(a); u)) is a nonnegative weak solution of (8.1) in the ball Br(a) satisfying \u\ < L. Then it holds that THEOREM
(8.5)
sup u < C(r) inf «, B,r(a)
B,r{a)
where C(r) is a positive number depending on p, r, b, M, 0, and n but independent of each weak solution u. From this one can show the local Holder continuity of the weak solution u. In the rest of this section, we shall study the equation (8.4) assuming that (8.6)
tj(x) = dist(x, M)ap,
(p > 1),
where M is a given closed set satisfying the property P(s) with s > 1 denned in Example 3 in §3. Therefore, u) has the propertis (S) and (P). Then we have
322 THEOREM 8.3. Let u be defined by (8.6) with M satisfying the property P(s) {s > 1). Let f e LQ(D), Q>\. Assume that 0 < a < 1 - %, and assume that u e WJ£(Br(a);u) is a nonnegative weak solution of (8.4) in the ball Br(a). Then u is locally Holder continuous in Br(a).
SKETCH OF THE PROOF. Before the proof we remark that there is a coun terexample which shows the upper bound of a in this Theorem is sharp ( i.e. a < 1 — 31 ). Let us set K(r) = re for a sufficiently small e > 0 and put V = u + K(r). Then the proofs of Thorem 3.1.10 and Theorem 3.1.15 in [15; Chapter 3] work in this case with minor changes, if we replace / by |/|[yrRTlP-1In fact in place of (3.14), (3.15) and Lemma 3.17 in [15], we can use (S), (P), Theorem 1 in [10 ; Chapter 3], and Lemma 8.4 below:
LEMMA 8.4. Let 6{x) = dist(x,M) and Br = Br(a). Let a,p,h and k be nonnegative numbers satisfying a > 0,1 < h < p, and 0 < 1/A; — 1/h = (1 - ap/h)/n. Assume that
(8-7)
JB M"|/| dxj^^
< C(r)rfc jf |V
Then it holds that
(8.8)
JB M'l/I dx-K^
f \
where n is an arbitrary number satisfying 0 < n < 1, and C(r) is a positive number independent of each
8.4. It suffices to put
of Young's inequality. END OF THE PROOF OF THEOREM 8.3. Let 6(x) = dist(x, M). Since / 6 L® and 0 < a < 1 - jj^, we have from Holder's inequality and Theorem 1
323
in [10; Chapter 3] with (p,q,a,0) being (h,k,ap/h,Q).
JB\
(8.9)
( j f Mfcdx)
\f\***dx\ < C ( r ) ^ ^ — r" / where (810)
/ IV^l"^"^
|Vv|*«^«fa, for all p € (^(B,.),
C(r) = C . - ^ ( J ^
| / | A ds)1"""
* C j f ^ r < +oc, if e is sufficiently small.
Therefore the assumption in Lemma 8.4 is now proved. Next we consider the equation (8.1) under the boundary condition (8.11)
I I - G E WO'P(D;U),
for G € W lj, (R n ;w) n C°(D).
Then we can show the boundary regularity for weak solutions combining our results with the arguments in the paper [8] by P.Gariepy and W.P Ziemer ( See also [15]). THEOREM 8.5. Let u satisfy (8.6) with M satisfying P(s) (s > 1). Let C be a positive number and XQ € 3D. Let u 6 WJ£(D;w) be a weak solution for the equation (8.1) with the boundary condition (8.11) satisfying |u| < C. If it holds that
,
u)
(812)
(l ( qr(j?r/4(x0)\AR"K \ * dr _ Jo {c(r)fBAxo)dist(x,M)<»dx)
r "+°°'
then (8.13)
limsup |u(i) - G(x0)\ = 0. z—>xo,x€D
Here we note that the necessity of the condition (8.12) is partly proved. We also state a most fundamental result for p-harmonic equation from the potential theory ( The proof is omitted.). THEOREM 8.6. Let u satisfy (8.6) with M satisfying P(s) (s > 1). Let K be a compact set in D. Assume that C^(K,D) = 0 and u is A-harmonic in D\K. Thenu is A-harmonic in D.
from this and the fact (3.4), we have
324 COROLLARY 8.7. Let u satisfy (8.6) with M satisfying P(s) (s > 1). Let u € WJ£(D;u). If a > 1 and u is A-harmonic in D\M, then u is A-harmnic inD. PROOF. Assume that a > 1. Then it follows from (3.4) that fl£_p(M D B) < CH\_p+op(M C\B) < +oo for any ball B. By Corollary 4.2, we get C%(M r\B,D) = 0 for any ball B C D. Then it follows from Theorem 8.6 that u is .A-harmonic in D.
References 1. D. R. Adams, Weighted nonlinear potential theory, Irani. Amer. Math. Soc. 397, N o . l (1986), 73-94. 2. D. R. Adams, N.G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Math. J. 33 (1972), 139-158. 3. L. Carleson, Selected problems on exceptional sets, Toronto-London-Melbourne: Van Norstrand Co., (1967). 4. G. Choquet, Theory of Capacities, Ann. Inst. Fourier S (1955), 131-395. 5. B. DiBenedetto, C 1 + a local regularity of weak solutions of degenerate ellptic equations, Nonlinear Analysis, Theory, Methods & Applications 7 N o . 8 (1983), 827-850. 6. E. B. Fabes, D. S. Jerison, C. E. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier(Grenoble) 33 (1980), 151-182. 7. E. B. Fabes, C. E. Kenig, R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. P.D.E. 7 (1) (1982), 77-116. 8. P. Gariepy, W. P. Ziemer, Behavior at the boundary of solutions of quasilinear elliptic equations, Arch. Rat. Mech. Anal. 87 (1977), 25-39. 9. D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, second edition, Springer, 1983. 10. T. Horiuchi, The imbedding theorems for weighted Sobolev spaces, J. Math. Kyoto Univ. 39 N o . 3 (1989), 365-403. 11. T. Horiuchi, On the relative p-capacity, J. Math. Soc. Japan 43 N o . 3 (1991), 605-617. 12. T. Horiuchi, The imbeding theorems for weighted Sobolev spaces II, Bull. Fac. Math. Sci. Ibaraki Univ. series A. No.33 (1991), 11-37. 13. V. G. Maz'ja, Sobolev spaces, Springer, 1985. 14. J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. 15. E. W. Stredulinsky, Weighted inequalities and degenerated elliptic partial differential equations, Lecture note in mathematics Springer 1074 (1984).
WSSIAA 3 (1994) pp. 325-344 © World Scientific Publishing Company
325
ON THE NULL-CONTROLLABILITY SET, MINIMAL TIME FUNCTION AND TIME OPTIMAL CONTROL OF FINITE DIMENSIONAL SYSTEMS SHOUCHUAN HU Southwest Missouri State Department of Mathematics Springfield, MO 65804 USA and V. LAKSHMIKANTHAM and NIKOLAOS S. PAPAGEORGIOU Florida Institute of Technology Department of Applied Mathematics Melbourne, Florida S2901-6988, USA
ABSTRACT In this paper we examine the null controllability set and the minimal time function for a class of finite dimensional systems with control constraints and in which the state variable enters linearly. In particular, we study how those items respond to changes in the data and we use the minimal time function to completely describe the null-controllability set. Finally we prove a "bang-bang" principle for the time-optimal controls for a class of nonlinear systems. 1.
Introduction In this paper, we study the properties of the null-controllability set and of
the minimal time function, for a class of finite dimensional control systems in which the state appears linearly. In particular, we concentrate our attention to the changes of the above items, as we vary the data describing our system and we also see how the minimal time function describes completely the null-controllability set. Finally, we prove a "bang-bang" property that the time optimal controls have, for a class of nonlinear systems. In the past, the minimal time function of linear systems (both finite and infinite dimensional), has been studied extensively. works of Lee-Markus 12 , Hajek 5,6 and Carja2.
We refer to the fundamental
The importance of this function is
326 emphasized in the book of Lee-Markus12 and in the paper of Hajek5, where the minimal time function is used to construct feedback controllers. 2. Preliminaries Let (fi,E,/i) be a a-finite, complete measure space and X a separable Banach space. nonempty,
We will be using the following notations:
closed, (convex)} and Puc\(X)
=
{AC X:
Pf,cJX) nonempty,
= {A C X: compact,
(convex)}. A multifunction F:Q-*Pj(X) is said to be measurable if and only if for all z 6 X, w-*d(z,f(w)) = inf{ \\z — x\\:x£
F(u)} is measurable. Other equivalent
definitions of measurability of a multifunction can be found in the survey paper of Wagner 18 .
By SlF we will denote the set of integrable selectors of F( ■); i.e.,
SF = {/ € L\X)\
/(ii>) 6 F(w)fi — a.e.}. This may be empty. It is nonempty if for
example F( •) is measurable and w-» | F(w) | = sup{ \\x\\:
x g F(w)} € L1 + .
This
18
follows easily from Aumann's selection theorem (see Wagner , theorem 5.10). Such a multifunction is called "integrably bounded". It is easy to check that SF is closed in Ll(X),
it is convex if and only if F( ■) is convex valued and it is bounded
if and only if F( •) is integrably bounded. Umegaki
7
15
and Papageorgiou .
For further details we refer to Hiai-
l
Using S F we can define set-valued integral for
F(-),
by setting / FH
1
C 2*>{0}. We define:
Um An = {y € Y-.lim d(y,An) = 0} = {y € Y: y = lim yn, yn 6 An, n > 1} and lim A„ = {yeY:Um = {y 6 Y: y = lim ynk,ynk
d(y,An) = 0}
€ ^n fc .»i < n, < ... < n t < . . . } .
Both sets are closed (maybe empty). definitions that HmA„ C UmAn.
Also it is clear from the above
If HmA„ = UmA„ = A, then we say that the A n 's
converge to A in the Kuratowski sense, denoted by An-*A (see Kuratowski 11 ). On Pf(Y) we can define a generalized metric, known in the literature as the Hausdorff metric by setting
327 h(A, B) = max[sup d(a, B\ sup d(b, A)} for ail A,Be
Pf(Y).
If Y is complete, then so is (Pf(Y),h).
space, a multifunction
G:Z-*Pf(Y)
I f Z i s a topological
is said to be Hausdorff
continuous), if G( ■) is continuous from Z into the metric space Suppose {A„, A)n
>
C Pf(Y).
continuous (h(Pj(Y),h).
We say that the A^s Hausdorff converge to
A (denoted by An-*A) if and only if h(An,A)-*0 as n-»oo (i.e., we have convergence in the Hausdorff metric). From corollary 3A of Salinetti-Wets 17 , we know that on P fcc (R m ) Hausdorff and Kuratowski convergence coincide. Also
if
A
is
a
nonempty
= R U { + oo} we
denote
subset the
of
a
support
Banach
function
of
space, A;
by i.e.,
€ A}, where by (•, •) we denote the duality brackets for the
Recall that the support function of a set A, describes completely the
closed, convex hull of A; i.e., convA = {x £ X: (xm, x) <
/ g L »(R
Recall that {/„}„>, C L\JR
+,R
m
) converges weakly to
m
+ ,R )< if for every K C R + compact {/„ | K}n>l m
/ | K in L\K,R ).
Also by ACloc(R
converges weakly to
m
+
,R ) we will denote the space of all locally
absolutely continuous functions; i.e., x e AC« (R + ,R m ) if and only if for every 6 > 0 x S AC([0,b],Rm). ra
withR xI
1
£ o c (R +
By Lebesgue's theorem AC^
(R + ,R ro ) can be identified
m
,R ).
3. The null-controllability set On the state space Rm we consider the following sequence of control systems defined on the positive time semi-axis; i.e., for t > 0: (• l and the limit system r 1 I
i n ( t ) = A n (t)x n (t) + fl(t,u„(<))a.e. -I u„(t) €U„(t)a.e.,un(-)=
measurable)
x(t) = A{t)x{t) + B(t, u(t))a.t. / u(t) G U(t)a.e.,u( ■) = measurable.
^*)-
328 For the above systems, at every time instant t > 0, we can define the nullcontrollability sets C„(t),C(t)
n > 1, which consist of all those initial states that
can be transferred to zero at time t > 0, along trajectories of (*)n and (*) respectively, using admissible controls. So if 5„(r) and S(T) are the fundamental matrices for (*)n and (*) respectively (see Lee-Markus 12 , p. 64), then the sets
CJt)
and C(t), n > 1 can be described as follows:
CW = {y G Rm: - y = /
Sn " 1{T]BJT,
u n (r))dr, un G L'^JR
+,
R"), «„(t) G Un(t)a.e.}
0 i
C(t) = {y G Rm: - y = / S " W , « M ) < f r , « G ^ ( R + . I T ) , ^ ) G tf(t)a.e}. 0 Let C n = U Cn(<) and C = U C(<), n > 1. These sets contain all initial «>o t>p states that can be steered to the origin by our sequence of systems. Then the minimal time functions are defined as follows: inf{t G R + : x G C„(r)}
for x G Cn
+ oo
otherwise
and T(x) = {
i n
^
t & R +: x e C
^
iotxeC otherwise
+ oo
We will make the following hypotheses concerning the data of control systems (*)„ and (*): CI^(R
{An,A}n^
g(B):
B n : R + x R*-»Rm are functions s.t. for every n > 1
+
, i ( R " ) ) and AnlA
in L l ^ ( R +,Jt(R™)).
H{A):
(1)
<-»Sn(t, w) is measurable,
(2)
u-»i? n (t,u) is continuous,
(2)
II Bn(t, u) || < an(t) + bn\\u || a.e., with {a n } n < , C L ' ^ J R + ) uniformly integrable and sup bn < oo,
(4)
for every t G T, B„(t~- y*B(t, •) uniformly on compacta in R*.
Remark:
Because
of
hypothesis
H(B)(£)
and
the
Dunford-Pettis
compactness criterion, by passing to a subsequence if necessary, we may assume that an%a in L 1 « ( R + ) and bn-*b. Then because of hypothesis H (B)(4) for every D CR
+
, X(D) finite (A = Lebesgue measure on R + ) , we have for all r/ G
329
J \\Bn(t,u)\\r,(t)dt*J \\B(t,u)\\V(t)dt D
D
= * / || B(t,u) || n(t)dt < Ja(t)V(t)dt D
+ Jb || u || r,(t)dt
D
D
=> || B(t, u) || < a(t) + b || u || o.e., with a( •) e i ^ J R +), b > 0. H(U):
Un,U:R (1)
+
-*Pkc(Rk) n > 1 are measurable multifunctions s.i.
sup
| {/„(<) | = sup
n>l
(2)
n>l
sup
|| t; || <
v g U (0
Un(t$U(t) a.e.
Because of hypotheses -ff(-B) and #({/) and from well-known properties of the finite dimensional Aumann integral (see for example Klein-Thompson 10 , corollary
18.1.10,
Cn(t),C(t)
m
p.
203
and
theorem
18.3.2,
p.
206),
we
have
that
6 Pkc(R ) for all t> 0 and all n > 1.
The following lemma will be useful in what follows: Lemma S.l: If hypothesis H(A) holds, then foraUt>0
Sn(t)-*S{t} in l(Rm)
as n-K».
Proof: Recall that the fundamental matrices S„( ■) of (*)n, n > 1 and S( •) of (*) satisfy the following matrix-valued Cauchy problems: Sn(t) = An(t)Sn(t)a.e. S n (0) = /
and S(t) = A(t)S(t) a.e. 5(0) = / .
Hence we have
j-t\\sn(t)\\
< \\sn(t)\\ < 114.(011-IIs B (0II «•«• t
=* || 5„(t) || < 1 + /
||An(5)||.||5n(3)||d5.
0 Invoking Gronwall's inequality, we get that ||Sn(<)||<exP||An||Ll([ot)il(Rra)),<>0.
330 But from hypothesis H(A),
we have that sup
|[ An || .
I D m
<
M^t) < oo with Afj( •) increasing in t. So for all n > 1 and all t> 0 we have: \\Sn(t)\\
<M,(t).
Next, observe that l | 5 . ( * ) - 5 ( « ) | | = \\jAn(r)Sn(r)dr-J
A(r)S(r)dr\\
0 < || j
An(r)Sn(r)dr-
0
0 j 0
t
t
+ | | / A(r)Sn(r)dr-J 0 i < /
A(T)Sn(r)dr\\
|| A„(r) - A(r) || • M,{t)dr + /
A(r)S(r)dr\\ 0 t || A(r) || • || Sn{r) - S(r) || dr.
Since by hypothesis H(A), An-*A in L1^ (R + ,i.(R m )), given e > 0, we can find no > 1 s.t. for n > UQ, / || j4 n (r) — J4(T) || • Af i(t)dt < e. So for n > n 0 , we have t II 5 n (t) - S(r) || < e + / || A(r) || • || 5„(r) - S(r) || dr. 0 A new application of Gronwall's inequality gives us II SJt) - S(t) || < eexp || A || ^ ^ R m , , , n > n 0 . Since e > 0 was arbitrary, we deduce that Sn(t)-*S(t) in i.(R m ) as n-*oo, for all t > 0. Q.E..P. Remark: mVeJR
It is clear from the above proof that we also have Sn( • )-*S( •)
+ ,L(R'»)). In our first theorem, we obtain a convergence (continuity) result for the
null-controllability set.
It says that the controllability set responds continuously
(in both the Vietoris and Hausdorff topologies), to variations of the data describing the system.
331 Theorem S.2: If hypotheses H(A),H(B)
and H(U) hold,
then for all t> 0, CJtfictt)
as n-K».
h
Proof: Let y € HmC„(t). Then by definition, we can find y„ € Cn{t) s.t. yn-*y in R
m
(for economy in the notation we denote subsequence with the same
index
as
sequences).
Let
Gn{t) = B(t, Un(t)) = \J
B(t,u)
and
G(t) =
« 6 UJt)
B(t,U(t))=(J
B(t,u).
Clearly because of hypotheses H(B)
and H{U),
we
u 6 U(t)
have that Gn{t),G(t) € Pk(Rm) for all n > 1 and all t > 0. Also let un(:T-*Rk k
UA:T-*R
i>l
be
U(t) = {uJt)} by
functions
s.t.
U„(t) = {u Jt)}
and
for all t > 0. Such sequences of measurable functions exist since
hypothesis"
theorem
measurable
and
4.2
Un(-),U(-) of
n>l
Wagner 18 ).
B(t,U(t)) = {Bfau.tt))},
are Then
measurable Bn(t,Un{t))
multifunctions
(see
= {B(t,un((t))}
and
for all t > 0. So once again theorem 4.2 of Wagner 18
tells us that Gn( •), G( •) n > 1 are measurable multifunctions.
Then from the
properties of the finite dimensional, set-valued integral (see Klein-Thompson 10 , chapter 18), we have that / S ^ ^ ^ G ^ d r , f S\r)G{T)dT e PkJ&m)- T h e n for o o every v' € Rm, we have t -(v',yn)<<7(v',J Sn-l(r)Gn(T)dT) 0 where by (•, •) we denote the inner product in R m . But from theorem 2.2 of Kandilakis and Papageorgiou 9 , we know that t t *(v',J Sn-\r)Gn(r)dr)= j a(Sn'-\ry,Gn(r))dr. 0 0 Using hypotheses H(B) and H(U), we can easily check that for all r £ [0, t] GJT)-*G(T).
But note that G„(T),G(T)
n > l are compact, connected sets.
So
17
corollary 3A, p. 23 of Salinetti-Wets , tells us that Gn(r)!+G(T)=>convG„(T)!+ convG(r)
as n-»oo=wr( • ,convGn(r)) = a{ ■ ,G„(T))-* m
n-K» uniformly on compact sets in R . SH—l{ry-*Sm-\Ty.
Hence a{Sn--\Ty,
But from lemma 3.1, we have Gn(r)}+
and so
1
from the extended dominated convergence theorem (see Ash , theorem 7.5.2, p. 295), we have that j v ^ ' - W , o
Gn(r))dT-> ja(S* o
" 1 (T)V*, G(r))dT as n-*oo. So
332 in the limit as n-*oo, we get that < - ( » • , » ) < /
t = v(v',J
0
S-\T)G(T)dr). 0
m
l
Since v" G R was arbitrary and JS (T)G(r)dT 6 Pkc(Rm)> o t - y e / s-\r)G(T)dT
w e
deduce that
o i
=>-y=f
g G L\[0,t},Rm),g(r)
S-\r)g{r)dr,
G G(r)a.e. on [0,*].
0 Let H(T) - {u 6 17(T):$(T) = 5 ( T , U ) } , T G [0,<].
From the definition of the
multifunction G( •), we see that H(T) ^ 0 for all T G [0, t]. Also GrH = {(T, U) G [0, t) x Rk. g(r) = B(r, u)} n GrU \ GrU |
- {(T, U) G [0, t) x R* • u G U(t)}.
Rk
GrU |
Rfc G
^ Rk, where
Because U( •) is measurable, 18
B([0,t]) x B(R*) (see Wagner , theorem 4.2). Also since B(T,U) is
Caratheodory (i.e., measurable in T, continuous in u), it is jointly measurable. Thus { ( T , U ) G [ 0 , ( ] X k
B([0, <]) x B(R ).
R*: g(r)=
B(T,U)}G
Therefore GrH G
Applying Aumann's selection theorem, we get u: [0, t}-tRk
measurable s.t. for all r G |<M],U(T) G # ( r ) .
u G L\[0,t],Rk),
B([0,t])xB(Rk).
Then
-y=
JS~\T)
B{T,u(r))dT,
o u(t) G U(t) a.e. =»y G C(t). So we have proved that for all t > 0 « „ ( < ) Q C(t).
(1)
Next let y G C(t). Then by definition, we have t
-y=
J S-\T)B{T,u(T))dT 0
with u( ■ ) 6 I ' i (R + ,R ), u(t) G U(t) a.e. Via a straightforward application of Aumann's
selection
theorem,
«„(r) || =d{u(r),Un(r))
we can produce
rG[0,r].
S^V^S"'^)
in
|| U(T) —
Then because of hypothesis H(U), | | u ( r ) -
U„(T) II -»0 a.e. on [0,t]. So B „ ( T , U„(T))-*B(T,u(r)) while
u„€l'([0,(],R'),
m
l(R )
(recall
a.e. on [0,t] (hypothesis #(fl)(4)), that
5 n " l(r) = S 1 ^ - r )
and
333 1
S
(T) = S( — r)). Therefore, through the dominated convergence theorem, we get
that t J Sn-\r)Bn(r,un(r))dT^J
t S-\r)B(T,u(r))dr.
0
0 l
Let -yn=
JSn- (T)B„{r,un(T))dT, o R . Thus we have proved that
n > 1. Clearly y„ € CJt)
and yn-+y in
m
C(t)C/imCn(t).t>Q.
(g)
From (1) and (2) above, we deduce that for all t> 0 C„(r)3C(<) as n-K». But recall that Cn(t),C{t) & Pkc{Rm).
So the Kuratowski and Hausdorff
modes of set convergence coincide. Hence we finally have for all t > 0 Cn(t)$C(t)
as n-K».
4. The minimal time function and the null-controllability set In this section, we will examine the variation of the minimal time function, as we perturb the data of the problem and then we will use it to obtain a complete description of the null-controllability set at every time instant. To this end we will need the following additional hypothesis: H0:
For every n > 1, 0 e B„(t, Un{t)) a.e. and 0 6 intB(t, U(t)) a.e. An important consequence of this hypothesis is that the multifunction Cn( •)
is expanding; i.e., if 0 < t, < t 3 , then C„(t,) C CJt2). controllability set C(t).
Similarly for the limit null-
To see this let Ln(t) = { u € Un{t): Bn(t,u) = 0}.
of hypothesis H0, Ln(t) ^ 0 a.e. Clearly GrLn € l(R the Lebesgue c-field of R + .
+
)xB(Rk),
So we apply Aumann's selection theorem and find
u„:R + -»R* measurable s.t. un(t) € L„(t) t > 0=>Bn(t, u n (i)) = 0 a.e. !/6C n (<,). i
W
R
+'
Then Rt
)'
u
by
Because
where 1(R + ) is
definition
i W e t f » ( t ) a.e.
-y= Set u j =
/5-1(r)Bn(T,uJ(r))dT, Xfot,]""+ X(«I.ia]«n-
Now let with
Clearl
7
uj 6 u
2 €
334 £ I ([0,*a],"*).«a(*)e !/„(*) «•«• on [0,r2] and
-y = j S~l(T)B{T,^{r))dT=>y G o Q ' a H Q ' i ) ^ ^ ( ' 2 ) M claimed. Actually in lemma 4.2 below, we will obtain a stronger property satisfied by C( ■). We will also need some auxiliary results. Lemma J^.l:
If (ft,E,/*) is a cr-finite measure space, Y is a separable Banach space, F:ft-»2 y .{0} is a multifunction s.t. GrF G E x B(Y), it has open values, and S1p ^ 0,
Proof:
then int J F(u)dfi(u) ^ 0. n Let g G S 1 p. By considering if necessary G(IJJ) = F(u>) — g(u>), we
may assume without any loss of generality that 0 G F(w) ft — a.e.
So 0 G
f F(w)d(i(u>). We will show that / F(w)dfi(u>) contains a neighborhood of n n the origin, establishing this way the lemma. To this end, let 7/:ft-»R + be defined by
t)(u) = d(0,Fc{u)),
where
Fc(u) = F{w)c.
Note
that
GrFc = {GrF)c=>
GrFc G £ x B(Y) and since Fc( •) is closed valued, from theorem 4.2 of Wagner 18 , we deduce that Fc( ■) is measurable, hence u>*r/(w) is measurable (see section 2) and also 77(a)) > 0 fi — a.e.
Since {w Gfi':v(u») > 0} =
U {w € fi: 77(01) > 75}, then n>l
there exists n > 1 s.t. /i{w G fi: 77(01) > H} > 0. Let fin = {u> G fi: 77(01) > jj}- We have /i(ftn) > 0 and for u G fin,0 < e < k, Be(0)CF(u)
(Bl(0) =
{y&Y:\\y\\<e)
=^(fi n )B t (0) C JF[u)d^y>)C
jF(u)dr(u,)
n
^
the last inclusion being a consequence of the fact that 0 G F(u>) fi — a.e.
Since
fi(£l„)Bt(Q) is a neighborhood of the origin, we conclude that 0 G int J F(u>)dft(u>). °
Q.E.D.
This general result about set-valued integration, allows us to improve our initial observation that the null-controllability multifunction C(t) is expanding. Lemma 4.2:
If A G I 1 ^ ( R
+
,i.(R m )),B:R + xR*-»Rm is a
map (i.e., B(t,u) \B(t,u)\
Caratheodory
is measurable in t, continuous in u),
+
),b>0>
335 U:R+-*Pkc(Rk)
is
a
measurable
multifunction
U | U(t) | = sup{ || v ||: v G U(t)} g L\JR
+
s.t.
) em«*
O e t n t B(t, {/(<)) a.e., then for t < <„ we Aowe C(<) C m
By definition, we have
t i
l
{y G R": - y = |
5 " >(T)B(T, «(r))
+,
R*), u(t) 6 U(t) a.e.}.
0 SoC(*,)= - / 5 - 1 ( r ) 5 ( r , t / ( r ) ) d r = o B(T,U(r))dT. Hence
-/S-^T^T.^T))^-'^"'(r) o t
C(t1) = C(t) + W(<1,t) 'i
1
with W ( t j , t ) = — f S
(T)B(T,1/(T))JT.
(*)'
Using the open mapping theorem, we
t
have
0 G /" S " l (r)mtS(T, t/(r))aV = /" m t 5 " \T)B(T, t t
U(T))dr.
But from the proof of theorem 3.2 we already know that T-*B(T, U(T)) is a measurable, measurable,
P tc (R m )-valued nl
Pfcc(R )-valued
multifunction too.
Then
=*-T-»S - 1 (T)B(T,[/(T)) = R{T) note
that
d(y,bdR(r)) > 0}, where bdR(r) denotes the boundary of R(T). 4.6
of
Himmelberg 8 ,
we
know
{(T,y)eGrR:d{y,bdR{T))>0}eB(R to
the
measurability
of
k
GrintR G 5 ( R + ) X B(R ).
R( ■)
that
T-*bdR(r) k
+
)xB{R ),
(see
is
is
intR(r) = {y G R(T): But from theorem
measurable.
Hence k
since GrR G B{R + ) x B{R ), due
Wagner 18 ,
theorem
4.2).
Thus lemma 4.1 tells us that intW(tut)
Therefore
^ 0 and so from
(*)' above, we conclude that C(t) C m
If A G LlioJR
+
,L(Rm)),B:R+
map (i.e., B(t,v)
xR*-4* m is a Caratheodory
is measurable in t,
continuous in
u), || B(t, u) || < a(t) + b || u || a.e. with o( ■) 6 ^ ( R +), 6 > 0, I/: R + -*Pkc(Rk) is a measurable multifunction s.t.
Mtf(t)|el 1 loe (R + ), t/ten for every x G C = U C(t), * G C{T(xj). <>o
336 Proof:
Let t„lT(x)
and xGC(< n ) for n > l .
Then
1
B(T, u„(r))dT, n > 1 with u n G L « (R +) and «„(<) € #(<) a.e.
-x=f
S~1{T)
Because of our
growth hypotheses on B( ■, •) and | U( ■) | and the Dunford-Pettis compactness criterion (see for example Dellacherie-Meyer4, theorem 11-25), by passing to a subsequence
if necessary,
L\[0,M),Rm),
with
tn<M
we may assume
that
for all n > 1.
gn( •) = B{ ■, u n ( • ))-*g in
Then
from
theorem
3.1 of
13
Papageorgiou , we have G Pkc(Rm) a.e.
g{t) G conv (Sn{gn{t))n >, C convB{t,U{t)) Then note that
*« T(x) \J S-\r)gn{r)dr- j S-\r)g(r)dr\ 0
0
*. <\JS~
\r)gn{r)dr
T{x) - j S~ \r)gn{r)dr
0
0
T(x)
T(x)
\j
S-\r)gn{r)dr-
J
0
T{x) S ~ \r)gn(r)dr
\ + \J
T(x)
<„ J
S~ \r)(gn(r)
- g(r))dr | .
0 and g„™g in L\[0,M],Rm),
Since tnlT(x)
S-\r)g(r)dr\
0
tn < I/
\+
we get that
T(x) S-l{T)g(T)drasn-«x>
S-\T)gn(T)dT->J
0
0
T(x)
T(x) x
=*> - x G J
S~ (T)convB(t, U(t))dt = J
0
0
S~ \r)B(t,
(see corollary 18.1.10, p . 203 of Klein-Thompson 10 ).
U(t))dt
Via a straightforward l
application of Aumann's selection theorem, we get u G L »
(R +, R*), «(<) G U(t)
337 r(x) a.e. s.t. - x = / 5 " ' ( T ) ^ ( ^ u{r))dT^ X E °
C{T(X)). Q.E.D.
Now we are ready for the convergence theorem concerning the minimal time functions of systems (*)n and (*). We have: Theorem 44:
If hypotheses H(A),H(B),H(U),H0
hold and xn->x in Rm,
thenT„{xn}*T{x)inR+. Proof:
First assume that x £ C\ i.e., the initial state x 6 Rm is not finite
time null-controllable. Then by definition T(x) = -f oo. If T n (x n }^ + oo as n-»oo, then this means that there exists Af > 0 and a subsequence {k} of {n} s.t. Tk(xk) < M < oo for all k =>xk 6 Ck(M) (recall Ck( •) is expanding). From theorem 3.2, we know that Ck(Mf*C{M)
as Jfc-*oo=« 6 C(M)=>
T(x) < M < oo a contradiction. Therefore Tn(xn)-> + oo as n-»oo. Now assume that x G C. Then T(x) < oo. We will first show that T(x) < limT„(x„).
Suppose not. Then /tmT„(x n ) <
T(x) and so we can find a subsequence {k} of {n} and a t < T(x) s.t. for all k Tk(xk)
x e c(t) = ^ ( x ) < i < T(x) a contradiction. Therefore we have T{x)
we will show that
limT„(xn) < T(x).
•
(1)
Again we proceed
by
If the desired inequality does not hold, we can find t > T(x) and a
subsequence {it} of {n} s.t. for all A: we have T(x) < t < Tk(xk). From lemma 0 we know that C(T(x)) C intC{t), while xk £ C(t). the strong separation theorem, we can find x*k € Rm, || xj || = 1 s.t.
So from
338 <7(X'k,C(i)) <
(Xlxk).
By passing to a subsequence if necessary, we may assume that x'k-*x" in m
R , || a;'|| = 1 .
Then because of theorem 3.2, we have cr(x"k, Ck(i))-*cr(x", C(i)),
while clearly (xj, xk)-*{xm, x). So in the limit as k-*oo, we get
(2)
But from lemmata 4.2 and 4.3, we have x G C{T(x)) C intC(t).
(3)
From (2) and (3) we get the desired contradiction. Thus li^Tn(xn)
< T(x).
(i)
From (1) and (4J above, we get that T„(x„)-*T(x) as n-*oo. Q.E.D. Remark
If An = A,Bn = B,Un = U for all n > 1, from theorem 4.4, we
deduce that T:C-»R + is continuous. Note that because of lemma 4.2, C C Rm is open. Our next result, completes the above observation, by telling us that as we approach the boundary points of C, the minimal time function becomes arbitrarily large, establishing this way the continuity of T from Rm into R
+
= R + U { + oo}.
So in what follows, we consider only the limit system (*), and for this we make the following hypotheses: H(A)1):AeLl,JR H{B\:
H(U)y H^_
+
,R"')
B: R + x R*-»Rm is a map s.t. (1)
t-*B(t, u) is measurable,
(2)
u-*B(t,u) is continuous,
(3)
|| B(t, u) || < a(t) + 6 || u || a.e. with a( •) e L ^ ^ R + ) , 6 > 0,
U:R+-*Pkc(Rk) is a measurable multifunction s.t. t-+ \ U(t) | € L1^ (R + ) 0€intB(t,U(t))a.e. Theorem 4-5:
If hypotheses H(A)U H(B)U H{U)X and H'0 hold, then for every x € bdC we have Urn T(y) — + oo.
339 Proof:
Suppose not.
^(j/n) < M for all n > 1. expanding,
we have
Then we have a sequence yn-*x and M > 0 s.t.
Recalling that the null-controllability multifunction is
yn G C(M) C intC(M + e ) C C
(see lemma
4.2) =>x 6
C(M) C C and since C is open we have a contradiction to the fact that x G bdC. Q.E.D. Now, we describe the null-controllability multifunction C(t) using the minimal time function T(x). So we have: Theorem 4.6:
If hypotheses H(A)UH{B)U
H(U\ and H'0 hold,
that (i) C(t) = {xeRm:T(x)
intC{t) = {x& Rm: T(x) < t},
(iti) bdC{t) = {1 6 Rm: T(x) = t}. Proof:
(i) Clearly we have C(t) Q{x 6 Rm:T(x)
On the other hand from lemma 4.3, we have x g C(T(x)) C C(t) for every 1 (E D C C. So equality must hold between C(t) and D. (H)
Recall that T( •) is continuous on C and C(t) C C.
part (i) above, we get that {x 6 Rm: T(x) < i) C intC(t).
So using
Suppose that the inclusion
was strict. Then there exists x e intC(t) s.t. T(x) = t. But C( •) is continuous (see the proof of theorem 4.6 below). So x G C(i') for some t' < t=>T(x) < t = T[x) a contradiction. (ill)
Since bdC(t) = C(t)intC(t),
from (t) and (H) above, we have
m
that bdC(t) = {xe R :T{x) = t). Q.E.D. We have a last observation concerning the null-controllability multifunction C(t).
Theorem
4-7: If hypotheses
HiA^HiB^HiU^
and H'0 hold and
y G intC(t), then there exists 6>0 t-t' Proof:
<S
s.t. y 6 mrC(t') for all 0 < t' < t,
andforallt'>t.
We already know from lemma 4.2 that the result is true for all
340 t'>t. Now we claim that t-*C(t) is /i-continuous (and also continuous in the Vietoris topology), from R + into Pkc(Rm). So let tn-*t and take y 6 HmC(tn). Then by definition, we can find y„ G C(t„) s.t. yn-*y in Rm (for economy in the notation we denote subsequences with the same index as the original sequences). We have: *. - V. = /
S - \T)B(T,
un(r))dr, « n 6 L\JR
+
, R*), un(r) 6 U(t) a.e.
0 So for every v' g R™, we have
*„ - ("*, y„) < /
*(S* - \r)v', G(r))dr, where G(r) = B(r, U(r)),
0 *-(v',y)<J
t
^-ye
J S-\r)G(r)dT 0
=j./i^c(t n ) c c(t).
(i)
On the other hand, let y e C(t). Then by definition t -y=
J
S-\T)B(r,u(r))dr
0
for some u € I 1 , (R + ) , u(t) 6 U(t) a.e. Set - y„ = / S- 1 ( T ) B ( T . u ( 7 '))aVtoe
Clearly
0
yn € C(<„) and y„-*y. So we have C(«) C limC(tn).
(2)
From (lj and (2) above, we deduce that C(tn)$C(t)
(3J 10
(i.e., we have continuity in the Vietoris topology; see Klein-Thompson ). m
C( •) is Pkc{R )-valued, we have from (3) above that C(tn)0C(t) =>C( ■) is A-continuous.
Since
341 From proposition 2.1 of DeBlasi-Pianigiani 3 , we have that t-*bdC(t) is hcontinuous from R + into P t (R m ). Suppose now that the conclusion of our theorem is not valid. Then we can find tn < t s.t. tn-*t and y £ intC(tn) for all n > 1. Case l i Suppose y g bdC(tn) for all n > r^ > 1. Then since bdC(tn)-*bdC(t) as n-»oo, we get y g bdC(t), a contradiction. Case 2l y i C(in) for all n > r^. Since C(tn) g Pkc{Rm)
n>l,we
have
d(y,C(tn)) =
d(y,bdC(tn))>0
while d(y,C(t)) = 0. Note
that
d(y, C(tn))-*d(y, C(t))
d(y,bdC(tn)y*d(y,bdC(t))
(since bdC(-)
(since
C{ ■) is
is A-continuous).
^-continuous) and
Hence d{y,bdC(t)) = 0
=>y £ bdC(t), again a contradiction. So there exists 6 > 0 s.t. for t' < t, t — <' < 6, we have y g intC(t'). Q.E.D. 5. Bang-bang property of time optimal solutions In this section we derive a "bang-bang" principle for the time optimal controls of nonlinear systems. First we obtain a general result about differential inclusions in R m , which incorporate nonlinear control systems in Rm.
So on R + x R m , consider the
following differential inclusion: x(t)eF(t,x(t))a.e.
(**)
We will assume that this set-valued dynamical system is "normal" in the following sense: "If
x:[0,<*}-»Rm is a time optimal solution for system (**), that transfers
K0 to Klt then for each t g (0, t'), x \. t. is the time optimal solution of (**) that transfers K0 to {x(t)} B .
342 This is a natural extension to nonlinear, set-valued dynamical systems, of the well-known concept of normality for linear control systems (see Lee-Markus ia , p. 76). We will need the following hypothesis on the orientor field. fft:
F: R + x Rm-+Pfcc(Rm) is an ^-continuous multifunction, with solid values (i.e., for all (t,x) 6 R + X Rm,intF(t,x) Theorem
5.1: If
hypothesis
Hx
fi 0).
holds,
system
(**)
is normal
and
m
x( ■) g ACe (R + ,R ) is the time optimal solution of (**), that transfers K0 g P fcc (R m ) to Kx in minimal time t', &S3 i ( t ) € M.F(M*)) a.e. Proof:
Let S(K0) be the trajectories of (**) emanating from the set K0.
Set R(t) = {y(t):y g S(K0) C AC([0,f],R m )}. From theorem 3.3 of Papageorgiou 16 , we know that R( ■ ) is ^-continuous. Suppose x(t) g intR(t), for some t g [0, t*], then from
lemma
3.1 of
Papageorgiou 14 ,
we get
that
there
ra
bt(x{t))
= {z 6 R • || z - z(r) || < 6} C intR(t') for all \t'-t\
y € S(K0)
exists
<S.
6>0
s.t.
So we can find
s.t. y(t') = x(t) for some r* < t, contradicting our normality hypothesis.
Therefore x(r) g bdR(t) for all t € [0,t*]; i.e., x{ ■) is an extremal trajectory of (**). Invoking theorem 3.2 of Papageorgiou 14 , we conclude that x(t) g bdF(t, x(t)) a.e. Q.E.D. Now consider the following nonlinear control system a.e, u(.)eLlu(R
*(*) = B{t,x(t))u(t)
+
,Rk), u(t) g U(t) a.e.
We need the following hypothesis on the control vector field H3:
ra
m
B:R+ xR -»Jt(R*,R )
is
a
continuous
map
s.t.
(**)'
B(t,x). for
(t,x) g R + xR™, B(t,x) is surjective, || B(t,x) \\ < Mx and (7:R + ->
every Pkc(Rk)
is an ^-continuous multifunction with intU(t) / 0 and | U(t) \ < M 2 for all <>0. As before x( •) g AC a (R + ,R m ), is the time optimal trajectory of (**)' that transfers set K0 g P fcc (R m ) to set Kv Theorem 5.2:
If hypothesis H2 holds, then
there
exists ugL°°(R + ,R*) s.t.
u(t) g bdU(t)
and u( •) generates the time optimal trajectory x( ■).
a.e.
343 Proof:
B(t,x)u € Pkc(Rm)-
Set F(t,x) = B{t,x)U(t) = (j
We have for all
u m
(<',x),( t ,*)eR + ir
*
h(F(f,x'),F(t,x)) = h(B(t',x')U(t'),B(t,x)U(t)) < h(B(t', x')U{t'), B(t', x')U(t)) + h(B(t', x')U{t), B(t, x)U(t)) < MWWM*))
+ M2 || B(t',x') - B(t,x) ||
=>(*, x)-*F(t, x) is /i-continuous on R + x Rra into P t c (R m ). Also since by hypothesis Ht, for every (t, x) € R + x R"1, B(t, x) is surjective, from the open mapping theorem, we have intB(t,x)U(t) ^bdB(t,x)U(t)
= B(t,x)intU(i) =
^ 0
B(t,x)bdU(t).
Furthermore, from theorem 5.1, we have that x(t) S bdB(t, x(t))U(t) = B(t, x(t))bdU{t) a.e. Recalling that t-*bdU(t) is measurable (in fact A-continuous) through
a
simple
application
of
A um arm's
selection
u( •) e L°°(R + ,R m ) s.t. u(t) € bdU{t) a.e. and x(t) = B(t,x(t))u(t)
theorem,
we
get
a.e. Q.E.D.
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R. Ash, Real Analysis and Probability, Academic Press, New York 1972.
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0 . Carja, On the minimal time function for distributed control systems in Banach spaces, J. of Optim. Theory Appl. 44 (1984), 397-406.
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F. DeBlasi and G. Pianigiani, Remarks on Hausdorff continuous multifunction and selections, Comm. Math. Univ. Carol 24 (1983), 553-561.
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C. Dellacherie and A. Meyer, Probabilities and Potentials, North Holland, Amsterdam 1978.
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O. Hajek, Geometric theory of time optimal control, SIAM Control and Optim. 9 (1971), 339-350.
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F. Hiai and H. Umegaki, Integrals, conditional expectations martingales of multivalued functions, J. Multiv. Anal. 7 (1977), 149-182.
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minimal
time
Jour,
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344 8.
C. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.
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D. Kandilakis and N.S. Papageorgiou, On the properties of the Aumann integral with applications to differential inclusions and control systems, Czechoslovak Math. Jour. 39 (1989), 1-15.
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E. Klein and A. Thompson, Theory of Correspondences, Wiley, New York 1984.
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K. Kuratowski, Topology I, Academic Press, London 1966.
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E. Lee and L. Markus, Foundations New York 1967.
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N.S. Papageorgiou, Convergence theorems for Banach space valued integrable multifunctions, Intern. J. Math, and Math. Set. 10 (1987), 433-442.
14.
N.S. Papageorgiou, On the attainable set of differential inclusions and control systems, J. Math. Anal, and Appl. 125 (1987), 305-322.
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N.S. Papageorgiou, Decomposable sets in the Lebesgue-Bochner spaces, Comm. Math. Univ. S.P. 37 (1988), 49-62.
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N.S. Papageorgiou, Properties of the solution and attainable sets of differential inclusions in Banach spaces, Radovi Matematicki 2 (1986), 247261.
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G. Salinetti and R. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Review, 18-33.
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D. Wagner, Survey of measurable selection theorems, SIAM and Optim. 15 (1977), 859-903.
of Optimal Control Theory, Wiley,
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Control
WSSIAA 3 (1994) pp. 345-360 © World Scientific Publishing Company
345
SINGULAR HOPF BIFURCATION PROBLEMS AND ROTATING-SLIDING SPIRAL FLOWS GEORGE H. KNIGHTLY Department of Mathematics, University of Massachusetts Amherst, Massachusetts 0100S, U.S.A. D. SATHER Department of Mathematics, University of Colorado Boulder, Colorado 80S09, U.S.A. ABSTRACT The problem of rotating plane Couette flow is investigated using singular evolution equations. It is shown that Hopf bifurcation occurs leading to a con tinuum of disturbance flows in the form of periodic waves that are supercritical and asymptotically stable. 1. I n t r o d u c t i o n . In a recent paper 1 0 the authors studied the problem of rotating plane Couette flow (RPCF), a problem that arises as a narrow-gap limit of viscous flow between two concentric cylinders rotating about a common axis and sliding parallel to that axis. It was shown that the problem of R P C F is equivalent to solving a time-dependent equation of the type
^+L0w-\(L as
+ i(M1+M2))w
+ B(w) = 0,
(to, A, 7) G X x R 2 ,
(t)
where A" is an appropriate Banach space, A is a "load" parameter (i.e., the Reynolds number), and 7 is a "structure' 1 parameter. Here LQ,L,M\ and M2 are Unear operators and B is a quadratic operator. It was shown that, for each fixed 7 near 7 = 0, there exists a unique branch of periodic orbits of (f) that bifurcates supercritically from w = 0 at some critical value Ac(7) and represents a periodic wave. Such a result suggests, but does not establish, the existence of a continuum of periodic orbits of (f) bifurcating supercritically from w = 0 and depending continuously on 7. Since for 7 near 7 = 0 the spectrum of the linear problem for (f) consists of eigenvalues of the form p — £ + 1777, such a continuous dependence result involves the study of "singular" Hopf-type bifurcation problems.
346 In the present paper, by carrying out a detailed analysis as 7 —> 0 of solutions of a "singular" time-dependent equation of the form -fd-^-+L0w-\(L
+ 'y(M1+M2))w
+ B(w) = 0,
(to, A, 0,7) € X x R 3 ,
(*)
we establish the continuous dependence upon 7 of a family of 27r-periodic orbits of (*) for R P C F . Such a result implies, in particular, that a continuum of periodic waves is present in R P C F whenever there are fluctuations (i.e., imperfections) in the basic flow, corresponding to variations in 7. Since such fluctuations are likely to be present in any experimental setup, a result of this type provides the existence of complex time-dependent flows and offers a simple explanation of the relatively rapid transition to turbulent-like flows observed in experiments on rotating-sliding spiral flows (e.g., see the discussion of and references to such experiments in Joseph 8 , Chap. VI). The use of a singular equation such as (f) to determine periodic solutions is similar to an analytical approach developed by Renardy 1 2 ' 1 3 to construct homoclinic and heteroclinic orbits of reaction-diffusion equations. The approach makes use of a generalization of the implicit function theorem that does not require, e.g., the continuity of DyF in a neighborhood of (0,0) when solving F(x,y) = 0, .F(0,0) = 0. Such methods were extended to Navier-Stokes type equations in Sather 14 and heteroclinic orbits in the Taylor problem were determined as solutions of a singular equation. The present paper shows among other things that singular equations can be used also to determine periodic solutions of a general class of singular Hopf-type bifurcation problems. The outline of the paper is as follows. In Section 2 we formulate the problem of R P C F as an equation of the form (f), discuss the steady state linear problem when the structure parameter 7 = 0 and establish the basic properties of the various linear and nonlinear operators in (f). In Section 3 we introduce the singular equation (*) and then reduce the problem of solving (*) to one of solving a standard nonsingular, finite-dimensional Hopf bifurcation problem. Standard splitting methods are used in Section 4 to derive the bifurcation equations and these equations are then simplified and solved to obtain a continuum of bifurcating supercritical, 27r-periodic orbits of (*), Using the equivariance properties of the equations for R P C F , one then sees that these asymptotically stable solutions are actually periodic traveling waves.
347 2. Formulation of the problem. The equations governing the disturbances of R P C F are given on p. 178ff of Joseph 8 . In dimensionless variables the fluid occupies the region Roo = \ {x,y,z)
:
—oo < x < oo, —oo < y < oo, —j < z < ^ > with lower wall at z = — i translating at velocity U = (cosx 0 iSinx 0 ,0), where the basic spiral angle x 0 is defined by cosx 0 = tov/U, sinx 0 =
Hv/U,
Q,U = ue/d, nv = a(fii - n2)/d, u = («?, + n 2 v ) 1/2 Here the dimensional quantities d, £/ c ,a(fti — f^) are, respectively, the distance between walls, the velocity of the lower wall in the x-direction and the velocity of the lower wall in the y-direction. In addition, the system rotates about the x-axis with angular velocity Q = Q2/U. It is shown in Joseph 8 how this "plane" problem is developed as a narrow gap limit (i.e., d/a —► 0 and Qu,ilv bounded) of flow between concentric rotating cylinders, where the inner cylinder of radius a also translates axially. A disturbance (ui,U2,U3,p) has spiral flow angle \ if in spatial coordinates (x',y',z) the disturbance is independent of the coordinate x', where the x'-axis makes the constant angle x with the x-direction. It is convenient to set ft-1sin(x-Xo)
7 =
and
A = 7lRe,
(2.2) 8
where Re = Ud/v is a Reynolds number and V? is the Rayleigh discriminant ; we always assume that ft, x and x 0 are such that Tt > 0. For purposes of the analysis it is convenient to assume that the "structure" parameter 7 in (2.2) is small and that 7 > 0. Thus, since 7 is a measure of (x — X0)i w e s e e ^ disturbance spiral flow solutions when the disturbance spiral flow angle x is close to the basic spiral flow angle x 0 • We shall see that 7 plays a role in the analysis similar to that of an "unfolding" or "blow-up" parameter. The disturbance spiral flow v = («i,«2,t;3) T with spiral flow angle x satisfies the equations 1 0 0 = Av + XMv + A 7 ( i - z ) j - - {v ■ V)»
1
(2.3a)
, dv
-vq-xn- -^, 0 = V • v, v =0
on
(2.36) z = ±-,
(2.3c)
348 where M = (m, ; ) with rrnj = 0 except for mi3 = mn = 1 and 77123 = —7. To study (2.3) we introduce the following function spaces. Let R = {(y, z) : 0 < y < 2n/a, - i < z < i } , o is specified in (2.10), and let Cg(R)
= {v:v
= v(y,z)
C0°#(.R) = | u € Cf(R)
€ C 0 0 ^ 1 x ( - i , i ) ) and — p e r i o d i c in y } ,
: v = 0 in a neighborhood of |z| = i j ,
J#(i?) = {«6Co^(iJ):V-t; = 0}. Let L 2 (fl) = {cl. (Cg(R)) in || • ||}, where cl. denotes closure and || • || is denned by the usual complex L 2 inner product
(U,V)=
/ U-V = V / UjVj. JR j^i-tR
The classical Sobolev spaces used are WR(R) = {cl. (Cj?(R)) || • Has (it) is defined by the inner product t
in || • ||H£(R)}I where
9l*lu
(«,»)^(I0 = E JR(Dkn,D"v),
D'u =
^ ^ .
\k\K.Tn
in L 2 (J?)} and Hi = {cl. (J#(R))
We now set H = {cl. (J#(R))
in % ( f l ) } .
o
Throughout the paper we will use the norm || • ||i for Hi defined by the inner product (u,w), = / Vti • Vv = V JR
/
VujVvj.
i^l-'H
Since elements of Hi satisfy a Poincare inequality, || • ||i is equivalent to || • Hgi/ R \ o
on Hi. In addition, we require the Hilbert spaces
/C = l$(R)r\H
and
V=^(R)nHu
where the norms of K and V are denoted by || • ||/c and || • ||D and are defined by the inner product for H2(.R). As in Iooss 4 ' 5 ' 6 we have compact imbeddings
v^n^H.
349 The Banach spaces used in the sequel are C m (
v is 27r-periodic in t . } ,
where the norm on €7"(X) is the usual sup norm on (—00,00). If m = 0 we write only HI • || I*. Finally, throughout the paper, C(Xi; X2) denotes the Banach space of bounded linear operators from a Banach space X\ to a Banach space X2. Let II denote the orthogonal projection onto "H in L2(R). Then 6 IIVp = 0 for all p € HL(R). If one now sets t = \'R.~13 in (2.3), one obtains the basic time-dependent equation for R P C F , ~ + L0v~X(L as Here Mi = Tl(-
+ B(v) = 0.
(f)
— Y D ( M i ) = V, B(w) = Il(w ■ V)w, w e P , /l
L0 = - n A
+ 1(M1+M2))v
0 \0
0 0\ /0 1 0 , V(L0) = V, L = U I 0 0 1/ \1
~s. t M2=-E[z— 1-n
V^/
/0 0 0 0
0\ 1 ,
0 0 0
1\ 0),V(L) 0/
= H,
V(M2)=V.
(2.4)
(2.5)
\o 0 0;
Note that the quadratic operator B is generated by the bilinear operator $ , $(u, v) = II(u • V)v,
u, v € 2?.
(2.6)
We seek periodic solutions of (t) in C(V) l~l C 1 (W) for real A and 7. The linearized time-independent equation for 7 = 0 associated with (f) is L0u - fiLu = 0,
u € P , fieR1.
(2.7)
Since LQ is positive and selfadjoint in H and since L is a bounded symmetric operator on H, the operator L^—fiL is selfadjoint in Ti. In addition, Lo bas compact resolvent (the imbedding V '-* H is compact). Thus, the characteristic values of (2.7) are real, isolated, and of finite multiplicity. In fact, problem (2.7) is equivalent to the classical problem for smooth u and q, periodic in y with period 2ir/a, obtained by
350 setting 7 = 0 and omitting the nonlinear terms in (2.3). The resulting problem is essentially the classical Benard problem whose solution is 1 Uj = eik»tpj(z), 2
iky
; = 1,3,
u2 = ia- ke
(2.8a)
2 ,ky
2
q = a~ e D
a = |*|,
(2.86)
where D2 = —— — a 2 , a prime denotes — and ipi,
D4
(2.9a)
2
D
(2.9b) at
2= ±-.
(2.9c)
One can show 7 for a > 0 that the eigenvalue problem (2.9) has a countable number of positive simple eigenvalues, 0 < fii(a) < 1*2(0) < • • •, depending continuously on a. Moreover, f*i(a) —► 00 as either a —► 0 + or a —+ 00. Calculations 1 (Fig. 2.2) suggest fii(a) takes its minimum at a unique positive value of a. We assume this to be the case, choose ag to be the minimizing value, and set l*o = m(a0).
(2.10)
If Af = Af(Lo — HQL) denotes the null space of LQ — fioL, then M C V and M is spanned by t/>o and
V>o = e"°V(*)
(2.11)
where
351 (Hi) the bilinear operator $ in (2.6) has the mapping properties $ : V x V —► AC and $ : A/" x Af —► AC fl ML. In addition, $ satisfies
||*(«,v)lk
«,«€©,
(2.12)
/or jome constant c\ depending only on R. Proof. The proof of (i) is immediate. Since N is spanned by {ipo,ipo}, the mapping property in (ii) of Mi on M is also immediate. Moreover, since M* = —Mi, if v € A/"x, then {M\v, ip0) = —(v,Miij>0) = 0 because M\\j>o G N. Note also that ||AfiU)||/c < C 2 | H | H ^ ( R ) < c2||to||i>,
w
eV,
by basic Sobolev theory and the definition of the norm on V. Thus, Mi € C(V; AC) and a similar argument shows also that M 2 € C(V; AC). To complete the proof of (ii), we write M 2 as M 2 = M 2 (1) + M 2 (2) where M2(1) = - I I (z^-
J. If ^ = /?,V>o + / M o
is in AT, then ( M ^ V . ^ o ) = -JRZ^-rp0
= -iaofix ( ^ ) /
4K*)l2
because each component of
(M<2)v, <M = A ( ^ ) y t 2 ^M*)*** = o. Thus, M ^ is orthogonal to ^o and similar calculations show that Mi^ is a^so orthogonal to V>oThe fact that $ : V x V —► fC and satisfies (2.12) is proved in Iooss 6 . To prove $ : A/" x A/" —► AC fl A/"x, it suffices to show ($(u, t>), u>) = 0 whenever u, t>, u> € A/". Note that ($(u, v), w) = I [II(u • Vw)] • u) = / (u • Vv) ■ w, JR
JR
where each term in the integrand depends on y via a product of three factors of the form e±,a°y. Since the y-integration is zero in each case, it follows that (<£(«, v), w) = 0 whenever u, v, w € A/-. This completes the proof of Lemma 2.1. Let A be the restriction of the selfadjoint operator LQ — noL to A/""1 with V(A)
= V D Af"1", i.e., A = (Lo — MO-COIA/"-1-- Then A is selfadjoint, has compact
352 resolvent, and the spectrum of A,
of finite multiplicity with pn > p\ > 0 (see Iooss 4 ).
3 . R e d u c t i o n t o a nonsingular Hopf bifurcation problem. Since we seek periodic solutions of (|) of unknown period 2ir/ry6, as usual in such problems it is convenient to fix the period as 2n. Thus, setting t = 70s in (f) with 7 > 0 and 0 < 9\ < 9 < 92, we look for periodic solutions of period 2ir in C^(U) D C^(W) of the "singular" equation 7 0 - ^ + L0w - X(L + 7 (Afi + M2))w + B(w) = 0.
(*)
The decomposition H = M © A/'"L introduced in Section 2 defines a natural decomposition C^(D) = 0#{M) © C^(2? n N^-). Thus, we seek solutions of (*) having the form u, = 7 ( u + 7„), u e C ^ ( j V ) , veC^(VnATL),
(3.1a)
2
A = /to + 7 r, r € R \
(3.16)
where HQ is defined in (2.10). Let S :H —+ M denote the orthogonal projection of H onto //. Then Q = I—S is the orthogonal projection of H onto Af-1. Substituting (3.1) into (*), using that u € N{LQ ~ HQL), dividing out a factor 7 2 , and recalling the definition of A in Section 2, we obtain 0 = 7 0 - £ + Av + 9^- - ^ 0 ( M i u + M 2 u) + B(u) at at + f\
— TLU — (io(Miv + M2v) + $(u,v)
+ 7 [ - rLv
- TMIU
- TM2U
- IT(MIV
+ $(v,v.) + M2v)
+ B(v))
}.
(3.2)
Projecting onto Afx and Af with Q and S, and making use of the various mapping properties in Lemma 2.1, we obtain the following equations in «A/'"L and A/": 0 = 70-jf + Av ~ ^oM 2 u + B(u) + iQH{v,«, T , 7 ) , (3.3) at 0 = 9— -poMiu-yS r X u + / i 0 M j v - $ ( u , v ) - $ ( v , « ) - 7 G ( t ) , « , r , 7 ) , (3.4)
353 where H is the quantity inside the bracket < the bracket
> in (3.2) and G is the quantity inside
in (3.2); in deriving (3.3) and (3.4) we have used, in particular,
QM-iu = M2U, and SMiv = 0, which hold because Mi : N —► K. n A T 1 and Mi : ML -+ ACfW-1-. For fixed (u, r ) € C^ (AT) x R 1 we first solve the "singular" problem (3.3) for the unique solution v = V(U,T,9, y)(t) in CL(2? f~l A/"x). Substituting this solution v = V(U,T,8,J) into (3.4), we then obtain a "regular" Hopf bifurcation problem on the underlying finite-dimensional space M, a bifurcation problem that we can solve by standard methods. The use of the singular equation (*) and the approach described in Renardy 1 2 ' 1 3 and Sather 14 allow us to solve (3.3) for v depending smoothly on 7 even though the usual implicit function theorem does not apply in CL (£>) in a neighborhood of 7 = 0, 7 > 0. We shall see that 9\ and #2 can be chosen to depend only on u)\ in (3.10) and a given value of ro; thus, since ui\ is fixed, the choices of $1 and 62 depend essentially only on ro. In view of this last remark, it is convenient to set 5(ro) = { ( u , r , 0 ) € C^(A0 x R 2 : \\\u\\\^
< r„, | r | < r„, *i < B < 62).
A proof of the following result is given in Knightly and Sather 11 (see also the proof of Theorem 5.2 in Sather 1 4 ). T h e o r e m 3 . 1 . Given ro > 0 there exist Ro > 0 and 70 > 0, depending only on r0, such that if (u,T,6,*y) € S(r0) x [0,y0), then in B = {v € C^(X> n Af-1) : IIMIIiJ < -Ro} one can solve (S.S) for a unique solution v = v(u,T,9,y)(t) in C^('DnA / '• 1 -)nC^(A /, - l -). The solutionv is analytic inu,r,d ami 7 on S(r0)x(0,-f0), continuous in 7 at 7 = 0, and satisfies
INII^^IIIulU^. Moreover, if v is considered as an element of CSL(P r\AfL), v = A _ 1 v 0 + 7V,
(3.5) then
v0 = noM2u - B(u),
(3.6)
1
where V = V(U,T,0,J) 6 C^(I>nA/"- ) is analytic on S(r0) x (0,70), continuous in 7 at 7 = 0, and satisfies
\\\V\\\v
(3.7)
If we now substitute t; = v(u,r, 8,7) from Theorem 3.1 into (3.4), instead of solving the singular equation (*) in CL(2?) we have reduced the problem to solving a nonsingular autonomous equation in CL(A/"), namely O^-iiQM1u
= 1F{u,T,0,1),
(u,r,0,7)€S(ro)x[O,7o).
(3.8)
354 Here F is given by F(u, T, 9,-y) = S \TLU + nQM2v - $ ( u , v) - $(v, u) - yG(v, u, r, 7 ) ] .
(3.9)
Note that, as an element of C^(A/"), F is analytic in U,T, 8 and 7 on 5(ro) x (0,7o) and continuous in 7 at 7 = 0. We show next that solving (3.8) in CL(Af) is essentially a standard Hopf-type bifurcation problem on the underlying finite-dimensional space Af. Note first of all that the linear operator Afi maps M into Af and satisfies M\ipo = - aoV'o, where V>o is denned in (2.11) and ao is the wave number in (2.10) associated with noThus, we have essentially a standard Hopf bifurcation problem provided that we choose the basic frequency such that, at 7 = 0, e'Vo belongs to the null space of the operator on the left in (3.8). If we set 6 = u>i + fw, ui € R 1 , and choose w
i = 2 ^°a°'
(3.10)
then u>i is the desired basic frequency. Moreover, for |w| < TQ and 70 sufficiently small, one can now choose 9l = — and 62 = 2u>i so that 9 € [#i, #2] for 0 < 7 < 70, where 70 depends only on r 0 and u>i. Substituting 8 into (3.8) and setting U(r0) = | ( u , r , u ; ) € C^(A0 x R 2 : |||u|||y> < r 0 , |r| < r 0> H < r 0 } , one obtains finally the basic Hopf bifurcation problem to be solved in CL(Af): du wi-j- - / i 0 M i u = 7 F(U,T,WI + 7 W , 7 ) - w —
(3.11)
where (U,T,U>,7) S ft(ro) x [0,7o). Note that, as an element of CSU.A0. the righthand side of (3.11) is analytic in u,r,u and 7 on R(r0) x (0,7o) and continuous in 7 at 7 = 0. 4. T h e bifurcation equations for periodic solutions. The Lyapunov-Schmidt method to solve (3.11) is outlined in Section III.l of Iooss 6 and may be described as follows. Let V be the projection operator on CL (A/-) defined by
Vu{t) =(J^J
\u(s)^0)e-iads\
e'Vo + (J^ J "(«(*),&)c"ds) c " " ^ , .
355 Then any real element in the range of V can be written as j3ip(t + fo), where /? € R+ (i.e., 1$ > 0 for nontrivial solutions), <po € R 1 is the phase, and i>(t) = c"^o + e-'Vo-
(4.1)
Since a periodic solution is determined only up to a phase shift and since the phase
fi
€ R\,
U € ( / - *>)<£ (AT)-
(4-2)
c Note that V—— = 0 and x( I — V)—r- = 0 because Yrb is a linear combination of e* 1 ' dt ' dt Thus, since w 1 ^ - / i 0 M 1 V = 0,
(4.3)
equation (3.11) can be written as a system wi—
- HQMIU
= {I -V)
\F(fiip + -yU,T,u}i + 7 W , 7 ) - 7 u ; —
Q = V F(fiif> + -/U, T,WI + yu, 7) - w/?-^-
(4-4) (4-5)
The following lemma establishes the compatibility condition under which u)l
\ ~j
l^oMi I has a bounded inverse (e.g., see Iooss 6 for a proof). L e m m a 4 . 1 . (i) If g € C^(jV), then the equation wj — - / J 0 M ! U = g
(4.6)
has a solution U £ CL (A/-) t/ and only if Vg = 0 M satisfied. (ii) IfVg = 0, then there exists a unique solution U of (4-6) such that VU = 0. If one defines the linear operator J by U = Jg, then J is a bounded linear operator from (I - V)
2||M||i,1,J
is determined from (4.1); in addition, E(ro) denotes
£(n>) = {(P,T,u) e K x R2 : P < P M . M < ro, M < r0}.
356 T h e o r e m 4.2. Given ro > 0 there exist R\ > 0 and 70 > 0, depending only on r0, such that if (0,T,U>,I) € E(r 0 ) x [0,70), then in Sx = {U € (J - 7>)C^(AT) : lll^lllp < Ri} one can solve (4-4) for * unique solution U = U(0,r,uj,y)(t). The solution U is analytic in 0,T,UI and 7 on £(ro) x (0,70), continuous in 7 o< 7 = 0, aiwiaatta/iM |||C?"|||^> < cs0. The bifurcation equations for periodic solutions are then obtained by substi tuting U = £/(/?, T,W, 7) into (4.5); the resultant complex-valued equations are now solved for r and w in terms of 0 and 7 by means of the following steps. It follows from (4.1) and the definition of V that the leading terms in (4.5) of order 02 must necessarily vanish because terms quadratic in ip generate factors e , m t with m even. In fact, it can be shown by general arguments (e.g., see p. 168 of Iooss 6 ) that the invariance of (3.11) with respect to translations in t implies that the bifurcation equations (4.5) are not affected by our phase shift assumption, 0 > 0, in (4.2), and that they are of the form 0 = V F{0rj> + yU(0, r,w,y), r, Wi + 7 ^ , 7 ) - u0 dt
(4.7)
,
0 /»(/?V,u,, 7 )e'Vo + A ( / ? V , w > 7 ) e - t o Thus, substituting (4.2) into (3.6), one sees from (4.5) and (3.9) that Q = VS
-0u-£
+ PrLip + HoM2uo - P$(rp,ti0) - 0 * ( u o , VO + 10fi
= PVS + P2 [#(*, A'1 Bty)) where f\ = fi(P2,r,u>,y),
+ * ( A'1 fl(V0, V)] + 7 / i
(4.8)
as an element of (?L(Af), is analytic in P2,T,U> and 7
on £2(7-0) x (0,70), continuous in 7 at 7 = 0, and satisfies |||/i|||p constant c$ depending only on ro; here ^(ro)
< Ce with the
is determined from E(ro) by replacing
p
VSLxj) = — Vrp = — A*o
VSM2A-lM2rl>
(4.9)
= id^'Vo - e-'tM, Ho
(4.10)
do^,
= d 0 (M 2 ^- 1 M 2 V'o, V-oKVo + do(M2A~* = do(bi + »62)c**0o + Mb
~
i(
ib2)e- tko,
Mito^e'Ho (4.11)
357 where 61,62 are real and do = HV'oll (recall that tpo is normalized in the H\ norm 1/2
°
not the H norm); we have also used here that V{L0' ) = H\ and (£021,22) = ( I , J ' 2 Z I , . L Q 2 2 2 ) = (21,22)1 for 21,22 e Hi both of which are well-known 2 . Finally, some recent work of the authors (see p. 14 of Knightly and Sather 1 0 ) shows that the coefficient of the quadratic term ff* in (4.8) is VS[*(rl>, A-lB(4>)) + ^(A-'B^),^)}
= -d0ce'Vo - 4>ce~^o,
(4.12)
where the constant c is the sum of six terms like ($(ipo,1l>o),A~1$(ipo, V'o)) and satisfies c > 0. Remark 4 . 3 . If one employs the scaling (3.1) to study the linearized problem for (f), one can show 10 that the critical characteristic value, A c (7), of Lo — X(L + 7(Mi + M2)) has the form A c (7) = ^ o - 6 i M o 7 2 + 0 ( 7 3 ) ,
(4.13)
where 61 is defined in (4.11). Collecting the above facts and omitting the trivial solution ft = 0, one sees that the equations in (4.8) reduce to two complex equations, namely 0 = -id 0 u> + 1*0^ + ^0(61 + ih) - c02 + ig{0*, T,u,-y),
(4.14)
and its complex conjugate. Here g is analytic in /?2,r,u> and 7 on £2(7*0) x (0,7o)> continuous in 7 at 7 = 0, and satisfies \g\ < C7 with the constant C7 depending only on r 0 . Note that the "solvability condition" for Hopf bifurcation is just (i^oiV'o) = HQ1 > 0. Thus, by the implicit function theorem, the real and imaginary parts of (4.14) can be solved for r and u in terms of ft1 and 7 to obtain the following result. T h e o r e m 4.4. Given ro > 0 there exists 70 > 0 depending only on ro such that, for j3 < p(r0) andj < 70, (4-H) has a solution (r(/? 2 ,7),u;(/? 2 ,7)) of the form r(/? 2 ,7) = d1+
7-3(7) + ^[Moc + f(/? 2 ,7)],
2
u,(/? , 7 ) = d2 + W 0 ( 7 ) + / ? W , 7 ) ,
(4.15a) (4-156)
that is uniformly bounded, analytic in /32, continuous in 7 up to 7 = 0, and unique /<""(/92,T,W,7)
in
£ = {09 2 ,T,w, 7 ) e E 2 (r 0 ) x [0,70): |r -
< k2-y0}.
358 Here d\ = — fJ^bi, d.2 = ^ 0 6 2 ^ , the constants ki and &2 depend only on r<j, and roil), f(-,j), uio(l), and £(-,7) ore of 0{y) as 7 -» 0 + , uniformly for fi < p(r0). R e m a r k 4.5. Note that c > 0 and the form of the solution r(/3 2 ,7) in (4.15a) ensure that for each fixed 7, 0 < 7 < 70, we have supercritical bifurcation with the unique turning point of the "parabola" at d\ + 7-0(7). Thus, in the following theorem, the asymptotic stability of the bifurcating periodic solution for e&ch fixed 7, 0 < 7 < 70, is a standard result 5 for supercritical bifurcation. The solution of the bifurcation equations (4.14) described in the last theorem leads to the existence of periodic solutions of (*) and, hence, periodic solutions of the original equation (f). T h e o r e m 4.6. Given r<j > 0 there exists 70 > 0 depending only on ro such that, for 0 < /? < p(ro) and 0 < 7 < 70, (f) has a unique branch of periodic orbits bifurcating from w = 0 at Xc(f) in (4.IS). The periodic orbit w(/3,i)(s) at \(02,j) has period 2^/7$ in s, and depends analytically on j3, and continuously on 7 up to 7 = 0. Moreover, w(0, 7 ) ( a ) = 7/3 [tf(7(wi + 7 u ; » + yW(0,7X7(^1
+ 7")*)],
A(/3 2 , 7 ) = ^ o + 7 M ^ 2 - 7 ) , w = « G 8 2 > 7 ) , * = « i + 7 « ( / ? 2 , 7 ) ,
(4-16a) (4-166)
where tj>(t) = exttp0 + e_,fV>o, wx = ^fioao, and r = r(02,y) and w = w(/32,y) are given by (4-15). The higher order term W(fi,y) belongs to [(I - V)C^(M)} © [C^XTlA/--1-)] and satisfies \\\W\\\v < c8. For eacA fixed 7, 0 < 7 < 70, the branch of periodic orbits w{fi,-f) bifurcates supercritically from A c (7), is asymptotically stable in H, and determines a wave w(P,y) = w(/?,7)(aoj/ + jds,z), periodic in aoy + j8s of period 2ir. T h e proof of Theorem 4.6 follows from Theorems 3.1, 4.2, and (4.2), since formally W = 0-1(U + v). The stabiUty of the bifurcating supercritical solution for each fixed 7, 0 < 7 < 70, follows from Remark 4.5. Finally, since the governing system of equations (2.3) is equivariant under translations in time and also in y, it follows as in §IV 3.3 of Iooss 6 that the bifurcating orbits are actually periodic waves of the indicated form. The existence of periodic waves as disturbance states for R P C F emphasizes the complicated nature of flows between rotating, sliding cylinders. The primary linear shear flow winds around the inner cylinder at the basic spiral flow angle, xo- The disturbance flows obtained here are periodic waves possessing spiral flow angles, x, close to xo • They consist of a finite number of spiral bands (or rolls) winding around the inner cylinder in the direction of x, however, the bands are not stationary but
359 rather move around the inner cylinder. In addition, within each band the fluid rotates (as in Taylor cells or Benard rolls) with counterrotation in adjacent bands. Thus, even for a fixed value of 7, the combined flows exhibit a quite complicated structure. R e m a r k 4.7. In the stability analysis of Hung, Joseph and Munson 3 (see also §51 of Joseph 8 ) it is shown formally that the critical disturbance of linear theory for R P C F is a periodic wave of frequency ^OQ sin(x — Xo)- Since Theorem 4.6 yields ■yO = -/x„ao7 + 7 V / ? 2 , 7 ) ,
(4-17)
one sees from (2.2) that the leading term of the frequency -yO agrees with that obtained previously 3 up to the factor £
360 References 1. S. Chandrasekhar, Sydrodynamic and Hydromagneiic Stability, Clarendon Press, Oxford, 1961. 2. H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269-315. 3. W. L. Hung, D. D. Joseph, and B. R. Munson, Global stability of spiral flow. Part II, 3. Fluid Mech. 51 (1972), 593-612. 4. G. Iooss, Theorie non lineaire de la stabilite des ecoulements laminaires dans le cas de "Vechange des stabilitis," Arch. Rational Mech. Anal. 40 (1971), 166-208. 5. G. Iooss, Existence et stabilite de la solution piriodique secondaire intervenant dans les probUmes devolution du type Navier-Stokes, Arch. Rational Mech. Anal. 4 7 (1972), 301-329. 6. G. Iooss, Bifurcation and transition to turbulence in hydrodynamics, Bifurca tion Theory and Applications, (L. Salvadori, ed.), Lecture Notes in Mathemat ics, vol. 1057, Springer-Verlag, New York, 1984, pp. 152-201. 7. V. I. Iudovich, On the origin of convection, J. Appl. Math. Mech. 3 0 (1969), 1193-1199. 8. D. D. Joseph, Stability of Fluid Motions. I, Springer Tracts in Natural Philos ophy 28, Springer-Verlag, New York, 1976. 9. G. H. Knightly and D. Sather, A selection principle for Binard-type convection, Arch. Rational Mech. Anal. 8 8 (1985), 163-193. 10. G. H. Knightly and D. Sather, Periodic waves in rotating plane Couette flow, Z. angew Math. Phys. 44 (1993), 1-16. 11. G. H. Knightly and D. Sather, Reduction of a singular equation of NavierStokes type to a regular Hopf bifurcation problem, Proc. Inter. Conf. on Advances in Geometric Analysis and Continuum Mechanics, August, 1993, to appear. 12. M. Renardy, Bifurcation of singular and transient solutions. Spatially nonperiodic patterns for chemical reaction models in infinitely extended domains, (H. Berestycki and H. Brezis, eds.), Pitman, London, 1981, pp. 172-216. 13. M. Renardy, Bifurcation of singular solutions in reversible systems and appli cations to reaction-diffusion equations, Adv. Appl. Math. 3 (1982), 384-406. 14. D. Sather, Transition solutions in the Taylor problem, Arch. Rational Mech. Anal. 121 (1992), 267-301.
WSSIAA 3 (1994) pp. 361-379 © World Scientific Publishing Company
361
State estimation in nonlinear systems Dedicated to Wolfgang Walter on the occassion of his sixty-six th birthday
H.W.Knobloch Abstract The paper is concerned with the problem of reconstructing inaccessible parts of the state x of a finite - dimensional system from observed output y: Find an estimate for x(t) given y{r),r < t. By an observer we mean a control system with input y whose state (or parts of it) provide an estimate which is asymptotically accurate. Under the assumption that x evolves according to a differential equation x — f(t, x) and that y is part of x concrete proposals for observers will be made in two situations. The results are new at least in the general non-linear and non-autonomous case. They will be obtained with the help of the advanced invariant manifold technique which has been developed in [1] for the purposes of control theory. Key words: Observability, Observer Design A M S Subject Classifications: 93C41, 93B07, 93B27
1
Introduction
The paper provides further examples of dynamical observers which can be constructed on the basis of methods which are developed in [1], chapter 4. The following topic will be considered. Given a differential equation (abbr.: de) x = f(t, x), x = (*,,..., x n ) T , • = d/dt.
(1.1)
Some of the components of x are declared to be the 'output' y = ( x m , . . . , x n ) T . Let x(-) be a solution of (1.1) which exists for all t > 0. Problem: Estimate x(t) given y(r) = ( i m ( r ) , . . . , X„(T)) for T < t.
To begin with we use the word 'observer' in the traditional way: This should be a control system with input y which is specified by writing down a de: x = /(<,x,j,),
(1.2)
and whose state x(t) provides the desired estimate for x(i). More precisely we speak of an observer if an estimate of the form Supported by Deutsche Forachungsgemeinschaft Kn 164-3
362 \\x(t) - x(t)\\ < C||i(0) - *(0)||e-^,< > 0,
(1.3)
holds true for any solution £(•) of (1.2) which satisfies the initial condition l|i(0)-*(0)||<*.
(1.4)
C,l?,6 are suitable positive numbers which do not depend upon x(-). In passing we note that in the linear case (f(t,x) = A(t)x) the proviso (1.4) is meaningless - due to linearity - and our definition of an observer coincides with the standard one. Actually in the second half of this paper an observer will be a more general system whose state dimension will be larger than the dimension of z. The inequality (1.3) has then to hold for respective components of x and x. So the precise problem to be discussed in this paper is this: Find conditions (on / and y) which are sufficient in order that there exists an observer having the properties (1.2) - (1.4) and describe these observers. Contributions to our problem can be found in [1], Sec. 4.4. They are special in two respects: It is assumed that (i) the bound 6 for the initial error is sufficiently small, (ii) / is t— independent, each solution of (1.1) approaches an equilibrium as t —> oo. As a consequence of (ii) the structure of the observers presented in [1] resembles the one which is familiar from the linear scenario, though this is not the result of a straightforward linearization procedure. Such a procedure by the way is out of ques tion since the equilibria of (1.1) need not be isolated - a situation which one encounters e.g. if parameter identification is treated within the context of state estimation. The present paper goes beyond [1] insofar as we assume nothing about the asymptotic behaviour of the solutions of (1.1) except boundedness (precise information can be found in Sec. 2, beginning with (2.24)). In this respect our contribution to observer design is truly nonlinear and nonlocal. It is however local in the sense that we take the bound 6 to be sufficiently small. We cannot build upon existing theory of linear observers and have to develop our own linear background independently. This is done in Sec. 2 and may be of interest in its own right. There we study time-variant linear systems from the viewpoint of what in the time-invariant scenario is sometimes called 'pole placement'. The analysis is rather elementary and straightforward. The reason that it has to be done in all details is easy to explain. For our purposes it is important to know explicitly the connection between exponential decay rate f) and the respective 'peaking' coefficient 7, cf. (2.5), (2.19). Our main results - Theorem 3.1 and Theorem 3.2 - are announced in Sec. 3. The proofs - which are given in Sec. 3 and 4 - are actually nothing else than a detailed descrip tion of the observer whose existence is stated in the respective theorem. Therefore our approach is fully constructive. The difference between the two theorems concerns the amount of information which is available for estimating the state. The hypotheses of
363 Theorem 3.1 imply that the output dimension is at least half of the state dimension whereas in Theorem 3.2 this requirement is weakened considerably. The prize which one has to pay is the amount of extra work which is involved in the proof of the second result. Whereas the crucial arguments used in the proof of Theorem 3.1 can all be found elsewhere - mainly in [1] - the proof of Theorem 3.2 requires a careful ad-hoc-construction of an invariant manifold (abbr.: imf.). This construction demon strates clearly that the theory of imf.'s as developed in [1], Appendix B, is better tailored to the needs of observer theory than, say, classical center manifold theory (cf. [3]). Useful information which can be extracted only from [1] concerns the freedom in prescribing the cross-section of an imf. (in the (2,z)-space) with some hyperplane t = const. (cf.(4.9)) and an estimate of Jacobian matrices associated with an imf. (cf. (4.16)). The latter one plays a crucial role in the stability analysis which has to be carried out. It allows decoupling and thereby reduction of a non-standard situation to a standard one and may be of interest in its own right. In order to assess the scope of our results the statement of each theorem is sup plemented by a short discussion of the linear situation. Considered as criteria for observability or detectability the statements obtained seem to be partially new in the time-variant case whereas the autonomous case can be handled directly via the Hautus'test. Hence the paper also provides illustrative examples of how standard criteria for time-invariant linear systems can be generalized to the time-variant case. To conclude this introduction we wish to state a version of the Gronwall lemma which is needed in the sequel. The proof is almost identical with the one for Proposition 1, Lecture 1, in [4]. One simply has to evaluate the integral in the first line of p. 3 more carefully. Lemma 1.1 Let <j>(t,r) be the principal matrix solution (i.e. the matrix solution sat isfying 4>(T,T) = / , abbr.: pms.) of the linear de. x = A(t)x and assume that an estimate IW<,r)||<7e-A<'-T>,0
satisfy
11^(011 < * + ce-",*>0, with constants d > 0,c > 0,/J > 0. Conclusion: The estimate IW,r)||<77e-(A-,')(<-T),0
364
2
Auxiliary results on linear time-variant systems
We deal in this section mainly with a problem concerning time-variant linear de.'s which we assume to be given as Vi = FiVl + Ki2y3 + Kay3,
. = 1,2,3,
(2.1)
or in compact form
y = Fyl + k( Jj ) , y = (yi,yi,y3)T.
(2.2)
y,- are vectors of dimension n< and we assume tij = n 3 .
(2.3)
The Fi are given matrices, the Kij should be constructed such that the pms. ( = principal matrix solution) of (2.2) satisfies an estimate to be explained below. The number of rows and columns of F t , K^ can be read off from the relation (2.1), note in particular that F3 is a square matrix because of (2.3). The elements of the matrices F, are supposed to be functions of t which are twice differentiate and bounded together with their first and second order derivatives on the set of all t > 0. Furthermore we assume that Fiit)-1 =: F(t) exists for t > 0, \\F(-)\\ < oo. (2.4) Here we have used the abbreviation ||F(-)|| for Sup<>o||F(t)||. Let
(2.5)
Find also a bound for the 'peaking factor' 7 = -y(fi). We proceed to a solution of this problem and take K12 = 0, Kyi = 0. The third of the de.'s (2.1) reads then J/3 = F 3 yi + K33V3
and can formally be solved for y\— because of (2.4): Vi = F(y3 - Kstys).
(2.6)
If this expression for t/i is substituted into the first of the de.'s (2.1) one obtains (note that Ku = 0!) Fy3 + Fy3 - FK^
- (Ffoa) y3 = F,F(y 3 - A^ya) + Kl3y3.
365
If we multiply the last relation from the left with F~l = F3 we arrive at a second order de. for y3 : y3 + Pys + Qy3 = 0 (2.7) where
P-^FzF-Ka-FzFtF,
Q :=-F^FK^Y
+ F^FK^-F3Kl3.
(2.8)
Next we turn to the second of the de's (2.1) and compute (y2 - F2Fy3y = F2yi + K22y2 + K-ays - (F2F)y3 - F2Fy3. Since we have F2yi - F2Fy3 = -F2FK33y3 rewritten as follows (y* - F2Fy3y =
/C M (J/J
- cf. (2.6) - the last relation can be
- F2Fy3) + {K22F2F + K^ - (F2F)' - F^K^.
(2.9)
We now are in a position to specify the matrices #33, K\3, Kn, K^. They should be chosen - one after another - in such a way that these matrix equations are satisfied where 0,Pi,02 are arbitrary real numbers and P, Q defined by (2.8):
K22F2F + Kj3 - (F2F)- - F2FK33 = 0.
(2.10)
I\,I2 respectively is the ni,n2—dimensional unit matrix respectively. In making the choice of K^ one has to take (2.3) and (2.4) into account. The de.'s (2.7),(2.9) assume then the form z = -0z,
ys + (A + ftMi + 0ift»s = 0,
z:=y2-F2Fy3.
(2.11)
Regarded as a system of first order de.'s (2.11) is equivalent to (2.2). To change from (2.11) to (2.2) one simply has to express the j/,- in terms of z,y3,y3. For later purposes we do this in matrix form (yi,y2,y3)T = ^(z,y3,2/3)r
(2.12)
and determine P and P - 1 explicitly from (2.6),(2.11): / 0 -FKaa P= \ h F2F V0 /,
F\ ( 0 I2 -F2F \ 0 , P-1 = 0 0 /, . 0 } \ F3 0 AT33 /
(2.13)
With the help of the matrix P(t) and the matrix solution Y(t) := dia9{e-»I2, ( _ ^ \
^
^
j)
(2.14)
366
of (2.11) we now can write down a formula for the transition matrix of the linear de. (2.2). It is correct under the proviso that the Kij have been chosen subject to the condition (2.10) and runs as follows.
(2.15)
V(O)-1 exists if 0j jt 02 (which we assume from now on). One finds
no)-. = ^ (/ „ _!_(_«
_/,J,.
(m
Next we specialize 0i = 0, 0j = 2/9 and assume from now on that 0 > 0, t — r > 0. We wish to estimate the norm of the matrix (2.15). To this purpose we first observe from (2.14), (2.16) that Y(t - r)K(0)-» = e-t-riiUrth,
( £ *
^
J
»))
(2.17)
and each C{ is a function of 0, t, r which satisfies the estimate 0 <
(2.18)
for 0 < T < t,0 > 0. Prom (2.15), (2.17) we obtain ||^,r)||<7(^)e-^-T)
(2.19)
with ^)
= \\P{t)diag(I2^^h
{
i^h))P{rr\\
(2.20)
and P is the matrix (2.13). In the sequel we use the symbol 0(1) to denote a quantity which is bounded independently from t, r, 0, provided we have 0 < T < i, 0 > 0. From (2.8), (2.10) and (2.13) we obtain the relations (note that Fi,F are 0(1)!) #33 = -30h + 0(1),
P=\
/ 0 h \0
3fiF + 0(l) * /,
F\ 0 \,P-1= 0 /
/O 0 \*
h 0 0
* \ h -Zfih + 0(1) J
(2.21)
where * means a block matrix of order 0(1). The product of the two first matrices appearing in (2.20) can be written as / 0 h 0
\
0(3a, +
* *
0(1)
(3a, +
J
367 The symbol 0(1/0) is used if a quantity can be represented as (l/0)O(l)). If this matrix is multiplied with P'1 (cf. (2.21)) from the right one arrives at a matrix which has the block 0(3(7! +
+ O(l).
(2.22)
For the definition o/||F(-)|| cf. (2.4). Remark. The conclusion of the lemma remains true if the state variable y consists of the components j/i, y3 only, e.g. if the underlying de. is of the form y\ = Fiyi + K13y3,
y3 = F3yi + K^y^
(2.23)
and it is assumed again that (2.4) holds true. We apply the transformation (2.6) which can be written in the form (yi,y3)T = P(y3,y3)T with
'-(-?" 0One arrives then at the de. (2.7), (2.8). If the 7^3 are specialized as we did before we obtain again the second order de. (2.11) for y3. We conclude this section by presenting a further lemma which is a straightforward apphcation of the foregoing results and prepares the stage for the constructions to be carried out in the next sections. It concerns an arbitrary (not necessarily linear) de. i = /(M).
(2-24)
We assume that the state variable x is partitioned as follows: * = (*,,x') T ,
x' = (x2,x3,x<)T,
(2.25)
the Xi being vectors of dimension n,-, t = 1 , . . . , 4. We assume n2 = n4
(2.26)
368 and put n' := nj + r»3 + n 4 ,
Dim x =: n = ni + n'.
(2.27)
Occasionally we partition the de. (2.24) according to (2.25): ii = fi(t,x),#
= f'{t,x)
or x; = fi{t,x),i
= 1,...,4.
(2.28)
The symbol Wt is used from now on in the same sense as in [l],cf. (B.5) and hypoth esis (i) of Theorem B.l.l. We assume in addition that there exists a bounded set M of the z—space which contains a 6—neighborhood of Wj for each t > 0,6 > 0 a fixed number. The /< should be sufficiently smooth vector fields on the set {t, x : t > 0}. Our first basic hypothesis will be formulated as follows (Hi)fi and their partial derivatives up to order 2 are bounded on the set Af:={t,x:t
>0,X6JV}.
(2.29)
The Jacobian matrix dft/dx2 (which is a square matrix because of (2.26), (2.28)) is invertible and the matrix F := (dfjdx,)-1
(2.30)
||F(.)|| = S u p | | ( 3 / 4 / a r 2 n i
(2.31)
is bounded on N. We put
where the Sup has to be taken over all (t,x) € Af. Note that each solution x(-) of (2.24) with x(0) € VV0 exists for all t > 0 and satisfies x(t)e
Wt CAT,t>Q,
(2.32)
see the definition of W( in [1]. For shortness we will denote the set of these solutions
t>yxLet x,ii,x' respectively be variables of the same dimension as x,Xi,x' respectively. Consider then the de. x = f(t,x),
x' = r(t,zuS>)
+ K(t,xui>)(*
Z
I*)-
(2-33)
For the definition of f'(t,x) see (2.28), K — K(t,x) is an arbitrary matrix of type n' x (n 3 + nt). Note that the argument in / ' and K on the right hand side of (2.33) is not t,x = (xi,x') but t,{x\,x'). It is immediately confirmed that - regardless what K is - the de. (2.33) possess an imf. which is given in terms of the equation x' = x'( i.e. £,• = xt for t = 2,3,4). The flow in this manifold is described by the first of the de.'s (2.33). We are interested in the question whether K can be chosen in such
369 a way that the theory of imf's as developed in [1], Sec. B.3, can be applied to the de. (2.33). This would imply in particular that we have the exponential decay-property ||x'(t) - x'(*)ll < 7.||£'(0) - x'(0)||e-^
(2.34)
for the x' and x' component of any solution of (2.33), provided the initial deviation Hx'(O) — x'(0)|| is sufficiently small. (2.34) becomes a consequence of [1], (B.51), after one has changed to new variables x, A' := x' — x'. The answer to this question is affirmative if certain real numbers a,@,-f,,~fw,G,H,cr satisfy certain inequalities (cf.[l|, Sec. B.3, in particular the lines following the proof of Lemma B.3.I.). Now in the present situation - after one has changed the variables by introducing the deviation A' := x' — x', cf. [1], Sec. 4.2 - the numbers H and
(2.35)
where
v = n<)v+ * ( « ) ( £ ) ,
y = (y*,y*,y*)T-
(2.36)
yi is of dimension r»j,» = 2,3,4, the matrix F' (of type n' x n') and K (of type n' x (n 3 + n,«)) are defined as follows
F(t) := (df'/dx')(t, x(i)),
K(t) := K(t, x(t))
where x(-) is any solution in the set x-
(2.37)
Note that F', K and hence
370 Lemma 2.2 Given the de. (2.24), ' e ' the symbols x',Af etc. have the previous mean ing, assume that hypothesis (Hi) holds true. Given also a number 0 > 0. Then one can find a smooth matrix K = K(t,x) of type n' x (r»3 + n*) such that the estimate (2.S5) is satisfied, where <j> is the pms. of the linear de. (2.36), (2.37). 7, = i(0) depends upon 0 only, in particular it does not depend upon the x(-) € x u>hich enters into the definition of F',K. Furthermore, given any number CQ > 11, one can take 7 (/J)
= co\\F(-)\\f)
(2.38)
(cf. (2.31)) provided 0 is sufficiently large. 'Sufficiently' depends upon Co, not upon
For later purposes we repeat the construction of the 'gain' matrix K(t, x). Choose 0 > 0 arbitrarily and write (df'/dx')(t, x) as (F(t, x), F(t, x)) where F and F respectively consist of the r»j first and the remaining columns of df'/dx' respectively. Put then K := K — F and determine K as described in the previous part of this section which was concerned with the de. (2.2). Time derivatives have to be interpreted as Lie-derivatives with respect to the time-dependent vector field / .
3
Construction of observers: The basic theorems
Given an ode. x = f(t,x).
(3.1)
We assume for the remaining portion of the paper that all hypotheses - in particular (2.26) and (Hi) - which were introduced earlier are satisfied, cf. the lines following (2.24). We also adopt the previous notation and partition x, / in the form x = ( « l t x') T , x' := (x 2 , x 3 , x 4 ) T , / = (fij'f,
f := (f2, / „ f 4 ) T .
(3.2)
The dimensions of x',x,- are denoted by n',rii, we also write /< for the n,—dimensional unit matrix. We introduce the observed output y = (x3,x4)T
(3.3)
and wish to study the problem of estimating the unobserved parts Xi(t),X2(t) of the full state given y(r) for r < t. We speak of asymptotic reconstruction of the state and this means that the difference A(*) := (ii(t) - xi(t)), (x 2 (0 - * 8 ( 0 )
(3.4)
between the estimated and the true parts £< and x,- of the state satisfy a relation ||A(0|| < C | | A ( 0 ) | | e - "
(3.5)
371 with some /9' > 0 and a constant C which may depend upon @ and the dynamics, i.e. the right hand side / and the set M" respectively. Some requirement concerning the size of A := A(0) = 11(5,(0) - x,(0)), (x2(0) - z 2 (0))|| (3.6) has to be added. In passing we note that in the linear time invariant case the relation (3.5) implies detectability. If /?' can be chosen arbitrarily large we have observability of course. The quantities x,- which provide the estimates for x,- will be defined in this and the subsequent section. They are components of the observer state. What an observer should be has been explained in Sec. 1: A control system with input y where y is now given by (3.3). The remaining portion of this paper is devoted to the explicit construction of such observers for the de. (3.1). We first consider the special case that the full state x coincides with (x^, £3, £4) i.e. the component x, is not present. The problem of observer construction has practically already been solved in the last part of the previous section. Note that we then have x' = x , / ' = / and the second half of the de. (2.33) reads therefore x = f(t,x)
+ K(t,x)(y-y),
y:=(x3,x4).
(3.7)
This is the observer for the situation under consideration. Note that the unobserved part of the state x consists of xj only and that x = (£2, x 3 , £4). The desired estimate (3.5) follows hence from (2.34). We formulate what we found so far as Theorem 3.1 Let x = (x2,£3,£4) T , assume that (2.26) and (Hi) is satisfied and let y = (x 3 , £4) be the observed output. Conclusion. / / the 'gain' matrix K is constructed following the instructions given at the end of Sec. 2 and if the number 0 is bigger than \\(df/dx)(-)\\ then the system (S. 7 ) has the properties of an observer. Proof. The arguments are borrowed from [1], Sec. 4.2 and have already been touched upon in the previous section (see the lines following (2.34)). The requirement is /3 > a where a is a bound for | | d / / d x | | (which is a finite number according to hypothesis (Hi)).fi can be made arbitrarily large - according to Lemma 2.2 - by a proper choice oiK. D Let us have a short look on the linear case:x = Ax,y — Cx. The hypothesis of the theorem can then be expressed in terms of the matrices A, C : C is of the form (0, / ) , / being the (n 3 + 14) x (n 3 + n 4 ) unit matrix. (Hi) means that the (n 2 x n 2 )-block in the lower left corner of A is an invertible matrix. That the system under these cir cumstances is observable - that is what the conclusion of the theorem implies - can then be verified directly, say by applying Hautus'test. We next turn to the more difficult case that the unobserved portion of the state consists of the two parts Xi,i2- As before we assume that the observed output is y = (£ 3 ) £4) T and that (2.26) and hypothesis (Hi) hold true. Before stating the next
372 theorem we introduce a further hypothesis. (Hj) There exist fixed numbers a' > 0, Y - which do not depend upon x(-) € x " that the matrix solution of the initial value problem X = ((dfl/dx1)(t,x(t)))X,
sucn
*(()) = /,
satisfies Il*(<)||<7'e-°''fort>0.
(3.8)
The following abbreviations will be used in the sequel (cf.(2.31): Sup\\(dfJdx2)-*\\
(3-9)
= F(.),
where the Sup has to be taken over all (t, x) 6 MTheorem 3.2 Let (2.26), hypotheses (Hi),(Ht) and the inequalities 22 X iX4
22 7 'xXiX4<«'
(3.10)
be fulfilled. Then the inaccessible components X\, x2 of the state x can be reconstructed from the output (S.S) via an observer Before turning to the proof let us have again a closer look on the linear time-invariant case: f{t,x) = Ax,y = Cx = (x3,x4), A,C constant matrices. (3.11) Clearly the conclusion of the theorem implies detectability. If we think of A as par titioned into blocks A^ according to the partitioning of x = (x1,xi,X3,xt)T we can express the hypotheses of the theorem in terms of the submatrices ;4y as follows. (Hi): det(A^) ^ 0 (note that A42 is a square matrix since n2 = n<), (#»):|k,nt||<7'e-*V>0,
(3.12)
Xi == Miall.X* := II^II.X = 11(^1,^)11.
(3.13)
That our theorem, if applied to (3.11), provides a sufficient condition for detectability can be seen directly. In fact, detectability follows already from these conditions det A4i ? OrfWAuAgAuW < a',
(3.14)
which are weaker than the hypotheses of the theorem. One observes that (3.12) plus (3.14) - by virtue of the Gronwall lemma, see Lemma 1.1 - imply that the matrix Au -
AuA^An
373 is Hurwitz. If one uses this information detectability of (3.11) is then easily established via the Hautus test. The above considerations can be extended with minor modifications to the linear time-variant case: f(t, x) = A(t)x, y = (z 3 , x 4 ) T , A, A, A bounded for t > 0.
(3.15)
The definition of \i, X*> X carries over if we replace 11v4.*^ 11 by 5«p||i4,j(/)|| where the Sup is taken over all t > 0. (3.12) has to be replaced by M(t,0)\\
4
< V e " a V := Pms. of ix = i4 u (0*i-
Proof of Theorem 3.2
In the course of the proof we will here and there assume that a suitable scaling Xj = Ax,, A a positive number which can be determined explicitly, has occurred at the outset of the construction, x 2 is then again denoted by x 2 . This means that eventually the component x 2 of the observer state has to be divided by A in order to obtain an estimate for x 2 . For the reader's convenience we write down the changes in the right hand sides of (3.1) which take place due to the scaling of x 2 : fi(xi,X2,x3,x4)
-> /,(xi,Axj,X3,x 4 ) if » ^ 2,
MX1,X1,X3,X4)
-» j/2(Xl,Ax2,X3,X4).
V4'1)
The variables and vectorfields x,x',x,-,y and / , / ' , / , • have the previous meaning. In addition we introduce new variables x,-,t = 1,2, and z = (z2,z3,zt)T respectively which have the same dimensions as x, and x' = {xi,x3,xA)T respectively. The proof will amount to a detailed analysis of the following system of de.'s for (x, xlyx2,z) =: £.
i = /(*,*), £j = fi(t,xux2,y)
+ Ki(t,Xi,X2,y)(x,
- z2)T,i = 1,2,
i = /'(i, i „ z) + K(t, xu x 2 , y)((z s , z4)T - y)T.
(4.2)
Some explanation is called for. The meaning of the first terms on the right hand sides of the last three de.'s is clear: One takes /; and / respectively and replaces XJ,XJ by Xi,x 2 and Xi,x' by £\,z respectively. Concerning Ki : We put K^x)
:= - ( d / i / d x , ) ( i , x ) ,
K2(t,x)
:= -{df3/dxt)(t,x)
- a'x,
and replace x, by x„ * = 1,2, (x 3 , x 4 ) by y.a' is the number appearing in (3.8). Concerning K : Given some number 0 > 0— to be determined later -
(4.3)
374 we choose then K{l,x) following the instructions stated in connection with Lemma 2.2 and replace Xi by x,-,i = 1 , 2 , (x3,x4) by y.
(4.4)
We next write down three observations, they are in dependent form the choice of Ki,K and can be confirmed simply by inspection. (i) The three last de.'s in (4.2) (i.e. the de.'s for Xi,x 2 , z) can be viewed upon as a system with input y, hence they - at least formally - constitute an observer for the first de.. (ii) If ((■) = (x(-),Xl(-),x2(-),z(-))
is solution of (4.2)
then x(-) is solution of x = f(t,x). If x(-) is solution of x = f(t,x), x(t),
(4.5)
then the quadruple ((•) which is given by
ii(t) := Xi(t),i
= 1,2,
z(t) = x'(t)
(4.6)
is solution of (4.2). (iii) I f £ ( ) = (x(-),Xi(-),x 2 (-),*(-)) is solution of (4.2) then the triple xi(-),xj(-), z(-) is solution of Xi = fi(t,xux3,y(t)) z = f(t,xuz)
+ / £ ( t , i i , £ a , y ( 0 ) ( £ j - *i), +
tf(t,x„x2,y(i))((z3,z4)T
» = 1,2
- y(t)f.
(4.7)
Given now a fixed solution of (4.2) which is of type (4.6). It will be denoted by £(•) henceforth and its components by x(f),
x,(0 = x,(i),» = l , 2 ,
z(t) = x'(t).
(4.8)
We also write y(i) := (x3(t),x4(t))T. The coordinates of £(0) satisfy the linear equa tion * = (* 2 ,x 3 (0),x 4 (0)). (4.9) This follows from (4.8) if we put t = 0. The equation (4.9) defines an (affine) manifold in the (xi,X2,z)-space. One can think of it as given in the form z = a(xi,x 3 ) though in the present situation s depends upon x 2 only. We wish to extend this manifold to an imf. z = S ( i , x i , x 2 ) for (4.7) (with y(t) in the place of y(t)) and refer to [1] and [2] for background material. The relevant result is contained in [2] where it has been shown (cf. Proposition 3.2) that a map S having the imf. property and satisfying the initial condition 5(0,xi,xj) = a(xi,x 2 ) (= the right hand side of (4.9)) exists and is of class C1 on a set of the form { i , x „ x 2 : t > 0,||x,- -
Xi(t)\\
<6,i=
1,2}
(4.10)
375 where 6 is some positive number. The hypotheses and the further conclusions of the proposition are the same as in [1], (B.ll) and (B.13). The involve numbers a,fi,"ia,fx,G,H, which are explained in [1], (B.7) - (B.IO). Since the statement of the proposition is a local one - valid for some sufficiently small number S cf. (4.10) - we need to compute these quanties along the special solution (4.8) only. Note that the number
(4-11)
0 > 0 arbitrary, 7, = 7(/9) (cf. (2.38)), according to Lemma 2.2, H = ||3/70*i(-)ll-
(4-12)
The arguments behind the symbols in | | . . . || are t,x(t) and one has to take the Sup over all t > 0. By choosing /3 sufficiently large one can fulfill the condition (B.ll) from [1] if 2-J^-G = 2co^-\\F(-)\\ ||^(.),K2(-)\\ < 1 (4.13) p—a p — or (for the definition of F(-) and K1,Ki respectively see (2.31) and (4.3) respectively). Note that the expression on the left hand side of (4.13) represents the second of the terms appearing in [1], (B.ll) (with a = 1) only, the first one has been neglected since it can be made arbitrarily small by choosing /? large enough. Now the right hand side of (4.13) can be simplified if one admits scaling of x 2 , cf. (4.1). In fact the quantities involved behave as follows: F(-) —► A - l F(-), K\ —* XKi,K2 —* K2. Hence for A sufficiently large the term ||F(-)|| ||/Ti(-), 7^(-)|| can be estimated by XiX* ( ^ (3.9), (4.3)) and it i s then clear that (4.13) becomes a consequence of the first of the inequalities (3.10) provided ft is sufficiently large. Thereby we know that - by choosing 0 sufficiently large - one can enforce the condition (B.ll) from [1]. Hence there exists a smooth mapping t,xux2-*
S{t,xux2) = z
(4.14)
having all properties listed in connection with (4.10). Following our practice of parti tioning z as (z2,Z3,Zi)T we write S = (S2,S3,S4)T.
(4.15)
We refer to [1] once more. Since by now we know that (B.ll) holds true we can apply (B.13) to obtain
\\dS/dzudS/di2\\
< 2 - A t f + 7 ( 0 ) e x p ( - ^ t ) = co\\F(.)\\ J »
376 where
B')
:=
{2^113/7^,11 + ^ e x p ( - ^ t ) } .
(4.16)
We are now in a position to compare our reference solution £(•) with neighboring solutions living also on the imf. z = S, i.e. having initial values satisfying (4.9). This becomes a standard problem if one has the subsequent lemma at one's disposal. Lemma 4.1 Given besides (4.8) a solution t —» {(t) = (£(t),xi(t),x2(t),z(t))T of (4-2) which has the same x-component as ((•). Assume that ||xj(t) — x,(t)|| ^ £ for all t > 0 and i = 1,2 where 6 is the number appearing in (4-10). Furthermore let ((0) satisfy the relations (4-9). Conclusion: The two t—dependent quadruples (xi,Xj,Xi,Xa) which are given as (Xl(t),£2(t),x,(t),x2(t))
an
(4.17)
are both solutions of the de. x{ = fi{t,Xx,x2,y £i = / , ( t , x u x 2 , y ( t ) ) + I
(t)),i = 1,2, [(x2 - x2) + S2{t,xux2)
-
S2(t,xux2)],
t = l,2, for the definition of S2{t,X\,x2)
see (4-H),
(4.18)
(4-15).
Proof. We first observe that the quadruples (4.17) are the projections of {(t) (cf.(4.8) and l(t) into the {xi,x2,Xi,x2)— space. On the other hand, if we project ((t) and ((t) respectively into the (xi,X2,z)—space we obtain the t—dependent vectors ( ^ ( O . X J W . X ' W ) and (x,(i),x 2 (r),z(t) = (* 2 (0,...))
(4.19)
respectively. They are both solutions of the de. (4.7) (for y(t) = y(t)) since ((•) and £(•) are solutions of (4.2) (see the observation preceding (4.7). Furthermore they both live on the imf. z = £l(t, i\, x2) since their initial values belong to the set (4.9). Hence we have identically in t x2(t) = S2(t,Xl(t),x2(t)),
z2(t) = 5 2 ( t , x , ( 0 , x , ( 0 )
and the conclusion of the lemma follows now simply from this observation: The right hand sides of the first two de.'s (4.7) and the last two de.'s (4.18) coincide along each of the vectors (4.19), note that we have z2(t) = xj(t),x,(f) = £,-(*) along (4.8). O The proof of Theorem 3.2 is now easily completed. Let x(-) be a solution of (3.1) and let y(t) := (x3(t),£i(t)) be observed at each time t > 0. We take the system (4.7) (with y(t) in the place of y(t)) as 'observer' and define the observer state as the solution of (4.7) having these initial values Xi(0) = initial guess for x,(0),» = l,2,z<(0) = Xj(0),» = 3,4 and
377 z2(0) = x2(0).
(4.20)
The last relation implies that the initial value of the observer state satisfies the relation (4.9). Hence if we take the quadruple consisting of true state and observer state and assume ||*.-(t)-*.-(OII<*.» = 1 . 2 , « > 0 , (4.21) we obtain a solution £(-) of (4.2) as considered in Lemma 4.1. Again we denote by ((-) the particular solution of (4.2) which is given in terms of the relation (4.8). We introduce the deviation A(<) := (A 1 (t),A 2 (t)) where A;(t) := Xi(t) - x,(<),t = 1,2. Our aim is to establish an estimate of the form ||A(*)||
(4.22)
for some constants C, a > 0 to be determined. Thereby we indeed have justified the name 'observer' for the system (4.7) (cf. Sec. 3, in particular (3.5)) and accomplished our task. We also see that the condition (4.21) can be enforced simply by taking ||A(0)|| sufficiently small. Now A(-) can be viewed upon as solution of a de. A = g(t, A).
(4.23)
To obtain g one has to subtract the first two from the last two of the de.'s (4.18) and replace x< by x,(t),« = 1,2 and x, by A, + x,(<). Clearly g(t,0) = 0 and the estimate (4.22) will be the result of a classical stability analysis of the equilibrium A = 0 of (4.23) (asymptotic stability in first approxima tion). We have (dg/dA)(t,0) = A(t) + R{t) (4.24) where A(t) is the matrix which is obtained from the last de.s (4.1) by first cancelling Si, then taking partial derivatives with respect to £,• and finally evaluating the ele ments of the Jacobian matrix at x,- = x, = x,(<),» = 1,2.R(t) is obtained by treating the remaining terms Ar,(t,x 1 ,x 3 ,y(t))(5 2 (<,x,,x 2 ) - S 2 (t,x„x 2 ))
(4.25)
in the same way. One finds - if one takes (4.3) into account -
Again one can influence the shape of A by assuming that xt has been scaled before starting the observer construction, cf. (4.1). Taking A sufficiently large we can bring the block in the left lower corner of (4.26) arbitrarily close to 0 whereas the diagonal blocks remain unchanged. Therefore, if hypothesis (3.8) is brought into play, we can
378
assume without loss of generality that the matrix solution P0(-) of the initial value problem P = A(t)P,P(0) = I (4.27) satisfies
IIPoWII < 7'e-°'V > 0,
(4.28)
where a" > 0 can be chosen arbitrarily close to a'. We next turn to the remainder term R(t) and remind the reader on the definition: The derivatives of (4.25) with respect to z,- have to be taken at £, = x< = x,(<). The estimate
11^(011
(4-29)
is therefore clear, in view of (4.16). Assuming again that Xj has been scaled with a sufficiently large factor A we can replace the first two terms in (4.29) by XiX* (cf(4.13) and the subsequent passage). Next we have to find out what the scaling of xj and the choice of a large factor A does to
a/V&i = (dh/dx1,df3/dxudfjdx1). The Sup—norm of this quantity tends (for A —» oo) to Xi as can be seen immediately from (3.9) and (4.1). So employing (4.16) once more we finally arrive at this relation which is true if A and /? are sufficiently large: P W I I < Co2xX:X4 + ebXiX40exp(-£=^)-
(430)
Here we already have replaced /?/(/? — a) by 1 which is legal if we take /? sufficiently large and increase Co slightly. Still we can assume that a := a" - 7'2coX XiX* > 0.
(4.31)
This is a consequence of hypothesis (3.10), second half. Note that a", Co respectively can be chosen arbitrarily close to a' and 11 respectively. We now combine (4.27), (4.28) with (4.30),(4.31) in order to estimate the matrix solution P of the initial value problem P = (A(t) + R(t))P, P(0) = /. (4.32) From Lemma 1.1 we obtain \\P{t)\\ < 7'7e- 5 '
(4.33)
where 7 = exp(7'coXiX4 j f ^ ) -
(4.34)
Remembering how (4.32) is related to our original problem, see (4.24), we see that the two relations (4.33), (4.34) indeed show that the equilibrium A = 0 of the de. (4.23) is exponentially asymptotically stable.
379
References [1] H.W. Knobloch, A. Isidori and D. Flockerzi, Topics in Control Theory, DMV Seminar Band 22, BirkhEuser Verlag, Basel, Boston, Berlin 1993. [2] H.W. Knobloch, Foundations of invariant manifold theory for ordinary differ ential equations In: Recent trends in differential equations (R.P. Agrarwal ed.), World Scientific Series in Applicable Analysis, 1: 365-392, Singapore 1992. [3] J. Carr, Applications of center manifold theory, Applied Math. Sciences 35, Springer-Verlag, New York, 1981. [4] W.A. Coppel, Dichotomies in stability theory, Springer LNM 629, SpringerVerlag, Berlin, 1978.
WSSIAA 3 (1994) pp. 381-397 © World Scientific Publishing Company
381
Linear Ordinary Differential Equations Q. Kong and Anton Zettl Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115, U.S.A.
Abstract This work is dedicated, in admiration, to Wolfgang Walter. It was inspired by dis cussions the second named author had with Professor Walter at a meeting in Gregynog, Wales in the summer of 1993. One of Walter's keen interests is in providing elementary proofs. It is very much in this spirit that this paper is written.
1
Introduction
Many results for ordinary linear differential equations are obtained by specializing nonlinear theorems to the linear case. Often the nonlinear proofs are highly technical and simpler proofs are available for the linear case. In addition it is frequently the case that the linear results can be obtained with weaker assumptions than those one gets by simply specializing nonlinear results. The aim of this paper is to give elementary proofs, under minimal conditions, of basic results for linear ordinary differential equations: continuous dependence on all parameters including the coefficients, compact dependence -in the sense of operator theory on Banach spaces, differentiable dependence on parameters - in the sense of Frechet derivatives on Banach spaces. We hope to persuade the reader that the natural spaces for the study of linear ordinary differential equations are L and Lix. Some of our results on compact and differentiable dependence, although elementary, seem to be new. N o t a t i o n . An open interval is denoted by (a, b) with - o o < a < b < oo. Thus [a, b] always denotes the compact interval with left endpoint a and right endpoint b. Let R denote the reals, C the complex numbers, and X either of them. Let N = { 1 , 2 , 3 , . . . } . For n, m e N, ^ • " • ( X ) denotes the collection o f n x m matrices over X. If m = n we also write £ " ( X ) , if m = 1 we write X". The default value of X is C, and we sometimes write E"-"1 for £ ^ ' m ( C ) . For any interval J of the real line, open, closed, half open, bounded or unbounded, by L(J) we denote the linear manifold of complex valued Lebesgue measurable functions y defined on J for which
£ \y(t)\dt = \} \y(t)\dt = \} \y\ < oo.
(1.1)
This work grew out of the July 1993 meeting on scientific computing and analysis held in Gregynog, Wales.
382 The notation Lioc(^) is used to denote the linear manifold of functions y satisfying y G L{[a,P]) for all compact intervals [a,/3] C J. (If J = [a,b], then Lloc{J) = L{J).) Also, we denote by ACix( J) the collection of functions y which are absolutely continuous on all compact intervals [a, /?] C J. For a matrix function P = (p„) from J into E^-m we use the notation P € Lloc(J) or P G /4Cioc(J) to mean that each pj, G Llx(J) or p^ G ^ ^ ^ ( J ) , respectively. The norm of a constant matrix or matrix function P is denoted by \P\.
2
Existence, uniqueness, and continuous dependence.
Definition 2.1 Let J be any interval of the real line, open, closed, half open, bounded or unbounded. Let P : J -¥ En,n; and F : J -> E"'™, n, m G N. By a 50/utton o/ «/ie equation Y' — PY -YF on J we mean a function Y : J —» E"'"1 which is absolutely continuous on all compact subintervals of J and satisfies the equation Y' = PY + F a.e. on J . Theorem 2.1 Let J be any interval. Let P : J ->• En'n; F : J -> £"' m , n,m G N. Assume P 6 LlBe(J)
(2.1)
F G L(0C(J).
(2.2)
and TAen every initial value problem (IVP) Y' = PY + F
(2.3) m
Y(u) =C, ueJ, Ce E"' (2.4) has a unique solution defined on all of J. Furthermore if C, P, F, are all real valued, then the solution is also real valued. Proof: Choose any ty G J. We prove that (2.3), (2.4) has a unique solution on [u, t\] if ti > u and on [tj, u] if ti < u. Assume ti > u. Let B = {Y:[u,t1]^En'm,
KGC([u, t l ])}.
1
Following an idea of Bielecki , we define the norm of any function V G B to be ||V||= sup (e-* r /> ( * ) l < i j |y(r)|} t6[u,(,| I
>
where A" is a fixed positive constant, K > 1. It is easy to see that with this norm B is a Banach space. Let T : B —> B be defined by (TY)(t) = C+ f'{PY -r F)(s) ds,
YeB,te
[u, U}.
(2.5)
383 Then for Y, Z G B \(TY)(t) - (TZ)(t)\ < f \P(s)\\Y(s) - Z(a)\ ds Ju
and hence t K ]p dr e- * / > ( . ) l < b | ( 7 T ) ( f ) _ (TZ)(«)| < ||V - Z\\ [ \P{s)\e- £ W ds
Therefore
< h\Y
- Z\\
||7r-rz||
By the contraction mapping principle we see that (2.5) has a unique solution, i.e., (2.3), (2.4) has a unique solution on [u, ti]. The proof for the case that tx < u is similar. We only indicate that the norm we need for this case is defined to be
||F||= sup {c*/.*i^)i-|y(0l}. This complete the proof. I Remark 2.1 Let J be an interval. For each P e Lix{J), each F € Lix{J), each u G J and each C G En-m there is, according to Theorem 2.1, a unique Y G ACioc(J) such that Y' = PY + F
onJ and Y(u) = C.
We use the notation Y = Y(-,u,C,P,F)
(2.6)
to indicate the dependence of the unique solution Y on these quantities. Below, if the varia tion of Y with respect to some of the variables t, u, C,P,F is studied while the others remain fixed we abbreviate the notation (2.6) by dropping those quantities which remain fixed. Thus we may use Y(t) for the solution when u,C,P,F are fixed or Y(-,u) to study the variation of the solution function Y with respect to u, etc. Theorem 2.2 Let J = (a,b),P : J -> E"' n and assume that P G Li^J) matrix solution of Y' = PY, on J.
and Y is
annxm (2.7)
Then we have rank Y(t) = rank Y{u),
t,ueJ.
(2.8)
Moreover, if m = n and given u G J, then (detV)(0 = (dety)(«)expf /'traceP(s)ds),
t G J.
(2.9)
384 Proof: The formula (2.9) follows from the fact that y = det Y satisfies the first order scalar equation : y' + py = 0 with p = trace P. To prove (2.8) for the general case let Y(u) = C and assume rankC = r. If r = 0 then Y(t) = 0 for all t 6 J by Theorem 2.1. For r > 0 let Ci, t = 1 , . . . , r, denote r linearly independent columns of C and construct a new nonsingular n x n matrix C* by adding n - r constant vectors to Ci,i = 1,... , r. Denote by }> the solution of (2.7) with y(u) = C", then by (2.9) rank^(t) = n, for t 6 J. From this and the uniqueness part of Theorem 2.1 it follows that for each t g J the (constant) n-vectors Yt(t), Y2(t),..., YT{t) are linearly independent. Hence rank Y{t) > r, for t € J. Now suppose that rank Y(t0) > r for some tf, in J. Then by repeating the above argument with u replaced by t0 we can conclude that rank Y{t) > r for all t 6 J. But this contradicts ranky(u) = r find concludes the proof. I Theorem 2.3 (Everitt and Race2 ) If for some u e J and linearly independent constant vectors C\,..., C n each initial value problem Y' = PY,
Y(v) = d,
has a solution defined on all of J , then P 6 Lix{J). then P is continuous.
t = l,...,n,
(2.10)
Furthermore, if each Y> is aC
1
solution,
Proof: Let Yt denote a vector solution satisfying Y(u) = d and let Y be the matix whose t-th column is Y{, i = 1 , . . . , n. By Theorem 2.2 Y is nonsingular at each point of J. Let P =
Y'Y-K
Let if be a compact subinterval of J. Then since Y is continuous and invertible on K it follows that Y~l is continuous and hence bounded on K. Also, V" is absolutely continuous on K since Y is a solution on J. Therefore P 6 Lioc(J). The furthermore statement is clear from the construction of P. I Definition 2.2 Let P : J -► En,n satisfy (2.1). Annxn matrix solution of (2.7) which is nonsingular at some point of J is called a fundamental matrix of (2.7) on J. It follows from Theorem 2.2 that annx n matrix solution of (2.7) is nonsingular at some point of J if and only if it is nonsingular at every point of J.
385 Given P : J ->• En,n satisfying (2.1), from Theorem 2.1 we know that for each point u of J there is exactly one fundamental matrix y of (2.7) satisfying y(u) = / n where /„ denotes the n x n identity matrix. For each fixed u e J let *(-,u) be the fundamental matrix of (2.7) satisfying *(u,u) = / „ . Note that for each fixed u in J, $(-,u) belongs to AC(J) if J is compact and P € L(J). In this case, u can be an endpoint of J. Note that *(t,u) = K«)>'- 1 («)
(2-11)
for any fundamental matrix y of (2.7). We also write $ = *(P) = (ft,.)?^, and *(P)(t,u) = *(t,u,P) .
(2.12)
Note that for any constant nonsingular n x n matrix C, $ C is also a fundamental matrix and that every fundamental martix has this form. Theorem 2.4 Let J = (a, 6), -oo < a < b < oo. Let P : J -> £"•"; F : J -► F n ' m and suppose i/ia« P G L ( J ) , FeL{J). (2.13) .Assume t/iat /or some u € J, C £ C", we hawe V = P y + F on J,
and K(u) = C.
(2.14)
T/ien |VWI<(|C| + / j F | j e x p ^ | P | j ,
a
(2.15)
Proof: Note that (2.14) is equivalent to y (t) = C+ f\p{s)Y(s)
+ F(s)) ds,
a
Case 1. u < t < b.
\Y(t)\<\C\ + \f(PY + F)\ < |C|+ A|P||K| + |F|)
-( l c l + /J F I ) + r ( I F I I y l ) ' u < ( < 6 Hence from Gronwall's inequality we get |V(*)| < (\C\ + [ |F|) exp ( j f |P|) < (|C| + £ |F|) exp ( j f |P|) , u < t < b.
386 Case 2. a
Jt
\C\+f\F\+n\P\\Y\) Jt
<(\C\ + £\F\j + JtU(\P\\Y\),a
|F|) exp ( j f |P|) < (|C| + j[" |F|) exp Q T |P|) , a < « < u.
Combining the two cases we conclude that \Y(t)\ < (\C\ + j [ ' \F\) exp (jf* |P| j , a < t < b.
(2.16)
This concludes the proof. I Theorem 2.5 Let J = (a, b), -oo < a < b < oo. Let P : J-f £"•"; F : J -> E"' m . tj Suppose that in addition to (2.1) and (2.2) with J = (a, 6) we have that PeL(a,c),
FeL{a,c)
(2.17)
for some c 6 (a, 6). *4s5ume t/iat for some u € J and C 6 En,m, Y is a solution of the IVP (2.3), (2.4) on J. Then K(a) = (lim Y(*) (2.18) exists and is finite. ii) Suppose that in addition to (2.1) and (2.2) with J = (a, b) we have that P€L(c,b),
FeL{c,b)
(2.19) n m
for some c £ (o, b). Assume that for some u e J and C e F ' , Y is a solution of the IVP (2.3), (2.4) on J. Then Y(b)=\im_Y(t) (2.20) exists and is finite. Proof: We establish the theorem for 6; the proof for the end point a is similar and hence omitted. It follows from (2.15) that Y is bounded on (c, 6), say by M. Let {6j : i € N} be any increasing sequence converging to b. Then for j > i we have
\Y(bj) - y(6,)| = I f' PY <M T |P|.
(2.21)
Hence {Y{bi) : i e N} is a Cauchy sequence and thus converges. I For equation (2.7) where F = 0, by Theorems 2.2 and 2.5 we have the following corollary.
387 Corollary 2.1 Let the assumptions and notations of theorem 2.5 hold. Let u € J. i) Under the condition i) of Theorem 2.5 with F = 0, rankK(a) = rankV(u). Moreover, given any C 6 En-m there exists a solution Y of (2.7) such that Y(a) = C. Thus initial conditions can be specified at a regardless of whether a is a finite or infinite end point and for each such initial condition there is a unique solution. ii) Under the condition ii) of Theorem 2.5 with F = 0, ranky(ft) = rankF(u). Moreover, given any C 6 En-m there exists a solution Y of (2.7) such that Y(b) = C. Thus initial conditions can be specified at b regardless of whether b is a finite or infinite end point and for each such initial condition there is a unique solution. Theorem 2.6 Let u,v e J = (a,b), -oo < a < b < oo, C,D 6 En-m, P,Q,F,G 6 L(J). Assume Y' = PY + F on J, Y(u) = C; Z' = PZ + G on J, Z(u) = D. (2.22) Then
\Y(t) - Z(t)\ < K exp U ' \Q\ J , a < t < b,
(2.23)
where
K = \C - D\ + I f \F\\ + M I /" \P\\ + fb\F - G\ + M f \P - Q\, and
M= |ci+ |F| exp , p|
(2.24)
(225)
( r ) (r - )-
Proof: This follows from Theorem 2.4 and Gronwall's inequality. I Theorem 2.7 Let J = [a,b], u e J, C 6 £> m , P,F € L(J). LetY = Y(-,u,C,P,F) on J. Then Y is a continuous function of all its variables u, C, P, F uniformly for t G [a, b]; more precisely, given any e > 0 there is a S > 0 such that ifv G J, D € £> m , Q,G,6 L(J) satisfy \u - v\ + \C - D\ + t \P-Q\+ Ja
I" \F-G\<
S,
(2.26)
Ja
then \Y(t,u,C,P,F)-Y(t,v,D,Q,G)\
for allt e [a, 6].
(2.27)
Note that in particular Theorem 2.7 yields that Y(t, u, C, P, F) is jointly continuous in C, P, F uniformly for (t,u) in [a,b] x [a,b]. Proof: The absolute continuity of the integral and (2.26) imply that the constant K in (2.24) can be arbitrarily small. The conclusion then follows from Theorem 2.6. I In view of the importance of the dependence of solutions on the "eigenparameter'' in the theory of boundary value problems we introduce the parameter z in Theorem 2.8.
388 Theorem 2.8 Let J be any interval, let Pk : J -»• £»(C) satisfy Pk e Lloc(J),k = 1,2; let ueJ,Ce £"' m . TAen /or each zeCthe IVP (2.3), (2.4) with P = Pi + zP2 has a unique solution defined on J, sayY = Y(t,u,C, Pi+zP2), and for eachfixedt,u,C, Pi, Pi the solution Y is an entire function of z. Proof: This follows from the.proof of Theorem 2.1 by Naimark in [2, Theorem 1, pp 51-54] coupled with the observation that each successive approximation is a polynomial in z and a result from complex analysis which says that the uniform limit of a sequence of analytic functions on a compact subset of C is analytic. I Theorem 2.9 Let J = (a, b), - c o < a < b < oo, let Pk, Fk : J -> £"(C), Ck e C, uk e J, k 6 N. Assume (i) Pk -> P0 as k -> oo locally in Lix(J)
in the sense that for each compact subinterval I of J we have J \Pk - Po\ ->• 0 as k -> oo;
(ii) Fk -> F0 as k -> oo locally in Lix{J) in the sense that for each compact subinterval I of J we have f \Fk - F0\ -> 0 as k -> oo; (Hi) Ck -y C0 € C as k -¥ oo; (iv) uk —► Uo G J
as k —► oo.
Then Y(;uk,Ck,Pk,Fk)^Y{;u0,C0,P0,F0)
as k -f oo
locally uniformly on J, i.e., uniformly (in t) on each compact subinterval of J. Proof: This follows from Theorem 2.7. I The next result is called the variation of parameters formula and is fundamental in the theory of linear differential equations.
389 Theorem 2.10 (Variation of Parameters Formula) Let J be any interval, let P e Lloc(J) and let * = *(-,u,P) be the fundamental matrix of (2.7) defined by (2.11). Let F £ Lioc{J), « e J and C e £>"". (Wote that F must benxm in this case.) Then Y(t) = *(t, u, P)C + /" $(«, s, P)F(s) rfs, ( £ J
(2.28)
is the solution of (2.3), (2.4). Note that if J is compact and P e L(J), F £ L(J), then Y € AC( J), and u can be an endpoint or an interior point of J. Proof: Just differentiate (2.28). I
3
Compact Dependence of Solutions on the given D a t a
Definition 3.1 >1 (not necessarily linear) operator T mapping a Banach space X into a Banach space Z is said to be compact if every bounded sequence {xn} in X has a subsequence {x„y} such that the sequence {T(xn>)} converges to a point in Z. Here we show that each one of the maps : C^Y(t,u,C,P,F),
P->Y(t,u,C,P,F), F->Y(t,u,C,P,F)
(3.1)
P^Y(-,u,C,P,F),
(3.2)
as well as each of the maps C^Y{-,u,C,P,F),
F^Y(-,u,C,P,F)
is compact. Recall that En-m = ^'""(C), the collection o f n x m complex matrices, with any norm, is a Banach space . Let H1 = {Y : J -> E"'m such that Y e AC(J) (and hence) Y' € L(J)}.
(3.3)
Then H1 is a Banach space with the norm :
m = /im^m
(3-4)
Theorem 3.1 Let J be an interval, t,« e J, P,F e L(J). Let Y = Y(t,u,C,P, F) be the solution of the IVP (2.3), (24). Then each of the maps defined by (3.1) and each of the operators defined by (3.2) are compact.
390 Proof: Since En'm is a finite dimensional space every bounded sequence has a convergent subsequence. The claim for the first three maps in (3.1) follows from this and the continuity properties given by Theorems 2.7 and 2.9. Assume for some M > 0 S = {P e L(J) : ! \P\<M<
oo}.
(3.5)
N, for all P e S .
(3.6)
By Theorem 2.5 there exits an N > 0 such that \Y(t,u,C,P,F)\< For s < t \Y(t,u,C,P,F)
- Y(s,u,C,P,F)\ < NM{t-s),
= \ f PY(-,u,C,P,F) PeS,
< (3.7)
and similarly for s > t. From this it follows that {Y(-,u,C,P,F), P € 5} is an equicontinuous family of functions. The compactness of the first operator in (3.2) follows. The proofs of the remaining two cases are similar and hence omitted. In fact we observe that this proof shows that the map T which maps the triple (C, P, F) in the space D = E"-m{C) x £"' n (L(J)) x E"' m (L(J)) (3.8) into H' defined, for fixed u in J, by T(C,P,F)
= Y(;u,C,P,F)
(3.9)
is compact. I
4
DifFerentiable dependence of solutions on t h e given d a t a including the coefficients
Theorem 2.7 shows that the solution of the initial value problem (2.3), (2.4) with P,F,Ce Lioc{J) depends continuously on all the given data. In this section we show that this depen dence is not only continuous but also differentiable. Definition 4.1 A map T from a Banach space X into a Banach space Z T-.X^Z
(4.1)
is differentiable at a point P e X if there exists a bounded linear map dTp.X^Z
(4.2)
391 such that for H 6 X \T(P + H)- T(P) - dTP{H)\ = o{\H\), as \H\ -» 0.
(4.3)
That is, for each c > 0 there is a 5 > 0 such that \T{P + H)-T{P)-dTp(H)\
for all H e X with \H\ < 5 .
(4.4)
// such a map dTP exists, it is unique and is called the derivative of T at P. A map T is differentiable on a set S C X if it is differentiate at each point of S. In this case, the derivative is a map: P -> dTP from S into the Banach space L(X, Z) of all bounded linear operators from X into Z, denoted by dT. The differentiability of Y =
Y(t,u,C,P,F)
with respect to t follows from the definition of solution. The differentiability of Y with respect to u follows from the representation (2.11). In fact this representation shows that Y has the same smoothness properties with respect to u as it has with respect to t. For fixed t, u, Y = Y(t, u, C, P, F) is a function of C, P, F mapping E"'m x E"'n x £"•*" into £"•"■. Is this mapping differentiable in C, P, F? In the following for any P : J -> £"•'" satisfying P e L(J), we denote \P\ ~ Jj \P(t)\dt. Theorem 4.1 Forfixedt, u, P, F the solution Y = Y(t, u, C, P, F) of (2.3), (2.4) is not only continuous in C as shown by Theorem 2.7 but is also differentiable in C. Moreover dYc = j£(t, u, C, P, F) = *(t, u, P).
(4.5)
Thus we have dYc(H) = $(t,u,P)H,
H G En'm.
Proof: This follows directly from (2.28). Thus this derivative is constant in C. I Theorem 4.2 LetJ = [a,b}. Fixt,u £ J,C e £^'m,anrf P,F e L(J). LetY = We have dYF(H) = ^(t,
u, C, P, F){H) = j f *(t, s)H{s) ds, H e L(J).
Y(t,u,C,P,F). (4.6)
Here the right side of equation (4-6) defines a bounded linear operator on the whole space L(J). The derivative dYF is constant in F. Proof: From the representation (2.28) we have Y(t, u, C,P,F + H) - Y(t, u, C, P, F) = / ' *(t, s, P)H{s) ds.
(4.7)
The conclusion follows from this equation and the definition of derivative. I Before stating the next Theorem we give two lemmas, these are of independent interest.
392 Lemma 4.1 Let P € L( J), u € J. Then for any t € J we have *(t,u,P)
= I+f
P+f Ju
P{T) f P{s)drds+ f P(r) f P(s) ['' P{x) dxdsdr + ■ ■ ■. (4.8) Ju
Ju
Ju
Ju
Ju
Proof: This follows directly from the successive approximation proof of the existenceuniqueness theorem: just start with the first approximation $o = /, then $ t = I + ftP, etc. I Lemma 4.2 Let P,H& L(J). Then for any t,u e J we have *(t, u,P + H) = *(t, u, P) $(*, u, S),
(4.9)
S = *" l (-, u, P) H *(•, u, P).
(4.10)
where
Proof: The proof consists in showing that both the left and right hand sides satisfy the same initial value problem. I Remark 4.1 If P commutes with the integral of H in the sense that P(t)(ftH) Ju
= (ftH)P(t)
forteJ,
(4.11)
Ju
then (4-tO) reduces to 9{t,u,P + H) = *(t,u,P) *(t,u, H).
(4.12)
This follows from the fact that (4-It) implies that $(-,u,P) H = H $(-,u,P) and hence S = H in (4.10). Theorem 4.3 Let J = [a,b]. Fix t,u £ J, C € £"' m , F e L(J). For P e L(J) let Y = Y(t, u, C, P, F) be the unique solution of (2.3), (2.4). Then the map P -* Y(t, u, C, P, F) is differentiable with respect to P and dYP = ^{t,u,C,P,F)
(4.13)
is the linear transformation from the space : L(J) —> E",m(C) given by dY^H)
= •(*, u, P) Q f •- 1 (r, u, P)ff(r)*(r, u, P) dr) C
+ £ * ( i , r , P ) Mt
F(r)dr, H e L(J).
(4.14)
For eoc/i ,/ized P e £(./), dKp w a bounded linear transformation defined on the whole space L{J).
393 Proof: Let Y(t, P) = Y(t, u, C, P, F). From the variation of parameters formula (2.28) it follows that for H 6 L(J) and S denned by (4.10) Y(t, P+H)-Y(t,
P) = *(«, P+H)C+ f *(t, s, P+H)F(s) ds-
Ju
= *(t, u, P) [*(f,u, 5) - /] C + / ' *(*, a, P) [*(*, r, 5) - /] F(r) dr = *(t, u, P) [jf S + j f S(x) j f S(j/)di/dx + j f 5(i) £ 5(j/) j T S(z) dzdydx + • • •] C +f «(t, r, P) [jf 5 + j f S(x) j f S(y) dydx + j f S(x) j f 5(») j f 5(z) dzdydx + • ■ •] F(r) dr. (4.15) Hence Y(t, u,P + H)-
Y(t, u, P) - *(t, u, P) ( £ S{r) dr] C- £ 9(t, r, P) Qf' S{x)dxj F(r) dr
= *(t, a, P) [jf S(x) j f S(y) dydx + j f S(x) £ S(y) j f S(z) dzdydx + • • •] C + j f *(*, r, P) [jf S(x) j f 5 ( y ) dydx + j f S(x) j f S(y) £ S(z) dzdydx + • • •] F(r) dr = E(H). (4.16) Noting that |*(t,u,P)| and |* _ 1 («,u,P)| are bounded on J, from (4.10) we have | 5 | < \kH\ for some fc 6 R. Then there exists an M > 0 such that |£(ff)| < M\C\ [\S\2 + \S\3 + ■■■}+ M\F\ [\S\2 + \S\3 + ■■■} < M\C\\kH\ [\kH\ + \kH\2 + \kH\3 + ■■■]+ M\F\\kH\ [\kH\ + \kH\2 + \kH\3 + • • • ] . (4.17) From this we get
ffi)Uo
as |lf|-»0.
(4.18)
From this we see that dYP exists, is given by (4.14) and is a bounded linear transformation on the space L( J). This completes the proof of Theorem 4.3. I Theorem 4.4 Let the hypotheses and notations of Theorem 4-3 hold and assume (4-U) holds. Then 1. H(t) *(t,u,P) = *(t,u,P) H(t),
t,ueJ.
(4.19)
2. the exponential law holds, i. e. 9(t, u, P + H) =*{t, u, P) $(t, u, H), t, u G J.
(4.20)
394 3. Formula (4-14) reduces to dYP(H) = *(t, u, P) (f
H(s) ds\ C + f *(t, r, P) (f
H{s) ds\ F(r) dr,
t,ueJ,
(4.21)
Note however that dYp is not the operator defined by the right hand side of (4-17) since H cannot be restricted to satisfy (4-H) in the definition of the derivative dYP. Proof: This follows from Theorem 4.3 and Remark 4.1. Remark 4.2 In the special case when P and H are constant matrices we have dYP(H)
= exp((t - u)P) (fexp{(u
+ f exp((t - r)P) (f
- r)P)H{r) exp((r - u)P) dr) C
exp((s - u)P)H(s) exp((s - u)P)ds) F(r) dr.
(4.22)
Note that if P and H are constant and satisfy (4.11 ) , then (4-18) reduces to dYp(H) = (t-u) exp ((t - u)P) HC + f\t - r) exp((« - r)P)HF(r) dr. But this reduction does not hold, in general, for constant matrices not satisfying (4-H)For fixed t,u,C,F replace P by P + zW and fix P and W. We know from Theorem 2.8 that the unique solution of Y = Y(t,u,C,P + zW, F) of (2.3), (2.4) with P replaced by P + zW is an entire function of z. What is
The next result answers this question. Theorem 4.5 Let J = (a,6), - c o < o < b < oo, t,u e J, C e £"'m, P,F,W e L(J); let Y = Y(t,u,C,P + zW,F) denote the unique solution of (2.3), (2.4) for each z e C. Then Y is an entire function of z and dY
dY
r'
* = -g; = *(*. u - p ) [*(«.u- 5 ) 1 c + 1 *(*. r > p ) *(*. r - s ) F(T)dr
where S is defined by (4-10) with H replaced by W.
( 4 - 23 )
395 Proof: By Theorem 2.10, Lemma 4.2 and the observation that $(t,u,zP) from (4.10) we have Y(t,u,C,P + [t^{t,r,P+(z
+ {z + h)W, F) - Y(t,u,C,P + h)W)F(r)dr-^{t,u,P
=
z$(t,u,P)
+ zW, F) = * ( t , u , P + (z + h)W) C + zW)C-
Ju
f $(t,r,P + zW) F{r)dr (4.24) Ju
= h(t,u,P)[${t,u,S)]C + h f'^(t,r,P)mt,r,S)]F(r)dr.
(4.25)
The conclusion now follows by dividing by h and taking the limit as \h\ -> 0.1 Theorem 4.6 Let J = (a, b), -oo < a < 6 < oo, t,u e J, C e £™'m, P, F G L(J), and z € C; let Y = Y(t,u,C,P + zW,F) denote i/ie untgue so/utton of (2.3), (2.4) for each W e L(J). Then Y is a differentiate function of W and
dYw =
§v= *(i'"'p) [*(t'"'s)] C + Il *(*'r'p) *(t'r's)F{r) dr
(426)
where S is defined in (4-10) with H replaced by W. Proof: By Theorem 2.10 and the observation that
+ z{W + H),F)-
Y(t,u,C,P+
zW,F) = $(t,u, P + z{W + H))C
+ f *(i, r,P + z(W + H) F(r) dr - *(*, u,P + zW)CJu
f *(*, r,P + zW) F{r) dr (4.27) Ju
Prom this and (4.10) we get that (4.27) equals =
u, S(H))-*(t,
u, 5(0))] C+£ *(t, r, />)[*(*, r, S(J/))-*(t, u, 5(0))] F(r) dr (4.28)
where S(70 = «- 1 (.,u,P)[z(W + This completes the proof. I
5
ff)]*(-,u,P).
T h e inverse problem
Notation. Given d n-dimensional vectors Yi, Y2,..., Vj we denote the n x d matrix whose tth column is Yi, i = 1,..., d, by
y = [yi,K2,...,yj
(s.i)
396 Above we started with a coefficient matrix P and, possibly, a nonhomogeneous term F and then studied the existence of solutions and their properties. Here we reverse this. Given a number of functions under what conditions are they solutions of a first order linear system? For the sake of completeness we state the theorem for both the direct and the inverse problems. Theorem 5.1 (i) Let 1 < d < n, P G Lloc{J), P : J -> £"•**. Assume that Yit i = 1 , . . . , d are vector solutions of Y' = PY.
(5.2)
rank[y I ,y 2 ,...,y (( ](t) = (i
(5.3)
for some t in J, then this is true for every t in J. (ii) Let Yi G ACioc(J), i = l,...,d,
are an n-vector functions, 1 < d < n . Assume that
rankfK!,..., Yd](t) = d Then there exists annxn of (5.2).
for each t G J.
(5.4)
matrix P G Lix{J) such that Yit i = 1,...,d, are solutions
Furthermore ifYi G C'(J),i = l , . . . , d , then there exists a continuous such P. Proof: Part (i) is contained in Theorem 2.2, so we only prove part (ii). If d = n take P = Y'Y~l where Y is defined by (5.1). If d < n, we construct an n x n matrix M = [Yi,..., Yi, Yd+l,..., Y„] as follows. For each d is a constant matrix with all components zero except one which is the number 1. For each t\ G J the matrix M so constructed is non-singular and by continuity det M(t) ^ 0 for all t in some neighborhood Ntl of tx. Take P(t) = M'(t)Af"1(t) for t in Nh . (5.5) Any compact subinterval of J can be covered by a finite number of such neighborhoods ^V(l and hence P can be defined on J. On points which are covered by more than one such neighborhood P is multiply defined, we just choose one definition, say the one determined by the lower numbered neighborhood. Clearly, P G Lix(J) and Yit i = 1 , . . . , d, are solutions of (5.2). This completes the proof of the first part of (ii). To prove the furthermore part we note that the constructed matrix P is piecewise continuous by construction. Thus to get a continuous P on all of J we alter the parts of the construction above which removes the multiply defined aspect as follows : On a subinterval which is covered by two of the neighborhoods Nt discard both definitions for M used above -just on this subinterval- then
397 connect the two remaining pieces smoothly together and keep M nonsingular on J. Then define a P according to the new M as in (5.5) for t € J. This results in a continuous P and completes the proof of Theorem 5.1. I
References [1] A. Bielecki, Une remarque sur la methode de Banach-Cacciopoli-Tikhonov dans la the>ie des equations differentielles ordinaires, Bull. Acad. Polon. Sci. Cl. Ill 4(1956), 261-264. [2] W. N. Everitt and D. Race, On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations, Quaestiones Math. 2(1978), 507-512. [3] M. A. Naimark, Linear differential operators, Ungar, New York, 1968. [4] J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Math. 1258, Springer-Verlag, Berlin, 1987.
WSSIAA 3 (1994) pp. 399-415 © World Scientific Publishing Company
399
SUPERCONVEXITY AND INTERNAL COMPLETENESS Dedicated to Wolfgang Walter on his 66th Birthday HEINZ KONIG Fachbereich Mathematik Unlversitfit des Saarlandes D-66041 Saarbrucken, Germany
The present paper wants to relate two classes of convex subsets of real vector spaces: On the one hand these are the superconvex sets, studied after Simons [1972] by the author [1986] and some of his former students, and 1n Kdn1g-W1ttstock [1990]. The other class consists of the convex sets which are complete 1n the internal metric (or part metric), defined in Bear-Weiss [1967] as an abstraction of the famous GleasonHarnack metric in the theory of complex function algebras and studied in Bauer-Bear [1969], Bauer [1970], Bear [1970], and Bauer [1972]. Both notions have the flavour of Interior completeness; for example, the open unit ball of a Banach space 1s both superconvex and Internally complete. But it seems that the two concepts have not been related so far. We shall prove that superconvexlty implies Internal completeness. In the opposite direction one cannot expect natural results but for those convex sets which coincide with their interior, that means which consist of one part. For these sets superconvexlty will be proved to be equivalent to internal completeness plus a remarkable property 1n superconvex analysis called Internal boundedness. 1. Superconvex Sets We shall not consider abstract superconvex spaces but restrict oursel ves to the more concrete superconvex subsets of real vector spaces. We summarize from Kbnlg [1986] and Kon1g-W1ttstock [1990] the basic notions and facts. Define Q to consist of the sequences t=(t.), of real numbers CO
t.iO VleK with i
£ t.=1, and P of the i=(t.). 1n Q with t,=0 for almost ■ II
1=1
400 all leW. Let E be a real vector space. A nonvoid convex subset X<=E is called superconvex iff there exists a map I:QxXW-»X, written I(t,x)= I (t-.xj with x=(x 1 ) 1 €X N , 1=1 ' ' ' ' with the properties <s
I(t,x)= E t,x, whenever teP, 1=1 ' ' CD
CD
CD
CO
I vf t , I (t?, x j ] = I f E t t ?p, x , ) VteQ and tp€Q VpeW, p=1 P 1 = 1 ' ■ 1=1>1 ' " that 1s a superconvex structure on X which extends the natural convex structure of X. The basic uniqueness theorem due to Rode asserts that the superconvex structure I on X 1s unique. There are obvious examples: Let 5 be a Hausdorff vector-space topology on E. A nonvoid convex subset XcE 1s called a-convex 1n I iff, for any teQ and x=(x.). 1n x , the series
E t.x. converges 1n I to some member N I(t,x) of X. One verfles that the map I:QxX —»X thus defined is a superconvex structure on X In the above sense. 1= 1
1.1 Proposition. Let I be a Hausdorff vector-space topology on E. A nonvoid convex subset XcE 1s a-convex 1n I Iff 1t'1s superconvex and bounded 1n S. Then the unique superconvex structure I on X is as descri bed above. The question arises whether all superconvex sets are obtained 1n this manner. The answer which follows is the main result of Kbn1g-W1ttstock [1990]. 1.2 Theorem. Let X<=E be convex such that L1n(X)=E and hence convfxu(-x)JcE 1s absorbent. Let |-|:E—»[0,®[ denote the Minkowski func tional for conv[xu(-x)J. If X is superconvex then 1)
|'| is a complete norm on E;
i1) X is a-convex 1n (the topology of) |*|; 111) for any Hausdorff vector-space topology 5 on E we have: X 1s a-convex 1n X *•* 3c|.|.
401 It follows from the open mapping theorem that there is a unique com plete norm topology on E in which X is a-convex. We conclude with two important special cases which are obtained via Hahn-Banach separation. 1.3 Proposition. 1) Let E be finite-dimensional and I be the unique Hausdorff vector-space topology on E. Then each nonvoid bounded convex subset X C E is cr-convex in I. 11) Let E with |-| be a real Banach space. Then each nonvoid bounded convex subset XcE which 1s closed or open is a-convex 1n |*|. 2. The Internal Metric for Convex Sets Let E be a real vector space and XcE be a nonvoid convex subset. For u.veX we form X(u,v):={0
for u*v we have: d(u,v)=0 iff X contains the half-line
{u+t(v-u):t*o}; i1)
1-d(u,w) i (l-d(u,v))(l-d(v,w)) Vu.v.weX;
iii) for fixed aeX the function d(a,«):X—»[0,1] 1s convex; hence 1f X is superconvex then d(a,-) is superconvex; iv) for fixed aex the function (l-d(-,a))~ :X—»]0,o] is convex. Proof. 1) 1s obvious, ii) We can assume that the two factors on the right are both positive; then fix 0
with x.yeX.
It follows that w=<x0u+(1-«)Px+(1-0)y = aPu+(1-a0)z with some zeX,
402 and hence that d(u,w)£1-<x0 or 1-d(u,w)£a|3. i11) 1s a simple verification and known from K6nig-W1ttstock [1990] Proposition 1.1. 1v) The assertion means that the subset A:=[d(-,a)<1]cx 1s convex and that the restriction [l-d(',a)J~ |A 1S convex. Now fix u.veA with d(u,a)<s<1 and d(v,a)
for some x,y£X;
and furthermore w:=(1-x)u+Xv with 0<X<1. One computes that a =
(1-X)(l-t)+X(i-s) ((1-t)(1-s)w+(1-X)s(1-t)x+X(i-s)ty):
note that the sum of the three coefficients 1s =1. It follows that d(w,a)<1 or weA; and more precisely a) i i aiw.a; s ^ d(w
Q-t)(1-s) (i-x)d-t)+x(i-s)
0_ or
1 s IzX l-d(w.a) 1-s
+
J_ 1-f
This Implies the assertion. We next define the symmetric function D=Dx:XxX—»[0,1] to be D(u,v)= =Hax[d(u,v),d(v,u)j Vu.veX. A S an Immediate consequence we have 2.2 Proposition. The function D=D X has the properties I) for u*v we have: D(u,v)=0 iff X contains the line {u+t(v-u):teR}; II) 1-D(u,w) £ (l-D(u,v)](l-D(v,w)) Vu,v,weX. From 11) we see that D 1s a semlmetrlc on X. In view of 1) 1t 1s a metric on X Iff X 1s lineless. Instead of D the earlier papers quoted above used the equivalent distance function lc-gr^g with values 1n [0,®]. The function D appears to be natural since d is a Minkowski type functio nal; the Idea to use D Itself comes from my former student Hartmut Schfifer. One defines X to be Internally
complete Iff 1t 1s complete 1n the
internal semlmetrlc D. 2.3 Historical Example: The Gleason-Harnack Metric for Function Algebras. Let M be a nonvold set, and let B(M,R) and B(M,C) consist of the bounded real- and complex-valued functions on M. We fix a complex subalgebra A<=B(N,C) which contains the constants and separates the points of M. The classical example 1s the disk algebra A on M={zeC:|z|*l}. De-
403 fine the Gieason, Harnack, and H.A.Schwarz functions G,H,S:MxM—»[0,1] for A to be G(u,v) = | Sup{|f(u)-f(v)|:feA with |f|si>, H(u,v) = I n f | ^ ^ : f e A with Ref>o|, S(u,v) = Sup{|f(v)|:f£A with |f|*1 and f(u)=0} Vu.veM. It is obvious that G 1s a metric on M and that H(u,v)H(v,w)£H(u,w) Vu.v, weM. Gieason proved 1957 the basic nontrlvial result that G(u,v)<1 1s an equivalence relation on M, and Bishop proved 1964 that G(u,v)<1 ** H(u,v)>0. The present author then 1967 established the formula ,2
H - fJ=er. izs. " " U+GJ " 1+s' in particular H and S are symmetric. For all this we refer to BarbeyKbnlg [1977] Chapter III and to Bear [1970]. The connection with the in ternal metric is due to Bear [1965] and Bear-Weiss [1967]: The real li near subspace ReAcB(M.R) contains the constants and separates the points of M. We form the usual state space S(ReA):={
compact subset of (ReA) . We have the
canonical map M—>S(ReA), defined to be x€M»—»x€S(ReA): x(Ref) = Ref(x) VfeA. Under this map the function d=ds(._ ... is related to the previous func tions via the formula
Since the latter expression 1s symmetric, the Internal metric D=DC,D„... A A
A A
OlKeAJ
itself becomes 0(u,v)=d(u,v) Vu.veM. 3. Superconvexity implies Internal Completeness Let as before E be a real vector space and X<=E be a nonvoid convex subset. We come to our first main result.
404 3.1 Theorem. If X 1s superconvex then 1t is internally complete. Note that X is lineless after 1.1 and 1.2. The two notions involved are both such that we can assume L1n(X)=E. Thus by 1.2 there exists a complete norm |*| on E such that X is a-convex in |-|. Let us fix |-|. We need some technical preparations. We fix a sequence (£,). of real 00
00
numbers 0<e.<1 Vlefl with £ e
for Oix1
xr^1 VreH,
which has an obvious inductive proof. Now for 1£p£q we form
oq:= ^E = / 1 p
"
a
q
for p=q
I (i-c )...(i-E
1. ) for p
For fixed pen then «pi« for q—*». We need two more formulas. 1) For lew we have e
iVi = H'-Vta+i = v r ° v
It follows that l=p
2) For 1sp£l we have lp
l
v
1>P
PP
I p p J l p
pJ
It follows that °° 1 £ £,a' = 1-a VpeW. 1 P P l=p We turn to the main part of the proof. Let (u,), be a sequence in X which is Cauchy in the internal metric D. It suffices to show that some subsequence of (u,), is convergent 1n D. We fix a sequence of real numbers (.£,)■, as above and can thus assume that D(u 1 ,u l+1 )<e 1 VieW. From d(u 1 ,u l+1 )<e 1 and d(u 1+1 ,u 1 )<e l we have
405 u, +1 = (1-cl)ul + e a
with a.ex,
u1 = ( 1 _ e i ) u i+i + e i b i w1tn b i e x Vle*1) In view of the technical preparations 1)2) we have elements x ,y eX VpeH with VP =
Z
-TIT1
fl P
1=P '""p
and
n
£
Vn-~-
P
1 =P
41^_ 0"t
b
p
n-
P
with series convergent 1n |-|. It follows that 1-ap a p l-ap
*p
x
£ 1-a ,. = — P - b + P±l
y y
1
P
"
a p
P
where we have used VP
+ (1
1
-
p
E
p+1'
oo]e a e P b =- £ -b +
i=p+i
a = 1 £ a D ( ~ D) D+<
-VXP'- V P
= (1 € )a - P p+i u P +
+E
WiaP
1
"
i
i-«p P
1-a ,. Btl (1-e )y 1-p
(
P)yP+r
f ° r 1*P+1- Thus we have
PVi
a
P + "-IHI'VI
+ (1-a
l;(v (lH, p ,y J r H ( 1 - E P } V I
P+i)Vi +
VP
=
ViVi +
(H
pn ] Vi'
" V P - ( 1 "Vi ) ( 1 -VVi)
p+t
where we have used a = (1-e„)°< .. Therefore there exist xex and y£E such p P P+i that x = a p U p +(1-a p )x p and up=(1-ap)yp+ccpy Vpe,<11) We combine these equations to obtain
*=
a
P(
(1
+ (1
- V wp'j ) ~VV p"p p' p' x-y = -(1+ap)(1-ap)y ♦ « p (1-« p )y p + (1"« p )x p , and hence |x-y| i (1-« )(2|y|+2M)
VpeW,
where M>0 is an upper bound for J-| on X. For p — w we obtain y=xex. 1i1) From the final equations 1n 1) we now conclude that
406 d(u ,x) * 1-ot and d(x,u ) * 1-« , and hence D(u ,x)*1-<x VpeW. In view of « tl for p — » the assertion folP P P lows. The above theorem implies, once one assumes standard results in the superconvex theory, some of the previous sufficient conditions for internal completeness. For example, combined with 1.3.11) it furnishes Bauer [1972] Korollar 3. 4. The Interior for Convex and Superconvex Sets Let as before E be a real vector space and XcE be a nonvoid convex subset. For aex and xex we have d(x,a)<1
<-» there exists x>0 with a-t(x-a)eX,
which has an obvious Interpretation. In explicit terms we have d(x,a)
I(X) of X (in the sense of convex analysis) to be
I(X) =
407 Of course this implies aei(x). But for aei(X) we have often d(a)=1. For example, for the open interval X: = ]0,<»[cR the above equivalence shows that d(a)=1 Vael(X)=X. Furthermore we see from 2.1.11) that 1-d(u) * (l-d(a))(l-d(a,u))
Va.ueX.
Therefore d(a)<1 for some ael(X) Implies that d(u)<1 for all uel(X). If this is true then X will be called internally
bounded.
The next theorem has been superconvex folklore for quite some time. 4.3 Theorem. If X is superconvex then it is internally bounded. Proof. Assume that d(a)=1 for some ael(X). Fix some (t,), in Q with t.>0 VleN and choose a sequence (x,), in X such that 1 °° d U ^ . a ^ W t , VleW; then form x:= I ( t ^ x j e x . i
l
l
1=1
i
i
By assumption d(x,a)<1; hence there exist 0
408 5. The Converse Theorem Let E be a real vector space and XcE be a nonvoid convex subset. We need a certain construction which alms at Bauer [1970] Theorems 1 and 2. It will be phrased 1n the present terms; but since the substance 1s known we shall omit the explicit verifications. 5.1 Remark. X is absorbent Iff L1n(X)=E and 0€i(x). Assume that X 1s absorbent. Lettf:E—►[0,<»[be the Mlnkowski functional for X, that is «(x) = Inf{t>0:xetX} VxeE. It 1s known that [»<1]=I(X)cXc[*si]. we define e:E-»[0,»[ to be 0(x)=Max(*(x),a(-x)) VxeE; thus e is a seminorm. One verifies that 6 1s the Mlnkowski functional for Xn(-x). 5.2 Remark. The following are equivalent. 1) 6 is a norm on E; 11) X 1s Uneless; and 111) X contains no line which passes through the origin. The next lemma relates 9 and © to the distance functions d and D for X defined 1n §2. We restrict ourselves to the case X=I(X). 5.3 Lemma. Assume that X 1s absorbent and satisfies X=I(X). For u,x£X we have 1J
1+»(-u)
11)
Hu%*ln-u)
cuu,x;
1-*(u)'
* d(x ' u) * Hu%*-H»y
11D e(x-u£lk(-u) * D < u ' x > * f S l U ) ' 1v) 0(u,x)(l-»(u)) s O(x-u) * wlull]
or
and he ce
"
^"ivalently (l+»(-u)).
From 5.3.1) and 11) with u=0 we obtain the 5.4 Special Case. For xex we have i ) d(0,x)=*(x); and 11) d(x,0) = ffyly
or equivalent!/ 3(-x) = t _ j ] j * ; | ] j .
409 5.5 Proposition. Assume that X 1s absorbant and satisfies X=I(X). Define c:= Sup tf(-x) and C:- Sup 0(x) e [0,<»]. xex x£X Then the following are equivalent. I)
d(0)<1, that 1s X 1s internally bounded;
II) c<®; III) C«», that is X is bounded in ©. If this is true then d(0) - — r , or equivalently c - . i,L. Furthermore *(-x)ic«(x) VxeE. Proof. The equivalence 1)**ii) and the relation between d(0) and c follow from 5.4. The equivalence 11)«-»111) is clear since X=I(X)=[9<1]. For the last assertion note that *(x)
>(-W iC. The next result is Bauer [1970] Theorems 1 and 2. It follows from 5.3, except that for the Implication —» one has to invoke 2.2.11) in order to see that for a sequence (x,), 1n X which 1s Cauchy for D the numerical sequence [ D C X - . O ) ] . is bounded away from 1. 5.6 Proposition. Assume that X 1s absorbent and satisfies X=I(X). Then X is internally complete *■» E is complete in ©. Furthermore D an 8 define the same topology on X. We come to the main result of the present section. 5.7 Theorem. Assume that X=I(X) and that X 1s Uneless. If X 1s 1) in ternally complete and 1i) internally bounded then 1t 1s superconvex. Proof. We can assume that 0ex=I(X); and then L1n(X)=E. Then X 1s ab sorbent by 5.1, so that we are 1n the former situation. It follows from 5.2 and 5.6 that © is a complete norm on E, and X-I(X)=[S<1] shows that X 1s open 1n ©. By 5.5 X is bounded in ©. Thus 1t follows from 1.3.11) that X is a-convex in © and hence superconvex.
410 We conclude with examples 1n order to show that the two assumptions 1) and 11) 1n the above theorem are Independent. 5.8 Example. Let E=R and X=]0,<»[. Then X is lineless and open, and hence X=I(X). We have seen 1n §4 that X is not internally bounded. To see that X 1s internally complete one could compute
D(u v)
' = M i f e VU'V£X-
But it is simpler to pass to X=]-1,<»[ and Invoke 5.6. 5.9 Example. Let E = C 1 ( [ 0 , 1 ] , R ) with the usual C 1 norm |-|1, and let XcE consist of the feE with |f|<1. Then X is convex and absorbent, and lineless (but not bounded!). Furthermore X is open 1n |-| , and hence X=I(X). In the former notations we have e=tf=|-| , the usual C
norm. Thus
X is internally bounded by 5.5, but by 5.6 1t 1s not internally complete. 6. Open Convex Sets in Real Banach Spaces The former results 5.2 and 5.6 have the consequence below, which has been implicit in the proof of 5.7 and 1n Bauer [1970] Theorems 1 and 2. 6.1 Proposition. Let XcE be a nonvold lineless convex subset of a real vector space E, and let also X=I(X) and L1n(X-X)=E. If X 1s internally complete then there exists a complete norm |-| on E such that X 1s open
in H It is therefore natural to ask: Assume that E with |-| 1s a real Ba nach space and XcE is a nonvoid Uneless convex subset which 1s open (and hence fulfills X=I(X) and L1n(X-X)=E). When 1s X Internally complete? Example 5.9 shows that this need not be true. We shall present an answer which has a certain relation to Bauer-Bear [1969] Section 3. 6.2 Theorem. Under the present assumption the following are equiva lent. 1) X 1s internally complete. ii) Each continuous linear functional
411 difference *>=oc-0 of linear functlonals a,/?eE which are bounded above on X (which of course Implies that ot,0€E'). Proof. We can assume that Oex. Then X 1s absorbent; thus we can form 9 and 9 as 1n §5. By 5.2 0 1s a norm on E. 0) There exists c>0 such that 6sc|»|. In fact, since X 1s open and Oex there exists e>0 such that for xeE:|x|se -» xex -» *(x)<1, and hence *^|*|
and also 0*£|*l-
D + 1 1 ) By 5.6 8 Is a complete norm on E. Hence by 0) and the open map ping theorem there exists M>0 with I'l^MS. Thus for
« * M | - | * M|
412 7. The Part Decomposition of Convex Sets Let E be a real vector space and X C E be a nonvold convex subset. In view of 2.2.11) the relation D(u,v)<1 1s an equivalence relation on X; the equivalence classes are called the (Gleason) parts of X. For aex let Gl(a)=Glx(a) denote the part of X which contains a. Thus Gl(a) = [d(a,-)<1]n[d(-,a)<1]
V a ex,
which combined with 2.1.i11)1v) shows that the parts of X are nonvold convex sets as well. In the sequel we have to return to the twofold no tation d=dx and D=D X . 7.1 Well-known Facts. 1) The parts of X are open 1n D x and hence also closed 1n D x < 11) For aei(x) we have Gl(a)=[d(a,')<1]=I(X) from 4.2. Hence 1f I(X) *0 then I(X) is a part of X. 111) X consists of one part Iff X=I(X). 1v) For nonvoid convex Acx we have d.£dv and D.*D V on A. If A 1s a
part of X then part of X 1s then d.=d„ and Itself one-part. A itself 1s one-part.
A X i 1s a Apart X of X then the set n
n A= x
on A. v) If A 1s a part of X then the set
1n X 1s Cauchy for D x then 1t 1s ultimately
v1) If a sequence (x,)-, 1n X 1s Cauchy for D x then 1t 1s ultimately 1n some part A of X. If (x,), converges to xex 1n D x then xeA as well. v11) X is internally complete Iff all parts of X are Internally complete. The main point in the present section will be to consider the parts of the subsets P and Q of 1 defined 1n §1. 7.2 Lemma. Let X=P or Q. For u,v£X then 1) dx(u,v)<1 «-♦ u£Rv for some real R>0; 1i) Dx(u,v)<1 «■♦ u*Rv and v^Ru for some real R>0. Here * means the componentwise partial order on 1 . Proof. We have to show 1). —►) Let d¥(u,v)
413 7.3 Proposition. 1) We have for aeP:
Gl (a)=Glp(a)- {u£P:a*Ru and u£Ra for some R>0} cp ;
for a€Q\P: Gln(a)= {u€Q:a£Ru and u£Ra for some R>0> <=Q\P. w H ) For a£Q the superconvex closure sconv(Gl_(a)J, defined to be the smallest superconvex subset of Q which contains Gl_(a), consists of the u€Q such that (*)
a*Ru for some real R>0, and u,=0 for all leW for which a,=0.
iii) We have for aeP:
GlQ(a)=Gl_(a)cp 1s superconvex;
for aeQ\P: Gl-.(a)cQ\P is not superconvex. y Proof. 1) follows from the above lemma, ii) Let HcQcl
consist of the
ueQ with (*). 1i.1) From 1) we see that Gl-.(a)cH. 11.2) An Immediate u 1 verification shows that H is a-convex 1n the 1 norm and hence supercon vex. Therefore sconv[GlQ(a)JcH. It remains to prove the converse Inclu sion. 11.3) To see this fix u€H; thus there exists 0<s<1 with u*(1-s)a. Then v:=-[u-(1-s)aj€Q; and we have v.=0 for all lew for which a,=0. Now from i) we see (1-s)a+seneGl-(a) for all ncW with a * 0 . u n It follows that 00
00
E vn((1-s)a+sen) = n=1
£ vn((1-s)a+sen) e sconv[GlQ(a)). n= 1 a
n*°
But the left side 1s =(1-s)a+sv=(1-s)a+fu-(1-s)aJ=u; hence we obtain uesconv(GlQ(a)J. H i ) The first assertion is obvious from i) and 11). In order to prove the second assertion we quote a well-known fact form the elementary theory of infinite series: For any a=(a,), 1n Q and real X>1 there exists a se quence (oOi of real numbers a ii View such that 00
a.foo and
£ a a.-X; 1=1 ' see for example Konig [1984] Aufgabe II.3.18. In the present context we 1
414 can take any X>1. Then u^jja.a.j. is a member of Q which 1s 1n sconvfGlQ(a)J by i1). But we have u*Gl_(a); since otherwise there were an R>0 with uiRa, that is «,sXR for all leW with a,*0, which 1s 1n contra diction to a£Q\P. 7.4 Consequence. We have I(P)=I(Q)=0. Proof, i) To see I(P)=0 we need to know that for each aeP there exists ueP with dp(u,a)=1, that 1s such that u^Ra for no R>0. This 1s ob vious. 11) To see I(Q)=0 we need 1n addition that for each aeQ\p there exists ueQ with dQ(u,a)=1, that 1s such that u^Ra for no R>0. But any uesconv[GlQ(a)]\GlQ(a) does this. Thus P<=l
1s an example of a convex set which 1s not superconvex, but
all parts of which are superconvex. On the other hand Qcl
1s an example
of a superconvex set of which some parts (1n fact, all nontrlvlal parts) are not superconvex. This has to be compared with the respective behav iour of Internal completeness, as expressed 1n 7.1.v11). It follows that beyond the frame of one-part convex sets no natural connection between superconvexIty and internal completeness can be expected. From the above even more horrible examples could be constructed. References Barbey Klaus and Heinz Konig [1977] Abstract Analytic Function Theory and Hardy Algebras. Lect.Notes Math.593, Springer Verlag. Bauer Heinz [1970] An Open Mapping Theorem for Convex Sets with only one Part. Aequationes Math.4,332-337. Bauer Heinz [1972] Intern vollstandlge konvexe Mengen. In: Theory of Sets and Topology (1n honour of Felix Hausdorff 1868-1942),pp.27-38, VEB Deutscher Verlag der Wlssenschaften. Bauer Heinz and H.S.Bear [1969] The Part Metric 1n Convex Sets. Pacific J.Math.30,15-33. Bear H.S. [1965] A Geometric Characterization of Gleason Parts. Proc. Amer.Math.Soc.16,407-412. Bear Herbert S. [1970] Lectures on Gleason Parts. Lect.Notes Math.121, Springer Verlag. Bear H.S. and M.L.Weiss [1967] An Intrinsic Metric for Parts. Proc.Amer. Math.Soc.18,812-817.
415 Konig Heinz [1972] SubUneare Funktionale. Arch.Math.23,500-508. Konig Heinz [1984] Analysis 1. UTB BlrkhSuser Verlag. Konig Heinz [1986] Theory and Applications of Superconvex Spaces. In: Aspects of Positlvity 1n Functional Analysis (Tubingen 1985), pp.79-118, Math.Studies 122, North-Holland. Konig Heinz and Gerd Wittstock [1990] Superconvex Sets and a-convex Sets, and the Embedding of Convex and Superconvex Spaces. Note di Mat.10 Suppl.2,343-362. Simons S. [1972] A Convergence Theorem with Boundary. Pacific.J.Math.40, 703-708.
WSSIAA 3 (1994) pp. 417-430 © World Scientific Publishing Company
417
HARDY INEQUALITIES OF FRACTIONAL ORDER VIA INTERPOLATION
Alois Kufner Mathematical Institute, Academy of Sciences Zitna25, 115 67 PRAGUE, CZECH REPUBLIC and
Lars Erik Persson Department of Mathematics, Lulea University S-971 87 LULEA, SWEDEN
ABSTRACT One-dimensional Hardy inequalities can be interpreted as continuous embeddings between (homogeneous) weighted Sobolev spaces and weighted Lp-spaces. Inspired of an inequality of Grisvard (see also Kufner&Triebel ) we want to create a scale of fractional order Hardy inequalities , where a Hardy type inequality is one endpoint estimate and where some trivial or well-known embeddings give the other endpoint estimate. We present an interpolation technique to obtain such scales of inequalities. Some examples of inequalites obtained in this way are proved and discussed. The relations to other similar results are pointed out.
1. Introduction First we consider the following inequality by Grisvard4 (see also Triebel19): I x-*P 1 f(x)IPdx
(1.1)
where C > 0 is independent of f, p > 1, 0 < A. < 1, X * 1 / p and f e C^(0,oo).
418 The results in this paper show in particular that this inequality can be interpreted as an "intermediate" inequality between the classical Hardy inequality (see Th. 330 in Hardy et al. 5 ) and the trivial embedding L c L . Therefore we may say that (1.1) is a fractional order Hardy inequality. For our discussions later we note that (1.1) can be rewritten as r
p
" " lf(x+tH(x)l p JL ^ 7 -xplfMiP, ^7f7'«*«>-frO' dx.
j x ^lf(x)l p dx£Cj
i
r-z
dt dx.
(1.2)
1+
0 | t | *P order) Hardy inequalities of the Moreover, some versionso^-x of fractional type ("""■ I tfxWM I p " 1 / w0(x) If(x)I p dx
Kufner& Triebel 8 . In this paper we work with functions f
belonging to less
restrictive function spaces than C^XO,**). We prove some weighted inequalities of the type in (1.3) but without some weighted L terms on the right hand side and not necessarily with the same parameter p on both sides of the corresponding inequality. We present an interpolation technique to create such scales of inequalities with a Hardy type inequality as one endpoint estimate and with a certain trivial or wellknown embedding as the other endpoint estimate. Some examples of inequalites obtained in this way are proved and discussed. The relations to other similar results are pointed out. This paper is organized as follows: Section 2 is reserved for some preliminaries. In Section 3 we prove a weighted version of (1.1) yielding for functions f E L-,(0,°O) and where we can even permit some negative values of the parameter X (see Proposition 1). In Section 4 we briefly discuss the close connection between interpolation and inequalities. In particular, these ideas imply a simple technique to prove inequalities and in particular fractional order Hardy type inequalities. In Section 5 we present and discuss a general fractional order Hardy inequality for differentiable functions on an interval [a,b], -°° £ a < b < ~>, and give some concrete examples of such inequalities in terms of classical function spaces (see Propositions 2 and 3). Finally, in Section 6 we present some concluding remarks and some possibilities to generalize
419
our results. In particular we point out some results obtained by using higher order Hardy inequalities (see e.g. Kufner7 and Opic&Kufner13) as one endpoint estimate and some results obtained by using descriptions from real interpolation in the cumbersome off-diagonal cases which can be done even in the parameter function case (see Maligranda&Persson12 and Persson15). Some concrete examples of such results are sketched or proved (see e.g. Proposition 4 and (6.5)).
2. Preliminaries a) Conventions. In this paper p, q, r, p', a, X, a and b denote real numbers satisfying 1 < p,q,r < «>, p'= p/(p-l) (p'= 1 if p = °° and p ' = <*> if p = 1), 0 < a,\ < 1 and -°o < a < b < <». C denotes a positive constant not the same at different occurences. w, w0 and w5 denote positive (weight) functions on the interval (a,b). b) On some function spaces. As usual Lp(w) = Lp(w;[a,b]) denotes the weighted Lebesgue space defined by the norm b
l|f||p,w = (Jlf(x)|Pw(x)dx)1/p, and with the usual supremum interpretation for the case p = °°. The weighted homogeneous Sobolev space W* = Wpfa.bl consists of all differentiable quasinorm|f||
functions
1 p;w
f on [a,b] with
the
corresponding
= ||f'|| p;w .The "intermediate" scale B° q = Bpq[a,b] of
homogeneous Besov spaces is defined by the quasinorm
|£|- q = (j(f*a V (t f » q f) 1/q ' where co (t;f) is the usual integral modulus of continuity C0p(t,f) = sup{ || f(x+h)-f(x) || L [ a b ] : x,x+h € [a,b] and Ih I < t}. It is well-known that there are several equivalent norms on the Besov spaces (see e.g. Taibleson's Thesis 18 and the books of Bergh-Lofstrom 1 and Triebel19). In this paper we shall use the remarkable fact that the Besov norm above is equivalent to the norm
420
If IISq = (j(J l«x4h)-«x)|Pdx) ,,/p -^ L ] 1/q
(2.D
p s
' [o -~ h q) for a n y 5 > 0 (f = 0 outside the interval (a,b)). This fact is a direct consequence of the theorem on page 189 in Triebel 19 (c.f. also Th.4, p. 421-422 in Taibleson 18 ). c) On Hardy type inequalities. Such inequalities have been intensively studied during the last decades (see e.g. the book of Opic&Kufner 13 ). In this paper we only need the following t w o examples (see e.g. Opic&Kufner 13 , formula (6.4), and Th. 1.14, respectively): (HI) Let g be a function on [0, °°). Then J w0(x) I Jg(t) dt Ip dt £ c p JWl(x) I g(x) I p dx O
x
(2.2)
0
if and only if sup (jw0(t) dt)(/(w 1 (t)) 1 " p 'dt) p " 1 = Co < o o . x>0
0
x
(H2) Let 1 < p ^ q < oo and let f be a differentiable function on the interval [a,b] such that f(a) = 0. Then
llfll^cllfllp,^, if and only if b
x
sup (Jw0(t) dt) 1 / q (J(w 1 (t)) 1 - p 'dt) 1/p ' = q < - . aixsb x
(2.3)
a
d) On Peetre s K-spaces (of interpolation): Let (AQ ,AJ) be a compatible Banach couple , i.e., AQ and A 1 are Banach spaces which both are embedded in a (large) Hausdorff topological vector space. Then the spaces (A^Aj)^ can be defined by using the Peetre K-functional which is defined for any f e AQ+AJ as follows:
K(t,f) = K(t,f,A0,A1) = inf {II f0 II A + t II ix I I A : f = f0 + iv f0 € A^f, e A^. The norm II • II.
of the space (A^Aj)^ is given by
*x*«> = ifIIx, q =[i(t"*K(t^)) q T J
and with the usual modification for the case q = °°.
(Z4)
421
A continous function p: (0,°°) -»(0,°°), is said to belong to the class T0 if p(t) = sup , . < <»> for every t > 0. s>o p(s) Introducing the Matuszewska-Orlicz indices a p and Pp of p e r o as log p(t) log p(t) <xpn = hm—; and Kp p„ = urn— , t-rfk logt ,-*- logt we say that p e r* if 0 < a < Pp < 1. Now, if we replace the function f* in (2.4) by a more general parameter function p = p(t) e T* , then we obtain the more general parameter function spaces (AQ/AJ) This 16 construction is well-studied (see e.g. Persson and the references given there). We also remark that an even more general construction, where the function norms ®x or <J> were replaced by a more general one, was introduced already in 1963 in a fundamental paper by Peetre14. All these spaces are (real) interpolation spaces in the sense discussed in Section 4.
3. A weighted version of (1.1) for f e LjCO/w) First we generalize (1.1) in the following way: PROPOSITION 1 Let p > 1 and X > - 1/p. Further, let i e L^O,-) and let w0 and w, be weight functions on (0,~>) satisfying t
-
sup (|W0(T) dx)(J(w1(x))1"p dx)p-1 < ~ .
(3.1)
Then J I f(t) IP Wl(t) dt < C ? J ' f ( t H ( T ) l W (t) dx dt, (3.2) 1+ o o o 1 1 ^ 1 *P x where W(t) = t P w0(t) + t^-^P w/t), C = ^P" 1 ' max(l,c p ) and cp is the constant from (2.2). Proof: First we use an idea by Grisvard4 (c.f. also p. 261 in Triebel19 ) and define 1* 1* g(t) = f(t) - - / f(x) dx = - J [f(t)-f(x)] dx. (3.3) ° to to
422 Obviously, g(°°) = f(°o) and t
t2
o
t
V
t *
/
and we conclude that f(t) =
gW
•g
Therefore, by using the inequality I a+b I p S 2P (I a I p+ I b Ip), the assumption (3.1) and Hardy's inequality (2.2), we obtain that J l«t)|Pw 0 (t)dt^2M/ lg(t)|Pwo(t)dt + j w 0 ( t ) l i ^ d x | p d t Is vO
0
0
t
^
/
2 2P"f i I g(t) I p w0(t) dt + c, JWl(t) I ^ I p dt) = 2P"1 J I g(t) I p W0(t) dt, r VO 0 t ) 0 where W0(t) = w0(t) + c t^w^t). Hence, if we denote W/t) = w0(t) + rPwj(t) and use (3.3), we find that J1 f(t) Ip w0(t) dt £ C /1 g(t) I p W^t) dt = ,t
-
= C J | —/ [f(t)-f(x)] dx | p W^t) dt. (3.4) o to Furthermore, by Holder's inequality and our assumption that 1 + Xp £ 0, we find i t
t
I -i [f(tH(x)] dx I p < r
0
t
^
p
t^ 1 11 f(tH(x) I p dx = 0 |M:I1+Xp
The estimate (3.2) follows by combining (3.4) with (3.5) and the proof is complete. COROLLARY 1 Let p > 1, X > - 1/p and a > Xp - 1. Iff e L^O,*"), then lf(tH(T)|P ]\{(t)\pta-^dt
where C = C a A p = 2P_1(1 + p/(a-Xp+l)).
(3.6)
423
Proof: The condition (3.1) is fullfilled for w 0 (t) = tP and Wj(t) = tP+P , p" > - 1 , and in this case we have W(t) = 2 tP+>-P and, according to the classical Hardy inequality, c = p/((3+l). Therefore, by denoting a = fJ + Xp, we obtain (3.6) with the constant 2 p (max(l, p / ( a - X p + D ) . However, it is obvious from the proof of Proposition 1 that (3.6) also holds with the somewhat smaller constant C a ^ . Remark If a <, X.p-1 and if f(0) > 0 and f is continuous at 0, then the integral on the left hand side of (3.6) diverges to infinity and, thus, the restriction a > Xp -1 is necessary. Moreover, according to Corollary 1 applied with a = 0, we see that (1.1) holds for - 1 / p <. X < 1/p for any f e L1(0,°°). Both of the restrictions - 1 / p £ X and X < 1/p are necessary.
4. Interpolation and inequalities The first interpolation proof of an inequality (Haussdorff-Young's inequality) was given already in 1926 by M. Riesz 17 . He wanted to find a simple proof of the Haussdorff-Young inequality and this was the main reason to prove the famous convexity theorem of Riesz. Nowadays it is well-known that also most of the other classical inequalities (e.g. those by Paley, Young, Holder, Minkowski, Beckenbach-Dresher, Clarkson, Carlson, Grothendieck etc.) can easily be proved by using interpolation techniques (see e.g. Maligranda&Persson 1 1 and the references given there). The aim of this section is to discuss in general terms why interpolation theory nowadays is a very powerful tool to prove and explain classical or new inequalities e.g. the fractional order Hardy inequalities studied in this paper. Let (A^Aj) be a compatible Banach couple. The Banach space A is called an intermediate space between A 0 and Aj if A 0 n A ! c A c AQ+AJ _
Now let (A^Aj) and (B^Bj) be two compatible Banach couples. Then the spaces A and B are said to be interpolation spaces with respect to (A^Aj) and ( B Q ^ ) if A and B are intermediate spaces and if, for any bounded linear operator T, such that T: A0-> B0 and T: Aj-> Bj it holds that T: A-> B. This can be written in terms of inequalities in the following way:
424
IF II Tf IIB <M 0 lflL
and II Tf IIB < M1II f IIA , then
II Tf IIB < GCMQ^M,)) II f II A ,
(4.1)
where G(x,y), x,y > 0, is a positive and bounded function. Nowadays there exist many methods to construct interpolation spaces but the most studied and applied methods (functors) are the (Lions-Peetre) real method (v)*. „ , 0 < X . < 1 , l £ q < » (see our Section 2 (d)) and the complex method [•/ \ , 0 £ X < 1. If we want to get the best constant it is suitable to use the complex method (where G(x,y) = x1-Xy*-) or some other exact interpolation method. For our purposes it is most suitable to use the real method (v)i q arid its generalization (,) to the parameter function case but similar statements hold also for the other interpolation functors. For the real method we also have G(x,y) = x1-*y*in (4.1) and thus (4.1) implies in particular the following technique to prove (fractional order) inequalities: If II Tf IIB < MQ II f IIA and II Tf IIB < M1II f IIA , then
llTfLRR,
SMoUM,»-llfLA
A
v .
(4.2)
The power of this technique depends heavily on the fact that nowadays we can identify the spaces (A^AA and (the parameter spaces) (A0,Aj) in a lot of cases of practical importance. See e.g. the books by Bergh&Lof Strom1, Brudnyi&Krugljak2 and Triebel19 and, in particular, the Bibliography of Maligranda10 induding more than 2000 references.
5. On a general fractional order Hardy inequality In view of the discussion in Section 4 we observe that the Hardy inequality (H2) can be interpreted as a bounded continuous embedding between the homogeneous Sobolev space wl/w,) and LJw,,) ,i.e., that for allfe Wptw,),
llfll.Lq(w , 0).SMjfll *<>
!, ,. Wp(w,) Therefore, if we can find Banach spaces B0 and AQ which are compatible
425
with L„(w0) and w l / w j ) , respectively, and if AQ is continously embedded in B0, i.e., if II f IIR < M, II f II . , then, according to (4.2) applied with the embedding operator, we find that the condition (2.3) implies the following general fractional order Hardy inequality :
llflL_ . , „ < M°0uM,MlflL ,, » ■ (B^WQ))^ i (A^w'pfw,))^
(5.1)
First we consider this estimate for the simple case when At, = B0 = L p (wj). The following facts are well-known: (a) For the diagonal case r = p x , l / p x = (l-X)/p + X/q it yields 1 ' 19 ( L p t w ^ w , , ) ^ px = Lpx(vtx) ,where w x = (w 1 ) p * ( 1 - W / p (b) The off-diagonal
(v,/*'*.
cases r * px are much more cumbersome to
15 12
handle but in the next section we will discuss this case too even for the more general parameter function case. (c) For the unweighted case w 1 = 1 it is well-known that 1 - 19 (LpWp^BpV (d) The spaces (Lp(w1),W1p(w1))^r are not possible to describe in terms of classical function spaces for the general weighted case. However, some descriptions of this kind are presented by Ditzian&Totik 3 , Ivanov 6 and Lofstrom 9 but we will not continue in this direction in this paper. In particular, by using the Hardy inequality (H2), our general fractional order inequality (5.1), (a), (c) and the description in (2.1), we obtain the following statement: PROPOSITION 2 Let I < p <, q < ~, l / p ^ = (l-X)/p + X/q, 0 < X < 1, and let f be a differentiable function on [a,b] satisfying f(a) = 0. // b
sup (jw 0 (t)dt) 1 / q (x-a) 1 / p ' =C, < ~ , a£x5b x
then , for any 8 > 0, b
8
b
JI f(x) I PHw 0 (x))^ / q dx < Cjr X P^( J I f(x+t)-f(x) IP d x ) ^ / p d t . a
0
a-t
(5.2)
426 Example 1 If p = q, then p x = p and (5.2) reads JI f(x) I PHw0(x))" dx <; C if J a
lf<X t) f(x)l
't "
O^a-t
^P+
dxl dt.
(5.3)
J
This is a "fractional" version of the Hardy inequality b
b
Jlf(x)l p w 0 (x)dx:£Cjlf'(x)l p dx. a
a
In particular, if a = 0, b = «>, 8 = oo and w0 = x"P, then (5.3) essentially coincides with the Grisvard inequality (1.2). Next we apply (5.1) in another simple situation namely when AQ = B0 = w i • We recall the following well-known facts1,19: (a)'
( W ^ w j j w , ) ) ^ = W ^ w , ) , where w x = (w 1 ) p * (1-W/p .
(c)'
( W ^ L q ^ p , = (Lq,Wq)x;Px = B ^ .
Therefore, by using again the Hardy inequality (H2), our general fractional order inequality (5.1), (a)', (c)' and the description in (2.1), we obtain the following statement (which, in a sense, is dual to the statement in Proposition 2): PROPOSITION 3 Let\ < p < q < °», 1/ P j l = (l-X)/p + X/q, 0 < X < 1, and let f be a differentiable function on [a,b] satisfying f(a) = 0. // X
sup (b-x)1/q (J(w1(t))1_pdt)Vp' =Cj
a
then , for any 8 > 0, 6
b
b
lt-*-P>r\ j | f(x+t)-f(x) I <> dx)^ / q dt <. C /1 f'(x) I PKw1(x))(1_Wpx/p dx. (5.4) 0
a-t
a
Example 2 If p = q, then p^ = p and (5.4) reads
O^a-t
t^1
J
a
This is again a "fractional" version of the Hardy inequality b
b
Jlf(x)l p d x S C j l f ' t o l P w ^ d x . a
a
427
Remark: Obviously, we can in the same way prove some variants of Propositions 2 and 3 for functions f satisfying f(b) = 0.
6. On some generalizations and concluding remarks First we recall that we have assumed that Wj = 1 in the version of Hardy's inequality which was an endpoint estimate of the inequality formulated in Proposition 2. However, the result in Proposition 2 can be used to obtain fractional order Hardy inequalities also for a general weight w, which can be seen in the following way: By using the substitutions X
x = g(y) with y = J(w,(t)) 1_p ' dt = g_1(x) a we can rewrite the Hardy inequality (H2) into the form A
A
(J I U(y) I q W0(y) d y ) V q < C (J I U'(y) I p dy) 1 / p 0
(6.1)
0
with U(y) = u(g(y)), W0(y) = w0(g(y)) g'(y) and A = Jj(w1(t))1~p dt • Now we can apply Proposition 2 with (6.1) as endpoint estimate and, finally, return to the initial case by making an inverse substitution. Our next goal is to point out generalizations in two other directions namely to get rid of the restriction r = p^ in Propositions 2 and 3 by also interpolating in the more complicated off-diagonal cases (see (5.1) for the case (b)) and to use the more general interpolation with a parameter function (see Section 2). Let p = p(t) € T*. Then, under the hypothesis implying (5.1), we can use the interpolation theorem in the case of interpolation with a parameter function (see e.g. Th. 2.2 in Persson 16 ), to obtain the following more general fractional order Hardy inequality: (6.2) Again we let AQ = B0 = L p and recall the following facts: (A) According to Th. 4 in Maligranda&Persson 12 it yields that (L p ,L q (w 0 )) pr = L?'(w2,wdx) with
428
w 2 = {wjV«rP\ w = ( w ^ P - ^ and
(6.3)
and where L|?(w2,wdx) is a generalized Lorentz space defined by the norm
here f® means that the rearrangement is done with respect to the measure wdx. (B) According to the well-known fact that K(t,f;L^wl) => to (t;f) we have tLp,W1p)p^=BPr, where B^ can be characterized in terms of the seminorm
Now, by using (H2), (6.2) and (A) and (B) above we obtain the following statement : PROPOSITION 4 Let 1 < p <, q < «° and r £ 1. Let p e B0 and let w, w 2 and cp be defined by (6.3). // f is a differentiable function on [a,b] satisfying f(a) = 0 and if b
sup (jw0(t) dt) 1/q (x-a)Up' = C, < - ,
aSxSb x
t/ien
For the case when r = px and p(t) = t* Proposition 4 coincides with Proposition 2. Moreover, an analogous generalization of our Proposition 3 can be proved in a similar way. Next we recall that the first order parameter X in Proposition 2 is less then one and that X = 1 corresponds to the Hardy inequality (H2). Therefore in order to be able to create fractional order Hardy inequalities with X > 1 we must use higher (but integer) order Hardy inequalities as one endpoint estimate. Such inequalities of the type
429 b
b
(JI f(x) 11 w 0 (x)dx) 1/q SCC/I f ^ W I pwk(x) d x ) 1 / p , a
(6.4)
a
with k = 2,3,... , are known for various classes of functions (see e.g. Opic&Kufner13 and Kufner7). In particular, by interpolating (with the real method) between (6.4) and the estimate equipped with the trivial embedding L c L we obtain, exactly as in the proof of Proposition 2, that the fractional order Hardy inequality b
(J I ftx) I * (w 0 (x))^ / q dx) 1/p * < C I f || B* a
(6.5)
P'Pl
holds, where the (semi)norm on the right hand side can be expressed in many different equivalent ways e.g. in classical terms similar to those in our Propositions 1-3. A similar generalization of our Proposition 3 can be done in an analogous way. Moreover, we can make the other generalizations described above also in this more general setting. We aim to develop these ideas further in a forthcoming paper .
References 1. J. Bergh and J. Lofstrom, Interpolation Spaces-An Introduction, Berlin-Heidelberg-New York, 1976. 2. Yu. A. Brudnyi and N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, North-Holland, 1991. 3. Z. Ditzian and V. Totik, Moduli of smoothness, SSCM 9, Springer Verlag, New York, 1987. 4. P. Grisvard, Espaces intermediaires entre espaces de Sobolev avec poids, Ann. Scuola Norm. Sup. Pisa 23 (1969), 373-386. 5. G. H. Hardy, J. E. Littlewood and G. P61ya, Inequalities, Cambridge University Press, 1988 (1934). 6. K. Ivanov, A Characterization of Weighted Peetre Kfunctionals, /. Approx. Theory 56 (1989), 185-211. 7. A. Kufner, Higher order Hardy Inequalities, Bayreuth Math. Schr. 44 (1993), 105-146. 8. A. Kufner and H. Triebel, Generalizations of Hardy's inequality, Conf. Sem. Mat. Univ. Bari, 156, 1978.
430
9. J. Lofstrom, Interpolation of Weighted Lp and Sobolev spaces on Intervals, Research report 4, Dept. of Math., Chalmers University of Technology, 1984 (ISSN 0347-2809). 10. L. Maligranda, A bibliography on "Interpolation of operators and applications" (1926-1993)", second ed., Dept. of Math., Lulea University, 1993. 11. L. Maligranda and L.E. Persson, Inequalities and Interpolation, Collectanea Math, (to appear) 12. L. Maligranda and L.E. Persson, Real interpolation of weighted L p - and Lorentz spaces, Bull. Polish Acad. Sci. Math. 51 (1989), 765-778. 13. B. Opic and A. Kufner, Hardy-type inequalities, Longman Scientific & Technical, Harlow, 1990. 14. J. Peetre, A theory of interpolation of normed spaces, Lecture Notes, Brasilia, 1963 (Reprint in Notas de Mat. 39, Rio de Janeiro, 1968). 15. L.E. Persson, Descriptions of some interpolation spaces in off-diagonal cases, Lecture Notes in Math. 1070, 1984, 213-231. 16. L.E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986), 199-222. 17. M. Riesz, Sur les maxima des formes bilin^ares et sur les fonctionelles Unbares, Acta Math. 49 (1926), 465-497. 18. M. H. Taibleson, On the Theory of Lipschitz Spaces of Distributions on Euclidean n-Space I. Principle Properties, Journal of Mathematics and Mechanics 13 (1964), 407-479. 19.H. Triebel , Interpolation Theory, Function Spaces , Differential Operators, North-Holland, VEB Deutscher Verlag der Wissenschaften, Berlin 1978; Russian transl., Mir, Moscow 1980.
WSSIAA 3 (1994) pp. 431-436 © World Scientific Publishing Company
431
Dynamic systems on time scales and Super-linear convergence of iterative process. S.Leela SUNY,College at Geneseo Geneseo, NY 14454 S.Sivasundaram Embry-Riddle Aeronautical University Daytona Beach , FL 32114 1. Introduction. The wellknown method of quasilinearization [2,3] has recently been generalized and extended [6,7] so as to apply to large class of functions. In this paper , we develop a method which isolates the ideas imbedded in the method of quasilinearization so that it can apply to a variety of problems. To do this , we shall utilize the dynamic systems on time scales [1,4,5] which unifies the theory of discrete as well as continuous dynamical systems. For this purpose , we shall first investigate existence results for dynamic systems on time scales in a sector generated by upper and lower solutions and then using such a result, we prove the superlinear convergence of monotone iterative process of dynamic systems on time scales, we have given examples to illustrate the generality of our results. 2. Preliminaries. Let T be a time scale with f0 £ 0 as a minimal element and no maximal element Since a time scale T may or may not be connected , we need the concept of jump operators. Definition 2.1 The mapping £,T|: T -> T such that £(r) = inf[seT:s>t) and r\(t) = s\ip{seT:s
432
Continuous =* rd-continuous. If T contains ldrs-points (left-dense and right scattered) then the first implication is not invertible. However, on a discrete time scale the two notions coincide. Definition 2.4 a mapping V:T —» X ( X is a Banach Space) is said to be differentiable at t e T, if if there exists an a e X such that for any e there exists a neighborhood V of t satisfying \V(&f))-V(s)-(£(t)s)a\ <; %(t) - s\ for all s in V. We denote the derivative of V by VA(t). We note that if T = R,a = —^ dt if T = Z,a = u(t+\)-u(t).
and
The following properties of the derivative are needed. (PI) If V is differentiable at t .then it is continuous at t; (P2) If V is continuous at t and t is right-scattered,then V is differentiable and V*(t) = -
Ji'W
Definition 2.5 Let h be a mapping from T to X. The mapping g.T—tX is called the antiderivative of A on 7" if it is differentiable on T and satisfies gA (r) = h(t) for ; e 7". The following known properties of the antiderivative are useful. (a) If h:T —» X is rd-continuous, then h has the antiderivative #:? ->\h(s)ds, s,teT. (b) If the sequence { X } ^ of rd-continuous functions T-> X converge uniformly on [r,s] to the rd-continuous function h then (\hn(t)dt)
-> \h(t)dt, in X.
Definition 2.6 The mapping g:T xX —»X is rd-continuous if (1) it is continuous at each (t,x) with right-dense or maximal t, and (2) the limits g(t~,x) = lim g(s, v) and limji>(r, v)exist at each(r,x) with left-dense / . (I.JO-K'".-0
y-M
The basic tool which is employed in the proofs is is the following induction principle, well suited for time scales. See [1,4]. Theorem 2.1 Let T be a time scale as before. Suppose for any t e T, there is a statement A(t) such that the following conditions are verified: (HI) A(t0) is true; (H2) If t is right-scattered and A(t) is true, then A{a(t)) is also true; (H3) For each right-dense t, there exists a neighborhood V such that whenever A(t) is true, /4(s) is also true for all s e V,s £ t\ (H4) For left-dense t, A(s) is true for all s e [r0,f) implies A(r) is true, Then the statement A(t) is true for all t e T.
433
In case T = N, Theorem 2.1 reduces to the well known principle of induction.
mathematical
3. Method of upper and lower solutions. Let T be a time scale with t = 0 is a minimal element and t = T > 0 is a maximal element such that T is not left scattered. Let us begin by considering the initial value problem for dynamic system on time scale «A =/(;,«), M(0) = «0, (3.1) where feCJTxR\R"]. Definition 3.1 A function veC^[r,/?"] which is differentiable is said to be an upper solution of (3.1) if v A >/(r,v), teT, v(0)>«0 and a lower solution of (3.1) if the reversed inequalities hold. We need the following result concerning upper and lower solutions proved in [5]. Theorem 3.1 Let T be the time scale as before, let v,w :T -»R" be the rd-continuous mappings that are differentiable for each t e T and satisfy v 4 (»)S/(r,v(0), w*(t)>f «,*«)), teT where f eC^lTxR",R"], f(t,x) is a quasimonotone nondecreasing in .x.and for each i , 1 <(, fXt,x)n'(t) is nondecreasing in xi for each re7".Then v(f0)£w(f0) implies v(r)< w(t), teT, provided / satisfies /(r,x)-/(f,>)SLXui.-y1.),
x>y.
(3.2)
ixl
If we know the existence of upper and lower solutions w,v such that v<w, we can prove the existence of a solution in the closed set £2 = [(f,«): v(f) < u < w(t),t e T]. In order to prove such a result we need the following. Proposition 3.1 Consider the system x 4 =/(r,x), x(t0) = x0,
(3.3)
where feCrd[TxR",R°].and f(t,x) is bounded everywhere. Then there exists a solution Jt(/) on T of (3.3). The proof follows by the usual procedure employing Schauder fixed poin theorem and we omit it. See [5]. Theorem 3.3 Let the differentiable maps v, w e C,d[T,R"], be lower and upper solutions of (3.1) such that v(t)<w(t) on T and / e C r J Q , « " ] with / being bounded on £2,and / is quasimonotone nondecreasing in xand fi{t,x)\i{t) is nondecreasing in xt for each t .Then there exists a solution u(t) of (3.1) such that v(t) < u(t) £ w(f) on T , provided that v(0)S« 0 <w(0).
434
Proof: Let P:TxR"-*R* be defined by ^(r,w) = max[v1.(f),min(j<j,vv1.(f))] for each / , then f(t,P(t,u)) defines an rd-continuous extension of / to TxR" which is also bounded since / is assumed bounded on TxCl. Hence by proposition 3.1 , there exist a solution u for u*=f(t,P(t,u)), M(0) = «0 onT. We shall show that v(f) S«(/)S w(f) on T so that u(t) is also a solution of (3.1). Fore>0and e = (1,1, 1) consider w. = w(f)+e(l + f)w.,(a(0), l ^ y ^ n , M,(a(0)£w.,(o(f)),i*y Then v(o(/))^(o(l),u(o(()Pw(o(()) and />(CT(f),«(a(f))) = w,(o(f)) Hence using quasimonotone property and w* £ / ; (a(r), P(o(t), u(a(r)))) = «f (o(/)) since >v^ (a(f)) > wA (a(f)), it follows that H£ (o(r)) > «A (a(f)) which imply by definition w«/g(0)-w. > (0 ii ; (a(0)-ii ; (r) ^ w«<(a(Q)-H>(Q n'(0 nV) ~ ^'(0 which imply «y(f) >M V(0 on [0,a(f)] .which contradict the assumption. Hence A(o(t)) is true. Let t be right dense and U be a neighbourhood of t, such that A(t) is true. We need to show that A(s) is true for s>t, sell. This follows by the proof of Theroem 1.3.5 in [5]. Let t be left dense such that A(s) is true for s< f.We need to show that A(t) is tue. Since v,u,w are all rd-continuous, it follows that limv. (,$■)< limK(5)
s-*r
j-»t"
v(t) £ «(/) 2 w(t) on 7", and the proof is complete. 4. Main result Consider the dynamic system on a time scale x*=f(t,x), x(0) = x0 (4.1) where feCja^R"], ft0 =[(t,x):a0(t)ZxZp0(t),teT] and a 0 ,p 0 such that a 0 (Osp o (0 on 7\ Let £l = [(t,x,y):a0(t)<x,y<.$Q(t),teT]. We are now in a position to prove our main result.
eCJT.R']
Theorem 4.1 Assume that there exists a function S e C^[ii, /?"] such that (i) c v P o e C ^ r . / T ] with o 0 (r)Sp 0 (r) and ctf£S(f,a 0 ,a 0 ), P„A >S(f,p0,P0); (ii) 5 is bounded on £2,5(f,x, v) is quasimonotone nondecreasing in x and 51(f,JC,3')(J.*(r)
435
is nondecreasing in x, for each 1 < i £ n, t e T and a 0 (t)
(iv) $(f,JC,,y)-S,.(f,.Xj,)>)££,£(■*,,.-Ja) w n e n e v e r *i -*2 and cioCO^y^PoCO.f e7"; .-i
(v) S(r,jr, x ) - S ( f , . y , z ) S N f x - ^ + W l x - z ) , whenever P 0 (f)>x>y >z>a0(f), teT, where Af,M are «xn nonnegativematrices. Then there exists monotone sequence {<x„(Oj which converges locally uniformly to the unique solution of (4.1) on T and the convergence is superlinear. Proof: Consider the IVP for dynamic system of =5(r,a„a 0 ), o,(0) = ^ (4.2) Now a0A < S(r, a,. a , ) . and by (iii), pJ > S(t, P0, P0) > S(t, P„, a 0 ). Hence by Theorem 3.3, there exists a solution a, (f) of (4.2) such that oc0 < a, < P0 on T. An application of Theorem 3.2 shows that a, is the unique solotion of (4.2). We note that, because of (iii), <xf = S(f, a,, a 0 ) < S(t, a,, a,). Consider the IVP a 2 4 =S(?,a 2 ,a,), a 2 (0) = x0 (4.3) Wehave af < S(t,a^a^, and as before by (iii), pA >S(f,P 0 ,p 0 )>S(f,p 0 ,a,). Hence by Theorem 3.3 there exists a solution a2(f) of (4.3) such that a, < a 2 < P0 on T. Again Theorem 3.2 shows that a 2 is the unique solution of (4.3). This implies a 0 < a , 2 a 2 < p o on T. Proceeding successively, we obtain a0^a,< £ a „ < p 0 on 7, where rx„ is the unique solution of < = 5 « , a „ , a „ . , ) , a n (0) = *0 on T. Employing standard arguments, we get limcc„(f) = x(t) locally uniformly and monotonically on T. See [5]. It is then easy to see that x(t) is the unique solution of x*=S(.t,x,x) = f(t,x). To show that a„ converges to x superlinearly, let P„+1 (0 = *(»)- a r i (0, so that p„+, (0) = 0 ,and pL=nt,x(t))S(t,a^(t),a,,it)) = S(t,x(tU«))-S(t,a„t](t),an(,t)). Hence by (iv), we get, noting that pn(t) > 0, PL (') * Wp„+1 (0 + Np'r (0, p„+, (0) = 0. Thus ,we obtain P^(0SW||* M (r,o(5))p|r(*)Aj, using the variation of parameter formula (see[l]), where Q>u(t,s) is the fundamental matrix solution of Xs = Mx, x(0) = I (identity matrix).
436 We then get ™«|p*i(')|s*'raK|p.W| l * a where K = N\ |<J>M(7",a(5))Aj|, and the proof is complete. To demonstrate the existence of such a function S(t,x,y) ,we now give sufficient conditions on f(t,x) such that S(t, x,y) satisfies the properties of Theorem 4.1 . Assume that (CO) a 0 ,p o eC r „[7\/?] with a 0 (O£p 0 (f)and cc 0 A /(r,P„) on 7; (Cl) /isboundedon Q = l(t,x):a0(t)<xZpo(t),teT], f(t,x)\i'(t) is nondecreasing in x for each teT and fx(t,x),fa(t,x) exists, rd-continuous on Q; (C2) /„(/,*) 2:0 onQ. Suppose that (CO) and (Cl) hold ,then in view of /„(f,x)>0, we get forx>y nt,x)2f(t,y) + fs(t,y)(x-y) and hence if we set S(.t,x,y)mf(t,y) + f,(t,yXx-y), then it is easy to verify that S(t,x,y) satisfies the conditions (i) to (iv) of Theorem 4.1 with a = 1. If on the other hand, (C2) is weakened to (C2)' fa(t,x) + tya(t,x)ZQ where ^^Crd[Q>R],^x,^a existsand rd-continuous and +a(t,x)Z0onQ; then we define
S(.t,x,y) = f(t,y) + [fll(t.y) + W,y)Kx-y)-W,x)
+ W,y),
it is not difficult to verify that this S(t,x,y) also satisfies all the conditions of Theorem 4.1 References (1) B.Aulbach and S.Hilger, " Linear Dynamic Process with In homogeneous Tune Scales", Nonlinear Dynamics Systems, Akademic-Verlag, Berlin 1990. (2) R.Bellman. Methods of Nonlinear Analysis, Vol II, academic Press, New York 1973. (3) R.Bellman and R.Kalaba. Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York 1965. (4) S.Hilger,"Analysis on Measure Chains.A unified approach to continuous and discrete calculus', Res. in Mathematics 18,(1990),pp 18-56. (5) B.Kaymakcalan, "Existence and comparision result for dynamic systems on time scale", J.M.A.A,172(1993) 243-255. (6) V.Lakshmikantham, S.Leela and S.Sivasundaram, Extensions of the method of quasilinearizations (to appear) (7) V.Lakshmikantham and S.Malek. Generalized quasilinearization, Nonlinear World, 1,(1994).
WSSIAA 3 (1994) pp. 437-448 © World Scientific Publishing Company
437
TWO INEQUALITIES RESEMBLING AN INEQUALITY OF
GABUSHIN
E.R. LOVE Department of Mathematics, The University of Melbourne Parkville, Victoria 3052, Australia
ABSTRACT The inequality of Gabushin concerned relates to Lebesgue Lp-, Lq- and Lrnorms of a function f and its kth and /th derivatives. The two inequalities of the title have this same general character, but k and / are not restricted to be integers, and there is also a weight function in the norms. To pay for this extra generality it is required that f and the derivatives concerned be non-negative. An important item in Gabushin's hypothesis is an inequality which may for the moment be expressed as A £ B, where A and B are functions of k, I, p, q, r. The very same inequality occurs in the hypothesis of one of the inequalities presented here, except that k and / may have non-integral values; while in the other inequality the hypothesis includes the reversed inequality, A <, B.
438 1. Introduction The inequality of V.N. Gabushin in question [4: Theorem 2] is as follows. Let p, q, r £ 1 and Q<,k
J lf(x)IPdx I 0 * J lf(0(x)|rdx r '
(G)
for all f such that iand ft'-1) are locally absolutely continuous in [0,<») and the integrals on therightare finite, it is necessary and sufficient that /-k-1/r+l/q ~ M/r+l/p •
nr a
„
p
k-1/q+l/p ~ M/r+l/p
(so that a + P= I), and L-V
V
I
This paper establishes, in Theorems 3 and 4, two inequalities which, in the case of unit weights w, closely resemble Gabushin's Inequality (G), at least in form. However, besides the appearance of weights, a major difference is that the derivatives occurring are of real orders K and X, not merely of integral orders k and / as in (G). Another major difference is the unfortunate restriction that the derivatives are required to be non-negative, whereas those in (G) are not. A lesser difference is that p, q, r are only required to be positive. On the other hand, the hypothesis (H) occurs in Theorem 3 with the integers k and / replaced by the real numbers K and X, but otherwise precisely the same. In Theorem 4, curiously, this hypothesis occurs with the inequality reversed. The special case jfc = l , / = 2 = p = q = ris also a special case of the HELP (Hardy, Everitt, Littlewood, Polya) Inequality; a particularly special case because of the non-negativity requirement on the derivatives. Generalized versions of that inequality have been studied extensively; see references [1, 2, 3, 5, 6] and bibliographies therein. Part of this note is concerned with making precise the oft-quoted properties of Riemann-Liouville fractional integrals and derivatives. When that has been done, the proofs of the inequalities amount to little more than applications of Holder's Inequality, sometimes in extended forms due to Jensen. 2.
Fractional integrals Let (a, b) be a fixed interval, with - » < a < b S « > .
The uses of the symbols a and p in the rest of this paper are unrelated to those in Gabushin's Inequality (G). If a > 0 and f is locally integrable in [a, b), I°f is defined as the function
a
T(a)
439 if this exists for almost all x in (a, b)asa Lebesgue integral. If a = 0 this definition is replaced by I°f(x)=f(x). By "locally integrable" we shall mean "locally integrable in [a, b)" throughout. Similarly for "almost everywhere". By "=°" we shall mean "equals almost everywhere in (a, b); and similarly for "£°" and other such expressions. We need extensions of some well known properties of these Riemann-Liouville fractional integrals (1). Lemmas 1 and 2 are mainly the real cases of Theorems A and B in [7: p. 387]. Lemma 1 // a > 0 and f is locally integrable, then I a f exists almost everywhere and is locally integrable. Ifiis continuous so also is Iaf. Iff is nonnegative so is Iaf. This lemma follows from well known properties of convolutions. Lemma 2 // a £ 0, P £ 0 and f is locally integrable, then lha{ =° I°+Pf.
(2)
If also I°f is continuous, or merely exists everywhere, then ="can be replaced by =. The classical proof of (2) depends on the following change of order of integration, supposing that a > 0 and P > 0: x
s
x
x
J (x-s)p_1ds J (s-tf-'fCOdt = J f(t)dt J (x-s^Cs-O^ds a a a t
(3)
Ha+p) aJ Eq. (3) is justified by absolute convergence of (4), which is the case for almost all x because (4) is a convolution of integrable functions. If however I°f(x) exists for all x, the needed absolute convergence for all x follows from x
x
J lf(t)l(x-t)a+)Mdt <. (x-a)p J IfCOKx-t/^dt; a a the last expression is finite because the integral (1) for I^x), being Lebesgue, is absolutely convergent Eq. (2) is obvious if either a or p is zero, or if both are. 3 . Fractional integrals of negative order locally integrablefunction $ (if any) such that
If a < 0, ^ f is defined as a
I^4> =° f.
(5)
r™ being defined as at (I) since -a > 0. It is well known that any such <> | is in a unique class of functions equivalent on (a, b). In the definition (5) there is no need to assume that f is locally integrable. This is implied in the definition, because, by Lemma 1, Y^Q is locally integrable.
(i) (ii)
For a of either sign, the phrase "I°f exists" shall mean, throughout, that: if a > 0,1°f(x) exists as in (1) almost everywhere; and if a < 0, a locally integrable $ exists as in (5). Lemma 3
/ / a £ 0 < P, fis locally integrable and I°f exists, then
The former equation is covered by the real case of Theorem 1 in [7: p. 393]. The latter equation is the real case of Theorem 2 in [7: p. 393]; and the next lemma is its corollary. Lemma 4
If a < P, f is locally integrable and I°f exists, then fif exists.
4. An inequality for fractional integrals It is convenient to use the letters p and q here although they are not the p and q occurring in (G) and the later theorems. Theorem 1
/ / 0 < X < \i, p > 1, p"1 + q"1 = 1, I^f exists and T*f(.x) £° 0,
then r^PfW
r(p.-x/P) Proof Let g = r^f; g is locally integrable by definition, and non-negative almost everywhere by hypothesis. Also f is locally integrable. Since X/p < X < \i, Lemma 4 gives that T*f and I_Wpf exist. Let v = \i-X > 0. By Lemma 1, I^i+^g exists, and by (1) X
rOi/q+v/pfl^'Pgto = J ( x - t ^ ' ^ V O d t a x
= J (x-t)<»l-lv<'g(t)1/''(x-t)(v-1)/Pg(t)1/Pdt a
441
(r
W
; J (x-o^-'gCDdt
r
V'P
J (x-tr'gcodt
so that I^+V/Pg(x) £ KI^g(x)1/<'Ivg(x)1/P,
(6)
where K=
r(n/o+v/P)
r(|i-x/p)
By Lemma 3, jM/q+v/p _ jH/q+v/p j - n f _o |(v-*i)/pf _ j-typj
and Ivg = I v I- >l f= 0 I v ^f = r x f ; while by (5) I^g =° f• The three functions on the left are non-negative, by Eq. (1), hence so are those on the right almost everywhere. Substituting these in (6) gives the required inequality. 5 . Fractional integrals and differentiable functions For positive integers n, let AC" -1 be the class of functions which have (n-l)th derivative locally absolutely continuous in [a, b) and which, with their first n-1 derivatives, vanish at a. Lemma 5 I
M
/ / n is an integer, 0 < a < P £ n and I n_p f e AC 1 " 1 , then
1
f e AC"" . Proof
By Lemma 4, InMIf exists. There is a locally integrable function $
such that the (n-l)th derivative of I ^ f is the locally absolutely continuous function
J
(7)
IOUS. If since l'ij>, I2<J>,.... I" -1 ^ are continuous. If nn == 1 this argument fails, but in that case is locally absolutely continuous and vanishes at a, so that (7) is immediate.
442 Since both sides of (7) are continuous, two applications of Lemma 2 give
Another application of Lemma 2 gives I"*?-0,), =« H ^ .
(9)
and since 1^4* >s locally integrable by Lemma 1, Fl^"*^ is continuous. Again by Lemma 1, since P^f is continuous so is lP""°ln""Pf in (8). Thus both sides of (9) are continuous, whence =° can be replaced by =. So (8) and (9) give I n-a f = InIP-o<((.
(10)
It follows that I n-a f and its first n-1 derivatives exist and vanish at a, and that the (n-l)th derivative is the locally absolutely continuous function I'l^ -0 ^. Thus I ^ f e AC"-1, as required. 6.
Fractional derivatives
Define D°f as follows. Let 0 < a £ n, where n is -1
an integer, and let P ^ f e AC" ; then D°f is the locally integrable function defined almost everywhere by
iysw^T^r^fcx).
(ID
dx Uniqueness of D a f up to equivalence, for various integers n i a, needs to be established. For this let m and n be integers such that m > n 2 o , and let I n - a f e AC" -1 . This makes In"°f continuous, so Lemma 2 gives that jm-Of _ jm-nin-af
C o n s e q u c n t l y pn-Of 6
jm-n
<x - 9 —-rf(x) = r- a f(x) dx m - n
AC
jm
jn
and - ^ mr r-°f(x) =-^-rn r ^ f f r ) , dx dx
as required. dn
When n = a, (11) becomes Dnf(x) =° — - f(x). Thus for any a > 0 the dx definition (11) can be rewritten D a f=°D n P w x f
(12)
for any integer n £ a and I ^ f € AC*"1. Lemma 6 / / n is an integer, 0
f**f e AC"-1, then D°f
443 Proof By Lemma 5,1""0^ e AC1"1. So there is a locally integrable 4» such n ,v <x that D ~'l " f = I'ft and by successive integrations of continuous functions vanishing at a, as for (7), In_af = r > Using Lemma 2, I " ^ =° I0"1^ =° I°ln<(i = I"!"-^ = I"f; and since the extreme members are continuous, InIa$ = I"f. Differentiation of this gives jn-ljo^ = o r n-l f
(13)
If n > 1 both sides of (13) are continuous, so that =° can be replaced by =. Repeating this process successively gives l'la<|> = l'f, and one more differentiation gives 1™$ =° f. If n = 1 the last equation is immediate from (13). By the definition at (5) (with a replaced by -a), the last equation gives that r"f exists and r ° f =° $ =° DT~"f = D'f, as required. 7 . An inequality for fractional derivatives The next theorem is little more than arestatementof Theorem 1 in the light of §5 and §6. Theorem 2 D ^ x ) £° 0, then
/ / n is an integer, 0 < X < | i S n , O < v < l , f e
DXvf(x)
AC"-1 and
(14)
with K _rao'-T(M.)
v
nn-Xv) Proof By definition of D^f, I ^ e AC""1. By Lemma 6, r^f exists and r ^ x ) =° D ^ x ) £° 0. Similarly by Lemma 6, r*f =° D*f
and
I_Xvf =° DXvf.
Theorem 1 with p_1 = v and q"1 = 1 - v now gives the stated results. 8 . First inequality of the title In the rest of this paper p and q are no longer Holder-conjugate indices. Also v is given a more specific meaning than in Theorem 2.
444
Theorem 3 Ifn is an integer, 0
i
, . *»».
X-K K ^ X -p-+7sq»
(15)
then \vft 'b V/q (b yi-vVpf b K q J D f(x) w(x)dx £ KM fWwW'^'dx J Dxf(x)rw(x)r/qdx
with K=
r(n)l-vrqi-\)v
(16)
r(u-K)
Remarks Observe the resemblance of (IS) to hypothesis (H) in Gabushin's Inequality (G) in § 1. In contrast with that inequality notice that p, q, r, a and b are less restricted here. Proof
Define positive constants s and t by
i = J(i-V) and } = a v ;
(17)
1 1 then the condition (15) is equivalent to - + - £ 1. The hypotheses of Theorem 2 are included in those of Theorem 3, so that (14) holds. This, together with Jensen's extension of Holder's Inequality [5: Theorem 22], gives
V/q
(r Xv
q
J D f(x) w(x)dx f
<JK J {f(x)Mx)}1-v{D*f(x)qw(x)}vdx ' b
as required
\ i/ q , / b
> l/qt
£K J {f(x)"w(x)}(1-v)sdx V» j
J {Dxf(x)''w(x)}v'dx \« J
= KJ J f(x)«,w(x)p''»dx
J D*f(x)rw(x)r/<>dx
,
445 9 . Special cases There may be some interest in the case in which K = 1, X.=p = q = r = 2, u, = n = 3. The derivatives occurring in Theorem 3 are now ordinary, not fractional, and Theorem 3 becomes:
/ / - » < a < b i " , w i s measurable and non-negative on (a, b), f' is locally absolutely continuous, f(a) = f'(a) = f"(a) = 0, f "(x) £° 0, then f b -\2 b b J f(x) 2 w(x)dx £ 4 J f(x)2w(x)dx J f "(x)2w(x)dx.
V»
J
»
(18)
a
Known as the Hardy-Littlewood Inequality when w(x) = 1 and (a, b) = (0, <») [5: Theorem 2S9], (18) has received much attention from later workers, (1,2,3,6]. They supposed f much less heavily restricted but w more so; in fact they required w to be increasing, because of certain counter-examples. It is perhaps surprising that w can be allowed so much more freedom when f is subject to the present restrictions. The value 4 for the constant on therightof (18) seems to have been encountered by several of those writers; but in [3: pp. 39 and 62] it was brought down to a value well below 3 when w(x) = x - a. Changing the value of \i does not visibly affect the inequality in Theorem 3; its effect is only to change the level of differentiability that f is permitted to have. Leaving the parameters K, X, p, q, r the same as in the above special case of Theorem 3, but increasing (I and n, the constant 4 in (18) becomes, by (16),
r(u-2) 2 _(u-i) 2 . rxn-i)4 (u-2) 2 '
K 4_r(u)
2
this decreases below 4 as \i increases above 3. Of course, this entails increasing n and therefore the order of differentiability of f. 10. Second inequality of the title This requires a lemma which must surely be known but for which I have found no reference. Lemma 7 / / - ~ S a < b i » , s > 0, t > 0, s"1 + r 1 S 1, f and g are measurable and non-negative on (a, b), and w is integrable and non-negative on (a, b) with integral W, then b / b \ i,, / b \m J f(x)g(x)w(x)dx <, w u l / t - 1 A J f(x)sw(x)dx J g(x)'w(x)dx a [? J [a J Proof Let F(x) = f(x),/r and G(x) = g(x),/r, where r_I = s"1 + r 1 . Using Holder's Inequality with indices s/r and t/r, the left side of the required inequality is b
J {F(x)w(x))r/,{G(x)w(x))lftdx
446 \ilt
( *>
y/t
( •>
<\ J F(x)w(x)dx
J G(x)w(x)dx
( b
WA j G(x)v(x)dx
= Wr/» J F(x)v(x)dx
,
(19)
V» where v(x) = w(x)/W, so that J v(x)dx = 1. Suppose that r > 1. Jensen's Inequality a gives that (19) is 1/t / b
f b ^ ^yi/l+rA
J F(x) r v(x)dx
l/t
J G(x)\(x)dx y
\l*f b
Z' b
= W Jf(x)' V»
W
-^dx
MA
Jg(x)'^dx J
\*
and the required inequality follows from this. In the remaining case r = 1, the inequality following (19) is an identity and Jensen's Inequality is not needed. In fact the whole lemma then reduces to a single application of Holder's Inequality, and W need not be finite. Theorem 4 TfO < K < X < |i, v = K/X, -°° < a < b £ <», w is integrable and non-negative on (a, b), r**f exists and V^t(x) S° 0, p, q, r are positive constants and (contrast (15)) 1-v v l — + ?S-,
. . that,s,
X-K
-p- +
K F
X ^q.
(20)
then /b M/q J b yi-vyi>(b ^ v/r K < p J r f(x) 'w(x)dx £ KWH J f(x) w(x)dx J I _x f(x) r w(x)dx
»
y
U
y
U
w«AKasm (16), r
W=Jw(x)dx
1
and
l_v
V
k =±-—— j .
(21)
Pro*/ The assumed existence of I"*f ensures existence of r*f and r*f by Lemma 4, remembering that this existence implies that f, as well as these three functions, is locally integrable.
447 Theorem 1, with its conjugate indices p and qreplacedby 1/v and l/(l-v), gives that
rXvf(x) £° Kf(x)1_vrxf(x)v
(22)
with
r(n-X) v . roi-Xv) ' roi-Xv)
K _r(n)'-
K =-
V
since Xv = K this value of K agrees with (16). By Lemmas 3 and 1,
r*f =° p - * m ^° o,
r*f =° P^r+f ;>° o.
As in (17) define positive constants s and t by i = J(l-v)
and
i = fv;
(23)
then (20) is equivalent to s"1 +1"1 £ 1. By (22), and then Lemma 7, b
b
J rKf(x)qw(x)dx £ Ki J f(x)q(1-v)rxf(x)('vw(x)dx a a
( 1 1
1
b
V* f b l v)5
£ ROW - "- " J f(x)i< - w(x)dx
f b \qd-vVp/' b =K W J f(x)Pw(x)dx J T x f(x) r w(x)dx q
qk
^ l/i
J r X f(x) qvt w(x)dx
W ,
(24)
making use of (21) and (23). The required inequality follows from (24) by taking qth roots of both sides.
448 REFERENCES 1
W.N. Everitt. On an extension to an integro-differential inequality of Hardy, Littlewood and P61ya, Proc. Royal Soc. Edinburgh, A 69 (1971/72), 295-333.
2
W.D. Evans and W.N. Everitt, A return to the Hardy-Littlewood integral inequality, Proc. Royal Soc. London, A 380 (1982), 447-486.
3
W.N. Everitt and A.P. Guinand, On a Hardy-Littlewood type integral inequality with a monotonic weight function, in General Inequalities 5, editor W. Walter (Birkhauser Verlag, Basel, ISNM 80,1987), 29-63.
4
V.N. Gabushin, Inequalities for norms of a function and its derivatives in Lpmetrics (Russian), Mat. Zametki 1 (1967), 291-298 [Math. Rev. 34 #6518].
5
G.H. Hardy, J.E. Littlewood, G. P61ya, Inequalities, Cambridge (1934).
6
M.K. Kwong and A. Zettl, Norm inequalities of product form in weighted Lpspaces, Proc. Royal Soc. Edinburgh 89 A (1981), 293-307.
7
E.R. Love, Two index laws for fractional integrals and derivatives, J. Australian Math. Soc. XIV (1972), 385-410.
WSSIAA 3 (1994) pp. 449-457 © World Scientific Publishing Company
449
Method of Generalized Quasilinearization for Second Order Boundary Value Problem S. Malek Department of Applied Mathematics Florida Institute of Technology Melbourne, FL 32901-6988 and A. S. Vatsala Department of Mathematics University of Southwestern Louisiana Lafayette, LA 70504
Abstract In this paper we extend the method of generalized quasilin earization to second order nonlinear boundary value problem to obtain two sided bounds for the solution. Further we prove the iterates converge uniformly and monotonically to the unique so lution of the boundary value problem and the convergence is quadratic.
1
Introduction.
Consider the boundary value problem -u" = f(x,u),
Bu(fi) = b,
(1.1)
where / € C[I x R,R] where Bu(fi) = oyx(/i) + ^u'(fi) = b^, p = 0,1. Here a 0 , a i > 0,/?0,/?i > 0,1 = [0,1]. It is known [[1],[2]] that the method
450
of quasilinearization yields sequence of approximate solutions to the bound ary value problem (1.1) which converge uniformally and quadratically to the solution of (1.1) . If / is uniformly convex in u, then the method of quasilin earization provides a monotone increasing sequence of approximate solutions that converge uniformly to the solution of (1.1). If f(x,u) is concave, one can prove a dual result which gives upper bounds See[[l],[2j] for details. The method of upper and lower solutions coupled with monotone iterative technique [[4]] also offers monotone sequences that converge, in general to the extremal solutions of (1.1). However the convergence is not quadratic since the assumptions are much weaker than requiring convexity. In this paper we extend the method of generalized quasilinearisation [[5]] to second order BVP under a less restrictive condition, namely f(x, u) +
2
Main Results.
Consider the nonlinear second order boundary value problem -u" = f(x,u),
Bu(p) = br
(2.1)
where Bu(fji) = a„u(/x) + ( - l ) " + 1 ^ u ' ( / i ) = 6„, a0,cti > 0, #>, A > 0, and ao + a x > 0, bp e R, f € C[I x R, R] and / = [0,1]. Let I0 denote the iterior of/. A function v G C2[I,R] is said to be a lower solution of (2.1) if -v"
Bv(n)<
&„,fi = 0,l
and an upper solution of (2.1) if the reversed inequalities hold in (2.1). Next we recall some known results in special forms which are needed for the main results. The first one is the following comparison theorem. Theorem 2.1. Assume that (i) / € C[I x R,R],f(x,u) /«<0;
is decreasing in u for each x € / , that is
451
(ii) v and ware lower and upper solutions of (2.1) on / ; then we have v(x) < w(x) on / provided Bv(fi) < Bw(fi). See [[3]] for proof. We note that the solution of (2.1) if it exists is unique if / u < 0 on / . If we know the existence of upper and lower solutions of (2.1) such that v(x) < w(x), then we can prove the existence of a solution u(x) of (2.1) in the closed set Cl = [(x, u) : v(x) < u < w(x), x 6 I] Next we recall precisely this result. Theorem 2.2. Assume that (Ai) v,w € C2[I, R] are lower and upper solutions of (2.1) such that v < w on / ; then there exists a solution u(x) of (2.1) such that v < u <w on I provided Bv(n) < Bu(fi) < Bw(ft). See [[3],[4]] for proof. We can now prove our main result using Theorems 2.1 and 2.2.Here we do not assume / ( x , u) is convex.However we assume that / ( x , u) +
Denoting f(x,u)
+
it is easy to observe
f(x, m) > F(*, Vi) + Fu(x, iu)(f|2 - 171) - tp{x, ija)
(2.2)
452
and
f{x,m) < F{*,V2) + F*(*,m)(ih - m) -
(2-3)
for any f/i, 172 £ Cl since Fuu(x, u) > 0 on fi. Let us now define the sequences {u n (x)} and {u>„(x)} as follows: -v" = -u;J,' =
g(x,v„-i;vn), Bvni(p) G(i,v n _i,u>„_i;u; n ), Bw„(/i)
= 6„ = b„
(2.4) (2.5)
where g(x,r)!;u) Gfc.Jji.ifeju)
= F{x,^) + Fu(x,rji)(u - J?I) - y?(z,u) = F ( « , 7a)+ F«(x J i/ 1 )(u-172)-¥>(*,«)■
(2.6) (2-7)
Setting v = vo,w = wo we shall first show that vi and tui the solutions of the boundary value problem of (2.4) and (2.5) exists for n = 1 such that vo < V\ < w\
< w0.
(2.8)
For that purpose it is enough to show that vo and too are lower and upper solutions of the boundary value problems (2.4) and(2.5). Since v0 and w0 are lower and upper solutions of (2.1) it is enough to show a(i) a(ii) b(i) b(ii)
f{x,v0) f(x,w0) f{x,v0) f(x,w0)
< > < >
g(x,v0;v0) g(x,v0;w0) G(x,v0,w0v0) G(x,v0,w0;w0).
By definition of g it follows that f(x,v0) = g(x,v0;v0) which proves a(i). Using (2.2) a(ii) follows with 772 = w0 and rji = to0. Using Theorem (2.2) V\ exists such that vo < vi < w0 ■ Also by definition of G it follows f(x, wo) = G(x, v0, w0; wo) which proves b(ii) and using (2.3) 6(t) follows with r)i = v 0 and j/2 = f o- Now using Theorem (2.2) u>i exists such that vo < Wi < wo . Further v\ and 101 are unique using Theorem (2.1) since gu Gu
= Fu(x,7)i)-
< 0 < 0
whenever J/I < u for J/I, u G il since /„ < 0 and y>„„ > 0 on Q. Now it is enogh to prove v\ and w\ are lower and upper solutions of (2.1). From (2.2), and (2.4) we get
453
-v" = F(i,t;o) + F u (i,v 0 )(u 1 -v0)-ip(x,vi)
< f{x,vx),
BvY(fi) = 6M.
Similarly using (2.3) and (2.4) we have -to"
= = = = >
F(x,w0) + Fu(x,v0)(wr-w0) -(p(x,wx) - f(x,Wi) + f(x,wi) F(x,w0) - -F(x,u>i) - Fu(x,v0)(w0-Wi) + f(x,u>i) [Fu(x, C) - Fu{x, u0)](u;o_ii;1) + f(x, wr) [Fuu{x,a)](( - uoXwo-ioi) + f{x,wi) f(x,wi), Bw2(fi) = bli
where toi < ( < wo and v0 < a < (. From Theorem (2.1) we get v\ < w\. This proves (2.8). Assume that for some n > 1, u*, to* are lower and upper solutions of (2.1), then we prove that f*+iand to*+i exists and are unique such that Vk
<
Vk+\
<
Wk+1
<
Wk.
For this purpose we prove u*and to* are lower and upper solutions of the boundary value problems (2.4) and (2.5) for n = k+l. Since u*,u>* are lower and upper solutions of the boundary value problem (2.1) we have -vl
< f{x, vk) = g(x, vk; vk),
Bvk(fi) = o„
> f{x, wk) > g(x, vk; to*),
Bwk(fi) = o„
and -wi
using (2.3). Now from Theorem (2.2) it follows that Vk+i exists such that Vk < Vk+i < Wk on fi. Also Vk+i is unique by Theorem(2.1) since 5„(x,u*;u* + i) =
Fu(x,vk)
-fu(x,Vk+i)
< 0.
Similarly using (2.3) we have -v'k<
f{x,vk)<
G(x,Vk,wk;vk),
Bvk(fi) = b„
and -Wk>
/(x,to*) =
G(x,Vk,wk;wk),
Bwk(n) = bll.
454
Using Theorem 2.2 Wk+i exists such that u* < Wk+i < lo*. Further Wk+i is unique since Gv(x,Vk,Wk',Wk+i) = F^x^^ — fuix^k+i) < Ofollows from the fact fu(x,Vk) < 0,
= = = = >
G(x,vk,wk;vk) = F(x,wk) + Fu(x,Vk)(wk+i - wk) -
where Wk+i < C < wk and Vk < o < CThis proves vn < vn+i < wn+i < wn for all n and vn,wn are lower and upper solutions of the boundary value problem (2.1). Hence we obtain by induction v
o
^
v
i
^
••• vn
<
u <
wn
<
... <
w-i < wo-
where u is the unique solution of the boundary value problem (2.1) since / , < 0 on ft. We can now use standard arguement [[4]] to show that {v n },{«;„} converges uniformly and monotonically to the unique solution of the boundary value problem (2.1). We note that the solution of (2.1) is unique since /„ < 0 on ft. Finally we show that the convergence of {u„}, {wn} to u is quadratic. For this purpose set pn = u — vn and qn = wn — u. Then it is easy to observe that Bpn(fi) = 0 and Bqn{fi) = 0. Therfore we can write pn(x)-
I G0{x,s)[f(s,u(s)-g(s,vn-1;vn)]d3 Jo
(2.9)
and 1n{x) = I G 0 (x,s)[G(s,t; n _i,t(; n _ 1 ;u; n )-/(s,u(s)]ds Jo
(2.10)
455
where
is the green's function given by
GQ(X,S)
1
C
G0{x,s) =
u(s)v(x)
1 tt(i)7;(a) LC
0< s <x <1 0 < x < 3 < 1.
H e r e u ( i ) = — x+l,v(x) = —(1—x) + l are the two linearly independent Po Pi solutions of —u" = 0. It is easy to observe that Go(x,s) is nonnegative. Therefore from (2.9) we get
0 < p„ =
/ G0{x, s)[f(s, u) - {F(s, u n _i) + Fu(s, u n _i)(v n - u n _i) JO
<
-
=
/ G0(x,s)[F(s,vn-i) Jo
=
+F»u(s,(ry
^~1'
-
vn)}]ds
I Jo +FUu(s,a)-
+
+Fu(s, vn_i)(u -
Fu(s,vn-1)(vn-vn^)
vn-i)
- {F(s,t; n _i) + Fu(s,u„_i)(u„
Go(x,s)[Fu(s,u)(u-v„)-
- v n _i)
456 =
J G0(x,s)[{fu(s,u)
+
+Fuu(s,*)iu-£-l)2)ds <
f1Go(x,s)Fuu(s,a){u~^-l)2ds Z\
Jo
=
£
G0(x,s)Fuu(s,a)^-d3
using the fact /„ < 0 and tpuu > 0. Here a and 6 are such that i>n_! < a < u and u n _i < 6 < u. Similarly one can get 0<wn-u
< J
G 0 (x,3)[F(s,io n _ 1 ) + i ; , u (s,t; n _ 1 )(u) n -t(; n _i)
-
f(s,u)]ds
j G (x,s)[F{s,wn-1)-F(s,u) Jo 0
+
Fu{s,vn-1){wn-wn-i)
-¥>u(s,<5i)(u-u; n )]
/ G0(x,s)[{Fu(s,cri) Jo
- Fu(3,vn^)}(wn-i
+{/u(s,u n _i) + y u (s,u n _ 1 ) -
- u) wn)}]ds
I G0(x, s)Fw(s, 7)(«ri - t;n_i)(u>n_i - u)ds Jo
<
/ G0(x,s)Fuu(s,'r)qn-i{qn-i + pn^)ds Jo where o\, S\,7 are such that u < o\ < u>n_i, u < 81 < wn and vn-i < 7 <
K
r1
Zl Jo
\G0(x, s)|ds] max[|u - vn^ \2] I
457
and ma.x[|u;n - u\] < K[ |G 0 (x,s)|ds]{max[|u> n _i-u| 2+|u>„_i - u\ \u - f„_i|]} / Jo > where \FUU\ < K. Now adding the two inequalities we get
max[|u — vn\ + \wn — u\] < K I \Go(x, s)\ds max[|u — u n _i| + \wn-i - u\]2, / Jo I which yields the desired result. Therefore the proof is complete. We note that if we look for one sided bound for the solution of (2.1) as in the initial value problem [[5]], then we have the usual type of quadratic convergence.
References [1] R. Bellman, Methods of Nonlinear Analysis, Vol. II, Academic Press, New York (1973). [2] R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Value Problems, American Elsevier, New York (1965).
Boundary
[3] S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems. [4] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston (1985). [5] V. Lakshmikantham and S. Malek, Generalized Quasilinearization, To appear in Nonlinear World.
WSSIAA 3 (1994) pp. 459-466 © World Scientific Publishing Company
459
Holder Inequality and Periodic Solutions of some Planar Polynomial Differential Equations with Periodic Coefficients Raul Manasevich Departamento de Ingenieria Matematicas, Universidad de Chile Santiago, Chile Jean Mawhin Institut math&natique, University de Louvain B-1348 Louvain-la-Neuve, Belgium Fabio Zanolin Dipartimento di Matematica e Informatica, Universita di Udine 1-33100 Udine, Italy Abstract Using HSlder inequality and degree arguments, this paper provides a new simple proof and some consequences of a recent existence theorem of Srzed nicki for periodic solutions of some planar non-autonomous polynomial dif ferential equations when the coefficient of the highest order term are timeperiodic. To Wolfgang Walter, for his 66th birthday
1
Introduction
In a recent paper [6] (based upon previous considerations in [5]), and in the mono graph [7], Srzednicki has observed that the solvability for every h was insured for the problem z' = z2 + h(t), *(0) = z(T). or the obviously equivalent one
y = z2 + h(t),z(0) = z(T),
(1)
460
a special case of Y = p(z) + h(t), z(0) = z(T),
(2)
with p a non-constant complex polynomial and 7 the complex conjugate of z. The existence result of Srzednicki which concludes to the solvability of (1) is proved by combining techniques of the theory of isolating blocks, results on the computation of the index of a plane vector field and the Lefschetz fixed point theo rem. In particular, in the obvious equivalent form replacing the unknown function by its conjugate, Srzednicki's Theorem 0.8 of [6] or [7] states that the problem * =
£
M O * * * + cT**. *(0) = z(T),
0
where the bu are continuous functions over [0,T], c ^ 0 is real, r ^ s and r — s < 1, has at least one solution. Srzednicki's proof is quite long and technical and, in [3], the second author has given a simpler proof of a slightly more general version of this result which uses only classical topological degree techniques and simple a priori bounds arguments, based upon the use of Holder inequality. Srzednicki has also proved in [6] [7] some existence results for the more general problem *= £ M*)*** + cWx; *(0) = z(T), 0
when the bu are continuous functions over [0, T\, c ^ 0 is real, r ^ s , r—s < — 1, and w is an integer multiple of ^r. The aim of this paper is to show that a slight extension of this result can again be proved by using simple topological degree arguments and Holder inequality. An interesting feature of the proof is how the exponents of the higher-order nonlinear term reflect in the linear problem to which the homotopy is made. We also show by an example that the existence result does not hold when r = 0 and s = 1 in the equation above, i.e. when the problem is linear.
2
A n Existence Theorem for Periodic Solutions
Let r,a be nonnegative integers, T > 0, u € R \ {0}, c € C, and bu : [0,T] —► C be continuous functions, where A; and I are nonnegative integers such that k + l < r + s. We consider the existence of a solution for the T-periodic problem *=
bu(t)-Zkzl + cei"trz;z{Q)
£
= z(T).
(3)
0
To motivate the homotopy introduced to prove this existence result, we observe that if z is a possible T-periodic solution of (3), then we have r-'e-"'?
=
£ 0
e-*"6 w (t)2 f c + - r z' + c|z|2',
461 and hence 1
Vi)
" [..-iwt^-r+ll ,
s-r+ldt1 =
'
s-r+i
--twt-rf-r+1 e
2
e-^6w(0lfc+'-V+c|z|2V
£ 0
If e - *"' is T-periodic, i.e. if we assume that T for some m € Z \ {0}, then the integral over [0, T] of the first term of the left-hand member is equal to zero and we can proceed like in the proof of Theorem 1 of [3] to obtain an a priori bound for z. But the argument works as well for the family of equalities —T^[e"^J_r+1] + A—^—e-^z-'+1 3-r + ldt a-r + l = A £ e-iutbki(t)zk+'-rz' + Xc\z\2', A e ] 0 , l ] , 0
which follows similarly from the differential equation if z is a possible T-periodic solution of the equation
J'-(I-A)——-z = A 3
r
+1
X] 6«(0*V + AceT**.
0
We shall therefore use this homotopy to prove the following existence theorem, which slightly extends Szrednicki's one by allowing c to take complex values. Theorem 1. Assume that r + s > l , r < s — 1, c ^ O and that LJ = Qp- for some m 6 Z \ {0}. Then tie problem (3) has at /east one solution. Proof. As suggested by the discussion above, let A € [0,1] and z be a possible solution of the problem Z'-(I-A)
l
".iy=A
£
&w(*)2*z' + A c e " ' r * ' , z(0) = z(T),
(4)
with w = ^Y1 f° r s01116 »n € Z \ {0}. For A = 0, problem (4) reduces to the linear one
which has only the trivial solution z = 0 if m ? n(s - r + 1), n 6 Z,
(6)
462
and has the family of solutions z(t) = ae-**nt'T, o € C, if m = n(s - r + 1)
(7)
for some n 6 Z \ {0}. Moreover, it is easily checked that the non homogeneous problem
* ~ (a-tTlJT* = h®'
Z{0) = Z{T)
'
(8)
has a unique solution for each h € C([0,T],C) if (6) holds, and is solvable if and only if [Th(t)e-3i*nt'Tdt
= 0,
(9)
Jo
if (7) holds. If (6) holds, we shall therefore apply Theorem 3.10 of [4] (or equivalently the abstract Theorem IV.5 of [2] specialized to (3)), and if (7) holds, we shall apply Theorem 3.15 of [4] (or equivalently the abstract Theorem IV. 13 of [2] specialized to (3)). In both cases, let A € ]0,1] and let z be a possible solution of (4). Then e-*-*r-y - (1 - A)—^—re-****—■* 8—r+ l =A
bu(t)e-*utI-rJ'J
£
+
\cZ-rTz',
0£*+l
a-r + ldt[e = A[
-<*V" r+1 +
!" 8
~
r +
L
£
T
J
M 0 e - < u , V + - r z ' + c|z|3'].
(10)
0
Letting | | M = WfaIMOI. (0
N . ( 0 < J < r + «-l),
k+l-j
and using u = ^ ^ and the version of Holder inequality given in formula (6.1) of [4], we get
MjfT|2(*)|*»,tt<
463
J L . *P £ w o r * ~ * + J ^ ^ £ |2(,l*"+1 * This immediately implies that
(^jfl*Wla,)*
(»)
where fio is the largest positive root of the equation
The inequality (11) then implies the existence of some r € [0, T] such that |*(T)| < Ro.
(12)
Now, we also deduce from (4) and (11) that
o^fc+j
T
y
°
+M/ r | zW |r+.
J
°
( s - r + l)T7o
||6«||d/ T |z(0l a '*) W )
^ 0£k+J
+7=7+1^
T
-70
/oT|z(t)|2M0* + |c|(i/ o T |z(t)| 2 'A)* < * i ,
(13)
where i?i depends only upon flo,|c|,r,a,o; and the ||&w||. Consequently, we have, using (12) and (13), l*WI = \Z(T) + £ z'is) ds\
(14)
for all t € [0,T\, and each possible solution z of (4). Therefore, the existence follows from Theorem 3.10 of [4] with fi = B{R) and R > Ro +TRX if (6) holds. In the case of condition (7), we have, according to Theorem 3.15 of [4], to study the properties of the mapping F : C - C,a ~ i fT
e-a-t/r^H^n^^/T
464
+
bkl(t)*ke2i*kn"Tale-2i*lnt/T
E 0
+ c e 2i»n(«-r+l)t/T a r e 2iirrnt/T a » e -2«irjnt/T]^
= - ^ 3 + 1
[^/ T & W We 2< * ( *-'- 1)nt/T Ala*a'+cira*
E 0<,k+Kr+M
1
(15)
Jo
Thus F has the form F(a) = c|a| 2 p a- r +
£
/? w aV,
0
for some /?« depending only upon the i«, n and T. Therefore, if F(a) = 0, we have |c||ar +J <
I^Ha|* + '.
E 0<*+«r+«
and hence M < Rz, where Ra is the largest positive solution of the equation Mj/r+* -
E
lAil***' = o.
0
Finally, if we define the mappings F, G and H from C into C by F(a) =
Put*0' + <&«'> G ( a ) = <&*'> H(a) = a ' " r .
E 0
then, for any R > R3, the Brouwer degrees (see e.g. [4], [8]) deg[F,B(R),0], deg[G,B(R),0] and deg[H,B(R),0] are defined and, for sufficiently large R, are all equal to a — r > 1. Here we have used the fact that the degree on a ball centered at 0 of a linear invertible holomorphic mapping on C is equal to one and the degree of an on such a ball is equal to n. Consequently, all the conditions of Theorem 3.17 in [4] are verified and the proof is complete. Corollary 1. Ifs > 2, w = ^ for some m € Z \ {0}, and c e C \ {0}, then the problem *= E bu(t)z*zl + ceiutz',z(0) = z(T), 0
has at least one solution. Proof. It suffices to take r = 0 in Theorem 1.
465 Corollary 2. Us > 2, u = ^ for some m 6 Z \ {0}, c* : [0,T] -» C, (0 < j < s—1) are continuous and c € C \ {0}, then the problem
* = £ c,(0** + «*-*«', z(0) = z(T), i=o
has at ieast one solution. Remark 1. When r + s = 1 and r < s - 1, i.e. when r = 0,s = 1, problem (3) reduces to the linear one zJ = b00(t)+ceimtz,
z(0) = *(T).
In contrast to the nonfineor situation covered by Theorem 1, this problem does not need to have a solution for each forcing term 6oo, as shown by the example * = ie2itz + ie", z(0) = «(2»), which has no solution. Indeed, if z is a solution, then u(t) = eltz(t) will be a 2ir-periodic solution of the equation tf = t(u - H) +1, so that, letting u = v + tty, ty will be a 27r-periodic solution of the equation -w' = 1, which is impossible. Remark 2. When too = 0 in (3) then the problem admits the trivial solution z = 0. Supplementary conditions to Theorem 1 have been given in [6], which insure the existence of a nontrivial solution. They follow immediately from the argument above by replacing the open ball B(R) in the space Cr([0,T],C) of continuous Tperiodic complex-valued functions over [0, T\ by the open annulus B(R) \ B[e] with sufficiently small e.
References [1] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Thins. Amer. Math. Soc. 329 (1992), 41-72. [2] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conf. in Math. n. 40, Amer. Math. Soc, Providence, 1979. [3] J. Mawhin, Periodic solutions of some planar non-autonomous polynomial dif ferential equations, to appear.
466 [4] N. Rouche and J. Mawhin, Ordinary Differential Equations. Stability and Pe riodic Solutions, Pitman, Boston, 1980. [5] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, J. Nonlinear Anal., to appear. [6] R. Srzednicki, On periodic solutions of planar polynomial differential equations with periodic coefficients, J. Differential Equations, to appear. [7] R. Srzednicki, A Geometric Method for the Periodic Problem in Ordinary Dif ferential Equations, S&ninaire d'analyse moderne No. 22, University de Sherbrooke, 1992 [8] E. Zeidler, Nonlinear Functional Analysis, vol. I, Springer, New York, 1986
WSSIAA 3 (1994) pp. 467-482 © World Scientific Publishing Company
467
Nonlinear discrete inequalities and stability of difference equations Rigoborto Medina* Departaineuto de Cieiicias. Iiistituto Profesional de Osoruo Casilla 933. Osoruo - Chile Manuel Pinto* Departaineuto de Mateninticas. Facultad de Cieucias Universidad de Chile. Casilla G53. Santiago - Chile
Abstract Discrete inequalities of Bihari-type with several nonlinear terms are considered. The stable properties of the estimations obtained allow us to get the stability of very general nonlinear difference systems. A.M.S. Subject Classification: 3JK10 Key words and Phrases: Nonlinear discrete inequalities, Difference systems, Stability.
1
Introduction.
The present paper deals with SOUK- nonlinear inequalities which enable us to find an estimation of the solutions of difference equations [1-10]. They also play a handy tool in the study of the stability, asymptotic behavior and other properties of the solutions of difference equations. Many results have been published recently concerning bounds of solutions of inequalities (see for example, [11]). •Research supported in part by Proyecto de Investigacion Iiistituto Profusional de Os oruo and Fondccyt 92-0148 'Research supported by Proyeclos DTI Universidad de Chile E 3063-9222, and Fondecyt 0839-93
468 Our object here is to pn-senl some inequalities and their applications to the study of the stability of perturbed difference system !/(» + [) = A(„)y(n)+g(n.y(„)),
(1)
where the matrix A(-) is defined for all n € No = N U {0}, y = g(u,y) is defined on the produc space N 0 x R'" and R'" denote the m-dimensional real euclidean m-space; }• is an m-vector. In section 3 we study the difference equation (1) where the linear system j-(ii + l) = A(n)x(n),
(2)
is asymptotically stable and I he perl in bation g : N 0 x R l
m
PGN
—► R'" satisfies (3)
i=i
with A,. ic, suitable functions. We prove that under certain conditions, the solutions y = y(».»o>//o) ot the perturbed system (1) such that |j/o| is small enough are defined on all of No and the zero solution of equation (1) is asymptotically stable. An important class of admissible functions «', is any polynomial system «<,(«) = » \
->-> l
(1 <»'<;»),
i.e. where g satisfies
M».0)i(«)i»r
(!)
;=i
for which we prove that Kq. (1) preserves the exponential stability of eq. (2). As oilier kind of results, we prove that if in (4) A,(») —» 0 as n —* oo, then all the solutions of the equation (1) tend to zero as n —» oo.
2
Discrete inequalities.
The discrete inequalities obtained here are discrete analogs of the integral inequalities established in [13]. Let .S'(N) be the vectorial space of real sequence 5+(N) C 5(N) the cone of uomiegative sequences. Define on .S'(N) the operators: /(«)(ti):= £ > ( * ) . f i > l ,
/(«)(0) = 0,
469 and (lie "differential" operator defined by £>(«)(«) := Ai/(/») = u(n) - i/(» - 1). H > 1, where i/ € * ( N ) . The integral operator operators verifying
/
is monotone, furtiier
D(I(u))(„)
/
and
D are linear
= ,,(„)
(5)
and I(D(u))(n)
= t,(n)-u(0).
(6)
Now, we consider the discrete inequality
«(»)<<- + DEAi(<-).r,(«(/'))],
(7)
t=i k=\
where 111) The functions «r, : [d.oo) -♦ [0,oo) (1 < f < />) are continuous and nondecreasing, Wj{u) > 0 for u > d and (r,+,/u>, (1 < i! < p — 1) are nondecreasing on {d, oc). //j) i/ : N —> [d, oo) and A, : N —► (O.oo) are functions, c is a constant such that c > d. We define the functions i) ]],(„) = /M". -^L.. tf > 0, u, > 0 function.
(1 < ? < />) and Wrx
their inverse
ii) fu(n) = ti and ifi = s, o <;,_, o • • • (,,, S,(M) = H'r'["'.(«) + Of], where a; > 0 is a constant. Thus, we can establish the next theorem.
(8)
470
Theorem 1 ([12-1JJ) Let J £ R. Assume such that '»->
adm) := £
foe
A,(A-) < /
/ / , ) and H3).
Let in € N
//,
-^-
(I < i < p),
(9)
W»m //»r functions
A„(*)],
for any n < in. Proof: We sketch a proof of this theorem which is by induction on p. We rewrite the inequality (7) in the following form: p
v(n) < c + £ Si(n - 1,«, «.',-), «=i
where St(n - l,«,u>,) = £ A,-(*)»i(u(*)),
(1 < «• < ;»)•
(10)
We define r,(n)
=
c+5'i(n-l.K,tt-i),
i'.(n)
=
5,-(ii-l,«,«r,-)(2
r = E?=i»'..
Hence by (5) AP(»)
=
E?=,A,(»-1)«-,(./(»-1))
<
A,(n - l)(n(«'(» - I)) + Ef=i A,(» - l)wj(t'(fi - 1)).
So - 7 ^ 3 T T T < A 1 ( » - l ) + ^A,(n-l)Hv(r(n-l)). where «\- = u\/tvt. But if W is a primitive of noiulecreasing function w we have
l/u>,
(11)
then for any
471 AIV(r)(»)
=
ir(r(»))-W'(i.(H-l))
_
fi")
<
-
Ji_
f(»)-t(n-l) _ Mlr(n-l)) ~
,. 2 > A.(n) u,(v(>i-l))'
because t' and w are noiulecreasing. So, by (11) and (12) we get A»F,(r)(») < A,(n - 1) + £ A,-(n - l)«v(t'(" - 1))-
(13)
By applying the operator / to inequality (13), we obtain /(A»',(r))(w) < /(A,)(n - l) + £ / ( A i H \ ) ( i > ) ( n - 1), or, by (6) H,(r(»))
<
ir,(r(l)) + UZ\ A,(fc) + Ef=j *.(»» - 1,», u>,),
<
H'i(c) + UZ\ A,(t) + E f . » S ( » - 1, »,*,).
(14)
Let u' = M',(t>) and c* = H',(r) + £ £ 7 * A,(*). then for » < m, from (14) we get
«'(») < c' + E #(" " 1V,,{,« ° "T1)' l'm2
where S\ is defined by (10) with i / = «' and tr, o IV,"1 instead of IP,-. Hence «'(») < «•' + E *.'+.(» - I V , « \ o IV,-1)-
(15)
»=2
Now, we verify the hypothesis of induction for p = 1. By (11), for p = 1 we gel
»r,(t'(»o)
472
v(v)<\\\-,[\\\(c)
+ YtXi(k)l
for n < m, where m € N is such) hat m _ 1
fX,
JS
JC
(]
tVi{»)
The last afirmation is the conclusion of the theorem for p = 1. Then now we can apply the inductive hypothesis to (15) to deduce the theorem in the same way that Theorem 1 [13]. So, the proof of Theorem 1 is complete. As a direct consequence we get: Theorem 2 Under conditions //,) and / / 2 ) . with 0 < Aj € fi(N), 1 < « < p, if c < (^"'(oo) then for any v € N , we hare «(")<*p(c)
00
We remark that if r*' ds / - T T = *>
0 <»'
07)
J4+\ «";(.«)
then pi,, (and *) is defined for all a. Thus (10) is valid for all c > 0. The dual condition to (17), namely
L JJ+
~n
= x
0 <'
08)
ll'i(s)
implies thai an}' ipi, (and cj) is defined for all u small enough. Then (JO) is valid if r is small enough. Moreover, (18) implies
Under the same conditions of
«(»») < c + £ A0(*>(A) + £ [ £ A,-(*-)«'i(«(*))l, 1= 1
t = l
Jr=l
473
wheiT X0 is a sequence of nonnegatires real numbers and t«'i(o«) < r,-(o)tr,-(«);
o > 1,
it > 0, r, € C([l,oo)).
Then for all n < m we have «(»)
<
U]sl(l
+
XoUniV-^W^^c))
+
El~jAo"(A-)rp(Ao(fr))Ap(^)],
when we use the same notations of the Theonm I with Ao(Jfc)r;(A„(*))A,-(*) instead of \j{k)
and Ao(t) = "*»,( 1+AoO'W,
*€N.
Proof: Consider the linear inequality «(»)
(19)
where
I=I
t=i
Introducing the abbreviation
i>(») = c(")+X>o( *•)«(*). k=\
we get ti < t' and At- = Ac + ,\0u < Ac + Aor. Multiplying (20) by (A 0 (»))
-1
we obtain
AWA„) = ^ - ^ i < V ( » ) A c ( n ) , A0(w)
Ao(n-l)
(20)
474
So, v(n) A 0 (»)
v{l) ^.^i , . , , . . . A 0 (l) ^
...
or n-l
v(n) < c(l)A 0 (n) + A0(n) £
Aj'(*)Ac(*).
Jt=i
This implies «(»») < c(l)Ao(n) + £ ( 0 ^ ( 1 + X0(j))]Ac(k).
(21)
fc-i
Since Ac(Jk) = Ef=i ^(fcJwjM*)), and by (21) we get u(»») < cAo(»0 + D D » t
+
i(l
+ A
o(»)]A,(fc)t»,(«(fc))
(22)
Multiplying (22) by AQ l ( n ) , then z — u/A 0 satisfies *(n) < c + X:iV(it)r < (Ao(Jb))A j (fc)u>.(r(*))], tml
and the result follows from Theorem 1. Theorem 3 Under the conditions of Theorem 1, suppose that there exist functions r, : [d, oo) —»fO,oo) such that w4(au) < ri(o)uu(u); a > d, u>d,
(1 < t < p).
(23)
Let n < m, m € N such that r M l . m ) := c ' V ^ c ) f ) A.(t) < f °
- £ r , ( l < » < p),
t/Aene (fi(l,c) is given by (8) with u = l onrf c*,(l,m, c) = o , ( l , m ) - r , ( c ) ' c - 1 instead a , ( l , m ) . If the function u satisfies the inequality (7), then «(»»)<
(24)
475
In fact, from (7) and (23) we obtain c-lu(„)
< 1+ £ £ >=1
A,-(fc)f"Ir.-(r)«v(r-,tt(fc))
* = l
and hence Theorem 3 follows at once from Theorem 1. Moreover we ran establish Corollary 2 Under Iht conditions of Theorem 3 assume that r,(r)/r-»0
if c^d
and A, € ^i(N) (1 < i < p). c' = c"(l\) such that
(1 < i < p)
Thin for every
I\ > 1
«(»)
(25) Hurt
(fists
(26)
for all u € N and c < r*. In fact, putting o, = J2V=i l-M*)l (1 < ' < />)• because of (25), there exists c = c" satisfying c - 1 • rt(c') • ev, < / "
-A-
(1 < i < p),
(27)
where the inequality is strict for i = p. Thus the functions
3
Stability
Consider I he equation y(n + \) = A(n)y(n) + g(»,y(v)ly(„0)
= yo,
(28)
where y € R m , A(n) is an »i x m nonsingular matrix, g : N0 x Bg —» Z?< and for all » e A/o fl»(».0) = 0 and AQ = {»0<"o + li»o + 2. • • •} (n 0 > 0 is a fixed integer). When g is small in the sense to be specified, one can consider Eq.(28) as a perturbation of the equation •r(n + l) = v4(»i);r(»)i
(29)
476
and the question arises whether the properties of stability of Eq. (29) are preserved for Eq. (28). The following theorem offers an answer to such a question. Assume that the following conditions are satisfied. A) Assume that there exist positive constants I\ and a such that |*(«, k)\ < A7—("-*> for
n > k > n0
where *(»,£) is a fundamental matrix of Eq. (29) satisfying * ( « , » ) = / (The identity matrix). B) The perturbation g = , y) satisfies:
M»^)I<EA.(»)«-,(M). pen,
(30)
where t') «'i (1 < * < p) verify condition (Ht) for d = 0 and they are " r,—inultiplicatives", i.e. there exist nonnegative functions r,- defined on (0, oo) such that irj(on)
« > 0, a > 0
(31)
and in addition / ' - ^ - : = oo Jo* »',(.«)
(!<»<;»)
(32)
D) A,- (1 < i < p) are positive .sequences such that eff(*+l)-r,(e~<'*r)'A1- € ^ and
A" £ A,(fc)c"<*+'> ■ r,(e-») < T
A -
(33
(1 < i < p), with a > 0 constant. We remark that by [12, 13] condition (32) implies the stability property w ( 0 + ) = 0(1 = 1,- •-,/>). In order to not interrunipt the next- proofs we state here the variation of constants formula for discrete in homogeneous system (28).
>
477
Theorem 4 ([14, Th. 4.6.I]) Tin solution y(»i,»0.tfu) of Eq. (28) satisfies the equation -i-l
y(n) = * ( » . ult)yu + £ when
$(H.H0)
* ( » , ; + i )fj(j,y(j)h
(34)
is the fundamental matrix of the Eq. (29).
Theorem 5 Assume condilions A) - D) . Then tin solution y(n) = 0 of Eq. (28) is asymptotically stable. Proof: By condition A, we have |*(«,»o)| < Ke-"l"-no),
a > 0,
K > 0.
Hence, by applying (30) and (31) to (34) we obtain \v(n)\
<
Kc-"^-"^\yo\
+ A T.U[Ul\
x
wA \y(k)\)]
<
A'e-"<"-"°»|i/o| + KZUlUZl
x
r,-(r-*)«-'.-(e'*ltf(*-)l))-
A,(*)e-*<-*-»
A,(A-)e-'<-*-,>
Thus we get, e"|y(n)|
<
AV-|jfo| + A' ZPM[LM
x
ri(<-»k)wi(e''k\y(k)\)}-
A,(*K ( f c + I )
Hence, by Theorem 1 we have |y(»)| <
(-an-W,:,[\VP(
+
AE2:,'-M<)e",H,,''V('-"l)j,
which is well defined for all H € A„0 if |j/ol ' s small enough. Then \y{n)\<(-""
•^(A'e—ljtol),
(36)
478
for |jto| small enough and all it € A'„0, where ,; (1 < t < p) is given by (8) with
k=i +
Since (32) implies
(37)
where A is a constant matrix we obtain: Theorem 6 Assume that conditions B, C and D an satisfied and in addition the matrix. A has all the eigenvalues inside the unit disk. Tltcn the zcw solution of Eq. (37) is asymptotically stable. Theorem 7 Let conditions A) - C) be satisfied, when the functions t < p) satisfy £ > ' * • r , ( e - * ) < oo;
« = l,2,--,p.
rj (1 <
(38)
In addition assume that ii)' A, : N^, —» [0,oo) such lhat X,(k) —» 0 as k —► oo. Then all the solutions y(n,n0,yi})
of Eq. (37) It nd to zero for n —» oo.
Proof: By (ii)', given c > 0 there is an m 0 6 A'„0 such that for all i = 1,2, • • •, p and it > vt0 > »0- Then we get t"n\y(»)\
|A,(»)I<~.
< AVTmoM»»o)l + t - - A - - E f = , [ E ? : , ' c < " ' + " x
r,-(f-*)«.v(«"*|»(fc)|)].
By introducing the new variable ii(n) = c""|j/(»)|. we sec that «(n)
<
Kti(m0) + s ■ A • £?=i[T.nkZ\ e"lk+1)
x
r,-(f-*)«'i(«(*))].
479 Using Theorem 1 again, we arrive at «(n)
li;l[H„(^.1(/u<(mo))
<
for n > m0. This implies \y(n)\
<
e-""H p -'[ir p (^_,(A'e'""o|y(mo)|)) (39)
+
c-A'E*;ie'
(t+,,
Tp(e-'»)],
thus, we can choose £ small enough such that the right member of (39) is defined for all n > in0. Moreover \y{n)\ < e-"» ■ fPiKt""\y(m0)\),
(40)
where yJj (1 < » < p) is given in (8) with o; = / w £ f * < f c + , > - r 1 ( e - < ' t ) . Ine<juality (40) implies the convergence of y(v) towards zero as n —► oo. Let M = in(e) given by M = m«.r{|A((A)|/l < * < / » , 1 < k < m0). Hence, using the inequality (35) for 1 < n < m 0 , we obtain |»(»)| <
e-'"'H'- , [Hp(Vp-.(A'e'"" 0 l»(mo)|))
+
MI
HI)
By (38) we can choose |y(i» 0 )| sufficiently small such that the right member of (41) is defined for all », 1 < »i < m0- Moreover l»(n)|<e—.v»,(AV-|»(m„)|) Then the result follows. Theorem 8 Assume that condition (A) holds, and the perturbation g(»,y) is defined on A'^ x R m such that
i»(».j/)i<EA,(»)| y r'.
P
eN
g =
480
whtix -)j > 1 (1 < i < p) air constants and the functions ait positives, such that
f> l (A•) £ '' fc < , -^ < oo,
A,- : N^ —» R
(l
Then the trivial solution of equation (28) is asymptotically stable. Proof: Let iv(u) = u' + 1 and c(s) = H'-'ni'Ja) + o], s > 0, o > 0. We will prove that if * is small enough, then there is a constant M > 1 such that <(*) < Ms.
(42)
In fact, if ( = 0 we have for all .•» > 0 «(*) = H - , [ l H * ) + o) = c°* If ( > 0. then ll'(w) = -u-'/C,
u > 0; W-l(v)
= (-(r)-l,\
v<0.
So we obtain <(,) =
H - ' [ H » + o.]
=
s[\ -
<
Ms.
ts'a]-1'1
for a sufficiently small, where M > 1 is a constant. Iterative application of (12) implies that for |(/ 0 | small enough, there exists a constant Mt > 1 such that y» p (A't'"-||) t o |)
(«)
By (43) we have |y(»,no)|<e—.^(AV-||to|). Then, the asymptotic stablity of the solution y(n,n0) follows by (43) and (44).
(44) = 0 of Eq. (28)
481
References [1] R. P. Agarwal and E. Thandapani, On discrete generalization of Gronwall's inequality, Bull. lust. Math. Acad. Sinica. 9(2), (1981). 235-248. [2] S. Mc Kee, Generalized discreta Gronwall lemmas, Z. Angew.Matli. Modi. 62(9) (1982), 429-134. [3] B. G. Pachpatte, Finite difference inequalities and their applications, Proc. Nat. Acad. Sci. India, Seel. A 43 (1973), 318-356. [4] B. (J. Pachpatte, On discrete inequalities related to Gronwall inequality, Proc. Indian Acad. Sci. Sect. A 85 (1977). 26-40. [5] J. Pojienda and J. Werbowski. On the discrete analogy of Gronwall lemma, Fasc. Math. 11 (1979). 143-154. [6] R. Rodhcffer and \V. Walter, A comparison theorem for difference in equalities, J. Diff. Eqs. 41 (1982), 111-117. [7] S. Sugiyama, Stability problems on difference and functional difference equations, Proc. Japan Acad. 45 (1969), 526-529. [8] En Hao Yang, On some new discrete inequalities of the Bellinan-Biliari type, Nonlinear Anal. 7 (11)(1983), 1237-11246. [9] En Hao Yang, On some new discrete generalizations of Gronwall's in equality, J. Math. Anal. Appl. 129(20) (1988), 505-516. [10] \J. Zamkovaja and B. I. Krjukov, The stability of nonlinear systems of differential and difference equations, Different ial'nye I'liivncniya 13 (1977), 756-757. [11] Z. 13. Tsalyuk, Some methods of stablisliiiig bounds for solutions of inequalities, Translated from Differentiarnye I'ravneniya, 24(2) (1988), 250-258. [12] M. Pinto, Des inegalitos fonctioniielles et quelques applications, Notas Soc. Math. Ch. Vol. XI (1992). 52-68. [13] M. Pinto, Integral inequalities of Bihari-type and applications, Funck. Ekvac. 33 (1990), 387-403.
482
[14] M. Pinto, Variationally stable differential systems, J. Math. Anal. Appl. 151 (1990), A" 1,254-260. [15] R. Medina and M. Pinto, On the asymptotic behavior of higher - order nonlinear differential equations. J. Math. Anal. Appl. 146 (1990), 128140. [16] V. Lakslunikanthani and D. Trigiante, Theory of difference equations with applications in numerical analysis, Academic Press (1988). [17] R. Medina and M. Pinto, Asymptotic representation of solutions of linear second order difference equations, J. Math. Anal. Appl. 165 (1992), A"" 2, 505-516. [18] II. Medina and M. Pinto, Asymptotic behavior of nonlinear second order difference equations, Nonlinear Analysis T.M.A. Vol. 19 (1992) N" 2, 187-195. [19] R. P. Agarwal, Difference Equations and Inequalities. Marcel Decker, New York, 1992.
WSSIAA 3 (1994) pp. 483-491 © World Scientific Publishing Company
483
MONOTONICITY A N D OLVER TYPE COMPARISON RESULTS FOR DIFFERENCE EQUATIONS
Malgorzata Migda and Jerzy Popenda Institute of Mathematics Technical University Poznan, Poland
Abstract. For linear m-th order difference equations (inequalities) sufficient conditions for positivity and monotonicity of solutions satisfying nonnegative initial conditions are provided. A comparison theorem of Olver type is also obtained.
As in differential equations comparison theorems play an important role in the study of difference equations. Such results can be found, for example, in [2,3,5], and in the recent monograph of Agarwal [1]. However, the results we present here extend the areas of applications compared to those contained in [4]. In this paper, we shall consider the m-th order linear difference equations of the form m-l
xn+m = £<*„+<,
n&N
(El)
t=0
where a' (» = 0,1, • • •, ro — 1) are real valued functions (sequences) defined on N. By JV we denote the set of nonnegative integers, and by R the set of real numbers. For any function z : N — ► R the forward difference operator A is defined by Az„ := zn+i — zn, n € N. We will consider such solutions of (£1), which satisfy the initial conditions xo>0, A*i>0, i = 0,l,••■,m-2
(C+)
484
and compare the solutions of (El) with the solutions of the equation m-l
yn+m = £ 6 nyn + .-,
neN
(E2)
i=0
where b' : N -> R. The proofs of our results are based on simple induction, and are similar to earlier known results and hence in most parts will be omitted. Throughout, we will assume that the void sum is zero. By induction it can be proved that if a', t = 0,1, • • •, m — 1 are nonnegative on N then every solution of (El) with nonnegative initial values Xi, i = 0,1,• • • ,m — 1 is nonnegative on N. If, furthermore, a™-1 > 1 for all n 6 N, then every solution of the problem (El), (C+) is nondecreasing also. Indeed, it follows from the fact that the equation (El) can be arranged as follows Ax n + m _! = «
- 1
- l)z n + m _ 1 + £ <*„+,•,
n£N.
i=0
Remark 1. Let us consider the question : does the condition a™-1 > 1 is necessary for the solution of (El) with initial conditions (C+) to be nonde creasing? It is evident that if all oj, > 0, for t < m - l , n € N and a™"1 G [0,1) then the solution of (El) with the initial conditions x, = 0, t = 0,1, • • •, m — 2, i m _j = a > 0 is nonnegative on N, but Ai m _i = (a™-1 - l)a < 0, so this solution is not nondecreasing at least at one point. Furthermore, because the functions Ek((o,
••■ , Zm-l)
=
Zk(o H
1"
Xk+m-2(m-2
are continuous and F*(0,•••,0) = 0, whenever we have positive coefficients **)•••> xk+m-2 and a positive constant at we can find positive £0 = a°, • • • > £m_2 = a ? - 2 such that ■f*( a t.---.ar - 2 ) = ]C a'kxk+i = " i 1=0
Taking a* < xt +m _!, denoting ek = ak-Xk+m-i,
and considering the function
Gk(t) = ^m+*-i{ + £k we can find positive ( = a™-1 such that for given
485 z m + t _ i , e* we have Gkia™-1) < 0. This allows us to construct an equation (£1), with a3n > 0, for n € N, j = 0 , 1 , • • •, m - 1 which for the given positive initial values satisfying conditions ( C + ) possesses a positive solution z on TV with the property that A z n < 0 for all n > m — 1. E x a m p l e 1. The above considerations show the necessary character of the condition a™ -1 > 1. On the other hand, solving the initial value problem x n + 2 = 2x n + 1 - nx n ,
ne
N
Xo = O/l > 0, I I = Ct2 > CCi, we obtain x-i = 2a 2 , x3 = 3a2, z 4 = 2a 2 , zs = — 5o/2, From this we see that for arbitrary a j > 0 we get z< < Z3, moreover Z5 is negative. Hence the assumption aJJ1-1 > 1 is not sufficient for the monotonicity of solutions of the problem (E\), (C+). Now we turn our attention to the sign conditions imposed on the coefficients 3
a for j < m — 1. T h e o r e m 1. Let aj? - 1 > 1 for all n 6 N and let a''' : N -+ R, i = 0 , 1 , • • •, m 2 be such that there exists a decomposition of the set S = {0,1, • • •, m — 2} into r disjoint subsets S j , • • •, S r each of them possesses one of the following properties (i) for every t 6 S/,, a'n > 0 for all n 6 N, or (ii) for j = minigs^ i, a3n < 0 for all n € TV and for all other s £ St, s ^ i> a n ^ 0, moreover 23, 6 s, aj, > 0 for all n £ N. Then every solution z = {z n }^_ 0 of (£1) with initial values x;, i = 0, l , " ' , m — 1 satisfying conditions (C+) is nonnegative and nondecreasing on N. Proof.
The proof is based on mathematical induction, and the following
rearrangement of the equation
(El)
A x n + m _ , = (
486
Suppose that x„ > 0, Ax n+ , > 0, i = 0,1, • • • ,m - 2 we get X„+m-l > I n + n - 2 > • ' • > X„ > 0, t = 0, 1, • • • , m - 2.
(2)
Let us suppose that the set 5* = {»i,- •,»*}, *i < *2 < • • • < »/t possesses property (ii), that is a% < 0 for all n e N, while aj,* > 0 for all n € N, and 3 > 1. For such a set, applying (2) we obtain T.i€s„
= <*»'*»+.> + • • • + a?- 1 *„+,»_, + <*xn+,„
> <• *„+,•, + • • • + <#-' x n+il + <* x„+<1 =
(E,gs4 < ) *»+.-, > 0.
The above estimate applied to all the sets with the property (ii) gives us Ax„+m_i > 0, because rest of the terms on the right hand side of (1) are nonnegative. Example 2. Let in (£1) ro = 2a + l, o j f > l , <£' + 1 >0, a**1 + a*'> 0, t = 0,1,---,* - 1. Here the terms of decomposition S; are 5,- = {2t — 2,2» — 1}, t = 1,2, • • •, a. Such type of equation is for example the following *n+s = x n+4 + n 2 x n+3 - nx n+2 + x n+1 + 2x n , n £ N. Here for S\ = {0,1}, «° = 2, a\ = 1 so condition (i) of Theorem 1 is satisfied, while for 52 = {2,3} we have a* = - n , a% = n2 which yield a% + a^ > 0, therefore condition (ii) holds. Hence Theorem 1 is applicable. In the equation x n + s = 2xn+4 + (n + 2)x n+3 + 2x n+2 - x B+1 - (2/(n + l))x„, the decomposition can be 5, = {0,3}, S2 = {1,2} or 5X = {0,2}, S2 = {1,3}. Remark 2. Analyzing the proof of Theorem 1 we can observe that the state ment remains true if the number r of subsets 5* and elements each of S* depend possibly on n. Such generalized theorem can be applied in the follow ing example.
487
Example 3. In the equation x n + 4 = 2x n+3 + x n + 2 + (-l) n x n + 1 + (-l) n + 1 x n ,
neN
we have Si(n) = {0}, S7(n) = {1,2} for n = 2k + 1, Sx(n) = {0,1,2} for
n = 2k,
keN.
This generalized theorem we have mentioned above has the following form. Theorem 2. Let cj, > 0 for all n G N and t G {0,l,---,m - 1} where c„ = a„
- 1 , cn = cn +an
Then every solution x = {xn}^=o °f (^)
, w
i = l , - - , m - l , n G W. ' ^ initial values x,-, t = 0,1, • • •,
m — 1 satisfying conditions (C+) is nonnegative and nondecreasing on N. Proof. Suppose that x„ > 0 and Ax n+ i > 0 for » = 0,1, • • • ,m — 2 to obtain Ax n+m _i
= c°x n+m _, + < - 2 x n + m _ 2 +
1- a°x„
> 4x„ + m _ 2 + o"- 3 x n + m _ 3 + • • • + a°nxn (3)
> ... >
From (3) the proof follows inductively. Corollary. The above two theorems allow us to compare two solutions of the equation (El) as follows : Let the assumptions of Theorem 1(2) be satisfied, moreover x1 = {x^}|JL0 and x2 = {x*}jJL0 be two solutions of (£1) such that x£ > xjj, and Ax} > Ax? for i = 0,1, • • •, m - 2
(4)
then x\ > x2n and Ax* > Ax* for all n £ N. Proof. By linearity of (E\) the sequence x = xl — x2 is a solution of (El). Furthermore, from (4) it follows that this solution satisfies conditions (C+),
488
therefore by Theorem 1(2) we get xn > 0 which yields xj, > x* and Ax„ > 0 and hence Ax^ > Ax*. Example 4. The solution of the equation Xn+3 = *n+2 + *n+l - Xn,
f» 6 N
satisfying conditions x0 = 0, Ax; = 1 for »' = 0,1 is x n = n for all nGiV. Therefore, by the Corollary we conclude that for any other solution wn such that wo>0 and Au>< > 1 for i = 0,1 we have u>„ > n, and Au>„ > 1 for all n e N. Remark 3. If in Theorems 1, 2 the initial condition inequalities (C+) are replaced by the inequalities xo<0, Ax,<0, t = 0,l,---,m-2
(C-)
then the solutions x of the equation {El) will be nonpositive and nonincreasing on N. If in Theorems 1,2 we replace both in the assumptions for the coefficients and in the initial conditions (C+),((C—)) the non sharp inequalities >,(<) by the sharp ones >, (<) then in conclusion we get the sharp inequalities. Now, we turn our attention to comparison theorems for (El) and (E2). We can formulate many types of such theorems suitable to the respective rearrangement of the equations (El), (E2). However, such results on using the transformation we have provided in Theorem 2 can be expressed as follows. Theorem 3. Let x = {x n }£i 0 and y = {yn}%L0 be two solutions of the equations (El) and (E2) respectively, and initial values Xo,•••,x m _i and Jfo) • • • t y?n-i satisfy conditions i o > y o > 0 , Axi > Ay, > 0 , » = 0,1,• • • ,m — 2. If <><>0
for neN,
i = 0,1,- • • ,m - 2,
(CC)
489 where
c°n = a:-i-i)C;l = c i +a;r- 1 < = 6r 1 -i, < = c1+
(5)
Proof. Nonnegativity of x and y as well as their monotonicity follows directly from Theorem 2. To prove (5) let us observe that from Ax n > Ay„ we get *n+i - y n +i > xn -
yn.
Let us suppose that for some k € N Xk>yk
and Aii > Ay,- > 0 for t = k, k + 1,- • • ,k + m — 2.
(6)
Notice that for any nonnegative constants a, /?, A, B, C, D such that a> 0 and A - B > C - D > 0 we have aA - 0C > aB - 0D.
(7)
From (El) and (E2) we obtain m-2
m-2
Axjt+m_i - Ay i + m _! = c2xfc+m_i - d?kyk+m-x + J^ aix t + i - J^ &iy*+i. (8) i=0
«=0
Now applying (7) with a = c°k, 0 =
B = xt+m_2, C =
yk+m-i, D = y*+m_2 we obtain c°kxk+m-i -
Ay t+m _i
> c°x t+m _ 2 - d?kyk+m-2 + aj , - 2 x t + m _ 2 m-3
-W^yk+m-i
m—3
+ $3 a*X*+' _ 12 b'kyk+i i=0
i=0 m-3
m-3
= 4**+m-2 - 4y t+m _, + £ 4xt+l- - £ Vkyk+i1=0
1=0
490
Following this way we obtain by (6) and (7) Azt + m _i - Ayt+m-i > c j - ' x t -
Furthermore, again from (6) for t = k we obtain *M-i-y*+i > xk-yk
> 0
and hence z^ +1 > y*+i, so all inequalities (6) are fulfilled starting now with k + 1 instead of k. Since for k = 0, (6) are exactly (CC), the mathematical induction is applicable and the theorem is proved. Example 5. Let us consider the following equations 2n+3 =
Xn+2 + Xn+i
- X n , Tl 6 N
and yn+3 = yn+i + —-ryn+i — T T " - ' n+1 n+1 Let initial conditions satisfy the relations
n N
^ -
yo = x 0 = 0, 0 < Ay< < Ax, = 1 for t = 0,1.
(9)
The solution of the first equation is x n = n. Because assumptions of Theorem 3 are satisfied the solution of the second equation with the initial conditions y,-, * = 0,1,2 such that (9) holds, is nonnegative, nondecreasing and bounded from above by n, that is y„ < n for all n G TV. Of course Theorem 3 can be applied for obtaining upper (like in this ex ample) or lower bounds for the unknown solution of one equation in terms of known solution of the second equation. Remark 4. A similar comparison theorems can be obtained for perturbed equations (El) and (£2) of the form m-l
x„ +m = 51 a'nxn+i + f{n, x„, • • •, x n + m _i), n G N t=0
and m-l
Vn+m = 5Z KVn+i + g(n,y„, • ••, y„+ m -i), n e N. ■=0
491
For this, to the assumptions of our theorems we need to add the conditions f>g>0
on the set
N x Rm
and both the functions / and g are nondecrasing with respect to their, second, third, ••-, (m + l)-th arguments for each n € N, then the conclusions of Theorems 1, 2 and 3 do not change. Remark 5. Comparison results like this given in the Corollary for two solu tions of the same equation can be given based on Theorem 3. Finally, we remark that Theorems 1 and 2 can be rather easily extended for inequalities. REFERENCES
1. R.P.Agarwal, Difference Equations and Inequalities : Theory, Methods and Applications, Marcel Dekker Inc., New York, 1992. 2. S.R.Grace, B.S.Lalli, C.C.Yeh, Comparison theorems for difference in equalities, J. Math. Anal. Appl. 113 (1986), 468-472. 3. F.W.J.Olver, Bounds for the solutions of second-order linear difference equations, Journal of Research of the National Bureau of Standards No.4 (1967), 161-166. 4. J.Popenda, On the solutions of finite difference equations, Fasc. Math. 22 (1991), 123-135. 5. R.Redheffer and W.Walter, A comparison theorem for difference inequal ities, J. Diff. Equat. 44 (1982), 111-117.
WSSIAA 3 (1994) pp. 493-500 © World Scientific Publishing Company
493
A UNIFIED FORM OF THE CLASSICAL MEAN VALUE THEOREMS
ZSOLT PALES 1 Institute of Mathematics and Informatics, Lajos Kossuth University H-40J0 Debrecen, Pf. 12, Hungary
ABSTRACT The main result of the paper obtains a Cauchy-type mean value theorem for the ratio of functional determinants. It generalizes Cauchy's and Taylor's Mean Value Theorems as well as other mean value theorems known for divided differences.
1. I n t r o d u c t i o n T h e a i m of t h e present note is t o offer a unified approach to most of t h e m e a n value t h e o r e m s known in elementary analysis. T h e m a i n result of t h e p a p e r reduces t o Cauchy's a n d Taylors's Mean Value Theorems in special settings. T h e Mean Value T h e o r e m of divided differences, moreover t h e Cauchy Mean Value T h e o r e m for divided differences d u e t o R a t z and Russel (see 3) are also easy consequences of this result. In order t o formulate this s t a t e m e n t , we i n t r o d u c e a n o t a t i o n which t u r n s o u t t o b e t h e extension of t h e notion of t h e Wronski d e t e r m i n a n t . Let [a, b] C R b e a n o n e m p t y closed interval a n d let t u j , . . . , wk : [a, b] —► R b e a system of k — 1 times d i f f e r e n t i a t e functions (k > 1). Let x t < • • • < x* b e a r b i t r a r y points in [a, 6]. T h e n , one can uniquely determine n 6 N , £i < • • • < £ n in [a, b] a n d k\,..., kn in N with k\ + • • • + kn = k such t h a t (Xl,...,xk)
= (&,...,fl,
...,
fci times
£ n , - . . ,£n)*„ times
In this case, we define
«;i({,)...ti»i* , - 1 ) «i) \Zl,...,Xfc,/
:
;
...
t^)...^":
1
'^)
:
i
w*Ki)-»»i* , " ,, «i) ••• »*«-)... »1*"_1,«-) 'Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1652.
494
where ur*' stands for the »th derivative of the function w. If xi = ■ • ■ = x* = £, then the above definition yields
which is known as the Wronski determinant of the system u>i,..., wi,. The class of functions / : [a, b] —> R that are k — 1 times continuously differentiable on [a, 6] and k times differentiate on the open interval ]a,b{ will be denoted by £>*([«, 6]). Now we are able to formulate the main result of this paper. Theorem 1. Let toi,..., io* € Dk([a, b]) be a system of functions such that
w(7)#o, .... *(?;:;;;?)*o
(D
for all ( € [o,6]. Let further f,g € Z)*([a, 6]), and /c< i x , . . . , ast+1 G [a, 6] tw'i/i *o < • • • < ** <"*<* *i < sjt+i- Then there exists ( G]xi,x*+i[ such that
W( / 7'--'7'{W u ' 1 '-- w *' * ")= w(wu...,m,9\w(m,.,m,
f \
(2)
The proof of this theorem is given in the next section. Now we list some of its consequences. Corollary 1. (Cauchy's Mean Value Theorem.) Let f,g e ^([a.S]) Then there exists £ €}a, b[ such that
Proof. Let k = 1, u>i = 1 and xi = a, Xj = b in Theorem 1. Then the statement follows immediately from Eq. 2. Q Corollary 2. (Tayior's Mean Value Theorem.) Let f e Dk([a,b]). Then, for all x €]o,6], there exists { €]o,s[ such that
/(*) = /(«) + /'(«)(* - a) + • • • + ^ ^ ( * " -)*-1 + ^ P ( x " a)*.
495 Proof. Let . , (x-aV-1 w.(j) = a_ iy
,
for
, *= !>•••.*
,
.. ^ a 9KX) =
(x-a)k j^j
•
Then, for all ( 6 [0,6],
therefore the relations in Eq. 1 are satisfied. Thus, taking Xi = • • • = X* = o and xt+i = x in Theorem 1, we obtain that there exists £ €]a, x[ satisfying ^,...,^W^..W*^ \ £ , •••, Li) \a,...,a,xj
=
^V.M^W^V..,»*,/Y \(, ...,(,tj \a,...,a,xj
(3) w
A simple computation yields that
"ft::: t'O - «■> - «•> - «■><—> - - £ $ < * - •>"• a,...,a,xj-
Jb! '
further
KT:::::fO=/"'«» - •feT?)- 1 Thus, Taylor's theorem follows from Eq. 3 at once. O Let u>i(x) = l,...,wk(x) = x* -1 ,t0A + i = x* for x € [a,b] and let a < xi < • • • < **+i < b with a < x* + i and x t < b. Then the ratio
K w(
wu...,wk, f \ Xi,...,Xk,Xk+iJ wu..
.,wk,wk+i\
\*U--lxk,*k+l)
is called the ibth order divided difference of / 6 Dk([a, b]) over the points x\,..., x*+i (c.f. 3, p. 45). Divided differences are usually denned in an inductive way in the literature, see e.g. 1, Sect. 3.17 and 2, Sect. 2.3. The proof of the equivalence of the above definition to the usual one can be found in 4, Theorem 2.51, p.47. Concerning divided differences, the following result is well known (c.f. 1, p. 274 and 4, (2.93)). Corollary S. Let f 6 Dk([a, b]) and a < xi < •■• < xk+i < b with xi < xk+i. there exists ( €]xi,x*+i[ such that
[x 1 ,...,x t+1 ]/=^p.
Then
(4)
496 P r o o f . Apply Theorem 1 with the function g(x) = u>*+1(x) = xk. Then we find that there exists £ G]xi, xk+x[ s u c n that [an,..., xk+l]f
= [^
^ ]/
*+i times On the other hand, a simple computation yields, that
*+i times Thus Eq. 4 is proved. D The following result, called Cauchy Mean Value Theorem, is due to Ratz and Russel 3. Corollary 4 . Let f,g € Dk([a,b)) such that gW({) ^ 0 for ( £]a,b[ and let a < xi < • • • < Xk+i < b with Xi < £fc+i. Then there exists { €]zi,£*+i[ such that
[X,,... ,«*+!],
W
,W(0"
P r o o f . Applying Corollary 3 for the function g first, we can observe that [xu...,xk+i]g^0. Hence the left hand side of Eq. 5 exists. Clearly, w/w1,...,wk,
[xi,...,xk+i]f [xu...,xk+1]g
_
f \ \ x i , , . . , xk,xk+ij g \ w/wl,...,wk, \Xl,...,Xk,Xk+lJ
Therefore, by Theorem 1, there exists £ 6 ) x i , x t + i [ such that
[Xl,...,xk+1]f [ z i , . . . , xk+1] g
whence Eq. 5 follows.
_
fwu...,wk,f\
W
\(,...,
(,()
/wu...,wk^\
=fW{()
gW (£) '
497 2. Proof of t h e Main Result Let / € Dk([a, b]) be an arbitrary function. We say that / vanishes k + l times in [a, 6] if there exist points xi < • • • < xn in [a, 6] with Xi < b, a < xn and natural numbers ki,...,kn with ki + • • ■ + kn = k + 1 such that f{j)(xi)
=0
for j = 0 , . . . , * , • - ! , » = ! , . . . , n .
(6)
For instance, the function f(x) = x(x — 1) vanishes twice in [0,1]. However the function f(x) = x* does not vanish twice in [0,1], but it does in [—1,1], (that is, all the zeroes of / should not be concentrated at the endpoints of the interval). Lemma 1 . If f,g £ Dk([a,b]) and f vanishes k+l vanishes k+l times in [a, b].
times in [a,b], then fg also
P r o o f . By the assumption, there are x\ < • • • < xn in [a,b] with Xi < 6, a < x„ and fci,..., kn € N with ki + • ■ ■ + kn = k + 1 such that Eq. 6 holds. Then, using Leibniz's Product Rule, one can check that (f9)U)(xi)
=0
Thus fg also vanishes k+l
for
j = 0,:..,fc.--l,
i=l,...,n.
times in [a, 6].
D
Lemma 2 . If f € Dk([a, b]) vanishes k + l times in [a,b], then f vanishes k times in [a,b]. P r o o f . We have Eq. 6 for some Xi < ••• < xn with Xi < b, a < xn and Jbi,..., k„ € N with ifci + • • • + ifcn = k + 1. If n = 1, then there is nothing to prove. Otherwise, by Rolle's Mean Value Theorem, there exist i j < & < x, + 1 such that /'(<,) = 0
for
i=l,...,n-l.
These equalities combined with Eq. 6 yield that / ' vanishes k times on [a, 6].
n
Lemma 3 . Let t o i , . . . , wi, € D*([o,6]) be a system of functions satisfying Eq. 1 for all ( € [a, 6]. For f € Dk{[a, &]), define
Wn{f; 0 := w(W£] ] ^ ^ ' { ) -n =
Woifi 0 := /(0,
0 , . . . , k.
Then the following recursive formula holds:
(w1,...,wn\y Wn(/;£) =
for all 1
dt
k and ( e]a, b[.
Wn-dfii) fwu...,wn\
(wu..nwn.l\
(7)
498 Remark. Using continuity, Eq. 7 can be verified also for £ = a and ( = b when n
«€]o,6[
(8)
is an nth order homogeneous linear differential equation for the unknown function / . Observe that, in this equation, the coefficient of fW is
<"•::::?)• Similarly, the equation
J
l' «l
w»-i(/;0
=0
(9)
is also an nth order homogeneous linear differential equation for the function / . In this case, the coefficient of / ' " ' is
K
t»,,...,u»»_A
Kt:::T)' Obviously, the functions toi,..., wn are solutions of both Eq. 8 and Eq. 9. Since we have Eq. 1, hence they form a fundamental system of solutions for both equations. Therefore the left hand sides of Eq. 8 is proportional to the left hand side of Eq. 9 with the coefficient indicated in Eq. 7. Thus Eq. 7 is proved. Q Lemma 4. Let tci,... ,u>* € Z?*([a,6]) be a system of functions satisfying Eq. 1 for all ( e [«,&]. Assume that the function f 6 £*([a,6]) vanishes k + 1 times tit [a, 6]. Then, for each 0 < n < Jb, the function Wn(f) defined in Lemma S vanishes k + 1 — n times in [a,b]. Proof. We prove by induction. If n = 0, then W 0 (/) = / , hence, in this case, there is nothing to prove. Let n > 1 and assume that W n -i(/) vanishes Jfc +1 — (n — 1) times. Then, applying Lemma 1 and Lemma 2, one sees that the function
*TO::?)JKt:::T)
499 vanishes k + \ — (n — 1) — 1 = k + 1 —n times. Thus, Lemma3 yields the statement.
□ Remark. If / G Dk([a, b]) vanishes k + 1 times, then, taking n = k in the above lemma, we obtain that W i ( / ) vanishes once in [a, 6], that is, there exists £ g]a, 6[ such that
K?:::::t0=°-
(10)
This statement is also a special case of Theorem 1 (which is going to be proved below). To see this, assume that /W(6) = 0
for
j = 0,...,*,-l, i = l,...,n.
holds for some points & , . . . , £ „ with ° < £n and £i < b and for ^ H Then, taking (*l,---,S*+l) = l&'"•'&,> •••. £n,-y,6») *t times *„ times in Theorem 1, we get
\-fc„= k + 1. (11)
\Xi,...,Xk,Xk+iJ
Therefore the right hand side of Eq. 2 turns out to be zero. Hence the left hand side is also zero (for all g € Dk([a, b])). Thus Eq. 10 follows. Now we are ready to prove the main result of the paper. P r o o f of Theorem 1. Let xi < • • • < x^+i be in [a,b] with xi < x t + i . Then there exist n € N , & < • • • , £ „ in [a, 6] and klt...,k„ e N with i H \-k„ = k+1 such that Eq. 11 holds. Define the function F : [a, b] —► R by
«*(*) «!(&)..• wi*1-1^!) ... F(x) :=
T„<*-!>, "'(ft)
Wk{x) iw t ((i)
/(*)
/(6)
*(*)
9(d)
/(*1-D(6)
wl((n)...w[k"-l\(n) »*«.) /«-) 9((n)
„(*»-!)
(6.)
■■■f(k"-1H(n) •• „(*»-! >«-)
It is obvious at once that F<%0 = 0
for
j=0,...,Jfc.--l, i = l,...,n,
therefore F vanishes A: + 1 times in [a, 6]. Thus, by Lemma 4, there exists ( G]£i,£n[ such that
K"r::::":f)-°-
<12>
On the other hand, expanding by the first column, we get
n.) = g,».W+(-I)"v,(:i:;;:;-g/W-(-.)^(:;:::::-f>(I,.
500
Obviously, for all i=l,... ,k,
"(Z:7:7)=°Therefore, applying the above expression for F, Eq. 12 reduces to Eq.2, which was to be proved. O 3. References 1. G. Aumann and 0 . Haupt, Einfuhrung in die retlle Analysis, Band II, W. de Gruyter, Berlin - New York, 1979. 2. 0 . Haupt and G. Aumann, Differential- und Integralrechnung, Band II, W. de Gruyter, Berlin - New York, 1938. 3. J. Ratz and D. Russell, An extremal problem related to probability, Aequationes Math., 34 (1987), 316-324. 4. L. Schumaker, Spline Functions, Wiley, New York - Toronto, 1981.
WSSIAA 3 (1994) pp. 501-504 © World Scientific Publishing Company
501
A GENERALIZATION OF PbLYA'S INEQUALITIES
JOSIP PECARIC Faculty of Textile Technology, University of Zagreb, Pierottieva 6, 41000 Zagreb, Croatia
and SANJA VAROSANEC Department of Mathematics, University of Zagreb, Bijenicka cesta SO, 41000 Zagreb, Croatia
ABSTRACT Some inequalities involving functions and their integrals and derivatives are considered. In fact, we give a generalization of some inequalities which are due to G. Polya.
1.
Introduction
In the famous book "Problems and Theorems in Analysis" by G.Polya and G.Szego 1 the following theorems of G. Polya are given. T h e o r e m 1. Let the function f : [0,1] —♦ R be nonnegative and If a and b are nonnegative real numbers, then
(f1X'+if(x)dXy > {i -(-^L-y) Jo
increasing.
fx>°f{x)dx fx*f{x)dx.
a + o +1
Jo
(i)
Jo
T h e o r e m 2. Let the function f : [0, oo) —» R be nonnegative and decreasing. If a and b are nonnegative real numbers, then /oo
(/ Jo
n x^f{x)dxf<{\-{——
(if improper integrals
exist.)
A
a + b +1
roo
too
f)
Jo
x*>f(x)dx
Jo
x»f(x)dx
(2)
502
H.Alzer a give a following extension of Eq. 1. Theorem 3. Let f : [a, b] —► R be nonnegative and increasing, and let g : [a, b] —» R and h : [a, b] —► R 6c nonnegative and increasing functions with a continuous first derivative. If g(a) = h(a) and g(b) = h(b), then (J\y/g(x)h(x))'f(x)dx)*
> jf* g'{x)f(x)dx j f h'(x)f(x)dx.
(3)
The aim of this paper is to present a generalization of Polya's inequalities and Alzer's result.
2.
Main Results
Theorem 4. Let f : [a, 6] —♦ R be nonnegative and increasing, and let X{ : [a, b] —* R, t = l,...,n, be nonnegative increasing functions with a continuous first deriva tive. Ifpi,i = l,...,n are positive real numbers such that ]C?=i ~ = 1 then
J\ii(*mi,piYf(t)dt
> n(/«5(o/(«)*) i / w .
(4)
Proof. The inequality Eq. 4 reduces to an equality for / = 0 and thus we may, without loss of generality, assume that f(b) > 0. Integration by parts yields
j\f[(^))1,K)'f(t)dt
=
= /(») n(*i(b)Y,Pi - /(«) n(*.-(«)),/w - f f[(Xi(t))1/pidf{t) t=l
Ja
i=l
«=1
> /(*) f[(*.(&))1/pi " /(«) fft*^)) 17 * - fl( f *<(t)df(i))1,Pi «=1
t=l
i=l
(5)
•/o
Holder's inequality is used in the last inequality. Now, let us consider Popoviciu's inequality 3 . £>,a,,...ajn > n c f ^ a f j ) 1 ^ t=l
j=\
(6)
i=l
where u>i > 0,u>2,...,to m < 0,
503
m = 3 , tox = f(b) > 0 , w2 = -f(a) , w3 = - 1 , au = (*•(&))* , a« = (x,(o)) 1/w and a3, = ( £ x,( <)#(<))" for i = 1,2,..., n. So, by using Eq. 6, Eq. 5 is greater than or equal to
n(/w*.-(*) - M*M - f'iMW" = »=i
and the inequality Eq. 4 is proven. □ Remark 1. For n = 2 , p\ = P2 = 2 we have Eq. 3 without conditions of equality in points a and b, so, in this case we have stronger result than in Theorem 3. Remark 2. If we set in Eq. 4 a = 0, 6 = 1 i,(<) =
[l 4-i+-+-/(t)
Jo
1 + E,=i «•
j=i •'o
Remark 3. If we set in (4): a = 0 , 6 = l , n = 2 , p i = p 2 = 2 , *,-(*) = <Su+1 , Sw+1 with u,v > 0 then we obtain Eq. 1. Xj(t) = t If we suppose that / is decreasing function, we get a reverse inequality, i.e. we have the following theorem. Theorem 5. Let / : [ o , 6 ] - » R i e nonnegative and decreasing, and let a:,- : [a, 6] —> R, * = 1 , . . . , n, be nonnegative increasing functions with a continuous first deriva tive and Xi(a) = 0 for alii = l , . . . , n . //p,- , i = I,... ,n are positive real numbers such that TV -*- = 1 then
f\fl(xi(t))lh"ynt)dt < n(/ 6 x:-(o/(W*-
(7)
The proof is very similar to the proof of theorem 4: in fact, instead of using inequality Eq. 6 we use Holder's inequality for discrete case 3 . For 6 —♦ oo we have
504
Corollary 1 Let f and z, , t = 1 , . . . , n be functions on [a, oo) with same proper ties like in the theorem 5. Ifpi > 0, i = 1,..., n, such that £?=i "T> then
r(iK*«c)),/K)7(o* < ii( r*;w/wA)i/w-
w
• I ./a
(Here, we suppose that the improper integrals exist.) Remark 4. If we set in Eq. 8: a = 0, x,(<) = f^+i then we have
where a,- > -jj-,» = 1,... ,n
r t^-^f{t)dt < rc-'fog+1)1/Pi f[( r t«»f(t)dty*. •/o
1 + E<=i «.•
(9)
i l j Jo
For n = 2 we obtain the result of Volkov * . Specially, if n = 2,pi = p 2 = 2 then Eq. 9 becomes Eq. 2.
References 1. G. Polya and G. Szego, Aufgaben und Lehrsatzt aus der Analysis, I,II, Berlin, 1925. 2. H. Alzer, An Extension of an Inequality of G. Polya, Buletinui Institutului Politehnic Din Iasi, Tomul XXXVI (XL), Fasc.l-4,(1990) 17-18. 3. D.S. Mitrinovic, J.E. Pecaric and A.M. Pink, Classical and New Inequalities in Analysis, Kluwer Acad. Publishers, Dordrecht, 1993. 4. V.N. Volkov, Utocnenie neravenstva Gel'dera, Mosk. Gos. Ped. Inst. im V.I. Lenina. Uden. Zap.-Neravenstva 460, (1972), 110-115.
WSSIAA 3 (1994) pp. 505-521 © World Scientific Publishing Company
505
Enclosures for Weak Solutions of Nonlinear Elliptic Boundary Value Problems Michael Plum Technische Universitat Clausthal, Institut fur Mathematik Erzstrafie 1, D-38678 Clausthal-Zellerfeld, Germany Dedicated to Professor Wolfgang Walter on the occasion of his 66th birthday Abstract For nonlinear second-order elliptic boundary value problems, a numeri cal method for proving the existence of a weak solution in an explicit H°neighborhood of some approximate solution w is presented. The method is based on a /7_i-bound for the defect of w, and on a suitable norm bound for the inverse of the linearization of the given problem at a>, which is obtained via eigenvalue estimates. All kinds of monotonicity or inverse-positivity assumpti ons are avoided.
1 Introduction This article is concerned with nonlinear elliptic boundary value problems of the form -Au + F(x,u) = 0 ( i € f i ) , u = 0 (x&dil)
(1.1)
or more precisely, with the corresponding weakly formulated problem uetfftn),
J[Vu-V
+ F(x,u)
(1.2)
Here, fi C IRn (with n > 2) is a bounded domain with Lipschitz-continuous boundary dil. The nonlinearity F is defined on IT x IR with values F(x,y) € IR. We assume that F and its derivative dF/dy are continuous, and that F satisfies the following growth condition: \F(x, y)| < C(l + \y\r) for x € H, y € IR.
(1.3)
Here, C > 0 and r > 1 are fixed constants, with the additional restriction r < " _ n if n > 3. We will establish a computer-assisted method for proving the existence of a solution of problem (1.2) within an explicit H°(n)-neighborhood of some approximate solution u> € Hf(tt), i.e., an existence and enclosure method. A well known approach which could be used to derive existence and enclosure state ments is the method of two-sided bounds requiring essentially the existence of upper and lower solutions (which have to satisfy certain differential inequalities) to ensure
506
the existence of a solution between them. A lot of guiding work in this field has been done by L.Collatz (e.g., [4]), J.Schroder (e.g.,[14]), and by W.Walter (e.g.,[15]). Unfortunately, the demand for the existence of upper and lower solutions implicitly restricts rather heavily the class of problems which can be treated by this method: An "almost" necessary consequence of this requirement is the inverse-positivity of the linear operator generated by linearization of problem (1.1) (resp. (1.2)) at the solution to be enclosed. Since this inverse-positivity is implied by the monotonicity of the nonlinearity F, Collatz classified the problems which can be treated by this enclosure method as "problems of monotone kind* [4]. In many relevant examples, however, the above inverse-positivity condition is not satisfied, so that new methods are required here. In [11, 12, 13] , we proposed a method which is based on the computation of an approximate solution w € Hj(il) (satisfying the boundary condition in an appropriate sense), of a bound for the Lj-norm of its defect —Aw + F(-,w), and of a bound for the (La —♦ Co)-norm of the inverse of the linear differential operator L constituting the linearization of the left-hand side of problem (1.1) at u>. This norm bound is obtained, under the assumption r» < 3, via an (Lj —* /fj)-bound for L~x, and an explicit version of the Sobolev imbedding Hj(Q,) <-t Co(ft~). The (Lj —» i/3)-norm bound is in turn reduced to the computation of an (Lj —* Lj)-bound for X -1 , which is obtained via eigenvalue bounds for L. In the present article, we will develop a corresponding existence and enclosure method for solutions of the weakly formulated problem (1.2). The approximate solution u need now only be in U? (ft), and its defect has to be bounded in the norm || • ||H_, of the dual space 2f_i(ft) consisting of all distributions which are bounded on //{"(ft). A bound for L~l is now required in the (H-\ —* If? )-norm. This new way of proceeding has two advantages, compared with the former approach: 1) The regularity assumptions on the boundary 3ft are weakened; its Lipschitz-continuity is sufficient. In particular, we may now allow domains with reentrant corners. 2) The smoothness requirements on w are weakened, so that a larger class of ap proximative methods for its computation is available. For instance, we may now use finite element methods with Co-elements, while our former approach requires Ci-elements. It should be regarded, however, that Co-elements provide less accu rate approximations u>, so that larger defect bounds and larger error bounds arise than for Ci-elements.
2 The existence and enclosure theorem Suppose that an approximate solution u» € frf(Cl) of problem (1.2) has been compu ted. Regarding that Aw is an element of the distributional space J?_i(ft) (endowed with the usual norm as the dual space of //{'(ft)), we assume that some constant 6 > 0 is known such that ||-Aw + F(.,w)HjfM<*,
(2.1)
507
i.e., 6 bounds the defect of w. We will describe the computation of such a 6 in Section 3. Suppose further that some constant K > 0 has been calculated such that HVUHL, < #||L[u]||w_, for all « €
tf?(n),
(2.2)
with L : H°(£l) —» H-\(tt) denoting the linearization of the left-hand side of problem (1.1) (resp. (1.2)) at w, i.e., L[u) := - A u + c • u , c(x) := ^ ( x , w(x)) (x €
ft).
(2.3)
To make sure that L maps in fact into /f_i(ft), and that L is bounded, we have to secure that / cuipdx < const • ||VU|U|VV>||L, for u,
(2.6)
and G(t) = o(t) for t -» 0. (2.7) Based on explicit Sobolev imbeddings, we will compute such a function G in Section 5. Theorem: Suppose that some a > 0 exists such that 6<1-
G(a).
(2.8)
Then, there exists a solution u of problem (1.2) such that
||V(«-W)|U,
508
Lemma 2.1: L : tfi(ft) -» H-i(fl) is one-to-one and onto. Proof: The weak boundary value problem « € #?(«) , / Vu • V
f M « ) -» L„(ft) \ \ u »-» J=,(-,w + u ) - f ( - , w ) - c - u J
w continuous. Proof: First we show that, for each measurable subset M c f ! , I I ^ M I I M " ) ^ ^ • (meas(M)p • (1 + ||u||'L(i) for all u € !„(«),
(2.11)
where C is a constant independent of u and M, and • f1 o := mm < [u
r
1
!
U
, n
A
> > 0. v
p
n]
In fact, (1.3) together with Holder's inequality provides, for u € L,,(M), II^(-.«+«)HM*O ^ +«riiMW) < C[||l|| M W ) + ||u; + u||^ ( w ) ] < C, [meas(M)i + |M|^, ( M ) + < C, [meas(Af)i + m e a s ( M ) ^ < C 2 .meas(Af)«.(l + ||u||^).
Ml^ti)] (MIAU)
+
NIM*))] (2 12)
509
Moreover, again by Holders's inequality, we obtain for u € L^M): ||c • U||L„(M) < ||c||r,,(M) • HIL„H^V)(M) < \\c\\Lr(M) ■ meas(M)i-i-t • ||u|| M M ) < C3-meas(M)^(l + ||«||J,J.
( 21 3 )
(2.12) and (2.13) obviously imply (2.11). Assuming now for contradiction that T is not continuous at some u € L„(il), we can find some e > 0 and some sequence («*) in L,,(ft) converging to « in iM(ft), such that ||^(u) - JFK)|| M n) > e for all k € JN.
(2.14)
Due to (2.11) and to the boundedness of (u*) in L^il), some t) > 0 exists such that, for each measurable subset M C ft satisfying meas(M) < 17, r W I l M " ) + II^UOIIMAO < ^
for all k € IV.
(2.15)
Furthermore, theorems from elementary measure theory imply the existence of a measurable subset M C ft such that meas(M) < tj , u and w are bounded on ft \ M, and some subsequence (u* •) converges uniformly to u onft\ M. Therefore, since F is uniformly continuous on compact subsets offt~xIR and c is bounded on ft\Af, ^"(t»k-) converges uniformly to T(y) on ft \ M. Consequently, some jo € JV exists such that ||^(u) - ^(u 4 i )|| M nvio ^ j f c f ° r j ~ jo(2.15) and (2.16) contradict (2.14).
(2 16)
-
■
Proof of the theorem: The given problem (1.2) is easily seen to be equivalent to the following problem for the error v = u — w: V € #?(ft) , L[v] = -F{v) - d[u],
(2.17)
with L from (2.3), 7 as in Lemma 2.2, and with d\u\ :- -Aw + F(-,w) 6 #_i(ft) denoting the defect of w. We are left to show that problem (2.17) has a solution v satisfying ||Vv||t, < a. Let n and u as in Lemma 2. Sobolev's Imbedding Theorem provides the compact ness of the imbedding E\ : #?(ft) *—► L^Q.), and moreover, the boundedness of the imbedding //{"(ft) •-► L„/(„_i)(ft), so that Holder's inequality yields / ■wipdx
< IMWMUwc-o ^ C||W||L.I|V V ||L,
(W €
L¥(U),V € fl?(fl)),
n i.e., the boundedness of the imbedding £ ? : Lv(il)«-» H-i(fl). Lemma 2.1 provides the existence of L~l : H-i(il) -» #?(fl), and (2.2) shows that L - 1 is bounded. Using Lemma 2.2, we obtain altogether that T : i/?(ft) -f tf?(n), Tv := -ZT 1 [EJFE^V)
+ d[u]\
(2.18)
510
is continuous and compact. Furthermore, T maps the (closed, bounded, convex) set £ > : = { « € #?(n) : ||VI;||L, < a} into itself, since (2.2), (2.18), (2.6), (2.1), the monotonicity of G, and (2.8) provide, for v £ D, WTV)\\L,
<
KUITV^H.,
= K\\T(v) + « „ _ , < Ar[||^( V )|| w ., + ||<*M||W_J
< A-JGdlVvHt,) + S] < K[G(a)
+s]
i.e., Tv 6 D. Consequently, Schauder's Fixed-Point-Theorem provides the existence of some fixed point v 6 D of T, which is obviously a solution of problem (2.17) satisfying ||Vv||t, < a. m
3 Computation of a defect bound 6 In this section, we will describe how a constant 6 satisfying the inequality (2.1) can practically be computed. The direct definition
(
/(Vw • V
can hardly be used numerically since a svpremum over an infinite-dimensional set has to be bounded from above. The following lemma converts this supremum into an infimum which is much better to handle numerically, with respect to the computation of an upper bound. The lemma is a special case of Goerisch's complementary variational principles (see, e.g., [8]), and it can be derived from these without difficulty. Here, we give a simple direct proof. Lemma 3.1: || - Aw + JF(-,«)||jr_I = inf {||Vu> - qfa : q € 1,(0)",div q = F(-,w)}
(3.2)
(with div q € H-\(Q) for q € Li(Q)n), and the infimum is attained at some q satis fying , J qpdx = 0 for all p € L2(il)n such that div p = 0. (3.3) a
Proof: For each
< l|Vu;-g|U,-||Vv>|U„
511
so that (3.1) provides "<" in (3.2). To prove ">" let
llVdli, < ||-Au, + F(.,u,)||„_,. Since Vip = Vw — q, we obtain ">" in (3.2) and, together with "<", that the infimum is attained at q. Finally, for all p € £j(fl)n such that div p = 0, / q-pdx = I V(w —
■
Lemma 3.1 shows that the inequality (2.1) is satisfied with 6 := ||Vu — g||j, for arbitrary q e Lj(ft)n satisfying div q = F(-,w), e.g., for q = q where «:=(9i,0l...,0)
J
F(I,X2).
..,x n ,w(t,x,,...,x„)) dt,
(3.5)
provided that the domain (1 allows the definition of such line integrals. However, the choice q = q will usually provide a constant S far from the optimal one, i.e., from the true value of || — Aw + F(-,W)||H_ 1 . TO obtain a better choice, we can proceed as follows. The optimal q is characterized as the solution of problem (3.3) (which is easily seen to be unique, under the side condition div q = F(-,u)). Problem (3.3) can usually not be solved in closed form, but it suffices to compute an approximate solution q € Lj(fl)n of problem (3.3) satisfying the equation div q = F(-,u>) exactly, because (3.2) then provides || - Aw + F(.,W)||W_1 < ||Vu, - ,11^ =: 6, (3.6) and this bound is "close" to the optimal one if q is an "accurate" approximate solution of problem (3.3). With QQ := {g0 € Ia(ft) n : div q0 = 0}, and with q € Lj(Q)n denoting some particular solution of div q = F(-,u>) (given, for example, by (3.5)), we can look for q in the form q = q + qo, where
forallp€Qo, (3.7) n which is often easier to handle, from the numerical point of view, than problem (3.3).
512 If n = 2 and ft is simply connected, we have
•-{(-&£)'■♦«*<4 so that problem (3.7) reads
*€*(«),/v*.V,WkIrH,^ n n L
dx for all n €
ffi(ft)
(3.8)
J
which is a weak Neumann boundary value problem for 4> and has a unique solution, up to an additive constant. With ip 6 #i(ft) denoting some approximate solution of (3.8), we obtain (3.6) with
To compute the desired .ff_i-defect bound we must therefore • calculate some particular solution q of div q = F(-,u) (e.g., via (3.5)), or (at least) compute qi,qu € £a(fl)n enclosing q, i.e., satisfying qt < q < qu (component- and pointwise), • compute an approximate solution V> of problem (3.8), • compute an upper bound for ||Vw — q — (—d^/dx^d^/dxiYWi, which all can be carried out by numerical means with the aid of interval analysis.
4 Computation of a bound K for L~l We will now describe how a constant K satisfying the inequality (2.2), with L defined by (2.3), can be computed. The following lemma shows that the computation of K is very closely related to the eigenvalue problem ti6flj(fl), Jcwpdx = \[VuV
"•-'■sjhrr provided that Xk ^ - 1 for all k € IV.
(4 2)
-
513 Proof: Let q := 2p/(p — 1), with p from (2.5). Holder's inequality and Sobolev's Imbedding Theorem provide, for u € £«(ft) and
cmpdx
< IMkNWMk < lk.
(4.3)
Thus, for u € i j ( ^ ) , the functional ip *-* fcwpdx is in i/_i(Q), so that the Laxn Milgram-Lemma provides the existence of a unique Tu € H^(fl) such that fv(Tu)-V
ML, -
sup
l\\V
= Hvfu + rtOiiL, = Ut=l
I
= [ E ( l + ^) 2 (Vu,Vu t )ij
> (m£|l + A*|)||Vu|U,
which yields the second part of the assertion. Remark Suppose that one is interested only in solutions of problem (1.2) which belong to some symmetry class (closed subspace) S C Hf(Q), that the approximate solution w belongs to 5, and that thefixedpoint operator used in the proof of our Existence and Enclosure Theorem (Section 2) preserves this symmetry class. Then, the estimate (2.2) is needed only for u € H°(ft) D S. Suppose further that some
514 subset J C IV is known such that, for Jfe € «/, the eigenfunctions u* belong to the orthogonal complement of 5. Then, the proof of Lemma 4.1 shows that (2.2) holds for u € .H?(ft) n S if in (4.2) the set JV is replaced by JV\ J. This may significantly reduce the constant K. According to Lemma 4.1, we are left with the computation of eigenvalue bounds for problem (4.1). First we consider the case c > 0. This condition, or the stronger con dition that F shall be monotone with respect to y (compare (2.3)), is often assumed if the method of upper and lower solutions (monotonicity methods) shall be applied. Lemma 4.2: In the case c(x) > 0 (x € ft), (2.2) holds true with K = 1. Proof: The assumption c > 0 provides that all eigenvalues of problem (4.1) are nonnegative. Since the eigenvalue sequence converges to 0 due to Lemma 4.1, we obtain K = 1 from (4.2). ■ The simplicity of the result K = 1 in Lemma 4.2 reflects the special (and simple) possibilities for obtaining existence and enclosure results if monotonicity assumptions are made. In the general case, one has to compute bounds for the eigenvalues of problem (4.1) neighboring —1, usually by numerical means. In this context, the methods by Rayleigh-Ritz and by Lehmann-Goerisch may be used. In the following, we briefly describe the application of these methods to our problem (4.1) under the additional assumption c(x) < 0 for almost all x 6 ft. (4.5) -1 By the simple transformation \i = —A , problem (4.1) is equivalent to u € #?(ft), J Vu • Vv> dx = p j \c\utfi dx for all
' ijml
, At := I / \c\v{Vj dx J M
Ml
/ ,j = i
M
Lemma 4.3: (Rayleigh-Ritz-method) Let Jix,...,JiM denote the eigenvalues of the matrix eigenvalue problem Ao( = pArf, ordered by magnitude. Then, /**
1,...,M.
515 Thus, upper eigenvalue bounds can be computed very easily. The computation of lower bounds is more complicated. The most accurate method which at the same time has the largest range of applicability is Goerisch's method [6,8] which generalizes Lehmann's method [9] in a famous way. Goerisch's method contains certain terms which may be chosen to optimize the results or to facilitate the application. Here, we describe one particular choice of these terms for the application of the method to problem (4.6). Let t»i,..., WM 6 Lj(ft)n denote vector functions satisfying div u>i = cvi (i = l,...,M)
(4.7)
in the distributional sense, and form the additional matrix Aj:=
I / w, • Wj dx 1 w
' IJ=I
M
Lemma 4.4: (see [8], Thm.6) Let g > 0 denote some constant satisfying, for some se{i,...,M), ?.<e
,
(4.8)
with Jlt from Lemma 4-S. Let K\, . . . , KM denote the eigenvalues of the matrix eigen value problem (At - QAM = K(AO - 2pAi + e^A^C , ordered by magnitude. Then, K, < 0, and k
-o
l-«t
fork =
l,...,s.
(4.8) requires the knowledge of a (rough) lower bound g for the (a+l)-st eigenvalue, in order to compute (accurate) lower bounds for pi,...,/*,. Such a g can often be read off from a comparison problem with known eigenvalues. In more complicated cases, one has to perform a homotopy method (which has been proposed independently in [7] and in [10]) for that purpose. The solutions to, of the equations (4.7) are of course not unique. Particular solutions tD; can be computed analogous to (3.5). Also the techniques for obtaining "good" functions w,- are very similiar to those used in Section 3: The optimal functions u>, are gradients of //{'(ftj-iunctions, because then we have Lehmann's setting for optimal eigenvalue bounds [9]. Consequently, these optimal Wi satisfy toi-pdx = 0 for all p € Lj(ft)n such that div p = 0, /• n under the side condition (4.7). This is formally the same problem as (3.3), and we may use the techniques developed in Section 3 to find "good'' approximations t5t to the optimal u\.
516
5 Calculation of a majorizing function G For the calculation of a monotonically nondecreasing function G : [0, oo) -♦ [0, oo) satisfying (2.6), (2.7), we first suppose that a function G : (0,oo) —» [0,oo) with the following properties can be found: (Gl) (G2) (G3) (G4)
|F(x,w(x) + y) - F(x,w(x)) - <<*)»! < G(\y\) for x € ft,y G IR, G is monotonically nondecreasing, G(i) = o(t) for t -► 0, for some r > 1 satisfying r < (n + 2)/(n — 2) if n > 3, the function G(t 1 /') is concave in t. Observe that (Gl) and (G4) together require, in particular, the growth condition (1.3). The easiest way to find such a function G is to use an ansatz
• <5(<) = £>,<"' with exponents r„ € (l, jj^§) (:= (1,°°) if n = 2) and with nonnegative coefficients or„ (which obviously satisfies (G2), (G3), (G4)), and to arrange the integer M, the exponents r„, and the coefficients or„ such that (Gl) is satisfied. In order to calculate a function G satisfying (2.6) and (2.7) we need, in addition to some function G satisfying (Gl) to (G4), the following lemma providing an explicit version of one particular imbedding in Sobolev's Imbedding Theorem. Lemma 5.1: Let q 6 (2,oo) ifn = 2, q € (l, £%] ifn > 3. Then, IMU,
(5.1)
C{q) : =
2^2 meas ( ft ) 1/ ' • M ~ *> * • * $ - v + !)] ,/ " * n = 2> C7 (« ) : = 7^2y m e a 8 ( f t ) , / '" 1 / : , + 1 / n *n^3Proof: In [5], Proof of Theorem 9.3, formula (9.10), it is shown that, for 1 < s < n,
I H«W(»-») k . ^ <- ^2 -„n -- 1,
SfeL
l n
(u€H?(n)).
Using the geometric-arithmetic mean inequality we obtain
n_1 Nk..,,.
mi
1/2
(u € # ? ( « ) ) .
(5.2)
For n > 3, we choose a := 2 and use, if q < 2n/(n — 2), the inequality a/0 2n/(n - 2) J\u\adx<meas(iiy-al0lj\u\0dx\
(0 < a < 0)
(5.3)
517
(with a := q, 0 := 2n/(n — 2)) on the left-hand side of (5.2), so the assertion follows. In the case n = 2, we choose s := 1, so that (5.2) provides
t'^til
\ 2
du dx\ dxi
(u € H°{H)).
Inserting, for some given u € ffi(n),|u|'/2 in place of u here, we obtain, using Schwarz's inequality
Iterating this inequality v = [q/2] times we obtain
Jwdx-{hY
l«a(«-2)a-(«-2(«'-1))a) (j\«r*A uvuii%.
We use (5.3), with a := q — 2i/ and fi := q, on theright-handside here, and divide by l/|u|«dx)
(j
|u|f
to obtain - ( ^ ) " ^ ( 9 " 2 ) J • • • (9 " 2(" " W ] * meaS(ft)J"/' • || Vu||%.
*)
Taking the 2i/-th root we obtain the assertion.
■
We now combine the explicit imbedding result obtained in Lemma 5.1 with the func tion G satisfying (Gl) to (G4), in order to obtain a function G satisfying (2.6), (2.7). Lemma 5.2: Let G satisfy (Gl) to (G4), and choose q € (1, oo) such that q € (2/r, 2) ifn = 2, q € [2n/(n + 2), 2n/(r(n - 2))] ifn > 3, with r from (G4). Then, with the constants C(- • •) from Lemma 5.1, G(t) := meas (fi)i/» . cfa)
■ G (meas(ft)-1/«r • C(qr) ■ t)
is monotonically nondecreasing and satisfies (2.6), (2.7). Proof: (G2) and (G3) immediately provide the monotonicity of G, and (2.7). Using (Gl), Holder's inequality and Lemma 5.1 we obtain, for u € Hi(il), ||F(., w + u ) - F ( . , u , ) - C - H l ^ <
<
sup
sup
hv
{liwnzj ilSduDlkllvlU,/^,,} < c(^)||G(|«|)|U,
(5.4)
518
Furthermore, (G4) implies that *(r) := [G(t1/*r)] is a concave function.(For smooth G, this follows by showing that $" < 0; for nonsmooth G, one can use appropriate mollifier techniques.) With v := \u\
[9(v)dx
meas(ft) J
< * (
^ r
fvdx)
\ meas(lJ) J
,
I
and thus, since tf (t>) = [Cx(|u|)]», HGduDHi, < meas(n)1/' . G (meas(ft)-1/" • | | u | | v ) , (5.5) and ||u||„ < C(9r)||V«||£a due to Lemma 5.1, so that (5.4) and (5.5) provide (2.6) for the asserted function G. m
6 Numerical examples In this final section, we report on some numerical examples which we used to test our existence and enclosure method. To compute an approximate solution u € #? (ft) for the given problem (1.2), we used a Newton-iteration. The linear boundary value problems occurring in each Newton step were solved approximately by &finiteelement procedure with triangular elements; on each element, the approximate solution is put up as a quadratic polynomial, so that 6 local knot variables occur, corresponding to the corners and the side-midpoints of the element. To keep the program code simple, we only used regular meshes (with all elements having the same size and shape), even for domains SI with reentrant corners (see example (6.3) below). To compute approximate eigenfunctions of problem (4.6) (which we used as basis functions v,- for the Rayleigh-Ritz and for the Lehmann-Goerisch method) we per formed a subspace iteration process with finite element approximations on the same mesh as above. Also the auxiliary problems (3.8), and the corresponding auxiliary problems occurring in the eigenvalue estimation procedure (see the remarks at the end of Section 4), were solved approximately on the same finite element mesh. If the nonlinearity F is a polynomial in all its variables (as in our examples), then all integrals needed (see, e.g.,(3.5), the Lj-norm at the end of Section 3, the matrices J4O, A\,AI in Section 4) can be calculated exactly since the integrands are polynomials on each element. To compute bounds for the matrix eigenvalues Jtk (see Lemma 4.3) and K* (see Lemma 4.4) we used a method (and a complete C++ program) by Behnke [3]. To avoid rounding errors we implemented the packages BIAS and PROFIL (providing, among other things, interval arithmetic) which have been developed by S.M.Rump and his coworkers at the Technical University of Hamburg-Harburg. These routines were used only where rigorous results are required, and therefore, not in the finite element approximation procedures. All numerical computations were performed on a 486 PC.
519 In our first example, we looked for a nontrivial solution of Emden's equation - A u = u2 on ft := (0,1) J , u = 0 on 3ft.
(6.1)
Since the determination of an appropriate starting approximation for the Newtoniteration is a nontrivial problem here, we considered the auxiliary problem - A u = u2 + A on ft := (0, l ) 2 , u = 0 on 3ft
(6.2)
for which we found, by path-following methods (starting from the trivial solution at A = 0), the bifurcation diagram (of approximate solutions) plotted in Figure 1, which lead us automatically to an approximation for a nontrivial solution of problem (6.1). For the approximation process, and for the computation of a constant K (see the remark following the proof of Lemma 4.1), we exploited the symmetry of the expected solution with respect to reflection at the axes xi = 1/2, x% = 1/2, xi = xj, xi = 1 — xj, and performed the computations only on the triangle | ( x i , x j ) : \ < xj < Xi
flrpfe
S5 25
Fig 1: Bifurcation diagram for problem (6.2)
Fig 2: Emden's equation
Figure 2 shows a plot of the approximate solution w of problem (6.1). Its maximum (attained at (1/2,1/2)) equals approximately 29.26. For three different mesh sizes (64 uniformly distributed triangular elements corresponding to 136 global number of variables, 256 triangles corresponding elements 6 K a to 528 global variables, and 1024 tri 64 0.5994 1.4683 1.1903 angles corresponding to 2080 varia 256 0.1524 1.4555 0.2337 bles), Table 1 contains the compu 1024 0.03834 1.4548 0.05646 ted defect bounds 6, the bounds K l for L~ , and the error bounds a for Table 1: Results for problem (6.1) ||V(« — oi)||tj provided by our Theo rem. The constant K was obtained, e.g. in the case of 256 elements, from the bounds A, € [0.4999,0.5001] , A2 € [3.195,3.201] for the first two eigenvalues of problem (4.6), which we obtained by the methods described in Section 4. (In particular, problem (4.1) has therefore one eigenvalue « - 2 , and the method of two-sided bounds is not applicable.) Properties (Gl) to (G4) (see Section 5) are satisfied for G(t) := t2.
520 The computed bounds are less precise than the bounds in [13] which we obtained by highly accurate hi-quintic rectangular elements, while here only quadratic triangular elements were used. On the other hand, we obtained the bounds (and the existence statement) here with a rather low number of unknowns (global variables), which saved a lot of computing time. In our second example, we considered the domain fi := (0, l ) 2 \ U j , 1) x (0, j]J which obviously has a reentrant corner so that the author's earlier approach [11, 12, 13] cannot be applied. The (parameter-dependent) equation is - A u - 56u = u 2 + A on fl , « = 0 on 5fl.
(6.3)
The factor —56 is chosen to make one eigenvalue of problem (4.1) less than —1 on the (expected) solution branch passing through (A = 0,u = 0), so that the method of two-sided bounds is not applicable. Again, we exploited the symmetry (with respect to reflection at the axis x\ = l — xj) of the expected solutions for the approximation process, and for the computation of K. For several values of A, we computed approximate solutions on a uniformly distribu ted finite element mesh with 768 triangular elements (corresponding to 1488 global variables), and carried out the existence and enclosure procedure. Table 2 contains the results. Figure 3 shows a plot of the negative of the approximate solution w for A = 16. Observe that u changes sign close to some parts of the boundary Oil.
A 2 4 6 8 10 12 14 16 Fig 3 : Problem (6.3), A = 16
IMU« 0.2402 0.4831 0.7290 0.9780 1.230 1.486 1.746 2.009
6 0.02270 0.04580 0.06932 0.09328 0.1178 0.1427 0.1681 0.1941
K 3.417 3.394 3.370 3.345 3.320 3.295 3.270 3.244
a 0.08000 0.1659 0.2588 0.3607 0.4744 0.6047 0.7615 0.9724
Table 2: Results for problem (6.3)
The rather low precision of the bounds is partly due to the low-degree elements (as in example (6.1)), but mainly to the effects caused by the reentrant corner. The results will certainly improve if local refinement techniques are applied, which we intend to do in future work.
521
References [1] R.A.Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] H.Bauer, Wahrscheinlichkeitstheorie de Gruyter, Berlin, 1978.
und Grundzuge der Mafitheorie, 3rd edition,
[3] H.Behnke, The Calculation of Guaranteed Bounds for Eigenvalues Using Com plementary Variational Principles, Computing 47 (1991), 11-27. [4] L.Collatz, Aufgaben monotoner Art, Arch. Math. 3 (1952), 366-376. [5] A.Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. [6] F.Goerisch and J.Albrecht, Eine einheitliche Herleitung von Einschlieflungssatzen fur Eigenwerte, in Numerical treatment of eigenvalue problems Vol 3 (J.Albrecht, L.Collatz, W.Velte eds.), ISNM 69 (1984), 58-88. [7] F.Goerisch, Ein Stufenverfahren zur Berechnung von Eigenwertschranken, in Nu merical Treatment of eigenvalue problems Vol.4 (J.Albrecht, L.Collatz, W.Velte, W.Wunderlich eds.), ISNM 83 (1987), 104-114. [8] F.Goerisch and H.Behnke, Inclusions for Eigenvalues of Selfadjoint Problems, in Topics in Validated Computations (J.Herzberger ed.), Series Studies in Compu tational Mathematics, North Holland (Elsevier), to appear 1994. [9] N.J.Lehmann, Optimale EigenwerteinschlieBungen, Num.Math ■ 272.
5 (1963), 246-
[10] M.Plum, Bounds for Eigenvalues of Second-Order Elliptic Differential Operators, ZAMP 42 (1991), 848-863. [11] M.Plum, Existence Proofs in Combination with Error Bounds for Approximate Solutions of Weakly Nonlinear Second-Order Elliptic Boundary Value Problems, ZAMM 71 (1991), T660-T662. [12] M.Plum, Explicit i/j-Estimates and Pointwise Bounds for Solutions of SecondOrder Elliptic Boundary Value Problems, J.Math.Anal.Appl. 165 (1992), 36-61. [13] M.Plum, Numerical Existence Proofs and Explicit Bounds for Solutions of Non linear Elliptic Boundary Value Problems, Computing 49 (1992), 25-44. [14] J.Schroder, Operator Inequalities, Academic Press, New York, 1980. [15] W.Walter, Differential and Integral Inequalities, Springer, Berlin and New York, 1970.
WSSIAA 3 (1994) pp. 523-535 © World Scientific Publishing Company
523
On Quadratic Differential Equations by Arthur A. Sagle and Klaus Schmitt 1 To Wolfgang Walter on his 66th birthday ABSTRACT: We discuss the use of the theory of nonassociative algebras in the study of quadratic systems of ordinary differential equations and provide an application to a system of differential equations which arises in the discretization of Boltzmann equations.
1
Introduction
Let 0 : RN x RN -» RN be a continuous bilinear mapping. If for x, y € RN we define the juxtaposition xy by xy = 0(x,y), this juxtaposition will yield a (nonassociative, in general) multiplication in the real vector space RN. Thus RN together with this multiplication will form an algebra, which we shall henceforth denote by A. In this paper we shall employ some of the properties of algebras to analyze systems of differential equations
d.i)
|-/W, where / : RN -> RN is given by f(x) = 0(x,x)
= x2
where 0 is as above. Such systems are called, for obvious reasons, quadratic systems. 'Supported by a grant from NSF
524
This approach to the qualitative study of the this special class of differ ential equations has a long history and goes back to Markus [6] and has since been discussed and used extensively (see e.g. [5] and [7]). The motivation for the present paper is a special quadratic system which was derived by Ulner and Rjasanow [3] in their numerical studies of Boltzmann equations, and we shall present here a detailed qualitative study of this equation by means of these algebraic methods. To make the paper somewhat self-contained we include a brief discussion of results about nonassociative algebras which are useful in the study of quadratic systems.
2
Algebras and quadratic differential systems
As above, we denote by A the algebra obtained upon introducing the multi plication induced by the form (3 in RN. Basic to the study of algebras are distinguished subspaces, namely subalgebras and ideals; we recall their definition. Definition 2.1 (i) A subspace Ai C A is called a subalgebra provided Ai is closed under multiplication or for short, A\ =AiAi
C A\.
(ii) A subspace A\ C A is called an ideal, whenever xy G A\,
yx G Ai,
Vj/ € A\,
Vx e A,
or for short AA\ c A\,
AiAC^Ai.
The following set of auxiliary results are useful in reducing a given quadratic system (1.1) into simpler (i.e. lower dimensional) subsystems. Lemma 2.1 Assume that A = Ai © A2, where the latter is a direct sum of two ideals. Then if we write x — xi+x2,
XitAi,
i = l,2,
525
the system (1.1) is equivalent to = x?
~dt
(2.1)
<
dxi • ~dt
*Z-o •
—
Proof: If x{t) is a solution of (1.1), we may write, for each t X(t) = X!(t) + X2(t),
Xi{t)eAi,
2 = 1,2,
where, since x(t) is differentiable, both Xi(t) and X2(t) are also, and since Ai, i = 1,2, is a subspace of A, ^ € A> * = 1> 2. Further using the multiplication rules, we have X2 = (Xi + X 2 ) 2 = x\ + XiX2 + X2Xi + x\.
Since A\ and A2 are ideals we must have that x^x2, and therefore We hence conclude that ^ = xj,
x2X\ e Ai
i = 1,2.
Lemma 2.2 Assume that A\ is an ideal in A and that A2 is a supplementary subspace which is a subalgebra, A = A\ © A2. Then the system (1.1) is equivalent to the system dx2 ~dt
-
dx\ = ~dT where x — x\ + x2,
Xi&Ai,
^2
(2.2) x\ + X i I 2 +X2X1,
i = l,2.
Proof: One proceeds as in the proof of the previous lemma, however one cannot conclude that X\X2 = x2x^ = 0, since A2 is not assumed to be an ideal, however, since A\ is an ideal these products reside there. Another lemma, which will be very useful in our later considerations is the following.
526
Lemma 2.3 Assume that A\ is an ideal in A and that .42 is a supplementary subspace (A = A\ © A2) which satisfies
A ^ M-
(2.3)
Then the system (1.1) is equivalent to
f ** = o <
dt
(2.4)
dx\ , , —- = x\ + xix2 + x2xi + xL v at where x = xi + x2,
Xj € Ai,i = 1,2.
Proof: The result follows immediately from the preceding lemma using the fact that A\ is an ideal and x\ € A\ for x2 € A2. Lemma 2.4 We note that the previous lemma tells us that X2(t) = constant and hence the second equation is of the form
- ± = x21+e(x1) + c, where £ : Ai —* Ai is a linear mapping and c e A2 is a fixed vector. Thus the system (1.1) has been reduced to a trivial equation (^j* = 0) and a sup plementary Riccati equation. Some further facts that are useful in the qualitative analysis of our system concern nilpotent and idempotent elements of the algebra. Definition 2.2 (i) An element p € A is called nilpotent (of index 2), whenever p2 = 0. (ii) An element p e A is called idempotent, whenever p2=p. We have the following elementary result.
527
Proposition 2.1 (i) Letp 6 Abe a nilpotent element, thenp is a stationary point for (1.1). (ii) Let p G A be a nonzero idempotent element. Then the subspace Rp is an invariant set for (1.1) and for every q G Rp, q ^ 0, the solution of (1.1) with initial value q blows up in finite time. Proof: If p2 = 0, then x(t) = p is the solution (initial value problems are uniquely solvable) of dx x2 ~dl x(0) = VSimilarly, for a € R, x(t) = ap, solves dx
x2
x(0) = ap, since (ap)2 = (ap)(ap) = a2p2 = 0. If p is a nonzero idempotent, we put x(t) = y{t)p, whe:re y : R —► R is defined by y
dt y(o) =
i,
i.e. i,(t\ =
1
.
then x'(t) = y'(t)p = y2p = y2p2 = (yp)2 = x2 x(0) = y{0)p = p.
528
On the other hand if q € Rp, then q = ap and the solution z(t) of dz
o
— = z\
/„x
2(0) = q = ap
is given by z(t) = ax(at), where x solves ^ = x 2 , x(0) = p, i.e. x(t) = y^p. Thus z{t) = jz^P> proving the second part of the assertion. Remark 2.1 The above also shows the scale invariance of quadratic sys tems, i.e. if x(t) solves (1.1) then for any scalar fj.,y(t) = fix(^it) solves (1.1). As a corollary of the above, one immediately obtains. Corollary 2.1 If A has a nonzero idempotent, then the trivial solution x = 0 of (1.1) is unstable. We next shall establish that the algebra A always must have either a nilpotent or idempotent element (c.f. [4]). Proposition 2.2 The algebra A has either a nontrivial nilpotent or a nonzero idempotent element. Proof: Assume A has no nonzero nilpotent element nor any nonzero idempotents. Let B = {x € RN : |x| < 1}. Then for any x € dB and any real numbers a, x2 ^ ax. First of all this is clear for a = 0 since no nonzero nilpotent elements exist. On the other hand, if there exists x € dB and a ( ^ 0) € R such that x2 = ax, then if we let y = ^x, we get y2 = (^x) 2 = ^jx 2 = 4 ; a i = £x = y, hence y is a nonzero idempotent. It therefore follows that for |x| = 1 and A G [0,1] Ax2 ± (1 - A)x ^ 0, which implies that M / > B, 0) = dB(id, B, 0) = dB(-id, B,0), where dB is the Brouwer degree of the mapping / , /(x) = @(x,x) = x2, and id is the identify mapping. Hence if N is odd, we have obtained a contradiction, since dB{id, B,0) = 1 and dB{—id, B,0) = —1.
529 If N is even we conclude that dB(f,B,0) = 1. On the other hand / is a smooth mapping (even) we hence may employ Sard's theorem to conclude that there exist x<> ^ 0 of arbitrarily small norm which are regular points for / . But for \XQ\ of small norm,
dB(f,B,0) = dB(f,B,x0). Since xo is a regular point of / and / is an even function we get (from the properties of the Brouwer degree) that dB(f,B,x0) — 0mod2, again a contradiction.
3 3.1
A quadratic equation arising from Boltzmann's equation T h e quadratic s y s t e m
In this section we shall employ some of the results discussed above to study a particular system of quadratic ordinary differential equations derived by 111ner and Rjasanow [3] in their numerical investigation of Boltzmann's equation (a quadratic integro-differential equation c.f. [2]). In particular the system of ordinary differential equations is given as follows. m ' dx-jf = E <*ij(4 _ X < X J)> 1 < * < ™ at j^f (3.1) -^-
= -aijixl
-
XiXj),
j ± i, 1 < j < m,
where a^- = etji, 1
1 < ij < TO,
then one obtains this to be a system of m(m -1)
530
equations and we let i e R " denote the vector whose components are given by £ l j • • • ,^m) a ; 12! • • • ) £ ( m - l ) m -
For i 6 R w w e denote by x2 the right hand side of equation (3.1) and define /?by
y)2-x2-y2,
2P{x,y) = (x +
and so 0 : RN x RN — ► RN is a continuous bilinear mapping which yields a commutative multiplication and we denote by A = (RN, /?) the thus defined (nonassociative) algebra. We observe the following elementary facts which provide invariant sets and stationary surfaces for the system. Proposition 3.1 (i) Let x(t) be a solution of (S.l), then for 1 < i < m we have for t in the interval of existence of x(t) Xi(t) + £ Xij{t) = constant.
(3.2)
j+i i-i
(ii) A point p e RN is a stationary point (nilpotent element) if and only if V%=PiP:,
l
(3.3)
The proposition suggests the introduction of new variables
!* = *< + £ * «
(3-4)
3=1
which will yield m trivial equations - r = °i at
1 < t < m.
We next wish to transform the other equations to a more tractable form. To do this, it is convenient to find a basis for RN which consists of nilpotent elements. Let us consider the following vectors in RN. For 1 < i < m, let e< be the vector with 0 in all entries except the 1th, where the entry is 1, and also 0 in the ij place. For 1 < i ^ j < m, we let
531
tij be the vector with entry 1 in place i and place j and —1 in place ij and 0 entries everywhere else. The collection Cii • • •, em, e n , . . . , e(m_i)m then is a basis for R^, as one may easily verify. We also observe immediately that each of these elements is nilpotent. We furthermore may compute the following multiplication table. Proposition 3.2 Let e\,... ,e m ,ei 2 ,... ,e(m-i)m be the basis elements for RN stated above. Then the following multiplication rules hold. (Here for convenience we set an = 0 J (i)
ejej = —2-eij,
l
(ii)
e{ejk = - - [ a ^ e y + a^e*],
l
_1 if (»,j')/(r,«) = 0, if (i,j) = (r,s). It immediately follows from this proposition that we may decompose RN as follows. Proposition 3.3 (i) Let A\ = span{ex,..., e m } and Ai = span{ei2,..., e( m _i) m }, then A = Ai © A2, Furthermore A2 is an ideal and A\ satisfies A\ C A2. (ii) The ideal A2 contains the following subalgebras . Ajk = span{ey, eik, ejk:l<j<j
m]
which are isomorphic to an algebra B = span{ui,u 1 ,u 3 } which satisfies the multiplication table
532
u2 z 0 z
1*1
0 z z
Ui
u2 u3
u3 z z 0
where z = aiUi + a2u2 + 0303 and z2 = -(aia2 + a\a3 + a2a3)z, i.e.
tijk = otijeij + aikeik + ajkejk,
l
satisfies zfjk = azijk. Thus Rzijk is an ideal in Aijk and it contains a nonzero idempotent and thus generates an invariant one dimensional subspace and each solution (^ 0) in this subspace blows up in finite time. Let us now specialize what has been derived in general to the case m = 3. 3.2
T h e c a s e m = 3, N = 6
It follows from the above considerations, that we may write A = Ai® A2, A\ = span{ei,e 2 ,e3}, A2 = span{ei2,ei3,e23} and A\ C A2, A2 is an ideal and A\ = Rz, where z = —z(auen
+ «i3ei3 + "23623)
(3.5)
and if we write y = 3/1 + 2/2,
2/1 e ,4i,
2/2 € A2,
the system of equations becomes
\djT-
= 0 (3.6)
533
where Ci = yi(0). Proceeding further we may write A2 = A2 0 Rz, where Ai is a supplementary subspace to Rz, and where, of course, A2, C Rz, since A\ = Rz. Thus let A2 = span{zi,z 2 ,z} and we may write 2/2 = ai^i + a2z2 + az and dy2 dt
da da2 da2 dt dt dt = c\ + 2ci(aiZi +a2z2 + az) + y\,
but y\
= a\z\ + a2z\ + a2z2 + 2a\a2z\z2 + 2a\az\z + 2a2az2z,
(3.7)
then since A\ C Rz, we have y\ =
n(aua2,a)z.
We hence obtain da\ ~dt
an + a2a\
% dt
= A + Aoa
da 37 < dt
(3.8)
= 7i +-y2a + n(ai,a2,a),
where \i is a quadratic polynomial in ai, a2 and a. We have thus succeeded
534
in reducing the six-dimensioned system to a trivial system of 3 equations &-0,
(3,,
a forced linear system of two linear equations (uncoupled) depending upon (3.8) ' da,\ — = a1 + a 2 a i (3.10)
^
= A + Ao,
*• at and finally a scalar Riccati equation da , x -r. = 7i +72a + M a i. a 2,a)dt
= j + 6{t)a + X(t)a2.
(3.11)
Thus all the complicated dynamics of the six-dimensional system is embod ied in the scalar Riccati equation (3.9). Further we know that all solution components (with respect to the basis ei, ei,e$, z\, 22, z) except the last are defined for all time, hence blow up in the system will only occur with re spect to the z component. Furthermore, since all constants and coefficients, 7,6(t),X(t) in (3.9) are completely determined by given initial conditions any singular behavior is completely determined, since Riccati equations are completely understood in this regard, see e.g. [1].
References [1] G. Bluman and S. Kumer. Springer, New York, 1989.
Symmetries and Differential Equations.
[2] C. Cercignani. The Boltzmann Equation and its Applications. Springer, New York, 1988. [3] R. Illner and S. Rjasanow. Numerical solution of the Boltzmann equation by random discrete velocity models. European J. Mechanics, 1993. To appear.
535
[4] J. Kaplan and J. Yorke. Nonassociative real algebras and quadratic dif ferential equations. Nonlinear Analysis TMA, 3:49-51, 1979. [5] M. Kinyon and A. Sagle. Quadratic dynamical systems and algebras. J. Diff. Eqns., 1993. To appear. [6] L. Markus. Quadratic differential equations and nonassociative algebras. Ann. Math. Studies, 45:185-213, 1960. Princeton, Univ. Press. [7] S. Walcher. Algebras and Differential Equations. Hadronic Press, Palm Harbor, FL., 1991. Department of Mathematics University of Hawaii Hilo, Hawaii 96720-4091 Department of Mathematics University of Utah Salt Lake City, Utah 84112-1193
WSSIAA 3 (1994) pp. 537-541 © World Scientific Publishing Company
537
AN INTEGRAL INEQUALITY OF EXPONENTIAL TYPE FOR REAL-VALUED FUNCTIONS SABUROU SAITOH Department of Mathematics, Faculty of Engineering Gunma University, Kiryu 376, Japan ABSTRACT For some real-valued and absolutely continuous functions f\XJ
on
X > a > 0, an integral inequality of exponential type is derived. Further more, the functions satisfying equality in the inequality are determined.
1. Introduction and Result In this paper, we shall obtain THEOREM. For real-valued and absolutely continuous functions f(x) on x > a > 0 satisfying lim /(z) = 0 (1.1) r-»o+0
and
/■CO
pco
Ja
^2xdx < oo,
(1.2)
the integral inequality 1 + a f°/'(*)V<*><** ^ ef"''^*"
(1.3)
is valid. For / ^ 0 equality holds here if and only if /fz) =
/log« I log*
on
(a,6)
on
(6,+oo)
for some point b > a. For such integral inequalities of exponential type for analytic functions, see [3] and [5]. For quadratic norm inequalities of exponential type for positive definite Hermitian matrices, see [4]. See also [6] for these inequalities.
538 2. Proof of Inequality First, note that . . min(z, y) Ka(x,y) = * 'a
(x,y>a)
is the reproducing kernel for the Hilbert space HK. which is composed of all realvalued and absolutely continuous functions f(x) on (a, +00) satisfying f°°f'(x)7xdx
< 00
and which is equipped with the inner product Uuh)HKm = /i(«)/»(o) + a Hf[{x)f2{x)dx.
(2.1)
Ja
This fact will be confirmed, directly. For the general properties for reproducing kernels, N. Aronszajn [1]. On the other hand, ka(x,y)=\0g^^.
(x,y>a) (2.2) a is the reproducing kernel for the Hilbert space Hk. which is composed of all realvalued and absolutely continuous functions f(x) on x > a satisfying (1.1) and I
f'(x)2xdx
and which is equipped with the inner product (/1,/»)»». = I"f[(x)f'2(x)xdx.
(2.3)
Ja
This fact also will be confirmed, directly. We now have the interrelationship between Ka(x, y) and ka(x, y)
Ka(x,y) = l + ktt(x,y) + ^
^
+ ... + ^f^
We denote the tensor product of n-times of H^ by (Hka)l =
Hk.®Hk.®-®HkQ.
+ ...
(2.4)
539 Furthermore, we shall consider the Hilbert space [(tft.)®], which is formed by restricting functions in {Hka)s> *° * n e diagonal set of (a, +oo) n = (a, +cc) x (a, +00) x •• • x (a, +00). Here, for any such restriction F € [(#*„)«],., the norm WF\\l(irt.)i]r is given by min ||C?||(jjkii)» for all G, the restriction of which to the diagonal set is F. See [1, Theorem 11, p. 361] for this argument. Then, n
II *«(*;>«) is the reproducing kernel for the Hilbert space (/ft,)® a n c ' ka(x, y)n (n ^ 2) is the reproducing kernel for the Hilbert space [(#*„)»]'• ([1, pp. 357-362]). Hence, from the identity HK.=
C a) Hk. e
[{Hk
-fv]r
ffi...
e
Kg*-)^f
e
...
(2.5)
which is induced from Eq. (2.4) (see [1]) and from the expansion
^.1+/(.)tKt...ta+.., l\
n!
we, in particular, obtain the inequality Eq. (1.3), as in ([3], [5]). 3. Proof of Equality Statement in Theorem We assume that for / € Hka(f ^ 0) equality holds in Eq. (1.3). As we see directly, note that
is the reproducing kernel for the Hilbert space H^i which is composed of all realvalued and absolutely continuous functions f(x) on x > a satisfying Eq. (1.1) and
rf,{x)2 {ii°*i)
dx<
°°
540 and which is equipped with the inner product
(/iJ 2 )* 4 2 =^°°/{(x)£(*)(£log|)
dx.
(3.1)
Hence, we, in particular, obtain the integral inequality f~(f(x)g(x)f
(1 log | )
dx £ l~f{x?xdxl~g'{x?xdx
(3.2)
for any functions /, g € Htm ■ Since, equality in Eq. (1.3) for / ^ 0 implies equality in Eq. (3.2) for / = g, we shall first determine the functions satisfying equality in Eq. (3.2) for First of all, note that for / = g equality holds in Eq. (3.2) if and only if (/(xi)/(x2),/-(xi,x2))(H4j|=0
(3.3)
for all F € ( # * J | satisfying F(x,x)=Q
on (o,+oo)
(3.4)
(cf. [2, (3.2)]). In particular, note that Mxi,yi)Mxi,y2)Mz2,y0)
MxTH^)
/„ . „ . ,„ . „x
(«<»i<W
/ , -v
(3.5)
€ (#0| and so, the functions F(xi,x,)
= t«(xi, yi)i a (xj, y2)
*.(xi,yi)fc.(xi.y2)*.(x2,yo) . / s
, , . . (o < yi < y2 < yo) (<>o)
*«(xi,yo) satisfy Eq. (3.4). Hence, from Eq. (3.3) we obtain, using the reproducing property of *«(xi,yi)*«(x 2 ,y 2 ) in (Hhm)%,
= /(yo)(/(yi)-log^log^ I a a
•/'0^)(i^Tji}
for
« < yi < »a < yo-
(3.8)
541
This identity implies that if / is not constant on (jfc, j/o) for some points 1/2 < Jto then /(j/i) is expressible in the form f(yi) = c log ^a
on o < yi < y2;
(3.9)
that is, if equality holds for / ( ^ 0) in Eq. (3.2), then f(x) is expressible in the form, for some point b > a and for some constant c f c log * on
/(*) = <
1 i
I c log* on
(a, b) )K1
\
(310)
(b, +00).
If equality holds for / ^ 0 in (1.3), then / is expressible in the form Eq. (3.10). Hence, we shall determine the constant c in Eq. (3.10) satisfying equality in Eq. (1.3) for c ^ 0, £. We have :
^{/v-'*-/v'j'} ^ 0.
(3.11)
Here, equality holds if and only if c = 1. Meanwhile, for c = | , equality does not hold, as we see directly. We thus have completed the proof of our results. REFERENCES 1. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 6 8 (1950), 337-404. 2. S. Saitoh , The Bergman norm and the Sxego norm, Trans. Amer. Math. Soc. 349 (1979), 261-279. 3. , Some inequalities for analytic functions with a finite Dtrichlet integral on the unit disc, Math. Ann. 340 (1979), 69-77. 4.
, Quadratic inequalities deduced from the theory of reproducing kernels, Linear Algebra Appl. 9 3 (1987), 171-178. 5. , Inequalities of exponential type for analytic functions with finite Dirichlet inte grals, Complex Variables 10 (1988), 171-177.
6.
, "Theory of Reproducing Kernels and its Applications,'' Pitman Res. Notes in Math. Series 189, Longman Scientific & Technical, 1988.
WSSIAA 3 (1994) pp. 543-548 © World Scientific Publishing Company
543
R E M A R K ON Q U A S I M O N O T O N I C I T Y
ALICE SIMON Departement de Mathematiques, Universite d'Orleans, BP 6759, 45067 Orleans Cedex 2, France BP 6759, 45067 Orleans Cedex 2, France and and PETER VOLKMANN Mathematisches InstitutVOLKMANN I, Universitat PETER
Karlsruhe,
Mathematisches Institut I, Universitat Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany ABSTRACT For a continuous right-hand side (of an ordinary differential equation) we show that a well known theorem on differential inequalities does not only follow from but also implies its quasimonotonicity.
Consider a real Banach space E. For a € E, p > 0 we use the notation S(a;p) = {x\xeE,
||x-a||<^}.
Let K C E be a wedge, i.e. K ^ 0 and A > 0, x € K, y € A' = > X(x + y) € K. We suppose K to be closed and such that
Int K t 0.
544 For x, y € E we write x
■<=► y — x & K,
x < y
f(t,x)
:D-*E
(where D C IR x E) is called giiast'monoione t'ncreastny with respect to x, if (*,*), (t,y) € A x < y, y> € A", p(x) = y(y) =► ? ( / ( * , x)) < *>(/(*, y)). For such functions the following property is well known (cf. [6]): (P) If v,w : [t0,
u'(t) = /(«,«(<))
(t0
(1)
also is equivalent to the quasimonotonicity of / . One might think of carrying over this result to Banach spaces E which are ordered by cones. But then one gets in trouble with the existence of solutions for (1). Nevertheless, for a wide class of
545 ordered Banach spaces, quasimonotonicity and continuity of the right-hand side of the differential equation in (1) imply the (local) existence of solutions (cf. [4]). More general situations remain as an open problem. Remark 2. To prove our theorem, we use the following fact, which is standard by now: Let g : [io,'o + v] r),p > 0), such that
x
S(a;p) -* E be a continuous function (t0 € 1R; a € E\
\\g(t,x)\\ <M
(t0
+ r,, \\x - a\\ < p).
Then, given a > 0, the initial value problem u(t0) = a,
u'(t) =
g{t,u(t))
has an a-approximate solution, i.e. there is a C1-function u : [to,T] —> S(a\p) satisfying u(t0) = a, | | u ' ( 0 - s ( * , u ( 0 ) | | < a (*o < * < T). In particular, one can take T = t0 + rj, where fj = min < TJ, -—-— >
(_ M + a ) Proof of the theorem. 1. Assume the theorem to be false. Then there are (to,xo), ('dJ/o) G D and y> € K' such that XQ < Vo,
v(f(to,x0))
>
(2)
but Choose 6 > 0 with
*< JrjMf(to,x0)-f(t0,y0)),
(3)
and put /z
= max{||/(
(4)
Furthermore, choose p € Int K such that x € E, \\x\\ < 1 ==> i < p.
(5)
2. According to the structure of D and the continuity of / at (to, x0)> (*0i S/o)> there is an tj > 0, such that ||/(*,*)-/(*o,xo)||<*
(t0
+ r,, \\x - x0\\ < r,),
(6)
546 \\f{t,y)
- f(to,yo)\\
<6
(t0
+ ri,\\y-y0\\
(7)
We consider the two initial value problems u(«o) = » o - ep,
u'(t) = /(*, u(0) - ep,
(8)
«(to) = Vo + ep,
u'(0 = /(*, u(<)) + ep,
(9)
where
0<eS
(10)
W
Observe that the functions on the right-hand sides of (8), (9) are defined on [to, t0 + 17] x S (x0 - ep; - J
and [t0, t0 + r)]xS
(y0 + ep; - J ,
respectively (this follows from (6), (7), and ||ep|| < r?/2), and they are bounded in norm by M = li + 6 + e\\p\\ (this follows from (4), (6), (7), (8), (9)). By remark 2 above, there are \ e-approximate solutions v,w : [to,T] —> E of (8), (9), respectively, where
»?/2 1 * = ™ ( % + « + e|MI + ' / 2 / f
r = <0 + >?,
(ID
Replacing in (11) e by its upper bound r?/2||p|| (cf. (10)) yields
^ : =™»{'/ > + , + ; / f + T ? / 4 || P ||}^. and therefore we have t\ := to + m < t0 + V = TNow we consider v, W only on the interval [to,
||u'(<)-/(i,t,(<)) + e p | | < ^ e
(12) (t 0 < t < «,),
(13)
and, similarly, w : [*o,«i] -♦ S (jto + ep; | ) C 5(y 0 ;»/), to(«o) = yo + ep,
||w'(0-/(*,w(0)-epll<2e
(14) (*o < * < *i)-
(15)
547 3. (5), (13), (15) imply t / ( 0 - /(*, w(0) + £P < 2 £ P
(*o < * < *i),
-«/(<) + /(«, w(0) + £P < 2 £ P
(*o < * < *i),
hence v'(t) - f(t,v(t))
< ~\ep
< \*P < v/(t) -
v'(t) - f(t,v(t))
< w'(t) - f(t,w(t))
f(t,w(t)),
that is (to < t < U).
Since furthermore (cf. (2)) v(t0) = x0 - ep < y0 - ep < y0 + ep = w(*o), the property (P) for / implies v(t)<w(t)
(<0<<<
(16)
4. Because of (12), (14) and t\ < t0 + 77, we can use (6), (7) for x = y = w(t), yielding
||/(*,«(0)-/(
v(t),
Wf(tMt))-nto,yo)\\<6.
From (13), (15) thus follows \W(t) - f(t0lx0)\\
<6 + e (||p|| + 0
,
\\w'(t) - /(< 0 ,ito)|| <S + e (\\p\\ + 0
,
ncncc \\w'(t) - v'(t) - (/(<„, »o) - /(
l|r(0ll<2« + e(2||p|| + l)
(*o
Integrating, we get w{t) - v(t)
=
w(t0) - v(t0) + I [W'(T) - v'{r)} dr
= yo-xo + 2sp + (t-to)[f(to,y0)-f(t0,x0)}
+ R{t),
(17)
548 where \\R(t)\\<(t-t0)[26
+
e(2\\p\\-rl)}.
We apply
!/o) - /(*o, *<>)) + ?(*(<))•
Now (16) implies
< 2e
We put t = ti and divide by t\ — t0: f(f(to,x0)-f(t0,y0))
<^ £ l
+
| M | . [2* + £ (2||p|| + 1)].
' i — 'o
This inequality is valid for every e satisfying (10). Passing to the limit £ J. 0 yields
9(f(to,x0)-f(t0,y0))<2S\\>p\\, which contradicts (3).
References 1. R.M. Redheffer, Gewohnliche Differentialungleichungen mit quasimonotonen Funktionen in normierten linearen Raumen, Arch. Rational Mech. Analysis 52 (1973), 121-133. 2. R.M. Redheffer and W. Walter, Flou>-invariant sets and differential inequalities in normed spaces, Applicable Analysis 5 (1975), 149-161. 3. R.M. Redheffer and W. Walter, Remarks on ordinary differential equations in ordered Banach spaces, Monatshefte Math. 102 (1986), 237-249. 4. A. Simon (A. Chaljub-Simon), R. Lemmert, S. Schmidt, and P. Volkmann, Gewohnliche Differentialgleichungen mit quasimonoton wachsenden rechten Seiten in geordneten Banachraumen, in General Inequalities 6, International Series of Numerical Mathematics 103, Birkhauser, Basel (1992), 307-320. 5. J. Szarski, Differential inequalities, Paristwowe Wydawnictwo Naukowe, Warszawa, 1965. 6. P. Volkmann, Gewohnliche Differentialungleichungen mit quasimonoton wach senden Funktionen in topologischen Vektorrdumen, Math. Z. 127 (1972), 157— 164. 7. W. Walter, Differential and integral inequalities, Springer, Berlin, 1970 (Ger man version 1964).
WSSIAA 3 (1994) pp. 549-558 © World Scientific Publishing Company
549
Isoperimetric Inequalities in a Boundary Value Problem in an Unbounded Domain by Rene Sperb Seminar fur Angewandte Mathematik and Institut fur Polymere ETHZ CH-8092 Zurich
Abstract In this paper a semilinear elliptic boundary value problem in the exterior of a finite domain is considered. An important example in applications is the PoissonBoltzmann problem. Isoperimetric inequalities for a functional of the solution are proven using optima] sub- or supersolutions.
1. Introduction Let ft be a bounded domain in IR , N > 3 and ft* the exterior of ft. We consider the boi boundary value problem A« = 7 J / ( « ) in ft* 1 u = 1 on 5ft I . u —» 0 as | i | —» oo J
(1.1)
Here i is a generic point of IR , 7 is a given positive parameter and the nonlinearity is assumed to satisfy /(O) = 0, /'(0) > 0, /"(«) > 0 for u > 0 .
(1.2)
The assumptions on /(u) guarantee that the solution decays to zero when | i | —♦ 00 at least as fast as the solution of the linearised problem. Two possible backgrounds for problem (1.1) are described in the following: (a) Poisson-Boltzmann problem: With this application in mind u represents the potential of a charge distribution. Important choices for /(u) are then / ( u ) = sin/fti or /(«) = u (Debye-Huckel approximation) . The parameter 7 then involves a number of physical constants (see e.g. Garrett & Poladian 2 ).
550 (b)
Reaction-Diffusion: In this interpretation u is the concentration of a reactant. The reactant is fed into ft and diffuses through #ft into the reaction region ft* where a simultaneous diffusion-reaction process takes place. The concentration inside fl is held constant by a continuous supply of reactant. An important case is again / ( u ) = u (e.g. for a first order degradation process). Another frequent choice is (Michaelis-Menten kinetics) ,. u , f[u) = , where o is a positive constant. v ' a+u
Especially with the second interpretation of problem (1.1) in mind one is mostly not interested in the values u(x), but rather in some functional of the solution. If (1.1) describes a diffusion reaction process then one has the relation 7
D'
where ifc is the reaction rate and D the diffusion coefficient. Very often then (see Aris l one is interested in a pure number characterizing the influence of diffusion in the chemical reaction. One such possible number may be defined as / |Vu|
(1.3)
Here da is the surface element of dil and |ft| the volume of ft. We will concentrate mainly on optimal estimates for TJ in terms of geometrical data of ft. Note that for 7 = 0 and the condition u(x) = 0(\x\i-N), the classical electrostatic problem and f
for |x| -» 00, N > 3, (1.1) is
\Vu\ds is up to normalization the electrostatic
capacity of ft. 2. Bounds derived from optimal sub- or supersolutions Suppose we can find a function y(x) with the properties (subsolution)
Au > 7 2 /(«) in"* u <
1
1 on 9ft, u(x) —» 0 for |x| —* 00 J
(2.1)
for the limiting case 7 = 0 we have to replace the boundary condition at infinity by u(x) = 0 ( | x | 2 - N ) • |x|, N>3(N=
Number of dimensions .
Then one has u(x) < u(x) for any x 6 ft*. For a supersolution u(x) one just has to reverse all inequality signs.
551 In the following we construct an optimal subsolution in two steps. First we define
^ ( A V f
= (iRYfM ^ pe(i,oo) 1
¥>(!) =
1.
(22)
J
The second important ingredient is the (exterior) distance function d(x) = min |x — j / | , i G ft*. We then set s(x)=
(l + H0d{x)y~N,
Ho =
%®{inri g
N>3.
(2.3)
Here
Ms,)
}'
where kj(y) denote the principal curvatures at a point y of dft. Thus H0 is the maximum of the mean curvature, which is assumed to be finite. Note that for the ./V-ball j / 0 = ^ . The next result will enable us to derive optimal bounds for n as defined in Eq. (1.3). Lemma 1 Let ft be a convex domain, i.e. kj > 0 for j'• — 1, ...,N— 1. Then the function u(x) = tp(s(x)), s(x) defined by Eq. (2.3) and where (p{s) = f>{p)> f{p) being the solution of (2.2) and s = -7^=7, is a subsolution to problem (1.1). In order to prove Lemma 1 it is convenient to prove first an auxiliary result stated as: Lemma 2 Let ft be a convex domain. satisfies As > 0 in ft*.
Then the function s(x) defined in Eq. (2.3)
Proof: We have AJ^
[2-N)HQ
A J
, (N-2)(N-l)
,rt
lr7Jl2
But one has (see e.g. Gibarg-Trudinger 3 , p. 355)
so that AS =
(l + H]0aY
{(N
~ 1}H ° ~ ° + H°d)M]
•
552 We now rewrite the term{(N — 1)Ho — (1 — Hod) Ad] putting the expression for Ad over a common denominator. Then we may write
Ad-A^ with
A{d) = £
J ■ Pi J-1,
B(d) = Z
;=0
P; ■ d' .
i=0
Here we have used the abbreviation Pj, where Po = 1 and Pj = sum of all products with j different factors Jk< and 1 <j, I < N — 1. With this notation we can write ^=(l[NHod)"B(d)
{(N-l)HoB(d)-(l
+
H0d)A(d)}
.
1(d)
The expression f(d) can be put into the form
1=0
where
§ HoPl{\--R—^-{l+\)Pl+x.
Cl =
We finally show that Ct > 0 for all t. It is clear for / = 0 and I = N — 1, noting that by definition Pi=
£
kj<(N-l)H0.
3-1
We consider Ct as a function of the N — 1 variables kj. Since Ct is symmetric in all variables the minimum is attained for values k, such that K\ = K2 — . . . = *N—1
=
"- •
and hence C^,...,fc w _ 1 )>C < (Jb o ,...,A : o ) = 0 , which implies that As > 0.
553 Remarks on Lemma 2: (a) For the special case 7 = 0 Lemma 2 shows that H0d(x))2-N
s(x) = (l +
is a subsolution in the "capacity problem", i.e. + H0d(x))*-N,
u{x)>(l
N>3,
(2.4)
and the equality sign holds in (2.4) if ft is the ./V-ball of radius R = y-. Inequality (2.4) was derived by Payne & Philippin * by a different method. (b) The convexity assumption is needed in order to ensure that Ad is finite for all x € ft*. Inequality (2.4) however holds without the convexity assumption, as shown by Payne & Philippin. Proof of Lemma 1: We have
At = g. A , + g,v„> and 1 1
((N-2)H0)> (\ + H0dy»-*]
Furthermore
dtp _ 1 ds N-2y
{{N-2)H0y ^-2
9]
N-i
w
df dp
and hence Au-72/(«)
>
^ 2
£J--7
fit)
■4i^S-*'. p"-l
d_ ( N-I ^ _ dp ^ dp
On 3ft we have p = 1 and thus u = y?(l) = 1 on 3ft as well as u(x) —► 0 as |x| —► 00. As a consequence of Lemma 1 we have Theorem 3 Let ft be a finite convex domain in IRN, N > 3. Denote by Ho the maximum of the mean curvature ofdil and by 770 the effectiveness of a ball of radius HQ1 (as defined bv £9. Eo. (1H)). satisfies 4i/ (1.8)). Then the effectiveness nofQ nofQ, satisfi<
^*mk V<
and the equality sign holds if ft is a ball.
(2J)
554 i
Proof: Since the subsolution a defined in Lemma 1 satisfies the boundary conditions required in problem (1.1) we have / \Vu\da < <£ |Vu|
(2.8)
But from Eq. (2.5) and (2.6) we find / \Vu\dv = -H0-^\ Jan dp 'f=1
■ \3tt\ .
(2.9)
From (2.2) it follows that (R = ff0_1) Vo = -ZT 7 Eqs.
Hl)l
•
(2-10)
(2.9) and (2.10) then imply inequality (2.7).
Remarks on Theorem 1 (a) An important case for applications is N = 3 and / ( « ) = w. Then the subsolution can be written down explicitly and one has at distance d from dtt
"M * rnb •
(2J1)
Inequality (2.7) then takes the form
„<M.
(,I2)
(b) We can also derive an explicit lower bound for rj from Lemma 1 in the case N = 3,/(u) = u. From the differential equation and the boundary conditions it follows that
ri=
w\iudx-w\i'udx-
We integrate over H" using parallel surfaces to dfl. One has for the surface area S(6) of a parallel surface at distance S from dCl (see e.g. Polya-Szego s , p.66) S(S) = 5(0) + 2M-6 + 4 x 5 s .
(2.13)
Here M is the Minkowski constant of dCl defined by M=
H da, Jan
and of course 5(0) = \dil\.
H = mean curvature,
(2.14)
555 Hence we have
^ifr^m+M+4d!»^
(2.15)
The evaluation of the integral gives after some routine calculation 1 f4?r V > ~ {— + A + K«o l « l | - A) e" • £,(/•)}
(2.16)
where the following abbreviations have been used: i/ =
flj7|n|,
M = -^-, Ho
A = 2Af//0-4x.
(2.17)
Ei(fi) denotes the exponential integral denned by f°° e~*
Inequality (2.16) is again isoperimetric since the equality sign holds if H is a ball. (c) One could also derive a version of Theorem 1 valid for plane domains SI. However then the limiting case 7 = 0 does not make any sense. In addition problem (1.1) seems less interesting in the two dimensional case. Next we derive an estimate for r; which is based on the construction of an optimal supersolution u. For this purpose we use as an auxiliary problem the "electrostatic problem" Ah
= 0
in
h = 1 h(x) =
0*
on dVt
(2.18)
0(\x\*-N)
In order to get an explicit inequality we restrict our attention to the special case f(u) = u. Theorem 4 Let f(u) = u in problem (1.1) and SI be a finite domain with boundary of class C2+c, but not necessarily convex. Let r = max |V/i|, h being the solution of (2.18), do
and C = wZl 9 \Vh\dc the "capacity" of dSl(wN = (N — 1) dimensional surface area Jan of the unit N-sphere). Then we have V>
1+
K N-2
*»j*(")l Ks^iK)!
C wN 7 2 |n|
(2.19)
556 where K = (N — 2) J an
The equality sign holds
ifilis
Proof: We use the solution of (1.1) with / ( u ) = u and fl the unit ball as an auxiliary function:
^ ( A V W ) ' = **
in (l.oo) 1
(22Q)
V(l) = 1. v(/>) -► 0 for p -» oo J The value of c will be chosen later. The solution of (2.20) is
We then set /> = h~T^s, h being the solution of problem (2.18). It is convenient to write
At this point we use a result of Payne & Philippin 4 stating that in N dimensions the solution h of (2.18) satisfies (the smoothness of dCl is used here!) |V/>|2 < T2 ■ A
2
^ .
(2.24)
Hence Au - 7 J « < r J —^ • /»^^T - 7 2 <^ , ah* but because of Eq. (2.22) this can be put into the form
if we choose c = ' Af ~ 2 ' 7 in problem (2.20). It is easy to see that G satisfies the boundary conditions. We therefore have /
|Vu|d<7 > /
\Vu\da .
(2.25)
557 The relations S-Ka-tiKhr)
c =A
(AT-2)7
^-ir1'
^^r>
1
P
=-7^T2
together with well known identities for Bessel functions then show after some routine calculations that inequality (2.25) leads to the statement of Theorem 2. Remarks on Theorem 2 (a) Inequality (2.19) is still not quite explicit. It is not hard to check that the function Kepi')
is increasing for s > 0. Hence we need an upper bound for r and a lower bound for the capacity C. It was shown by Payne & Philippin * that r<{N-2)H0.
(2.26)
For the capacity C one has the classical result of Poincare-Szego (see 6 ) that C>{N-Q(»B)V.
(2.27)
Inequality (2.19) therefore implies the weaker but still isoperimetric inequality f
7
K*?(%)\
(N-2)
which for TV = 3 reduces to
'^(? + ^k) O"'-
*»>
(b) For N = 3 the supersolution u can be written as U(*)«A(x)exp{-2(^-1)}.
(2.30)
An explicit upper solution ~H(x) for h(x) would then lead to an explicit upper bound for C(i). However for general domains it seems difficult to give an optimal choice ofX(x).
558 3. Concluding Remarks (a) The method of integration over level surfaces (see Sperb 7 ) could also be used in order to derive an optimal inequality for the "effectiveness" n. One can show then that for given TV-volume of fi n is a minimum for the TV-ball. In order to keep this note at a reasonable length we mention this result without proof. (b) A number of additional bounds could be proven by exploiting the fact that the function P : = | V u | 2 - 2 7 s F ( U ) + /?u, where F(u) = / f(v)dv and f) is chosen appropriately, is non positive in fi*. Here Jo techniques described in Sperb 7 can be used. (c) The case N = 2 has been excluded here since it is of less interest in applications. It is however not hard to extend Theorem 1 (but not Theorem 2) to this case. Also for the remarks (a),(b) above the two-dimensional case is no exception.
References 1. R. Aris: The mathematical theory of diffusion and reaction in permeable catalysts, Clarendon Press, Oxford 1975. 2. A.J.M. Garett & L. Poladian: Refined derivation, exact solutions and singular limits of the Poisson-Boltzmann equation, Annals of Phys. 188, (1988), 386-435. 3. D. Gibarg & N.S. Trudinger: Elliptic partial differential equations of second order, Springer, Berlin (1983). 4. L.E. Payne & G.A. Philippin: Some isoperimetric inequalities for capacity, polar ization and virtual mass, Applicable Analysis 23, (1986), 43-61. 5. G. Pdlya k G. Szego: Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton (1952). 6. G. Szego: Uber einige Extremaleigenschaften der Potentialtheorie, Math. Zeitschrift 31, (1930), 583-593. 7. R.P. Sperb: Maximum principles and their applications, Academic Press, New York (1981).
WSSIAA 3 (1994) pp. 559-571 © World Scientific Publishing Company
559
On functions whose gradients have a prescribed rearrangement Giorgio Talenti Dipartimento di Matematica dell'Universita,viale Morgagni 67/A, 1-50134 Firenze, Italy. e-mail address: [email protected] Abstract Real-valued functions u defined in euclidean n-dimensional space R" ate detected that render the Lorentz L(p, 1) norm a maximum and obey the following constraints: the support of u has a prescribed measure and | Vu |, the length of the gradient of u, has a prescribed rearrangement. 1 Introduction Consider a cylindrical rod made up of a number of plastic materials and subject to torsion. Suppose the rod has a given length; suppose the number of the materials, and the quantity and the plastic yield limit of each material are given. Then the rod withstands the largest twisting moment if and only if its cross-section is a disk and the materials are arranged in concentric annuli — the softest material innermost, the hardest material outermost, the other materials orderly in between. Recall from the theory of plasticity that the moment in question is proportional to the integral of a real-valued function « — the stress function — and the relevant data are stored in the distribution function of | Vu | — the length of the gradient of u. See [Ar] for details. The above assertion can be derived from the following theorem. Theorem 1.1. Let M be a nonnegative decreasing right-continuous defined in [0,oo[ that decays fast enough at the infinity,
function
and let V be a number
larger than or equal to A/(0). Consider any real-valued function u defined in R n that is nice enough and satisfies the following conditions: (i) the distribution function of | Vu | is M, i.e., m{x £ R": | Vu(at) | > t) = M(t)
(1.1)
for every nonnegative t; (ii) the support of u has measure V, i.e., m{x e R": | u(x) | > 0} = V.
(1.2)
Then If | J Rn
u(x)dx <(n+l)-1K-^n\'X'\vM/n-(V-M(t))i+l/n\dt. J0 L
(1.3) J
560 Equality holds in (1.3) if and only if either u or —« is a dome function
(i.e., a
nonnegaiive radial function whose restriction to its support is concave). Here n is any positive integer, m stands for n-dimensional Lebesgue measure, «B = T B / 2 [ r ( n / 2 + l ) ] - 1
(1.4)
— the measure of the unit n-dimensional ball. A full proof of theorem 1.1, including a derivation of the "only if„ clause, was given in [AT]. Related proofs appeared in [Ar] and [GN]. Theorem 2.1 from [GN] implies the following theorem. Theorem 1.2. Let M, V, u as in theorem 1.1. Then sup | « | < /c" 1 /" f °°(M(t))Vndt.
(1.5)
Equality holds in (1.5) if u is any spire function (i.e., a nonnegaiive radial function whose restriction to any ray is decreasing and convex). Motivated by the preceding results, one may consider the variational problem I u I = maximum, under conditions (i) and (ii) above
(1.6)
— i.e., the problem of rendering IIu H a maximum within a class of functions « whose support has a prescribed measure and such that | Vu | has a prescribed rearrangement. Here II II stands for a norm in some Banach function space. According to usage, we say that
two functions
are equidistributed, or a
rearrangement of each other, if they have the same distribution function. Theorems 1.1 and 1.2 give a complete answer if I II is either the norm in Ll(Rn) or the norm in I°°(R n ). Theorem 3.1 from [ALT] claims that, if || || is the norm in L P (R"), then a solution to problem (1.6) actually exists and is radial (provided p is suitably related to dimension n and the decay of distribution function M at infinity). A characterization of maximizers is left out in [ALT], however. Further investigations about problem (1.6) are in [FP], section 3, and [Po]. In this paper we display the geometry of solutions to problem (1.6) in the case where H II is (equivalent to) a norm in Lorentz space L(p, 1). Our main result — theorem 2.1 below — includes both theorem 1.1 and theorem 1.2. The present
561 paper may be viewed as an approach to problems from the calculus of variations in which side conditions constrain a rearrangement. Problems of such a sort are worked out, e.g., in [ALT], [Bu], [EST], [FP], [LS], [Mc]. Recall that, if u is a real-valued measurable function defined in R", the decreasing rearrangement of u — denoted u* throughout — is the real-valued function defined in [0,oo[ that is nonnegative, decreasing, right-continuous and equidistributed with u («* can be represented thus u*(s) = inf{t > 0:m{i € R": | u(x) | > t} < s} for every nonnegative s). If 0 < p < oo and 1 < q < oo, the Lorentz space L(p,q) is the collection of all real-valued measurable functions u defined in R" such that
{I
oor 1. /,p
s
u*( S )
2M
— the usual modification has to be made if q = oo. The left-hand side of the above L(p, p) = Lp(Rn)
inequality is equivalent to a norm of u in L(p,q).
if p > 1,
L(Pi l) C i ( p , r) if q < r — the inclusion is strict and the embedding is continuous. For the sake of brevity we shall abuse notations and set
Thus II u ||£,
| u(x) | p dx\l/p
.= [J
fluj II
, the norm of u in I P (R"); moreover
>p»/ff-l/r - l / « r l / r | 'L{p,q)
"
I
(18)
"L(p,r)
if q
562 conditions: (i) | Vu | and \ Vw \ are rearrangements of | Vu | , i.e., M(t) = m{x e R n : | Vv(x) \>t}
= m{x£
R": | Vu>(x) | > t}
(2.1)
for every nonnegative t; (ii) the support of v and the support of w have the same measure as the support ofu,
i.e.,
V = m{x£ R": | v(x) \ > 0} = m{x g Rn: | w(x) \ > 0};
(2.2)
(Hi) v and w are radial and radially decreasing; moreover the restriction of \ Vv \ to the support of v is radially increasing, while | Vw | is radially decreasing — in other words, v is a dome function and w is a spire function. Assertions:
S u L i) * I" L i)
«/ » > i -^ P > »/(»-D;
(2-3b)
furthermore
i» i«,, ) =^TirC^ , / ' + 1 / "-^-«<") , " + , / i^ <*•> |"U.1,'^TTTC(M(,,)'/'+I'"J'-
(24b)
A proof of theorem 2.1 is based on the following lemma. Lemma 2.2. Let u be a real-valued function
defined in R n . Suppose u is
sufficiently smooth — e.g., Lipschitz continuous — and decays at infinity — i.e., the measure of {x g Rn: | u(x) | ><} is finite if t is strictly positive. Then: (i) the following inequality
T- [
«** 1 {x 6 R":|ti<*)| >«'(*)}
I Vu(x) | dx > i«i/" 5 1 - 1 /" f - 4«!(,)
holds for almost every positive s; (ii) the restriction
(2.5)
of u* to every compact
subinterval o/]0,oo[ is absolutely continuous. Proof of Lemma 2.2. The basic ingredients involved here are a coarea formula and the isoperimetric theorem in R". Federer's coarea formula claims that if u is Lipschitz continuous and / is integrable, then
563
f
/(x)|V«(x)|<*x=f
dt\
f{x)Hn_x{dx).
The isoperimetric theorem in R" claims that if E is a measurable subset of R" and the measure of E is finite, then
^„_,(3£;)>n4/"[m(£)j1-1/n Here Hn_^
stands for (n-l)-dimensional Hausdorff measure. A proof of Federer's
coarea formula is in [Fe], a treatment of the isoperimetric theorem is in [Ta]. The following technicalities play a role. The distribution function of u, fi, is defined by fi(t) = m{x e R": | u(x) | > t}
(2.6a)
for every nonnegative t and obeys H(t-) = m{x € R": | u(x) | > r}
(2.6b)
for every positive t — /i is nonnegative, decreasing and right-continuous, u*, the decreasing rearrangement of u, is nonnegative and equidistributed with u; therefore the l-dimensional measure of {s > 0:u*(s) > t} = fi(t) for every nonnegative t and the l-dimensional measure of {a > 0:u*(s) > t} = fi(t—) for every positive t. u* decreases monotonically; hence the former equation implies /•(«*(*)) < *,
(2.7a)
li(u*(s)-) > s
(2.7b)
whereas the latter implies for every nonnegative s. Observe that m { x € R n : u * ( a ) > | u(x) | >«*(&)}
(2.8)
if 6 > a > 0. In fact, the left-hand side of (2.8) equals /i(u*(6)) — fi(u*(a)—) by reason of (2.6a) and (2.6b); moreover /i(u*(6)) - fi(u*(a)—) < b — a by reason of (2.7a) and (2.7b). The proof goes ahead this way. We have f \Vu(x)\dX>nKl/nal-l/n[u*(a)-u*(b)] n J{i6R :u>)>|u(r)|>u*(6)} if b > a > 0. In fact,
(2.9)
564 the left-hand side of (2.9) =
f
J
by Federer's coarea formula fk
u (t)
Hn_,{XeRn:\u(x)\=t}dt
>
by the isoperimetric theorem in R"
f " ( 0 ) nKlJn\m{x L i u'(b)
€ Rn: I u(x) \ > t}] 1 " 1 /" dt
>
J
f" ° nKlJn [m{x G Rn: \ u(x) | > u^a)}] 1 " 1 /" dt J u*(4)
>
the right-hand side of (2.9) — the last inequality follows from equation (2.6b) and inequality (2.7b). Inequality
(2.9)
implies
assertion
(i)
immediately
—
the
involved
derivatives exist almost everywhere since u* decreases monotonically and [0,oo[9s—► [ \Vu(x)\dx J{x€R n :|u<*)| >«•(«)} increases monotonically. Assertion (ii) is a straightforward consequence of inequalities (2.8) and (2.9). D Proof of theorem 2.1. Let ip and tp be nice real-valued functions defined in ]0,oo[ such that ¥>00>0,
*/>(s) > (1 - 1/n) ]
(2.10a)
sip(s) < (1 - 1/n) ]rl>(t)dt 0 for every positive s. We are going to prove , oo f oo f(s)u*(s)ds< vK«)«*(s)
(2.10b)
and xfts) > 0,
»oo
il>(s)u*(s)ds< '0n Notice these facts.
(2.11a)
.oo
I tjis)w*(s)ds. Jn
(2.11b)
565 (i) We have
"•M-O-tfW dt = 0
if
0<3
if
s>V.
(2.12)
Indeed, the very definition of u* implies that the support of u* is [0, V], and lemma 2.2 informs that the restriction of «* to every compact subinterval of ]0,oo[ is absolutely continuous. (ii) The following inequalities -4£{s)<
- f f i ^ ds
,, *
., { - T - f
l/nVl/nl « « „ ' V i/n {
\Vu(x)\dx)
if, o», , . , <" J {* 6 R":|u(*)| >«'(,)}
(2.13a)
|V«(x)Mxl(2.13b) )
hold for almost every positive s. This assertion follows from lemma 2.2. Indeed,
f
J{reRn:0<|u(i)|
[
| Vu(x) | dx =
| Vu(x) j das— f
' {z € Rn: | u(r) | > u » }
1 {z € R": |u(*)| > 0}
| Vu(x) | dx,
since theorem 3.2.2(c) from [Mo] ensures that either {x £ Rn: | u(x) | = Constant} has measure zero or Vu vanishes almost everywhere in such a set. Hence (2.13a) is a consequence of (2.13b). The latter is a copy of (2.5). (iii) The inequality \
|Vu(x)|dx<
J{i€R":0<|u(i)|)}
|Vu|*(t)dr
(2.14a)
\Vu\*(t)dt
(2.14b)
JO
holds for every s such that 0 < s < V, the inequality [ J{z6R":|u(*)|>u»}
| V u ( x ) | d x < f* Jo
holds for every positive s. Indeed, a generalization of theorem 378 from [HLP] y^**
r
' E
f(X)dx<
ME) J
ns)ds
0
if / is a nonnegative and E is measurable. On the other hand, formulas (2.6a) and (2.6b), and inequalities (2.7a) and (2.7b) ensure that
m { i g R n : 0 < \u{x)\
if 0 < s < V, and m{x € R n : | u(x) \ > u*{s)} < s if s > 0. The proof goes ahead this way. .00 J
tp(s)u*(s)ds 0
=
by formula (2.12)
<
by the first inequality in (2.10a) and inequality (2.13a)
< ["(
iJl
iSv(t)dt){-d\
<
J
0 < « <
\Vu(x)\dx\
integrations by parts
' l
0
JU{zeR n :0<|u(x)|
by the second inequality in (2.10a) and inequality (2.14a)
f I ^T7.{ SV ' (S) - (1 " ^ H i C ' ^' *W 4 *^ =
J
integrations by parts
lI{^^0'}|v»i'(v-*""=
567
CMO^-v^fr-hThus we have shown r9(s)u*(s)ds< \V^s)\\V\Vur(V-t)
J*]ds.
(2.15a)
Parallel arguments, that will be omitted here, show
[°V)u*(-)«fc< [ V ) ( [ V | V U | ' ( 0
^ J j i .
(2.15b)
The very definitions of v and u> give if | x | < (V/«„) 1 / n ,
| Vv(x) | = | Vv |*(V - * n |x|")
if | x | > ( V / « „ ) 1 / n ,
= 0
(2.16a)
and |Vu>(x)| = \Vw\*{Kn\x\n),
(2.16b)
as well as | Vv | * = | Vio | * = | V« | *.
(2.17)
We deduce the following representation formulas v(x)=\
|VuP(V-s)
J'
if HKtV/g"",
«n|X,«
if i x l ^ C V / ^ ) 1 / " ,
= 0
(2.18a)
and w(x) = f /c n |x|
| V« | *(a)
?',
if | x | < (V//c„) 1 / n ,
n
= 0
if | x | > ( V / / c „ ) 1 / n .
(2.18b)
= 0 if « > V ,
(2.19a)
We deduce consequently v*(s)=\V\Vu\'(V-t)
*'
if 0<s
u;*( a )= [ " i V u H O
* ■
if 0 < S < V , = 0 if
and a
>V.
(2.19b)
568 Inequalities (2.11a) and (2.11b) follow via inequalities (2.15a) and (2.15b), and equations (2.19a) and (2.19b). Observe that, if n = 1 or n > 1 and 0 < p < n/(n—1), and
(2.20a)
then conditions (2.10a) are satisfied; if n > 1 and p > n/(n—1), and ip is given by Ha) = sl/p-\
(2.20b)
then conditions (2.10b) are satisfied. Coupling (2.11a) and (2.20a) results in inequality (2.3a)-, coupling (2.11b) and (2.20b) results in inequality (2.3b) — equation (1.7) comes here into play. Equation (1.7) and equations (2.19a) and (2.19b) yield Equation (1.7) and equations (2.19a) and (2.19b) yield "
«L(p,l)
\v\u
n
n =-^[^
1/p+1/n 1
- ivur ( 3)cf,
Recall that any «nonnegative integrable function / H£(p,l) n
can be recovered as the
Recall that any nonnegative integrable function /
can be recovered as the
superimposition of the characteristic functions of its level sets, in formula
r°°
/= X, dt n here x stands for characteristicJ 0function {x€Rand :/(*)>i} the integral is a la Bochner. Thus — here \ stands for characteristic function and the integral is a la Bochner. Thus |Vu|*= y dt, (2.21) since the very definition of decreasing rearrangement and the definition of M ensure that {s > 0: | V « | * ( s ) > t ) is, for every positive t, precisely the interval [0,M(t)[. Formula (2.21) gives f Viy _ s ) l / P + l / n - l | V u | *(a) ds = - r i p f °° [ v ^ P + l / n _ {V-M{t))1'p+1'n] o js + n J o L
J
[ ^ l / p + l / n - l i V u i * ( a ) ds
J
o
Equations (2.4a) and (2.4b) follow. The proof is complete. D
=
1 _ f°° ( M ^ l / P + l / n
js + h J o
d t
J
dt,
569 3 Remarks An analog of theorem 2.1 cannot hold verbatim if Lorentz space L(p, 1) is replaced by Lebesgue space L p (R n ). The following are apropos examples. (i) Let n = 2 and consider the function u defined thus w(x) = 1 - | x | 2
if | x | < l , = 0
if | x | > l
(3.1)
— a dome function. An inspection shows that M, the distribution function of | Vu | , obeys M(t) = TT(1 — t 2 /4) if 0 < t < 2, = 0 if t > 2; and that V, the measure of the support of u, equals it. Then | Vu \ *(s) = 2^1 — s/it if 0 < 5 < TT, = 0 if s > it. Formulas (2.18a) and (2.18b) yield v = u,
(3.2a)
w(x) = arccos( | x | ) - | a ; | N j l - | z |
2
i f | z | < l , =0
if | x | > 1.
(3.2b)
Formulas (3.1), (3.2a) and (3.2b) give [ \v(x)\pdx = it/(p + l), \ \w(x)\Pdx J R2 J R2 Hence numerical analysis shows that
2-p-lit]{9-sine)pd0. 0
=
(3 3a)
rl £ P ( R y n ^ t f ) > *
-
if and only if p> 2.871649....
(3.3b)
Now consider the function u defined thus u(x) = ( l - | x | ) 2
if | x | < l , = 0
if | x | > l
(3.4)
2
— a spire function. This time M(t) = ir(l - 1 / 2 ) if 0 < t < 2, = 0 if t > 2; V = it; | Vu|*(s) = 2 ( l - J s / 7 r )
if 0 < S < T T ,
=0
if s > f .
Hence formulas
(2.18a)
and (2.18b) give v{x) = arccos( \x\)+ = 0 if
| ar | \|l - | x | 2 + 2(1 - | x | ) if
|x|
| x | > 1;
w = u. Formulas (3.4), (3.5a) and (3.5b) imply that the inequality Formulas (3.4), (3.5a) and (3.5b) imply that the inequality holds :/ and only if holds :/ and only if
(3.5a) (3.5b)
570 p > 2.777883....
(3.6b)
Note that the right-hand side of (S.Sb) differs from the right-hand side of (8.6b), and both differ from n/(n—1), the index appearing in theorem 2.1. (iii) Let {Pi,P2,...,p<} be a decreasing sequence of / positive numbers. Consider the problem (u(x)) dx = maximum,
(3.7)
JRn
under the following conditions: u is real-valued and Lipschitz continuous, and m(support of «) = «„, \Vu\*=ipiX.
(3.8)
.. ,.,_,
„_,.
(3.9)
— the right-hand side of the last equation is the step function that takes the value pt- at every point from [«„(»—l)/- ,/cni/
[ and takes the value 0 at every point
from [K„ ,OO[. Clearly, competing functions include Lipschitz continuous functions u having the following properties: u is radial and radially decreasing,
(3.10a)
«(z) = 0 if | x | > 1 ,
(3.10b)
|V«|=£?,Y
(3.10c)
;=i {(>-i)r'< | i | n < » r 1 } — here iqx, q2,..., qA is any permutation of lpt, p2, • •., pA, and the right-hand side of equation (3.10c) is the piecewise constant function that takes the value qi where (i—l)/ - 1 < | a r | n < i 7 - 1 and takes the value 0 where | z | > 1. Theorem 3.1 from [ALT] ensures that these competing functions include a solution to problem (3.7), (3.8) and (3.9). Let 1 = 7 and assume / p , , p 2 ,..., pA is {100, 85, 70, 55, 40, 25, 10}.
(3.11)
Computations show the following. If n = 2, the Lipschitz continuous function specified by (3.10a), (3.10b) and (3.10c) satisfies (3.7) if {?i,? 2 .••••«/} i s {10, 25, 40, 70, 100, 85, 55}.
(3.12)
If n = 3, the Lipschitz continuous function specified by (3.10a), (3.10b) and (3.10c)
571 satisfies (3.7) if {ft, ?2 »•••)?(}1S {40, 100, 85, 70, 55, 25, 10}.
(3.13)
Notice that neither (3.12) nor (3.13) is a monotone rearrangement of (3.11). In other words, :/ n = 2 or n = 3 the solution of problem (S.7), (S. 10a), (3.10b) is neither a dome nor a spire function. References [ALT]
A.Alvino & P.L.Lions & G.Trombetti, On optimization problems with prescribed rearrangements. Nonlinear Analysis T.M.A. 13 (1989) 185-220.
[Ar]
G.Aronsson, An integral inequality and plastic torsion. Arch. Rat. Mech. Anal. 7 (1989) 23-39.
[AT]
G.Aronsson & G.Talenti, Estimating the integral of a function in terms of a distribution function of its gradient. Boll. U.M.I. (5) 18B (1981) 885-894.
[Bu]
G.R.Burton, Variational problems on classes of rearrangements multiple configurations for steady vortices. Ann. Inst. Henri Poincare 6 (1989) 295-319.
[EST]
A.Eydeland & J.Spruck & B.Turkington, Multiconstrained variational problems of nonlinear eigenvalue type: new formulation and algorithms. Math. Comp. 55 (1990) 509-535.
[Fe]
H.Federer, Geometric measure theory. Springer-Verlag (1969).
[FP]
V.Ferone & M.R.Posteraro, Maximization on classes of functions with fixed rearrangement. Differential and Integral Eqs. 4 (1991) 707-718.
and
[HLP]
Hardy & Littlewood & Polya, Inequalities. Cambridge Univ. Press (1964).
[Hu]
R.Hunt, On L(p,q) spaces. Einsegnement math. 12 (1966).
[GN]
E.Giarrusso & D.Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations. Nonlinear Analysis T.M.A. 8 (1984) 289-299.
[LS]
P.Laurence & E.Stredulinsky, A new approach to queer equations. Comm. Pure Appl. Math. 38 (1985) 333-355.
[Mc]
J.B.McLeod, Rearrangements and extreme values of Dirichlet norms. Unpublished.
[Mo]
Ch.B.Morrey, Multiple integrals in the calculus of variations. Springer-Verlag (1966).
[Po]
M.R.Posteraro. Un'osservazione sul riordinamento del gradiente di una funzione. Rend. Ace. Sci. Fis. Mat. Napoli (4) 55 (1988) 10 pages.
[Ta]
G.Talenti, The standard isoperimetric theorem. Pages 75-123 from Handbook of Convex Geometry, P.M.Gruber & J.M.Wills editors, Elsevier (1993).
differential
WSSIAA 3 (1994) pp. 573-592 © World Scientific Publishing Company
573
A Free Boundary Problem with Strong Absorption V. W E C K E S S E R *
Universitat Karlsruhe, Mathematisches Institut I, D-76128 Karlsruhe, Germany
1. Introduction In this paper we investigate the following implicit free boundary problem with strong absorption (0 < T < oo) «*rr = «« + « p
on G „
«(t, s(t)) = 0
and
u(0,z) = u0(x)
ux(t, s(t)) = 0
forte(0,T], (P)
on [0, So],
Bu(t, 0) := au(t, 0) - bux(t, 0) = a(t)
on (0,T].
Here the set G, := {(t, x) : 0 < t < T,0 < x < s(t)} is denned with the function s : [0,T] —► [0, oo). We split the parabolic boundary of G, in three parts Ta := {(0,x) : 0 < x < s(0)}, T 0 := { ( t , 0 ) : 0 < t < T} and T. := {(t,s(t)) :0
574
and one cruical assumtion is the local Lipschitz contiunity of / . They also considered only the increasing problem without initial conditions (50 = 0). Our methods are closely related to those of Thompson and Walter, in particular in the increasing case. Yet the loss of Lipschitz continuity in ur brings additional difficulties. G.G. Sackett was the first, who used the method of lines for free boundary pro blems (see [8, 9]). He treated only the case u„ = ut. G.H. Meyer published several papers on this topic for linear problems, see [3, 4, 5]. Free boundary problems for the underlying differential equation (often in the setting of "dead core problems") has been studied by various methods. But always other boundary conditions are imposed. This implies that only very few results carry over to our problem; compare [1, 22, 21], We use the following definition: The class Z(G,). A function u is in Z(G,), if u is defined and continuous on £?„ u« is continuous and bounded on G, U T, (or on ^ , \ r „ , if 6 ^ 0 in the operator B), and UtfVxx are continuous on G,. We look for solutions (u, s) of (P), such that u belongs to Z(G.) and is positive on G,. The function s should at least be continous on [0,T] and positive on (0, T). We call a solution smooth, if t*t is continuous on G, U T, and satisfies ut = 0 on T,. 2. Qualitative properties We will use a generalized second derivative of a function u : I C K. —* 1R denoted by [u"j with the properties (i) For functions (p € C2(7) we have [
is a generalized second derivative. The class ZV(G,). A function u is in ZV(G,), if it has all properties of Z(G,) except that u** may not exist on G„ but [u n J is defined on G, and [«„] £ 1R. Theorem 1. Given a continuous, positive function s on (0,T). We assume that u belongs to ZV(G,) and satisfies [«M]
- «t < d(t, x)ux + c(t, x)u
on G,.
(1)
Furthermore we suppose Bu > 0 on To and u > 0 on Ta U T,. Then we have: (MPo) (Nagumo) If all presumed inequalities hold with the strong ' <'- or ' >'-sign, the inequality u > 0 holds on 7J.; no assumptions on c and d are necessary.
575 ( M P ) Ifsnp(d — c) < oo holds, we have u > 0 on G,. This is especially true, if a. i n f c > — oo and s u p d < + o o . (2) o. a, ( S M P ) If (2) holds and c and d are on G, locally bounded, there exists a to & [0, T] with u = 0 on G, n {{t,x) : t < t0} and u > 0 on G, n {(t,x) : t > t0}. (The case to = 0 reads u > 0 on G,). ( B M P ) (Hopf) Assume u(t*,0) = 0 and c lows ux(t',0) > 0. and d bounded in a ux(t',x') <0.
the assumptions of (SMP) hold. If for t" > t0 we have and d are boundad in a neighbourhood of (t*,0), it fol Analogously follows from x* = s(t') and u(t',x') = 0 (c neighbourhood of(t*,x*)) and D~s(t') < oo the inequality
Remark. (MP) means minimum principle, (SMP) strong minimum principle and (BMP) boundary minimum principle. Proof. The proof of (MP 0 ) is well known [14, §24]. For the proof of (MP) one uses the auxiliary function g(t, x) := e^' - * with /? > 1 + sup d — inf c and (MPo). The proof of (BMP) can be addapted from section 2 in [12]. A proof of (SMP) can be found either in [15] or [12]; only a few changes are necessary. ■ Remark. The continuity of u in the points (0,0) and (0,«o) can be replaced by the boundedness of u in a neighbourhood of these points, cf. [14, 28.IX] and [10]. As usually a comparison theorem follows from Theorem 1. Comparison Theorem 2. w e Z{G.) hold
Given a positive function 3. For u € ZV(G.) [uxx] — ut < ttp
on G, and
wxx — wt > to*
on G,.
and
Furthermore we assume Bu > Bw on To and u> w on T0 U T,. TTien we have u > w on G,. Proof. The function ip = u — w satisfies [fxz] -ft<^P-WP r
p
with z(t, x) := (u — w )/(u — w)iiu^w z > 0 and we can apply (MP) to
= Z(t, l)(fi and z(t, x) = 0 if u = w. Obviously holds u
Bounds on the solution. Let (u, s) be a solution of (P). Then the following holds (a) There is a constant C > 0, depending only on T, max a(t) and maxu 0 (x), such that u(t,x)
576 For the proof let w(t, x) := ce*~*. Then we have w„ — wt = 0 and 0 < u p = "** - ««• Futhermore we obtain Bw(t,0) = ce* > a(r), if we choose c > m a x a ( t ) , and w > 0 = u holds on T,. Choosing a bigger c, such that c > maxe'uo(x) holds, it follows w(0,x) = ce~x > Uo(x). Comparison Theorem 2 implies u(t,x) < ce*~x < ceT =: C. m As in (12] let u'(t) := max{u(r, 0) : 0 < r < t} and M := {t € [0, T] : u*(t) = u(i,0)}. Under the additional assumption «o(x) < u 0 (0)
/or 0 < x < So
we have (b)u(<,x)
onG..
The proof of [12] carries over, since it uses statements (b) and (c). Theorem 4. Let be u0 € C'fO, s0] and uj < 0 on [0, s0]. If we have a > 0, suppose a is increasing. Let (u,a) be a solution of (P) with u» G C ( Z J , \ { ( 0 , 0 ) , ( 0 , S O ) } ) . Then we have ux < 0 on G,. Proof. Define (p := — u s . We differentiate the differential equation and get Vix — ft=
pup~1'P
on G,.
577 Futhermore we have ip = 0 on T, and ¥>(0, x) = —u'0(x) > 0 for x € (0,3 0 ). In the case a = 0 we have
578
Proof. The function
\v\r
On
G„,
v(t,o-(t)) = 0 on (O.T], Bv(t, 0) < Bw(t, 0) on (0, T] and v{0,x) < u(0,z) on [O,s0], especially o"(0) < so, fcoM. Then we have v < u on G*ff. Tjf in addition the inequality v > 0 Zio/fis on G a , r7»w implies a < s on [0,T]. Proof. The function u satisfies [«,*] = u< + up on ST. Hence for
°n
with « between u and |v|. Here we choose [u^] = uIX in the decreasing case (we assumed that u is smooth in this case) and [u^] = D+(ur) (right derivative of ux) in the increasing case. Then we obtain [u^] = u„ on S*r\r, and [uxx] = 0 on T, in the increasing case. The assumptions imply Btp(t, 0) > 0 on (0, T] and
579 Theorem 10 (Continuous dependence for the decreasing problem). Let (u, s) be a smooth solution of (P) with u» > 0 on G, (e.g. the assumptions of Theorem 6 hold). Furthermore let (v,ff) be a smooth solution of a neighbouring problem o a = »| + f(t, x, v, vx) with data (/, a, vo) instead of (zr, a, u<>). LetO < c < 1 < C and 0 < d < 1 < £ . (a) If f(t,x, z,q) = zp and t*o = VQ holds, the inequalities a{ct) < a(t) < a(Ct) imply the estimates u(ct, x) < v(t, x) < u(Ct, x)
and
s(ct) <
(b) Additional to the assumed inequalities on a and a we suppose ±z>>f(t,x,z,q)>±z' for z > 0, q € 1R and d^> uo ( | ) < t>o(x) < D& uo ( - ^ ) on [0, oo), where «o and t>o <"*e extended by 0. Then it follows that d^
u (c t, 2) < v(t, x) < D^f u (Ct, i )
and
d^f s (ct) < a(t) < D^i s (Ct). Remark.
The estimates from above and below are independent from each other.
Proof, (a) Following [12] we define w(t,x) := u(^t,x)
1 j > wt + xif wxx = -wt + v? i 7 I < to, + u?
and obtain
for 7 < 1, for 7 > 1,
Bw(t, 0) = 0(71) and w = wx = 0 on {(t, s(ft)): 0 < t < T/-y}. Applying Theorem 8 (compare the remark there) to (wi,v) and (v,u) 2 ) with w\(t, x) := u(ct,x) and wi(t,x) := u(Ct,x) the assertion follows. (b) We study the function w(t, x) := 6^? u(~(t, x/6) with 7,6 > 0. We get w
" =^
+^ ~
V
= ^f* + ?*'
For the boundary operator B we calculate Bu>(i,0) = =
8^-l(a6n-buI) S^[a(1t)+a(6-l)u(it,0)}.
580
Now let 7 = C > 1 and S = D > 1 and obtain
Bw(t,0)>a(Ct)>a(t), w = wz = Q
for x = I>^» s(Ct),
«>(0,x) > v0(x). Theorem 8 implies w > v and 7 = c and 6 = d.
s(Ct) >
Theorem 11 (Continuous dependence for the increasing problem). Let be a = 0 and (u,a) a smooth solution o/(P). Furthermore let (v,o~) be a smooth solution of a neighbouring problem « « = Vt + f(t, x, v, vx) with data (/, a, Vo) instead of(z',a,u0). For 0 <
\z>>j{t,x,z,q)>±z> for z > 0, q € JR. and d^i uo ( J ) < v0(x) < D*h uo ( J ) on [0, oo), where u0 and vo are extended by 0. Then follows
d
M^£)- v(<,x) -^ u (^) and
Proof. Using the comparison function w(t, x) := 6*=pu(t/82, x/S) with 6 > 0 the proof is very similar to the previous one. ■ To proof the continuity of the free boundary in the decreasing case, we use the following lemma (Lemma 4.6 in [1]). Lemma. Let Xi < xj and 0 < tt < t2, Q := [h,h] x [x^xj] and Q0 := [ti,ti) x [xi,Xj]. Assume that the function u with u € C(Q) and ux, uxx, ut 6 C(Qo) satisfies the differential equation «II
= «« + up
on Q0
581 and in addition u(t, x) > 0
and
ux(t, x) < 0
on Qo-
Then we have u(t2, x) > 0 for x € [xi, xt).
Theorem 12 (Continuity of the free boundary). Suppose (u,s) is a solu tion o / ( P ) with ux < 0 on G,. If the free boundary s is monotone decreasing, it is continuous on [0,T]. Proof. For t > 0 no jumps of s can occure because of the continuity of u on ^ , and the preceding lemma. To show the continuity for t = 0 we choose e > 0. Since u is continuous on ^ , , there exists a. 6 > 0 with u(t,x) > u(0,ao —e)/2 > 0 for 0 < t < 6 and |x — s o + e | < 6. This implies so — e < s(t) < s(0) = So on [0,6). m 3. The increasing problem Now we will apply the method of lines to the problem (P) to show existence (cf. [6, 7]). We discretize problem (P) in the direction of t and obtain a system of ordinary free boundary problems (P/,). Therefore we divide the interval [0, T] by N + 1 points: 0 = to <
on(0,sn), and
u'n(sn) = 0,
(P h )
Bun(0) = au n (0) - K ( 0 ) = «n := <*(tn). Solutions are nonnegative, real numbers sn and functions u n with u„ € C [0, sn] for n = 1,...,N, that satisfy the previous equations. We agree to extend u„ by u n (x) := 0 for x > sn, hence «„ in C^O, oo). The (N+ l)-tupel ((u„, «n)) n = 0 is called a solution of (P&). We need the concept of the maximal solution for differential equations of second order (see [18, 9.VI and 9.VIII]).
582
Theorem and Definition 13. Assume the function g = g(x, y) is continuous on G C R 2 and strongly increasing in y. Then the initial value problem y" = g(x, y)
with y(() = r,0, y'(0 = IJ,
(3)
with (£,»?o) € G possesses a non-continuable solution y* (called maximal solution) with the following property. If a twice differentiate function v satisfies v" < g{x, u), then the inequalities v(() < T;0 and v'(£) < »/i imply v(x) < y'{x)
and v'(x) < (y*)'(x) for
x>(,
and the inequalities v({) < t)o and v'(£) > »;i imply v{x) < y'(x)
and v'(x) > {y')'(x) for x < Z,
as long as both functions are defined. The usual proof for first order equations (cf. [2, 18]) carries over by using a • comparison theorem for second order equations (cf. [11, Theorem 2.1]). Example. The initial value problem y" = \y\> with !/(0) = y'(0) = 0 with 0 < p < 1 possesses on H the maximal solution
.(.) = fl.|* w i t h ^ = ( | L ^ ) ^ . We use the following assumption. Assumption [I]. The function w0 belongs to C2[0,s0] and satisfies tt'0 < 0 and u —u o o ^ 0 on [0,so] and furthermore Uo(z) > z(x — so) (with z from the preceding example) in a small left hand side neighbourhood ofso, i.e. u0 lies above the maximal solution ofu'o = uj with u0(s0) = u'0(s0) = 0. We suppose that a is increasing and Q(0) > Buo(0). Assumption [I']. Additional to [I] we assume that the function a is Lipschitz continuous on [0, T] and the condition Bu0(Q) = a(0) holds. Now our main objective is to prove, that under assumption [ I'J the problem (P) has a solution. We will do this in several steps and use the method of lines.
583 Theorem 14. Assume [ I ] . Then a solution ((«n,5 n )) n = 0 of (Ph) exists for every h := T/N. The solution is unique and satisfies 0 < s„_i < sn and 0 < u n _i < u n on [0, sn]. More precisely there exists m € No such that sn-i = sn and u n _i = u n
for n <m and
sn-i < sn and u„_i < u n on [0,sn)
for n > m.
The inequalities u„ > 0 and u'n < 0 AoW on [0, s„) /or n > 1. The proof will be done by induction. Let be n > 1. We define w(x;s) maximal solution of
as the
w" =
^. + sga(w)\w\p-^zl^l mthw{s;s) = w'(s;s) = 0, (4) h h where sga(z) - 1 for 2 > 0, sgn(0) = 0 and sgn(z) = - 1 for z < 0. For a € [0,s„_i) we have 0 > —un-i/h on [0,s] (induction assumption). From w"(s;s) = —un-i(s)/h < 0 follows the existence of a J < s near s with u> < 0 and to' > 0 on [s,s). Hence 0 is a strong supersolution on [0, j] and this implies tu < 0 and w' > 0 on [0,s). Especially we obtain Bw(0;s) < 0, and a solution for these s is impossible. From u„_j < u„_i follows for s > a„_i on (0, s n _i)
w
" - - x ^ -sgnH H?=° - "--> - ""T""1 - "'-1-
(5)
Obviously this inequality holds also in the case n = 1. Finally, since u„_i is extended by 0 for z > 3„_i, this inequality holds on (0, s). For a = s n _i follows u> < u n _i and to' > «(,_! (« n -i is a maximal solution; cf. Theorem 13). Especially we get a n _i = Bu„_i(0) > Bui(0;s n _i). If w should be a solution, we need Bw(0;sn-i) = a„, i.e. a n = ctn-i- This implies w = u n _i, but this is only possible in the case u„_i = u„_ 2 , and by induction of this conclusion follows the first case for n < m of the theorem. In the case Q„ > an-i or u n _i ^ u„_j, a solution of (4) is impossible for s = s n _i. We notate Bw(Q; s„_i) < a n . Now let be s > s n _i. Then is 0 a solution of (4) on [s„_i, s], whence w > 0. Together with to" = w/h + wP >wf follows w(x) > z(s — x) and to'(x) < z'(s — x) on [s„_i,s], where « denotes the maximal solution of the example. Now we choose s n _i < s < s. Then we obtain to(.s; J) > 0 and w'(s\ J) < 0 and this implies w(z; s) < w(x; s) and w'(x; s) > w'(x; s) on [0, s]. It follows that Bw(Q; s) is monotone increasing in s and, as long as w is positive on [0, s), even strong increasing and continuous depending on s. We are going to prove, that Bu>(0; s) —» oo as s —► oo. Define n _
\ r
2
( x )
forx€[0,sn_!).
584
Here donotes ri(z) = z(x—sn) (z from the example) and rj is the solution of the initial value problem r" = (r-u n _!(0))//» withr 2 (s n _,) = r^s^) and r 2 (s„_i) = r[(sn-i). Comparison theorems yield to > r and to' < r' as long as r > 0 holds; the last condition is true, if we choose s so large, that ri(s„_i) = z(s„-i — s) > u„_i(0) and hence ra > un_i(0) holds. We end up with Bto(0;s) > £r 2 (0) > Bri(s„_i) = Bz(sn-i — s) and Bz(sn-i -s) = P\sn-i - s\ w ( a|s n _! - s\ + &r^— J -»oo for s —► oo. Hence there exists a smallest s„ with Sto(0;s„) = a„. Obviously Bw(0; s) ^ an for s £ sn. We define
„ („\._ / u>(z;5»)
M*) - 1 0
for x
e [o,s„]
for x > Sn
We will show u„ > «„_i. For sn > s„_i we have u„(s„_i) > 0. Assume there exists a ( € [0,a„_i) with u„(£) = u„_i(£). This implies u'n(() > ud_i(0- Assume in the first case un(£) = un_j(£); this yields u n (s n _i) < u„_j(sn_i) and u„_! > 0 on [0,s n _i), a contradiction. Assuming «(,(£) > «(,_!(£) it follows un(0) < un_i(0) and uj,(0) > «Jl_1(0), hence Bun(0) < i?un_j(0). This contradicts the monotonicity of a. Since u n is positive on [0, s„) it follows u" > u£ > 0 on [0, s„), hence u'n < 0 there. Corollary 15.
[/nrfer t/»e ossumpit'on [I] the following estimate
(t1)'
u ^ r <»>»
Sn < <
/lo/ds, uAere fl is from the example. Proof. Theorem 14 implies u" > u£ and hence un(x) > z(sn — x) with z from the example. It follows a(tn) = Bun(0) > fair Usn + This implies the assertion.
b^—\ ■
The next step is to prove a-priori-bounds on the solutions of (Pj,). We need the following theorem.
585 Theorem 16. Assume that for every positive So there exist numbers Lo,C > 0 withf(t,x,z,p)> -L0z + L0p-C forO
on (0, sn),
for x € [sn, S] and
Bvn(0) = an with 0<sn<S(S
the inequalities
vn(x) < wn(x) := w(nh,x) hold. The theorem holds in the special case f(t,x,z, 4.5 in [11] can be addopted.
on [0,s n ]
q) = zv. The proof of Corollary
Theorem 17. Assume [I] and that ((un,sn))0 is a solution o/(Pj,). Then there exist positive constants S and Co independent of h, such that sn< S
and
0 < u„ < C 0
on [0, sn]
holds for all h < ho (h0 > 0 sufficiently small). If b > 0 holds in the boundary operator B, then there exists a constant C\ > 0, also independent ofh, with -
on [0,sn].
If[V] holds (the case b = 0 is possible), there exists a constant Ci > 0 (independent ofh) with 0 < Sun < C2 on [0,3 n ]. Proof. Theorem 14 yields u n > 0, u'n < 0 and 6u„ > 0. Let be
586 Then Corollary 15 implies sn < S for n = 0 , . . . , N. Choose a K with K > a(T) and «o(x) < Ke~* on [0,5]. The preceding theorem gives u n < Kt*1 =: Co for h < I / 7 (7 from the theorem). A d u ^ . £u„ > 0 implies < > 0, hence u'n(0) < u'n(x) < 0. We obtain «'„(0) = ^(o«n(0) - an) > i(auo(0) - a(T)) =: - C , . Ad 6un. We define for n > 1 «n := *«n =
u„ - u„_i r .
and for r» = 0 let be u0 := u0' — u{J and Su0 := «o. Applying S to the differential equation yields for n = 2 , . . . , TV on (0,00) < = Svn + " " ~ / t U "- 1 > *v n . Obviously the assumption implies v" > Sv\. If A denotes the Lipschitz constant of a we obtain Bvn(0) = 6an < J4 (notice Suo(0) = Oo). Choosing a K with A" > A and Kt~x > v0 = 6u0 Theorem 16 gives the existence of C%. ■ By using the differential equation we easily get the conclusion 0<<
+ C$=:C3
on[0,5„].
Corollary 18. Assume [F]. Then there exists a constant Ct, such that for all solutions ( ( u n , 3 n ) ) 0 o / ( P h ) with h
*„-i <
CihT?
hold. Proof. Theorem 14 implies sn — s n -i > 0. This gives uj| > u* and hence u n (x) > z(sn — x) on [a„_i,s„] (z from the example). Theorem 17 gives 0 < u„(5„_,) = h6un{sn.-i)
< hC3.
So we obtain z(sn — s„_i) < hC3 and the assertion follows with C\ = (C^//?)"?.
■
Suppose ( ( u n , 3 n ) ) 0 is a solution of (PA) for a sufficient small h. To every t € [
S {t)
:=
AunOO + a - A K . , ^ ) ,
:=
AVi + (l-A)v
587 Suppose the assumptions of Theorem 17 and Corollary 18 are valid. With 5 from Theorem 17 we define QT:=[0,r]x[0,5]. Obviously the functions Uh are on QT uniformly Lipschitz continuous and the Sh are on [0,T] uniformly Holder continuous with exponent (1 — p)/2. The theorem of Arzela-Ascoli gives a subsequence hk —♦ 0, such that Ukk converges uniformly to a function u on QT and Slik converges uniformly to a function s on [0, T\. Then is « Lipschitz continuous on QT and s Holder continuous on [0,T). Since for all h the inequality u n (x) > z(sn — x) on [s„-i, sn] holds (z from the example) the function u is positive on G,. It is not hard to see, that ux exists and is bounded on QTWe now give a precise formulation of our main theorem. Existence Theorem 19. Assume [Y] and b > 0 in the boundary operator B. Then the constructed pair (u,s) is a solution of (P), i.e. s is continuous on [0,T] and increasing and u € C(G,), ux € B C ( G , \ r a ) and uxx,ut € C(G,). The sequence (Uh,Sh) converges uniformly to (v.,s) as h = T/N —* 0. The proof of Theorems 5 and 6 in [13] can easily be adopted. The necessary estimates are given in the preceding theorems. The convergence of the sequence (U , Sk) (and not only of a subsequence) follows from the uniqueness of the limit; cf. Theorem 9. ■ 4. The decreasing case Applying the method of lines straight forward to the decreasing case yields the following problem. Assume u n > 0 satisfies «J| = Sun + u£ with u n (s n ) = u'n(sn) = 0 and we have sn < sn-\. Obviously « n has a minima at sn with tij^s,,) = 0 and this implies ■u"(sn) > 0. On the other hand the differential equation gives u"(s n ) = -un-i(sn)/h < 0. We have to modify the ordinary differential equation and investigate u'Z = 6un + u>n + h
on(0,sn).
Talking about (P/,) in this section we always have this equation in mind. Assumption [D]. We suppose a — 0 and b = 1 in the boundary operator B. The function «o belongs to C2[0,So] and satisfies u 0 < 0 and u0' - uj < 0 on [0,so]. The function a is monotone decreasing and u o (0) < a(0) holds. Theorem 20. Under the assumption [D] there exists a solution (( u n,«n)) n = 0 ° / (Ph) for all h := T/N up to a N' < N, such that the inequalities u„(0) > h2 hold for n = 0 , . . . , N" — 1 and ujy.(O) < h2. The last inequality may be violated in the case N' = N. The solution is untgue and satisfies sn < s„_i and 0 < u„ < u n _! It holds u„ > 0 on [0,sn) and u'n < 0 on (0,s„).
on [0, sn].
588
The proof is very similar to that of Theorem 14. As the argument at the beginning of this section shows, the ordinary free boundary problem has in the case u*_i (0) < h2 no solution for n = k. To show the assertion on uj, we differentiate the differential equation and get for v := - <
«"- v (I+ P «r 1 ) = %-><>. Obviously we have v(0) = a(t„) > 0 and v(«„) = 0. Theorem 3 in [17] gives t; > 0 on (0,*„). ■ Theorem 21. Suppose [D] and that ( ( U „ , J „ ) ) 0 is a solution to (P&). 7Y»en tftere exist positive constants Co anrf Ci independent of h, such that 0 < «n < Co and - C , < «'„ < 0 /to/ds on [0,s n ]. 7/uo(0) = — o(0) holds and h < ho for a sufficient small ho, then there exists a Cj > 0 (independent of h) with -C% < 6u„ < 0
on[0,s„_i].
Proof. Theorem 20 implies u n > 0, uj, < 0, 6*u„ < 0, sn < s0 =: S and un(x) < u„(0) < uo(0) =: CoAd uj,. For vB := —uj, we have
Let be U := max(—u0) + a(0). Then wn := U = const, satisfies w0 > v0, ton(0) > a(0) > ot(tn) = vn(0) and u>j( = 6wn. Theorem 4.4 in [11] can be adopted to this case and implies t>„ < wn, i.e. — u'n < U =: C\. The existence of Cj can be shown as in Theorem 17. ■ The differential equation gives for u" with C3 := max{Ca, CJ + 1} and ft < 1 the bound \<\
+
(l-\)un.1(x)
589 We define sn = 0 and un = 0 for n with N' < n < N. Let be QT:=[0,T}x[0,so}. We choose a sequence A* — ► 0, such that Ukk converges uniformly to a continuous function u on QT- We denote the sequence (Ukk) by (Uk). Obviously all Uk and hence u are Lipschitz continuous on QT under the assumptions of Theorem 21. Let be s(t) := sup{x : u(t,x) > 0}. From Sun < 0 follows that u decreases in t and hence s is decreasing. Obviously u is positive on G,. Theorem 22. Assume [D']. Then the above constructed pair (u,s) is a solution of (P), i.e. s is continuous and monotone decreasing on [0,T*], u € C(G7), ux € BC(G~,\ro) and Uxx,ut € C(G,), and u is positive on G,. Here we define T" := sup{< € [0, T] : s(t) > 0}. The sequence (Uk) converges uniformly to u on (7, for k —♦ oo. Proof. Obviously u(0, x) = Uo(x) and u = u, = 0 on T, hold. The positivity of u on G, is also clear. Now fix a i 0 . Because Uk(to,x) is uniformly Lipschitz continuous for x € [0, so], there exists a subsequence ki —* oo, such that Uk'(to,x) converges uniformly to ux(t0,x) on [0,so]. The equation Uk'(t0,x) = t/*'(to,0) + /„r Uk'{t0,0^ holds. Since we can go to the limit under the integral, we obtain u(t 0 , x) = «(t 0 ,0)+/ o r Ux(to, {) df. Hence ux exists on {to} x [0,So], and we have ux(to,x) = ux(to,x). Because -Uk'(t0,Q) = A0a(<„) + (1 - XoMU^) holds with t0 = A0t„ + (1 - A0)<„_i, we get ux(to,0) = — a(to) (a is Lipschitz continuous). Furthermore we obtain, that ux is uniformly Lipschitz continuous in x (with constant C). This gives — Cs0 < ux < 0 on Qr. It is a similar calculation to the proof of Theorem 5 in [13] to show that for all V 6 C0~(G?) / / [(v*x + ft)u ~ ¥uV] dx dt = 0 holds. It follows from standard results for parabolic equations that u is a strong solution (up is Holder continuous on QT), i.e. u r , u„, ut € C(G.) and the differential equation holds in a strong sense on G,. Since ux is uniformly Lipschitz continuous in x we obtain u r € C(G^\r„). Since u is positive on G, (SMP) implies ux < 0 on G,. Obviously 5(0) = s0 holds. The continuity of 3 follows from theorem 12. ■
590 5. Example
The above figure shows a numerical computation, base on the method of lines, for the following problem: u0{x)
=
a =
±(x - l ) 4 6 = \
with s 0 = 1, and
«(') = i s 0 + 3<). The computation was done up to T — 1.0 with h = ^ . Acknowledgement. The author of this paper would like to express his deep appreciation to his dissertation advisor, Professor Wolfgang Walter, for suggesting this research topic, for valuable discussions and a lot of support in the recent years.
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592
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