SINGULAR BOUNDARY-VALUE PROBLEMS FOR ORDINARY SECOND-ORDER DIFFERENTIAL EQUATIONS I. T. Kiguradze and B. L. Shekhter
UDC 517.927
This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.) Introduction The systematic study of initial and boundary-value problems for an ordinary second-order differential equation
,,"= f(t,,,,,,')
(0.1)
having singularities with respect to the independent variable or one of the phase variables has only a thirtyyear history, although such problems began to arise quite long ago in applications. For example, as early as the beginning of the century, in a paper of Emden [74] devoted to the equilibrium of a sphere formed from a polytropic gas, there arose a singular Cauchy problem ,," =
t
- ,,',
u(o) = c 0 > 0 ,
u'(o) = o.
Nevertheless for a long time mathematicians limited themselves to the study of singular problems of a specific type and did not make any attempts to work out more or less general methods of investigation. In the well-known monograph of Sansone ([51], p. 349) in particular it is stated that because of the absence of any general theory of solvability of the singular Cauchy problem, the existence of a solution of the abovementioned problem from the paper of Emden can be established "only through direct study" of the corresponding differential equation. At present the singular Canchy problem has been studied with considerable completeness not only for Eq. (0.1) but also for differential equations and systems of higher orders [11, 12, 49, 59]. The theory of singular boundary-value problems for Eq. (0.1) is also quite far advanced. The present work is devoted to an exposition of the fundamentals of this theory using the example of two-point problems and the problems of bounded and monotone solutions. In the first chapter (w167 we consider two-point problems of the form
uCa+) =
c1,
u c'-1) (b-) = c~,
(0.2,)
where i E {1,2} and - o o < a < b < +or The basis of w is formed by the results of [7, 16, 61, 82, 83], which are concerned with the existence and uniqueness of solutions of the linear differential equation u" = pl ( t ) u + p2(t)
' + p0 (t),
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105-201, 1987. 2340
0090-4104/88/4302-2340512.50 9 1988 Plenum Publishing Corporation
satisfying the boundary conditions (0.2,) and (0.22). Here we do not exclude the case when all the functions pj : (a, b) -+ It, (j -- 0,1, 2), fail to be summable on the closed interval [a, b], having singularities at its endpoints. w167and 3 are devoted to the study of two-point boundary-value problems for Eq. (0.1) when f : (a, b) x I{ 2 --* R has nonintegrable singularities with respect to the first argument at the points a and b. In w we consider the case when Eq. (0.1) is comparable to a linear equation in a certain sense. The study of boundary-value problems for regular equations of such type dates from the papers of Picard [92], Tonelli [102], and Epheser [75]. Of subsequent investigations we note [26, 47, 68]. The existence and uniqueness theorems we give for solutions of the problems (0.1), (0.2,), (i = 1, 2), are a certain modification of the results of the papers [15, 82]. In addition, in w we discuss the question of nonuniqueness conditions for solutions of the problems mentioned. Following [28, 48, 60], we establish criteria for existence of at least a given number of solutions of these problems and study their oscillation properties. When uniqueness does not hold, such properties are frequently of interest from the point of view of applications (cf., for example, [86]). In w we study equation (0.1) with, right-hand side rapidly growing on the phase variables. The fundamental :results on the solvability of boundary-value problems for such equations in the regular case are due to S. N. Bernshtein [3], Nagumo [88], and Epheser [75]. These results have been generalized in various directions by many authors (cf. [4, 8, 30, 56] and the literature indicated in these works). In [13, 14] an approach to the study of singular two-point boundary-value problems is suggested based on a priori estimates of the solutions of one-sided differential inequalities. We adhere to this approach in our exposition of the results of w In w we consider the question of the existence and uniqueness of a solution of the equation (0.1) satisfying the conditions
u(a+)=O,
u(i-')(b-)=O,
u(t)>0
fora
(0.3,)
where i E {1, 2}. Here we admit the possibility of singularities of f : (a, b) x (0, +oo) x R --* t t on both the first argument (at the points a and b) and the second (at the point 0). Such singular problems are often encountered in applications. For example, the study of the equilibrium of a membrane in the absence of chain stresses on the boundary reduces to finding the solution of the boundary-value problem u"-
t2 32u 2 ,
u(O+)=O,
u(1-)=O,
u(t)>O
forO
(cf. [44, 45, 52]), and in the theory of the boundary layer of a viscous incompressible fluid there arises the problem u"--
1-t
- -t$,
u(O+)=O,
u(1-)=O,
uCt) > 0
forO
(cf. [67, 70, 71]). Yu. A. Klokov and A. I. Lomakina [23] have studied the problem (0.1), (0.3,) under the assumptions that f(t,x,y) - x -~ fo(t,x,y), A > 0, and f0 : [a,b] • It+ x It -+ It is a continuous function. Taliaferro [100] and Bouillet and Gomes [69] considered the problem (o.3,) for the equation u" = h(t)u -~ . In the case when the singularity of the function f on the second argument, in general, is not a pole, the problems (0.1), (0.31), (i = 1, 2), were studied by A. G. Lomtatidze [35, 39]. It is his results that are explained in w The second chapter (w167and 6) is devoted to problems on an infinite interval. In w which is based on the results of [14, 77, 81], we study the following problems about bounded and monotonic solutions:
st(t) <<.u(t) <. s2(t) f o r a < t < b , u{i-1)(a+)=c, sl(t)<~ uCt)<<.82(t) f o r a < t < b
(0.4) (0.5,)
and .c,-,) (o+) = c,
. ( t ) .> o,
< o
for t > o,
(0.6,)
2341
where i E {1, 2}, - c o ~< a < b ~< +co, and sk : (a, b) --+ R, (k = 1, 2), are preassigned functions satisfying the inequality sl (t) ~< s2 (t) for a < t < b. The problem (0.1), (0.61) was first considered as early as the end of the last century by Kneser [85], who established criteria for its unique solvability in the case when f(t, z,y) =- f(t,z). Thirty years later Thomas [101] and Fermi [76], in studying the distribution of electrons in a heavy atom, arrived at the boundary-value problem
"`,
= t-1/:t'`s/,,
"`(0+) = 1,
"`(+oo) = O,
which served as the impetus for a series of papers by Italian mathematicians on problems of Kneser type (cf. [51], pp. 3?6-380). Using the results of Bernshtein and Nagumo mentioned above, Hartman and Wintner [77], Opial [90], and Klokov [21, 22] established some quite general conditions for the existence of solutions of both the
problem (0.I),(0.61)and the problems (0.I),(0.4)and (0.I),(0.51). Recently problems of the form (0.1),(0.4)and (0.1),(0.51)have again attractedattentionin connection with the development by A. A. Logunov and A. A. Vlasov [32]of a new approach to describinggravitational interactionin Minkowski space. In particular,the followingproblems are posed in [32]: U" = - - 2 " ` t
t
"` -- 2
"`12
2
+ "`(~ - 2) + _ _ ( ' ` - 5 " `
tt(0+)=2,
lim
t~+oo
"`(t) t
=1,
~0,
+
"`(t)>2
fort>0
and
u"
_
2"`
b- t
"`
r~
"`2
['u" -{- (1
u(0+)=2,
_
u 2 ~2] 2"` ~--,j + --(1-
(b - Q 2
"`(b-)=0,
u2
"~--) +
0<'`(0<2
3"`
~-
b2
u2
[(1 -- ~--) (1 - .~_)],/2
'
for0
where b3 = 2r 2 and r /> 2. E. I. Moiseev and V. A. Sadovnichii [40-43] have proved the solvability of these problems, and the uniqueness of the solution of the first one has been established and its asymptotic representation in a neighborhood of +co has been found. L. A. Lepin [31] has shown that the solvability of the problems just mentioned can be obtained from general theorems on the existence of solutions of the problems (0.1), (0.4) and (0.1), (0.51). The problem (0.1), (0.62) also has interesting applications. For example, in the theory of capillary phenomena it appears in the form ,," = ,-,,(1 + , , " ) ' / ' ,
,,'(~+) = ~,
"`(+~) = o
and
r = C1+ "`" )'/' (,.'` t l+eiT~,, ), ~'(a+)=c,
"`(+co)=o,
where r > 0, a > 0, and c < 0 [80]. Problems of this type have been studied by A. D. Myshkis and G. V. Shcherbina [46, 64-66]. w is devoted to the particular boundary-value problem ~" = - ~ - ~ ' + ~ - I ~ l ' s g n ~ , t r = o, ~ ( + c o ) = o,
(0.7) (0.8)
where "I and A are real numbers with A > 0. This problem arises in nonlinear field theory in the study of elementary particle interactions. In addition, it is encountered in a variety of other areas of physics, in 2342
particular in the statistical theory of the nucleus and in nonlinear optics (cf., for example, [1, 2, 9, 87, 89] and the literature mentioned in these works). Equation (0.7) has been especially intensively studied in the case "7 = 2. In this case the transformation ~J
=
gA
A~t
,o.yl
-
t
brings (0.7) into the form
r
1
= , -
sgnv.
(0.10)
If v : (0, + c o ) - + t t is a solution of (0.10), then ,(0+)
= 0,
,(+co)
= 0
(0.11)
and, in addition, lira I~(t)l < +co, t--*O
(0.12)
t
then the function u : (0, +co) -+ It defined by relation (0.9) is a solution of the problem (0.7), (0.8) with "7=2. As Nehari has shown [89], for 1 < ), < 5 the problem (0.10), (0.11) has a positive solution on (0, +co), which for A ~< 4 satisfies (0.12), while for A = 5 this problem has no nonzero solutions. However, the method applied ill [89] attests that for all ), >/5 there are no such solutions (cf. also [63, 96]). Nehari's result was strengthened by Ryder [95], who showed that for 1 < A < 5 the problem (0.10), (0.11) has a solution with any preassigned number of zeros in (0, +co) (for), E (1, 4) the analogous assertion was established by V. P. Shirikov [63]). However, the question whether the solutions constructed in [89] and [95] satisfy the condition (0.12) for A > 4 remained open. A positive answer to this question was given by Sansone [96]. In this way it was established that if '7 = 2, 1 < ), < 5, and l E {0, 1,... }, then the problem (0.7), (0.8) has a solution with exactly l zeros in (0, +co). In the general case, i.e., when "7 may differ from 2, the problem (0.7), (0.8) was studied by Kurtz [87], who established (although the proof was not completely correct for i ~ 0) that for any nonnegative integer l there exists a solution having exactly l zeros on (0, +co), provided 1 ~< "7 ~< 2 and 1 < A ~< 3. In w we study solvability and properties of the solutions of the problem (0.7), (0.8) for all real '7 and positive A, following the paper [99]. In the present paper we use the following notation:
R+ = [0, +co),
R"=.R•
(,=2,3,...);
u(t+) and u(t-) denote respectively the right- and left-hand one-sided limits of the function u at the point t; C(I), where I C R is the set of continuous functions u : I --, R; C ~([t~, t2]) is the set of functions u : [t~, t2] --* R that are absolutely continuous together with their first derivative; /:,([tl,t2]) is the set of summable functions u : [t,,t2] --* R; C~oc (I) and L~oo(I), where I C R is an open or half-open interval, is the set of functions u : I --* R whose restrictions to any closed interval [t~, t~] c I belong to the class ~1 ([tl, t2 ]) and L([tl, t2 ]) respectively; K([t~,t2] • D), where D C R " , n E { 1 , 2 , . . . } , (R t = R) is the Carathdodory class, i.e., the set of functions f : [tl,t2] x D -* R such that f(.,x~,...,x,): [tt,t2] -* R is measurable for all ( x z , . . . , x , ) E D, f(t,.,...,.) : D ~ R is continuous for almost all t E [tl,t2], and
sup {IfC-,xl,...,x.)l : ( x l , . . . , x . )
e
D0} E L([tl,t2])
for any compact set Do C D;
2343
K~or (I • D), where I C R is an open or half-open interval and D c R " , n G {1, 2 , . . . }, is the set of functions f : I • D --* R whose restrictions to the set [t~,t~] • D belong to K([tl,t2] • D) for any closed interval [tx,t2] C I; g~ • 2) is the set of functions f : ( t l , t ~ ) x R 2 ~ R for which the mapping t ~ f(t,x(t),y(t)) is measurable for any x,y G C((tl,t2)). We shall call a function u : (a, b) --. R a solution of Eq. (0.1) in the interval (a, b) if it belongs to C11or((a, b)) and satisfies (0.1) almost everywhere on this interval. PROBLEMS w
Chapter 1 ON A FINITE INTERVAL
Linear Equations The present section is devoted to the linear singular differential equation
(1.1)
u" = p1(t)u + P2Ct)u' + po(t) under boundary conditions of the following two types:
u(a+) =
cx,
u ( b - ) = c2
(1.21)
u'Cb-) = c2,
(1.22)
and u(a+) = c,,
where c r R, (j = 1, 2). Before stating the restrictions imposed on the functions pl, p2, and p0, we introduce the transformation a : L,oc ((a, b)) --, C((a, b)) defined by the equality
a(p)(t) = exp
[/i
]
per) dr .
2
If a(p) e L([a, b]), we set
,,,(p)(O - ~(P)(O
oCp)(Odr ,,(p)Cr) dr
and
a2(p)(t)
-
1 f[ a(p)(r) dr
a(~(t)
on(a,b). In studying the problem (1.1), (1.21) we shall assume that
Pj,P2 e L,or
a(p~) e L([a,b]),
pjcrl(p2 ) e L([a,b])
(j : 0 , 1 ) ,
(1.31)
(j = 0,1).
(1.3~)
and in studying the problem (1.1), (1.22) we shall assume that Pj,P2 e L,or ((a,b]),
a(p2) e L([a,b]),
pja2(p2) e L([a,b])
Relations (1.31) hold if, for example,
Ip~(Ol< [(t_,=)(b_O],§
(i=o,1),
6
Ip2(t)l -< ~ + ~
6
+ b--~'
and relations (1.32) hold if A
Ip~(t)l -< (t-a),+~
2344
(i =0,1),
Ip2(0J -< ~ + - $- - a
for a < t < b,
where ), > 0 and 0 ~ 6 < 1. 1.1. T h e H o m o g e n e o u s E q u a t i o n . In this subsection we study the behavior of solutions of the singular homogeneous equation u" : p, (t)u + P, Ct)u' (1.4) near points where its coefficients have singularities. In what follows we shall pay special attention to certain classes of such equations having no nontrivial solutions under the boundary conditions u(a+) = 0,
u(b-)=O
(1.,.51)
u(a+) = 0,
u'(b-) = 0,
(1.52)
or for these classes also play an important role in the study of nonlinear boundary-value problems. In studying Eq. (1.4), we shall assume that either
Pl,P2 E L,or
a(p2) e L([a,b]),
pliYl(p2) E L([a,b]),
(1.61)
Pl,P2 e LIoc (Ca, b]),
6rCp2) e L(la, b]),
plO'2 (p2) e L([a,b]).
(1.62)
or
LEMMA1.1. Suppose conditions (1.61) hold. Then 1. Equation (1.4) has a solution u satisfying the initial conditions . ( a + ) : o,
with ~(t) = o
(/;
lim .'(t) t-.. a(p2)Ct) = 1,
o(p,)C~)d~
)
as
t-,
(1.7)
a;
(1.8)
2. any solution ~ of this equation linearly independent of u has a finite nonzero limit fi(a+); 3. for any bounded continuously differential function v : (a, b) --+ R lira inf u(t)lv'(t)l ,--o ~(p2)(t)
_
(1.9)
o.
PROOF: Let tk e (a,a--{--b-), (k = 1,2,...), and tk ---~ a as k --+ q-oo. F o r e a c h n a t u r a l n u m b e r k w e define the function uk on the interval [tk, b) as the solution of Eq. (1.4) under the initial conditions u(tk) = 0,
u'Ctk)=a(p2)(tk).
Then
u~(t) =o(P2)(t)[l + / i Pl(r) aCP2)(r)uk(r) dr]
fortk<<.t
(11o)
From this equality we easily obtain
t
w~(O ~ 1 +
fa
IvlC~)l~C~)~~Cv~)O)d~
for t~ ~ t < b,
where .
2345
Thus by the Gronwall-Bellman Lemma (cf., for example, 118], p. 49)
~a t
luk(t)l<~AT
o(p2)(r)dr f o r t k < ~ t < ~ T < b
(k=l,2,...),
(1.11)
provided
AT = exp [f. ~ [pl (t)[a~ (p~)(t) dr]. Therefore, according to (1.10),
IoCp,)(t) uLCt) 1
~AT
~'
IPxC~)lox(p~)Cr)dr f o r t k ~ t ~ T < b
(k=1,2,...).
(1.12)
We extend the function uk to [a, tk) by setting u~(t) = u~(tk). Then, taking account of the inequalities +c~1 and (o,I " t + ~ are uniformly bounded obtained it is not difficult to obtain the result that the sequences (uk)t= ~-tJt=l and equicontinuous on each closed subinterval of the interval (a, b), so that, without loss of generality, we may consider them to be uniformly convergent on these closed subintervals. It is obvious that the function
uCt) =
for a
lira uk(t)
k-*+oo
< t
<
b
is a solution of Eq. (1.4) and, in view of inequalities (1.11) and (1.12), satisfies (1.7) and (1.8). If h is an arbitrary solution of Eq. (1.4) linearly independent of u, then
~(t) = ~lU(t) § ~2U(t)
O'(P2)CT) u2(T ) dr for a < t ~ a*,
where aj = const, (j = 1,2), a2 # 0, and a* E (a,b) is such that
u(t)>O
fora
Consequently fi(a,) = a2 nm ~(p2)(t)
t-~.
u,(t)
- a2.
Finally, we shall assume that for some continuously differentiable function v : (a, b) -* R equality (1.9) does not hold. Then there exist numbers 6 and a0 E (a, b)such that ~(t) > o, and so
_~'(t)
I~'(t)l > ~---;:~-
u~r)
oo
I~Ct)l/>
for a < t ~ o~,
~(~o)
Iv'Cr)ldr-I~(~o)1/> 61n
uCt)
Iv(~o)l
for ~ < t < ~o,
which is impossible if the function v is bounded. The lemma is now proved. Using an elementary cha~ge of the independent variable in Eq. (1.4) we can obtain the following assertion from Lemma 1.1. LEMMA 1.1'. Suppose conditions (1.61) hold. Then 1. Equation (1.4) has a solution u satisfying the initial conditions
u(b-) : 0,
2346
lira o(p2)(t) u'(t) ----1, t-.b
(1.131)
with u(t) = 0
o(p2)(r) dr
)
as
t -+ b;
2. any solution ~ of this equation linearly independent of u has a finite nonzero limit ~(b-); 3. for any bounded continuously differentiable function v : (a, b) ~ t t liminf u ( t ) l v ' ( t ) l
,-.b
- o.
oCp2)Ct)
We remark that if conditions (1.62) hold, then the coefficients of Eq. 1.4 have no nonintegrable singularities at the point b. Consequently we can arbitrarily prescribe the values of the solution and its first derivative at this point. In particular, we shall need the solution of this equation under the initial conditions
u(b-) = 1,
u'(b-) = 0
(1.132)
below. We note also that, as shown in [83], if the second of conditions (1.22) is replaced by lim
u'(t) Cp2)Ct) - c2,
then the problem (1.1), (1.22) can be considered even in the case when the functions Pl, P2, and P0 are not integrable at the point b. 1.2. T h e G r e e n ' s F u n c t i o n . As in the regular case, there exists a close connection between the unique solvability of ~he boundary-value problem (1.1), (1.2d) and the absence of nontrivial solutions of the homogeneous problem (1.4), (1.5,), (i = 1,2). We now turn to the study of this connection. DEFINITION 1.1. Let i E {1,2}. A function ,~ : (a,b) x (a,b) -* R is called a Green's function of the problem (1.4), (1.5d) if for any r e Ca, b): 1. The function u(t) = ~(t, r) is continuous on (a,b) and satisfies the boundary conditions (1.5i); 2. The restriction of u to the intervals (a,r) and (r,b) is a solution of mq. (1.4); 3.
1.
LEMMA 1.2. Let i E {1,2); let the conditions (1.6i) be satisfied, and let the problem (1.4), (1.5i) have no nontriviaI solutions. Then there exists a unique Green's function ~ of this problem and
_ ~(t,r) =
u2(t)Ul(r) u2(a+)a(p2)(r) utCt)u2(r)
for a < r < t < b, (1.14) for a < t < r < b,
-2(a+)o(p2)(r)
where ul and u2 are solutions of the initial-value problems (1.4), (1.7) and (1.4), (1,13i) respectively. PROOF: As we verified above, under the hypotheses of the lemma there exist solutions ul and u2 of the initial-value problems (1.4), (1.7) and (1.4), (1.13i). Since the boundary-value problem (1.4), (1.5i) has no nontrivial sohltions, u2 (a+) ~ 0. It is clear that the function .~ given by equality (1.14) satisfies the conditions I and 2 in the definition of a Green's function. We shall show that it also satisfies condition 3 of the definition, i.e., that w(t) = u2 (a+), where wCt) = C x ( t ) u 2 ( t ) - u ~ ( t ) u l ( t ) for a < t < b.
o(p2)Ct) Indeed by L e m m a 1.1 we have w(a+) = u2(a+), and by Liouville's formula w(t) = const. Thus ff is a Green's function for the problem (1.4), (1.5i). Uniqueness follows from the unique solvability of this problem. The l e m m a is now proved. The basic proposition that we shall prove relating to the problems (1.1), (1.21) and (1.1), (1.22) amounts to the following.
2347
THEOREM 1.1. I f i E {1, 2} and conditions (1.3,) hold, then the problem (1.1), (1.2,) is uniquely solvable ff and only ff the homogeneous problem (1.4), (1.5,) has no nontriviM solutions. If the latter holds, then the solution u of the problem (1.1), (1.2,) can be represented by Green's formula
~ab u(t)=uo(t)+
~(t,r)po(r)dr
(1.15)
fora
where Uo is the solution of the problem (1.4), (1.2,) and ~ is the Green's function of the problem (1.4), (1.5,). PROOF: If the problem (1.4), (1.5,) has a nontrivial solution, then the problem (1.1), (1.2,) either has no solutions or has infinitely many solutions. Assume that (1.4), (1.5,) has only the trivial solution. Then obviously (1.1), (1.2,) has at most one solution. In addition, by Lemma 1.2 there exists a Green's function ~ of the problem (1.4), (1.5,), and since the solutions ul and u2 of the initial-value problems (1.4), (1.7) and (1.4), (1.13,) are linearly independent, there also exists a solution u0 of the boundary-value problem (1.4), (1.2,). By Lemmas 1.1 and 1.1', a positive constant A can be selected such that
lu,(t)l .< ,4
//
o(p2)(Od
,
].
lu=(t)l .<
for
a < t <
b.
Therefore, according to (1.3,) and (1.14) the integral on the right-hand side of (1.15) exists. Direct substitution confirms that the function u defined by the equality (1.15) is a solution of the problem (1.1), (1.2,). The theorem is now proved. 1.3. C o m p a r i s o n L e m m a s . Together with (1.4) we consider the equation V" = ql (t)v A- q2 (t)V',
(1.16)
a(qz)L([a,b]),
(1.171)
where either ql,q~ELior
qla1(q~)EL([a,b]),
or ql,q~ E L,oc((a,b]),
o(q2) E L([a,b]),
qla~(q2) E L([a,b]).
(1.17~)
In this subsection we present two comparison lemmas, i.e., propositions that make it possible to draw certain conclusions regarding the properties of Eq. (1.16) on the basis of the properties of Eq. (1.4). LEMMA 1.3. Let i E {1,2}, let conditions (1.6,) and (1.17,) hold, and let u be a solution of Eq. (1.4), with u(t) > o
ql(t) />pl(t),
fora
0, [q2(t)--p2(t)]u'(t) >10 for a < t < b.
(1.18) (1.19)
Then the solution v of Eq. (1.16) under the initial conditions v(a+) = 0 ,
lim - v'(t) - - aCq2)Ct)
1
(1.20)
t-~'~
satisfies the inequalities
v(t)>0
fora
v('-l)(b-)>0.
(1.21)
if, in addition, u'(t) > O f o r a < t < b ,
2348
(1.22)
then also v'(t) > 0
fora
(1.23)
PROOF: By Lemma 1.1 there exists a solution v of the problem (1.16), (1.20). Set wCt) 1 [uCt)v'(t) -vCt)u'(t)] for a < t < b. Then
w'Ct)-
1
{[ql(t)-pl(t)]u(t)v(t)+[q2(t)-p2(t)]u'(t)v(t)}
and, by (~1.18) and (1.19), the function w is nondecreasing in a right-hand neighborhood of the point a. Thus there exists a finite or infinite limit w(a+). On the other hand, according to Lemma 1.1 lim inf vCt)lu'Ct)[ = O.
Hence wCa+) =
Ca+) i> 0.
(1.24)
Suppose t* C (a, b) and v(t*) = 0. Then without loss of generality we may assume that v(t) > 0
fora
In such a case w'(t) >10 for a < t < t* and w(t*) < 0, and this contradicts (1.24). Thus, v(t) > 0 f o r a < t < b .
(1.25)
Consequently
w'(t) >>.0 for a < t < b
(1.26)
and the (possibly infinite) limit w(b-) exists. If v(b-) = 0, then, according to (1.25) and Lemma 1.1', lim v'Ct) < 0, t--.b a(q~)(t)
lim inf v(t)lu'(t)l - 0 .
t--,~
a(q2)(t)
But since, by (1.18), u(b-) > 0, we find from these relations that w(b-) < 0, which, by (1.26), contradicts (1.24). Therefore v(b-) > 0. Using this inequality, in the case when i = 2, we easily obtain v'(b-) > O. Finally, if (1.22) holds, then, since w(t) >~ 0 for a < t < b, we find from (1.18) and (1.25) that (1.23) holds. The lemma is now proved. We remark that when the coefficients of the equations (1.4) and (1.16) are continuous on the closed interval [a, b], pz (t) - q2 (t), and i = 1, Lemma 1.3 is essentially the known comparison theorem of Sturm ([56], p. 394). LEMMA 1.3'. Let i E {1,2}, and let relations (1.61), (1.17i), and (1.18) hold, where u is a solution of Eq. (1.4). Further suppose v is a solution of the problem (1.16), (1.20), and
ql(t) >>.pl(t),
[q2(t)-p2(t)]v'(t) >>.0 for a < t < b.
Then inequalities (1.21) hold. Moreover if condition (1.23) does not hold, then condition (1.22) also cannot hold. To verify this proposition it suffices to repeat the proof of Lemma 1.3 only replacing the factor 1/a(q2 ) (t) by 1/o(p2)(t) in the equality defining the function w.
2349
1.4. T h e Sets V1 ((a, b)) a n d 172((a, b)) a n d t h e i r S t r u c t u r e . We shall consider the classes of equations (1.4) whose coefficients pl and p~ satisfy the inequalities
pj,(t) <<.pj(t) <~p#,(t)
for a < t < b ( j : l , 2 ) ,
(1.27)
imposing on the vector-valued function (P11, P13, P31, P22 ) certain restrictions guaranteeing that there are no nontrivial solutions of the problem (1.4), (1.51) or (1.4), (1.53). Such an approach goes back to the work of Vallde Poussin [103]. In that paper necessary and sufficient conditions are established on the nonnegative constants 11 and 12 in order that the problem (1.4), (1.51) be uniquely solvable when the functions Pl and p~ are continuous on the closed interval [a, b] and
Ip#(t)l < t# f o r . < t < b (3" = 1,2). In the next subsection we shall obtain Vallde Poussin's result as a corollary of a more general proposition. In what follows a* (p21, p22 ) and a~ (p21, p22) respectively denote a(p) and al (p), where the function p is defined by the equality { p~l (t) f o r a < t < ~ a + b p(t) : a+b 2 ' P22(t) f o r - - ~ < t < b . In addition it will be convenient for us to set a~ (P21,P22)(t) = a2 (p21)(t). W e note that if
pj,,, e Lloc((a,b)),
a*(p21,p22) e L([a,b]),
pl,,,a~(p3,,p23) e L([a,b])
( j , m = 1,2)
(1.281)
and pD(t) < pj2(t)
for a < t < b
(3" = 1,2),
(1.29)
then for any measurable functions Pl,P2 : (a, b) ~ R satisfying inequalities (1.27), relations (1.61) hold; and if instead of (1.281) we have
pj,n ELioc((a,b]),
a(P21)EL([a,b]),
pl,na2(p~l) EL([a,b])
(3",rn=1,2),
(1.283)
then relations (1.62) hold. DEFINITION 1.2. Let i E {1, 2}. W e shall say that the vector-valued function (P11,P12,P21,P22 ) : (a, b) --* R 4 belongs to the set V~((a,b)) if conditions (1.28i) and (1.29) hold and, in addition, for any measurable functions PI,P2 : (a,b) -, It satisfying inequalities (1.27) the problem (1.4), (1.5i) has only the trivial solution. It turns out that the sets V1((a,b)) and V2((a,b)) decompose into certain subsets which, in turn determine the classes of equations (1.4) with particular oscillationproperties. DEFINITION 1.3. Let i E {1,2}. We shall say that the vector-valued function (p,p~l,p2~) : (a,b) ~ R s belongs to the set V~o((a,b)) if
Pl,P21,PZ2 ~- L,oe ((a,b)),a*(p21,p2~) E L([a,b]),
p1~ (p21,p22) ~- L(la, b])
(1.301)
plo'2(P21) E n([a,b])
(1.303)
f o r i = I; PI,~21,P22 ~ Lloc ((a, b]),
r
E n([a,b]),
for i = 2; the inequality P21 (t) ~ P22(t) for a < t < b
(1.31)
holds; and for any measurable function P2 : (a,b) --*R satisfyingthe inequality P21(t)~
fora
(1.32)
the solution u of the initial-value problem (1.4), (1.7) has no z e r o s in the interval Ca, b) and u ('-1) (b-) > O. Obviously V2oC(a,b)) c V,0 C(a,b)). DEFINITION ] . 4 . Let i 9 {1,2} and let k be a natural number. We shall say that the vector-valued function ( p n , p n , p 2 1 , p n ) : (a,b) --+ 114 belongs to the class Vi~((a,b)) if the conditions (1.28,) and (1.29) hold and, in addition, for any measurable functions p,, P2 : (a, b) -4 11 satisfying the inequalities (1.27) the solution u of the initial-value problem (1.4), (1.7) has exactly k zeros in the interval (a, b) when i = 1 and exactly k - 1 or k zeros when i = 2, and ( - 1 ) k u ('-*} (b-) > 0. It is clear that V~((a,b)) c v i ( C a , b)) ( i = 1 , 2 ; k = 1 , 2 , . . . ) . Moreover, if i 9 {1,2} and
(P1,P21,P22)9
V~o ((~ b)),
then, by Lemma 1.3,
(p,,p, p2x,p2,) 9 v~((,,, b)) --* 1t such that Pa~(pn,pn) 9 L([a,b])
for any measurable function p : Ca, b) and p(t) >>.pl(t) for a < t < b. (As it happens, this is precisely the reason why we defined V~0((a, b)), in contrast to V~k((a,b)), (k = 1, 2 , . . . ), as a set of three-dimensional rather than a set of four-dimensional vector-valued functions.) The following proposition shows that each set V~((a, b)), (i = 1, 2), consists entirely of vector-valued functions that either can be obtained from the elements of V~0((a, b)) by adjoining components satisfying the conditions indicated above or belong to some set V~k((a,b)), k 9 {1,2, ,.. }. THEOREM 1.2. Let i 9 {1,2} and
(p,l,pl=,p=,p=)
(1.33)
9 v,((,,, t,)).
Then either (pn,pn,p22) 9 Vio((a,b)), or there exists a natural number k such that
(p11,p,2,p21,p22) e v~, ((,,, b)). To prove this theorem and also to meet the needs of subsequent exposition we shall need some lemmas on a priori estimates. LEMMA 1.4. Let i E {1,2} and
(Pn,Pn,Pn,Pn) 6 V/((a,, b)).
(1.34)
Then there exist positive constants c and 6 such that for any measurable functions px,p2 : (a, b) --~ R satisfying the inequalities (1.27) the estimates
luCt)l < c rf' a(p,,)('r)d2,
N'(t)l
,,-(p2)(t)
hold and, in addition
t
~<1+c
fa Ip., ('r) lo'2(p:, ) (r) d,
luC'-:") (b-)J/>
fora
(1.35)
6,
where u is a solution of the problem (1.4), (1.7). PROOF: If the functions P, and p~ satisfy the hypotheses of the lemma and u is a solution of the problem (1.4), (1.7), then, as can easily be verified by taking account of Lemma 1.1, the representation
u'(O =~(p2)(t) 1 +
P'(~)~;(p2)(,)dT
for a < t < b
(1.36)
2351
holds, from which it follows that
t
w(t)<<.l+e*
fa IPl(r)lwCr)alCP2)(r)dr
where
,,,(t) = I,,(t)l and
[//
(gtb r =
-1
~a 2 0"(p22)(~') d~']
a(p2)(r)dr
fora
]1
b
+ [fa+br
,
-1
d~r]
2
Setting c=exp
[/.' (Ip11(01 + Ipx2(01)o; (pn ,p22)(0 dr] c"
and applying the Gronwall-Bellman Lemma to the last inequality, we obtain the first estimate of (1.35), after which we obtain the second using (1.36). Suppose now that no constant 6 satisfying the hypotheses of the lemma can be found. Then for any natural number n there exist measurable functions q j ( . ; n ) : (a,b) --+ R , (j = 1,2), such that
PjlCt) <~ qj(t;n) < pj2(t)
for a < t < b
(j = 1,2)
(1.37)
and
I= c'-11
(b-;n)l ~< 1,
(1.38)
n
where u(.; n) is the solution of the initial-value problem
u" =qx(t;n)u+q2(t;n)u',
u(a+)=0,
lira
-'Ct)
t-~, a(q2(.;n))(t)
= 1.
If ~j(t;n) =
+b q j ( r ; n ) d r for a < t < b ( j = 1,2), 2 ~ (.,. n ) ) +oo then, according to (1.37) the sequences (qj , = l , (J = 1, 2), are uniformly bounded and equicontinuous on each closed subinterval of (a, b). Consequently by the Arzel~-Ascoli Theorem they can be assumed uniformly convergent on each such closed subinterval without loss of generality. Let 15j(t): lim ~j(t;n) f o r a < t < b (3"=1,2). .---~-I-oo Then, by (1.37),
p j l ( r ) d r <. ~ j ( t ) - ~ j C s ) <~
pj2Cr)dr
for a < s < t < b
(j = 1,2),
i.e., the functions/~1 and ~52 are absolutely continuous on each closed subinterval of (a, b), and the functions pj : (a,b) -* R , (j = 1,2), defined by the equalities pi(O : ~;(0
satisfy (1.27). 2352
for a < t < b
Set
u'(t;n) vCt;n) : aCq2(.;n))Ct)
(n--1,2...).
fora
As follows from what was proved above,
~ a*(p21,p2~)(r)d1, t
I-(t;n)l < ~
Iv(t;n)l~
(Ip11(r)l+lpl,(r)l)o,(p,x)(r)dr
fora
(n : 1,2,...).
Therefore, using the analog of the equality (1.36), it is not difficult to verify that the sequence (u(.; n ) ) + ~ +oo is uniformly bounded and equicontinuous in the interval Ca, b), and the sequence (v(-; n)),= 1 is uniformly bounded and equicontinuous in the interval (a, b - e), where 6 is an arbitrary number from the interval (0, b - a) if i = 1 and ~ = 0 if i = 2. Thus without loss of generality we may assume that they converge uniformly on these sets. By the Krasnosel'skii-Krein Theorem [27], (cf. also L e m m a 2.3 of [18]), the function u0 : (a,b) --~ R, where Uo(t) = lim u(t;n) f o r a < t < b , n--*-b oo
is a solution of the problem (1.4), (1.7). On the other hand, according to (1.38), (i-1) uo (b-) = 0, which contradicts (1.27) and (1.34). The lemma is now proved. LEMMA 1.4 t. Let i E {1, 2) and let (1.33) be satisfied. Then there exist positive constants c and 6 such that for any measurable functions pl, p2 : (u, b) -+ R satisfying inequalities (1.27) the estimates
]pl(r)] [/ ' a(p2)(8)ds]'-' dr
lu(t)l < c [Zb ~(p,)(,) d, ]2-, ' ,,(p,.)(t) Ir < 2 - i + c / ' oCp2)(,)
fora
hold and, in addition, where u is a solution of the problem (1.4), (1.13,). This assertion is proved exactly like the preceding one. PROOF OF THEOREM 1.2: We shall carry out the reasoning for the case i : 1. The case i : 2 is handled in a completely analogous manner. Suppose the theorem is false for i : 1. Then there exist measurable functions pi (-; A) : (a, b) --+ R, (3"= 1,2; A = O, I), such that
Pjl it) <~Pj (t; A) ~< Pj2 (t)
for a < t < b,
the solution u(.; I) of the initial-value problem
u":p~(t;A)u+p2(t;A)u', uCa+)=O,
,z(t)
lim =1 ,--.,, ,,(p, (.; ,x))(t)
(1.39)
has exactly k~ zeros in the interval (a,b), and k0 # kl. For any A E (0,1) we define the function u(-; A) to be the solution of the problem (1.39), where pj(t;A) = Apj(t;1) + ( 1 - A)pj(t;O)
for a < t < b
(j = 1,2).
2353
Applying Lemma 1.4, it is not difficult to show that
=(t; A)
=(t: x0),
--*
aCp~ (-; ~))(t)
='(t;A0)
as ~ --* ~0,
a(P~ ('; ~0))(t)
and the first relation holds uniformly on t ~ (a, b) while the second holds uniformly on t E (a, b - e], where E (0, b - a) is arbitrary. Since according to (1.33) we have u(b-; A) # 0 for 0 ~< A < 1, it follows from this that if )t, A0 E [0, 1] and )t is sufficiently close to $0, the functions u(.; ),) and u(.; ~0) have the same number of zeros in the interval (a, b). Let ~* be the upper bound of the set of ~ E [0, 1] for which u(.; ~) has exactly k0 zeros in (a, b). Then by our assumption A* < 1, and so for all A* < ), < 1 the number of zeros of u(.; A) in Ca, b) must differ from k0. This, as we have shown above, is impossible. The theorem is now proved. We now consider in somewhat more detail the sets Vlo((a,b)) and V2o((a,b)). THEOREM 1.3. Let i E {1,2}. If
(Pl,P21,P22)E ~/0 ((a, b)),
(1.40)
then for any measurable functions ql, q2 : (a, b) --~ R such that qlai(q2) eL([a,b]),
ql(t)>~pl(t),
p21(t)<~q2(t)<.p2~(t)
fora
(1.41)
the solution u of the initial-value problem u"=qx(t)u+q2(t)u',
u(a+)=0,
lim a(q2)(t) u'(t) - 1 ,-~
(1.42)
has no zeros in the interval (a,b), and tt('-1) (b-) > O. Moreover if the relations (1.30i) and (1.31) hold and for any measurable functions q, q~ : (a, a) --. It satisfying (1.41) the problem (1.42), (1.5~) has no nontrivial solutions, then (1.40) holds. PROOF: The first part of the theorem follows directly from Lemma 1.3, as we have already noted above. Let us prove the second part. Suppose (1.30~) and (1.31) hold and the problem (1.42), (1.51) has no nontrivial solutions when the coefficients of Eq. (1.42) are measurable and satisfy (1.41). Then, in particular,
(pX,pl + Ipll,pn,p
2)
v CCa, b)).
But since the solution of the problem
=Ca+)=0,
lira
u'Ct)
tr(p2,)C t) - 1
has no zeros in the interval (a, b), this relation, according to Theorem 1.2, implies (1.40). The theorem is now proved. In order to bring o u t c e r t a i n other properties of the sets Vlo((a,b)) and V2o((a,b)) we introduce the following definitions. DEFINITION 1.5. Let c E (a, b]. We shall say that the vector-valued function (pl ,P2) E (a, b) --* R 2 belongs to the set V~'0((a, b), c) if conditions (1.61) hold and the solution tt of the problem (1.4), (1.7) satisfies the inequalities u'(t)(c - t) >l O f o r a < t < b , u(b-) > O. (1.43) DEFINITION 1.6. We shall say that the vector-valued function (Pl,P2) : (a, b) --+ R 2 belongs to the set V~o((a,b)) if conditions (1.62) hold and the solution u of the problem (1.4), (1.7) satisfies the inequalities u'(t)>0 2354
fora
u'(b-)>0.
It is clear that V~o((a,b)) C V~o ((a,b),b). THEOREM 1.41. If c E (a,b], (1.44)
(p,,p2) c V:o (Ca, b),~), and the functions P2x, pm ELloc ((a, b)) are such that p21(t) = p2(t), pro(t) >l p2(t) f o r a < t < e , pn(t)<<.p2(t), pm(t)=p2(t) f o r c < ~ t < b ( i f e < b ) ,
(1.45)
and
0.(Pro) C LC[a, b]),
Plo"1(P'/2) ~ L([a, b]),
(1.46)
then
(p,,p2,,p22) c Vlo ((a, b)).
(1.47)
Pl(t) ~<0 f o r a < t < b ,
(1.48)
Moreover/f (1.47) holds with then there exists e E (a, b] such that (1.44) holds, where p2(t)
J" P n (t) Pro(t)
for a < t < e, for c ~< t < b (if e < b).
(1.49)
We remark that if c < b, then (1.46) follows from (1.44) and (1.45). PROOF OF THEOREM 1.4x : The first part of the theorem follows directly from Lemma 1.3. Suppose (1.47) mad (1.48) hold. If the derivative of the solution u0 of the initial-value problem
u":pl(t)u+pn(t)u',
~'Ct) = 1 lim ,-, 0.(p,,)Ct)
u(a+)=0,
is positive in the interval (a,b), then (Pl,PZ) E V~o((a,b),b ). And if it has at least one zero in this interval, then there is a point e E (a, b) such that I %(t) >0
fora
and
I u0(e )=0.
Assume (1.49) and denote by u the solution of the problem (1.4), (1.7). Then by (1.47)
fora
u(t)>0
u(b-)>0.
On the other hand u(t) = no(t) for a < t < e, and so u'(e) = 0. Therefore we obtain (1.43) from (1.36) and (1.48). The theorem is now proved. Analogously we can prove THEOREM 1.42. If
(pl,p2) ~ V2'o((a, b)),
(1.50)
and the function p C Lloe ((a, b)) is such that
p(t) >~ p2 (t)
for a < t < b,
then (Pl,P~,P) E V2o((a,b)).
(1.51)
2355
Moreover, if (1.48) and (1.51) hold, then (1.50) also holds. To conclude this subsection we note that when C1.47) holds, Eqs. (1.42), whose coefficients satisfy (1.41), are nonoscillating on the closed interval [a, b], i.e., have no nontrivial solutions under the boundary conditions u ( t l + ) = 0,
u ( t 2 - ) = 0,
for any tl,t2 E [a, b], tl < t2. There is an extensive literature on regular nonoscillating equations (eL, for example, [73, 94]). But if (1.50) holds, then Eqs. C1.42) with qj ELloe{(a,b]), qj(t) >lpj(t) f o r a < t < b have no nontrivial solutions satisfying the boundary conditions
(3"=1,2),
~(tl+) = 0, ~'(t2-) = 0 for any tl, t2 E [a, b), tl < t2. These facts follow from Lemma 1.3. 1.5. S o m e V e c t o r - V a l u e d F u n c t i o n s B e l o n g i n g to t h e Classes V~((a, b)). In this subsection we shall present a variety of effective criteria guaranteeing the inclusion (1.33). LEMMA 1.51. Suppose relations (1.301) and (1.31) hold and suppose there exists a point d E (a,b) such
that /dp-~ (t) / t exp [ / " P21(s) ds] dr dt <~X,
/'
p~ (t)
]
(1.52)
p~, (s) ds dr dt <<.1,
exp
where p-~ is the negative part of the function pl, i.e., p~ (t) = Ipl (t)l - pl (t)
2 Then the inclusion (1.47) holds. PROOF: Suppose the contrary, namely that there exist b0 E Ca, b], a measurable function p2 : Ca, b) --+ R satisfying inequality (1.32), and a solution u of Eq. C1.4) such that
uCa+) = O, u(bo-) = O, u(t) > O f o r a < t < b 0 . Then
u'(t) > O for a < t < to, where to is some point in the interval (a, b0). Assume that to E (a, d]. By Lemma 1.1, /~=sup Obviously
{
u'(t)exp
[/o
]
u'(to) = O,
p2(r) dr :a
}
<+co.
/ / [/ ] [/~ ]L [/o ]
u(t)=
u'(r)dr<<.U
p2(s) ds dr fora
exp
1o
and the strict inequality holds on a set of positive measure. Thus, according to (1.32), the equality u'(t)exp
p2(r)dr
=
pl(r)u(r)exp
p2Cs)ds dr
for a < t ~< to,
and the first of inequalities (1.52), we obtain
< ~
p7 (t)
exp
p21 (s) ds dr dt <
~,.
Consequently to ~ Ca, d]. In a completely analogous manner we can show that to ~ (d, b0). The contradiction so obtained proves the lemma. The reasoning just given also implies the following proposition. 2356
LEMMA 1.52. Suppose relations (1.62) hold and
f f p~'(t) f f exp [ f ' p 2 ( s ) d s ] drdt <<.l, where p~ is the negative part of the function pl. Then the inclusion (1.50) holds. For any
ll, 12 E R set ds + s2 , I(11,12)= ~o+~ ll +12s
if ll +12s + s2 > O f o r a l l s ~ > 0
and I(11,12) = +cr otherwise. LEMMA 1.6~, Let 11,12 E R, and let the function 9 : (a, b) --* (0, +oo) be absolutely continuous on each closed subinterval of the interval (a, b) and summable on the closed interval [a, b]. Then the condition
~ab I(li,12) >1
g(t) dt
(1.53)
is necessary and sufficient for the relation g, ( - l l g Z , - 1 2 g + g ) e V~o((a,b),b ).
(1.54)
PROOF: Set
pl(t) = --llg2(t),
p2(t) = -12g(t) + ggl~---~ )) for a < t < b.
Then
a(p2)(t) = _t~+b \ exp u~,--5-J
[L 12
-
g(r) dr 2
]
for a < t < b.
Hence condition (1.61) follows. For any constants 11,12 E I t the equality
+oo (t)
~at
ds lx + 1 2 s + s 2
=
g(r)
defines a continuous function z : (a, b0) ~ R , where bo : s u p
{
te(a,b):
/
g(r)dr
)
9
Moreover z'(t) : -g(t)[ll + 12z(t) + z2(t)]
for a < t < b0
and in a sufficiently small right-hand neighborhood of the point a
Therefore the function u : (a, bo) ~ R such t h a t
uCt) = exp
[f 2
]
+b g(r)z(r) dr ,
is a solution of Eq. (1.4), a n d u(a+) = O. If (1.53) holds, t h e n b0 = b and z(t) > 0 for a < t < b. Consequently u'(t) > 0 for a < t < b, i.e., (1.54) holds. But if (1.53) does not hold, t h e n there exists a point c E Ca, b0) for which z(t) < 0 for c < t < b0 and so u'(t) < 0 for c < t < b, so t h a t (1.54) cannot hold. T h e l e m m a is now proved. In exactly the same way we can prove
2357
LEMMA1.62. Suppose lz, l~ 6 It. and the function g : (a, b) ~ (0, +oo) is absolutely continuous on each closed subinterval of the interval (a, b] and summable on the interval [a, b]. Then the condition b
I(ll,l,)
fa
>
g(t) dt
(1.55)
is necessary and sufficient for the relation r
(Ixg',-12g + g ) 6 V~o((a,b)). As Opial [91] has shown, in the case of nonnegative lx and 12 relation (1.55) follows from the inequality l l h 2 + 212h < 7r2,
where
b
h= 2
fa
g(t) dt.
Setting g(t) = (t - a) -~
for a < t < b,
we obtain the following corollary of Lemmas 1.61 and 1.62. COROLLARY. Let A 6 [0, 1) and 11,12 6 R. Then the condition
I(lx 12) >1 ( b - a ) Z - x '
(1.56)
1-),
is necessary and sufficient for the vector-wMued function (PI,P2) : (a,b) ---} R 2 to belong to the set V~o((a,b),b), where Pl (t) = -tl
(t - a ) - 2 ~ ,
p~Ct) = -t2(t
- a) -~
-
A t-a
- -
for a < t < b,
and if the inequality (1.56) is strict, the condition is necessary and sufficient for the function to belong to % CCa,b))LEMMA 1.7. / r e 6 (a,b), l j , m j 6 R, (j = 1,2), and the function g is the same as in Lemma 1.6,, then the inclusion (Pz,P2) 6 V:o((a,b),c ), (1.57) where
p,(t) = - l x g ' (t),
v,(t) = -12g(t) + 9'(t) g(t---ff for a < t ~< e,
Pl (t) = --17~1g 2 (t),
p2(t) = m2g(t) + 9'(t) g(t---~ for c < t < b,
holds if and only if mz >1O, I(l,,12) = and
(1.58)
j[cb /(lr'/'t 1 , I'1"12) >
2358
g(t)dt,
gCt) dr.
(1.59)
PROOF OF NECESSITY: Let u be a solution of the problem (1.4), (1.7). Then, according to (1.43), we have (pl,p2) e V~0((a,c),c ) \ V~0 ((a,c)). Hence, taking account of Lemmas 1.61 and 1.62, we obtain equality (1.58). In addition, it follows from (1.36) and (1.43) that ml is nonnegative. If (1.59) ,does not hold, then, denoting by b0 a point of the interval (c, b] for which
I(ml,m2) :
fe b~g(t)dt,
carrying out the change of variable t = b0 + c - t' in (1.4), and again applying Lemmas 1.61 and 1.6~, we verify that u(bo-) = 0. This, however, contradicts (1.57). Consequently, inequality (1.59) holds. The necessity is now proved. Sufficiency is proved analogously. COROLLARY. Ire E (a,b), $i E [0, 1), ll,m1 E R, (3" = 1,2), then the inclusion
(Px,P2) e V~o((a,b),c ), where pl(t) = -ll(t-
a)
plCt) = - m l ( b - t )
,
t -$1 a
p (t) =
l~ (t - a)-X'
for a < t ~< c,
$2 -2x~ , p2(t)= b------t + m 2 ( b - t ) -x2
forc
holds ff and only ff ml >>.O, i(11,/2) -- (C-- a) 1-)q 1 - A1
I ( m l , m 2 ) > ( b - c) 1-~2 '
1 -
$2
To prove this it suffices to set g(t)
J" ( b - c ) - ~ 2 ( t - a ) -~' (c a) -x' ( b - t) -x=
fora
In conjunction with Theorems 1.41 and 1.42, Lemmas 1.61, 1.6~, and 1.7 make it possible to establish effective conditions for vector-valued functions to belong to the sets V10 ((a, b)) and V20 ((a, b)). For example, the following assertion follows from Theorem 1.41 and Lemmas 1.6t and 1.7:lflx,12 E R, 12 >~ O, and the function g is the same as in Lemma 1.61, then the inclusion gS
(-llg2,-lzg
+
,12g + g ) E VloCCa, b))
holds ff and only ff 2I(11,12)
> b -
a.
We remark that for g(t) ---- 1 this assertion is the theorem of Vallde Poussin [103] already mentioned. LEMMA 1.8. Suppose i C {1,2}, k is a natural number, the function g is the same as in Lemma 1.6i, and l u , 11~, and 12 are constants with 112 <~111, and 12 >~O. Then the condition
(2k + 1-i)I(112,-12) <
~
b
g(t) dt < (2k +3-i)I(111,12)
(1.60i)
is necessary a~d sufficient for the inclusion ( - l l l g ~ , - l 1 2 g 2 , - 1 2 g + ~,12gg
gl
+ g ) E Vik((a,b)).
(1.61,)
2359
PROOF: Let i = 1. For e a c h ] 6 {1,2} choose a nonnegative integer nj and points tj-,, (n = 0 , 1 , . . . ,nj +1) such that the relations a = tj0 < t#l < .-. < t~-,;+l = b,
ft~ti"+= g(t) dt=I(l,j,(-1)J+ll,)
(n=0,...,nj-1)
if nj /> 1,
(1.62)
n
~and
f~] g(t)dt < I(lli,(-1)S+ll,).
(1.63)
ni
Denote by vj : (a, b) --* R, (j = 1, 2), the solutions of the initial-value problems
v" =qlj(t)v+q,#(t)v',
vCa+) = 0 ,
lim v'(t) t--.= a(q,# )(t) - 1,
where
qlj(t) --- -lxjg 2(t) f o r a < t < b , g'(t)
q2j(t) = (-1)#+"/2g(t) + g(t-'--ff for t#, ~< t < t#,+l,
(n = 0 , 1 , . . . , n # ) .
(1.64)
Then it is not difficult to establish, using the reasoning applied in the proof of Lemma 1.7, that the points t j , , (n = 1 , . . . , nj), with n even, and only those points, are the zeros of the function v# in the interval (a, b), and (-1)"v~(t)>0 fortj,
-lllg' (t)
pl(t)
-ll, g' (t),
ip,(t) _ a'Ct) 9(t)
l g(t)
for a < t < b.
Then it follows from (1.64) and (1.65) that
(--1)J[qlj(t)--pl(t)]>~O,
(-1)#[q,#(t)-p,(t)]v;(t)>~O
fora
(]=1,2).
(1.66)
By (1.62), (1.63), and the first part of inequality (1.601), we have n2 i> 2k. On the other hand, according to (1.66) and L e m m a 1.3',the solution u of the problem (1.4), (1.7) has at least one zero in each of the intervals (t2,,t=,+2 ), where n E (0, n~ - 2] is an even number. Thus u vanishes at least k times in the interval (a, b), and, again applying (1.66) and L e m m a 1.3',we obtain the relation nl ~> 2k. In this way, taking account of (1.62), (1.63), and the second part of inequality (1.601), we verify that either l't1 2k + 1 and :
g(t) dt < I(11,12),
ft[ "1
or nl = 2k. In both cases, as we have shown above, Vl (b-) # 0. Therefore, by (1.66) and Lemma 1.3, tt has at most k zeros in the interval (a,b) and u(b-) # O. Thus (1.611) holds, i.e., the lemma is proved for i = 1. Reasoning analogously, we can consider also the case i = 2. In Lemma 1.8 it is admissible to set, for example,
gCt) = ( t - a ) - x * ( b - t ) 2360
-x"
if i = 1,
and
gCt) = ( t - a ) -x
if i = 2 ,
where ~1, )t2, )t E [0, 1). In the case when i : 1 and g(t) - 1 Lemma 1.8 was established by E. L. Tonkov [55]. 1.6. E x i s t e n c e a n d U n i q u e n e s s T h e o r e m s . By Theorem 1.1 the question of the unique solvability of the inhomogeneous problems (1.1), (1.21) and (1.1), (1.2~) reduces to the question of whether the corresponding homogeneous problems have no nontrivial solutions. We shall reformulate this theorem below in terms of the sets Vl((a,b)) and V~((a,b)) for the problems (1.1), (1.5,) and (1.1), (1.52), and establish as well some a priori estimates for the solutions which will be used in the next section. THEOREM 1.51. Let
(g11, g12 , g21, g22 ) E Vl ( (a, b) ) . Then there exists a constant Co such that for any measurable functions pj : (a, b) --~ R (3" : 0,1, 2) satisfying the inequalities gjl(t)
co
]u (i-1) (t)l ~< [tr*l(g21,g22)(t)lY-1
f'
9*(t,r)lpo(r)ldr
for a < t < b
(j = 1,2),
where 1
~*(t,r)=
b
#(s) d8
~(~ / 1 fa t #(s) ds ;N
9
tz(r) dr
for a < r ~
#(r) dr
for a < t < r < b,
and #(t) : o* (g,1, g,, ) (t). PROOF: By Theorem 1.1 the problem (1.1), (1.51) has a unique solution u and
~ab u(t) :
~(t,r)PoCr)dr
for a < t < b,
where ff is the Green's function for the problem (1.4), (1.51). From this, by Lemmas 1.2, 1.4, and 1.4', it easily follows that there exists a constant Co satisfying the hypotheses of the theorem. THEOREM 1.6x. Let
(gl,g21,g22)
E Vlo((a,b)).
Then there exists a constant co such that for any measurable functions pj : (a, b) -+ R, (j = 0,1, 2), satisfying the relations pl(t) /> gl(t),
g,l(t) <<.p,(t) <<.g,z(t)
for a < t < b
(1.68)
and PlO"1(P2) e L([a, b]),
poa~ (g21, g2~ ) e L([a, b]),
the prob]em (1.1), (1.51) has a unique solution u with
CO
[u (s-l) (t)l ~< [O.l(g)(t)]j_ 1
fab ~*(t,r)p~Cr)dr
fora
(j = 1,2),
where the function ~* is as in Theorem 1.51,
p~(t)
=
l[Ip0(t)l
-
po(t)sgnu(t)]
for a < t < b,
(1.69)
2361
and g22 (t) g(t) = g21 (t)
for a < t ~< a + b 2 ' for -a- < + t b< b .
(1.70)
2
PROOF: The existence and uniqueness of a solution u of the problem (1.1), (1.51) follow from Theorems 1.1 and 1.3. Let v be a solution of the problem
v" = gt (t)v + p2(t)v' - p~ (t), vCaq-) = O, vCb-) = O.
(1.71)
According to Lemma 1.2 and Theorem 1.3 the Green's function .q of the equation v" = 91 (t)v + pz(t)v'
(1.72)
under the conditions (1.71) is negative in (a, b) • (a, b). Therefore b
v(t) = --
fa
,~(t,r)p;(r)dr>>.O
fora
(1.73)
We shall show that lu(t)l .< v(t)
for a < t < b.
(1.74)
Indeed in the opposite case there are points tl E [a, b) and t2 E (tl, b] for which N(t)l >
v(t) for tl
< t < t2,
lu(ti)l =
v(ti)
( / = 1,2).
(1.75)
We define the function z : (tl,t2) ~ (0, +co) by the equality
zCt) = NCt)l-
vCt).
Then z is a solution of the equation z" = gl(t)z + p2(t)z' + ~(t), where, by (1.68) and (1.69), /~(t)
=
[Pl (t)
- -
gl (t)]luCt)l + p~ (t) + P0 (t) sgn u(t) >~ 0
for tl < t < t2.
If ~ is the Green's function for the equation (1.72) under the boundary conditions v(t +) = 0 = v ( t ; ) = 0, then ~(t,r)<0 fort1 < t , r
ftl~'2t~(t,r)l~(r) dr < 0
for tl < t < t2,
which contradicts (1.75) and thus guarantees (1.74). According to Lemmas 1.2, 1.4, and 1.4' it follows from (1.73) and (1.74) that there exists a constant dl independent of the collection of functions pj, (3" = 0,1,2) such that lu(t)l <~ d,
2362
~ ; ( t , r ) p ~ ( r ) d z <~ dl
~*(t,r)p~Cr)dr
for a < t < b,
(1.76)
where
1
~;(t,r)
1/
~(P2) ( r )
I
~
a(p2)(r)
r a(p2)(s)ds ~ a(p2)(r)dr
for a < r ~< t < b,
a(p,)(s)ds
for a < t < r < b.
//
a(p2)(r)dr
Let us now estimate u ~. Let to E (a, b) be an arbitrary point at which 0. Then because of (1.51) there exists tl E (to, b) such that
u(t)u'Ct ) > 0
for
Set
to < t < tl,
u~(to) # O. Suppose u(to)u'(to) >t
(1.77)
u'(tl) = O.
b
jft~(p2)(r) dr
w(t) = r
f o r a < t < b.
Since w is a solution of the equation
b
w' = p2 (t)w + [pl (t)u(t)
+ po (t)]
ft
a(p2)(r)
d r - u'(t)a(p2)(t),
upon applying (1.68), (1.69), and (1.77), we obtain
,,,(to) <,,(p2)Cto) {~t[ l [lal(O,,(Ol+p;(O],,(p2)C1 0 frb ~(p2)(s)dsdr +,,(tl) } In addition, by (1.76), we have
L' o(p2)(r)dr f'
luCt)l <~ dl f:O aCp2)(r) d r
,9; (to, r)p; (r) dr
for to <<.t < b.
Therefore, taking account of the estimate
ft[1 a(p2)(r) p~(r) f f a(p2)(s)dsdr <~ ftO a(pz)(r)d 1 r f f ,9;(to,r)p~(r)dr,
we easily verify that Ir where
d2 =dl
[s
d2
~< al(a)(t)
~ab,9* (t 0,r)p0(r)dr,
Igl(t)la;(g21,g22)(t)dt +
]
a*(g21,g22)(t)dt + 1.
a
Making obvious changes in the reasoning just given, one can show without difficulty that this same inequality holds also for to E (a,b) at which u(to)u'(to) < O. Thus, one can set Co = max{d1, d2}. The theorem is now proved. For the case of boundary conditions (1.5z) Theorems 1.51 and 1.61 assume the following form.
THEOREM 1.52.
Let (911, g~2, g21, g22 ) E V2 ((a, b)). Then there exists a constant Co such that for any measurable functions Pi : (a, b) --+ It, (j = 0,1, 2), satisfying inequalities (1.67) and the relation poa2 (g21) E L([a,b]) the problem (1.1), (1.52) has a unique solution u with Co luCi-1) (t)! ~< [tr2(g21)(t)]i_,
f f `9,
(t,r)lP0(r)ldr
for a < t < b
(j = 1,2)
2363
where
for a < r <<.t < b,
~* (t,r) =
1
t
o(g21)(~)
o(g21)(~) ds
fora
THEOREM 1.62. Let (gl,g21,g22) E V2o((a,b)). Then there exists a constant co such that for any measurable functions pj : (a,b) ---}R, (3" = 0,1,2) satisfying inequalities (1.68) and the relations
Pl o2 (P2) E L([a, b]),
poor2(gzl) E L([a, b]),
the problem (1.1), (1.52) has a unique solution u with
co f b #' 9 I,, ci-1) (t)l ~< [a2Ca2,)Ct)]i-1 (t,r)PoCr)dr
for a < t < b (J = 1,2),
where the functions .~* and p~ are as in Theorems 1.5z and 1.61 respectively. Using the results of the preceding subsection, we can obtain as corollaries of Theorems 1.51, 1.52, 1.61, and 1.62 some quite easily verifiable criteria for unique solvability of linear boundary-value problems. w
Equations Comparable with Linear Equations
We now turn to the consideration of boundary-value problems for the nonlinear singular differential equation u"= f(t,u,u'). (2.1) As regards the boundary conditions, in this section it will be convenient for us to consider them to be homogeneous: uCa+) = 0, u (i-1) (b-) = 0. (2.2i) Throughout the following we assume that
(2.31)
f E Kioc((a,b) x R2).
In addition, when studying the problem (2.1), (2.2z) we shall sometimes impose on f the more stringent condition
f E Kloc ((a,b] x R2).
(2.32)
We shall systematically use the transformations a, ai, a~ and the sets V~((a, b)) and Vik ((a, b)) introduced in the preceding section. 2.1. A u x i l i a r y P r o p o s i t i o n s . We begin with the case when equation (2.1) is quasilinear, i.e., has the form
~" = p1(t)~ + p2(t)~' + p(t,~, ~'),
(2.4)
where
p ~ K~oc (Ca, b) • R2),
IPCt,x,y)l ~< P0Ct) for a < t < b,
x,y e R.
(2.5)
For such equations the question of the solvability of boundary-value problems is solved rather simply using Schander's fixed-point theorem. LEMMA 2.1. Suppose i E {1,2}, conditions (1.3i) and (2.5) hold, and the problem (1.4), (1.5,) has no nontrivial solutions. Then the problem (2.4), (2.2i) is solvable. PROOF: Consider the set of continuously differentiable functions w : (a, b) ~ R such that w and w'ai(p2) have finite limits at the points a and b, and denote by B the Banach space of these functions with the norm I1,~11 = sup{l~(t)l
2364
+ I~'(t)l=~(p,)(t)
: = < t < b}.
If ~ is the Green's function for the problem (1.4), (1.5i), then, according to Lemmas 1.2, 1.4, and 1.4', IO/-'~(t,r)] c*ai(p,)(r) o v -1 <" [o,(p~)(t)]i-,
fora
t#z
(j=1,2),
where c* is some constant. Therefore, by (1.3~) and (2.5), the continuous transformation H defined by the equality
s
H(,,,)(t)=
9(t,r)p(~,,,,(~),,,,'(~))d~
fora
maps the space B to a compact subspace of itself. By Schauder's theorem there exists u E B such that u(t) =
9(t,~)p(~,,~(~),,,'(r))d~
for a < t < b.
It now follows from Theorem 1.1 that u is a solution of the problem (2.4), (2.2~). The lemma is now proved. We now present two more propositions that we shall need in what follows. LEMMA 2.2. Let a
(n=l,2,...),
lima,
,--*+co
=a,
lim bn =b,
,-.-, +00
(2.6)
and let u. : [a.,b.] -+ R, (n = 1 , 2 , . . . ) , be a sequence of solutions of the equation (2.1) such that for any [tl,t2] c (a,b) sup{l~.Ct)l + I~'.(t)l : t e [ a . , b . ] f " ] l t l , t , ] , ,, = 1 , 2 , . . . } < +oo. Then it is possible to choose a subsequence ttun= J,~=1~+00 of the sequence ( u . ) .+00 = l which is uniformly convergent ttl +00 along with ( . . ),.=1 on each closed subinterval of(a, b) and whose limit is a solution of(2.1) on this interval. PROOF:
Set
Zjn(t)
:
{
u~ -1) (a.)
for a ~< t < a . ,
tt~ -1) (t)
for a,~ < t < b,,
u~ -1) (b,)
for b, <~ t ~< b.
(3" : 1,2;n = 1 , 2 , . . . ) ,
In view of (2.31) and the assumptions made the sequences (z~, ),=1,+00 (J = 1, 2), are uniformly bounded and equicontinuous on each closed subinterval of (a, b). Therefore, according to the Arzel~-Ascoli Theorem, we may, without loss of generality, consider them to be uniformly convergent on each such closed interval. Let u ( t ) = lira Zl,(t) f o r a < t < b . .---*+oo
Then
u'(t)=
lim z2, it)
.---~ +00
fora
If t is an arbitrary point of the interval (a, b), then according to (2.6), for all sufficiently large n
z,.Ct) = z , .
+
+b /(T,~,.(~),~,.Cr))d~. 2
Passing to the limit as n -+ +cr in this equality, we obtain
~'(t)=~'
+
+b I(~,~(r),~'(~))d~. 2
Consequently u satisfies (2.1) almost everywhere in (a, b). The lemma is now proved. REMARK. It is easy to see that the proposition just proved remains valid even when a = - c o or b = +oo or both. The following propositions are consequences of Lemma 2.2
2365
LEMMA 2.3,. Let vj : Ca, b) ~ R + , (3" = 1,2), be continuous functions, let v,(a+) = v , ( b - ) = 0, and let conditions (2.6) hold. Further suppose that for any natural number n Eq. (2.1) has a solution tt, : [a,, b,] --, R satisfying the inequalities lu(nj-l) (t)l ~< vj(t)
for an ~< t ~< bn
(s = 1,2).
(2.7)
Then the problem (2.1), (2.2,) is solvable. LEMMA 2.32. Suppose condition (2.32) holds, vj : (a,b) --+ R + , (3" = 1,2), are continuous functions, O, and the sequence of points a,,, (n = 1, 2,... ), of the interval (a, b) converges to a. Further suppose that for any natural number n Eq. (2.1) has a solution u,, : fan,b) ~ It satisfying the inequalities 111 ( a + ) =
[tt~ -1) (t)l ~< vi(t)
for a, < t < b
(j = 1,2),
with u: (b-) = O. Then the problem (2.1), (2.22) is solvable.
2.2. E x i s t e n c e T h e o r e m s . THEOREM 2.1. /f i E {1,2},
(2.s)
(gxx,g,2,g2x,g22) E Vi((a,b)), and on the set (a, b) • R 2 the inequalities
(2.9)
I f ( t , z , y ) - g, (t, z, y)z - g2(t,z,y)yt ~ go(t) and gjl(t) <<.gj(t,z,y) <~gi2(t) hold, where gj e K~
(2.10)
(3" = 1, 2)
• RZ), (3" = 1,2), and
goa~ (g21 ,g22 ) E L([a, b]),
(2.11)
then the problem (2.1), (2.2,) is solvable. PROOF: Let the constant e0 and the function ~* be as in the conclusion of Theorem 1.5i. Set
co
f'
vj(t) = [o*(g21,g22)(t)]J-I
~*(t,r)go(r)dr
for a < t < b
( j = 1,2)
and b-a
-a,,=a+
bn=~
3n '
~ b
-
b,
b-a
3----~-' i f i = l ,
(n=1,2,...).
(2.12)
if i = 2 ,
Define the functions ~ : (a, b) • R z --+ R, (n = 1, 2 , . . . ), by the relations
~. (t,~, y) = ~ (t)xo (t, I~I + lyl)[fCt, x, y) - g,i (t)x - 921(t)y],
(2.13)
where
r
=
1 0
1 _
Xo(t,z) =
2 0
2366
for t e [a., b,,], for t r [aN,b,], for z ~< r0(t),
z__y__
ro(t)
for ro(t) < z < 2ro(t), for z/> 2ro (t),
(2.14)
(2.15)
and r0Ct) = v l ( t ) + v2Ct) + 1.
(2.16)
By inequalities (2.9) and (2.10) it is clear that on (a, b) • 1t'
[
roCt)(gi2(t )
I~,(t,z,y)l ~< r (t) [g0 (t) + 2
gil.
j=l
Thus by Lemma 2.1 for any natural number n the equation ,," = g,, Ct),, + g~, (t),,' + ,,. (t,,,,,,')
with the boundary conditions (2.2i) has a solution u , . By (2.13) u , is also a solution of Eq. (1.1), where
for a < t < b
p~(t):gilCt)+r po(t) = r
ltt.(t)l+lu:Ct)l)[giCt, u.Ct),u'(t))-gjlCt)]
(j=1,2),
(t)xo(t, lu, (t)l + lu" (t)l)[f(t, u,(t), u: (t)) - gl (t, u,,(t), u" (t))u, (t) - g2(t, u,(t), u" (t))u" (t)].
Therefore, taking account of the choice of the functions vl and v2 and the inequalities
gjx(t)
(j--- 1,2),
Ip0(t)l ~
which follow from (2.9) and (2.10), we verify the estimates (2.7). From this, according to (2.14)-(2.16), it follows that u , is a solution of Eq. (2.1) on [a,,, b,,], (n = 1, 2 , . . . , ). It remains only to apply Lemma 2.3~. The theorem is now proved. THEOREM 2.2. I f i E {1,2}, (gl,g,l,g2=)
E
V~o(Ca, b)),
(2.17)
conditions (2.3~) and (2.11) hold, and on the set (a, b) • R 2 the inequalities [f (t, x, y) - gl (t )x -- g2 (t, x, y)y]
sgn x >i
--go Ct)
(2.18)
and
g,, (t) <<.g2(t,x,y) <<.g,,Ct), hold, where g2 E K~
(2.19)
b) x R ' ) , then the problem (2.1), (2.2,) is solvable.
PROOF: Let CO
vi(t)--- [~,(g)(t)]J-1
-Lb~*(t,~)go(r)d'r
for a < t < b
( / = 1,2)
where the function .q* is as in Theorem 1.5~, co is a constant for which the conclusion of Theorem 1.6i holds, and the function g : (a, b) --, R is given by equality (1.70) in the case i = 1 and is identically equal to g22 in the case i = 2. If (2.12) and (2.14)-(2.16) hold and on (a,b) • R 2 e.Ct,=,y) = r215
I=1 + lyl)lfCt, =, y) - g, (t)= - g=~ (t)y]
(-
= 1,2,...
),
then, as follows from Lemma 2.1, for any natural number n the equation
u"= gl (t)u + g=, Ct),~ + g,1 (t),~' + ~,. (t,,~, u') 2367
has a solution u, satisfying the boundary conditions (2.2i). Then u , is also a solution of Eq. (1.1), where
for a < t < b p,(t) = gl (t), p, (t) = gn (t) + r (t)Xo (t, lit. (t)l + Itt" (t)l)[g, (t, tt. (t), it" (t)) - g n (t)] and
p0 (t) = r (t)xo (t, lu. (t)l + I" (t)l)[/i t, u. (t), u',, (t)) - g, (t)tt. (t) - 92 (t, u. (t), tt~n(t))utn (t)]. But, according to (2.18) and (2.19),
gn(t) <.p2(t) <~gn(t),
l[Ipo(t)l-po(t)sgnu,(t)]
<~go(t)
for a < t < b.
Thus estimates (2.7) hold, u , (n = 1 , 2 , . . . ) are solutions of Eq. (2.1) on the closed intervals [a,,b,], and, according to Lemma 2.3i, the problem (2.1), (2.2i) is solvable. The theorem is now proved. We note that, whereas under the hypotheses of Theorem 2.1 the rate of growth of the right-hand side of Eq. (2.1) on the phase variables is linear, under the hypotheses of Theorem 2.2 this rate of growth, in general, can be anything. For example it follows from Theorem 2.2 that the problem (2.1), (2.2i) is solvable, i 6 {1,2}, where
fCt,x,y ) = g(t)x2.+l y2,,= + go(t), n and m are arbitrary natural numbers, g : (a,b) --* 1t.+ is a function in Ltor L,or ((a, b]) for i=2, and
for i = 1 and in
f=
a(t - a)(b - t) 2-i Ig0(t)l dt < +oo.
2.3. U n i q u e n e s s T h e o r e m s . THEOREM
2.3. If i e {1,2}, condition (2.8) holds, a n d on (a,b) x R 2 g n ( t ) l x l - x2[ ~< [f(t, xl,y) - f(t, x2,y)]sgn(x, - x2) <<.gn(t)]xl - x2[, g21 (t)lYl -- Y2 ] ~< [f(t, x, yx ) -- f ( t , x, y2)] s g n (Yl -- Y2) ~< g22 (t)lYl -- Y~. ],
(2.20)
then the problem (2.1), (2.2i) has at most one solution. PROOF: Let ttl and us be solutions of the problem (2.1), (2.2i). Set uCt) = , , l C t ) - u2it)
(2.21)
for a < t < b.
Then u is a solution of the problem (1.4), (1.5,), where
fCt, u, Ct),u'x(t))- f(t, u2Ct),u'l(t)) Pl (t) ~--
uCt) g~, Ct)
and
if uCt) = 0,
{ f(t, u 2 ( t ) , u ~ i t ) ) - f(t, u2it),u'2(t))
p~Ct)
=,(t)
:
if ,(t) # 0,
if u'it ) # O, if u'(t) = O.
g21 (t)
It follows from (2.20) that
gil(t)
(j=l,2),
and so, by (2.8), u(t) =- O. The theorem is now proved. It is not difficult to verify that a consequence of Theorems 2.1 and 2.3 is the 2368
COROLLARY. Suppose the hypotheses of Theorem 2.3 hold and in addition (2.22)
dt < +oo.
f b If(t,~176 Then the problem (2.1), (2.2i) has a unique solution.
The following proposition shows that by strengthening the restriction (2.8) it is possible to eliminate from Theorem 2.3 the requirement that the function f be Lipschitz on the second argument. THEORV.M 2.4. If i E {1,2}, condition (2.17) holds, and on Ca, b) x R 2
[.f(t, xl,y) - f(t, x2,y)]sgn(xl - x2) >/gl(t)lxx - x2l, g21 (t)]yl - Y21 ~< [f(t, .T,,y1) - - fit, x, y2 )] sgn (Yl - Y2) < g22 (t)lyx - Y2 l, then the problem (2.1), (2.2i) has at most one solution. PROOF: For definiteness we shall consider the case i = 1. Let u~ and it2 be solutions of the problem (2.1), (2.2x). Assume (2.21) holds. Using (2.23) it is not difficult to verify that there exist measurable functions 9z, h : (a, b) -4 R, such that
ggx(t)<<.g2(t)~g22(t),
h(t)>.O
fora
(2.24)
and tt is a solution of the equation u" = (ga (t) + h(t))u + g2(t)u'
(2.25)
in the interval (a, b). Suppose the function u is different from 0 at some point to E (a,b). Then there exist tl C [a, t0) and t~ C (to, b] for which u(t)~0 fortl
< t2r t < t 2
(?!, =
1,2,...),
tin --4 tj
as n ---r +co
(3" ----1,2)
hold. According to (2.17), (2.24), and Lemmas 1.2 and 1.3, for any natural number rt the problem
v"=gl(t)v+g2(t)v',
v(tl,)=0,
v(t2,,)=0
(2.27)
has no nontrivial solutions, and its Green's function .qn is negative on the set ( t l , , t2n) • (tin, t2n ). On the other hand, h'om (2.25) and Theorem 1.1 we have the representation t2n
u(t)=cl,,vl(t)+c2,,v,(t)+
9,,(t,r)h(r)u(r)dr
fortlr, <<.t<~t2n ( n = l , 2 , . . . ) ,
d tin
where vx and v2 axe arbitrary linearly independent solutions of (2.27) and
er = (-1) su(t2")v3-r
1(tl.
-uCtl,)vs-r
(t2. ) - vl Ct2.
(J = 1,2).
Ctl. )
(We remark that here, in general, one cannot use Green's formula on the whole interval (tl ,t2), for the endpoints might coincide with the points a and b, at which the integrability properties of h are unknown.) Thus, by (2.24) and (2.26), lu(t)[ < ICl,,V,(t)+cz,,v2(t)l
fortln ~
in = 1 , 2 , . . . ) ,
and since %, ~ 0 as n --. +oo, (j = 1,2),the equality u(to) = 0 follows from this estimate and contradicts the hypothesis. The theorem is now proved.
2369
COROLLARY. Suppose the hypotheses of Theorem 2.4 hold, along with the inclusion (2.3i) and inequality (2.22). Then the problem (2.1), (2.21) has a unique solution. 2.4. N o n u n i q u e n e s s T h e o r e m s . We assume throughout this subsection that
f(t,O,O) = O f o r a < t < b and that for any point to e (a, b) a solution u of Eq. (2.1) satisfying the initial condition
,,(to) -- o,
,,'(to) = o,
is identically zero. THEOREM 2.5. Suppose i 6 {1, 2}, k and m are natural numbers with k # m,
(2.28)
(g,~,gl,,g,l,g~,) e ~((a,b)), (hn,hn,h,l,h22) E ~m((a,b)),
(2.29)
and that on the set (a, b) • R 2 the inequalities (2.9), (2.10) hold, while
(2.30)
IfCt, x,y) - hxCt, x , y ) z - h,(t,x,y)yl <~ho(t, I~1) and
hjl(t) <.hj(t,x,y)<~hj,(t)
(3"= 1,2)
(2.31)
where gj,hj E K~ b) • R ' ) , (j = 1,2), (2.11) holds, and the function ho : Ca, b) x R+ ~ R+ is nondecreasing on the second argument and lira if. b ho(t,x)a~ (h21,hn )(t) dt = O.
(2.32)
z--40+
Then for any integer I such that min{k,m} ~< l < max{k, m},
(2.33)
the problem (2.1), (2.2i) has at least two solutions with exactly I zeros in the interval (a,b). PROOF: Assuming the contrary and applying the reasoning used in the proof of Lemma 1.4, we easily verify that there exists a number z > 0 for which
(g;1,g;2,g;i,g;2) 6 Vik((a,b)), whenever 9 gj~(t)=
{ g]~(t)
for a <
t ~< b - ~,
min{gi~(t),hil(t) } forb-r (j=1,2), g~ { max{gi,(t),hi2(t)} for a < t < a + e , 2=
g~'2(t)
fora+e~
In an analogous way we can also change the vector-valued function (h11, hz2, h n , h n ) in a neighborhood of a and b, without taking it outside the class V~,~((a, b)). This circumstance makes it possible for us to assume without loss of generality that
gjl(t)=h]l(t) 2370
forb-e
gj2(t)=hj2(t),
fora
(j=1,2),
(2.34)
where ~ ~ (0, b - a~ "T/"
+CO be a sequence of points of the We shall specify an integer l satisfying inequalities (2.33). Let (an)n=~ +0o a sequence of points of the interval ( b - ~, b) converging to interval (a, a + e) converging to a and (bn)n=l b. Further suppose the functions vi, (3" = 1,2), are as in the proof of Theorem 2.1 and
to(t) - v~(t) + v,(t) + 2
for
a <
t <
b.
By (2.32)
i' where •
~ (0,1) is a certain number and do is a constant that is chosen for the vector-valued function
(hll,h1~,h21,h2~) according to Theorem 1.5~. Since Eq. (2.1) has no nonzero solution vanishing on (a,b) together with its derivative, it is possible to choose for each natural number n a point rn E (0,x0) such that for any solution u of this equation and any point to E [a,, b,,] the inequality
I,,(to)l + I,,'(t0)l -< r. implies the inequalities
N'(t)l < 1 for a n < t < b n .
lu(t) I < x0, Assume that (2.14) and (2.15) hold,
I
X,,(z)
rn
for z .< ~-,
0
=
2z
-:
_
1
for
aio(t)= (gj2(t) gr { h i 2 (t) hr (t) = h~'x (t)
.),
(n = 1,2,
ur , ,
""
fora
and define the functions fn : (a, b) • R 2 --* R, (n = 1, 2,. .. ), by the equalities
g,0 (t)z + g20 (t)y + r + [yl)[f(t,z,y) f o r . < t < b, ixl+ lu[ ~> r.,
/.(t,z,y) = { hlo(t)x + h2o(t)y + r fora
- g,0 (t)z = g20 (t)y]
(t)Xn (Ix] + lyl)[f(t,x,y) - hlo ( t ) x - h2o(t)y]
I~1+ lul < r--
It is easy to see that fn ~ K~or ((a,b) • R 2) if i = 1 and f,~ E Kioc ((a,b] x R 2) if i = 2. Fix an arbitrary natural number n. If 7t and 72 are positive numbers with 7t sufficiently large and 72 sufficiently small, and ul0 and u20 are solutions of the initial-value problems
u" = glo(t)u + g=o(t)u',
u(a+) = 0,
u n = hxoCt)u + h2oCt)u',
u ( a + ) = O,
C(t)
,-.o o(g,o)(t) - " t l lim
and lira
ut(t)
,-.~ -(h~o)(t)
then ulo and U~o are also solutions of the equation
u " = f . (t, u,u').
(2.36) 2371
We now write this equation in polar coordinates, i.e., we carry out the change of variables u = p cos ~o,
u' = p sin ~o.
(2.37)
We obtain the system 1 p' = f . ( t , pcos~o, psin~o) sin~o + ~psin2~o, (2.38)
~~~ --_ 1.in (t,p cos ~o,psin Io) cos to - sin 2 to. P We shall understand a solution of this system to be a vector-valued function (p, ~o) : (a, b) --+ R 2 whose first component is positive. Then with each solution of the system (2.38) the transformation (2.37) associates a definite nonzero solution of Eq. (2.36) and to each nonzero solution of (2.36) a solution of (2.38), unique up to an additive multiple of 2~r in the second component. Let (Pl0, ~10) and (p20, ~20) be the solutions of the system (2.38) corresponding to u~0 and u20 with
< ~;0(t) < ~
(y = 1,2)
in some neighborhood of the point a. Then, taking account of (2.28) and (2.29), it is not difficult to verify that ~(,~" - 2 ) - ~ k < ~ 0 ( t ) < ~~'(i - 2) - ~-(k- I) and
~~r( ,.- 2) - ~.~ < ~,0 (t) < lr ( i - 2 ) - r ( m -
1)
in some neighborhood of the point b. Choosing a e (a, am) and fl~ E (b=, b) (v = 1, 2 , . . . ) from these neighborhoods so that fl~ --* b, as v ~ +oo, consider the set of solutions of the system (2.38) under the initial conditions p(a) = z, ~ ( a ) = ~Ol0(a), (2.39) where p20 (a) <~ z <~ plo(a). Since, by (2.34), we have p20(a) = ~10(a), it follows from Kneser's theorem on the structure of the set of reachable points (cf., for example, [56], p. 15);that for any natural number v there is a solution of (2.38) ( ~ , ~,,) satisfying the equality ~(~,,) = ~(,-
2) _ ~l.
It is clear that if the solution fi~, (v = 1, 2 , . . . ) of Eq. (2.36) corresponds to (~v, ~ ) , then ~,,(a+) = o,
,~v~c,-~)(Z~) = o,
fly (t) > 0 in a small right-hand neighborhood of the point a and fly has exactly I zeros in the interval (a, fly). ~ +co The properties of the function f,, make it possible to distinguish a subsequence of the sequence (u~)~= 1 that is uniformly convergent in the interval (a, b). Its limit u , is obviously a solution of the problem (2.36), (2.2i) having exactly l zeros in the interval (a, b) and positive near the point a. Suppose that to E [a,,, b,~] and I=.(t0)l + I,,'(t0)l ~< r.. Then, by definition of the constant r,,, I=.(t)l < =o,
I='~(t)l < x
(2.40)
for a . < t < b..
Consequently un is a solution of the equation ~" = h l 0 ( 0 ~ + h~0 (t)~' + r 2 1 5 2372
+ I,;I)[fCt, u,u') - hlo(t)u
- h~o (t)u'].
From this, ax:cording to (2.30) and (2.31) it follows (cf. the proof of Theorem 2.1) that u,, is a solution of some equation (1.1), where the measurable functions po,pl,p2 : (a, b) ~ R satisfy the inequalities
hil(t) ~pr
< hr
( i = 1,2),
Ip0(t)l ~< h0(t,z,),
and
x. = sup{N.(t)l : a~ < t < bn}.
(2.41)
But then, in view of the choice of the constant do, we have
~. <<.dof bho(t,~.)~,(* h 21,h=)(t)dt,
(2.42)
which contradicts (2.35) since, as follows from (2.40), zn ~< x0. Thus lu,(t)l + lu" (t)l > r, f o r a n <~t <<.b,, and u,, is a solution of the equation
~"
=
glo (t)u + 920 (t)u' +
r
(t)xo(t, I=l + lu'l)[f(t, =, =') - g l 0 (t)= -g20 (t)='].
From this, just as in the proof of Theorem 2.1, we obtain (2.7). Thus, the solutions u , , (n = 1 , 2 , . . . ), we have constructed satisfy Eq. (2.1) on the closed intervals +oo [a,,, b,, ], and there is no loss of generality in assuming that the sequence (u,,),,=l converges uniformly in the interval (a,b) to a solution u of the problem (2.1), (2.2,). Consequently, from (2.30), (2.31), and (2.34), we obtain the inequality (2.42), where x , is defined by the relation (2.41). From this it follows that u differs from the identically zero solution; for otherwise x~ ~< x0 for sufficiently large n, contradicting (2.35). We denote by l* the number of zeros of the function u in the interval (a, b). Obviously l* ~< I. Let l* < l, and let ti= E (a,b), (j = 1 , . . . ,l), be the zeros of the solution u= enumerated in increasing +~ , (j = 1,. ..,t), order. Then, since there is no loss of generality in assuming that the sequences (tin)n=1 converge, we need only consider three possibilities: a) tp~ --+ t*, ti+l,, --+ t* as n ~ +co, where t* S (a, b), and J E { 1 , . . . , 1 - 1}; b) tx, --+ a as n --+ +oo; c) t,, --, b as n --+ +co. In case a) we have u(t*) = u'(t*) = 0, i.e., u(t) = 0, and this contradicts what has been proved. If b) holds, then for any n there exists a point ~-,, E (a, tl,~) at which u' (r,,) = 0. But, as is easily verified, (hxx,hx,,h21,h22) e V2((a,r.)) for sufficiently large n. Therefore, using (2.30), (2.31), Lemma 1.2, and Theorem 1.1, we have [u.(t)l <<.d
ho(r, lu,,(r)l)a~(h21,h22)(r)dr
for a < t < r . ,
where d is a constant independent of n (the details omitted here can easily be reconstructed from [60]). On the other hand h0 is nondecreasing on the second argument, and, by (2.7),
sup{lu,(t)[:a
as n --+ +cr
Thus the inequality so obtained contradicts (2.32). In an analogous way we can show that the case c) also cannot hold. Thus l* = l, i.e., u is a solution of the problem (2.1), (2.2~) with exactly I zeros in (a, b). Replacing the second equality in (2.39) by ~o(a) = ~o10(a) + r and repeating the reasoning, we show that this problem has another solution that, in contrast to u, is negative near the point a. The theorem is now proved. By a slight modification of the proof of the preceding theorem and an application of Theorems 1.61 and 1.62, we verify the following propositions.
2373
THEOREM 2.6. Let i ~ {I,2}, let m be a natural number, and let the inequalities (2.18), (2.19), (2.30), and (2.31) hold on the set Ca, b) x R ' , where g,,hj e K~ x R ' ) , (3" = 1,2). Let conditions (2.11), (2.17), and (2.29) hold and let the function ho : (a, b) x R+ --+ R+ be nondecreasing on the second argument and satisfy (2.32). Then for any integer I E [0,m) the problem (2.1), (2.2~) has at least two solutions with exactly I zeros in the interval (a, b). THEOREM 2.7. (a, b) x R ~. Let
Let i E {1,2}, let k be a natural number, and let inequalities (2.9), (2.10) hold on the set [f(t,x,y) - h l ( t ) x - h2(t,x,y)y]sgnx >1 -ho(t,[xl)
and
h21 (t) ~< h2(t,T.,y ) < h22 (t), where gj,h, e K~
x It'), (3" = 1,2). Let conditions (2.11) and (2.28) hold and
(h1,h21,h22)
E
Vio((a, b)) ,
and let the function h0 : (a, b) x R + --, R + be nondecreasing on the second argument and satisfy (2.32). Then, for any integer I E [0, k) the problem (2.1), (2.2~) has at least two solutions with exactly I zeros in
the interval (a, b). As an example of an equation to which the results of this subsection apply, consider the equation
u" = g(t) sin u,
(2.43)
where the function g : (a, b) -4 R is such that
~ b(b
-
a)(b -
t) 2-'
IgCt)l
dt
<
-Foo,
and i E {1, 2}. We denote by m the number of zeros of the solution of the linear problem ,;' = g(t),,,
,~(a+) = O,
u'(a+)
= 1
in the interval (a, b) (such a solution exists by virtue of Lemma 1.1). Then by Theorem 2.6 for any integer l ~ [O,m) the problem (2.43), (2.2,) has a solution with exactly l zeros in (a,b). Taking account of the propositions of Subsection 1.5, we can obtain a variety of corollaries of Theorems 2.1-2.7 by replacing the requirement that the vector-valued functions belong to the sets V~((a, b)) or ~k ((a, b)), (i = 1, 2; k = 0 , 1 , . . . ) by effective conditions guaranteeing that these requirements are satisfied.
We also apply the comparison method described in w167and 2 to two-point boundary-value problems for two-dimensional singular differential systems [97]. A. G. Lomtatidze [33, 34, 37] used this same method to study the three-point boundary-value problem
u(a+) =
O,
u(to) = u(b-),
for Eq. (2.1), where to e (a,b). In conclusion we note that questions of numerical solution of singular two-point boundary-value problems are beyond the scope of the present work. We refer the reader interested in these questions to the articles of Jamet [79] and G. S. Tabidze [53,54]. w
E q u a t i o n s w i t h N o n l i n e a r i t i e s of B e r n s h t e i n - N a g u m o
Type
In this section, as in the preceding, we study the boundary-value problems
u ' = f(t,u,u'), u ( a 3 L) : Cl,
2374
u (i-1) ( b - ) -~- c 2
(3.1) (3.2)
where i 6 {1, 2}, el, e2 E It. In addition we assume throughout that
f e Kloc((a,b) x ITS). The cases when
f E Kloc(Ca, b] x It=) will be explicitly stated. 3.1. Lemmas
For any r 6 It+ we set
on A Priori Estimates.
f 0 sgny
T/r (y)
/
DEFINITION 3.1. A function w : I t --+ (0, +cr
for lyl-< r, for lYl > r.
is called a Nagumo function if it is continuous and
fo+~176 dY = fo+~176 dY -- +cr w(y) co'(=--y)
(3.3)
LEMMA 3.1. Let ro and r be nonnegative numbers, h : [a, b] --+ It+ a summable function, and w a Nagumo function. Then there is a positive constant r* such that for any number j 6 {1, 2}, any closed interval [h,t2] c [a,b], and any function u e Cl([tl,t2]) the inequalities
(-1)i-lu"(t)n,(u'(t)) <~w(u'(t))(h(t) + N'(t)l) for tl <~ t <~ t2, lu(t)l ~< ro
(3.4)
for t, ~< t ~< t=,
(3.5)
and
I='(tr
(3.6)
< r
imply I~"(t)l < r* PROOF:
for tl ~< t ~< t2.
(3.7)
Let 12i(y)= f f
dz ) w((_l)iz
(i=0,1),
12(y)=min{12o(y),12,(y)}.
In view of (3.3) the function 12 : R+ -* R+ has an inverse 12-1 : It+ --* It+. Set ,*
(/:
= n-'
)
hCt) dt + 2r0 + n0(r) + n l (r) .
(3.S)
Suppose the lemma is false. Then there exist [tl,t2] c [a,b], j = {1,2}, t* e (t,,t2) and a function u 6 ~1 ([tl,t=]) satisfying the inequalities (3.4)-(3.6) and such that
I='(t')l > r*.
(3.9)
For definiteness let j = 1. Then, by (3.6) and (3.9), for some t. 6 (t,,t*) and i 6 {0,1} we shall have (-1)'u'Ct)>r
fort,
I,,'(t,)l=r.
According to (3.4)
I~'(t)l' ~((-1)'l~'(t)l)
<.hCt)+(-1)'u'(t)
fort,
Integrating both sides of this inequality from t, to t* and taking account of (3.5), we obtain
t* a(tu'(t*)l)
~ab h(t)at+N(t*)-~(t,)l<.ao(r)+al(r)+
h(t)~+2ro.
,I t .
From this, in view of (3.8),
l='(t*)l< r*, which contradicts (3.9). The lemma is now proved.
2375
LEMMA 3.2. Let r0 and r be nonnegative numbers, h : [a, b] --, R+ a summable function, and w a Nagumo function. Then there exists a positive constant r* such that for any closed interval [tl ,t2] C [a, b] and any function u E Cl([tl,t2]) the inequalities (3.5),
u"(t)n,(lu'(t)l)sgnu(t ) >. -w(u'(t))(h(t)
+ lu'(t)l)
for tl ~< t ~< t2
(3.10)
and u(t,)rl,(u'(t,)) >>.0,
u(t,)~,(u'(t2)) <<.0
(3.11)
imply the estimate (3.7).
PROOF: Choose a positive constant r* such that the assertion of Lemma 3.1 holds. Let u ~ Cl([tl,t2]) and let conditions (3.5), (3.10), and (3.11) hold. If to ~ (tl,t2) and u'(to) ~ O, then two cases can hold: either (3.12) u(to)u'(to) >1 O, or
(3.13)
u(to)u'(to) < O.
Suppose (3.12) holds. Then according to (3.11) there is a point t* E (t0,t2] such that uCt)u'Ct ) > 0 for to < t < t*,
lu'(t*)l < r.
Then (3.10) implies the inequality for to < t < t * .
u"(t)rl,(u'(t)) >>.-w(u'(t))(h(t) + lu'Ct)l)
From this, in view of the choice of the constant r*, we obtain lu'(t0)l ~< r*. In a completely analogous manner one can show that this inequality holds also in the case when (3.13) holds. The lemma is now proved. LEMMA 3.3. Let a < a < ao < bo < ~ < b, and let ro a n d r benonnegativenumbers, h : [ a , b ] - + R + a sumrnable function, and w a Nagumo function. Then there is a positive constant r* such that for any points tl e [a,a], t2 E [/~,b] and any function u 9 Cl([tl,t2] ) the inequalities (3.5), u'(t)Tlr(ul(t))sgn(t--ao)<xw(u'(t))h(t)+lu~(t)l)
fortE(tl,ao)U(bo,t2)
(3.14)
and
u"(t),Tr(lu'(t)l)sgnu(t ) >>.-w(u'(t))(h(t) + lu'(t)l) for ot < t
(3.15)
imply the estimate (3.7).
PROOF: Without loss of generality we may assume that { r ~ m a ~
2ro ao-a
~
2ro ) ~-bo
.
Choose a positive number r* so that the assertions of Lemmas 3.1 and 3.2 hold. According to (3.5) there exist points t, E (a, a0) and t* E (b0, ~) such that
lu'Ct,)l < ao---~2r0 a < r,
lu'Ct*)l < -----~o fl 2r0 < r.
By the choice of r*, the inequalities (3.5), (3.14), and (3.16) imply the estimate
lu'(t)l < r*
for t e [tl,t, lU[t*,tz ],
and the inequalities (3.5), (3.15), and (3.16) imply the estimate lu'(t)l <~ r*
The lernma is now proved. 2376
fort, <<. t <~ t*.
(3.16)
LEMMA 3.4. Let )~ e [0,1), let/z >/0, h2 >~ 0, r0 >1O, be constants. Let ho E Llor ((a,b)) and hl e L([a,b]) be nonnegative functions with ho summable with respect to the-weight (t - a)(b - t) -~ . Then there exists a nonnegative function r* E C((a, b]) [7 L([a, b]) such that (b - t) -~ r* (t) -+ 0 as t -* b, and the estimate ~< r*(t)
Ir
for tl < t ~< t~
(3.17)
holds whenever [tl,t~] C [a, b] and the function u e Cl([tl,t2]) satisfies, together with (3.5), the conditions [ t_-L--~+ ~ b#-~-t + h a ( t ) ] lu'(t)l-h,
u"(t)sgnu'(t) > ' - h ~
lu'Ct)l 2 f o r t l < t
(3.18)
and
u'(t,) = o. PROOF: Set
(/:
l = exp and
r*(t) = l ( b - t ) ~ ( t - a )
-~
(3.1o)
hi (r) dr + 2h2 ro
~
)
b (b-r)-"(r-a)~ho(r)dr.
In view of the summability of h0 with weight (t - a)(b - t) -~ it is clear that r* e C((a,b))NL([a,b]). Let t be an arbitrary point of the interval (tl,t2) at which u' is different from zero. Then, by (3.19), there exists a point t* E (tl,t2] such that
u'(s) •0
fort < s < t * ,
u'(t*) = 0 .
(3.20)
Therefore from (3.18) we have
for t < s < t*
~> -ho(s) - gCs)lr
Ir and
]u'(t)] <~ ftC exp ( f , ~ g ( s ) d s ) h o ( r ) d r ,
(3.21)
where
g(S)-s_a
+
~ b-s
+hiC~)+h=lu'(s)l.
However, according to (3.5) and (3.20),
g(s)ds=Alnr-a+#ln t-a
-
+
hl(s) d s + h , ] u ( r ) - u ( t ) ] < < . A l n r - a t-a
+/~ln
b-t b-r
+ lnl,
by virtue of which we obtain [u'(t)[ ~< r*(t) from (3.21). The lemma is now proved. Analogously we can prove LEMMA 3.5. Let )~ E [0,1), let ro ~ O, r ~ O, and h2 >1 0 be constants, and let ho E L~or and hi E L([a, b]) be nonnegative functions with ho summable with respect to the weight t - a . Then there exists a nonnegative function r* e ((Ca, b])~L([a,b]) such that estimate (3.17) holds whenever [tl,t2] c [a,b] and the function u E Ol([tl,t2]) satisties, together with (3.5), the conditions
u" (t)rl,(u'(t)) >~ -ho(t) -
t - a + hi (t)
lu'(t)[- h2lu'(t)l'
for tt < t < t2
a~,2d
lu'Ct,)l < r.
Lemmas 3.3 and 3.5 imply
2377
LEMMA 3.6. Let a < a < ao < b0 < / ~ < b, A E [0,b - a), let r0 >1 O, r >>.O, h2 >1 0 be constants and h o e Lloc((a,b)) and h I e L([a,b]) nonnegative functions with ho summable with respect to the weight ( t - a ) • ( b - t ) . Then there exists a nonnegative function r* E C ( (a,b) ) n L([a,b]) such that estimate (3.17) holds whenever tl E (a,a], t2 E [/~,b) and the function u E Cl([tl,t~]), satisaes, together with (3.5), the conditions
u"(t)rl, Cu'(t))sgn(t-ao) <~ho(t) + ( t - a ) ( b - t )
+ hi(t) I~"(t)l + h~l~'Ct)l ~
for t e (tl,ao) U(bo,t2)
and ,.,"(t)~,.(I.~'(t)l) sgn,~(t)/> -~(t)
3.2. L o w e r a n d U p p e r problems of the form (3.1), be quite convenient. These [14, 30, 75]. The definitions
-
[
]
(t- a)(b- t) + h~(t) I'~'(t)l- h~l,~'Ct)l"*
for ~ < t < Z.
F u n c t i o n s . T h e S c o r z a D r a g o n i L e m m a . In studying boundary-value (3.2i), (i : 1, 2), the concepts of upper and lower functions have turned out to concepts were introduced by Nagumo [88] and have been further developed in presented below are taken from [14].
DEFINITION 3.2. A function s : (a, b) ~ 1t is called a lower (resp. upper) function of Eq. (3.1) if 1) s is locally absolutely continuous 1 and s' admits the representation s'(t) : a(t) + a(t), where a : (a,b) --* R is locally absolutely continuous and a : (a, b) --* R is a nondecreasing (resp. nonincreasing) function whose derivative is equal to zero almost everywhere; 2) the inequality
f(t,s(t),s'(t)) <~s"(t)
(resp. f(t,s(t),s'(t)) >1 s"(t))
holds almost everywhere on (a, b). DEFINITION 3.3. Let i e {1, 2} and let s be a lower (resp. upper) function of Eq. (3.1) having finite limits s(a+) and s ('-1} (b-) with 8(a7t-) < c1,
8(i-ll
(b-) ~< c2 (resp. n(aq-) /> Cl,
3 (i-1)
(b-) >/c2).
Then s is called a lower (resp.upper) function of the problem (3.1), (3.2,). The following lemma holds; it is a simple modification of the Scorza Dragoni Theorem (cf. [51], p. 110). LEMMA 3.7. Let i E {1,2} and suppose there exist a lower function Sl and an upper function s~ of the problem (3.1), (3.2i) such that Sl(t ) • s2(t ) for a < t < b (3.22) and
If(t,x,y)l<.f*(t)
fora
slCt)<.x<.s2(t),
yER,
(3.23)
where f* E L([a,b]). Then the problem (3.1), (3.2i) has a solution u satisfying the condition slCt) ~
fora
(3.24)
PROOF: Set
wCt.,)
=sup{IfCt.x.y)-fCt.x.z)l" Ixl+lyl < 1+ ~
Is~i-`l (t)l. ly-zl < 9
j,k=l
1That is, absolutely continuous on each closed subinterval of (a, b). 2378
for a < t < b,O <<.p <~l.
It is clear that w : (a, b) • [0,1] --* It+ satisfies the Carath~odory conditions, that it is nondecreasing on the second argument, that w(t, 0) = 0, and
lfCt, si(t),s:,(t))-f(t,
si(t),y)l<w(t,
ly-s:,(t)l ) f o r a < t < b ,
ly-s:,(t)l<.l
(j=l,2)
(3.25)
We construct a function ] : (a, b) x R 2 --* R, by setting
f(t, s2(t),y)+w(t, ](t,x,y) =
f(t,x,y) f(t, sl(t),y)-w(t,
-x-s2(t)+ 1)
forx>s2(t), for 81(t) ~
s , -sx(t)-x ~ --- x + 1 )
X ~<
a2(t),
(3.26)
for x < ,sl (t).
By inequality (,3.23) and the summability of the function f* it is clear that ] is measurable on the first argument and ,continuous on the second and third and that sup{I]C.,x,Y)l
:
x,y 9 R}
9
L([a,b]).
Thus if we carry out a change of variable in the equation
u ' = ](t,u,u')
(3.27)
to make the boundary conditions (3.2~) homogeneous and apply L e m m a 2.1, we verify that the problem (3.27)~ (3.2i) has a solution u. To finish the proof of the lemma it remains to show that u satisfies condition (3.24). According to Definition 3.2, for each k 9 {I, 2} we have
81k(t) = Otk(t) "JV (--1) k-llyk(t), where ak : (a,b) ~ t t is locally absolutely continuous and ak : whose derivative is zero almost everywhere. Let
zk(t)
(a,b)
--* I t is a nondecreasing function
= (-1)k[u(t) - sk(t)].
Then
zk(a+) ~ 0 (k = 1,2),
(3.28)
(4-1) (b-) ~< 0 (k-- 1,2), z~
(3.29)
and
z~(t):fkCt)-l-akCt)
(k=1,2),
(3.30)
where ~'k(t) -- ( - 1 ) k [ u ' ( t ) - o~ (t)]. Now assume that condition (3.24) does not hold. Then by (3.28) there exist j 6 {1,2} and tl 6 (a,b) such that z$.(tl) > 0, z;.(tl) > 0, (3.31) with a 1 continuous at the point tl. Since aj is nondecreasing and fj is continuous, conditions (3.29)-(3.31) guarantee that there exists a point t2 E (tl, b] such that zj(t)>0, z;(t)>0 f o r t , < t < t 2
2379
and
,; (t,-) =o.
(3.32)
Without loss of generality we may assume that
~i(t)
o
fort1 < t < t 2 .
1 § zj(t)
Then, according to (3.25) and (3.26),
(
fj(t) = (--1)J [u" (t) -- sj (t)] = (--1)J[f(t, sj(t),ul(t)) - s~(t)] + w t, 1 + zy(t)
)
>1 (-1)J[f(t, sj(t),s~(t)) - s~ (t)] - w(t,z~(t)) -t- w(t, z~(t)) >1 0 for t I < t < t 2 . Consequently fj is nondecreasing on [tz, t=). Therefore we conclude from (3.30) that z~ is also nondecreasing on the indicated interval and 2~;(t,--) ~ z~Ctl) > 0.
But this contradicts equality (3.32). The lemma is now proved. 3.3. E x i s t e n c e T h e o r e m s . THEOREM 3.11. Let sx and s= be lower and upper functions for the problem (3.1), (3.21) satisfying inequality (3.22). Further suppose
fCt, z,y)sgn[Ct-ao)y]
~< ~Cu)lh(t) + lul]
for t E Ca, ao)U(bo,b),
slCt) <<. z <<. s2(t),
lYl>r
(3.33)
and
f(t,z,y)sgnz
>1 -oa(y)[h(t) § lyl]
for ~ < t < ~,
sic t) ~ X ~
8,(t),
I~1 > r,
(3.34)
where r E R + , a < a < ao < b0 < /~ < b, h : [a,b] -* R+ is a summable function and oJ is a Nagumo function. Then the problem (3.1), (3.21) has a solution satisfying condition (3.24). PROOF: Set
r0 : sup{Isl(t)[ § [s2(t)[: a < t < b}.
(3.35)
Obviously without loss of generality we may assume that r/> max
,
2r~ / ~=bo
.
(3.36)
Denote by r* a constant for which the conclusion of L e m m a 3.3 holds and consider the equation (3.37)
u" = x ( t , u ' ) f ( t , u , u ' ) , where
•
1
lY[
2
p~)
0
for I~1 ~ p(t), for p(t) < lYl < 2p(t),
for I~1 ~ 2p(t),
and
p(t) = r* § Is~(t)l § Is~(t)l. Let
tx. E (a,~), 2380
t2n E (/~,b)
and Cjn E [Sl(tjn),S2(tjn)]
(j = 1,2;r~ : 1 , 2 . . . ) ,
with tl, ~a,
t2. ~ b ,
c1. ~c~.
asn~+oo
(j=l,2).
By Lemma 3.7 for any natural number n Eq. (3.37) has a solution u, such that
un(ti=)=ci, (]=1,2), sl(t) <<.u,(t) <~s2(t) fortl. <.. t <<.t2.. (3.38) In view of (3.33)-(3.35) and (3.38) for any natural number n the function u(t) - u,(t) satisfies inequalities (3.5), (3.14), and (3.15), where ti = t l , , t2 = t2,. Therefore by virtue of the choice of the constant r* ]u~(t)[ ~< r* for tin ~< t <~ t2, (n = 1 , 2 , . . . ) (3.39) and consequently each u , is a solution of Eq. (3.1) on the closed interval [tin, t2, ]. On the other hand lu,(t)-cl~t<~r*(t-a), [u,(t)-c2,]<<.r*(b-t) fort1, <~t<<.t2, ( n = l , 2 , . . . ) . (3.40) According to Lemma 2.2 inequalities (3.39) and (3.40) guarantee the existence of a subsequence I t . m )re=l, +co +er uniformly convergent together with (I/t, . ),,,=1 on each closed subinterval of (a, b), whose limit uCt) =
lim
ra--*+oo
u.. (t)
is a solution of Eq. (3.1) on (a,b). In view of (3.38) and (3.40) it is clear that u satisfies the conditions (3.21) and (3.24). The theorem is now proved. REMARK. It is easy to see that the assertion of Theorem 3.11 holds also in the case when instead of (3.34) the inequality
f ( t , z , y ) s g n [ ( t - t o ) y ] ~> - w ( y ) [ h ( t ) + lyl] for ~ < t < ~, holds, where t0 is a point of the closed interval [a, ~].
sl(t) .< 9 .< s2(t),
lyl > r,
THEOREM 3.12. Let Sl and 82 be lower and upper functions for the problem (3.1), (3.22) satisfying the inequality (3.22). Further suppose f 6 Klor ((a, b] • R 2) and
f ( t , z , y ) s g n y >~ -w(y)[h(t) + lYl] for a < t < b, sl(t) ~< x ~< s2(t), [Yl > r, where r E R + , h : [a, b] ~ It+ is a summable function, and w is a Nagumo function. Then the problem (3.1), (3.22) has a solution satisfying condition (3.24). THEOREM 3.21. Let 81 and 82 be lower and upper functions for the problem (3.1), (3.21) satisfying inequality (3.22). Further suppose A f ( t , x , y ) = sgn [(t - a0)y] ~< ho(t) + [(t - a)(b - t) + hx(t)]lYl + h2Y 2
forte(a, ao)U(bo,b ),
s2(t) <<.z<~ s2(t),
lYl>r
and A
f ( t , x , y ) s g n x >~ -ho(t) - -[ (t - a ) ( b - t) + hi (t)] lYl
-
h2Y 2
forar, where a < a < ar < b0 t O, h2 >>.O, and ho 6 Llor and h, E L([a,b]) are nonnegative functions with ho summable with respect to the weight ( t - a ) ( b - t). Then the problem (3.1), (3.21) has a solution satisfying condition (3.24). THEOREM 3.22. Let sl and s2 be lower and upper functions for the problem (3.1), (3.22) satisfying the inequality (3.22). Further suppose f 6 K,or (Ca, b] • R 2) and
f ( t , x , y ) s g n y >>.-ho(t)
-
[t -A- a + h l ( t ) ] [ y l - h 2 y
2
fora
sl(t)<,.x<,.s2(t),
]yi>r,
where A 6 [0,1), r /> 0, h2 /> 0, and ho 6 L,or ((a,b]) and hi e L([a,b]) are nonnegative functions with ho summable with respect to the weight t - a. Then the problem (3.1), (3.22) has a solution satisfying condition (3.24). The theorems just stated are proved in a manner similar to the proof of Theorem 3.11. The only difference is that instead of Lemma 3.3 we apply Lemmas 3.1, 3.6, and 3.5 respectively.
2381
THEOREM 3.3. Let 31 and 32 be lower and upper functions for the problem (3.1), (3.2~) satisfying inequality (3.22), with either i = 1 and cl = c2 = 0 or i -- 2, cl = 0 and f E Kloc ((a, bI • R2). Further suppose fCt, x , y ) s g n z >/-wCy)[h(t) + lYll for a < t < b, slot) ~< x ~< s2(t), lYl/> r, (3.41) where 9 E R+, h : Ca, b) --* R+ is a summable function, and w is a Nagumo function. Then the problem (3.1), (3.2~) has a solution satisfying condition (3.24).
PROOF: We shall carry out the proof for the case i = 1. Let r0 be the number given by equality (3.35) and r* > 0 the constant occurring in Lemma 3.2. By +oo (k = 1, 2) such that Definition 3.3 there exist sequences (tk,,)~=l, a < tin < t2n < b,
and 1. 2. 3.
(n = 1 , 2 , . . . ) ,
lim tin = a, n-*+oo
lim t~, = b n---~+oo
for any j E {1, 2} one of the following three conditions holds: slCtj~)<~O<~s2(tjn) (n=1,2,...); slCtj~)>0, (-1)J-ls~(tjn)>0 ( n = 1, 2 , . . . ) , lirn,,__.+oo 81 (tin) : O; S2(tjtt) < 0, (--1) j - l s ~ C t j n ) < 0 (tt : 1 , 2 , . . . ) , lim~-.+oo s2(tj,,) = 0. Let 0 in case 1, cj, =
slCtj~)
in c a s e 2 ( j = l , 2 ; n = l ,
32 (ti.)
in case 3.
2,...),
By Lemma 3.7 for any natural number n Eq. (3.37) has a solution u~ satisfying conditions (3.38), where X and p are the same as in the proof of Theorem 3.11. It is obvious that un(tln)u:Ctln) ~ 0 ,
unCt2n)u:Ct2n) <<.0 (n----1,2,...).
(3.42)
By (3.35), (3.38), (3.41), and (3.42), for any natural number n the function u(t) - u=(t) satisfies inequalities (3.5), (3.10), and (3.11). Therefore by the choice of the constant r* estimates (3.39) hold. It remains only to repeat word for word the last part of the proof of Theorem 3.11. THEOREM 3.4. Let c2 = 0 and let 31 and 32 be lower and upper functions for the problem (3.1), (3.22) satisfying inequality (3.22), where for some sequence b, E (a, b), n = (1, 2 , . . . ), converging to b we have (n = 1 , 2 , . . . ) .
s~(bn-)<<.O<<.s~(b=-)
(3.43)
Further suppose
f(t, ,y)sgn
-h0(t)-
+~
+ hlCt)] lyl-
h .y 2 for a < t < b
Sl(t) .< x .< s2(t),
y E R,
(3.44)
where ~ E [0,1),/z/> 0, h2 /> 0, and ho e Lloc ((a,b)) and hi E LC[a,b]) are nonnegative functions with h0 summable with respect to the weight/'unction (t - a)(b - t) -~ . Then Eq. (3.1) has a solution satisfying conditions (3.24) and u(a+) = cl, ~ i m ( b - t ) - " u ' ( t ) = O . (3.45)
PROOF: According to conditions (3.43) and (3.44) and Theorem 3.22 for any natural number n Eq. (3.1) has a solution u,, : Ca, b,, ] --* R such that
,,'(b,,) =o, 2382
lCt) .<
.<
for
< t .< b,,
(3.46)
and
"
u~(t)sgnu~Ct) >i - h ~
t
[t-~ a + b-~ +
hlCt)]lu'Ct)l-
I r Ct)l ~< r* (t)
for a < t <<.bn,
h21"'Ct)12
for a < t ~< bn.
By Lemma 3.4 (3.47)
where r* E C((a,b])NL([a,b]) is a function independent of n with limt-.b (b - t)-~ r*(t) = O. According to Lemma 2.2 it follows from (3.46) and (3.47) that there exists a solution u : (a, b) --* R satisfying, together with (3.24), the inequalities
lu'(t)l<~r*(t),
lu(t)-exl<
r'Cr)dr
fora
From this it is clear that u also satisfies conditions (3.45). The theorem is now proved. By the same method one can prove the assertion of Theorem 3.12 in the case when f r Klor ((a, b] • R 2) but c2 = r = 0 and inequalities (3.43) hold for some sequence (bn)+~ C (a, b) converging to b. We remark that the singular problem (3.1), (3.21) has also been studied in the case when the right-hand side of Eq. (3.1) is discontinuous on the phase variables [62]. Criteria for solvability in a generalized sense of the problem (3.1), (3.22) were established by Tsepitis [57]. Results for three-point problems analogous to those presented in this section have been obtained by Lomtatidze [38]. It is appropriate to mention at this point the papers of Vasil'ev and Lomakina [5], Gaprindashvili [6], Kiguradze [19], and Lepin [29] devoted to singular boundary-value problems for differential equations of higher orders and systems of second-order differential equations.
3.4. U n i q u e n e s s T h e o r e m s . THEOREM 3.51. Let the function f be nondecreasing on the second argument and satisfy for every r E R+ the condition
[fCt, x, y l ) - f ( t , x , y2)]sgn(yl-y2) >1 -lr(t)lYl-Y2[
for a < t < b,
Izl < r,
lYil < r
(j = 1,2), (3.48)
where lr E Llor ((a,b)). Then the problem (3.1), (3.21) has at most one solution. PROOF: Suppose the problem (3.1), (3.21) has two distinct solutions Ul and u2. Then without loss of generality we may assume that the function u(t) = Ul (t) - u2 (t) is positive at some point of the interval (a,b). Since u(a+) = u ( b - ) = 0, there exist tl e (a,b) and t, e (tl,b) such that u(tx) > 0 ,
u'(t) > 0
fortx ~ < t < t 2
and
(3.49)
u' (t2) = 0
Since f is nondecreasing on the second argument, we obtain from (3.48)
u"(t) = f ( t , ux(t),u~(t)) - fCt, u2Ct),u~(t)) + f(t, u2Ct),u~(t))- f ( t , u2(t),u~(t)) ~> - t r C t W ( t )
where r = max
for tl < t < t2,
}
ujCt)l + I,,)(t)l): tl < t < t2 . "j----1
Then,
u'(t2)>/u'(tl)exp(-ft:21r(r)dr)
> O,
which contradicts condition (3.49). The theorem is now proved. The following proposition is proved in an analogous manner.
2383
THEOREM 3.52. Let the function f be nondecreasing on the second argument and satisfy condition (3.48) for any r e a + , where t, e L,oo ((a,b)) and lim s u p /Jtt o l ' ( r ) d r < + c ~ t-.b Then the problem
(3.t), (3.22)
fora
has at most one solution.
THEOREM 3.6. Let # E R + , and let the function f be nondecreasing on the second argument and satisfy condition (3.48) for any r E R + , where l, E Lloc((a,b)) and liminf(b-t) -~exp
t--*b
(fl) -
lr(r) dr
>0
fora
Then the problem (3.1), (3.45) has at most one solution. w
Equations Having Singularities with Respect to a Phase Variable
In this section we consider the question of the existence and uniqueness of a solution of a differential equation u"= f(t,u,u') (4.1) satisfying the conditions u(a+)--O,
u ('-x) (b-) -- 0,
u(t)>O
fora
(4.2,)
where i E {1, 2}. Throughout the following it is assumed that f E K]oc((a,b) x (0,+co) x R ) and f(t,x,y)<~O
fora
x>0,
yER,
(4.3)
and the possibility is admitted that f may have singularities on both the independent variable (at the points a and b) and on the first phase variable (at the point 0). A typical representative of the class of equations considered in this section is the Emden-Fowler equation u" = h(t)u -x , (4.4) where )t > 0 and h E Lloc ((a, b)) is a nonpositive function. Special cases of (4.4) are the equations mentioned in the introduction t2 v," (4.5) 32u 2 and
ut'--
1--t
(4.6)
u
4.1. O s c i l l a t i o n L e m m a s . In this subsection we present some propositions on the oscillation properties of the equation v" = g(t)v, (4.7) where the function g : (a, b) --* R_ is summable either with weight (t - a) (b - t) or with weight t - a. These properties will be needed for the exposition that follows. We denote by v~ the solution of Eq. (4.7) satisfying the initial conditions
vCa+) =0,
r
1.
The values of ve and v P 8 at the point b will be taken as their left-hand limits. An immediate corollary of Lemmas 1.51 and 1.52 is 2384
LEMMA 4 . 1 .
Let i E {1,2} and
~
b(t - ~ ) ( b - t)~-i Ig(t)l dt ~< ( b - ~)2-i,
Then vg(i-1) has no zeros in (a, b]. Integral criteria for vg and v~ to have at least one zero in (a, b] were established by Korshinkova [24, 25] and Lomtatidze [36]. We present two lemmas from [36]. LEMMA 4.21. Let g : (a, b) --+ R _ be summable with respect to the weight (t - a)(b - t) and
1 t-a
t(~'-a)'(b-r)lg(r)ldr+-gi-~
(r-a)(b-r)'lgC~)ldr>~b-a
fora
(4.8)
Then vg has at least one zero in (a, b]. PROOF: Suppose, to the contrary, that vg(t) > 0
fora
Then according to Theorem 1.1 and the nonpositivity of the function g we have
(r
vg(t ) = 6(b - a)(t - a) + ~b -- at
- ~)lgCr)lv~Cr) dr + ~
(b -
r)lg(r)l,~ (~)
dr,
where 6 = vg(b)(b - a) -2 > 0. Setting
(t-a)(b-t)
r=inf
:a
and taking account of condition (4.8), we find from the last inequality that r/> 6 + r. The contradiction so obtained proves the lemma. Analogously we prove LEMMA 4.22. Let g : (a, b) ~ It_ be summable with weight t - a and
~a
b( r
-- a) 219(r)1 dr
>~
b- a
(4.9)
Then vg' has at least one zero in (a, b]. 4.2. E x i s t e n c e a n d U n i q u e n e s s T h e o r e m s . THEOREM 4.11. Suppose for any 6 E (0, 1) there are summable functions ljs : [a, b] --* R + , (j : 0,1), and constants I~s E [0,b - a), 126 >/0 and r6 ~> 0 such that
[f(t,x,y)[ <~ ( t - alob(t) )(b-t)
+ [ ( t - a ) (#tb - t )
+ l i t ( t ) ] ly[+l~ty2 fora
1
6~
lYl>rt.
(4.10)
Further suppose that for some 60 E (0, 1) the inequalities f ( t , ~ , y ) .< g(t)~
for a < t < b,
O < x < ~o, y E R
(4.11)
2385
and
f ( t , z , tl)>>-h(t)z
fora
1 b0'
z>
yER
(4.12)
hold, where g and h : (a,b) -~ R_ are summable with weight (t - a)(b - t), v~ having at least one zero on (a,b] and vh having no zeros on (a,b]. Then the problem (4.1), (4.21) is solvable. PROOF: By Theorem 1.1 the equation
v" = h(t)v has a solution v such that
1 vCa+) = vCb-) = -~'
v(t) >
1
-~o
for a < t < b.
In view of (4.12) it is obvious that v is an upper function for Eq. (4.1). According to L e m m a 4.1, for any sufficiently small r > 0 we have
fora
v,a(t) > 0
We denote the least upper bound of the set of such r by c0. Since v~ has at least one zero in (a, b], it is clear that ~ C (0,1] and vtoa(t)>0 fora
w(t) = e0V, o,(t) < 8o for a < t ~< b. Then by (4.11) w is a lower function for Eq. (4.1). On the other hand since
wCa+)-'w(b-)=o,
w(t)>0,
w"Ct)~
fora
there exists a point to E (a, b) such that
w'(t) >1 0 Let
for a < t <<.to,
w'(t) <. 0 for to ~< t < b.
1
~/= ~ min{to - a,b - to},
t~,,
tl,, = a + ~
(4.13)
= b
n
l't
and
w,(t) =
w(tln)
for a ~< t ~< tl,,,
wCt) w(t2,)
fort1,
Cn=1,2,...),
Taking account of inequalities (4.3) and (4.13), we easily conclude that, for each natural number n, w, is a lower function for Eq. (4.1), that
w(t) <~ w~+x (t) <~ w~(t) < v(t)
for a < t < b
(n= 1,2,...)
(4.14)
and that lira w , ( a ) - n---~-~ OO
lira w , ( b ) = 0 .
(4.15)
lrt.---~ -4- o o
If p = sup{v(t) : a < t < b} and b = min{wlCa),wl(b), 1}, then obviously on the set { ( t , z , y ) : a < r
t < b, wl(t) <~ z <<.v(t), [Yl > rs} inequality (4.10) holds. Therefore, according to Theorem 3.21, Eq. (4.1) has a solution ul such that ul(a+)=wlCa), 2386
ulCb-)=wlCb),
wlCt)~
fora
The function Ul, being a solution of Eq. (4.1), is simultaneously an upper function of Eq. (4.1). On the other hand, by (4.14) w2 (t) ~< Ux(t) for a < t < b. Again applying Theorem 3.21, we verify that there exists a solution u2 of Eq. (4.1) such that u2(a+)=w2(a),
u2(b-)=w2(b),
w2Ct)<~ u2(t)<<, ul(t)
fora
q-do of solutions of Eq. (4.1) such that Continuing this process, we obtain a sequence (Un)n=l Un(G-lL) = Wn({~),
Ion(t) <~ un(t) <~ un-I (t)
u n ( b - ) = wn(b),
for a < t < b
(n : 2,3,... ).
(4.16)
From (4.14) and (4.16) we find w(t)<<.u,~(t) < p
(n=1,2,...).
fora
(4.17)
1
Let [t.,t*] be an arbitrary closed subinterval of (a,b) and/~ = min
{w(t.),w(t*),~ }.
Then according
to (4.10) and (4.1~) for each. we have 2
los(t) lu"(t)lm,(iu'(t)l) <<. ( t - ~ - t )
+
[
/~, (t-a)(b-t)
]
q-ll, Ct) I4.(t)l+12, lu'(t)l 2 f o r t , < t < t*. (4.18)
By Lemma 3.6 inequalities (4.17) and (4.18) guarantee that the sequence vt.,,' , J~+oo , = l is uniformly bounded on +oo contains a subsequence It,, t* ]. Consequently the hypotheses of Lemma 2.2 axe satisfied. Therefore (u,,),=l (u,,~)k+~ that is uniformly convergent together with ( u ~ ) + ~ on each closed subinterval of (a,b) and
uCt) = k--*+oo lim u.~Ct) is a solution of Eq. (4.1). O n the other hand, by (4.16), u(t) i> w(t) > 0
for a < t < b
and limsupiu(t)l ~< w,(a),
t-~ a
limsupluCt)t ~<w, Cb) (n = 1,2,...).
t--*b
From this, according to (4.15) it follows that uCa+) : uCb-) : O. Consequently u is a solution of the problem (4.1), (4.21). The theorem is now proved. In an analogous manner we prove THEOREM 4.12. Suppose for any ~ E (0, 1) there exist summable functions ii6 : [a, b] -+ R+, (3" = 0, 1), and constants #3 E (0, 1], 126 >1 0 and r6 >~0 such that
if(t,x,y)l<, lo6(t___~)[ #6t_a+ll~(t)]iyI+I2~y= f o r a < t < b ,
d~<x<~,
lYl>r6"
Further suppose that for some 6o E (0,1) inequalities (4.11) and (4.12) hold, where g and h : (a,b] --~ R_ are summable with weight t - a, v#' h a v i n g at l e ~ t one z e r o in (a,b] a n d v'~ h a v i n g no z e r o s on (a, b]. Then the problem (4.1), (4.25) is solvable. The proof of the following theorem is analogous to that of Theorem 3.51.
2~r~ is the function introduced in w
2387
THEOREM 4.21. Let the function f be nondecreasing on the second argument and suppose that for any r E (1, +co) the function satisfies the condition
lfCt, z , y , ) - f(t,z,y~)l <~l,(t)lyl -y~t
1
fora
-~<~
lYil~
Cj=1,2),
(4.19)
where l, ELloc ((a,b)). Then the problem (4.1), (4.21) has at most one solution.
THEOREM 4.22. Suppose the function f is nondecreasing on the second argument and for any r E (1, +co) satisfies the condition C4.19), where 1, E Lloc CCa,b]). Then the problem (4.1), (4.2~) has at most one solution. To conclude this section we consider the equation (4.20)
u " = fCt, u),
where f E Kloe(Ca, b) )< (0,+col),
and f ( t , z ) <<.O f o r a < t < b ,
z>0.
C4.21)
Set
f; (t) -- m.
{[/'(t,x)l : b ~< x ~< ~ }
for 5 E C0,1).
Theorems 4.11 and 4.12 assume the following form for Eq. (4.20). COROLLARY 4.1. Let i E {1,2}, f,b(t-a)(b-t)2-if~(t)dt<+co
forbE(0,1)
(4.22)
and for any ~0 E Co, 1) Iet the following inequalities be satisfied: f(t,z)<<.g(t)z
fora
and
f ( t , z ) >>.h(t)z
fora
0<x<~0,
(4.23)
1 z > ~0'
(4.24)
where g and h: (a,b) ---+R_ are summable with weight (t - a)(b - t) 2-' , ve(i-1) having at least one zero on Ca, b] and vh(,-1) having no zeros on (a,b]. Then the problem (4.20), (4.2i) is solvable.
THEOREM 4.31. Let f be nondecreasing on the second argument and for some Zo E (0, 1) let rues {t E (a,b) : f(t, zo) < 0} > 0.
(4.25)
Then for the problem (4.20), C4.21) to have a unique solution it is necessary and su~cient that
f
b(t-a)(b-t)lf(t,x)ldt
< +co
for z > 0.
(4.26)
PROOF: We first prove sufficiency. Since f is nonpositive and nondecreasing on the second argument, we have, for any b E (0,1), f;(t) = lf(t,6)l. Therefore condition (4.221 follows from (4.26). 2388
According to (4.25) and (4.26) there exists a number 6o E (0, zo) such that the function g(t) = ~1o f ( t , xo) satisfies condition (4.8) and the function h ( t ) = 6of(t, zo) satisfies the condition
/
b(t - a)(b - t)lh(t)l dt < b - a.
On the other hand, in view of (4.21) and the monotonicity of f in the second argument, it is clear that g and h are nonpositive and that inequalities (4.23) and (4.24) hold. It now becomes obvious, by virtue of Lemmas 4.1 and 4.21 and Corollary 4.1, that the problem (4.20), (4.21) has a solution. The uniqueness of the solution follows from Theorem 4.21. We now turn to the proof of necessity. Suppose the problem (4.20), (4.21) is solvable and u is a solution of it. Then lim inf(t - a)lu'(t)l = 0, lim i n f ( b - t ) N ' ( t ) l - - o. (4.27) t--*a
t--*b
For an arbitrarily given x > 0 we choose a0, b0 E (a, b) such that =(t) < 9
for t e (a, a0l(.J[b0,b).
Then
f~O (r-a)lf(r,x)ldr =
-
f~ tl~( T - a ) l f ( ~ , u ( r ) ) l d r
~
/~176
(~ - ~),,"(,-) d,- = (t -.),~'(t)
- ,~(t) - (o~ - .),~'(o~) +,~(o~)
for a < t < a 0
and
ft
t(b - 01fCr, x)l dr
<. (t - b)u'(t) - u(t) + (b - bo)u'(bo) + u(bo)
forbo < t < b.
0
Therefore, by (4.27) we find
~
'~
<
+~,
f0
b(b - r ) l f ( r , x ) l dr < +oo.
Inequality (4.26) follows from this, since f(.,x) E L,oc ((a,b)). The theorem is now proved. In an analogous manner one can prove THEOREM 4.32. Suppose f is nondecreasing on the second argument f(., x) E L~oe((a, b]) for x > 0 and for some Xo E (0,1] condition (4.25) holds. Then in order for the problem (4.20), (4.22) to have a unique solution it is necessary and sutt~cient that
f , ~ ( t - a)lfCt, x)l dt < + c ~
for x > 0:
A consequence of Theorems 4.31 and 4.3~ is the following result of Taliaferro [100]. COROLLARY 4.2. Let i E {1,2}, )t > O, and let the function h E L~oc ((a, b)) be nonpositive and different from zero on a set o f positive measure. Then the condition
f
~(t - a)(b - Q 2-` IhCt)l dt < +cr
2389
is necessary a n d
sufficient for the problem (4.4), (4.2,)
to
have a unique solution.
From the last proposition, in particular, it follows that problems (4.5), (4.2i) and (4.6), (4.2i), (i -- 1, 2), have unique solutions for a = 0, b = 1 [45, 70, 71]. We note that problems analogous to those considered in this section for higher-order equations have been studied by Kvinikadze [10]. Chapter 2 PROBLEMS w
ON AN INFINITE INTERVAL
P r o b l e m s on B o u n d e d a n d M o n o t o n i c S o l u t i o n s In this section we consider the differential equation
u"= fCt, u,u')
(5.1)
and we study the following problems on bounded and monotonic solutions:
slCt) ~ u(t) ~ 82(t) f o r a < t < b , U(/-1)(a-~):C, s l ( t ) ~ u ( t ) ~ s 2 ( t ) f o r a < t U{/-1} (0-~-) -----C,
where i E {1,2), - o o ~ a < b ~ +r inequality
U(t) ~ 0,
u'(t) ~ 0,
(5.2)
(5.3,)
for t ~ g + ,
(5.4,)
c E R, and ,k : (a,'b) ~ R, (k -- 1,2), are functions satisfying the slit) ~<,$(t)
fora
(5.5)
As was already noted above, closely connected with (5.1), C5.2) and (5.1), (5.3,) are the boundaxy-value problems posed by Logunov and Vlasov [32]: U" ------2--Ul
t
Ul2
+ uCu
-
U -- 2
u 2) + - ( u - '
lim uCt) =1,
u(0+)=2,
t-*+oo
+
uCt)>2
t
~2u
(5.6)
),
fort>0
(5.7)
and
b-t - -
p~
u" -I-( 1 -
uS -u(O-{-) = 2,
u2~2l 2u ( 1 - ~-~) -i(1 ~-, j + (b~)2 ~" u(b-) --- O, O < u(t) < 2 f o r 0 < : t < b ,
~-)(1-
p2/j
'
(5.8) (5.9)
where bs = 2p 2, p i> 2. Special cases of the problems (5.1), (5.4,), (i = 1,2), axe the Thomas-Fermi problem
[~6, 1011 1 3 u" = t-2 u 2 ,
u(0+) = 1,
(5.10)
u(+oo) = 0
(5.11)
and the problem from the theory of capillary phenomena [80]
u"----(1 + u") s/2 (ruu'(0+) = c,
au'
~
Ct + .) +.v"i-~---~~~,' 1 /
u(+oo) = 0,
(5.12) (5.13)
where r > 0, a/> 0, a > 0, and c < 0. 5.1. T h e p r o b l e m s (5.1), (5.2) a n d (5.1), (5.3,), (i = 1,2). Throughout this subsection we assume that the condition f E K, oc ((a,b) • R 2) holds. 2390
THEOREM 5.1. Let s, be a lower function and s2 an upper function for Eq. (5.1). Suppose also that one of the following three inequalities holds I
f(t,x,y)sgnx>>.-w(y)[h,(t)+h~(t)lyl] f(t,x,y)sgn[(t-t0)v]
fora
/> -w(y)[hxCt) + h2(t)lYl]
sl(t)<~x<<.s~(t),
for a < t < b,
[yl>r,
sl(t) <<.x <<.s2(t),
(5.14)
lyl > ~ (5.15)
or
( - 1 ) i-1 f ( t , x , v ) s g n y
>1 -w(v)[hl(t) + h2(t)lYl]
for a < t < b,
where i 9 {1,2}, to 9 (a,b), r 9 R + , h, 9 Llor the problem (5.1), (5.2) is soluble.
sl(t) <~ x <<.s2(t),
IVl > r,
(5.16i)
h2 9 C((a,b)) and w is a Nagumo function) Then
PROOF: We shall carry out the proof for the case when condition (5.14) holds. The other cases are considered analogously. Let (an)+=~ and (b.)+_~ be decreasing and increasing sequences satisfying the conditions
a < a n < b ,
(n=1,2,...),
lim an = a ,
lim b. = b .
n--r + o o
n--r + o o
For any natural number n choose points a.' 9 (a, an) and b.' 9 (b., b) such that
s,(an) + (t --an)sl(an+).< ' s2(an)+ (t --an)4(a.+) 81
(b.)-~-(t
-- b n
)8~ (b.-)
fora,~
~< s2 (b.)+ (t- bn)s~(b.-)
Set
fn(t,~,v) = sk, Ct) =
for t 9 [a.,bn],
f(t,x,y) 0
(5.17)
for t r [a,,b.],
{ *~(an)+ (t- an)4Ca.+)
f o r a n ~
sk(t) sk(bn) + (t - bn)s'k(bn- )
foran < t < b n forb, ~
(k = 1,2),
and c,. = s,.(a'),
c2. = S,n(b').
The functions Sl, and s2, are lower mid upper functions of the problem
,,"=fn(t,,,,,,'), ,,(e+)=cln,
,,(b'.-)=,2..
Since, in addition, conditions (5.14) and (5.17) hold, it follows from Theorem 3.1, that for any n the problem in question has a solution un satisfying the inequalities I
!
sx.Ct)~
It is clear that u , is a solution of Eq. (5.1) on [a,,b,] and
sl(t) <~ u,,(t) <<.s2(t)
for a, ~< t ~< b,
(n = 1 , 2 , . . . ) .
(5.18)
s Regarding the concepts of Nagumo function and upper and lower functions for Eq. (5.1), cf. Definitions 3.1 and 3.2.
2391
Let It,, t*] be an arbitrary closed subinterval of (a, b) and let no be sufficiently large that 5 = rain{t, - ano, bno - t*) > O. We introduce the numbers
po=max{Isl(t)l+ls~(t)l:t,-~<~t<~t
* +6},
2po p = --~---+r.
According to (5.141 and (5.18) for any n >/no there exist points tin E ft, - ~, t, ] and t~n E [ t ' , t* +/i] such that I~n(t)l < p0 for t,n ~< t ~< t2n, lu'(txn)l ~< P, N'(t2n)l ~< p and
u~(t)rl,(lu~(t)l)sgnun(t) >>. -Wo(u'(t))[h(t) + N'(t)l]
for tin < t ~< t~n,
where
Wo(y) = w(y)max{1 + h2(t):t, < t ~< t*},
h(t) = Ihl(t)l.
By Lemma 3.2 the estimates
for t, <. t <. t*
lu'~(t)l
(n=no,no+l,...),
(5.19)
follow from these inequalities, where p* is a constant independent of n. According to Lemma 2.2 estimates (5.18) and (5.19), in view of the arbitrariness of [t,,t*] C (a,b), guarantee that there exists a solution u : (a, b) --, R of Eq. (5.1) satisfying condition (5.2). The theorem is now proved. THEOREM 5.2.
If(t,x, yl) -
Suppose that for any p E It+ the function f satisfies the condition
f ( t , x , y2)l ~< g, Ct)lyx - y~l
for a < t < b,
sl(t) <<.x <<.s2(t),
lY, I < p
(i = 1,2), C5.20)
where g, e L, oc ( (a, b) ). Further suppose there exist to e (a, b) and locally summable functions 11: (a, b) --* R+ and l~ : (a, b) -* R such that
fCt, z x , y l ) -
f(t, x2,y~) >1 ll(t)Czx - x2) +l~Ct)(yl - y 2 ) for a < t < b, Sl(t) <<.Z2 <<.Xl <. s2(t),
(Yl --y2)(t--to) >~0 (5.21)
and lim inf ,-..
s~(t) - s l {t) = l i m inf s2(t) - Sl (t) = 0, 1 + It(t)l ,-~b I + t(t)
{5.22)
where
l(t) = ftl dr ft~ exp ( f ~ 1 2 ( f ) d f ) l x ( ~ ) d ~ 9 Then the problem {5.1), {5.2) has at most one solution. PROOF: Let Ul and u2 be arbitrary solution of the problem under consideration. Set
uCt) = ul (t) - u2Ct). We first prove that
u(t)u'(t) <~0 for to ~< t < b. Suppose, to the contrary, that for some tl E [to, b)
u(tl)u'(tl) > O. 2392
(5.23)
Without loss ,of generality we shall assume that u(t~) > 0 and u'(tl) > 0. We denote by b0 the least upper bound of the set of t2 E (tl, b) for which the inequalities > 0,
> o
hold on the interval [tx,t2). Then by (5.21) we have
u"(t) >1ll(t)u(t) +l~(t)u'(t)
for t, < t < b0
and
u'(t)>/6[exp(/il2(f)d~)+ where ~ = min{u(tl),
u'(ta)}
ftlexp(/tl2(~)df)ll(,)d,]
fortl
<.t
> 0. From this, by the definition of b0, it is obvious that bo = b and
s2(t) - 81(t) >1u(t) >18lo(t)
for tl ~< t < bo,
where /o(t) = 1 + ftl [exp ( f t [ / 2 ( f ) d r ) + f / e x p
12(f) d~)ll (~) d~] dr.
(jft
However, this last inequality contradicts condition (5.22), for 1+
lim ~
,-.b
l(t) to(t)
< +co.
The contradiction so obtained proves inequality (5.23). The inequality
u(t)u'(t) >10
for a < t <~ to
(5.24)
is proved analogously. We now show that
(5.25)
=(t0) = 0 . Suppose the contrary. Then by (5.23) and (5.24) we have =0.
(5.26)
On the other hand we may assume without loss of generality that
u(t) >l O, u'(t) <<.O
for t o ~ < t < b .
(5.27)
Then, in view, of (5.21),
f(t, ux(t),u~x(t)) - f(t, u2(t),u~ (t)) >10
fort0 ~ < t < b .
In addition, according to (5.20), there exists a nonnegative function g 6 L~o~([to, b)) such that
f(t,u~(t),u~l(t)) - f(t, u2(t),u~(t)) >1g(t)u'(t)
for to ~< t < b.
Therefore u"(t)>g(t)u'(t)
forto ~
From this, by (5.26), there follows the inequality u'(t) >/0
for to ~ < t < b , 2393
according to which we find, from (5.21) and (5.27), s,(t)-st(t)
>>.u ( t ) = 6 o
and
&o/t(t)
forto <<.t < b,
where 50 = u(to) > 0. Consequently 11 (t) -- 0, l(t) = 0 for to ~< t < b and l i m i n f s2(t) - st(t) t>/~o > O, t--,b 1 + l(t)
which contradicts condition (5.22). This proves equality (5.25). However it follows from the conditions (5.23)-(5.25) that u(t) :- O. The theorem is now proved. REMARK. Inequality (5.15) follows from (5.21), where w(y) = 1 + [y[, h2(t) = 0, r : 0, and
hi (t) : IZ2(t)l + [f(t, sl(t),O)l + If(t, s2(t),o)l. Therefore if s 1 als.d s 2 are lower and upper functions of Eq. (5.1), then by Theorem 5.1 the condition (5.21) guarantees that the problem (5.1), (5.2) is solvable. Using the results of w by reasoning analogous to that used in proving Theorems 5.1 and 5.2, we can prove the following propositions on the existence and uniqueness of solutions of problems (5.1), (5.3i), (i = 1,2). T H E O R E M 5.31. Let a > - o o , and let st and s2 be lower and upper functions for Eq. (5.1) having finite
limits sk(a+), (k = 1,2), with s l ( a + ) <. e <. s2(a+). Further suppose that either condition (5.161) holds or e = 0 and condition (5.14) holds, or there exist points a E (a,b), ao E (a,b), bo E (ao,b), and fl C (bo,b) such that
f(t,z,y)sgn[Ct-ao)y] <<. w(y)[hl(t) + h2(t)lYl]
for t e (a, ao)U(bo,b ),
Sl(t)
< .~ <
S2(t),
lYl > r,
and
f(t,x,y)sgnx >1 -wCy)[hlCt) + h2(t)ly[] for a < t r, where r e R+, h, e Llor ([a,b)), h, E C([a,b)), and w is a Nagumo function. Then the problem (5.1), (5.31) is solvable. THEOREM 5.32. Let a > - o o and let sl and s2 be lower and upper functions for Eq. (5.1) whose derivatives have finite limits s~ CAW), (k = 1,2), with s~ Ca+) >1 c >1 s~(a+). Further suppose that f E Klor ([a, b) x R 2) and either condition (5.14) holds or condition (5.162) holds, where r e It+, h, e L,or ([a, b)), h~ e C([a,b)) and w is a Nagumo function. Then the problem (5.1), (5.32) is solvable. THEOREM 5.41. Let a > - o o and let sl and s2 be lower and upper functions of Eq. (5.1) having finite limits sk(a+), (k = 1,2), with sl(a+) <. c <. s2 (a+). Further suppose that either f ( t , z , y ) s g n y l>
hoCt) t-a
[ t - a'~ +hlCt)]lY[-h2Ct)y 2 fora
s, Ct)~<x~<s2Ct),
lYl>r,
or there exist points a E (a,b), ao E (a,b), bo E (ao,b) and ~ E (bo,b) such that
hoCt)
. f( t , z , y ) s g n [ C t- a o ) y] <. ~ - a +
[
1
t - a + hl(t)J lyl + h2(t)y 2 for t E ta, ao)U(bo,b),
Sl(t) < z < s2(t),
lyl > r,
Sl(t)
lYl>r,
and f(t, x, y) sgn z / >
2394
ho(t)t_a [ ~t-~+hiCt)]lyl-h2Ct)y2
for,~
where )t e [0,1), r e R + , hk e L]oc([a,b)), (k = 0,1), and h, E C([a,b)). Then the problem (5.1), (5.31) is solvable. THEOREM 5,,42. Let a > - c r and let 81 and 82 be lower and upper functions for Eq. (5.1) with
limsuplsk(t)l < +oo t--ta
(k = 1,2)
and suppose for some sequence a,, E (a,b), (n = 1 , 2 , . . . ) converging to a we have s'l(an-t-)>.O>/s~(an-F )
(n = 1 , 2 , . . . ) .
Further suppose f(t,z,y)sgny
<~ ( t - a ) U h o ( t ) + [ t - l~ a
+hl(t)]]y]+h2(t)y 2
fora
sl(t)<~z<~s2(t),
yER,
where ~ ~ R+, hk ~ L,o~([a,b)), (k = 0,1), and h2 ~ C([a,b)). Then Eq. (5.1) has at least one solution satisfying the conditions
sl(t) ~
lim(t-a)-~u'(t)=O, THEOREM 5.51. Let a >
--oo
(5.28)
fora
and let the function f satisfy the condition
f(t, z l , Y l ) -- f ( t , z2,y2) ~ ll(t)(zl -- z2) + 12(t)(yl -- Y2) for a < t < b,
sl (t) ~< -2 < -1 ~ ,2 (t),
Y2 ~'~ Yl;
(5.29)
with l, E Llo,: ((a,b)), (k = 1,2), 11 nonnegative, and for some to E (a,b)
liminf s2(t) - sl(t) = 0, t-.b l(t, to) w h ere
l(t, to)
(5.30)
=/i
Then the problem (5.1), (5.31) has at most one solution.
REMARK. If 81 and 82 are lower and upper functions of Eq. (5.1) having finite limits sk (a+), (k = 1, 2), slCa+) <~ c ~< s 2 ( a + ) ,
fa t ( r - a ) l f ( ~ , s ~ ( ~ ) , 0 ) l d T < + ~
for a < t < b
and 12 (t) = -t --- +a1 o 2 (t), where ,~ E [0, 1) and 1o2 ELloc ([a, b)), then, according to Theorem 5.41, condition (5.29) guarantees that the problem (5.1), (5.31)is also solvable. THEOREM 5.52. Let a > - c ~ and let the function f satisfy condition (5.29); here Ik E Lloe ((a,b)), (k = 1,2), 11 is nonnegative, and for some to E (a,b) equality (5.30) holds, where l(t, to) = 1 +
f; f; (1 dr
exp
)
12(~)df ll(r
I~ + Further suppose that for any p C R+ condition (5.20) holds, where gp ~ L,or ([a,b)), (resp. gp(t) = t---Z-~a gop(t), U ~ It+, gop ~ L,or ([a,b))). Then the problem (5.1), (5.32) (resp. the problem (5.1), (5.28)) has at most one solution.
5.2. T h e P r o b l e m s (5.6), (5.7) a n d (5.8), (5.9). The existence theorems presented in this subsection are due to Moiseev and Sadovnichii [40, 43], and the proofs of these results are taken from [311.
2395
THEOREM 5.6.
The problem (5.6), (5.7) is solvable.
PROOF: Let a2(t) = t + 2,
I(t'z'Y)=-~Y+x(..
-
sl(t) =
t2 2+~-
for 0 < t ~< 4,
t
for t > 4,
2) + - - z
x-' + ~-z
for
s~(t)<~ ~< ~
(t)
and { f ( t , sz(t),y) f ( t , z , y ) = f(t,sl(t),y)
for z > s~ (t), for z < sl(t).
(5.31)
Then f E Kloc ((0, +oo) • It2) and inequality (5.14) holds, where a : 0, b = +oo, w(y) -- 1, hl (t) : 0, 2 h2 (t) = - ~ , r = 0. On the other hand it is easy to verify that 51 and 52 are lower and upper functions of Eq. (5.1) satisfying inequality (5.5). By Theorem 5.1, the problem (5.1), (5.2) has a solution that obviously is also a solution of the problem (5.6), (5.7). The theorem is now proved. REMARK. It is proved in [43] that the solution u of the problem (5.6), (5.7) is unique and admits the asymptotic representation 2
2 lnt
1
uCt)=t+l+~+]-~-+O(-~)
ast--++oo.
In addition it follows directly from (5.6) and (5.7) that
(t~u'(t)) '>0, u'(t)>O fort>O. THEOREM 5.7.
The problem (5.8), (5.9) is solvable.
PROOF: We first consider the case when p > 2. We set sl (t) = O, s2 (t) = 2, and define the function f by the equalities 3x
f(t'z'Y)=b--t
p'
--
-~-2~] 1/2
b2
z' Y' + ( 1 - pzZ~' 2x ( 1 - ~ - ) + ~J + (b _--~)z for
mad (5.31). It is cleat that f E Kloc ([0,b) • I t ' ) ,
l(t)
9
and 82 are lower and upper functions of Eq. (5.1) and 4 6 2 inequality (5.161) holds, where a = 0, r = 0, 0J(y) = 1 + lYl, hi(t) - (b_t)2 + -~ and h2(t) - p2 - 4 " 81
Therefore a~cording to Theorem 5.31 Eq. (5.8) has a solution u satisfying the conditions
u(O+) = 2,
O ~ u(t) ~< 2 f o r 0 < t < b .
(5.32)
It is clear that
u(t) > 0
for0
(5.33)
for in the contrary case for some to E (0,b) we would have u(to) = uw(to) = O, but only the zero solution of Eq. (5.8) satisfies these initial conditions. In view of (5.32) and (5.33), lim inf(b - t)luw(t)l = 0 t---,b
2396
and
[,, ,,..,,,] [1(2 ,, ,,') (P'-u2t) 1/2+v ~/-~:u-~)
=
~F
(b-t)2(1-b'~'/'lu(t) ~-, j
p2
~>6u(t)>0
for0
1 ( 2 - b~) (O2 - 4) 1/2 > 0. From this it follows immediately that where 6 = ~,'(t) <0
anduCb-)=O.
for0
Consequently u satisfies conditions (5.9). It is somewhat simpler to prove that the problem (5.8), (5.9) is solvable in the case p = 2 by taking as lower and upper functions sl (t) = 2 - t and s2 (t) = 2 - 2 -6 t 7. REMARK. From the reasoning presented above it is clear that an arbitrary solution of the problem (5.8), (5.9) is a decreasing function. 5.3. T h e P r o b l e m s (5.1), (5.4,), (i =1,2). We shall study Kneser's problems (5.1), (5.41) and (5.1), (5.42) under the assumption that
f E Kloc((0,+oo) x R+ xR_) and
f(t,O,O) = 0 ,
f(t,x,O)>~O
x>O.
fort>0,
(5.34)
THEOREM 5.81. Suppose either
f ( t , x , y ) <<.w(y)(hl(t) + h2(t)lYl)
for t > 0,
0~<x~r0,
y~-r,
or there exist numbers a 9 R+ and/~ 9 (a, +co) such that f ( t , x , y ) <~w(y)(hl(t) + h2(t)lY[)
O <. x <. ro,
for 0 < t < ~,
y <. - r
and
f(t,x,y)
>1 -wCY)(hl
(t) + h=(t)lYl)
for t > ~,
0 < = ~< r0,
y ~ --r,
where ro > O, r >>.O, hi EL]oc (R+), h2 9 C(R+ ), and w is a Nagumo function. Then for any c E [0, r0] the problem (5.1), (5.41) has at least one solution. PROOF: Extend f to (0, +or) • R 2 using the equalities
f(t,z,y)=f(t,z,O)
fort>O,
z~>O,
y>O
x
yER.
and
f(t,x,y)=f(t,O,y)
fort>0,
Then, in view of (5.34),
/ e K~oo((0,+oo)xPJ) and
f(t,x,y)~O
fort>0,
x~>0,
y~>0.
(5.35)
On the other hand sl(t) = 0 is a lower and s2(t) = r0 an upper function of Eq. (5.1). Let c E [0, r0]. Then according to Theorem 5.31 Eq. (5.1) has a solution u satisfying the conditions u(0+)=c,
O <<.u(t) <<.ro f o r t > 0 .
If u'(to) > 0for some to > 0, then by (5.35) we find
,,'(t)/> ,,'(to)
and ,'o/> ,,(t)/> ,,'(to)Ct-~o)
for t > to.
The contradiction so obtained shows that u satisfies conditions (5.41). The theorem is now proved.
2397
COROLLARY 5 . 1 .
Let
f(t,x,y) <. pt ~-2 lyl ~ for
0 < t < a,
0 ~< x <~ to,
Y ~< - r
(5.36)
and
f(t,z,y)
>>. --W(y)(hl(t) § h=Ct)lYl)
for t > O,
0 < = < to,
y < -r,
where a, p, to, and r are positive numbers, A 9 C1, +oo), hi 9 L,or ((0, +eo)), h, 9 C((0, § Nagumo function. Then for any c 9 [0,r0] the problem (5.1), (5.41) has at least one solution.
(5.37) and w is a
PROOF: Choose c > 0 and ~ 9 (0,a) so t h a t
" z(t) dt > ro, where
(5.38)
1
.(t) = [~ + , t ~-I ] , - , . Without loss of generality we may assume that ~C") > r.
We set rl = z(O),
a(y) =
l
1
for y t> - r i ,
2- y rl 0
for - 2rl < y < - r l ,
(5.39)
for y ~< - 2 r l ,
](t,x,y) = { a(y)f(t,x,y)
f(t, =,y)
for 0 < t
(5.4o)
for t >1~,
and consider the differential equation
.,t= ](t,.,.').
(5.41)
Since ri > r, the inequalities
](t,=,y) < ~(y)(L (t) § L(t)lyl)
for 0 < t < ~,
0 ~< z ~< ro,
y ~< - 2 r i
and
](t,=,y) >1 - ~ ( u ) ( L ( O
+ ~,,(Olyl)
for t > o,
o < = < ro,
y < -2ri
foUow from (5.3~) and (5.4O), where/,i(t) = 0,/,,(t) = Ih~(#)l for 0 < t < # and/,,(t) = Ih~(t)l for t > #, (k = 1,2). According to Theorem 5.8i if c E [0, r0] the problem (5.41), (5.4i) has a solution u. We shall show that
ut(t) >>.- z ( t )
for 0 < t
(5.42)
Suppose, to the contrary, that for some to E (0, ~)
~'(to)< -~(to).
(5.43)
We denote by tl the least upper hound of the set of s E (to, a) for which the inequality
,,'(t) < - , ( t ) 2398
(5.44)
holds
on
the interval it0,4
view of (5.36) and (5.40), lu'(t)l ' > / - p t ;~-= I~,'(t)l:'
for to < t < t,.
From this, taking account of (5.43), we find 1
I,,'(ti)l/>
[pt; -1 + I,,'(to)l '-~'
-
pt~o-1 ],-.x > z(tl).
Therefore it is clear that tl = a and inequality (5.44) holds on [to, a]. On the other hand, according to (5.38) arid (5,.44) ro > uCto) - u C a ) = -
fo
u'Cr)dr >
/;
zCr) dr > ro.
The contradiction so obtained proves inequality (5.42). But it follows from (5.40) and (5.42) that u is a solution of Eq. (5.1). The corollary is now proved. COROLLARY 5.2. Let f
6
Klor (R+
x
R+
f(t,x,y)>~O
x
R _ ) and let fort>0,
x~>0,
y~<-r,
where r >/0. Then there exists a positive number r0 such that for any c e [0,r0] the problem (5.1), (5.41) has at least one solution. PROOF: Let r I be an arbitrary positive number and h(t) = sup { I f ( t , z , y ) l :
0 ~< :r ~< n ,
- 2 r , ~< y ~< 0}.
(5.45)
Choose numbers fl > 0 and r0 6 (0, r,) such that ro + f f hi (t) dt < r,, and define the function ] by means of equalities (5.39) and (5.40). It is clear that ](t,x,y)=O f o r O < t < ; ~ , o ~ = < r 0 ,
(5.46)
y<-2rl
and ] ( t , z , y ) >>.O f o r t > 0 ,
0~<x~
y <~ - 2 r l .
Therefore according to Theorem 5.81 the problem (5.41), (5.41) has a solution u whenever c 6 [0, r0]. In view of (5.4,) and (5.45)
min{lu'(t)l : 0 ~< t ~
I',"(t)l ~< h(t) for 0 ~< t ~< ft.
From these inequalities we have, by (5.46) lu'(t)l < rl
for 0 ~< t ~
According to the estimate just obtained it follows from (5.40) that u is also a solution of Eq. (5.1). The corollary is now proved.
2399
T H E O R E M 5.82. Let f E Kloc (R+ x R + x R_), and let f ( t , z , y ) >/ -w(Y)Ch I (t) + h2(t)lYl)
for t > 0,
0 ~< z ~< to,
y ~< - r
and f(t,z,y)>~(t)>/O for0
{
=
ror~
forf~a0 >,~< t at ~<
is an upper function for Eq. (5.1). It now follows from Theorem 5.32 that the problem (5.1), (5.42) is solvable. The following theorem has a proof analogous to that of Theorem 5.81. THEOREM 5.9. Suppose either
+ hx(t) lYl + h2(t)y2
f ( t , z , y ) <<.-----~ +
fort>0,
O~x~ro,
y <<.-r,
or there exist numbers a E R+ and/~ E (a, +co) such that
f(t,z,y)<<.,----i-+
+h~(t) [y[+h2(t)y'
forO
O<.z<.ro,
y<<.-r
and
f ( t , z , y ) ~> ho(t) t
[A--+hl(t)]lYl-h2(t)y2 Lt
fort>a,
]
o<~z<~ro,
y<<.-r,
where ro > O, r >1 O, hk ~ n~or (R+), (/~ = 0,1), and h2 ~ C ( R + ) . Then for any c ~ [0,ro] the problem (5.1), (5.41) has at least one solution. THEOREM 5.10. Let f(t,z,y)>>.a(t,z)>~O
fort>0,
O~
y~
(5.48)
where ro > o and the function a E Kloc (R+ x R+ ) is nondecreasing on the second argument and for 0 < z ~ r o .
fo ~176 ta(t,z) dt = + c o
(5.49)
Then for any solution of Eq. (5.1) satisfying the inequalities
O~u(t)~ro, u'(t)~Ofort>O,
(s.5o)
we h a v e
u(+co) = 0 ,
and
lim tu'(t) =0.
t.-., +co
PROOF: Set 13(t)
=
uCt) - tu'Ct),
z = u(+co).
According to (5.48) and (5.50) 13(0 t> o,
r
~ o
for t > o
(s.sl)
and
/o
',~(,,z) d,
~< 13(0-1-)
fort
>o.
(5.52)
In view, of (5.50) and (5.51), tu'(t) ~ 0 as t ~ +co; and, in view of (5.49) and (5.52), z = 0. The theorem is now proved. 2400
T H E O R E M 5.111. Let the function / satisfy inequality (5.48), let it be nondecreasing on the second argument, and let
for t > O,
f ( t , z , yl) - f ( t , z , y2) >1 l(t)(yl - y2)
0 <~ z <<.to,
- r o <<.tys <<.tyl <<.O,
(5.53)
where ro E R + , a E/floe (R+ • R+ ) and I ELioc ((0, +oo)). S u p p o s e also that either
exp
l(r) dr
dt = +oo
for to > 0,
(5.54)
or a is nondecreasing on the second argument and satisfies condition (5.49). Then for any e E [0, r0] the problem (5.1), (5.41) has at most one solution.
PROOF: Suppose the contrary: for some e E [0, r0] the problem (5.1), (5.41) has two distinct solutions ul and us. Then by (5.48) 0~
tu~(t)>>.-ro
fort>0
(k=1,2).
(5.55)
On the other hand we may assume without loss of generality that for some to > 0 the function u(t) = ul (t) - us(t) satisfies the inequalities
(t0) > o,
> o.
(5.56)
Since f is nondecreasing on the second argument we obtain from (5.53), (5.55) and (5.56) that u(t)>u(to)+u'Cto )
r (/o) exp
l(r)dr
ds
fort>t0.
./to
Consequently equality (5.54) does not hold. Equality (5.49) also cannot hold, for otherwise, according to Theorem 5.10, we would have 0 = u(+oo) > u(to) > 0. The theorem is now proved. From Theorems 5.9 and 5.111 we obtain COROLLARY 5 . 3 . Let the function f be nondecreasing on the second argument, and let 1
f(t,x,y)>>.O
fort>0,
O<~x<<.ro,
y~<0,
f0
t f ( t , ro,O) d t < + o o
and r~
l
f(t,z, y1)-f(t,z, y2)>>.-Lt +lo(t)J(y1-92)
fort>O,
o<<.z<<.ro, -ro <~ty2 <~tyl <~O,
where )~ E [0,1), the function lo : R+ -~ R+ is measurable, and
f0 ~176 lo (t) dt < +oo. Then for any e E [0, r0] the problem (5.1), (5.41) has one and only one solution.
According to Theorem 5.10 the problems (5.10) (5.11) and (5.10), (5.41), where e = 1, are equivalent. Therefore it follows directly from Corollary 5.3 that the problem (5.10), (5.11) has a unique solution.
2401
THEOREM 5.112. Let the function f E Klo~ OR+ x R+ x R _ ) be nondecreasing on the second argument, let f(t,z,y)>/a(t,z)>.O f o r t > O , z~>O, y~
fort>0,
If(t,x, yl) - f ( t , z , y2)l <~lp(t)(yl - y 2 )
0~
- p <~ y2 <~ yl <~o,
where lp EL]oc (R+) and the function a E K~oc (R+ • R+) is nondecreasing on the second argument and satisaes condition (5.49). Then for any c R_ the problem (5.1), (5.42) has one and only one solution.
Under the hypotheses of the proposition just stated it follows directly from Theorem 5.82 that the problem (5.1), (5.42) is solvable. As for the uniqueness of the solution, it is proved by reasoning analogous to that carried out in the proof of Theorem 5.111. It follows in particular from Theorem 5.112 that the problem (5.12), (5.13) has a unique solution. Solvability criteria for Kneser's problem for higher-order nonlinear differential equations are found in [17, 20]. Analogous problems for systems of differential equations have been studied by Hartman and Wintner [78], Coffman [72], Chanturiya [58], Kiguradze and Rachunkova [84], and Rach~nkov~ [50, 93]. O
w
9
O n a B o u n d a r y - V a l u e P r o b l e m Arising in N o n l i n e a r Field T h e o r y
In the present section we shall study the boundary-value problem ." :
t
+ . - I.I
u'(O+) = O,
u ( + o o ) = O.
(6.1)
(6.2)
We shall assume here that q and ~ are real numbers with ~ > O. Before stating the main result we shall agree to call a nontrivial (i.e., nonzero) solution of Eq. (6.1) oscillating if it has infinitely many zeros and nonoscillating otherwise. PROPOSITION. a) The problem (6.1), (6.2) has a nontriviaI solution if and only if one of the following conditions holds: q+3 I.q>l,l<~< q-l; II. 0 < q ~ < l , A > l ; III. q = 0 , A > l ; IV. q < O , ~ > l ; V. q > O , ~ < l . b) All nontriviM solutions of the problem (6.1), (6.2) are nonoscillating if I or II holds, have no zeros if III or IV holds, and oscillate if V holds. c) ha cases I and II for any nonnegative integer I there exists a solution of the problem (6.1), (6.2) with exactly I zeros on (0, +oo); in case III the problem has a solution that is unique up to sign, and in cases IV and V for any Uo E (-1,1) it has a solution u such that u(0+) = u0. The proposition just stated is a corollary of Theorems 6.1-6.6 proved below. Along the way we shall also establish some other properties of solutions of the problem (6.1), (6.2). 6 . 1 . P r e l i m i n a r y R e m a r k s . If u : J --, R , J c (0, + o o ) is a solution of Eq. (6.1), we shall denote by V,
the function defined on J by the equality
v~(t) = ~u'2Ct) + A--~lu(t)l~+l - 2u~(t).
(6.3)
For what follows we shall need some simple propositions which we state in this subsection as lemmas for convenience in later exposition. 2402
LEMMA 6 . 1 .
Any solution u : (a, b) ~ R , 0 < a < b < +oo, of F,q. (6.1) can be extended to the interval
(0,+oo). PROOF: For A ~ 1 the lemma follows from the global existence theorem (Wintner's Theorem). Let u : (a~,b) ~ R be a solution of Eq. (6.1), where A > 1. Since V'(t) = -7-u'2(t),
(6.4)
t
from the inequality 1 A+I
27 A + I
-
1
z2 >
for x/> 0,
-2
A> 1
(6.5)
we obtain 1_
Iv~(t)l ~< ~-~-2(vu(t ) + 2)
for a < t < b.
Thus according to a differential inequality theorem (cf., for example, [18], p. 48), the function V, is bounded in the interval (a, b). It follows from this that u can be extended beyond that interval. The lemma is now proved. LEMMA 6 . 2 . Let u : (0, +oo) ~ R be a solution of Eq. (6.1) with u(to) = e and u'(to) = O, where to 6 (0,+cr and e 6 { - 1 , 0 , 1}. Then that solution is stationary (i.e., u(t) = e). PROOF: It is clear that it suffices to consider the case when A < 1 and e = 0. In this case we have from (6.3) and (6.4)
IV'(t)l-< for all t sufficiently close to to. Thus V,, (t) = 0. The lemma is now proved. LEMMA 6 . 3 . Let u : (0,+oo) ~ R be a nonstationary solution of Eq. (6.1). Then the function V~ : (0, +cr ~ R is strictly increasing i f 7 < O, identically equal to a constant i f 7 = O, and strictly decreasing i f 7 > O. This proposition follows directly from (6.4) and L e m m a 6.2. LEMMA 6 . 4 .
If u : (0, +oo) --+ R is a nontrivial solution of the problem (6.1), (6.2), then V,(+oo) = u ' ( + o o ) = 0 ,
sgnV,,(t) = sgnT.
PROOF: According to L e m m a 6.3 the limit V,(+oo) exists. Thus V~(+oo) = C(+oo) = 0. It remains only to apply L e m m a 6.3 again. 6.2. T h e Case 7 > 0, A > 1. W e note that this case is the most important from the point of view of applications.
THEOREM 6 . 1 . Let 7 > 0, A > 1 and let u : (0, +oo) ~ R be a nontrivial solution of the problem (6.1), (6.2). Then u is nonoscillating, uCO(t)~CL1)~et-~/2e -t
ast~+oo,
(in0,1)
(6.6)
and 11 : 12:13 = [7 + 3 - A ( - / - 1)]: [2(A + 1)]: [(7 + 1)(A - 1)],
(6.7)
where e is a nonzero constant and I1 =
/o
t~u2(t) dt,
I2 =
/o
t
lu(t)l
TM
dt,
& =
/o
t~u"(t) dt.
A direct consequence of (6.7) is the
2403
COROLLARY. For "7 > 1 and A/> (7 + 3)/("7 - 1) the problem (6.1), (6.2) has no nontriviaI solutions. PROOF OF THEOREM 6.1 : If the solution u of the problem (6.1), (6.2) is oscillating, there exists a point to E (0,+co) such that 0 < lu(to)l < 1, u'(to) : 0 and so V,(to) < 0 also, which contradicts Lemma 6.4. Thus u is nonoscillating. Further, by (6.2) and Lemma 6.4,
xu'(t) for all sufficiently large t, which, as is easy to verify, implies the inequality
lu(t)[ < Ke -'12
for t > 0,
where K is some constant. Consequently by Mattell's Theorem (cf., for example [18], p. 107), the linear differential equation
r = _2r t
+, _ lu(t)l
-I
has a fundamental system of solutions (~, v0) such that ~(')(t) ~ t -~/n
e', ~(i)(t)N
( _ l ) i t - ~ / 2 e-t
as t --* +co,
(i=0,1).
(6.8)
Since u is also a solution of this equation and satisfies (6.2), there exists a nonzero constant c for which u(t) =- C6o(t). Relation (6.6) follows from this. Finally, substituting the solution u in Eq. (6.1), multiplying both sides of the equality thus obtained by t 7+1 u'(t), and then integrating them over the interval (0, +oo), taking account of (6.2) and (6.6), we arrive at the equality (7 -{- 1)It
2('7 A ++ II) /2 + ( 7 - I)13 = 0.
(6.9)
The same procedure, only multiplying by tTu(t) gives the equality It
--/2+
13 = 0.
(6.10)
Relation (6.7) follows from (6.9) and (6.10). The theorem is now proved. THEOREM 6.2. / f e i t h e r 7 > l a n d 1 < A < (7 + 3)/(7 - 1 ) o r 0 < 7 ~< 1 a n d A > 1, then for any nonnegative integer I the problem (6.1), (6.2) has a solution u : (0, +co) --~ R with exactly I zeros. +co +co PROOF: Let I E {0, 1,... }. We construct sequences (t~),=l and (T,),= t such that the relations
1 O
l~
(n:1,2,...)
and t.~0,
T,--*+co
asn--*+co.
Then according to Corollary 4 of [98], for any natural number n Eq. (6.1) with the boundary conditions u'(t.) = o,
u(r.) = o
has a solution u , with exactly I zeros in (t,, T~). Lemma 6.1 makes it possible to assume that u~ is defined on (0, -t-oo). By Lemma 3 of [98] there exists a constant r/such that
1,2,...), 2404
where r~ are points of the closed interval [1/2,1]. Thus from (6.3) and Lemma 6.3 we have
Vu.(t)>~V..(T.)>O
for0
(n=1,2,...)
(6.11)
and
V,,.(t)<<.v..Cr.)<<.#
(n=l,2,...),
fort~>l
(6.12)
where 1 2+
=
1
_x+l
9
By (6.12)
lu,(t)l+lu'~(t)l<~ro
fort~>l
(6.13)
(n=l,2,...),
where r0 is some constant. We shall prove that there exists a constant r such that
lun(t)l<<.r
fort/>tn
(n=l,2,...).
(6.14)
Indeed, according to (6.3)-(6.5), 27 1 V~. (t) >/ ---~-(V,. (t) + 5)
fort>0,
(n:l,2,...).
Hence for any natural number n by the differential inequality theorem we have 1
1
1
0 < V,. (t) + ~ ~< t-~-(tt + {) and so
1
l-'(t)l< ~-v/2/z+l
for 0 < t <~ 1,
forO
If . / < 1, then, taking account of (6.13), we obtain (6.14) from the last inequality with r=ro+
v
+l 1-"/
But if " / = 1, then by the same inequality we find lu,(t) I < r0 - X / ~ + l i n t
for 0 < t
~< 1,
(n : 1 , 2 , . . . ) ,
and, as follows from (6.1), in this case we can set r = r0 +
fo
(1 + r0 - V~-/z + 1 lnt) x dr.
Let us now assume ~r > 1. Using the reasoning applied in deducing (6.9) and (6.10), we verify the equalities (7 + 1)11,
2(7 + 1) 12. + (71)I,. = 2(t~n+1Vu,, (in) - V.,. (1)) ,~+I
and
/i. - 1 2 . + ls. = u~.(1)u.(1),
(6.15)
where n E { 1 , 2 , . . . } and I.,
=
x2. =
t
lu.Ct)l T M
tit,
Is,, =
t~u'~2Ct)dt.
2405
Consequently 3 + ff - A('7 - 1) I~. = 2(I1. - t~+1V.n (t.) -t- V(u. (1)) -t- ('7 - 1)u~ (1)un (1). ),+1 On the other hand, according to HSlder's inequality,
T21(~+1) I1. ~<-2. These relations, by (6.11)-(6.13) and the hypotheses of the theorem, imply that there exists a constant Q such that
I2. ~
(n=1,2,...).
Iln + Isn <<. r~ + Q
(6.16)
We now fix a natural number n and denote by vn and vo the solutions of the linear equation
v" = --7--v' + v, t
(6.17)
satisfying the conditions
vnCt.)=l,
v~nCtr,)=0,
vo(1)=0,
!
Vo(1)=1.
Since !
v.(t)>l,
v. C t ) > 0
(6.18)
fort>t.,
we have vnI! (t) <~ vn (t) and, by the differential inequality theorem
v . ( t ) < ~ ( 1e.-
t
t. + e t . - t ) ~<2
fort.~
(6.19)
Furthermore
vo(t) < O,
vto(t) > 1
forO
and
vo(t)
.
1 [1 "~- 1 .
Consequently t "~-1 Ivo(t)l < ~
.
1 ftl( 1 tT-1 + t~':l .
'(/' 1+
1) s~-I
.
IvoC~)l,~ ds
)
Vo(S)s~ ds ]
for 0 < t ~< 1.
By the Gronwall-Bellman Lemma (cf., for example, [18], p. 49) we obtain d I~o(t)l < t~----r for 0 < t ~< 1,
(6.20)
where -~ - 1
2(-~ -
1)
We remark that (6.17) has no nontrivial solutions satisfying the boundary conditions v'(t,) = 0, v(1) = 0. Thus, by Green's formula 1 ) u, Ct) -- v.(1
un (1)v,, (t)
root)
~,,.,0-)~-" lu, (r)l ~'-1 ,.,,, 0) d~- ,.,, (t)/1 vo(O~-' lu,.,(Ol~'-I u, (0 drl
2406
fort.~
Hence, in view of (6.13) and (6.18)-(6.20),
I,,.(t)l -< 2to + 2d
r~l,,.Cr)l ~ d~ +
rl,,.Cr)l ~ dr
for tn ~< t ~< 1.
(6.21)
Choose a positive number e satisfying the inequality
<
3 + f f - - ) t ( f f - - 1)
(6.22)
2
By the hypotheses of the theorem the right-hand side of this inequality is positive and so such a choice of e is possible. Set ~/-1 vj-- 2 e(j-1) (j=l,...,k),
rv~_~+ Q Q1 = ro +
- 1 '
2Q~d Qi+l
(6.23)
= 2to + 2 - ~ - (A - 1 ) ~
(i = 1,...,
k),
where k is the smallest natural number such that 7-1
k~>--
2~
We remark that according to (6.13), (6.16), (6.23), and the Cauchy-Schwarz inequality
lu.(t)l
~< lu.(1)l +
lu'.(r)ldr <. r +
is.
r-" dr
Qlt -~1
for ta ~< t ~< 1.
Assume that k > 1, j E { 1 , . . . , k - 1}, and I = . ( t ) l -< Q i t - ~
for t . . <
t .< 1.
Then by (6.21)-(6.23),
iu,(t)l<~2ro+2Qid(t-~y-_~ ~t v~-x~r ~< 2r0 +
dr+~t
r 1-~vi d r )
2Q~dt-~{ +~
r 1-"-(~-1)v~ dr ~< Qr
t -vj+l
f o r t , ~
It follows from this that
N.(t)l < Qkt -~*
for t, ~< t ~< 1.
Again applying (6.21)-(6.23), we obtain
I,~.(t)l < Qk+l
for t . ~< t ~< 1.
This, by (6.13) and the inequality Qk+l > r0 means that (6.14) holds with r = Qk+l. Thus the existence of a constant r for which estimate (6.14) holds is proved for all admissible values of ff and A. As follows from (6.14) and the definition of u , , lu'(t)l ~< (1 +
r)~t
for
t >~t,
(n = 1 , 2 , . . . ) .
(6.24)
2407
Thus without loss of generality we may allow that there exists a solution u : (0, +co) --+ R of Eq. (6.1) +co such that u'(0+) = 0, u and u' are respectively the uniform limits of the sequences (u,),= 1 and (~. J. . j'~+oo .=l on each closed subinterval of (0, +co) and, in addition, unCtn) --~ u(O+) as n --+ +co. Since by (6.11) 1
A-I
(n = 1,2,... ),'
(6.25)
the solution u is not identically zero. We remark that the number of zeros of u on [0, +co) cannot exceed l; for otherwise u,, for sufficiently large n, would have more than l zeros. Consequently u is of constant sign in some neighborhood of +co. If the solution u is not monotonic for sufficiently large values of the argument, there exists a point e (0, +co) for which u'(t) -- 0 and u"(t)u(t) i> 0. According to (6.1) in this case, lu(t)l ~< 1 and therefore V,(t) < 0. The last inequality contradicts (6.11) since Vu~ (t) ~ Vu (t)
as n ~ +co
(6.26)
uniformly on any closed subinterval of (0, +co). Thus u is monotonic in some neighborhood of +co. In view of (6.14) the limit u(+co) exists and is finite. On the other hand by Lemma 6.3 the limit V, (+co) also exists. Thus, by (6.1) and (6.3), u'(+co) = 0 and u(+co) E {-1,0,1}. But if iu(+co)l = 1, then V,,(t) < 0 for sufficiently large t, which, as we have already seen, is impossible by (6.11) and (6.26). Consequently u satisfies the boundary conditions (6.2). It remains to be verified that u has exactly l zeros in (0, +co). By Theorem 6.1 there exists a number K such that lu(t)[ < K e - t
for t > o.
According to the theorem of Mattell mentioned above, Eq. (6.17) has a fundamental system of solutions (~, ~0) satisfying (6.8). Without loss of generality we may assume that ~'(1) : 0 .
(6.27)
Then
mt -712 et <<.~(t) <. M t -~12 et
fortf>l,
(6.28)
where m and M are positive constants. Let a E (0, 1) and aA > 1. Choose the number To such that the following inequalities hold: To~2~+1 / A+I V~/CA _ 1) ,
u(To)u'CTo)
lu(To)[<(1-~)
1/(x-1)
(6.29)
and - 1)-F 1) LV~e-t < 41 i (ro "/(A Jr 1)(A
for t/> To,
(6.30)
where
L = --M r o e + K m
t ~12 e( ' - = ~ ) t dt).
(6.31)
J1
Obviously
u.CTo) :.CTo) < O, I .(To)l < (1
(6.32)
and I-.(t)l < Ke -t 2408
for 1 ~< t < To,
(6.33)
whenever n ~> no and no is a sufficiently large natural number. Fix n E {no, no + 1,... } and denote by So the smallest zero of u , in the interval (To, Tn]. As follows from (6.11) and (6.32),
"`,,(t)u'(t)
lu,,(t)l~<(1-o,')'/c~'-l)
forTo <~ t <<.so.
Hence
"`.1 2 (t)/> ,~2 "`.2 (t)
for To ~
and, according to (6.33), we obtain: I'`.(t)l ~<
I'`,(T0)le-'C'-To) < Ke-"
for To ~< t ~< So.
(6.34)
We remark that Eq. (6.17) has no nontrivial solutions satisfying the boundary conditions v'(1) = 0, V(So) = O. Therefore
"`.(t)
,,'(1) =
1
t
~I(1) o,(t)
eCr)~ I'`-(r)l ~-1 "`-(~) d~
+~(t)
f'~
~,(r),"
I'`,,(,)P-'
u , ( r ) dr]
for l ~ < t < s o ,
(6.35)
where
~, (t)
=
~(so)~o(t) - ~oCso)~(t).
Because of (6.27) and (6.28)
I~ (1)1-- 10(8o)~(1)1/>
m o o ~/2 e ~176 I~(1)1-
On the other hand, by Liouville's formula v~ (So) = v(1)~ (1)So "t 9
Thus, taking account of (6.13), (6.28) and (6.33)-(6.35), we have I,,'(so)l <
(6.36)
Lso~/~ e-'~ ,
where L is determined by equality (6.31). Assume so < Tn. Then from the definition of un it follows that there exists a point 61 E (so, 7',) satisfying the conditions I,,,,(s,)l = 1, o < I'`,(t)l < 1 for so < t < sx. (6.37) According to (6.1)
u,(t)u~Ct)
> 0
for So < t ~< 61.
(6.38)
Let r be a minimum point for the function I"'~1 on the closed interval [So,S1]. By (6.3) and (6.11), t2 "`. (t)>,,~.Ct)
2
A-1
2
~ + l l , , . ( t ) l ~+' >~-~--~--iu.(t)
for so <<.t <<.sl,
(6.39)
i.e.,
I'h(~')l > I~"(~)1 ,X-- 1 ~ i'`h (,.)l(,. __~o)V/~__--1 V/"~L-~--iu By (6.20) So r
1 2
2409
Further, from (6.1), (6.37), and (6.38), we have
al,," (t)l/> -~1~,"(t)l
for So ~< t ~< st.
Therefore
I,,~ (,-)1 (,soy, 1 I-'(so)l ~>- ~- > 2--~" By (6.30), (6.36), and (6.37), there exists s E (So,Sl) such that +
ff~~(~-1)1 .'(so)
(6.40)
(6.41)
:..(s).
Taking account of (6.40) and the inequality
I,,.(s)l ~ I,," (,-)lCs - so), we obtain V//~.+_ 1 , ()k- 1.1)I-" (so)l > ~l,,.(so)l(s-
so).
Thus, according to (6.29), So 1 s 2 As follows from (6.38), (6.39), (6.41), and Le~ma 6.3,
,~-1 u~ ' (t) > u .2 (s) -~ + 1 = lu~Cso) = 2 V u
(6.42)
.(so)> ~V,,,,(t)
for s~
Hence
v&, (t)=
t " (t) < -~-1E.. (t)
_ 2 u, ~
for s <. t <. s l ,
2t(~
and, by the differential inequality theorem
r.. (s,) < -
s[
81
r.. (s)
',/(A-l) ~" , ] 2s($ + 1) u. Cr)dr .
(6.43)
In view of (6.30), (6.36), and (6.41), there exists s, e (8,81) such that
I~.(~,)1
: 5'1 51 ~
for s~ ~< t < s,.
According to (6.13) 1 81 --82 ~ 2ro
Thus, applying (6.30), (6.36), (6.42), (6.43), and Lemma 6.3, we obtain Vu.(,Sl) <: - 1- [ L 281 - q e-2so
sx
?(%-- 1). ] 16ro (~ -I- 1)J ~ O,
which contradicts (6.11). Consequently so -- T,, i.e., for n/> n0 the solution un has no zeros in (To,Tn). On the other hand, using (6.24) and (6.25), we conclude that for some to > 0 the zeros of u~, (n = 1, 2,... ), also cannot belong to (tn, to). Thus u has exactly I zeros on (0,-t-oo). The theorem is now proved. 6.3. The O t h e r Cases. For ~ ~< 0 and )~ > 1 the question of the solvability of (6.1), (6.2) is quite easily settled.
2410
THEOREM 6.:3. Let q = 0, A > 1, and Uo : (0,+oo) ~ :R be a solution of Eq. (6.1) under the initial conditions u ( 0 - b ) : (A2+_1') */{x-1) - -
,
= 0.
Then uo is monotonically decreasing, uo(-boo) = 0 and the problem (6.1), (6.2) has no nontrivial solutions distinct from Uo and - u 0 .
PROOF: According to L e m m a 6.3 V~o (t) - V~o (0-b) = 0. From this it follows that u0 is decreasing and u0(+oo)
= 0.
Further, if u is a nontrivial solution of the problem (6.1), (6.2), then, by L e m m a 6.4, V~ (t) = 0, i.e., ]u(0-b)[ = u0(0+). The theorem is now proved. THEOREM 6 . 4 . Let ff < 0 and ~ > 1. Then a nontrivial solution of the problem (6.1), (6.2) does not vanish on (0,+oo), and for a n y t o 6 It+ and u0 6 [-1,1] \ {0} there exists a solution u of this problem such that u(to+) = uo, u(t)u'(t) < 0 for t > to. (6.44) PROOF: The first part of the conclusion of this theorem follows from L e m m a 6.4. If to e 1%+ and Uo 6 [-1, 1] \ {0}, then, by Theorem 5.81 and L e m m a 6.1, Eq. (6.1) has a solution u : (0, +oo) --+ 1~ satisfying the conditions (6.44). Obviously u(+oo) = u'(-boo) = 0. By L e m m a 6.3 V~,(t) < V~(+oo) = 0. Hence
(~ ~-1) 1/(A-l) luCt)[ <
- -
for t > o,
and, solving Eq. (6.1) as a linear equation in u', we verify that u'(0-b) = 0. The theorem is now proved. REMARK. Setting to = 0 in (6.44), we obtain a solution that is monotonic on all of (0, -boo]. But for 7 < 0 and A > 1 the problem (6.1), (6.2) can also have nonmonotonic solutions. For example, it is easy to verify that if ~/< - 1 , to > 0, and uo = 1, then the solutions of (6.1), (6.44) are not monotonic on (0, -boo). We now consider the case when :~ ~< 1. From L e m m a 3 of [98] and L e m m a 6.4 above one can deduce LEMMA 6 . 5 .
IrA < 1 then all nontrivial solutions of the problem (6.1), (6.2) are oscillating.
THEOREM 6 . 5 . Let ",/> 0 and )~ < 1. Then for any Uo 6 ( - 1 , 1) there exists a solution u of the problem (6.1), (6.2) such that u(0+) = Uo. Moreover if u is a nontrivial solution of the problem (6.1), (6.2), then u is oscillating and u(0-b) 6 ( - 1 , 1 ) \ {0}. PROOF: Fix u0 e (-1,1) \ {0}. According to the corollary of Theorem 5.1 of [18], Eq. (6.1) has a solution u : (0, -boo) -+ R satisfying the initial conditions = uo,
u ' ( o + ) = o.
If to > 0 and lu01 < lu(t0)[ < 1, then Vu(to) >~ Vu(0-b), which contradicts L e m m a 6.3. Therefore lu(t)l < luo[ < 1 for t > 0, the function V~ defined by equality (6.3) is positive exists and is finite. Consequently
(6.45)
on (0,-boo), and by L e m m a 6.3 the limit ~ = Vu (-boo)
0 < Z < vu (t) < v , ( 0 + )
for t > o.
(6.46)
Suppose u is not oscillating. Then by (6.1) and (6.45) u is monotonic in some neighborhood of -boo and u(-boo) = 0. This, according to Lemma 6.5, contradicts our assumption, so that u is oscillating.
2411
Set
T=
2T M
(6.47)
~x (1 - u~-;~)"
From (6.46) we have
u'2(tn) > 2~ (n=1,2,...), where T ~
tl < t2,<
(6.48)
and
"''
u(t) TtO forth < t < t n + l
.(t.) =0,
(n----l,2,...).
Denote by "ln and ,~, the points of the semiaxis R+ satisfying the conditions ,1. < t. < s2.,
.,2(si.)
= ~,
.,2 (t) > #
for , 1 . < t < , 2 .
(Y : 1, 2; n = 1, 2 , . . . ) .
(6.49)
According to (6.1) and (6.45),
I-'(t.)l < lu'(sln)[ + tn - 81n (n = 1 , 2 , . . . ) . Therefore, taking account of (6.48) and (6.49), we obtain s,. - ,1. > t. -,1.
~> v ~ ( v ~ -
(6.50)
(n = 1 , 2 , . . . ) .
1)
Further, from (6.45), (6.46), and (6.49), it follows that
1 -I'('i") [~,+1 f12 ~---$--i _
<
(sJ")2 1"2
_
< I'~(si")l
(i = 1,2;n = 1 , 2 , . . . ) .
Since u' vanishes only once on [a2,, '1.+1 ], we have I~(t)l > ~-
2
for s , . < t < ,1.+1
(6.51)
(n = 1 , 2 , . . . ) .
We remark that by (6.46) U"(t) ~< 2Vu(O-{-)< 1 for t > 0.
Consequently, from (6.1),(6.45),(6.46),and (6.51)we have l,,"(t)l> Hence by (6.49)
(1- u~-x)- T-
2~+1 (1 - "I-;~ )
for s2,, ~< t ~< 81n-I-1
~A
V~ > 2-'T~T(1 - ul-;~ )[~- - 8.y.+2-.y I (J -- 1,2;n : 1 , 2 , . . . ) , where ~/n e [s,,,Sln+1 ] and u'Cr/n) = O. Thus 2 ~+I
81n+l -- 82n
~'(1--,,A-~)
(n : 1 , 2 , . . . ) .
According to (6.45) and (6.49), V/~(82n - 8 1 . ) <
l-(s2.)-,,(Sl.)l
< 2
(n = 1 , 2 , . . . ) ,
i.e., 82n --81n
2412
2
<:~
(n=
1,2,...).
(n = 1, 2 , . . . ) .
Thus, 31n+l
--
where
(•
$1n < ~
= 1, 2 , . . . ) ,
24+2 ~
(6.52)
2
3~(1--Ug -~) + V~" From (6.4), using (6.49), we obtain:
.~(t) Vu(t) < Vu(0q-)-fl~ Z
INS2"
n-- 1
for t > 0,
(6.53)
Sin
and re(t) : m a x { n : S2,~ ~< t}. By (6.50) and (6.52),
s2n - sin /> Sln
v~Cv~-1)
(n=1,2,...).
311 -[- ( n - - 1 ) ~
-}-oo
If 3 > 0, then the series ~].,=1 (s2, - s l , ) / s l , diverges and, according to (6.53), V~(t) < 0 for sufficiently large t, which contradicts (6.46). Thus 3 = 0, i.e., u satisfies the boundary conditions (6.2). Now consider an arbitrary nontrivial soluton u of the problem (6.1), (6.2). According to Lemmas 6.3 and 6.5, u is oscillating and u(0+) r 0. If we assume that lu(0+)l > 1, then from (6.1) we obtain the relation u(t)u' (t) > 0 for t > 0, contradicting the definition of u. The equality lu(0+)i = 1 is also impossible, since it leads to the identity ]u(t)l - 1 (cf., for example, Theorem 5.2 of [18]). Consequently u(0+) E (-1,1) \ {0}. The theorem is now proved. To complete our study of all the values of q and A we are interested in, it remains only to prove the following proposition. THEOREM 6.6. If either A < 1 and ~t <. 0 or A = 1, then the problem (6.1), (6.2) has no nontrivial solutions.
PROOF: Let )~ < 1 a•d "7 ~< 0 and let u : (0, +co) --, tt be a nontrivial solution of the problem (6.1), (6.2). By Lemma 6.5 u is oscillating. Then by Lemma 6.2 the function Vu defined by equality (6.3) assumes positive values in any neighborhood of +oo. This, however, contradicts Lemma 6.4. For A = 1 it suffices to consider the general solution of Eq. (6.1), which in this case can be written out explicitly. The theorem is now proved. A survey of papers devoted to problems of the type (6.1), (6.2) can be found in [1]. LITERATURE CITED 1. I. V. Amirkhanov and E. P. Zhidkov, "Some questions of existence and qualitative behavior of particlelike solutions," Khzp. fiz. kut. int~z. [Publ.], No. 82, 165-180 (1979). 2. N. K. Balabaev, V. D. Lakhno, and A. M. Molchanov, "Excited self-consistent electron states in homeopolar crystals," Preprint, Scientific Center for Biological Research, Akad. Nauk SSSR, Pushchino (1983). 3. S. N. Bernshtein, "On the equations of the calculus of variations," Usp. Mat. Nauk, 8, No. 1, 32-74 (1940). 4. N. I. Vasil'ev and Yu. A. Klokov, Foundations of the Theory of Boundary-Value Problems for Ordinary Differential Equations [in Russian], Zinatne, Riga (1978). 5. N. I. Vasil'ev and A. I. Lomakina, "On a two-point boundary-value problem with nonsummable singularity," Differents. Uravn., 14, No. 2, 195-200 (1978). 6. G. D. Gaprindashvili, "On a boundary-value problem for systems of nonlinear ordinary differential equations with singularities," Differents. Uravn., 20, No. 9, 1514-1523 (1984). 7. N. V. Gogiberidze and I. T. Kiguradze, "On the question of the nonoscillatory character of singular linear second-order differential equations," Differents. Uravn., 10, No. 11, 2064-2067 (1974).
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8. V. V. Gudkov, Yu. A. Klokov, A. Ya. Lepin, and V. D. Ponomarev, Two-Point Boundary-Value Problems for Ordinary Differential Equations [in Russian], Zinatne, Riga (1973). 9. E. P. Zhidkov and V. P. Shirikov, "On a boundary-value problem for ordinary second-order differential equations," J. Vych. Mat. i Mat. Fiz., 4, No. 5, 804-816 (1964). 10. G. G. Kvinikadze, "On a singular boundary-value problem for nonlinear ordinary differential equations," In: Ninth International Conference on Nonlinear Oscillations," Vol. 1, 166-168, Naukova Dumka, Kiev (1984). 11. I. T. Kiguradze, "On the Cauchy problem for ordinary differential equations with singularities," Soobshch. Akad. Nauk Gruz. SSR, 37, No. 1, 19-24 (1965). 12. I. T. Kiguradze, "On the Cauchy problem for singular systems of ordinary differential equations," Differents. Uravn., 1, No. 10, 1271-1291 (1965). 13. I. T. Kiguradze, "On a priori estimates of the derivatives of bounded functions satisfying second-order differential inequalities," Differents. Uravn., 3, No. 7, 1043-1052 (1967). 14. I. T. Kiguradze, "On some singular boundary-value problems for nonlinear ordinary second-order differential equations," Differents. Uravn., 4, No. 10, 1753-1773 (1968). 15. I. T. Kiguradze, "On a singular two-point boundary-value problem," Differents. Uravn., 5, No. 11, 2002-2010 (1969). 16. I. T. Kiguradze, "On nonoscillating conditions for singular linear second-order differential equations," Mat. Zametki, 6, No. 5, 633-639 (1969). 17. I. T. Kiguradze, "On monontonic solutions of nth-order nonlinear ordinary differential equations," Izv. Akad. Nauk SSSR, Set. Mat., 33, No. 6, 1373-1398 (1969). 18. I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilisi University Press, Tbilisi (1975). 19. I. T. Kiguradze, "On the solvability of the Vall6e-Poussin boundary-value problem," Differents. Uravn., 21, No. 3, 391-397 (1985). 20. I. T. Kiguradze and I. Rach~lnkov~, "On the solvability of a nonlinear problem of Kneser type," Differents. Uravn., 15, No. 10, 1754-1765 (1979). 21. Yu. A. Klokov, "A method of solving a limiting boundary-value problem for a second-order ordinary differential equation," Mat. Sb., 53, No. 2, 219-232 (1961). 22. Yu. A. Klokov, Boundary-Value Problems with Condition at Infinity for the Equations of Mathematical Physics [in Russian], RIIGVF, Riga (1963). 23. Yu. A. Klokov and A. I. Lomakina, "On a boundary-value problem with singularities at the endpoints of the interval," Latv. Mat. Yearbook, 17, 179-186 (1976). 24. N. L. Korshikova, "On the zeros of solutions of higher-order linear equations," In: Differential Equations and Their Applications, 143-148, Moscow State University Press, Moscow (1984) 25. N. L. Korshikova, "On the zeros of the solutions of a class of nth-order linear equations," Differents. Uravn., 21, No. 5, 757-764 (1985). 26. M. A. Krasnosel'skii, "On a boundary-value problem," Izv. Akad. Nauk SSSR, Ser. Mat., 20, No. 2,
241-252 (1956). 27. M. A. Krasnosel'skii and M. G. Krein, "On the averaging principle in nonlinear mechanics," Usp. Mat. Nauk, 10, No. 3, 147-152 (1955). 28. M. A. Krasnosel'skii, A. I. Perov, A. I. Povolotskii, and P. P. Zabreiko, Plane Vector Fields, Academic Press, New York (1966). 29. A. Ya. Lepin, "Existence of a solution of a nonlinear boundary-value problem for an nth-order ordinary differential equation with a singularity at the endpoints," Latv. Mat. Yearbook, 4, 215-230 (1968). 30. L. A. Lepin, "Generalized solution and solvability of boundary-value problems for a second-order differential equation," Differents. Uravn. 18, No. 8, 1323-1330 (1982). 31. L. A. Lepin, "The method of lower and upper functions for second-order differential equations on open and half-open intervals," Proceedings of the Extended Sessions of the Seminar of the Vekua Institute for Applied Mathematics, 1, No. 3, 81-84 (1985). 2414
32. A. A. Logunov and A. A. Vlasov, Minkowski Space as the Foundation of the Physical Theory of Gravitation, Moscow State University Press, Moscow (1984). 33. A. G. Lomtatidze, "On a boundary-value problem for linear differential equations with nonintegrable singularities," Proc. Vekua Inst. Appl. Mat., 14, 136-144, (1983). 34. A. G. Lomtatidze, "On a singular boundary-value problem for linear second-order differential equations," In: Boundary-Value Problems, Perm. Polyt. Inst., 46-50 (1984). 35. A. G. Lomtatidze, "On solvability of boundary-value problems for nonlinear second-order ordinary differential equations with singularities," Proceedings of the Extended Sessions of the Seminar of the Vekua Institute for Applied Mathematics, 1, No. 3, 85-92 (1985). 36. A. G. Lomtatidze, "On oscillation properties of solutions of second-order linear differential equations," Proceedings of the Seminar of the Vekua Institute of Applied Mathematics, 19, 39-53 (1985). 37. A. G. Lomtatidze, "On a boundary-value problem for second-order nonlinear ordinary differential equations with singularities," Differents. Uravn., 22, No. 3, 416-426 (1986). 38. A. G. Lomtatidze, "On a singular three-point boundary-value problem," Proc. Vekua Inst. Appl. Mat., 17, 122-1134 (1986). 39. A. G. Lomtatidze, "On positive solutions of singular boundary-value problems for nonlinear secondorder ordinary differential equations," Differents. Uravn., 22, No. 6, p. 1092 (1986). 40. E. I. Moiseev and V. A. Sadovnichii, On the Solution of a Nonlinear Equation in the Theory of Gravitation Based on Minkowski Space [in Russian], Moscow State University Press, Moscow (1984). 41. E. I. Moiseev and V. A. Sadovnichii, "A study of the solution of a nonlinear equation of the theory of gravitation," Dokl. Akad. Nauk SSSR 282, No. 4, 845-847 (1985). 42. E. I. Moiseev and V. A. Sadovnichii, "The solution of a boundary-value problem for a nonlinear eqaution of the thi.~ory of gravitation," Dokl. Akad. Nauk SSSR, 284, No. 4, 835-837 (1985). 43. E. I. Moi!seev and V. A. Sadovnichii, On Boundary-Value Problems for a Nonlinear Equation of the Theory of Gravitation, Moscow State University Press, Moscow (1986). 44. N. F. Morozov, "On the analytic structure of the solution of the membrane equation," Dokl. Akad. Nauk SSSR, 152, No. 1, 78-80 (1963). 45. N. F. Morozov and L. S. Srubshchik, "Application of Chaplygin's method to the study of the membrane equation," Differents. Uravn., 2, No. 3, 425-427 (1966). 46. A. D. Myshkis and G. V. Shcherbina, "On a limiting boundary-value problem not satisfying the Bernshtein condition and have applications in the theory of capillary phenomena," Differents. Uravn. 12, No. 6, 991-998 (1976). 47. A. I. Perov, "On a two-point boundary-value problem," Dokl. Akad. Nauk SSSR, 122, No. 6, 982-985 (1958). 48. A. I. Perov, "On a boundary-value problem for a system of two differential equations," Dokl. Akad. Nauk SSSR, 144, No. 3, 493-496 (1962). 49. A. I. Perov, "On the singular Cauchy problem," Proceedings of the Seminar on Functional Analysis of Voronezh University, 7, 104-107 (1963). 50. I. Rachunkova, "On the Kneser problem for systems of nonlinear ordinary differential equations," Soobshch. Akad. Nauk Gruz. SSR, 94, No. 3, 545-548 (1979). 51. G. Sansone, "Equazione Differenziali nel Campo Reale," Vol. 2, Zanichelli, Bologna (1949). 52. L. S. Srubshchik and V. I. Yudovich, "The asymptotics of the equation for large bending of a round symmetrically loaded lamina," Sib. Mat. J., 4, No. 3, 657-672 (1963). 53. G. S. Tabldze, "On approximate solution of a two-point singular boundary-value problem for a secondorder ordinary differential equation," Differents. Uravn., 10, No. 5, 851-859 (1974). 54. G. S. Tabidze, "On numerical solution of a two-point singular boundary-value problem," Proc. Vekua Inst. Appl. Mat., 17, 153-179 (1986). 55. E. L. Tonkov, "On a second-order periodic equation," Dokl. Akad. Nauk SSSR, 184, No. 2, 296-299 (1969). 56. Philip Hartman, Ordinary Differential Equations, Wiley, New York (1964). O
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