Boundary-value problems for systems of ordinary differential equations

t'-'

,~ +

qk(t) dt + ~ " ~ l l y , ollo

]

( m = 1,2,...).

(5.16)

i=1

--

PROOF: According to L e m m a 5.2 there exists a number b E (0, 1) such that the functions and functionals 15~k and ~50,, (i, k : 1 , . . . , n) given by equalities (5.10) satisfy the condition (5.9). For each i E { 1 , . . . , n} we introduce the transformation

(//)1/ (I)["

hi(zx,...,zn)(t)--~oi(Zl,...,zn)exp

a~ (r) dr

+

exp

a~(s)ds

Z

k~i,k=l

on the set

C([a,b];R~ ), where a~(t) = p.(t)sgn(t - t,). Let Zio(t) ~ 1

(i---- 1 , . . . , n ) ,

r/= 2

P,,Ct,

exp i=l

and let (z~,~)~=l, (m :- 1, 2 , . . . ), be the sequence of vector-valued functions given by the equalities Z / r e ( t ) - - - - - h ~ ( Z l m _ l , o . . , z n . _ 1 )(t)-~-~/

(i:

1,...,n;m

(5.17)

= 1,2,...).

It is easy to see that

l~
for

Consequently

a<.t<.b,

(i=l,...,n;m=l,2,...).

n

p- = ~

I1,,,,,11o

(,-,, = 1,2,...)

i=1

is a nondecreasing sequence of positive numbers. We shall show that p=

lim

Suppose the contrary, i.e., that p,~ - , +oo as m --* +oo. h , ( x , , - 1 , . . . , x - - - 1 )(t), and , ,

= - r/ -. P,,,

(5.18)

p,~ < + c ~ .

Let us set xi,,~ (t) -

1 P~

, , . (t), ~ , . (t) =

Then

lim , . = o,

(5.19)

n

~11~,-I1o = 1

(re=l,2,...),

(520)

i=1

and the sequence of vector-valued functions (~i~)i~x, (m = 1, 2 , . . . ), is uniformly bounded and equicontinuous. Therefore lim

m--*+oo

where (xi),"-_l E

sup~i~(t) =

uniformly on [a,b]

(i = 1 , . . . , n ) ,

C([a,b];R~_). On the other hand, we find from (5.17)

xd,.,,,.(t)~i~(t)-{"-~ 2282

~i(t)

for

a~t~b

(i=l,...,n;m:l,2,...)

(5.21)

and zim(t)<
-~-t]rrt_l,...,~nm_

1

-{-t}m_l)(t )

a ~ t ~< b

for

(i = 1 , . . . , n ; m : 1 , 2 , . . . ) .

In view of (5.19)-(5.21) it follows from these inequalities that n i=l

and

9 i(t) ~<xi(t)

(i = 1,...,n),

for a ~ t ~ < b

where (i = 1 , . . . , n ) .

9,,(t) = h , C ~ , . . . , ~ , ) C t )

Therefore n

[x~(t)

-

ai(t)zi(t)] sgn (t

-

t,) = k#i,k=l n

<

E

P'kCt)xk(t)

for

a
(i=l,...,n)

k~i,k= l

and xiCti) _7_ ~ t ~ 0 i ( . ~ l , . . . , : ~ n )

<~ ~ 0 i ( ~ : 1 , . . .

,~rt)

(i :

1,...,/g).

n Consequently ( ~)~=t is a nonzero nonnegative solution of the problem (5.12), (5.13). But this contradicts the condition (5.9). The contradiction so obtained proves inequality (5.18). Let It

(y,o)in=t CC([a, bl;R:),

q0 = Ellyio[[c > 0 , i=1

and let 7,~ E R + , (!/im)~'=t E O([a,b];R~), and q~ E L([a,b];R+), (m = 1 , 2 , . . . ) be arbitrary sequences satisfying inequalities (5.14) and (5.15) for each m. We set

f,,, = E 6 "~-k 7k +

"

( f : qk(t) ) dt

~,oCt) = t,

~ , m ( t ) - u,~,Ct)

k=l

+r

llu,o l b , i-1

(i= 1,...,n).

Taking account of the inequalities (ra = 1 , 2 , . . . ) , we find from (5.14) and (5.15) that (i = 1 , . . . , n ; m = 1 , 2 , . . . ) . In view of (5.17) we obtain from this the result that

~i,,~(t)~zi,7,(t)

for

a~t~b

(i=l,...,n;m=l,2,...)

and i=1

i=1

Consequently we have estimates (5.16). Since p is independent of (yio)in=l, these estimates will also hold in the case when Yio (t) ~ 0, (i = 1 , . . . , n ) . The lemma is now proved.

2 2 8 3

U(h,...,t,,).

5.2. O n t h e Set LEMMA 5.4. Let

n

~oo,(~,...,~,) = ~ t,,llx, ll..,

li,. e R + ,

(i,k = 1,...,n),

P,k e L~'([a, bl;R+ )

k=l

for 1 <<.# ~ +r

-

1 2 + - = 1, and let the eigenvalues of the matrix kt i/ 2

lip,, II,.,,'

b a)

A=

-

i,k= l

be less than one in absolute value. Then condition (5.1) holds.

PROOF: Let (x,),"_ x be an arbitrary nonnegative solution of the problem (4.3), (4.4). Then 9 ,Ct)

.< ~ l , kll~,ll~- + k=l

p,,Ct)~,(~)

for

a ~< t ~< b

(i = 1 , . . . , n ) .

k=l

According to the Minkowski and HSlder inequalities it follows from this that 1

I1~,11.~,,'< (b-a)~ ~t,,,,,ll~,ll.. +

"

'=1

IIn,,ll.,-.,. '=1

v

[/: I// I*~(*)1~

dr 2

at

] -v

( i = 1,...,n).

On the other hand, by Wirtinger's inequality ([62], p. 409) 1

2

Therefore ( E - A)r ~< 0,

(5.21)

where r = (]]Xi]]L~)i~=l and E is the identity matrix. Since the eigenvalues of the matrix A are less than one in absolute value, it follows from this last inequality that r = 0. Consequently the problem (4.3), (4.4) has no nonzero nonegative solution. The lemma is now proved. LEMMA 5.5. Let

~0,(~1,...,~.) = ~,(s,),

~, ~ [a,a],

~, # t ,

(i = 1 , . . . , , ~ ) ,

let Pi, (t) = Pi, = const, p , < 0, p,, /> 0 for i # k, and let the real parts of the eigenvalues of the matrix (Pik)in,=l be negative. Then condition (5.1) holds.

PROOF: Set

a.=O,

a~.-lp,,Pi, I

for

i#k

and

A=(ai,)i~k=x .

It follows from the restrictions imposed on the matrix (pi,)i",k=l that the eigenvalues of the matrix A are less than one in absolute value. n Let (X '),=1 be an arbitrary nonnegative solution of the problem (4.3), (4.4). Then 9 ,(t) <~ 6,(t)~,(s,) + (1 -6,(t)) ~ k=l

2284

o~, II~,llc for a<~t<<.b ( i = l , . . . , n ) ,

where &i(t) =

expCp,,It

- t,I). Since s i r

ti we find from this t h a t n

=,(~,) < ~a,~ll=~llc

(i= 1,...,n)

i=l and

n

II=,llc < ~ l l = ~ l l c

(i= 1,...,n).

k=l

Consequently the vector r = (11=,11c),"=1 satisfies inequality (5.21), f r o m which we obtain r = 0, i.e., xi(t) = O, (i =: 1 , . . . ,n). T h e l e m m a is now proved. w

Proofs of the Theorems

PROOF OF THEOREM 4 . 1 : T h e necessity is obvious, for if (x~)~"__~is a solution of the p r o b l e m (4.1), (4.2), then conditions (4.5)-(4.7) hold for the vector-valued functions ax (t) = a2(t) = (xi(t))~= 1 . Let us prove the sufficiency. For any i C { 1 , . . . , n} we set

{ Otli(t) Xi(t,8) =

8

< 0tli($),

for

8

for

~li(t) < 8 < rv,i(t),

a,i(t) for s > a,iCt), 2,(u)(t) = xi(t,u(t)), ~(t, x l , . . . , z , ) = fi(t, Xl(t, Zx),...,• and

~ i ( X l , . . . , 2 : n ) ~-~ r T h e n on [a, b] • R " we have the inequality n

~ I.,,r

< q(t),

d=l where

qCt) : m a x

{"Y]~lf~(t, Xl,...,x,)l:

Otl(t) ~ (Xj)jn_--I ~ ot2Ct)

}

/=1 and q E L([a, b]; R + ). On the other h a n d , in view of (4.7) we have on C ( [ a , b ] ; R " )

~li(ti) ~ ~ i ( X l , . . . , ~ , )

~ Ol21(ti)

(i = 1 , . . . , n ) .

Since in addition the p r o b l e m

dxi dt

=0,

xi(t,) = O,

(i = l , . . . , n )

has only the zero solution, by Corollary 2.1 the differential s y s t e m

dx i dt = ~(t, X l , . . . , ~ n )

(i = l , . . . , n )

(6.1)

n possesses a solution ( i)i=x satisfying the b o u n d a r y conditions

x,(ti)

:

~,(zl,...,x~)

(i

=

1,...,n).

(6.2) 2285

To prove the theorem it suffices to establish that

ali(t) <~xi(t) <~ot,i(t)

for

a ~< t ~< b

(i = 1 , . . . , n ) ,

(6:3)

for each solution of the problem (6.1), (6.2) satisfying these inequalities is simultaneously a solution of the problem (4.1), (4.2). Suppose condition (6.3) is violated, i.e., for some k E {1,2}, i E { 1 , . . . , n } and T0 E [a,b] the function

u(t)

= (-1)* [zi (t) - a~i(t)]

satisfies the inequality

=(To) > o.

ti.

According to (6.2) and (4.7) we have To # exists q E [t,, TO) such that

For definiteness we shall suppose that To > tk. Then there

u(t)

'~Cn) = 0,

> 0,

for

T1 < t • T0.

(6.4)

On the other hand, by (4.6) we have u' (t) = ( - 1 ) ~ [fi(t, X1 ( X l ) ( t ) , . . . , Xi-1

(Xi-1)(t), Olki(t), Xi+I ( X i + l ) ( t ) , . . . , Xn (xn)(t)) - ot~i (t)] <~ 0 for

r, < r < r o ,

which contradicts conditions (6.4). The contradiction so obtained proves the theorem. PROOF OF THEOREM 4.2: Choose a positive number p so that the conclusion of Lemma 5.1 holds. Set

Po = P(7+ f=~q(t) dt), 1

x(s)

p,(t) =

pii(t)sgn(t -

t,),

=

for

Isl
2- Is_l

for

P0 < s < 2p0,

0

for

Isl/> 2p0,

(6.5)

P0

qi(t, xl,...

,x,) = X

Ixkl

[fi(t, xl,...

,x,) -

pi(t)xi]

_

(i = 1,... ,n). (6.6)

and

r Then

= x(

I1~,11o)~,(=1,...,~,)

( i = 1,...,n).

(6.7)

k=,

sup{~tq, C',zz,...

" ew' ,:~.)1:( x ~),=,

e L([a, b]; R+ )

(6.8)

i=1

and sup { ~ [ffi(Xl,...,x,=)[:(xk)~=a E i=1

c([a,b];R")}<+oo.

(6.0)

It is clear that the problem

dxi

dt =P'(t)x"

2286

x,(t,)

= 0

(i = 1 , . . . , n )

(6.10)

has only the zero solution. Taking account of the conditions (6.8) and (6.9) along with this circumstance and applying Corollary 2.1, we verify that the problem d~i

(i = 1,...,n),

dt = pi(t)zi + q i ( t , z , , . . . , z , , )

ziCti) = f f i ( z x , . . . , z , )

(6.11) (6.12)

(i=l,...,n)

is solvable. Let (z~)~L1 be an arbitrary solution of this problem. In view of the inequalities (4.8), (4.9) and equalities (6.5)-(6.7) it will also be a solution of the problem (5.2), (5.3). Therefore, by the choice of p estimate (5.4) holds, i.e.,

I1=~ I1o -<
q(t)

=

~t

y~ If~(t,o,...,o)l,

w = Y~ I~,(o,...,o)1.

i=1

i=1

Consequently all the hypotheses of Theorem 4.2 hold, which guarantees the solvability of the problem (4.1), (4.2). It remains only for us to prove that it has no more than one solution. 'n n Let (Xxi),=l and (~ ,,)i=l be arbitrary solutions of the problem (4.1), (4.2) and y, Ct) = Izx,(t)

(i = 1,... ,n).

- ==,(t)l

According to (4.12) and (4.13) (Yi)~=i is a nonnegative solution of the problem (4.3), (4.4). Therefore in view of (4.10) we have y~(t) = 0 (i = 1,... ,n). The theorem is now proved. Corollaries 4.1-4.4 follow directly from the theorems just proved and Lemmas 5.4 and 5.5. We shall now show that Remark 4.1 holds. Since condition (4.10) is violated, the problem (4.3), (4.4) rt . has a nonzero nonnegative solution (X i)i=l For each i 6 { 1 , . . . , n } we shall denote by Yl the solution of the Cauchy problem

Y~= [Pii(t)yi+ ~

PikCt)zk(t)]sgn(t-- tl),

y, Ct,) ==,(t,).

kgi,k=l

It is easy to see that

y~(t)>>.z~(t) for

a~
(i=l,...,n)

and the vector-valued function (~/i)~"--1 is a nonzero solution of the linear homogeneous problem dy/

dt

'~ = 2/~'kCt)Yk

(i = 1 , . . . , n ) ,

k=l

y, Ct,) = 6 , ~ (u,,...,y,)

( i = 1,...,~),

where ~i E [0,11,/~i, (t) = pii(t), ifiik E L([a,bl;R), and IP~k(t)l ~< p,k(t) for a < t < b, i # k. According to Remark 1.1 there exist numbers ci (i = 1 , . . . , n) such that in the case when n

f, Ct,=,,...,=,)

= ~,~,~(t)=~, k=l

~,C=l,...,=,)

= ~,~o,(=,,...,=,)

+ ~,

(i = l , . . . , n ) ,

2287

the problem (4.1), (4.2) has no solution, although fl and ~i (i = 1 , . . . , n ) (4.13).

satisfy inequalities (4.12) and I

PROOF OF THEOREM 4.4: Suppose the hypotheses of Theorem 4.2 hold and the problem (4.1), (4.2) has the unique solution x ~ = (Xi)i=l" 0 n By T h e o r e m 3.1 and Definition 3.3, to prove that the problem under consideration is well-posed it suffices to establish that x ~ is strongly isolated at an arbitrary radius r > 0. Choose p ~ (0, + c o ) such that the conclusion of Lemma 5.1 holds. Let po = r +

I1~~

(/:)

+ p ff +

q(t) dt

,

i=1

and let p~, q~, and if,, (i = 1 , . . . , n ) ,

be the functions and functionals given by equalities (6.5)-(6.7). Let

~ be the Kronecker symbol, P ( t ) = (Sikpi(t)),~k=l ,

ICx) = (xiCt,))'~=l,

lo(x) = 0

and ?(z) = -(~,(z~,.

. . ,~)),~=~.

Then

(fiCt, xl,...,x,))~.= 1 = P(t)x + (,(t, xl,...,x.))n=, (x, Ct,) - ~5,(xa,...,x,~))~'=, = l(x) + [(x)

for for

a ~< t ~< b,

IIx- x~

IIx- x~ <

~< r,

r

and conditions (6.8) and (6.9) hold. According to (4.10) the problem (6.10) has only the zero solution, i.e., the matrix-valued function P satisfies the Opial condition with respect to the pair (/,10). On the other hand the quasilinear problem (6.11), (6.12) has no solution except x~ for, as was noted in the proof of Theorem 4.2, each solution of this problem is simultaneously a solution of the problem (4.1), (4.2). Consequently x ~ is strongly isolated at radius r. The theorem is now proved.

PROOF o r Ta~.OREM 4.5: According to Theorem 4.3 the problem (4.1), (4.2) has the unique solution (x~)l'=x . On the other hand, it follows from condition (4.12) that for arbitrary i ~ { 1 , . . . , n}, c0 ~ R, and zk ~ C([a, b];R"), (k ~ i;k = 1 , . . . , n ) , the Cauchy problem du d'--[ = fi(t, z l ( t ) , , . . .

,zi-1 (t),u,z~+l ( t ) , . . . , z ~ ( t ) ) ,

,,(t,) = , o

has a unique solution defined on the whole interval [a,b]. Therefore for any (X,o),~=~ e C ( [ a , b ] ; R " ) there exists a unique sequence (x,~)~=x, (m = 1 , 2 , . . . ), such that for each natural number m and i e { 1 , . . . ,n} the function x ~ is a solution of the problem (4.15), (4.16). By (4.12) and (4.13), for any m the functions

y~, (t)= Ix,(t)- x~,(t)l

(i-- 1,...,n)

satisfy inequalities (5.14) and (5.15), where q~ (t) = 0 and ~,,, = 0. Therefore according to Lemma 5.3 estimates (4.17) hold, where r0 ~ (0, +co) and 6 ~ (0, 1) are numbers independent of m. The theorem is now proved. trY.MARK 6 . 1 . The process described above for constructing the solution of the problem (4.1), (4.2) is stable in a certain sense, h fact suppose the hypotheses of Theorem 4.5 hold. Then for any (~0)~:a ~ C([a, b]; R") (q~)':=a e L([a,b];R") and ('~,~)~'=~ e R ", (m = 1 , 2 , . . . ) , there exists a unique sequence of vectorvalued functions (~ir~)i= i n E 8 ( [ a , b ] ; R " ) , (m = 1,2,. . .), such that for any natural number m and any i E { 1 , . . . , n} the function Xin, is a solution of the Cauchy problem

9 ~(t) =/,(t,~,~_~

2288

(0,.-.,~,,-~-~ (t),~:,~ (0,~:,+,~-~ (t),...,.~.~_~ (0)+ q,,,,(O, 9 ,,,, (t,) -- ~,, (~,,,_~ ,..., ~,,,~_~ ) + ,~,,,,.

On the other hand, by L e m m a 5.3, trt

tg

i=,

k=0

n 9 where ( X ira)i=: 1 (m = 1 , 2 , . . . ) is the sequence occurring in Theorem 4.2,

,7o = ~--~ I1=:,o - ~:,ollo,

,7,,, --

i=1

I*.,(t)l dt

"~,,,,I +

,

(m : 1 , 2 , . . . ) ,

i=1

and p 9 (0, +.c~) and $ 9 (0,1) are numbers independent of m, q~,,,, and '7i,,,. From what has been said it follows in particular that if lim r/~ = O, Ytt-*-4- co

then

IIx,~-~,~llc

lim

m--,+oo

=0

Chapter TWO-POINT w

Multidimensional

(i=1,...,,~). 3

PROBLEMS

Differential Systems

In this section we establish criteria for solvability of the boundary-value problem

d~i

(i = 1,...,n),

= fiCt, X l , . . . , x n ) dt (z,(a))~= 1 9 S,,

(x,(b))i~=z 9 S=,

(7.1) (7.2)

where - c o < a < b < +c~, n/> 2, f = (f~),~l 9 K([a,b] • R " ; R " ) , and Sj C R " , (j = 1,2), are nonempty sets. 7.1. S t a t e m e n t o f t h e E x i s t e n c e T h e o r e m s . We first introduce the following DEFINITION 7 . 1 . (~/i)in__z 9 C([a,b];R n) is called a Jower [resp. upper] vector-valued function of the differentia/system (7.1) if there exists a set I0 of measure zero such that for all i 9 { 1 , . . . , n} and t 9 [a, b] \I0

f, Ct,=,,...,=,) ,< ~;Ct) [resp. f i ( t , x i , . . . , x , )

=, = .~,(t),

for

/>'7~(t)

for

xi ="/iCt),

~ <..~(t)

(k # i;k = 1 , . . . , , ) ,

xk >~"TkCt) (k # i;k = 1 , . . . , n ) ] .

For any ~ = ('Y'),~=I : [a,b] --* R " we set

M.,Ct) = {(=,);'=1 9

"=1 < ~lCt),...,=,,

M"Ct) -- {(=,);'=1 9

"=, > ~,(t),...,=,,

M= = M=

=

{(t, zx,...,z,):t {(t, xz,...,=,):t

< ~,=(t)}, >'~,,Ct)},

9 [a, bl, (xi)~'= , 9 M=(t)}, 9 [a,b], (=,),"=x 9 M=(t)}.

We shall say that the solution ( = i ) ~ l of the system (7.1) passes through the set D c [a,b] • R " if there exists to E [a,b] for which (to,zz(to),...,x,(to)) E D.

2289

If S c R n, we take X / ( a , S) to denote the set of all solutions of the system (7.1) maximally extended to the right and satisfying the initial condition !

(~,(~))~=~ 9 s. A solution (x,)i%l of the differential system (7.1) defined in the interval [a,t*) c [a,b) will be called singular if n

lim Z

Ix'(t)] = +oo.

t-,t*

i=1

THEOREM 7.1. Let $1 be a bounded continuum and S2 a closed set, and suppose there exist a lower vector-~lued

f u n c t i o n ~ = (~,),~=~ of the system (7.1) and an upper vector-~lued function ~ =

(~,),~=~

such that a) X ! (a, S~ ) contains both a solution passing through M~ and a solution passing through M# ; b) X! (a, St) contains no singular solutions not passing through Ma U M~ ; c) S2NM~(b) = $2 NM#(b) = ~ ; d) $2 has nonempty intersection with each continuum whose intersections with both M~ (b) and M ~ (b) are nonempty. Then the problem (7.1), (7.2) has a solution (xl)i~=x satisfying the condition (xi(t))~.=l ~ M ~ ( t ) U M # ( t

)

for

a<~ t<. b.

(7.3)

7.2. A u x i l i a r y P r o p o s i t i o n s . LEMMA 7.1. (cf. [71, 75]). Suppose there exists a function q E L([a,b];R+ ) such that the inequality n

~l:,(t,~.~,...,~.,,)l < q(t)

(7.4)

i=I

holds on [a, b] x t t ~ . Then for any continuum S C R n the set S0 = {Czi(b))~:l : (z,)~.=~ 9 X , Ca, S)} is also a continuum. n PROOF: We prove first that the set So is closed. Let (Cik)i=l (k = 1 , 2 , . . . ) , be an arbitrary convergent sequence of points of this set and

lim c~k = ci

(i = 1 , . . . , n ) .

k~+oo n Then there exists a sequence (X ~k)~=l, (k = 1 , 2 , . . . ) , of solutions of the system (7.1) such that

(xik(a))~.:, 9

(k= 1,2,...)

(7.5)

and xik(b)=ci,

(i: l,...,n;k=

l,2,...).

Because of (7.4) this sequence is uniformly bounded and equicontinuous. Therefore without loss of generality we may assume it converges uniformly. It is obvious that xi(t) =

2290

lim xi~(t)

k-*+oo

(i=l,...,n)

is a solution of the system (7.1). On the other hand, since S is closed, it follows from (7.5) that (x~)'~=1 9 Xy(a,S). Therefore (c,)'~=1 = (x,(b))~= 1 9 So. Consequently So is a closed set. To complete the proof of the lemma it remains for us to establish that So is connected. Suppose the contrary. Then there exist nonempty closed sets S (i} , (j = 1, 2), such that

So =

U

13 80c') =

Therefore

S = S (I)U S(2)' where S (j) , (j = 1,2), are such that

s~j)

= {(=,(b))'t=,

:

(=,),"--_, 9

X~(a, sCJ))}

(j =

1,2).

(7.6)

From the fact that S0(j} , (j = 1, 2), is closed, it follows that S(D is closed, (j = 1, 2). On the other hand, since S is a continuum S (1)[']S (2) # 0 . Let (7.7)

" 9 S(*) N S(2) C : ( c i)i=1 and let X! (a, c) be the set of solutions of the system (7.1) satisfying the initial condition

=,(a) = c,

(i = 1,...,,,)

and Q = {(x,(b)),~ 1 : (x,),~ 1 9

Xf(a,c)}.

Then according to (7.6) and (7.7) Q C So,

Q N S~i) ~ 0

(j = 1,2).

But this is impossible, since by Kneser's Theorem ([62], p. 28) Q is a continuum. The contradiction thus obtained proves the lemma. LEMMA 7.2. Suppose S1 is a continuum and there exist a lower vector-valued function a = (c~)i~l of the system (7.1) and an upper vector-valued function/~ = (3i),"_-1 satisfying conditions a), c) and d) of Theorem 7.I. Suppose in addition that each function fi has a partial derivative

f'(t,=l,...,=.)

]~ 9

=

a.~(t,=.l,... a=,

,=,)

(7.s)

K([a,b] x R " ; R )

and inequality (7.4) holds on [a,b] x It", where q 6 L([a,b];R+ ). Then the problem (7,1), (7.2) has a solution satisfying the condition (7.3). PROOF:

We first show that an arbitrary solution (z~)~=l 6 X l ( a , $1) passing through M g satisfies the

condition

(x,(b))~=l 6 M/JOb).

C7.9)

Suppose the contrary. Then there exist (x~)'~=1 9 X f ( a , SI) , to 9 [a,b), tl 9 (t0,b], and k 9 { 1 , . . . , n } such that =~(t) > ~,(t) for to < t < t, (i= 1,...,n) (7.10)

2291

and

xk(tl) = flt(tl).

(7.11)

Setting u(t) = xk (t) - ~k (t), according to the definition of an upper vector-valued function and conditions (7.8) and (7.10), we find

= [/(t, x l ( t ) , . . . , ~ C t ) ) -/(t,~l(t),...,~_l ( t ) , ~ ( t ) , ~ + 1 ( t ) , . . . , ~ , ( t ) ] + [f(t,x~(t),...,zk_~ (t),~k(t),xk+l (t),...,z,(t))--Z2(t)]/> -g(t)u(t)

r

for

to ~< t ~< tx,

where

9(t) = m ~ x ( I f ~ ( t , ~ l ( t ) , . . . , ~ _ ,

(t), s , ~ + ~ ( t ) , . . . , ~ ( t ) ) l

: ~(t) < s < ~(t)}.

From this it follows immediately that U(tl) > ~ e x p ( - ~ t i ' g ( r )

dr)u(to ) > 0 .

But this contradicts equality (7.11). This proves the inclusion (7.9). In a completely analogous manner we can prove that an arbitrary solution (.T~)~=1 n 9 X f ( a , S 1 ) passing through the set M~ satisfies the condition (x,(b))~'=l 9 M~(b). By Lemma 7.1 the set

Slo = {(xi(b))~= 1 : (x,),~ 1 e XI(a, Sl)} is closed. On the other hand, according to condition a) and the property established above for solutions passing through the sets M~ and M ~ , we have

Slo~M,(b)#O,

SloNM~(b)#O.

Because of the condition d) it follows from this that Sl0 ~ S2 # 0. Consequently the problem (7.1), (7.2) is solvable. It remains only to note that because of the restriction c) imposed on the set S~ an arbitrary solution of the problem under consideration satisfies condition (7.3). LEMMA 7.3. Let S C R'* be a nonempty bounded closed set; let M C [a, b] x R n, and suppose the set

n 9 R~: (t,Xl,...,xn) 9 M ( t ) - - {( x '),:1 is open for any t 9 [a, b]. Suppose also that X! (a, S) contains no singular solutions not passing through M. Then there exists a positive number r such that for any (xi)i"=l 9 X l ( a , S ) and bo 9 (a,b] the estimate n

I=,(t)l

<

~ for

a <~ t ~< b0

(7.12)

i=l

holds provided (x, (t)),"__1 ~ M(t) P ROOF:

a ~< t ~< b0.

Let n Ix, l: (~,),=1 e s

r0 = max i--1

2292

for

(7.13)

and

If,(t,=,,...,=,,)l:

f* (t, r) : max {

I=,l
i=1

9

i=1

Choose a0 E (a, b] such that

L

.o f*(t, ro + 1)dr < 1.

(7.14)

" E X~, (a, S) admits the estimate Any solution ( x i),=x

I=,(t)l < ~o +

~,

/=X

I=,(~11 d~ i=1

9 n on its interval of definition. Hence because of (7.14) it follows that the interval of definition of (x.)~=l contains [a, ao] and n

~.l=,(t)l

< ro + l

for

a ~ t < ao.

i=1

Suppose now that the lemma is false. Then, according to the choice of o~, for each natural number m there exist b,~ e (a0,b] and (x~,~)~*=x e X / ( a , S ) such that (xi~(t))i~=l ~ MCt) [xi, (t)[ < r0 + 1 for

for

a ~< t ~< b,~,

(7.15)

~

(7.16)

a ~< t ~< a0,

i=l

Ixi,,~(b,)l > m.

i=X

Let

(t)

f =,,,, (t)

for

,, <.

I

for

b~ ~ < t ~ b ,

x~(b~)

y(t) = sup

t < b,,,,

I=,,~ (~)1: ~ < 9 < t,

m ~> 1

i---I

and t* = s u p { t e [ a , b ) : v ( t ) < + 0 0 } Because of (7.16) ao ~< t* ~< liminfb,~ ~ - - * - { - OO

and

n

limsup ~ m--*+oo

IX,,,, (t*)l = +co.

(7.17)

i=l

From the definition of t* it follows immediately that the sequence ( ~,~)~=1, (m = 1, 2 , . . . ), is uniformly bounded and equicontinuous on each closed interval contained in [a, t* ). Without loss of generality we may consider this sequence to be uniformly convergent on each such closed interval. It is easy to see that the vector-valued function (x,(t))~.=x = lim (xi,~(t))~.= x for a <. t < t * m---* + c o

is a solution of the differential system (7.1) and, since S is closed, " ( = ,),=, e xI(a,s).

2293

On the other hand we have from (7.15)

(xd(t))'~=1 r M(t) since

M(t)

for

a~
is open for any t 9 [a, b]. From this it is clear that

sup

r* =

< +oo,

Ix,(t)l : a ~< t < t* i=1

for X ! (a, S) contains no singular solutions not passing through M . Let t. 9 (a0, t*) and the natural number m0 be such that

ft[" f*(t,r* and

+2)dt < 1

n

~--~ Iz,. (t.)l

for

< r* + 1

m > m0.

i=1

Then

TI

y~l~,=(t)[
for

t, <<.t<~t* ( r e = m 0 + 1 , . . . ) .

i=1

But this estimate contradicts condition (7.17). The lemma is now proved. '/.3. P r o o f o f t h e E x i s t e n c e T h e o r e m . According to Lemma 7.3 there exists a positive number r such " 9 Xf(a,S) and b0 9 (a,b] estimate (7.12) holds whenever that for any ( x i)i=1

(xi(t))~.=1 ~ M~(t) UM~(t )

for

a ~< t ~< b0.

(7.18)

Without loss of generality we shall assume that

r > sup

Ix, l: (~,),% 9 sl

{n

and r>max

(7.19)

/=1

~(l~,(t)l

+ I/~(t)l) : a ~< t < b

} .

(7.20)

(i = 1 , . . . , n )

(7.21)

i=1

Let X(U)

f u rsgnu

for for

lull r,

~(t,x,,...,x,,) = fi(t,x(zl),...,X(X~)) and

"71i(t) =min{a,(t),~,(t)},

"/2i(t)=max{ai(t),~i(t)}

(i= 1,...,n).

We set

~- ~(t)

for

~l,(t) .< ,, < -~,,(t),

for

u < "y, (t),

for

u > '72i (t),

for

~. (t) ~< u ~< "~2~(t),

r

for

u < "~1i(t),

r it, -~., (t))

for

u > "72i(t).

=

~,(t,~,,Ct)) ~,(t,~,,Ct)) ~,(t) -,~

r

2294

=

if a,(t) # ~i(:t) and

1

= r

~,(t,=)

if a,(t) = ~i(t). For any natural number m we introduce the functions

gim(t,Xl,...,Xn) = m

f

zi4"

~(t, xl,...,Xi-l,tt, Xi+l,...,xn)du,

flm (t, X l , . . . , X n ) = girn (t, X l , . . . , X n )

"~- [/i (t, 2~1,..., 2~i-1 , ~ i ( t ) , 2 ~ i + l ,. .. ,Xn) -- g'm (t, X l , . . . , X i - 1 ,~i(t), X,+I,... , X ~ ) 1 9 9 , ( t , X , ) + [~ (t, X l , . . - , ~ ' - 1 ,~,(t),~,+~ , . . . , ~ ) - g,, (t, ~ 1 , . . . ,~,-~, ~ (t), ~,+1 , . . . ,~)]r and consider the differential system

dxi

dt = fi,~(t, xi,...,xn)

It is clear that the functions that

Of~,~(t, xl,...,x,), Oxi

f~(t, Xl,...,x,_l,ai(t),xi+l,...,x,)--

(t'= 1 , . . . , n ) .

(7.22)

(i=l,...,n),belongtotheclassK([a,b]•

~(t, xl,...,Xi_l,aiCt),X,+l,...,xn)

(i=l,...,n)

(7.23)

(i= 1,...,n).

(7.24)

and that

fd,nCt, xl,...,Xd_l,~i(t),xi+l,...,x,)

=- ]~(t, Xl,...,Xd-l,fli(t),X,+l,...,x,)

In addition the conditions lim t'/1,--"4 -~- r

f~Ct, xl,...,x,,)= ]~(t, xl,...,z,)

(i=l,...,n)

C7.2s)

and

~lf,.~(t,~,,...,~)l

< q(t)

(m = 1,2,...)

(7.26)

i=1

hold on [a, b] • R " , where

q(t)

= 3max { ~lf,(t,~l,...,~,)l:

1~11 < ~ , . . . , 1 ~ 1 < ~ 9

/=1

Because of (7.20) and (7.21) it follows from the equalities (7.23) and (7.24) that c~ and/~ are lower and upper vector-valued functions of the differential system (7.22). Therefore according to Lemma 7.2 the problem (7.22), (7.2) has a solution (x~,~)~=l such that

(x,,~(t))d~=l ~ M~(t) UM~(t )

for

a ~ t ~ b.

(7.27)

The fact that the set $1 is bounded and the inequalities (7.26) guarantee that the sequence (x~)~'=l is uniformly bounded and equicontinuous, (rn = 1, 2 , . . . ), so that without loss of generality this sequence can be considered uniformly convergent. In view of (7.25) and (7.26) and the fact that the sets $i, (j = 1,2), are closed, the vector-valued function (~,Ct))~'=l

=

lim

m--*+co

(~,mCt))~=l 2295

is a solution of the differential system

dxi = ~(t, x l , . . . , x , ) (i=l, .,n) dt "" with the boundary conditions (7.2). On the other hand it is clear from (7.27) that (x~)~'=l also satisfies condition (7.3). It remains for us to show that (xi)i~x is a solution of the system (7.1). According to (7.21) in order to do this it suffices to establish that n

Z

Ix,(t)] < r for

a <~t <<.b.

i=1

Suppose the contrary. Then because of (7.2) and (7.19) there exists bo e (a,b] such that t~

It

Ix,(t)l < r for

a .< t

< bo,

Z Izi(b~

i=1

--

r.

i=l

According to (7.21) the restriction of (xi)i~x to [a, b0] is a solution of the system (7.1). However (xi)~'=l also satisfies condition (7.18). Therefore, because of the choice of r, estimate (7.12) holds. We have now obtained a contradiction, which proves the theorem. 7.4. Corollaries of t h e E x i s t e n c e T h e o r e m . Consider the case when the boundary conditions (7.2) have the form xi(a) =~oi(x,(a)) ( i = l , . . . , n - 1 ) , ~o(xl(b),...,x,(b))--O, (7.28) where ~, 6 C ( R ; R )

(i = 1 , . . . , n -

1),

p 6 C(R";R).

(7.29)

COROLLARY 7.1. Suppose there exist a lower vector-valued function a = (ezi)i"_-i of the differential system (7.1), an upper vector-valued function 13 = (/3,)i"__1, and a positive number r such that

~,(u) < a,(a) for ~i (u) /> B, (a) for u / > r ,

u~<-r, (i=l,...,n-1),

(7.30)

and W(xl,...,x,) #0

for

(7.31)

(x,),n=l 6M~(b) U M~(b ).

Suppose further that the system (7.1) with the initial conditions (7.32) 9 ,(a) = ~,C~.(a)) (i= 1,...,~1) has no singular solutions not passing through Ms 13 M a 9 Then the problem (7.1), (7.28) has a solution (z,),=l satisfying condition (7.3). PROOF: Without loss of generality we may suppose that r > max

{l~.(a)l, la.Ca)l}.

We set {~o~(u)

#i(u) =

wi(r) + 1 ~oi(-r)-I

r_u r

for

- r ,< u ,< r,

for

u > r

for

u<-r,

(i = 1 , . . . , n -

1),

(7.33)

U

$1 ={(xi)i~x 6 R " :xi = ~ i ( x , )

(i=l,...,n-1)},

and

s2 = {(x,)L1 e R " : ~ ( X l , . . . , ~ , ) = 0}. In view of (7.29)-(7.31) the sets $1 and $2 are respectively a bounded continuum and a closed set for which conditions a)-d) of Theorem 7.1 hold. Therefore the problem (7.1), (7.2) has a solution (xi)L 1 satisfying the condition (7.3). It follows from (7.3), (7.30), and (7.33) that x,(a) E [-r,r]. Consequently (xi)L 1 satisfies the boundary conditions (7.28). The corollary is now proved. 2296

Suppose there exist positive numbers rk, (k = 1,..., n) and a function q E L([a, b]; R + )

COROLLARY 7.2.

such that

~oi(u)sgnu/> r/ (i = 1 , . . . , n - 1) for I=1 r,, ~9(Xl,...,~n) #0 for fi(t, xt,...,x,)sgnxl>lO ( i = l , . . . , n )

xksgnxx > r k

( k = 1,...,n), for a~ r k

(k=l,...,n)

(7.34) (7.35) (7.36)

and the inequality

~fi(t,xl,...,zn)sgnzi<.q(t)(l+~-~lz, i=1

I)

(7.37)

i=l

holds on [a,b] • R " . Then the problem (7.1), (7.28) is solvable. PROOF: In view of (7.34)-(7.36) a(t) = (--ri)n=l and fl(t) = (ri)~= 1 are lower and upper vector-valued functions of the differential system (7.1) for which the conditions (7.30) and (7.31) hold. On the other hand according to (7.37) each solution of the system (7.1) satisfying the initial conditions (7.32) is nonsingular since on its interval of definition it admits the estimate n

]Xi(t)[ ~ (i~__1 Ixi(a)l) exp ( / t

q(r)dr).

i=1

The corollary is now proved.

Suppose

COROLLARY 7.3.

(7.38)

liminf ~oi(u) sgn u > ro (i = 1 , . . . , n - 1), lul-.+ao ~O(Xl,...,xn)#0 for x, sgnxn >r0 ( k = l , . . . , n - 1 )

(7.39)

and the inequalities

rl[Xi+l I < fi(t, xl,...,xn)sgnxi+l

[

<<. r=l=,+ll

.

fn(t, x l , . . . , x n ) s g n z , ~< ho(t, xl)+hl(Xl)E(1

(i

=

+ I=,1)

i=2

1,...,n-- 1),

(7.40)

.1

E ( 1 +1=,1) '-1

(7.41)

i=2

and

f,(t, xl,...,x,-1,0)sgnx,-1 ~>0 for I=,-11 > to, (7.42) hold on [a, b] '< R", where ri, (i = 0,1, 2), are positive constants, ho E K([a, b] x R; R+ ) and ht E C(R; R+ ). Then the problem (7.1), (7.28) is solvable. P ROOF: To simplify the proof we shall assume in addition that f , has a partial derivative Ofn (t, Xl , . . . , zn) axn belonging to the class K([a,b] x R'~;R). Then according to (7.42)

f,,(t, xx,...,x,,)sgnx,,_l / > - h ( t , = , , . . . , = , ) l = , l

for

I=,-11>/to,

(7.43)

where h E K([a, b] • R ~ ; R + ). We note that it is easy to get rid of this restriction using the technique employed in the proof of Theorem 7.1. We set fli (t) -- - a i (t) = r0, (i = 1 , . . . , n - 1), and/3, (t) -= an (t) - 0. Then by (7.40) and (7.42) a = (~i)~=l is a lower vector-valued function of the differential system (7.1) and 9 n fl = (/3/)i=1 is an upper vector-valued function. Inequalities (7.30) and (7.31) follow from (7.38) and (7.39), where r is a sufficiently large positive number.

2297

By Corollary 7.1 the solvability of the problem under consideration will have been established if we show that the system (7.1) has no singular solutions satisfying the initial conditions (7.32) and not passing n through M,~ I..JMa 9 Suppose the contrary. Let there exist a singular solution (x i)~=1 of the system (7.1) defined in the interval [a, t* ) c [a, b) and not passing through M~ U Ma. Then according to (7.40) lim sup I=.(t)l = +oo,

(7.44)

t-*t*

and either

p = sup

Iz,(t)l : a ~< t < t*

< +c~,

(7.45)

i=1

or there exists a point to E [a, t*) such that

x,,-1 (to)x.(to) > O,

(7.46)

I=--1(t0)1 > to.

Assume first that (7.45) holds. Then from (7.41) we have

Ix.C01' < [t0(t) + tllx.Ct)l](l+ Ix.C01)

for

a < t < t*,

(7.47)

where lo e L([a,b];R+) and 11 e R+. In view of (7.44) there exist points tl e [a,t*) and t2 e (tl,t*] such that x~(t)x.(tl) > 0 for tl ~ t ~ t2 (7.48) and (7.49)

Ix.Ct2)l > plC 1 + Ix.Ctl)l), where Pl = exp

[L'

1

lo(t) d t + 2P/1 . rl j

By inequalities (7.40), (7.45), and (7.48)

f~i2 ix~(t)ldt ~< 1 /12

x I,,-1 (t) dt = 1 Ix,,_1 (t,) - x,,-1 (tl)l < -2p- . rl rl

Therefore we find from (7.47) that

Ix,,(t2)l < exp

[/? lo(t)dt + tl

Ix.(t)ldt 1

]

(1 + Ix.Ctl)l) < a1(1 + I=.(tl)l),

which contradicts the condition (7.49). Thus it is proved that inequality (7.45) cannot hold. It remains to consider the case when inequalities (7.46) hold. According to (7.40) and (7.43) ,'11~,+1 (t)l <~ x~(t)sgnxi+x (t) <~ "'1=,+1 (t)l

for

to < t < t*

(i = 1,...,n-

and !

x~Ct)sgnx,-1C t)/> -9(01=,C01 where gCt) = h ( t , x l ( t ) , . . . , x , ( t ) ) .

to < t < t*,

From this, taking account of (7.46), we find

I=.(t)l~>exp

2298

for

(fl) -

gC~)a~-Ix,,(to)l>o

for to<~t
1)

(7.50)

and

x,~(t)x,,_l (t) > O,

Ix.-1 (t)l > ~o,

Ix.-1 (t)l' > 0 for

to ~< t < t*.

(7.51)

By (7.50) and (7.51) there exists a point t, e [to,t*) such that

xi(t)#O and xi, (i = 1 , . . . , n -

for

te[tl,t*)

(7.52)

(i = l,. .. ,n),

1), is monotonic on [t,,t*). Further (7.53)

lim IZl(t)l < +oo. t-*t*

Indeed, in the opposite case we would obtain from (7.50) (i = 1 , . . . , n -

lim xi(t) sgnx,(t) = +co

t-*t*

1).

n But this is impossible, since ( i)i=1 does not pass through M~ U M a . Setting ,,

=

'/= ,

IZo = 1 + max

{IxlCtl)l,...,lx,(tl)l}

and

#,(t) : ~o + max {1=,(01 : t l

.N< r ~

t}

(i

=

1,...,n),

and taking account of (7.50) and (7.52), we obtain

=,_',

=i(r)x~(r) d~" ~
x~Ct)=x~(tl)+2 1

(~)=,§ (~) d~

rl

~< #g + 2r---![#i+x it) -/z0] Ixi-1 (t) - xi-1 (h)[ rl

2r2

< / ~ -{- - - [ / Z i +

1 (t) -- #0]//,i--1 (t)

rl

~<~#g -- 2 # 0 # i _ 1 (t) -~-

2r2

rl

#i+1 ( t ) # / - 1 (t)

12 ~< -/z02 q- ~r3/~/+l (t)/~i-1 it)

for

tl ~< t < t*

(i = 2 , . . . , n -

1)

and consequently 9.

#2(t) • r3~i+ 1 (t)~i_ 1 (t) for tx ~< t < t* (i : 2 , . . . , n - - 1). It follows immediately from this that n--i

i-1

/~i(t) ~< r~i-l}(~-i) [/~l (t)] ~-i- [p, (t)] ~-1

for

tl < . t < t "

(i=l,...,n-1).

However, in view of (7.53) Therefore

i-1

#iCt) < r[t~n(t)] n-1

where r

n 2

= r3

for

t 1 ~< t < t*

(i = 1 , . . . , n -

1),

(7.54)

#1 (t*--). 2299

From (7.41) and (7.54) we find

~tn(t) <<.2tto +

/,i[

Ao(r) + 11 ~--~(I+ Iz, Cr)l)

g . ( r ) dr

"

for

tl ~ < t < t * ,

i=2

where

X0(t)

=nrmax{ho(t,u): I"1 <

d,

11

=nrmax{hl(u): I=1 ~< r}.

From this and the Gronwall-Bellman Lemma [27, p. 49] follows the estimate gn (t) ~< ~/0 exp 11

(7.55)

+ 1=2

tl

where r/0 = 2/~0 exp

1o (r) dr . ( I ' ) Because of Young's inequality 1 , - r ) -1 + 2i_ 2 1~_ I (t*- r)i_ 2 (l + l~, (r) l) for tl ~< r < t*. Ix(l+lz,(r)l)~----~-1 ~<~(t

Therefore t

11

ft

1

1

1

t* - t

rt i - 1 I (1 + I~, (r) l) ,--r dr .< ~ In t* - t-----~+2 ~ (b-a)~n i-1 ~]

+ 2 A1

-r)'-'l~,Cr)ldr

(r

for tl ,< t < t*

(i = 2,...,,~).

(7.56)

According to (7.50) and (7.52)

I=~(r)l dr ~< ~2

I=~ (r)ldr ~< r2#l(t*-) < + c r

sup{(t*-t)lx2(t)l:tl

<<.t
and

(r - ~),-2 Ix, Cr)l dr .< r2

(t' - r) '-~ x',_x (r)

+(i-2)~2

f'

<~ r2(t*-t)'-21Z,_l(t)l+r2(t * -tl)'-21z,_x(tl)l

(t'-r)'-~[~,_l(r)ldr

for t l ~ < t < t *

(i=3,...,,~).

1

Applying induction, we now verify that

~['(t*-r)~-21z,(r)ldr < + ~

( i = 2,...,n).

From these inequalities and the estimates (7.54)-(7.56) it follows that there exists a number rl E (1, +cr such that i-1 #,(t)~
#.(t)<~rloexp[2nAlriCb-,) 1/2] 2300

for

t, ~ < t < t *

and lim sup Iz.(t)l ~< tt.Ct*-) < +oo,

t~t"

which contradicts condition (7.44). Thus the system (7.1) cannot have any singular solutions not passing through M~ IJ M r and satisfying the initial conditions (7.32). The corollary is now proved. In conclusion let us consider the problem (7.1), (7.28) for n = 3, i.e., when it has the form

dxi

: fiCt, x l , x 2 , x s ) (i = 1,2,3), dt x/(a) =PiCx3(a)) ( i = 1 , 2 ) , p(xlCb),x2(b),xn(b)) = 0 . COROLLARY 7.4.

(7.57) (7.58)

Let

~olCu)u ~> 0, ~(~1,~,~)#0

~o2(u)sgnu/> r0 for ~ 1 ~ > 0 ,

for I=1 ~> r, ~2sgn~s > r 0 ,

(7.59) (7.60)

and let the inequaliffes

0 < f, ( t , ~ l , ~ , ~ )

sgn~2 < g, Ct,~=)(x + Ix, I),

(7.61)

0 ~ f2(t, Zl,Z2,X3) s g n x 3 ~ g2(t, X l , Z 3 ) ( l -F I~1),

[

O :~ f3(t, xl,:r.2,:r.3)sgnxl ~< lhl(t, xl,x,)+h2(z~,z,)~_,ly~(t,

(7.62)

zl,Z,,zs)l (1 +

I~1),

(7.63)

i=l

and

~I2(t,~,, ~ , , ~ s ) ~ , - f, ( t , ~ , , ~ , , ~ , ) ~ ,

< I,(t, ~ 1 , ~ , ~ , ) ~ 1 ,

(7.~)

be satisi~ed on [a, b] • R 3 , where r0 r 0, r > 0, ~/> 0, gl e g ( [ a , b] • R; R+ ), g2 and hi e g ( [ a , b] • R 2 ; R + ), h2 E C ( R 2 ; R + ) . Then the problem (7.57), (7.58) is solvable. PROOF: Let /~2(t) - - a 2 ( t ) = r0, ~dCt) - ai(t) =- O, (i = 1,3). According to (7.59)-(7.63) a = (ai)i~l and fl --- (fli)d=l 3 are lower and upper vector-valued functions for the differential system (7.57) for which conditions (7.30) and (7.31) hold with n = 3. If the system (7.57) with the initial conditions

9 ,(a) = ~,(~sCa))

(i = 1,2)

has no singular solutions, then by Corollary 7.1 the problem (7.57), (7.58) is solvable. It remains for us to consider the case when the system (7.57) has a singular solution (xi)i=l s defined 3 passes through in the interval [a,t*) C [a, b), and to prove the corollary it suffices to establish that (x i)i=l M~ [.J M ~ . We shall first show that lira sup Ix2(t)l = +oo. (7.65) t.-.,t* Suppose the contrary. Let p~ = sup{t~(t)l : a ~< t < t*) < +~o. Then from (7.61) we find

pl = sup{ l~ l(t)l : a < t < t * } < ( 1 +

I~l(a)l)exp

[/'

]

91(t, ~, (t)) dt < +oo.

In view of the boundedness of 11 and 12 lim Ixs(t)l---t.--,t*

-t-oo.

(7.66) 2301

By this fact it follows from inequalities (7.61)-(7.63) that there exists a point to E [a, t*) such that

xs(t) ys O, xiCt) # 0 ,

x~(t)xi+l (t) >10 for

to ~ < t < t *

(i=1,2)

and Ix (t)l'

a0(t) +

)11Z

Ix:(t)l (1 + Iz,(t)l)

for to

t < t*,

i=l

where ~0 = hl(',a~l('),z2(-)) E L([a,t*],R+) and )~1 =sup{h2(xl(t),x2(t)):a

[x3(t)[ ~ (1h-[xs(to)[)exp [ ftl Ao(r) dr + A1

Z

~< t < t*} < +oo. Therefore

xi(r ) dr

i=1

<~ (1 + Ixs(t0)[) exp

If'

:~0(r) dr + Al(m + 02)

tl

]

for

a~
which contradicts condition (7.66). This proves equality (7.65). According to (7.64) ['~xl(t)- 2xl(t)xs(t)]' <<.0 for a<<.t
7x](t) <~Co + 2xl(t)xs(t)

for

a ~< t < t*,

(7.67)

where c0 : fizZ(to) - 2xl (to)xs(to). We shall show that x~ is nonzero in some left-hand neighborhood of t*. Indeed, if this were not the case, then according to (7.62) and (7.65) there would be a sequence tk e (a,t*) converging to t*, (k : 1,2,...), for which xs(tk) = 0 ( k = 1 , 2 , . . . , ) , lim 1=2(t,)l = +oo. k--*+oo

But this contradicts inequality (7.67). By conditions (7.61)-(7.63) and the constancy of sign of x2 in a left-hand neighborhood of t* there exists a point to E [a, t*) such that

xi(t) # 0 (i : 1,2,3),

' Xz(t)xsCt ) >10 for to ~ < t < t * .

Taking account of these inequalities, we find from (7.65) and (7.67) that

lim [x2(t)sgnxz(t)] = +oo,

t--. t

t~m [xl (t)xz(t)] : -]-00.

From this it is clear that (Xi)i__l 3 p a s s e s through the set M~ U M~. The corollary is now proved. 7.5. T h e U n i q u e n e s s T h e o r e m . THEOREM 7.2. For each i E { 1 , . . . , n ) let the function fi have partial derivatives f~(t, x l , . . . , x ~ )

:

c3fi(t'Xl"" "'Xn), (i : 1, ... ,n), and Oxi f~i E K([a,b] x R " ; R )

and

f~ e K([a,b] x R" : It+)

(j # i,j = 1,...,n).

(7.68)

Suppose further that the functions ~oi, (i = 1,... , n - 1), are increasing and the function ~ satisfies the condition ~o(yl,...,yn) > ~ o ( z l , . . . , x n ) for Yi > x i ( i : X , . . . , n ) . (7.69) 2302

Then the problem (7.1), (7.28) has at most one solution. PROOF: Suppose the contrary. Let the problem (7.1), (7.28) have the two distinct solutions (xi)i~=~ and (Yi)i~1. Then

~,.,(,~) # x,,,(,,,),

for otherwise we would have x, (a) = y, (a), (i = 1 , . . . , n), which is impossible since the Cauchy problem for the system (7.1) has a unique solution. For definiteness we shall assume that

y,,(,,,) > x,,(,O. Then yiCa) =gai(y,(a)) > p i ( x , Ca)) =xiCa)

(i=l,...,n-1),

since ~oi, (i := 1 , . . . , n - 1), are increasing functions. Consequently, in a certain right-hand neighborhood of a the inequalities y~(t) > xiCt) (i = 1 , . . . , n ) (7.70) hold. On the other hand, because of (7.69) these inequalities cannot hold on the whole interval [a, b]. Therefore there exist k E { 1 , . . . , n} and to E (a, b] such that the inequalities (7.70) hold on [a, to) and uCt0) = O, where

(7.71)

u(t) =: yk(t) - xk(t). According to (7.68) and (7.70)

u'(t) = fk(t, yl(t),...,y,(t)) - fk(t, xl(t),...,x,(t)) >~ fk(t, xl(t),...,xk-1 (t),yk(t),xk+l (t),...,x,(t)) -- fk(t, xt(t),...,x,(t)) >>.g(t}u(t) for a < t < t 0 , where

g e L([a, to];R). From this we find u(to) >>-u(a)exp [ f t ~ g(r) dr] > 0 ,

which contradicts the equality (7.71). The contradiction thus obtained proves the theorem. w

Two-Dimensional Differential Systems

8.1. S t a t e m e n t o f t h e P r o b l e m . Consider the differential system

dxi dt = fi(t, xl,xz)

(8.1)

(i = 1,2)

with boundary conditions

(~l(a),~,(~)) e s,,

C~lCb), ~,(b)) e S,,

C8.2)

where

], e K([a, b] • R';R) and

(i = 1,2),

Si C R 2 , (i = 1, 2), are continua.

Let Sil and Si2 be the projections of the set S~ respectively on the interested in the case when one of the following three conditions holds: infSi2 = - o o ,

S12 bounded,

~2

and

$22

supSi2 = +co

infSi, : - c o ,

bounded,

(i = 1,2),

supSi, = +co

infSix = - c o ,

Oxx and Ox2 axes. We shall be (8.3)

( i = 1,2),

supSil = +co

( i = 1,2)

(8.4) (8.5)

2303

For what follows it is convenient to introduce the following definitions. DEFINITION 8.1. We shall say that the vector-valued function (ff,,ff2) E C([a, b];R 2) belongs to the set A* (f,, f2) (resp. to the set A. (f,, f2)) if there exists a set of measure zero I0 such that for all t C [a, b] \ I0 and x~ E R the inequalities [fl (t, "], (t), x2) - ~]~(t)11272 - ~2 (t)] >t 0 and

f2Ct, q, Ct),"12(t)) >17~(t) hold. DEFINITION 8.2. w E C(R; (0, +r

(resp.

is called a

o_

f2(t,q,(t),72(t)) <<.7~(t))

Nagumo function if

ds _ fo +~ ds

.,(s)

,,,(s)

-

+oo.

(8.6)

8.2. L e m m a s o n A P r i o r i E s t i m a t e s . LEMMA 8.1. Suppose r, E R + , r, e (r,,-~-oo), I C R e cCR; C0, + ~ ) ) ~ d

~r r2 ds .,(s--5 > IlhollL + IlhlllL, 1

is an interval, ho e L([a, b]; It+ ), hi e L ( I ; R + ) ,

/_-rl

ds ~,(s--5 < IlhollL + Ilhlll~.

(8.7)

t'2

Then for any t, E [a,b), t2 E (t,,b], and to E [t,,t2] an arbitrary vector-valued function (x,,x2) e C([t,,t2]; I x R) satisfying the system of differential inequalities

zi(t)x2(t)>~O,

x'2(t)sgn[(t-to)x2(t)]<~[ho(t)+h,(xl(t))lx'lCt)l]w(z2(t))

for

tl


(8.8)

and the condition 1~2(to)l ~< r,,

(8.9)

admits the estimate Iz2(t)l < r2 for t, ~< t ~< t2.

(8.10)

PROOF: We first prove that if to < t2 then

Ix2(t)l < r2

for to < t < t2.

(8.11)

Suppose the contrary. Then because of (8.9) there exist numbers sl such that ~z2Cs,)=r,, o z 2 C t ) > 0 for s, < t < s 2 ,

E [to,t2), s2 E (sl,t], and a E ( - 1 , 1 } ~z2Cs,)=r,.

Therefore we find from (8.8) that

a

,,

w(u)=

,

w--~x~-~)dt<<,

ho(t)dt+ ',,C,,) h, Cu)a~
which contradicts condition (8.7). The contradiction thus obtained proves (8.11). Analogously we show that Iz2(t)l < r2 for t, .< t .< to. Consequently estimate (8.10) holds. The lemma is now proved.

2304

LEMMA 8.2. Suppose a ~ ao < b0 ~< b, ri e R + , (i ---- 0,1), I C R is an interval, ho e L([a,b];R+), hi E L(I; R+ ), w is a Nagumo function, and b E g([a, b] • R+ ;R+ ) is a function that is nondecreasing on the second argument, and f bo 6(t, rl) dt > ro.

(8.12)

0

Then there exists a number r2 6 (rl, +oo) such that any vector-valued function (Xl,X2) e 6'([a, hi; I x R) satisfying the system of differential inequalities

9 ~(t)sgn~,Ct) ~> bCt, l~,Ct)l) for a < t < b, x~Ct) sgnx2(t)/> -[hoOt) + hi (xlCt))lx~ (t)l]-(~?(t)) for a < t < bo, x~(t)sgnx2(t) ~< [ho(t) + h, Cx,(t))lz~(t)ll~Cx=(t)) for ao < t < b,

(8.13) (8.14) (8.15)

and the condition lxl(bo)

-

xl(ao)[ ~< to,

(8.16)

admits the estimate

I=,(t)l < r2

for

(8.17)

a < t < b.

PROOF: By (8.6) there exists a number r~ 6 ( r l , + e o ) such that inequalities (8.7) hold. If we assume that I~,(t)l ~> r l

for

a0 < t < bo,

then according to (8.12) and (8.13) we find bo

IXl(bo)- xl(ao)] >1

b(t, rl) dt > to. 0

But this contradicts inequality (8.16). Consequently for some point to C [a0, b0] we have

I~(to)l < r,. On the other hand, inequalities (8.8) follow from (8.13)-(8.15), where t, = a and t2 = b. Therefore by Lemma 8.1 estimate (8.17) holds. The lemma is now proved. LEMMA 8.3. Suppose rl e R + , I C R is an interval, ho e L([a,b];R+), hi e L ( I ; R + ) , and w is a Nagumo function. Then there exists a number r2 6 (rl, +oo) such that any vector-valued function (Xl, x2) E C([a, b]; I x R) satisfying the system of differential inequalities

x'~(t)x2Ct)>~o,

x'2Ct)sgnx2Ct)<[hoCt)+hlCx~Ct))lx'lCt)l]wC~2Ct)) for

a
and the condition

Ix,~Ca)l < r,., admits the estimate (8.17).

PROOF: Choose a number r2 6 (r1,+cr so that inequalities (8.7) hold. Since (Zl, x2) satisfies the system of differential inequalities (8.8) and condition (8.9), where tl = to = a and t2 = b, it follows from Lemma 8.1 that estimate (8.17) holds.

2305

LEMMA 8.4. Suppose c 9 Is, b], I C It is an interval, ho 9 L([a, b]; R+ ), hi 9 L ( I ; i t + ) , and w is a Nagumo function. Then there exists a number r2 9 (rx, +or) such that any vector-valued function (Xl,Xz) E C([a,b]; I • It) satisfying the system of differential inequalities

x~(t)x2(t)>~O,

x'2(t)sgn[(t-c)x2(t)]<~[ho(t)+hl(Xl(t))lx'l(t)l]w(x2(t))

for

a
and the boundary conditions Ix~(a)l < rl,lx2(b)[ <~rl, admits the estimate (8.17). PROOF: Let r2 9 (r~, +oo) be chosen so that inequalities (8.7) hold. Then by Lemma 8.1 estimate (8.17) holds; for the restriction of the vector-valued function (Xl, x2 ) to the closed interval [a, c] (resp. to the closed interval [c,b]) satisfies the inequalities (8.8) and (8.9), where tl = to = a, and t2 = c (resp. tl = c and to = t2 = b).

8.3. Existence Theorems. THEOREM 8.11. Suppose conditions (8.3) hold and there exist (Otl,a2) E A . ( f x , f 2 ) A* (fl, f2) such that

~l(t) ~< Zl (t)

for a < t < b,

(8.18)

{(xi,z2) :Xl < alCa), x, > ~ ( a ) } n 81 = {(x,,z2):Xl > ~l(a),x 2 < {(Xl,~,) :~1 < ~ , ( b ) , ~ < ~ ( b ) } n s ~

and (fl1,fl2) 9

z,(o)}

= {(~l,X~):~1 > ~ l ( b ) , ~ > ~ ( b ) } n s ,

=~

(8.19)

=0.

(8.20)

Suppose further that

fx (t,=l,=~)sgn=2 ~ aCt, I=~1) for a < t < b ,

f2(t,xl,x2)sgnx2

al(t) ~<Xl ~>. -[h0(t) + h11fl(t,=l,=2)l]w(=2) for a < t < bo, ~1 (t) < ~1 < 81(t), ~, e It,

(8.21) (8.22)

and

for ao < t < b,

f2(t, xl,X2)sgnx2 <~ [ho(t) +hl(lfl(t,xl,x2)l]W(Xz)

al(t) <<. xl ~<~lCt),

x2ER, (8.23)

where a <~ao < bo < b, h o e L([a,b];R+ ), hi E R+ , w is a Nagumo function, and 6 E g([a,b] • R + ; R + ) is a function that is nondecreasing on the second argument with bo

5 ( t , p ) d t = +oo.

lim p---~ + o o

(8.24)

o

Then the problem (8.1), (8.2) has a solution (Xl,X2) such that

~1 (t) < Xl(t) < Zl(t)

for

a < t < b.

(8.25)

PROOF: Let ro = 1 + II~,ll + Ilfllll,

i=

(-ro,ro),

hi(u) --- hi.

According to (8.24) there exists a positive number rl satisfying inequality (8.12). Choose r2 E (rl,-t-cr that the conclusion of Lemma 8.2 holds. 2306

so

Set

](t,~l,Z,)

:

ACt, /~l(t),x2) fl(t, Zl,272) fl(t, ot1(t),272) 2

for for

~1 > /~l(t), ~ (t) < 271 < al(t), 271 < O~l(t),

for

{1

(8.26)

r=r2 + ~--:~(ll~,llc + II~llc), i=1

x(s) =

aCt, s)

for

0~<s~
2- -

for

r<s<2r,

0

for

s/> 2r,

f2(t,[31(t), u)l : Iu -~2(t)[

= s + m a x {If2 (t, Bl (t),/~2(t))-

{ X(lX2[)f2(t,[31(t),x2) +aCt, ]2(t, xl,z2) = X(lx~l)f2(t, zl,z2)

(8.27)

zl - ~1 ( t ) )

1+

J71 -- /~1 (t)

~l(t) - 271

x(Iz21)A (t, m (t),z2) - act, \

/

~< s},

for

271 > Zl(t)

for

a l ( t ) ~< Zl ~< #1 (t),

for

271 < O~1(t),

(8.28)

(8.29)

H, = { ( - r , z2) : - r < z2 < ~2(~)} [.J {C271,-r) : - r < 271 < ZlCa)}, //2 = {(r,272):/~2(a)~< 27, ~< r} U{(271,r): al(a)~< 271 ~< r}, D = I-r, r] • I-r, r], 2

= Csl D) U H,

(8.30)

i=1 and consider the problem d27i dt : ~(t'271'27~)

(i = 1,2), (271(b),x2(b)) E 82.

(271(a),272(a)) E $1,

(8.31) (8.32)

Because of equalities (8.26) and (8.29) it follows from the conditions a = (81,82) E A,(fl,f2) and /~ = (/~1, ~2) E A* (fl, f2) that a and ~ are lower I and upper vector-valued functions of the differential system (8.31). The system (8.31) itself has no singular solutions. Because of (8.3), (8.19), and (8.30) the set $1 is a bounded continuum, and

~INM.(a) r ~,

$1 nM~ca) # ~.

On the other hand it follows from (8.3) and (8.20) that the set $2 satisfies conditions c) and d) of Theorem 7.1. According to this theorem the problem (8.31), (8.32) has a solution (271,272) satisfying the condition

(271(t),272(t)) ~ M~(t) UM~(t )

for

a ~< t ~< b.

(8.33)

We shall prove that

27,(t) <~[31(t)

for

a <<.t <~b.

(8.34)

Suppose the contrary. Then there exists to E Ca, b) such that

27x(t0) > al(t0). 1Cf. Definitions 7.1 and 8.1. 2307

W e denote by t, the exact lower bound of tl 6 (a, to) for which

xl(t)>ax(t)

for t1~
Because of (8.33)

9 ~(t) > a~(t), Since a 6 A * ( A , f 2 )

x,(t) ~< Z,(t)

for

t, < t .<< to.

(8.35)

we find from (8.26) and (8.35) that

X~l(t) - a~l (t) = fl (t, al (t),x,(t)) - a~ (t) <<.0

for

t, < t < to

and

9 l(t,)-

a~(t,)t> ~ , ( t 0 ) - a , ( t 0 ) > 0.

From this and from the definition of t, it follows that t, = a. Thus

x2(a) < a,(a).

=i Ca) > al (a), However, according to (8.19) and (8.30)

(~i,~,) r 21

for

zl > a,(a),

~, < a2(a).

Therefore

(8.38)

9 ,(a) = a, (a). Without loss of generality we may suppose to sufficientlyclose to a that

Ix~(t)l <,',

Ix,(t)-a~(t)l

< 1

xlCt)-al(t) +xl(t)-al(t)

for

a
Taking account of these inequalities we obtain from (8.27)-(8.29) that

~;(t) - a;(t) = s

(

271(t)--=~-~it) al(t) )

al(t),~,(t)) + g t, 1 u

- a;(t)

, ~I - a~ (t) >1 f,(t, al(t),a, Ct)) - a, Ct) + 1 + xl(t) - a l ( t ) > 0 for

a < t < to.

But this is impossible because of (8.35) and (8.36). This proves estimate (8.34). Analogously we prove that x l C t ) > / ~ l ( t ) for a < t < b . Consequently condition (8.25) holds.

By (8.2x)-(8.23) and (8.25)-(8.29) the vector-valued function (xl,x2) satisfies inequalities (8.13)(8.16). Therefore according to the choice of r, we have

[x,(t)l < r2 ~< r

for

a ~< t ~< b.

(8.37)

By the estimates (8.25) and (8.37)it follows from (8.26)-(8.31) that (xl,z2) is a solution of the problem (8.1), (8.2). The theorem is now proved.

2308

THEOREM 8.12. Suppose the conditions (8.3) hold, the sets Sxi and $2+1 are bounded, and the inequalities

~(t,l==l) < fl(t,xl,X2)sgnx2

<~ g(t,

lx21)wo(zx)

(8.38)

< [ho(t)+ h,(=,)lfl (t,=1,=2)1]~,(=2),

(8.39)

and

IA(t,=,,=,)l

hold on [a,b] x R 2, where ho e L([a, b]; R+ ), hi e L ( R ; R + ) , w0 and w are Nagumo functions, g and E K([a, b] • R+ ; R+ ), g being nondecreas/ng on the second argument, and ~a b lira p--*+oo

$(t,

p) dt

= +oo.

Then the problem (8.1), (8.2) is solvable. PROOF: Let b0 = b,

a0 : a , let ro and

rl

I = R,

be positive numbers such that r0 To \ 2 ' 2 }/

SilC

(i = 1,2),

(s.40)

and let inequality (8.12) be satisfied. Without loss of generality we may assume that t~(t, p) = ~(t, rl)

for

p > T1

(8.41)

and that the function g is nondecreasing on the second argument. Choose r2 E (rl, +oo) such that the conclusion of Lemma 8.2 holds, and choose rs E (to, +co) so large that

o

woCs---)>

gCt, r2)dt,

,,

~

~o(s)

>

g(t, r2)dt.

(8.42)

Let r = "2 + rs and let X be the function given by equality (8.27). Set

]1 (t, =1, =2) = (1 - x(l=, I+ 1=21))~(t, I==1)sgnx2 + x(l=, I+ I== I)A (t,=,,=2), LCt,=i,==) = x(l=ll + [=21)f2(t,=l,=2)

(8.43) (8.44)

and consider the boundary-value problem (8.31), (8.2). Because of (8.38), (8.39), and (8.41) it follows from (8.43) and (8.44) that ]t and .~ satisfy the inequalities

~(t, I==1) < L (t,=,, ==)sgn=, < g(t, 1=2 I)o~o(=,),

ILCt,=,,==)l-< [noCt)+ hi (=,)1], (t,=l,==)l]~,(==) and

(8.45) (8.46)

2

I],(t,=l,=2)l < f* (t),

(8.47)

i=1

on [a, b] x 1%2 , where

{~,lY, Ct,=,,=2)l : I=11+ 1=21-< 2T}. 2

f* (t) = max

i=1

2309

Let X(a, ,.ez) be the set of solutions of the system (8.31) satisfying the condition

(=,(~),~,(~)) e s,. Then by Lemma 7.1 the set

= {(xl(b),x,(b)): (x,,x2) e X(a, S1)}

Slo

is a continuum. According to (8.3) and (8.40) there exist

(=,~,=~ ) e x(,,s,)

(k = 1,2)

such that

r0 IXlk (a)J < ~-,

(--1)kX2k(a) > r +

b

i

f*(t) dt

(k = 1, 2).

Taking account of inequalities (8.12), (8.45), and (8.47), we find

f*(r)dr > r

(--1)kx~k(t) /> (--1)kx2k(a)-

for

a<.t<.b

(k = 1,2)

and

(--1)kx,k(b) /> (--1)kx,k(a)

+

~ab

&CT,Ix=C01)dr ~>

i b

6(~,r,)d~-I=,*(~)1

>

r0

(k : 1, 2).

Consequently the continuum Sx0 has both a point situated left of the strip

r0 x, eR}, D={(x,,x2) e R 2 : - - ~ro< x , <-~-, and points situated right of this strip. On the other hand, by (8.3) and (8.40) $2 C D,

infSn =-oo,

sup S n = +oo.

Therefore

S,oN 82 0, i.e., the problem (8.31), (8.2) is solvable. Let (x,, x2) be an arbitrary solution of the problem (8.31), (8.2). We shall show that it is also a solution of the problem (8.1), (8.2). By (8.43) and (8.44) in order to do this it suffices to establish that Ix,(t)l + I=~(01 < r

for

~ < t < b.

(8.48)

From (8.40), (8.45), and (8.46) it is clear that Cab,x2) satisfies the inequalities (8.13)-(8.16). Therefore, because of the choice of r2 it admits the estimate (8.17). From (8.17), (8.40), and (8.45) we have

Ix2(a)l

<

r0,

I=~(01'


for

a
Hence taking account of (8.42) we find I~-, (t)l < r3

for

a .< t .< b.

Consequently estimate (8.48) holds. The theorem is now proved. Theorems 8.2~, (k = 1, 2), and 8.3 given below contain criteria for the solvability of the problem (8.1), (8.2) in the cases when the sets S/, (i = 1,2), satisfy conditions (8.4) and (8.5). We omit the proofs of these theorems, since they are analogous to the proofs of Theorems 8.1k, (k = 1, 2). The principal difference is that Lemmas 8.3 and 8.4 are applied instead of Lemma 8.2. 2310

THEOREM 8.21. Let (~1,~2) e A , ( f l , f 2 ) and (~1,~2) ~ A * ( f l , f 2 ) and let conditions (8.4) and (8.18)(8.20) be satisfied. Suppose further that on the set

{(t, x l , x 2 ) : a < t < b,

-l(t) ~< 2;1 ~< oL2Ct), -T2 e l~}

(8.4~)

the inequalities f l ( t , z l , x 2 ) s g n z 2 >>-0

(8.5o)

f~(t, xl,x2) sgnx2 ~< [ho(t)+ hill1 (t, xl,X2)l]wCx~)

(8.51)

and hold, where ho E L([a,b];R+ ), hi E R+, and w is a Nagumo function. Then the problem (8.1), (8.2) has a solution (xl, x2) satisfying the condition (8.25).

THEOREM 8.22. Suppose the conditions (8.4) hold, the set 821 is bounded, and the inequalities 0 <. fl(t, x2)sgnx2 <<.9(t, 1~21)~(~1)

(8.52)

h ( t , xl,x2)sgnz2 <~ [hoCt) + hlCzl)lh(t,z,,x2)ll~(z2)

(8.53)

and hold on the set [a,b] • R 2, where g e K([a,b] • R + ; R + ) , ho e L([a, bl;R+ ), h 1 e L(R;R+), and wo and are Nagumo functions. Then the problem (8.1), (8.2) is solvable.

THEOREM 8.3. Suppose (o/1,o~2) E A . ( f l , f 2 ) , (~1,~2) E A * ( f l , f 2 ) , and conditions (8.5) and (8.18)(8.20) hold. Suppose further that the inequalities (8.50) and f2 (t, X 1 ,~2 )sgn [(t -- C)X 2 ] ~ [ho(t ) + h I 12"1(t, Xl,X 2) II~(:~:~),

(8.54)

hold on the set (8.49), where c E [a,b], ho E L([a,b];R+ ), h I ~ P•+, and w is a l~agumo function. Then the problem (8.1), (8.2) has a solution (xl,z2) satisO/ing the condition (8.25).

To conclude this section we consider the case when the boundary conditions (8.2) have one of the following three forms.

x1(a) ----@i(x2 (a)), 9 2(a) = ~91(Xl(a)), :g2(a) = ~Ol(Zl(a)),

Xl(b)---- V)2 (x2 (b)), x1(b) = ~2(xu(b)), z2(b) = ~o2(Zl(b)).

(8.55) (8.56) (8.57)

From Theorems 8.1k, 8.2~, (k = 1, 2), and 8.3 the following propositions follow immediately. COROLLARY 8.11. Let (O~1,~2) e A , ( f l , f 2 ) and (fll,132) E A*(fl,f2). Let

~l(~)~>.lCa)

~2(u)>~al(b)

for for

~>-2(a),

u
~1(~)<~1(~)

for ~ < ~ 2 ( ~ ) , ~o2(u)<~fll(b) for u>/32(b),

(8.58)

and let the inequalities (8.18) and (8.21)-(8.23) hold, where a ~ ao < bo ~ b, ho E L([a,b];R+ ), hi E R+, w is a Nagumo function, and ~ 6 K([a, b] x R + ; R + ) is a function that is nondecreasing on the second argument and satisfies condition (8.24). Then the problem (8.1), (8.55) has a solution (Xl,X2) admitting the estimate (8.25).

COROLLARY 8.12. Let sup {l~,(u)l : u e a } < +o~

(i : 1,2)

(8.59) 2311

and let inequalities (8.38) and (8.39) hold on the set [a,b] x R 2, where ho 9 L([a,b];R+ ), hi 9 L ( R ; R + ) , wo and to are Nagumo functions, and g and 6 belong to K([a, b] x R + ; R + ), 6 being nondecreasing on the sdcond argument with

lim

/'

p--* + oo

6(t, p) dt = +oo.

Then the problem (8.1), (8.55) is solvable. COROLLARY 8.21. Suppose (Oil,Or2)9 A , C / 1 , f 2 ) , (/~1,/~2)9 A * ( / 1 , f 2 ) ,

~,[~,(a)]

.<< ~ , ( a ) ,

(8.6o)

,,,:,,.[al(a)] /> a , ( a ) ,

and conditions (8.18) and (8.58) hold. Suppose in addition that inequalities (8.50) and (8.51) hold on the set (8.49), where ho 9 L([a,b];R+ ), hi 9 R+, and w is a Nagumo function. Then the problem (8.1), (8.56) has a solution (xz,x2) satisfying the condition (8.25).

COROLLARY 8.22. Suppose suP{l~,l(u)l:u 9

} <+oo

and inequalities (8.52) and (8.53) hold on the set [a, b] x R 2 , where g 9 K([a, b] • ;R+ ), ho 9 L([a, b]; R+ ), hi 9 L ( R ; R + ) , and w0 and to are Nagumo functions. Then the problem (8.1), (8.56) is solvable.

COROLLARY 8.3. Suppose (81,82) 9 A . ( A , / 2 ) , (/~1,/92) 9 A * ( f l , f 2 ) ,

~2 [~~ (b)] ~< au(b),

~:~[/~', (b)] ~
and conditions (8.18) and (8.60) hold. Suppose further that inequalities (8.50) and (8.54) hold on the set (8.49), where e 9 [a,b], ho 9 LC[a,b];R+ ), hi 9 R+, and w is a Nagumo function. Then the problem (8.1),

(8.57) has a soIution (x~,x~) satisfying condition (8.25).

8.4. Uniqueness Theorems. THEOREM 8.4. Let fl and ]'2 have partial derivatives on the phase variables belonging to the class K([a,b] x R 2 : R), and suppose there exist a 9 {-1,1} and a set of positive measure I C [a,b] such that O"

Ofk(t, xl,x2) >1 0 O,~X3_k

for

a < t < b,

(xt,x2) 9 R 2

(k = 1,2)

(8.61)

and ~

Oxs-k

~ o

for

t 6 I,

Cxt,x2) 6 R 2

(k = 1,2).

(8.62)

Suppose further that for each i 6 {1,2} arbitrary points (x,,x2) and (y,,y2) of the set S~ satisfy the inequality (--1) i-lO"(2:1 -- Yl)(X2 -- Y2) >/ 0. Then the problem

(8.1), (8.2)

has at m o s t one solution.

PROOF: Suppose the problem (8.1), (8.2) has two distinct solutions (Xl,X2) and (Y,,Y2). Then either XlCa) # ,1(a) or x~(a) # y2(a). For definiteness we shall assume that 9 1 Ca) > yl(a).

(8.63)

Set ,~1 (t) = Xl (t) - yl (t),

,,2(t) = o[x2Ct) - y, (t)].

In view of (8.61) and (8.62) u~Ct) : gil(t)Ul(t) ~- gi2(t)u2Ct)

2312

({: 1,2),

(8.64)

where gik 9 L([a,b];R), (i,k = 1,2), and gks-k(t) ~>0 for

a
gks-k(t) > 0

for

t9

(k=1,2).

(8.65)

On the other hand, according to the restrictions imposed on S~, (i = 1, 2), and inequality (8.63),

Ul(.) > o ,

-2(~)/>o

(8.00)

and

(8.67)

Ul(b)u 2(b) ~.~ 0. By (8.65) and (8.66) it follows from (8.64) that U1 (t) > O,

us(t) >>. o

for

a • t ~< b and

us(b) > O,

but this contradicts inequality (8.67). The contradiction so obtained proves the theorem. COROLLARY 8.4. Suppose fl and fs satisfy the hypotheses of Theorem 8.4 and the functions (-1) i-1 er~gi, (i = 1,2) are nondecreasing. Then each of the problems (8.1), (8.55); (8.1), (8.56); and (8.1), (8.57) has at most one solution. Analogously to Theorem 8.4 one can prove THEOREM 8.5. Suppose fl and fs have partial derivatives on the phase variables belonging to the class K([a,b] • It2;I{.) and inequality (8.61) holds for some a 9 {-1,1}. Suppose further that there exists i 9 {1, 2} such that arbitrary distinct points (xt, xs) and (yt, Ys) of the set S~ satisfy the inequality (-1) i-laCxx - yl)(xs - Ys) > O, and arbitrary points

(Zl, Z2) and (y,, Y2) Of the

set Ss-~ satisfy the inequality

(-1)/a(xx - Yx)(xs -Yz) /> O. Then the problem (8.1), (8.2) has at most one solution. COROLLARY 8.5. Suppose fl and fs satisfy the hypotheses of Theorem 8.5. Suppose further that there exists i E {1, 2} such that (-1) '-1 a~o, is increasing and (-1)'a~s-~ is nondecreasing. Then each of the problems (8.1), (8.55); (8.1), (8.56); (8.1), (8.57) has at most one solution.

Chapter 4 PEItIODIC w

AND BOUNDED

SOLUTIONS

P e r i o d i c S o l u t i o n s of M u l t i d i m e n s i o n a l Differential S y s t e m s

In the present section we establish criteria for existence and uniqueness of an w-periodic solution of the differential system dzi - f, Ct, x l , . . . , x , ) (i = 1 , . . . , n ) (9.1) dt and we indicate a method of constructing it. It is assumed that w > O, the functions f~ : It • It" --* 1%, (i = 1 , . . . , n ) , are periodic on the first argument with period w, i.e.,

f, Ct + W , ~ l , . . . , x , ) = f,(t, X l , . . . , ~ , )

Ci = 1 , . . . , , ) ,

(9.2)

and their restrictions to [0, w] • It" belong to the class K([0, w] • I t " ; i t ) .

2313

The symbol L~ denotes the set of w-periodic functions p : R --* R whose restrictions to [0, w] belong to the class L([0, w]; R). If p E Lw and

//

1

g(p)(t,r) =

1

p(t) dt r O, we set

-exp(fo~

for

0~
(0.3)

- exp ( foW P(s) ds) t-l exp ( f/ p(s) dS + fo~ p(s) ds)

for

t
9.1. O n t h e set U --W~ .....""

DEFINITION 9.1. Let cri E {--I, 1}, (i = 1 , . . . , n). We shall say that the matrix-valued function (P~k)~,k=~ belongs to the set U~ 1.....o, ifp~k E L ~ , (i,k = 1,...,n),p~k(t) I> 0 f o r t E t t , i ~ k , and the system of differential inequalities

n

i ,yi(t) ~< ~--~pik(t)yk(t)

(i= 1,...,n)

O:

(9.4)

k=l

has no nonzero nonnegative w-periodic solutions. LEMMA 9.1. //'

(p,, ),".,=l e v:, .....

(9.5)

then fo ~ p,(t) dt < 0

(i = 1, ... ,n).

(9.6)

PROOF: Assume that for some j E { 1 , . . . , n}

fo ' p~i (t) dt >>.O. Choose a function p E Lw such that

p(t)<<.pir ) for t 6 R

and

f0 t~ p(t)dt=O.

Then the vector-valued function (Yi)i"--1,where

y,(t) - 0

for i ~ i ,

ui(t) =exp

(/o') a~

p(r)dr ,

is a nonzero nonnegative w-periodic solution of the system (9.4). But this contradicts the hypothesis (9.5). Thus inequality (9.6) is proved. LEMMA 9.2. ha order for

condition (9.5) to hold it is necessary and sufficient that (P;(~Oo,),~=l)E U(tl,...,t,),

(0.7)

where2 ti

--

1-.,

----~

W

(i

~90i(Yl,...,yn) = yi(w -- t,) ~Cf. Definition 4.1. 2314

~

1, . . . ,.), (i= 1,...,n),

(9.8) (9.9)

and e is the restriction of the matrix-valued function (Pik)ink=1

tO

[0, ca].

PROOF: Suppose condition (9.5) holds. Then by Lemma 9.1 inequalities (9.6) hold. Therefore for each i E {1,..., n} the periodic problem dlt

-~ = trlp, i(t)u,

u(O) = u(w)

(9.10)

has only the zero solution and its Green's function gi admits the representation gi(t,r) =a~g(trlp,,)(t,r)

(i= 1,...,n),

(9.11)

where g is the transformation given by the equality (9.3). We now suppose that condition (9.7) is violated, i.e., the problem n

x:(t)sgn(t--ti)<<.Zpik(t)zk(t)

for

0
(i=l,...,n),

(9.12)

k----1

9 ,(t,) ~ ,~,(ca - t,) has a nonzero nonnegative solution ( i)i:1

(i = 1,...,n)

Because of (9.8) there exist qi C L([O, cal;R+ ),

ct, C R+,

(9.13)

(i = 1 , . . . , n )

(9.14)

such that n

z:(t) =ai ~-~pik(t)zk(t)--aiqi(t)

(i= 1,...,n)

k=l

Xi(ca) = xi(O) + triOti

(i = 1,... ,n).

Hence, taking account of (9.6), (9.11), and (9.14), we find w

x,(t)=--a~g(aipi,)(t,O)-

g(aip, i)(t, rJq~(r)dr+y~(t)<~y~(t)

for

O<<.t<~w

(i=l,...,n),

where

u,(t) = ,,, ff

(i = 1 , . . . , n ) . k#

--1

It isclear-that y,(o) =y,@)

(i = 1 , . . . , n )

and l't

o,u~(t) = p,,(t)u,(t)+

E k~i,~'=- I

n

pit(t)zk(t) ~El~t(t)yk(t) for 0 < t < ~

Ci=l,...,n).

k= I

Consequently the periodic extension of (Yi)i~x to R is a nonzero nonnegative w-periodic solution of the system (9.4). But this contradicts condition (9.5). The contradiction thus obtained proves that (9.7) follows from (9.5). The fact that (9.5) follows from (9.7) is obwious; for the restriction of an arbitrary w-periodic solution of the system (9.4) to [0,w] is a solution of the problem (9.12), (9.13). The lemma is now proved.

2315

LEMMA 9 . 3 . Let p~ 6 L~, (i,k = 1 , . . . ,n), pik(t) >1 0 for t 6 R, i # k, and let the inequalities (9.6) be satisfied. Further suppose that the eigenvalues of the matrix S = (sik )i~k=l , where

sil = 0,

8,k = max

{/0 ~ g(aipii)Ct, r ) p i ~ ( r ) d r : O < x t < . w }

for

is~k

are less than one in absolute value. Then condition (9.5) holds. PROOF: Let (Yi)~*=l be an arbitrary nonnegative w-periodic solution of the system (9.4). Because of (9.6) and (9.11)

y,(t)<

for 0.<
(i=l,..,n).

k#i,k=l

Therefore (l[y, ll~

< S(lly, l l ~

Hence it follows that [[y~[[c = 0 (i = 1 , . . . , n), since the eigenvalues of the matrix S are less than one in absolute value. Consequently condition (9.5) holds. The lemma is now proved. LEMbIA 9 . 4 . Let P = (p~)~.k=l be a constant matrix and P~k >10 for i # k. Then in order for condition (9.5) to hold it is necessary and sut~cient that the real part of each eigenvalue of P be negative. PROOF: Suppose condition (9.5) holds. Then according to L e m m a 9.1 p, < 0

(i = 1 , . . . , n ) .

(9.15)

Set

~,, = 0 ,

~,~ = -p,~/p,,

for

i#k,

S=(s,~),:~=i.

For the real parts of the eigenvalues of P to be negative it is necessary and sufficient that r(S) < 1,

(9.16)

where r(S) is the spectral radius of the matrix S. Consequently we need to prove that (9.16) holds. Suppose, on the contrary, that r = r(S) /> 1. 9 n Because S is nonnegative, it has a nonnegative eigenvector (y~)i=l corresponding to the eigenvalue r. It is clear that n

O= p,y, + r -1

~ k•i,k=l

n

P,kYk <~ ~ P , kYk

(i=l,...,n).

k=l

Consequently (Y~)~=I is a nonzero nonegative w-periodic solution of the system (9.4). But this contradicts condition (9.5). Hence (9.16) is established. Suppose now that the real parts of the eigenvalues of the matrix P are negative. Then inequalities (9.15) and (9.16) hold. On the other hand,

/o ~ g(~ip.)(t, ~)Pik dT= --P,k/P. = 8,k

for

i =~ k.

Therefore according to L e m m a 9.3 condition (9.5) holds. The lemma is now proved.

2316

LEMMA 9.5. Le* 0'1 = 0"2 . . . . . Un, P~k E L~, (i, k = l , . . . , n ) , and let P~k (t) >. 0 for t C R, i # k. Then in order for (9.5) to hold it is necessary and sufficient that the differential system n

dyi = Z p , k(t)y" dt

(i=l,...,n)

(9.17)

k=l

be asymptotically stable on the right. PROOF: For definiteness we shall assume that cr~:l

(i = 1 , . . . , n ) .

(9.18)

The case when ai = - 1 , (i = 1 , . . . , n) is considered analogously. Let { 'TP,~ (t) for i # k , Pik (t'~/) = p,(t) for i = k, let Y~ be a fundamental matrix of the differential system tg

dyi = Z p i , ( t ; , 7 ) y t dt

(i:l,...,rt),

(9.19)

k=l

satisfying the condition Y~(0) = E, where E is the identity matrix, and let r(~/) be the spectral radius of the monodromy matrix Yu (w). Then

Y~ (t) ~> 0 for t >/o,

~/t> 0

(9.20)

and r E C(R+ ;(0, +oo)). Suppose now that the system (9.17) is asymptotically stable on the right. Then

r(1) < 1.

(9.21)

Consider an arbitrary nonnegative w-periodic solution y : (yi)~n=l of the system (9.4). Taking account of (9.18) and (9.20), we find y(t) <<.Y1 (t)y(O) for 0 ~< t ~< w and y(0) ~< Y1 (w)y(0). Hence by (9.21) it follows that y(t) - O. Consequently condition (9.5) holds. It remains for us to prove that inequality (9.21) follows from (9.5); for (9.21) is equivalent to the asymptotic sCability of the system (9.18) on the right. Suppose, on the contrary, that (9.5) holds but

r(1)/> 1. By Lemma 9.1 inequalities (9.6) hold, according to which r(0) < 1. Therefore there exists -7 E (0,1] such that =

1.

Let c E R ~ be an eigenvector of the matrix Y~ (w) corresponding to an eigenvalue equal to one. T h e n (y, Ct))L1 =

r.,Ct)c

is a nonzero nonnegative solution of the system (9.19). It is obvious that it also satisfies the system (9.4). But this is impossible in view of the condition (9.5). The contradiction so obtained proves the lemma.

2317

9.2. L i n e a r S y s t e m s . Consider the linear differential system !$

dxl = E a ~ t ( t ) x k + bi(t)

dt

( i = 1,...,n),

(9.22)

k=I

where

ai~ e L~, THEOREM 9 . 1 . 9 "" <

nm

bi e Lw

(i,k = 1,...,n).

(9.23)

Suppose there exist natural numbers m and hi, (j = 1 , . . . ,rn), such that 0 = no < ni <

--n~

a/k(t)=O

(i=nj_,

+l,...,nj;k=nj+l,...,n;j=l,...,rn-1)

(9.24)

and for a/l (t, x l , . . . , z , ) 6 R x R " the inequalities "i

ni

2

Xi

i,k=ni_ 1 +1

(i = 1 , . . . , m ) ,

(9.2s)

i=n/_ 1 +l

hold, where aj e { - 1 , 1}, (3" = 1 , . . . , m ) , a e L~, and o~ a(t) dt > 0.

(9.26)

Then the system (9.22) has one and only one w-periodic solution. PROOF: By Theorem 1.1 and condition (9.23) in order to prove Theorem 9.1 it suffices to establish that the homogeneous periodic boundary-value problem

dxl n at = E a i k x k

(9.27)

(i= 1,...,n),

k=l

x,(O) = xi(w) has only the zero solution. Let

(Xi)in__l be

(9.28)

(i = 1,...,n)

an arbitrary solution of this problem. Set "i

,,jCt) =

=,2 (t)

~

(.i = 1,

,m)

/=n$'_ 1 +i

Because of (9.24) and (9.25) Itl 1

nl

=,(t)=;(t) = 2,,,

!

,,,,,, (t) = 2o, i = no + '

~

a,,(t)=,=~ >~ 2a(t),,,(t)

for

0 < t < w

i,k=no + l

Therefore in the case at = 1 we have

u,(t)exp (2 ft ~ a(,) d~-) .< ~, (w) = ~, (0)

for 0 .< t .< ~,

and in the case al = - 1 we have

u,(t)exp (2 fo t a(,) d , ) .< u, (0) = u, (w)

for

0 .< t .< w.

Hence according to (9.26) it follows that Ul (t) -

o.

Starting from this identity and taking account of conditions (9.24)-(9.26) we prove by induction that ui(t ) = 0, (j = 1 , . . . , n ) . Consequently xi(t) =- 0, (i = 1 , . . . , n ) . The theorem is now proved.

2318

THEOREM 9.2. Let adaii(t) <<. p,i(t),

I~(t)l < p,,(t)

for

t e 1~

(9.29)

(i r k;i,k = 1, . . , ~ ) ,

Iet at E { - 1 , 1}, and suppose condition (9.5) holds. Then the system (9.22) has a unique w-periodic solution (x~)7=i a n d n

I=,(t) - = , , (t)l < r0~"

for

0 ~< t < w

(m = 1,2, ..),

i=1

where (=,0),"=1 e C([0,wl; R n) is an a r b i t r ~ y

p,(s) d,)=,,,,_, (w +

-

vector-~lued

t,)

f/ (I')[ exp

runction,

p,, (,-)=,.,,_, (,-) + b,(,-)] dr

p , (s) ds

(m : 1,2,...),

kff -1

ti - 1__~-ai w~, (i = 1, . . . , n), and r0 > 0 and 6 e (0, 1) are numbers independent of m. PROOF: Let t~ and ~o0~, (i = 1 , . . . , n), be the numbers and functionals given by equalities (9.8) and (9.9), n

,,,, (t)=, + b,(t)

f,(t,=,,...,=.) = ~

(i

=

1,...,n),

k=l

a=0,

b=w

and

~o,(t,=,,...,~.)

= =,(w

-

t,)

(i = 1,...,r,).

In view of (9.23) the periodic extension of the solution of the problem (4.1), (4.2) to R is an w-periodic solution of the system (9.22), and conversely the restriction to [0, w] of an w-periodic solution of the system (9.22) is a solution of the problem (4.1), (4.2). On the other hand, by inequalities (9.29) and Lemma 9.2 conditions (4.12), (4.13), and (9.7) hold. If we now apply Theorem 4.5, then Theorem 9.2 becomes obvious. REMARK 9.1. In Theorem 9.2 the condition (9.5) is essential and cannot be weakened; for, as will be shown below, if p~k 6 L~, (i, k = 1 , . . . , n), P~k (t) /> 0, for t 6 R, i r k, and

(p,,),%, r u:, ..... ~-,

(0.30)

then there exist bi e L~, (i = 1 , . . . , n) and functions aik 6 L , , (i,k = 1 , . . . , n ) , satisfying inequalitites (9.29), and such that the system (9.22) has no w-periodic solutions. In view of (9.30) the system (9.4) has a nonzero nonnegative solution (yi),n_-i 9 We set n

p,(t)

=

- 1 -Ip,,(t)l,

ho,(t) =-p,(t)y,(t) + ~_.p,,(t)y,(t)

(i = 1 , . . . , n ) .

k=l

Then for any i E { 1 , . . . , n} the differential equation

=;(t) = ,,,p,(t)=,(t) + ,,,ho,(t) has a unique w-periodic solution xi and

y,(t) < =,(t)

for t e R. 2319

Therefore

0 <<.hoi(t) = aix~(t) - p i ( t ) x i ( t ) <~hi(t)

(i = 1 , . . . , n ) ,

where n

hi(t) = - p i ( t ) x i ( t ) + ~-~pik(t)z~(t). k=l

Consequently

(2~i ) in= 1 is

a nonzero w-periodic solution of the differential system (9.27), where a . ( t ) = aipi(t) + airli(t)[p.(t) - piCt)],

~ (t) = o,~i(t)p,~ (t) for i # ~, hoi(t)/hi(t) for hi(t) > O, rli(t) = 0 for h,(t) = 0 . and 0 ~< r~ (t) ~< 1 for t e R . According to Remark 1.2 there exist bi E Loj, (i = 1 , . . . , n ) , such that the system (9.22) has w-periodic solutions. On the other hand it is obvious that the functions aik satisfy inequalities (9.29). By Lemmas 9.3-9.5 Theorem 9.2 has as a consequence the

no

COROLLARY 9.1. Suppose inequalities (9.29) are satisfied, where al 6 { - 1 , 1 } and Pik E Lo~, (i, k = 1 , . . . , n). Suppose further that one of the following three conditions holds: n 1) the eigenvalues of the matrix (8 ik )i,k=l, where sii = 0 and sik = max

{/:

gC~ipi, )Ct, T)pi~ (~) dr: o <

t ~< w~ for i # k, are less than one in absolute value;

)

2) plk (t) -- Plk = const, (i,k = 1 , . . . , n ) and the real part of each eigenvalue of the matrix (Pik)i~,k=l is negative; 3) al = a2 . . . . . a. and the differential system (9.17) is asymptotically stable to the right. Then the conclusion of Theorem 9.2 holds. THEOREM 9.3. Suppose

aiaii(t) <~p,,(t),

O <<.aia, k(t) <~pik(t)

for

te R

(9.31)

(i # k ; i , k = l , . . . , n ) ,

ai e {-1,1}, and condition (9.5) holds. Then the entries of the Green's matrix (g,k)in,~=l of the periodic problem (9.27), (9.28) admit the estimates

aig,(t,r) >>.g(aii )(t, r) > 0,

akgik(t,r) >~0 for

i ~ k,

(9.32)

where g is the operator given by equality (9.3). PROOF: Let bi E L~ and

aibi(t) >10 for

tER

(i= 1,...,n).

(9.33)

By Lemmas 5.1 and 9.2 there exists a positive number r such that any nonnegative solution system of differential inequalities

(Xi)in__l

of the

n

aix~(t)<<.Epik(t)xk(t)+aibi(t)

for

0
(i=l,...,n)

(9.34)

k=l

satisfying the boundary conditions (9.28) admits the estimate

xi(t) < r

2320

for

0~
(i=l,...,n).

(9.35)

Set X0(S)=

0

for

s
s r

for for

O~<s~ r,

(9.36)

and consider the differential system

dxi dt = aii(t)zi +

n

(i = 1 , . . . , n ) .

~ , (t)xo(~) + b,(t)

E

(9.37)

k~i,k=l

By conditions (9.5) and (9.31) and Lemma 9.1

(,7,o~J,),,k=l e _~ and o,

//

a . (t) dt < o

(i = 1,...,n).

(9.38)

Consequently the homogeneous problem d~

dt

=~,(t)z,,

z,(O)=zi(w)

(i=l,...,n)

has only the zero solution. Therefore according to Corollary 2.1 the problem (9.37), (9.28) has a solution 2:

n

('),:1"

Because of (9.3), (9.33), and (9.38)

x, Ct)=aijo g(a~i)Ct, r)

E

a~(r)x~

dr~O

for

O<~t~<w

Ci--1,...,n).

(0.39)

k~i;k=l

Because of inequalities (9.31) and the nonnegativeness of the vector-valued function (z~)~=l, it is clear that it satisfies the system (9.34). Therefore by the choice of r estimates (9.35) hold. Taking account of these estimates and (9.36), we verify that (xi)~=l is a solution of the problem (9.22), (9.28), and consequently

gikCt, r)bk(r)dr

x, Ct) =

(i=l,...,n).

(9.40)

k=l

From (9.39) and (9.40) we have

gik(t,r)b~(r)-g(a~i)(t,r)aibi(r) dr >10 ( i = 1 , . . . , n ) . Since these inequalities hold for arbitrary functions bi E Lw, (i = 1 , . . . , n), satisfying the conditions (9.33), the functions gik, (i, k = 1 , . . . , n), admit the estimates (9.32). The theorem is now proved. COROLLARY 9.2. If conditions (9.5), (9.31) and (9.33) hold, then thesystem (9.22) has a uniquew-periodic

solution and it is nonnegative.

2321

9.3. Nonlinear

Systems.

9 ~ THEOREM 9.4. Suppose that for a11 t 9 [0,w], (x,)~=1 E R n, (Yi)i=l 9 Rn the inequalities

If,(,,z l , . . . , z , , ) -

~ p , kCt, z,,...,x,)zk

<~q t,

k=l

"

ni

ai ~

I~

(i=

1,...,n),

k=l

ni

Plh(t, x l , . . . , z n ) Y i Y k >la(t)

i,k=l

~

y~

(j= l,...,m),

i=ni_x + 1

and

1,1

Ipi~Ct,z~,...,:~,,)l ~< b(t), i,k=l

hold, where 0 = no < nl < "" < n,~ = n, a t 9 { - 1 , 1 } , b 9 L~, pik (t, z l , . . .

,zn) - 0

Pit 9 K ( [ 0 , w l x R " ; R ) ,

(i,k

= 1,...,n),

a and

(i=ny-1 +l,...,ny;k=ny+l,...n;j=l,...,m-1), fo ~ a(t) dt > O,

the function q 9 K([0, w] x R+ ; R+ ) is nondecreasing on the second argument, and p--+~ lim p-1 ~0~ q(t, p) dt = 0.

Then the system (9.1) has at least one w-periodic solution. PROOF: We denote by S the set of matrix-valued functions (a~k)i~k=l 9 L([0,w];R "• ) satisfying conditions (9.24)-(9.26) and n

(01 < b(t)

for

0 < t < w.

As was shown above, 1 for any (a~)in, k=, 9 S the problem (9.27), (9.28) has no nonzero solution. On the other hand 9 s, provided (xl)i"_-1 9 C([0, w]; R " ) 9 It is also easy to see that if {Y) , aik ),,k=l E S

(j = 1 , 2 , . . . )

and f t aik (Y) ( r lim Jo /--.+oo

f0 t aik(r) dr

uniformly on [O,w] (i,k = l , . . . , n ) ,

n then (ad"k)i,k=1 E S. From what was said above it follows that the matrix-valued function (Pik)/",k=1 satisfies the Opial condition with respect to the pair (1, Io), where l ( x l , . . . , x , ) = (xi(w) - x,(O))~= 1 and 1 0 ( x l , . . . , x , ) - 0. Therefore according to Theorem 2.1 the problem (9.1), (9.28) has at least one solution (x,)'~=1 . Because of (9.2) the periodic extension of (xi)~=l to R is an w-periodic solution of the system (9.1). The theorem is now proved. Taking account of (9.2), we easily verify that a consequence of Theorem 4.1 is 1Cf. the proof of Theorem 9.1. 2322

THEOREM 9.5. The system (9~1) has at least one to-perlodic solution if and only ffthere exist a~ E {-1,1} (i = 1 , . . . , ~ ) , and (~,~)L-, e C([~ R") such that

O~li(t ) < O~2i(t)

for 0 < t < to ($ = 1 , . . . , n ) ,

(--1)~ai[~ki(0)--~(to)]/>0 ( k = l , 2 ; i : l , . . . , n ) and the inequalities ~
(--1)kai[fiCt, x , , . . . , x i _ l , a k i ( t ) , x ~ + , , . . . , x n ) - - a ~ ( t ) ]

hold on theset { ( t , x , , . . . , x , )

:0 <~ t <~ to, a,,(t) <~ x, <~ a,i(t)

(i= 1,...,n)}.

By Lemma 9.2 we obtain the following propositions from Theorems 4.2 and 4.5. THEOREM 9.6. Suppose the inequalities Pt

aifi(t,x,,...,zn)sgnxi ~<~-~p,~Ct)lzkl+q(t ) (i= 1,...,n),

(9.41)

k=l

hold on R • .R", where a l e {-1,1}, q E Lw, and (pik )i~,~=l satisfies condition (9.5). Then the system (9.1) has at least one oJ-periodic solution.

THEOREM 9.7. Suppose the inequalities

a~[f~(e,xl,...,=,)- f~(t,y,,...,u,)]sgn(=,-~)

<~ ~-~ p,,

(t)lzk -y,I

(i = 1 , . . . , n ) ,

(9.42)

k=l

hold on R x R n, where a i e {-1,1} and (Pik)in,k=l satisfies condition (9.5). Then: a) the system (9.1) has a unique to-periodic solution (=,)L ; b) for any tXo,),=, e C([0,to];R n) there exists a unique sequence (= ',~),=1 e 6~([o,o~];a - ) (m = 1 , 2,. .. ) such that for each natural number m and i E {1,..., n} the function xi,~ is a solution of the Cauchy problem

dxi,, (t) - f, Ct,=,~-~ (t),...,=,-,,,,-, dt

(t),=,~Ct),=,+,,,_~

(t),...,=,~_~

(t)),

~im

Xira - 1

2

to )

and n

l=,(t) -

=,~ (t)l <

,'o~" fo, o < t < to (m = 1,2,...),

i=I

where ro > 0 and ~ E (0, 1) are numbers independent of ra.

By Lemmas 9.3-9.5 Theorems 9.6 and 9.7 imply the COROLLARY 9.2. Suppose inequalities (9.41) (resp. (9.42)) hold on R • R ", where ai E {-1,1}, q E L~, and pik E L~, (i, k -- 1 , . . . , n), satisfy one of the conditions 1)-3) of Corollary 9.1. Then the conclusion of Theorem 9.6 (reap. Theorem 9.7) holds. COROLLARY 9.3. Suppose the inequalities n

aifi(t,x,,...,x,)

<<.~-~pik(t)x.k +q(t)

(i---- 1 , . . . , n )

(9.43)

k=l

2323

and

oifiCt, zl,...,z,-lO, xi+l,...,z,)

>~0

(i=l,...,n)

(9.44)

I

hold on It x I t S , where a, e {-1,1}, q E L~, and (pik),n,k=I satisfies condition (9.5). Then the system (9.1) has at least one nonnegative w-periodic solution.

PROOF: Suppose S 0 s

X(S)

for for

s < 0, s~>0.

According to Theorem 9.6 and conditions (9.5), (9.43), and (9.44) the system of differential equations dxi

dt --

a~(1 + IP,,(t)l)[z, - X(~,)] + f, Ct, xCz,),...,X(Z.))

(i = 1 , . . . , n )

n n has an w-periodic solution (x~)~=~. It remains for us to show that (X i)i=x is nonnegative; for each nonnegative solution of the latter system is also a solution of the system (9.1). Suppose the contrary. Then there exist numbers i e { 1 , . . . , n } , tl e [O,w), and t2 e (tl,w] such that

x,(tl)=zi(t2)

and

xi(t) < 0

for

tl < t < t 2 .

But this is impossible, since by (9.44) we have

o~=:(t) = -(1 + Ip,,(t)l)x,(t) +aifi(t,x(zlCt)),...,x(x,_l

(t)), 0, x(z,+l

(t)),...,X(Zn(t))) > 0 for tl < t < t2.

The contradiction so obtained proves the corollary. COROLLARY 9.4. Suppose inequalities (9.42) and (9.44) hold on It • I t " , where ai E { - 1 , 1 , } arid (pik)in,k=l satisfies condition (9.5). Then the system (9.1) has one and only one w-periodic solution and it .is nonnegative. w

Periodic

S o l u t i o n s of T w o - D i m e n s i o n a l Differential S y s t e m s

Consider the differential system d~ci

dt - fi(t, xl,x2)

(i = 1,2),

(10.1)

where the functions fi : t t x It2 __. It, (i = 1,2), are periodic with period w > 0 in the first argument and their restrictions to [0, w] x Itz belong to the class K([0, w] x Itz;it). First of all we introduce the sets P* (fl, f2; w) and P. (fl, f2 ; w), in terms of which we state the existence theorem proved below for a periodic solution. DEFINITION 10.1. We shall say that a vector-valued function (71,72) E C([0, w]; It2) belongs to the set P* (fl, f2;w) (resp. the set P, (fl, f2;w)) if 71(0) =71(w),

72(0) ~< 72(w)

(resp.

72(0) >~ 72(w))

and the conditions 7~(t)

=

fl(t,"h(t),72(t))

and

~(t) ~< f2(t,71(t),~2(t)) hold almost everywhere on [0, w]. 2324

(resp. ~(t) /> f2(t,~l(t),~2(t))

THEOREM 10.1. Suppose .fl is nondecreasing on the third argument and that there exist (al,a2) P , ( f l , f 2 ; w ) and (~/l,fl~) E e ' ( f l , f ~ ; w ) such that al (t) <<.~l (t)

for

E

O <<.t <<.w.

Suppose further that

fl(t, xz,X2)sgnx2 >/6(t, 1~21) for 0 < t < w, ~z (t) < Zl < Zl(t), z, e It, f2(t, zl,Z2)sgnz2 >>.-[ho(t) + hxlA(t,x~,x2)l]~(z2) for o < t < bo, ,~, (t) < z, < ,8, (t),

(10.2) z2 E R, (10.3)

and

z2eR, (10.4) where 0 ~< ~o < bo ~< w, ho ~ Z([0, w]; R+ ), hi ~ R + , ~ is a Nagumo function, and ~ ~ K([0, w] • g + ;It+) is a function that is nondecreasing on the second argument with f~(t, xl,Z2)sgnx2 <.[hoCt)+hzlA(t, zi,x2)l]~(x2)

for ao < t < w ,

oq(t)~
bo

lim p--,,'+ oo

6(t, p) dt = +oo. o

Then the system (10.1) has at least one w-periodic solution.

COROLLARY 10.1. Suppose fx is nondecreasing on the third argument and there exist constants el E R and e~. E [Cl,~+c~) such that fl(t, ei,O)=O, (-1)'f2(t, ci,O)>>.O for 0 < t < w (i=1,2), fl(t, xz,z2)sgnz2>/SolZ2[ x for 0 < t < w , ez~~ -[ho(t) + h11~21~](1+ Iz21) for 0 < t < bo, ex ~< zx ~< c2,

x2 E R ,

.f2(t,x~,x2)sgnx2

:~2 e R,

and ~< [ho(t) + hllX21;'](l+ Ix21)

for

o.o < t < w,

~1 <~ xl < ~2,

where o ~< ao < bo ~< w, ho ~ L([O, w]; R+ ), So > o and ~ > o. Then the system (10.1) has at least one w-periodic solution.

Consider the auxiliary problem

(~1,z2) E w, zt(O) = xl(w), z2(o) = z2(w),

(lO.S) (10.6)

where W c C([0,w;R 2) is a compact set. For the proof of Theorem 10.1 we need the following LEMMA 10.1. For the problem (10.5), (10.6) to be solvable it sumces that the set W possess the following properties: 1) Zl(0) = Xl(W) for any (zl,z2) E W; 2) there exist ( z . ~ , z , , ) and (z~,z;) E W such that :,,., (t) ~< :~i (t)

for

o..
:~.2(0)~>:~;.2(w),

:~;i(o)..<:~i(w);

2325

3) ilc (2~1,X2) and (Yl,Y2) e W, Xl(0) -- yl(0), and xl(t ) <~ yl(t) lCor0 ~< t ~<. o.~, the/l ~g2(0) ~.~ V2(0) and ~ ( ~ ) >- y,C~); 4) ir (~,,~,) . d ( y , , v , ) e w , ~(t) .< v, Ct) ro~ o <. t <. ,,,, . d c e [xlCO),vl(O)], then the~e exists (zi,z2) z W such that zlCa) =C adld Xl(t ) ~. zlCt ) ~< y~Ct) :orO <<. t <~ w. PROOF: The case x,1 (0) = x~ (0) is trivial since we then have from 2) and 3)

9 .,(~) .< ~.,(o) .< ~;(o) .< ~;@) .< ~.,(~), i.e., x;(0) = x~(w) and (x'~,x~) is a solution of the problem (10.5), (10.6). Thus without loss of generality we may suppose that x, 1 (0) < x~ (0),

X,2(0) > X,2(W) ~;(0) < ~;@).

(10.7) (10.8)

Let

w~

for

0~
We shall show that there exists t/~ (X,l (0), z; (0)) for which x2(0) > x~(w)

for

(Xl,X2) E W ~ and

X,l(0) ~< Xl(0) ~ r/.

(10.9)

Indeed, in the contrary case, by the compactness of W ~ and property 4) there exists a uniformly convergent sequence (ylk ,Y~k ) E W ~ (k = 1,2), such that Y2k(O) ~
k-*oo

Hence by property 3) we obtain x,2(0) ,< W(0) ,< y2(w) ,< x,~@), which contradicts inequality (10.7). Let t/* be the least upper bound of the set of ~/E (x,1 (0), x~ (0)) for which condition (10.9) holds. By properties 3) and 4) and inequality (10.8) there exist uniformly convergent sequences

(~l~,X~)

and ( y ~ , y ~ ) 9 W ~ (k = 1,2,...)

such that xl~ (O) < ~?* <~ ylk (O),

x2,(O) > x2,@),

xlk Ct) ~
V,,(O) .< V,,@)

O~
and lim xlk(O)=

k-~+oo

lim Ylk(O)---O*.

k--*+co

Setting

x,(t)=

lim x,k(t),

k--*+oo

y~(t)=

lim y~kCt) (i--1,2),

k--*+oo

we shall have, according to 3), Consequently (Xl, x~) satisfies the boundary conditions (10.6). The lemma is now proved. 2326

PROOF OF THEOREM 10.1: We shall prove the theorem under the additional assumptions that

A(t,:rl,.e2) >ACt, xl,Z2) for ~2 > x2

(10.10)

and the function fl has a partial derivative on the second argument whose restriction to [0, w] x R 2 belongs to the class K([0, w] • R2;R). We note that it is easy to get rid of these restrictions by applying the technique used in the proof of Theorem 7.1. According to Lemma 8.2 there exists a positive number r such that any vector-valued function (Xl, x2) E C([0,w] x R2;R) satisfying the inequalities

al(t) ~<x~(t) ~<~l(t) for O~ ~Ct, l~:2Ct)l) for 0 < t < r x~Ct)sgnx2(t) >1 -[hoCt)+ hllz~(t)lko(z2(t)) for 0 < t < bo

(10.11) (10.12) (10.13)

x~Ct)sgnx2(t) <~ [hoCt) +

(lO.14)

and

hIl~:~(t)l]~o(~2(t))

for -o < t < ~,,

admits the estimate Ix2(t)l.~< r

for

0 < t < ~,.

(lO.15)

For any c E [at(0),/~I(0)] we denote by Wc the set of solutions of the system (10.1) satisfying the boundary conditions

Zl(0) = ~:1@) = and the inequality (10.11). According to Corollary 8.11

We#O, since ( a t , a 2 ) E Set

A.(fl,f2)

and (~1,~2) E A*(J't,f2).

W=

[.J

W0.

By (10.2)-(10.4) any vector-valued function (Xl,X2) e W satisfies inequalities (10.11)-(10.13). Consequently it admits the estimate (10.15). Therefore it is obvious that W is a compact subset of the space

c([0,~];R2). We shall now show that the problem (10.5), (10.6) is solvable. To do this it suffices to establish that W possesses the properties listed in Lemma 10.1. Let

(z;,z~) ~ W~,c0) We show first of all that

z;(o) .< Z2 (o).

(lO.16)

Suppose, to the contrary, that the inequality

x~(t) > Z2(t) holds on some interval [0,t0) C [0,w]. Then according to condition (10.10) and the differentiability of .71 on the second argument we shall have

[z~ (t) - ~1 (t)]' = .fl (t,z~ (t), z~ (t)) - / 1 (t,~l (t),~2 (t)) > A(t,z~(t),~2Ct))

- fl(t,~l(t),~2(t))

>~ gCt)lzlCt) - ~l(t)]

~or 0 < t < to,

2327

where g 9 L([a, b]; R). In view of the fact that x, (0) =/~1 (0) we find from the last inequality that

z~ (t) > #x (t) for 0 < t < t 0 . But this is impossible since (x~, x~) 9 W. The contradiction so obtained proves inequality (10.16). In exactly the same way we can prove that

x~ (~) i> #2 (w) and

9,~(o) i> ~(o),

~,~(~,) .< ,~(w),

provided

(z,l,z,2)

e w:,(o) .

Therefore 9 ; (o) .< #~ (o) <. #: (w) .< ~1 @) and

x,,(o) ~> ,~:(o) ~> ~:(w) i> ~.,(w). Consequently W possesses the properties 1) and 2). Analogously it is proved that it also possesses property 3). As for property 4) for W, it follows immediately from Corollary 8.11; for in view of (10.10) each solution of the system (10.1) defined on the interval [0,w] belongs simultaneously to the sets A*(fl,f2) and A , ( f l , f ~ ) . Thus we have proved that there exists a solution (xl,x2) of the problem (10.5), (10.6). It is clear that the periodic extension of (Xl, x2) to R is an w-periodic solution of the system (10.1). The theorem is now proved. THEOREM 10.2. Let fl and f2 have partial derivatives on the phase variables whose restrictions to [0,w] • R ' belong to the class K([0,~,] • R ' ; R ) . Suppose further that the inequalities

aA(t, Zl,Z2) >~o, af,(t,z~,z:) >o, 0X 1

hold

on R

x

R 2 and

aI:(t, zl,z2) >o,

0~r2

(lO.17)

~X,

either O f , (t, Zl, 2~2) ~ 0

0x, or along with (10.17) the condition

aI2(t, zl,x2) <.o Ox:

holds. Then the system (10.1) has at most one w-periodic solution. PROOF: Suppose the theorem does not hold. Then there exist w-periodic solutions (xx, z2) and (Yl, Y2) of the system (10.1) such that either the functions u,(t) = xi(t) - y i ( t )

(i = 1,2)

satisfy the inequalities ulCto) ~O,

us(to) >~0, ul(to)+u2(to) >0,

(10.18)

for some to 6 [0,w) or

ttl(t) > 0 ,

tt2Ct) < 0

for 0 ~ < t < w .

(10.19)

a f, (i' k = 1, 2), Because of the restrictions imposed on ~-xk'

try(t) =m,(t)u,(t) +m2(t)u2(t) (i= 1,2), 2328

(10.2o)

where

g~k E Lw, (i,k = 1,2), the inequalities gll (t) /> 0,

g12 (t) > 0,

g21 (t) > 0,

(10.21)

hold on It, and either gll (t) -- 0,

(10.22)

g22 (t) ~< 0.

(10.23)

or, along with (10.21), we have If inequalities (10.18) hold, then we find from (10.20) and (10.21)

ui(t) > 0

for

to < t ~ < w

(i=1,2).

Hence by the equalities ui(0) =

ui(w) (i = 1,2)

(10.24)

it follows that > 0

(i = 1,2).

Taking account of these inequalities we obtain from (10.20) and (10.21) that

ux(t)>ul(O),

u2(t) > 0

for

O~t~co,

which is impossible because of (10.24). Thus we have proved that inequalities (10.18) cannot hold. Inequalities (10.19) also cannot hold; for if they did, we would obtain from (10.20) and (10.21) that either u~(t)=g12(t)u2(t)
u'2(t)>g22(t)u~(t)>lO for 0 < t < w , according as (10.22) or (10.23) holds. But each of these inqualities contradicts the conditions (10.24). The theorem is now proved. w

Bounded

Solutions

In this section we study questions of existence, uniqueness, and so-called a-stability of the solutions of a differential system dxi dt = f / ( t ' z l " ' " z " ) (i = l , . . . ,n), (11.1) satisfying one of the two conditions sup

Ix,(t)l : t e R

< +co

(11.2)

Ixi(t)l : t E I t +

< +co.

(11.3)

i=l or

sup i=l

It is assumed throughout that the functions fi, (i = 1,... ,n), are defined on t t x It" and their restrictions to [a,b] x It'* belong to the class K([a,b] x I t n ; i t ) for any a E It and b E (a, +co). If a = (a,),~l , where a E {-1,1}, then we denote by N+ (a) (resp. N _ ( a ) ) the set of i E {l,...,n} for which ai = 1, (resp. ai = -1).

2329

U.1. 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 11.11. Suppose there exist numbers trl 6 {-1,1}, (i : 1 , . . . , n ) , and vector-valued functions 1, 2), absolutely continuous on each tinite interval, and such that

(Otk/)~t=l : R ---r R n , (k

-----

sup{la (t)l : t e R }

+co ( i = 1,...,n;k= for t 9 R (i = 1 , . . . , n ) , <

ali(t) <. a~i(t)

1,2),

(11.4) (11.5)

and the inequMitites (-1)~a~[f~(t,z~,...,Zi_l,ak,(t),z,+l,...,z,)-a'~(t)]

<~0

(i= 1,...,n;k=

1,2)

(11.6)

hold on the set {(t, Z l , . . . , X , ) :t e R , t~li(t) ~<xi <~ ct~i(t)

(i= 1,...,n)}.

then the problem (n.1), (11.2) is solvable. PROOF: By conditions (11.5) and (11.6) for each natural number m the differential system (11.1) has a solution (x,,,)i=l defined on f-re, m] and satisfying the conditions 2 9

n

1 z,,(~-m) = ~[aliC-m) + ~ ( - m ) ] a:j,,~(m)

=

1 ~[ctxj(m) + a2j(m)]

for

i 6 N+(a),

for j 9 N+(tr)

and ali(t)~
for

-m<~t<~m

( i = 1 , . . . , n ; m = 1,2,...).

(11.7)

Extend each zi,,, to all of R using the equalities f :vim( - m ) / zi,,~ (m)

(t)

for t < - m , for t > m.

It is obvious that the sequence ( ira)i=,, (m = 1,2,... ,), is uniformly bounded and equicontinuous on each finite interval. Therefore we can distinguish in it a subsequence (z~,~.)~'=1, (u = 1,2,... ,), that converges uniformly on each finite interval. According to (11.4) and (11.7) n

(:gi)n=l --"

lira (xi,~)in_-1

v-*d-co

is a solution of the system (11.1) defined on R and satisfying condition (11.2). The theorem is n o w proved. Analogously we prove the THEOREM 11.12. Suppose there exist numbers ai 9 {-1,1}, (i = 1 , . . . , n) and vector-valued functions (at~)i"_, : R ~ R " , (k = 1, 2), absolutely continuous on each finite interval, and such that

s u p { l a k , ( t ) [ : t e l t + } < +c~ ( i = 1 , . . . , n ; k = 1,2), a,,(t) <~ a2,(t) for t 9 R+ (i = 1 , . . . , n ) , and inequalities (11.6) hold on the set

{(t,zl,...,z,):teR+,ali(t)~
2330

Then for any ci e [ali (O),azi (0)], (i 6 N+ (a)), where a = (ai)i~=l, there exists at least one solution of the system (11.1) satisfying along with (11.3) the conditions S z,(O) : c,

for

i 6 N+ (a).

(11.8)

THEOREM 11.21. Suppose the inequalities n

~,/,(t,~l,...,~.)sgn~, .< ~ p,~l~l + q(t)

(i = 1,... ,n)

(11.9)

k=l hold on R • R " , where ~q 6 {-1,1}, Pik = const, p, < 0, pi~ ~ 0 for i # k, the real parts of the eigenvalues of the matrix (P~k)i~k=l are negative, the function q : R --~ R+ is summable on each ~nite interval, and sup

{f+'

q(r) dr: t e R

}

< +oo.

(11.10)

.It

Then the ,roblem (n.1), (11.2) is solvable. To prove this theorem we shall need the following LEMMA 1 1.1. Suppose Plk >t 0 for i # k, Pii < 0, and the reM parts of the eigenvalues of the matrix P = (pik)~.k-1 are negative. Then there exists a positive number r such that for any a 6 R, b E [ a + 1, +co), q e L([a,b];R+ ), and ai 6 { - 1 , 1}, (i = 1 , . . . , n ) , an arbitrary nonnegatlve solution (u,),=l" of the system of differential inequalities n

aiu:Ct) <. E p , kukCt)+q(t) k----1

for

a
(i=l,...,n)

(11.11)

admits the estimates n

Z "i(t) < r(PO + Pl ) i=I

for

a • t < b

(11.12)

q(t) dt

(11.13)

and u,(t) dt <. r Po +

,

i=1 where

p0=

~

~,(~)+

deN+ (,7) 6=min{1, b-a},

and

~

~,(b),

,,:(,),:,, n

ietr (~,)

p,=max

{c

q(r) d r : a < ~ t ~ b - 6

.It

} .

PROOF: Setting

( IP,, r~,,,I

for i e N+ (~), ti=

b

for

iEN_(a),

sik = / O

for

i # k,

for

i=k,

and ~, = m a x {,,,(t) : a < t .<
(i = 1,... ,,~),

5If N+ (a) = O, conditions (11.8) drop out.

2331

we find from (11.11) that

ui(t) <. exp(p, lt - t,I)u,(t, ) +

p,,

exp(-piilt - ri)uk(r) dr + vi(t)

k~i,k=l n

<.ui(ti)+~sik'~k+vi(t)

(i = 1 , . . . , n),

a
for

(11.14)

k=l

where v,(t) = ]f~ exp(pii It - rl)q(r ) dr I. Let i E N+ (a), t E [a, b], and let m be the integer part of the number t - a. Then

v,(t)=exp(p,,it-a))[L f

exp(-pi,(r-a))q(r)dr

+

/.'

exp(-pii(r-a))q(r)dr

+ra

Ja+k-1

k=l

]

e x p ( - p i i k ) + exp(-pii (t - a)) pl <~ nipi,

-N<exp(p. (t - a)) '-~=1

where ni = [exp(IP,

l) -

1] -1 exp(lpii l) + 1. Analogously we show that vi(t)..
for

a
also in the case i E N_ (a). Taking account of these estimates, we obtain from (11.14) that

( E - S)(~li),n=l <~ (ui(ti) -F pln,),n=l, where S = (s,k)i,k=l and E is the identity matrix. On the other hand the spectral radius of the matrix S is less than one, since the real parts of the eigenvalues of the matrix P are negative. It therefore follows from the last inequality that (7i)~=1 <~ (E - S) -1 (ui(ti) + P, ni)~'=l. (11.15) 9

1"1

In view of (11.11) n

u~(t) ~< - o , lp, I- ~

' u,(t) + [p,l-Xq(t) + ~

s,kuk(t)

a
for

(i=l,...,n).

k=l

By integrating these inequalities from a to b we obtain

u,(t)dt <. a~lp,,I-l(u, Ca) -u,(b)) + lP,,I -x

q(t)dt +

sik

uk(t) dt

(i= 1,...,n)

k=l

and

b ui(t) dt

<<. Ooui(t,) + 7o

q(t) dt

i=1

where no = max { IPii 1-1

.. i ~-~

+S

uiCt) dt

i=1

, i=1

1 , . . . , n}. Therefore

u,(t) dt

<~ (E - S) -1 i=1

(

noui(t) + To

/:

q(t) dt

.

(11.16)

i=1

Estimates (11.12) and (11.13)follow from (11.15) and (11.16), where r = I I ( E - S ) -1

II(E,"_-i n, +nn0 / \

is a number depending only on Pik, (i, k = 1 , . . . , n). The lemma is now proved. 2332

]

9 n PROOF OF THEOREM 11.21: Let a = (ai)i=l, and let r > 0 be the number occurring in Lemma 11.1. n According to Corollary 4.2 for any natural number m the system (11.1) has a solution (X ~=)i=1 satisfying the boundary conditions

xi,~(-m)=O

for

ieN+(a),

xj,~(ra)=0

for

jEN_(a).

(11.17)

In view of (11.9) for each m the vector-valued function ( i)i=l = (Ixi= I)L, is a nonnegative solution of the system of differential inequalities (11.11), where a = - m , b = m. Therefore by Lemma 11.1 and conditions (11.10) and (11.17) n

Y~[xir~(t)[~
for

(11.18)

-m~
i=l

where r0 = r s u p ( f : +l q ( r ) d r : t E R}. We shall suppose that xi,n(t) = xi,,,(-rn) for t < - m and xi,+ (t) = x,,~ (m) for t > m. Since (x~,~)i~l, (m = 1,2,... ), is uniformly bounded and equicontinuous on each finite interval, it contains a subsequence ( x + , ~ ) , ~ , (v = 1, 2,... ), uniformly convergent on each such interval. Because of (11.18) (xi(t))~=x = lim (xi,~. (t))~'=1 for t e R v---*-t- o o

is a solution of the problem (11.1), (11.2). The theorem is now proved. THEOREM 11.31. Suppose the inequalities t'1

ai[fi(t, x l , . . . , z ~ ) -

-yi) ~<~ pik[xk -v,l

f+(t, yl,...,yn)lsgn(z,

(11.19)

k=l

hold on R x R " , where a+ E {-1,1}, p+k = const, pii < 0, Pik >t 0 for i # k, and the real parts of the eigenvalues of the matr/x (pi~)+~=~ are negative. Further suppose [fi(r,O,...,O)ldr:tER

sup .It

/

<+c~

(i=l,...,n).

(11.20)

Then the problem (11.1), (11.2) has a Unique solution. PROOF: Inequalities (11.9) and (11.10) follow from (11.19) and (11.20), where n

q(t) = ~-~'~If+(t,0,...,0)l. i=1

Consequently all the hypotheses of Theorem 11.21 are satisfied, which guarantees the solvability of the problem under consideration. It remains for us to prove that it has at most one solution. Let r be the number occurring in Lemma 11.1, let (xi)i~=l and (Y+)i"=l be arbitrary solutions of the problem (11.1), (11.2), and let - , ( * ) -- I~,(t) - y,(t)l

(i -- 1 , . .

,n)

and

r, = sup

, , ( t ) : t e tt i=1

Because of (11.19) n

o,,,~(t) < ~ p , k , ~ ( t )

for

t 61%

(i

=

1,...,n).

k=l

2333

Therefore according to Lemma 11.1 we have

It follows from this that there exist sequences tl,~ E ( - o o , - 1 ) and t~,~ E (1,+oo), (m = 1,2,... ,), such that lim tl,, = - e o , lira t2r~ = + c ~ and lim 6,~ = 0 , (11.21) ~rl,--+ + o o

m--*-{-~

ft~--.* q- o o

where gm

=

+

uCt,.)].

i=l

Applying Lemma 11.1 again, we find r$

ui(t)~
for

tl,~ ~
(re=l,2,...).

i=I

From these estimates it follows by (11.21) that u~(t) =_--O, (i = 1 , . . . ,n). The theorem is now proved. From Corollary 9.1 and Theorem 11.31 follows the COROLLARY 11.1. Suppose the conditions of Theorem 11.31 hold and, in addition,

fi(t + w , x , , . . . , x , ) - -

fi(t, x l , . . . , x , )

(i=l,...,n),

where o: > O. Then the problem (11.1), (11.2) has a unique solution and it is w-periodic.

The following propositions are proved analogously to Theorems 11.2x and 11.31. THEOREM 11.2=. Let a~ E { - 1 , 1}, a = (ai)~=l ; let the inequalities (11.9) hold on the set It+ x It", where P~k = const, p~ < O, P~k >>-0 for i ~ k, the real parts of the eigenvalues of the matrix (Pik)~,k=l are negative, the function q : It+ ~ It+ is summable on each finite interval, and sup

{f'+'

}

q(r) d r : t E R +

<+c~.

,It

Then for any el E It, ( i E N+ (a)) the system (11.1) has at least one solution satisfying the conditions

(11.3) and (II.8). THEOREM 11.32. Let ai E { - 1 , 1}, a = (O'i)i=i, let the inequalities (11.19) hold on the set R+ • It'*, where Pik = const, pii < 0, P~k ~> 0 for i ~ k, and the real parts of the eigenvalues of the matrix (p~k)~,k=l are negative. Further suppose n

sup

{V'

.

IS,(r,O,...,O)ldr:teR+

}

<+c~

(i=l,...,n).

,It

Then for any ci E It, (i E N+ (a)) the system (11.1) has a unique solution satisfying conditions (11.3) and (11.8). 1 1 . 2 . a-Stability.

DEFINITION 11.1. Let oi E {--1, 1}, a : (ai)i=l, and (x,),=1 0 n be a solution of the system (11.1) defined n in an interval I. The solution (x~ is called a-stable if for any ~ > 0 there exists 5 > 0 such that for any segment [a, b] C I and numbers

e, E [x~

2334

- 5i,x~

+ 5]

(i = 1,...,n),

(11.22)

where t~ = a + -1----~a(i b - a), the system (11.1) has at least one solution satisfying the b o u n d a r y conditions xi(ti) : ei

(i = 1 , . . . , n ) ,

(11.23)

and for each such solution the inequality tt

[ x i - x , (0t ) ] < r

for

a~
(11.24)

i=1

holds. We note that if I = R and ai . . . . . a , = 1 (resp. al = . . . . a , = - 1 ) , then a-stability is equivalent to uniform Lyapunov stability to the right (resp. left). In general uniform stability does not follow from a-stability, l~br example the zero solution of the s y s t e m dxi

dt = - a i z ,

(i=l,...,n)

is a-stable, yet it is Lyapunov unstable both to the right and to the left if ai # ak for some i and k 6 {1,...,n}. THEOREM 1 1.4. Let a~ 6 { - 1 , 1 } , a = (a~)'~=~ ; let I C t t be an interval, and let inequalities (11.19) hold on I x R " , where Pik = const, Pii < O, Pik >- 0 for i # k, and the real parts of the eigenvalues of the matr/x (p~k )~,,=~ are negative. Then an arbitrary solution of the system (11.1) defined on I is a-stable. PROOF: Let r be the positive number occurring in L e m m a 11.1, and let (x~)i=x 0 , be some solution of the system (11.1) defined on I. For an arbitrarily given r > 0 set 6 =

(11.25)

nr+ 1

Let [a, b] c I, and let c,, (i = 1 , . . . , n), be numbers satisfying the conditions (11.22). According to Corollary 4.4 the problem (11.1), (11.23) has a unique solution (x,),~=, defined on [a,b]. Because of (11.19) and (11.22) the vector-valued function .

= (Ix, -

0

I)?_-1

is a solution of the system of differential inequalities r~

for

a
(i = 1 , . . . , n)

k=l

and ~eN+ (o)

ieN_ (~)

According to L e m m a 11.1 ~-~ui(t) ~< rnb

for

a ~< t ~< b.

i=l

Estimate (11.24) follows from this by (11.25). Consequently (x~

is a-stable. The theorem is now proved.

2335

COROLLARY 11.21. Suppose the hypotheses of Theorem 11.3 hold. Then the problem (11.1), unique solution and it is a-stable.

(11.2) has a

COROLLARY 11.22. Suppose the hypotheses of Theorem 11.31 hold. Then for any c~ E R, (i e N+ (a)), the system (11.1) has a unique solution defined on R+ and satisfying the conditions (11.3) and (11.8), and it is a-stable. LITERATURE CITED 1. N. V. Azbelev, ~On some trends in generalizations of the differential equation," Differents. Uravn., 21, No. 8, 1291-1304 (1985). 2. N. V. Azbelev and V. P. Maksimov, UA priori estimates of solutions of the Cauchy problem and solvability of boundary-value problems for equations with retarded argument," Differents. Uravn., 15, No. 10, 1731-1747 (1979). 3. N. V. Azbelev and V. P. Maksimov, ~Equations with retarded argument," Differents. Uravn., 18, No. 12, 2027-2050 (1982). 4. N. V. Azbelev and L. F. Rakhmatullina, "Functional-differential equations," Differents. Uravn., 14, No. 5, 771-797 (1978). 5. M. T. Ashordiya, '~On a many-point boundary-value problem for a system of generalized ordinary differential equations," Soobshch. Akad. Nauk Gruz. SSR, 115, No. 1, 17-20 (1984). 6. M. T. Ashordiya, "On a nonlinear boundary-value problem for a system of generalized ordinary differential equations," Soobshch. Akad. Nauk Gruz. SSR, 118, No. 2, 261-264 (1985). 7. M. T. Ashordiya, ~On the structure of the set of solutions of the Cauchy problem for a system of generalized ordinary differential equations," Proceedings of the Vekua Institute of Applied Mathematics, Tbilisi State University, 17, 5-16 (1986). 8. D. G. Bitsadze and I. T. Kiguradze, "On well-posedness for boundary-value problems for systems of ordinary differential equations," Soobshch. Akad. Nauk Gruz. SSR, 111, No. 2, 241-244 (1983). 9. D. G. Bitsadze and I. T. Kiguradze, "On the stability of the set of solutions of nonlinear boundary-value problems," Differents. Uravn., 20, No. 9, 1495-1501 (1984). 10. N. I. Vasil'ev, "Some boundary-value problems for a system of two first-order differential equations. I," Latvian Mathematical Yearbook, 5, 11-24 (1969). 11. N. I. Vasil'ev, "Some boundary-value problems for a system of two first-order differential equations. II." Latvian Mathematical Yearbook, 6, 31-39 (1969). 12. 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). 13. R. V. Gamkrelidze and G. L. Kharatishvili, "Extremal problems in linear topological spaces," Izv. Akad. Nauk SSSR, Ser. Mat., 33, No. 4, 781-839, (1969). 14. Sh. M. Gelashvili, "On a boundary-value problem for systems of functional-differential equations," Arch. Math., 20, No. 4, 157-168 (1984). 15. Sh. M. Gelashvili and I. T. Kiguradze, "On a method of numerical solution of boundary-value problems for systems of ordinary differential equations," Soobshch. Akad. Nauk Gruz. SSR, 115, No. 3,469-472 (1984). 16. G. N. Zhevlakov, Yu. V. Komlenko, and E. L. Tonkov, "On the existence of solutions of nonlinear ordinary differential equations with linear boundary conditions," Differents. Uravn. 4, No. 10, 18141820 (1968). 17. M. A. Kakabadze, "On a problem with integral conditions for a system of ordinary differential equations," Mat. Gas. 24, No. 3, 225-237 (1974). 18. M. A. Kakabadze, "On a singular boundary-value problem for a system of ordinary differential equations," Dokl. Akad. Nauk SSSR, 217, No. 6, 1259-1262 (1974). 19. M. A. Kakabadze and I. T. Kiguradze, "On a boundary-value problem for a system of ordinary differential equations," Differents. Uravn., 7, No. 9, 1611-1616 (1971). 20. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, New York (1982) 2336

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4, 57-65 (1970). 24. I. T. Kiguradze, "On a boundary-value problem for a system of two differential equations," Tr. Tbilissk. Univ., 1(137)A, 77-87 (1971). 25. I. T. Kiguradze, ~On periodic solutions of a system of ordinary differential equations with singularities," Dokl. Akad. Nauk SSSR, 198, No. 2, 286-289 (1971). 26. I. T. Kiguradze, "On some nonlinear boundary-value problems (II)," Bul. Inst. Politehn. Iasi, 18, No. 3-4, 95-107 (1972). 27. I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilissk. Univ. Press (1975). 28. I. T. Kiguradze, "On a periodic boundary-value problem for a two-dimensional differential system," Differents. Uravn., 13, No. 6, 996-1007 (1977). 29. I. T. Kiguradze, "On periodic solutions of a system of nonlinear ordinary differential equations," Usp. Mat. Nauk, 39, No. 4, 137- 138 (1984). 30. I. T. Kiguradze, "On two-point boundary-value problems for systems of nonlinear ordinary differential equations," In: Ninth International Conference on Nonlinear Oscillations. Vol. 1 [in Russian], 168-173, Naukova Dumka, Kiev (1984). 31. !. T. Kiguradze, "On many-point boundary-value problems for systems of ordinary differential equations," Proceedings of the Extended Session of the Seminar of the Vekua Institute of Applied Mathematics, Tbilisi State Univ., 1, No. 3, 54-60 (1985). 32. I. T. Kiguradze, "On periodic solutions of systems of nonautonomous ordinary differential equations," Mat. Zametki, 39, No. 4, 562-575 (1986). 33. I. T. Kiguradze and N. R. Lezhava, "On some nonlinear two-point boundary-value problems," Differents. Uravn., 10, No. 12, 2147-2161 (1974). 34. I. T. Kiguradze and B. Puzha, "On some boundary-value problems for a system of ordinary differential equations," Differents. Uravn., 12, No. 12, 2139-2148 (1976). 35. Yu. A. Klokov, ~Uniqueness of the solution of boundary-value problems for a system of two first-order differential equations," Differents. Uravn., 8, No. 8, 1377-1385 (1972). 36. Yu. V. Kornlenko, "A two-sided method of constructing a periodic solution of a system of ordinary first-order differential equations," In: Problems of the Modern Theory of Periodic Motions [in Russian], 29-38, Izhevsk (1981). 37. M. A. Kraznosel'skii, Translation along Trajectories of Differential Equations, American Mathematical Society Translations, Vol. 19 (1968). 38. M. A. Kraznosel'skii and S. G. Krein, "On the averaging principle in nonlinear mechanics," Usp. Mat. Nauk, 10, No. 3, 147-152 (1955). 39. J. Kurzweil and Z. Vorel, "On continuous dependence of the solutions of differential equations on a parameter," Czechosl. Mat. J., 7, No. 4, 568-583 (1957). 40. A. Yu. Levin, "Passage to the limit for nonsingular systems 3~ = A, (t)X," Dokl. Akad. Nauk SSSR, 176, No. 4, 774-775 (1967). 41. N. R. Lezhava, "On solvability of a nonlinear problem for a system of two differential equations," Soobshch. Akad. Nauk Gruz. SSR, 68, No. 3, 545-547 (1972). 42. A. Ya. Lepin, "Application of topological methods to nonlinear boundary-value problems for ordinary differential equations," Differents. Uravn., 5, No. 8, 1390-1397 (1969). 43. A. Ya. Lepin and A. D. Myshkis, "On an approach to nonlinear boundary-value problems for ordinary differential equations," Differents. Uravn., 3, No. 11, 1882-1888 (1967). 44. A. Ya. Lepin and V. D. Ponomarev, UContinuous dependence of the solution of boundary-value problems for ordinary differential equations," Differents. Uravn. 9, No. 4, 626--629 (1973).

2337

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dz

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