Boundary-value problems for systems of ordinary differential equations

B O U N D A R Y - V A L U E P R O B L E M S F O R S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS UDC 517.927

I. T. Kiguradze

This arti[cle contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems. Introduction The theory of boundary-value problems for systems of ordinary differential equations, has basically been a creation of the last quarter-century. It was during this time that the method of a priori estimates was largely developed, making it possible to establish criteria for a wide class of nonlinear problems with functional [8, 9, 16, 17, 19, 34, 42-44, 53, 54, 56, 69, 70, 82], many-point [18, 21, 22, 27, 31, 48, 55, 73, 74, 78, 80], and two-point [10-12, 23, 24, 26, 28, 30, 33, 35, 41, 45, 47, 57, 58, 62, 63, 65, 66, 72, 79, 87, 89] boundary conditions to be solvable and well-posed. The present work is devoted to an exposition of the fundamentals of this theory. In the first chapter (w167 we study boundary-value problems of the type d2;

d--t = f ( t , x ) , =0,

(0.1) (0.2)

where f : [a, b] • R " -+ R " is a vector-valued function of the Carath6odory class, and h is a continuous transformation from the space of continuous vector-valued functions into R " . The linear case is considered in w In w we present sufficient conditions for existence and uniqueness of the solutions of the nonlinear boundary-value problem (0.1), (0.2) generalizing the results of Conti [69, 70] and Opial [82]. In w we study the connection of the solution of the problem (0.1), (0.2) with the solutions of a problem closely related to it in a certain sense (0.1')

dt = 0.

(0.2')

In the case of the Cauchy problem, i.e., when h ( x ) ___ hCx) -

(t0) - c,

this question ]has been studied in considerable detail. Here the Krasnosel'skii-Krein Theorem [38] deserves special mention, along with a variety of interesting modifications and generalizations of it [13, 39, 49--51, 59, 68, 88] in which it is asserted that the solutions of problems (0.1), (0.2) and (0.1'), (0.2') are close to each other when the integral of ] - f is small. Translated from Itogi Nauki i T'ekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 3--103, 1987. 0090-4104/88/4302-2259512.50

9 1988 Plenum Publishing Corporation

2259

In contrast to the Cauchy problem the existence of an isolated solution of the general problem (0.1), (0.2) in the nonlinear case does not even guarantee that the problem (0.1'), (0.2') can be solved, no matter how small f - f and h - h are. For example, as is easy to verify, the problem dx d-t = z2'

z(O) = z(1)

has only the zero solution, while the problem d__xx= x2 + e, dt

x(O) = x(1)

has no solutions for any ~ > 0. It is precisely this fact that explains why in many papers the connection between the solutions of the problems (0.1), (0.2) and (0.1'), (0.2') is studied under the a priori assumption that both the initial and the perturbed problem are solvable (cf., for example, [12, 44]). Attempts to find conditions for nonlinear boundary-value problems to be well-posed have led to the concept of a strongly isolated solution [8, 9]. It is in fact the papers [8, 9] that are the basis for w where it is proved that problems with strongly isolated solutions are well-posed and the analogs of the abovementioned theorems of Krasnosel'skii-Krein type hold for them. In the second chapter (w167 we study many-point boundary-value problems for the system (0.1) which are generalizations of the Cauchy-Nicoletti problem [78, 80]. Following [15, 31, 34], we establish criteria that are optimal in a certain sense for their solvability and unique solvability. These criteria have the nature of one-sided restrictions on f. Here we also propose a method of approximate solution. The third chapter (w167 is devoted to the two-point boundary problems

e s,,

e S,

(o.3)

for the system (0.1), where S~ c R" (i = 1, 2) are nonempty closed sets. The exposition in this chapter relies on the results of [23, 24, 26, 30, 33], in which, in turn, the ideas of Bernshtein and Nagumo are developed. The existence theorems proved here cover the case when the components of the vector-valued function are rapidly growing on the phase variables. In the fourth chapter (w167 we investigate those questions of the theory of periodic and bounded solutions that are immediately connected with boundary-value problems. Naturally many problems of this vast theory, quite rich in results, do not appear here (cf. [36, 37, 46, 52, 60-62, 86]). Sections 9 and 11 are based on the results of [29, 31, 32, 34, 81, 83], and w is based on the results of [28]. Boundary-value problems for differential systems with nonintegrable singularities [18, 22, 25, 27, 64, 73, 74, 87] are beyond the scope of this work, as are initial- and boundary-value problems for functionaldifferential systems and generalized differential systems in the sense of Kurzweil (cf., for example, [1-7, 14, 67, 76, 77, 84]). Throughout this work we shall adhere to the following notation: R =

R + = [0,

n R ~ is the n-dimensional real Euclidean space of vectors z = ( i)i=l with norm

n

I111 =

Ix, I; i=1

=

R "•

is the space of real n • n matrices X =

e

0};

n (Xik)i.k=l

i,k=l

2260

with norm

" " if x = (xi)i=l E R " and X = ( x ik)i.,=l 6

R "•

,

then

I~1 = Ci~,l),=~,

I~,1 + ~,

[~]+

2

ixl

/=1

=

We denote the determinant of X by det X and the inverse of the matrix X by X - 1 ; E is the identity matrix. X ( t + ) and X ( t - ) denote respectively the right- and left-hand one-sided limits of the mapping X at the point t; C([a,b];R") and C([a,b];R n• ) are the spaces of continuous vector-valued functions x : [a,b] --~ R n and matrix-valued functions X : [a, b] --* 1%"x" ;

I1~11~ = m~',{ll~Ct)ll

: ,* < ~ < b};

q([~,b];R~.) = {~ e C([a, bl;a") : ~(t) e a ~ for ~ .< t .< b}; C([a, b]; R " ) is the set of absolutely continuous vector-valued functions x : [a, b] ~ R " ; L"([a,b];R ~) and L"([a,b];It "• ), where 1 0 such that the conclusion of this lemma is true. According to (2.14) there exists a positive number pl such that p0

[ll/,(p)ll /: +

1

aCt'p) dt < P for p/> p,.

(2.15)

We set

q(t,x) = f ( t , x ) -- P(t,x)x, 1 X(r) =

2 0

forO~ p0. We set P ( t , x ) =O,

q(t,x) = f ( t , x ( t , x ) ) ,

" l(x) = x(to),

/0(x)-0,

[(x) = - g ( x ) ,

where X is the function given by equality (3.13). It is obvious that conditions a), b), and d) of Definition 3.1 hold. On the other hand, because of the choice of p0 any solution of the problem (3.3), (3.4) belongs to the ball U(0; p0) and therefore is simultaneously a solution of the problem (3.1), (3.2). But the latter problem has no solution except x ~ Consequently condition c) of Definition 3.1 also holds, i.e., x ~ is strongly isolated at radius r. The corollary is now proved. For g(x) = const we obtain from what has been proved the following COROLLARY 3.4. (of. [38, 39]). Suppose to ~ [a,b], c E R " , and the system (3.1) has a unique solution defined on the interval [a, b] and satisfying the initial condition (3.18). Then the problem (3.1), (3.18) is well-posed.

COROLLARY 3.5. Suppose there exist a solution x ~ of the ~roblem (3.1), (3, 2) and a positive number r such that If(t,x) - fCt, x~

- P ( t ) ( x - x~

~< Q(t)lx - x~

for

a < t < b,

I1~- x~

< r

(3.19)

and

Ih(x)-t(x-x~

~O for

a 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)] '7~(t)

for

xi ="/iCt),

~ ~"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) 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


m.

i=X

Let

(t)

f =,,,, (t)

for

,, 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 m0.

i=1

Then

TI

y~l~,=(t)[ "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 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 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

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)) ~o,

x'2Ct)sgnx2Ct) / ~ 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

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 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~>.O. Choose a function p E Lw such that

p(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) ~ 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) >.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) 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) >.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 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(w),

:~;i(o).. 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) ,~:(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) ~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)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) >-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 +

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+

}

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~