Structure of the set of solutions of multivalued operator inclusions

Thus, having constructed k e under condition (4.3) for G=co(pr0G), valued operator f - k e and the corresponding equation

we arrive at a single-

O-~-f (x)-- ke (x), xEU. F o r o p e r a t o r s f - k, w h e r e fs162 c o n d i t i o n (f--k)-1(O)cUthe t o p o l o g i c a l

(4.4)

and k:U § Y i s c o m p l e t e l y c o n t i n u o u s , u n d e r t h e d e g r e e deg 2 ( f - k, U, 0) i s d e f i n e d [ 1 2 ] .

2.4.10. Definition. L e t f G~0C~(U) be a p r o p e r m a p p i n g , l e t s e m i c o n t i n u o u s and f - i m p e r m e a b l e m-mapping, and s u p p o s e c o n d i t i o n

G:U § Kv(Y) be an u p p e r ( 4 . 3 ) h o l d s f o r G. Then

the topological characteristic deg (F, U, 0) is defined with values in :Z2 of the set of solutions of inclusion (4.1) by the equality

deg(F, U, O)--~--deg2(f--k~,

U, 0),

where ~ is sufficiently small. It can be shown that this definition is good. It admits a number of generalizations in correspondence with the various versions of the degree of Fredholm mappings and also to the case where G is a generalized acyclic m-mapping. 2.4.11. (4.1).

THEOREM.

If deg (F, U, 0) ~ 0, then there exists

x.%U

satisfying the inclusion

The theorem thus gives a principle for solvability of the inclusion (4.1) with Fredholm principal part. The topological characteristic is preserved under homotopies of the operators f and G preserving their basic properties; this makes it possible to compute the topological characteristic in concrete situations by homotopying the inclusion ( 4 . 1 ) t o a simpler inclusion~ it is also possible to formulate a number of principles of the existence of solutions for inclusions (4.1) by modeling analogous principles for purely Fredholm and impermeable multivalued vector fields. 5.

Structure of the Set of Solutions of Mu!tivalued Operator Inclusions

In this section connectivity and acyclic principles of the set of solutions are proved for abstract operator inclusions. We remark that the connectivity principle proved in the work is a natural development of the topological scheme of M. A. Krasnosel'skii and A. I. Perov although in the work [15] it was asserted that this scheme is in principle not applicable to inclusions with multivalued operators. The results of this section were obtained by B. D. Gel~man. 5.1. Connectivity Principle of the Set of Solutions of an Inclusion with a Multivalued O__perator. Let Y be a Banach space, let U be a bounded open set, in Y, and let F:~ § Kv(Y) be a completely continuous m-mapping. 2.5.1. THEOREM. Supposey(i--F, aU)~-0 and for any e > 0 and any point xIGFixF there exists a completely continuous m-mapping F~,x,:U-+Kv(F)such that !) Fe~xl is a multivalued e-approximation of F; 2) the set Fix Fe,xl is either empty or belongs to an s-neighborhood of the point x I. Then the set FixF

is connected.

Proof. The fact that the set FixF is nonempty follows from properties of the rotation of multivalued vector fields. We shall prove that this set is connected. For this we suppose otherwise. Then the set FixF can %e represented in the form of a union of two nonempty nonintersecting closed sets N O and N I. We denote by U 0 and U I nonintersecting neighborhoods of them lying in U. By properties of the rotation of multivalued vector fields

(~--F, OU)=~(i--F, 0U0)-l-~(~--F, 0Ul). H e n c e , one o f t h e numbers y ( i - - F , 0U0), y ( i - - F , 0U1) i s n o n z e r o . S u p p o s e , t o be s p e c i f i c , y ( i - - F , OUo)~=O. We c o n s i d e r an a r b i t r a r y p o i n t xlGNI. I t i s n o t h a r d t o show t h a t t h e r e exists e0 > 0 such that

that

minp(x,F(x))~eO. a:~ 0Uc

Let 0 < g < e0/3 ; we consider the m-mapping Fe,xz satisfying the conditions of the theorem. It may be assmned with no loss of generality that FixFe,x,n U 0 = ~ for otherwise e can be decreased. Now it is not hard to prove that the fields i - F 0 and i - Fe,xl are linearly

2799

homotopic on 8U 0. Hence, 7(i--F, OUo)=y(i--Fe,x;, OUo)-'/=O. We obtain a contradiction, since the field i - FE,xl must not have singular points in U0, but, on the other hand, by the properties of the rotation there must exist a singular point belonging to U 0. This contradiction proves the theorem. This theorem is a generalization of the connectivity principle of Krasnosel'skii-Perov to the multivalued case. However, it is more general than the theorem known earlier even in the case of a single-valued completely continuous mapping. 2.5.2. COROLLARY. Let f:U § Y be a completely continuous single-valued mapping such that v(i - f, 3U) ~ 0. If for any e > 0 and any point ~Xl~Fixf there exists a completely continous mapping i/e,x,:O~Y such that ][f (x)--fe,x~(X)II<~ for any xGU, and Fixf~,xlc_U~(xO, where u~(xl)-~-{x'IIlxl-x'II<8}, then Fixf is a nonempty, connected set. It is convenient also to use the following theorem to investigate the structure of the set of fixed points of specific classes of m-mappings. 2.5.3. THEOREM. Suppose F:U + Kv(Y) is a completely continuous m-mapping. Suppose there exists a sequence {Fn} , Fn:U + Kv(Y) of completely continuous m-mappings satisfying the following conditions:

l)l Fn(X)DFn+I(X )

for any x~V;

F,,(x)=F(x)

for any X~U;

2) ~ n=l

3)

the

set

F i x Fn i s

connected

for

a n y n = 1, 2,

....

Proof. From the complete continuity of the m-mappings F and F n it follows that Fix F and FixF n are compact, and F i x F c F i x F n § n for any n = I, 2, .... We suppose otherwise: suppose the set FixF is not connected; then there exist two open nonintersecting sets U0, U 1 in Y such t h a t F i x F c U o U U1, FixF~Ui=/=O,-i~-0~ 1. Since t h e s e t s FixF n are connected, for any n there exists a point Xn6FixF ~ such thatxn~UoUU I. It may be assumed with no loss co

of generality that x n + x0, whereby, as is easily seen, x0@nFixFn,

i.e.,

xoEFixF.

This

n=l

contradicts the fact that

FixF~UoUUI.

Hence, the set FixF is connected.

As an application of these theorems we prove that the set of fixed points for one class of m-mappings is connected. Let U be a bounded open region in a Banach space Y, and let F:U + Kv(Y) be a completely continuous m-mapping. Suppose also that F is a multivalued Lipschitz mapping, i.e., h(F(xl), F(x2)i
i) fxo (x)6F

(x) for any x6U;

2 ) ]I XO-- f,Y. (2r ~/~111 XO~'X II f o r a n y x6U'. Proof. Let XoGFixF, and suppose k I satisfies the inequality k < k I < i. We consider the function ~(x)~P(xo, Fix)); it is easy to see that this function is continuous. Let ~(x)= (I+~)~(;), where 0 < $ < (k I - k)/k. We set Fl(X)~F(x)~ It is easy to see that Fl(x)sAGfor any x ~ U and is a convex closed set. It can also be shown that F1(x) is a lower semicontinuous m-mapping, and hence by Michael's theorem there exists a continuous section fx0:U + Y of the m-mapping F I. Since Fi(x)cF(x), this means that condition i is satisfied. We shall show that condition 2 is satisfied:

(x0,

(x))<

(x) = (i

Since x0GFixF, it follows that p(x0, F(x))~h(F(xo), ~) kp (x0, x)
x0, F (x)).

F(x))~
Hence,

p(x0,

f,o(X))<(l+

Let F:U ~ Kv(Y) be a completely continuous m-mapping satisfying the i) [z(F(XO, F(x2))<[[xl--x2l[ for any x~, x2@U; 2) ~](i--F,OU)~O. Then

the set Fix F is nonempty and connect~xl.

2800

(x) = 0 +

Proof. That FixF is nonempty follows from the properties of the rotation of multivalued vector fields. Let x l c F i x F ; we consider the m-mapping F~(x)=(1--8)F(X)@sx~I 0<~
h (F~ (y~), F~ (Y2))"-- ~ ((1 - - ~) F (Yx), (1 - - ~) F (Y2)) < (1 - - ~)1{y~-- Y2 II H e n c e , b y Lemma 2 . 5 . 4 t h e r e fixed point of this mapping.

existsfx,:Oi~F, fx,(x)EF~(x)

for

any xGU,

and x 1 is

the

unique

Then

/ ; , (x)cF~ (x)cu~, (f (x))+ U~H;,II(O)c U~+~ll;,H (F (x)), where

gl

~ ( N -t- I),

=

N=

max.llUil.

y~F(U)

Thus,

/~1 (x)6U~(N+11~,II+II(F (x)). Suppose ~0 > 0 is an arbitrary number. For 0<~<~0/(N-~-l]x1[l-7'1 ),the mapping fx I constructed on the basis of ~ is the mapping fe,xl of Theorem 2.5.1, which proves the assertion. 5.2. Acyclicity of the Set of Solutions. with integer coefficients are considered.

In this subsection reduced Czech cohomologies

Let Y be a Banach space, let X be a metric space, and let #:X + C(Y) be some m-mapping. We shall study the inclusion

0 ~ (x). We denote the set of solutions of this inclusion by N(~, 0). 2.5.6. THEOREM. Suppose there exist an acyclic paracompact topological space V and an m-mapping S:X • C(Y) such that the following conditions are satisfied: i) ~ ( x ) =

U S(x,~

for a n y

xCX;

v6g 2) the inclusion 0ES(x, v) has a unique solution for any

O~S(,~(v) v)is a continuous acyclic for any x6N(~, 0).

3) the mapping ~:V + X such that 4) the set

{v'O~S(x,

v)} is

vfiVi closed mapping;

Then the set N(~, 0) is acyclic. Proof. We consider the single-valued mapping a:V § X; by the assumptions made this mapping is continuous and closed, and the set

~,' (X) ={~ I0es (x, v)} is nonempty and acyclic for any XGN(~,0). Since e(V) = N(~, 0), a!! the conditions of the Vietoris-Begle theorem are satified, and hence the homomorphism ~*:Ifi(N(~,0);Z)-+ffi(V,Z) is an isomorphism for i = 0, 1 ..... Acyclicity of the set N(~, 0) now follows from acyclicity of the space V. We shall prove an approximation theorem which it is convenient to use in the study of concrete classes of inclusions. 2.5.7. LEMMA. Let X be a paracompact Hausdorff topological space, and let {Ni}i~=0 be closed subsets in X satisfying the following conditions:

1)

No~N~,i>~l;

2) f o r a n y n e i g h b o r h o o d Nn c U f o r n ~ > n o ; 3) the sets

N~,i~l,

U of the

set

No there

exists

an i n d e x

n o = no(U)

such that

are acyclic.

Then the set N O is also acyclic. Proof. We consider a confining sequence {U~}~E~ of neighborhoods of the set N 0. homomorphism of the cohomology groups

The

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li_mmH --~ (U~, Z)J-~H " (No, Z) i n d u c e d by t h e i m b e d d i n g map j(z:No-~U(z i s t h e n d e f i n e d i n a n a t u r a l way. By t h e r i g i d i t y p r o p e r t y o f Czech c o h o m o l o g y t h e o r y , t h e homomorphism j * i s an i s o m o r p h i s m . There then exists a s e t Ni s u c h t h a t U s ~ ]'l ~~r , i ~J, No-

Therefore,

t h e homomorphism j ~ i s e q u a l t o j2oj], w h e r e

/~n (Us, Z) J-~Ftn (N,, Z) ~-~2H ~ (No, Z). .~'r

'

S i n c e t h e s e t Ni i s a c y c l i c , t h e homomorphism j ~ i s z e r o . H e n c e , t h e homomorphism .]~ i s a l s o zero. Th e l i m i t homomorphism j * i s t h e n a l s o z e r o . S i n c e j * Ss an i s o m o r p h i s m , i t f o l l o w s t h a t //n(N0, Z ) ~ 0 , i . e . , t h e s e t NO i s a c y c l i c . Y its

2.5.8. Definition. An m - m a p p i n g F:X + C(Y) i s c a l l e d c o m p l e t e p r e i m a g e FI ~ i s a c o m p a c t s e t i n X.

proper

if

for any compact set

G c

We r e m a r k t h a t i f X i s a c l o s e d s u b s e t i n Y and F:X + K(Y) i s a c o m p l e t e ! y c o n t i n u o u s m - m a p p i n g , t h e n t h e m - m a p p i n g O:X + K(Y), ~(X) = x - F ( x ) i s a p r o p e r m - m a p p i n g . 2.5.9. THEOREM. L e t X be a m e t r i c s p a c e , and l e t ~:X § C(Y) be a p r o p e r , u p p e r s e m i continuous m-mapping. S u p p o s e t h e r e e x i s t s a s e q u e n c e On:X ~ C(Y) o f u p p e r s e m i c o n t i n u o u s m-mappings satisfying the following conditions: i)

(D~(x)~CP~+1(x) for

any x6X;

2) for any number g > 0 there exists an index n o = n0(~) such that for n>.~t~0 the graph FX(~ n) is contained in an g-neighborhood of the graph FX(~); 3) the set N(~n, 0) is acyclic for any n = i, 2, .... Then the set N(~, 0) is also acyclic. Proof. From the upper semicontinuity of the m-mappings ~ and ~n it follows that the sets N(~, 0) and N(~n, 0) are closed, and N(q), 0)~N((~LI, 0 ) f o r any n = i, 2, .... It is also not hard to show that N((D, 0 ) = ~ N (CDn,0)o

Indeed,

suppose x0G ~ N((D,,, 0);

n=I

n=l

then 0s for any n = i, 2 ..... We consider a sequence cm ~ 0, sm > 0; then by condition 2 there exist points x m, x m + x 0 and Ym; YmE(1)(xm), i~YmN<s~, i.e., Ym + 0. Since ~ is an upper semicontinuous

m-mapping,

it follows that OC(D(x0), and hence x0~N(cD, 0).

Let U be an arbitrary open neighborhood of the set N(~, 0). We consider the graph l'x\u(q)) of the m-mapping over the set X \ U. We shall now show that there exists a positive n~unber s o such that U~~ is the empty set. For this we suppose otherwise, i.e., suppose there exist sequences of Positive numbers gn, gn + 0 and points xn@X such that p(Xn, % \ U ) < s n andyn~cD(xn), where IIy,~ll<~n. Then the sequence Yn + 0, and, since the m-mapping ~ is proper, the set qg-l({yn})9 is compact in X. Hence, it may be assumed with no loss of generality that x n + x 0. Since the set X \ U is closed, it follows that xo@X\U; however, by the upper semicontinuity of the m-mapping ~ we find that O~(D(x0). This contradicts the fact that )V ((D,0)cU' Let ~0 > 0 b e such a number; then by condition 2 there exists an index n o such that for Iience, Fx\u(On)~(xx{o})=~5, i.e., N((Dn,O)cU. The assertion of the theorem now follows from Lemma 2.5.7.

t~>~: Fx\u(On)mUso(FX\u((D)).

As an application of Theorems 2.5.6 and 2.5.9 we consider the following assertion. 2.5.10. THEOREM. Let X be a complete metric space, and let r ~ Kv(Y) be a proper upper semicontinuous m-mapping. Suppose for any r > 0 there exists a completely continuous m-mapping Gc:X + Kv(Y) satisfying the conditions

1)

max II YI]'< e f o r any x6.X;

~o~(x)

2) t h e

i n c l u s i o n O~(x)@G,(x)~-v

has a unique

solution

for any v, [[vll~e.

Then t h e s e t N(~, 0) i s n o n e m p t y and a c y c l i c . Proof. We s h a l l p r o v e t h a t t h e s e t N(~, O) i s n o n e m p t y . L e t {e~}, e~>0, be a s e q u e n c e of numbers tending to zero. Then there exist sequences {Gn} and {Vn} satisfying the conditions of the theorem such tha~ the inclusion O6qO(x)+G~(x)+v~ has a unique solution x n. Let

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0 = Yn + Zn + Vn, where 9~6~(x~}, z~G~(x~).

Then ILZnll § 0 and llVnll + 0, and hence Hyn! + 0.

We consider the set~:1({yn}). Since the m-mapping ~ is proper, this set is compact; hence, it may be assumed with no loss of generality that the sequence {Xn} converges to a point x 0. By the upper semicontinuity of the m-mapping ~ we then find that 0C~(x0) which implies that the set N(~, 0) is nonempty. We shall now prove that the set N(~, 0) is acyclic. For this we again consider an arbitrary sequence of positive numbers {En} + 0. We set B~Y, B ~ = {xl]]x[[~e~}. The m-mapping ~ : X - + C ( Y ) , ~(x)=@{x)-FG~(x)+B~ is then defined where G n is an m-mapping satisfying the conditions of the theorem and constructed on the basis of sn. We shall show that the sequence {~n} satisfies the conditions of Theorem 2.5.9. In order to prove acyclicity of the set N(~n, 0) we use Theorem 2.5.6; for this we consider the m--mapping Sn:X x B n + C(Y) defined by the relation

It is obvious that ~n{x)=

U

Sn(x, ~,

and the inclusion

O@Sn(x, ~> has

a unique solution for any

v~B n

VGBn.

We denote by ~n:Bn § X the mapping assigning to v~B n the solution of this inclusion. We shall show that ~n is a continuous mapping. We consider sequences {Vm}CBnI'V~-+~o. Let x m = ~m(Vm). Then there exist zmEOn(xm) and ymG~(x:m) such that 0 = Ym + Zm + Vm. Because of the complete continuity of G n it may be assumed with no loss of generality that zm § z 0. Then the sequence -Ym = Zm + Vm, i.e., the sequence Ym is convergent, and hence, since the m-map ~ ping ~ is proper~ the set ~iZ({ym}) is compact. It may thus be assumed without loss of generality that the sequence {Xm} is convergent andx0--1im xm. By the upper semicontinuity of the m-mappings ~ and G n we find thatz0EG~(x0)~-2Zoi-%~(xo),

i.e., there is the inclusion

OG~ (Xo)+ O. (xo) + ~, and hence ~n(V0) = x0, which implies the continuity of the mapping ~n" We shal I prove that the mapping ~n is closed; for this it suffices to show that if a sequence {v~}c~_B~ and x m = ~n(Vm) converges to x0, then from the sequence {Vm} it is possible to extract a convergent subsequence. We t a k e ! y ~ ( X m ) a n d z~,GG~(x~) such thatO~y~+z~+v~. Since under the action of an upper semicontinuous m-mapping with compact images the image of a compact set is compact, it follows that the sets ~({x~}) and G~({Xm}) are compact. The sequences {Yn} and {Zm} thus contain convergent subsequences; then also contains a convergent subsequence.

the subsequence

{Vm}

The set {v]O6Sn(Xo, v)} is convex for any xo6N(~, 0), since ~(x0) , Gn(x0) and B n are convex sets. Thus all the conditions of Theorem 2.5.6 are satisfied, i.e., the set N(~n, 0) is acyclic. We shall now show that the m-mapping ~ and the sequence {r satisfy the conditions of Theorem 2~ Condition 1 of this theorem is obvious, condition 3 has already been proved, and it remains to verify condition 2. This condition follows from the inclusion

~ (x) = 9 (x) + o~ (x) -= B~ c where e ~ 2~ n.

(x) + ~

(~ c uo (~ (x))0

The theorem is proved.

2.5.11. COEOLLARY. Let U be a bounded open region in a Banach space Y, and let f:U § Y be a completely continuous, single-valued operator satisfying the following conditions: i) on the set 8U the operator f has no fixed points, and the rotation y(i - f, 8U)=~0; 2) for any s > 0 there exists a completely continuous operator f ~ : U - + F such that]l f (x)-f~(x)][<e for any x ~ J , and the equation x = fs(x) + v has in U no more than one solution for each ~, ][V[I-~<~, Then the set of fixed points of the operator f is an acyclic set. The proof of this assertion follows from Theorem 2.5.10 if for ~ we take the mapping i - f and for Gg the mapping f~ - f. Thus, under the conditions of the Kraseosel'skii-Petrovconnectivityprinciple (see, for example, [30]) the set of fixed points is acyclic. A similar assertion was proved by different methods in [71].

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