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)II0, 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

2802

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-~