ON A LOCAL DEGREE FOR A CLASS OF MULTI-VALUED VECTOR FIELDS IN INFINITE DIMENSIONAL BANACH SPACES

This paper is devoted to the development of a local degree for multi-valued vector fields of the form f − F . Here, f is a single-valued, proper, nonlinear, Fredholm, C-mapping of index zero and F is a multivalued upper semicontinuous, admissible, compact mapping with compact images. The mappings f and F are acting from a subset of a Banach space E into another Banach space E1. This local degree is used to investigate the existence of solutions of a certain class of operator inclusions.


Introduction
An important part of the theory of multi-valued mappings is devoted to inclusions of the type f (x) ∈ F (x), where f is a single-valued mapping and F is a multi-valued mapping. Such inclusion types can be found in different branches of mathematics, e.g., optimal control problems, mathematical economics, game theory.
For the solution of such problems we often employ topological invariant methods. In particular, we use the theory of topological degree, the rotation of a vector field, etc.
In this paper we study the following case: f is a nonlinear Fredholm mapping and F is an admissible, compact, multi-valued mapping. For this purpose we introduce the concept of the local topological degree for mappings of the form f − F. We thus generalize the results given by the authors in [2].
For the construction of the local degree we study the degree of an admissible * -pair (p; q). This degree is defined for mappings which are acting from a finite-dimensional manifold to a finite-dimensional vector space. This process generalizes the construction of the coincidence index which has been introduced by Kucharski [13].
A topological degree for mappings f − F in the case when F is a compact multi-valued mappings with convex images has been constructed in the article of Borisovich [3]. When the multi-valued mapping F is condensing relatively to the mapping f, a topological degree has been constructed in the papers [4] and [5] .
The local degree constructed in this paper is employed to investigate the existence of solutions of a class of operator inclusions. The mapping p induces the homomorphisms p * of theČech homology groups with compact support and coefficients in Q. For more information about theČech homology groups, the reader is referred to [17]. Let M n be a n-dimensional manifold, R n a n-dimensional topological vector space and X a topological space. Let p and q be two single-valued mappings of the form : We shall give a construction of a local degree for this class of pairs of single-valued mappings.
Consider an admissible * -pair (p; q). Then we have and we can thus deduce the following diagram: Using Theorem 1.2, we confirm that p * is an isomorphism and we can thus define the homomorphism (p; q) * = q * • p −1 * : H n (M n ; M n \K) −→ H n (R n ; R n \θ).
Suppose now that M n is oriented with a fixed orientation O ∈ Γ(M n ; Q). Let O K be the fundamental class of the compact set K and let O θ be the fundamental class of the zero θ ∈ R n . The homomorphism (p; q) * transforms O K into O θ multiplied by an integer γ θ ( p; q). Thus, . Definition 1.4. The integer γ θ (p; q), defined above, is called the "local degree" of the admissible * -pair (p; q).
In some particular cases we will denote this degree by γ θ [(p; q), M n ] . We now consider some properties of this degree. Let (p; q) be an admissible * -pair and K 1 a compact subset of M n such that We consider the mappings where ∼ p and ∼ q are the restrictions of p and q, respectively.
The proof of this fact is a consequence of the commutativity of the corresponding diagram. Definition 1.6 (Homotopy). Two admissible * -pairs (p 0 ; q 0 )and (p 1 ; and the following diagram is commutative: where χ i (x) = (x, i), i = 0, 1.
The proof of this proposition is an analogue of the same theorem in [13]. Proposition 1.8 (Product). Let (p 1 ; q 1 ) and (p 2 ; q 2 ) be two admissible *pairs of the forms is also an admissible * -pair and The proof of this proposition is a consequence of the formula of Kunetta and the corresponding commutative diagram.
Let M n be an oriented manifold of class C 1 , L n−1 an oriented submanifold M n of class C 1 and (p; q) an admissible * -pair with , the restriction of (p; q) on L n−1 , i.e., For this purpose we shall prove the following. Proof. This is a consequence of the fact that there exists a neighborhood U of the null selector of the normal bundle on L n−1 and a homeomorphism h of U in some neighborhood of L n−1 in M n such that h(x, 0) = x for every x ∈ L n−1 (see [14]). Because every line bundle on an oriented manifold is a product bundle [12], we can easily deduce the lemma. So, using the the excision property, we have Consider the admissible * -pair (p 1 ; q 1 ) : Then γ θ [(p; q); W ] = γ θ (p 1 , q 1 ). Choose an orientation in R n−1 and R 1 such that the product of the orientations coincides with the orientation in R n . Let us consider the restriction of the orientation of R 1 on (−ε, ε) and define an orientation on V such that the product of these orientations coincides with the orientation on O(V × (−ε, ε)). Let us calculate γ θ (p 1 ; q 1 ). Consider the following admissible * -pairs: From the product property we have γ θ ( Consider the following diagram: where f (y, t) = y for every (y, t) ∈ ∼ X × (−ε, ε) . So we obtain a commutative diagram with the group's homologies induced by the above diagram. This is a consequence of the fact that ∼ p ×id is Using the commutativity of the corresponding diagram we obtain: This is a consequence of the construction of the local degree.
For the proof see [10].
We finish this section with the following proposition.
It is easy to see that in the case p = id : M n −→ M n the local degree γ θ (p; q) coincides with the degree of Dold [10].

Degree for a multi-valued vector field in a finite-dimensional manifold
Let M n be an oriented manifold with a fixed orientation O ∈ Γ(M n ; Q), and let R n be a n-dimensional topological vector space.
Let Φ be an upper semicontinuous multi-valued mapping with nonempty compact images such that Φ : M n −→ K(R n ), where K(R n ) is the set of all nonempty compact subsets of R n . The properties of multi-valued mappings can be found in [8], [1].
Definition 2.1. The multi-valued mapping Φ is called an "admissible multivalued vector field" if there exists an admissible * -pair (p; q) such that In this case the pair of single-valued mappings (p, q) is called a "selected pair" of Φ and we use the notation (p, q) ⊂ Φ.

an admissible multi-valued vector field. Then the set of integers
Let us now give some properties of this degree.

Proposition 2.3. If Φ is an admissible multi-valued vector field with acyclic images then
The proof of this proposition can be deduced from the theorem of Vietoris and the commutativity of the corresponding diagram.
The proof of this proposition is a direct consequence of the definition of Deg θ (Φ, M n ).

an admissible upper semicontinuous multi-valued mapping such that
is a compact subset of M n . Then Φ 0 = Ψ(·, 0) and Φ 1 = Ψ(·, 1) are admissible multi-valued vector fields and This is a direct consequence of the property of homotopy invariance 1.7. Proposition 2.6. Let M n be an oriented manifold of class C 1 and L n−1 an oriented submanifold in M n . Let Φ : M n −→ K(R n ) be an admissible multi-valued vector field such that .
This proposition is a consequence of the property 1.9.

A degree for a multi-valued vector field perturbed by a Fredholm mapping
Let E and E 1 be two Banach spaces, and let U be an open bounded domain in E. Let f : U −→ E 1 be a single-valued, proper, continuous mapping such that the restriction f | U is a nonlinear Fredholm mapping with index zero of class C 1 . We note that f ∈ Φ 0 C 1 . The definition and properties of nonlinear Fredholm mappings can be found in [9] Let F :U −→ K(E 1 ) be an upper semicontinuous compact multi-valued mapping. We shall consider the following multi-valued mapping: The multi-valued mapping Φ is called a "multi-valued vector field generated" by F . We suppose that f (x) / ∈ F (x) for every x ∈ ∂U. Consider the set K ⊂ U defined by We shall build and study a degree for this class of multi-valued vector fields. we start with the following lemma. We can consider now the multi-valued mapping Using the theorem of Sapronov [16] concerning the decomposition of space we see that there exists a direct decomposition of is finite dimensional subspace) and a neighborhood U (K p ) of K p such that π • f : U (K p ) −→ Y p is a submersion in the elements of K p . The mapping π is the natural projector which activates in parallel to ∼ Y p .
Let us consider also the finite dimensional subspace . This is a n-dimensional oriented manifold of class C 1 and n = dim ∼ Rp, which is a consequence of the fact that f ∈ Φ 0 C 1 and Similarly, we can built an oriented manifold M n of class C 1 and a finitedimensional approximation Φ p = f − F p of the multi-valued vector field Φ such that Φ p : M n −→ K( ∼ Rp) and K p ⊂ M n .

Lemma 3.3. The multi-valued mapping Φ p is an admissible multi-valued vector field.
The proof is a natural consequence of the construction of Φ p .

is called the "local degree" of the multi-valued field
We should note that, in the case when F is a multi-valued mapping with acyclic images, Deg θ (f − F, U ) is a singleton: where ∼ t is the projector of the graph Γ M n (F ) on M n , ∼ r is the projector of Γ M n (F ) on E 1 and p is the projector of Schauder. We shall prove that the local degree Deg θ (f − F, U ) of Definition 3.4 is well-defined.

Lemma 3.5. The local degree
and consider the sequence of subsets connecting them: . Evidently T j−1 is a submanifold of class C 1 oriented in T j , and Consider the sequence of * -pairs (l j ; ϕ j ) : where M j = l −1 (T j ) and l j , ϕ j are respectively the restriction of l and ϕ =

using the property of the restriction we have
In the case when This space satisfies the condition of the theorem of Sapronov [16], and , so we can use the above part of this proof. for every x ∈ D.
We can define two multi-valued mappings F p 1 = p 1 • F and F p 2 = p 2 • F. 1, 2, are manifolds such that M n 1 and M n 2 are two oriented submanifolds of M n 0 .
• F be admissible multi-valued vector fields on the manifold M n 0 . Let (l, q) ⊂ F be an admissible * -pair: Let Z 0 = l −1 (M n 0 ). Then we can consider the * -pairs (l; ϕ 1 ), (l; ϕ 2 ) such that: we see that the mappings We must prove now that | . Let us consider the sequence of subspaces connecting the subspaces R p 1 and R p 0 : So we have the sequence of oriented submanifolds of class C 1 : . . , s. Then we have the admissible * -pairs (l j , ψ j ), j = 1, 2, . . . , s, defined by: where l j is the restriction of l on Z j and ψ j the restriction of ϕ 1 on L j .
The nonlinear Fredholm mappings f are also admissible multi-valued vector fields. For this class we can define a local degree Deg θ (f, U ). On the other hand, a degree, γ θ (f, U ), for nonlinear Fredholm mappings has been defined in [6]. We can prove that As in the case of the usual degree, the local degree Deg θ (Φ, U ) can be used to solve the existence problem for the inclusion θ ∈ Φ(x). For example let us give the following proposition. Let U be an open bounded subset of E, f : Proposition 3.11. Let f ∈ Φ 0 C 1 be proper, let F be a compact upper semicontinuous admissible multi-valued mapping. Suppose that

Some applications of the local degree
In this section we shall consider the existence solutions of a class of operator inclusions. For this purpose we consider the following hypotheses. Let Let us consider the following system: (1) , where ε > 0 is a given number. The solution of the system (1) on the interval [0, h] is defined by a pair of continuous single mappings (x(·), y(·)), where x, y : [0, h] −→ R n are such that 1. y(·) is an absolutely continuous mapping, y (t) ∈ F (t, x(t)) for almost all t ∈ [0, h] and y(0) = 0; 2. f (t, x(t)) = εg(t, y(t)) for every t ∈ [0, h] . We shall study the existence of the solution of the system (1). It is simple to see that the problem (1), with the right choice of mappings f and g, is a generalization of the Cauchy problem and a large class of boundary value problems for the differential inclusions. Let In the same way, from the mapping g we can define g : Let us now consider the multi-valued mapping : Proposition 4.1. The operator K is upper semicontinuous and has nonempty compact convex images, i.e., For the proof of the proposition see [8], [1]. We can consider now the following operator inclusion: The solution of the inclusion (2) is a continuous mapping u 0 : Proof. Let (x 0 (·), y 0 (·)) be a solution of System (1). Then f (t, x 0 (t)) = ε g(t, t 0 y 0 (τ ) dτ ), where y 0 (τ ) ∈ F (τ, x 0 (τ )) for almost everywhere τ ∈ [0, h] . So, we can deduce that f (x 0 )(t) ∈ ε (g • K) (x 0 )(t), and this means that x 0 is a solution of the problem (2).
Let us now consider the solvability of the problem (2). For this purpose we shall make the following hypotheses.
1 In [15] we can find some properties for functions f t which satisfy the above assumption. The proof of this proposition is a consequence of the fact that the Fréchetderivative f (u) of the operator of superposition f is an isomorphism for every u ∈ U. Proof. To show that f is an injective mapping, suppose that there exits a function y ∈ f (U ) such that f (x) = y, f (x 1 ) = y and, for some t 0 ∈ [0, h] , we have x(t 0 ) = x 1 (t 0 ). Then, using Condition 3, we can deduce that f (t 0 , x(t 0 )) = f (t 0 , x 1 (t 0 )). But y(t 0 ) = f (t 0 , x(t 0 )) = f (t 0 , x 1 (t 0 )), from which we obtain the desired contradiction.
The mapping f −1 is continued because f is a local homeomorphism.  Proof. Let Γ K be the graph of the multi-valued mapping K, and let (t, r) be the pair composed of the natural projectors t : Γ K −→U and r : Γ K −→ C [0,h] . Then the pair of single-valued mappings (t, g • r) is a selected * -pair of the multi-valued mapping G. On the other hand, t −1 (x) is acyclic because this set is convex. Theorem 4.7. Let the above conditions on the mappings f , g, F be satisfied. Assume further that the following conditions hold.
Proof. Let us prove that the operator inclusion (2) has a solution in U. From the fact that the single-valued mapping f is continuous on the compact subset [0, h] × ∂W and is not null in [0, h] × ∂W, there exists δ > 0 such that U δ (θ) ∩ f ([0, h] × ∂W ) = ∅. Then for every u ∈ ∂U we have f (u) ≥ δ. Since the multi-valued mapping G is upper semicontinuous and compact there exists N > 0 such that z ≤ N for every z ∈ G( U ).
Let r = min The proof of this theorem is similar to the preceding one.