On a Local Degree of One Class of Multi-Valued Vector Fields
by
M. N. Benkafadar

In this work we will study the inclusions of the type f(x) in F(x), where f is a Fredholm mapping, F is an admissible compact multi-valued mapping. For this purpose we introduce the concept of the topological local degree for mappings of the forms f - F which are called multi-valued vector fields generated by F. The mappings f and F are defined from a subset of a Banach space E in another Banach space E₁.

This form of inclusions can be found in different branch of mathematics i.e. optima control problems, mathematical economics, game's theory.

For the solution of this form of problems we often use topological invariant methods in particular the theory of topological degree, the rotational of vector fields and others.

Let E and E₁ be two Banach spaces, U be an open bounded domain in E.

Let f : U --> E₁ be a single proper continuous mapping such that the restriction f|_U is a non linear Fredholm mapping with index zero of class C¹.

Let F:[`U] --> K(E₁) be an upper semi-continuous compact valued mapping.

The mapping F defined above is called admissible multi-valued mapping if there exist a topological space X and two single mappings p : X --> [`U] and q : X --> E<sub>1</sub> which verify the following conditions: p surjective, q o p<sup>-1</sup>(x) subset F(x) for every element x in [`U], p^-1(x) is acyclic for every x in [`U].

We suppose that f(x) notin F(x) for every x in \partialU.

The constructed local degree from this class is denoted Deg_\theta(f-F, [`U]).

We prove in particular the following results:

Proposition If Deg_\theta(f-F, [`U]) =/= {0}, then there exist x₀ in U such that \theta in f(x₀)-F(x₀).

Let U be an open bounded subset in E, f :[`U] --> E<sub>1</sub>, F :[`U] --> K(E₁).

Proposition Let f in \Phi₀ C¹ and proper, F be a compact upper semi-continuous admissible multi-valued mapping. Suppose that: ||f(x)|| >= ||y|| for every y in F(x); x in \partialU; ||f(x)|| =/= 0; for every x in \theta in f(x₀)-F(x₀); Deg_\theta(f[`U]) =/= 0.

1 Borisovitch Y. G. Modern approach to the theory of topological characteristics of non-linear operators Lecture Notes in Mathematics N 1453 1990 21 - 50
2 Borisovitch Y. G., Zviagine V. G., Sapronov Y. I. Non-linear Fredholm mappings and Leray-Shauder theory Usp. Mat. Nauk 1977 32 3 - 54 (in Russian)
3 Benkafadar N., Gel'man B. D. On a local degree for multi-valued mappings with a principal part a Fredholm mapping VINITI N 3422-28 (in Russian)
4 Benkafadar N., Gel'man B. D. On a local degree of one class of multi-valued vector fields

Date received: August 14, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaj-68.