Continuous Approximations of Multivalued Mappings and Fixed Points

Abstract

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition \(F(x) \cap (x + T_X (x)) \ne \emptyset\), where TX(x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.