Continuous Approximations of Multivalued Mappings and Fixed Points

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 $F(x)\cap(x+T_X(x))\ne\varnothing$, where $T_X(x)$ 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.