An Existence Result for a Class of Extended Inclusion Problems with Applications to Equilibrium Problems

Abstract

Let $X$ be a real reflexive Banach space, $K\subset X$ a nonempty, closed and convex set, and $F:K\times K\to 2^X$ (the family of all the subsets of $X$) be a multi-valued mapping. In this paper, we consider the following extended inclusion problem: find $x^{\ast }\in K$ such that $K\subset F(x^{\ast },x^{\ast })$. Under suitable conditions, we establish an existence result for the extended inclusion problem. As applications, we give some existence theorems for equilibrium problems.