An existence result for a class of extended inclusion problems with applications to equilibrium problems (Q2494617)

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 \(x^*\in K\) such that \(K\subset F(x^*,x^*)\). Under suitable conditions, we establish an existence result for the extended inclusion problem. As applications, we give some existence theorems for equilibrium problems.