Composite Algorithms for Minimization over the Solutions of Equilibrium Problems and Fixed Point Problems

Abstract

The purpose of this paper is to solve the minimization problem of finding $x^{\ast }$ such that $x^{\ast }=\arg {\min }_{x\in \Gamma }{\Vert x\Vert }^2$, where $\Gamma$ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. We first present two new composite algorithms (one implicit and one explicit). Further, we prove that the proposed composite algorithms converge strongly to $x^{\ast }$.