Strong Convergence of an Extended Extragradient Method for Equilibrium Problems and Fixed Point Problems

Abstract

Abstract. In this paper, we introduced a new extended extragradient iteration algorithm for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a monotone and Lipschitz-type continuous mapping. And we show that the iterative sequences generated by this algorithm converge strongly to the common element in a real Hilbert space.

On common solutions gradient algorithms, strong convergence theorems and their applications

“…Recently, variational inequality and fixed point problems have been considered by many authors; see, e.g., [3,7,12,13,14,30,31] and the references therein.…”

Splitting methods for monotone operators and bifunctions

“…The equilibrium problems are general which include saddle point problems, variational inequality problems and complementarity problem as special cases. Recently, convergence theorems of solutions to the equilibrium problems were established; see [2,8,12,13] and the references therein.…”

Splitting methods for a convex feasibility problem in Hilbert spaces

“…It can be described as follows: D 1 , D 2 , · · · , D N , where N denotes some positive integer, are finitely many closed convex nonempty subsets of a Hilbert space with D := ∩ N i=1 D i = ∅. Convex feasibility problem is to find a solution in D. Closely related subjects of the problem are variational inequality problems, zero point problems, fixed point problems and equilibrium problem; see [1,3,4,7,8,9,10,11,12,13,14,15,17,18,19,20,23,24,26,27,28,29,30] and the references therein.…”