A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces

Jong Kyu Kim,Won Hee Lim

doi:10.1186/1029-242x-2013-128

Abstract

In this paper, we introduce a new algorithm for finding a common element of the set of fixed points of N strict pseudocontractions and the set of solutions of equilibrium problems with a pseudomonotone and Lipschitz-type continuous bifunction. The scheme is motivated by the idea of extragradient methods and fixed point iteration methods. We show that the iterative sequences generated by this algorithm converge strongly to the above mentioned common element under some suitable conditions on algorithm parameters in a real Hilbert space. And also, we consider the variational inequality problems as an application. MSC: 46H09; 47H10; 47J25; 65K10

Highlights

1 Introduction Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product ·, · and the norm ·, and let f be a bifunction from C × C into R such that f (x, x) = for all x ∈ C
We introduce a new iterative scheme for solving problem ( . )
In Section, we prove the convergence theorems for the algorithms which are defined in Section as the main results of this paper

Summary

Introduction

Marino and Xu [ ] constructed an iterative algorithm for finding a common fixed point of N strict pseudocontractions Si Chen et al [ ] introduced a new iterative scheme for finding a common element of the set of common fixed points of a sequence of strict pseudocontractions {Si}. The authors proved that the sequences {xk}, {yk} and {zk} converged strongly to the same point x*, under certain conditions on {αk} and {rk}, such that x* ∈ PrSol(EP(f ))∩F(S) x , where S is a nonexpansive mapping of C into itself defined by.

Sk tk

Hence we have

It follows that

The function F is called