An extragradient-like approximation method for variational inequalities and fixed point problems

Lu-Chuan Ceng,Qamrul Hasan Ansari,Ngai-Ching Wong,Jen-Chih Yao

doi:10.1186/1687-1812-2011-22

Abstract

AbstractThe purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.1991 MSC: 47H09; 47J20.

Highlights

Let H be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and || · ||, respectively, and let C be a nonempty closed convex subset of H
We propose and study an extragradient-like iterative algorithm that is based on the extragradient-like approximation method in [11] and the modified Mann iteration process in [20]
We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method in [11] and the modified Mann iteration process in [20]:

Summary

Introduction

Let H be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and || · ||, respectively, and let C be a nonempty closed convex subset of H. The following result shows the strong convergence of the sequences {xn}, {yn} generated by the scheme (3.1) to the same point q = PF(S)∩Ω f (q) if and only if {Axn} is bounded, ||(I - Sn)xn|| ® 0 and lim infn®∞ 〈Axn, y - xn〉 ≥ 0 for all y Î C.

Objectives

Results

Conclusion