Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems

Yonghong Yao,Yeong-Cheng Liou,Jen-Chih Yao,Mu-Ming Wong

doi:10.1186/1687-1812-2011-53

Abstract

Abstract In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. Mathematics Subject Classification (2000): 47H05; 47H09; 47H10; 47J05; 47J25.

Highlights

Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||
It is well known that the variational inequality problem VI(C, A) is to find u Î C such that
Korpelevich’s extragradient method has extensively been studied in the literature for solving a more general problem that consists of finding a common point that lies in the solution set of a variational inequality and the set of fixed points of a nonexpansive mapping

Summary

Introduction

Let H be a real Hilbert space with inner product 〈· , ·〉 and induced norm || · ||. Let C be a nonempty closed convex subset of H. Korpelevich [6] proved that the sequence {xn} converges strongly to a solution of V I(C, A).

Results

Conclusion