Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general system of variational inequalities, and a fixed point problem

Lu-Chuan Ceng,Jen-Chih Yao,Sy-Ming Guu

doi:10.1186/1687-1812-2013-313

Abstract

Abstract In the literature, various iterative methods have been proposed for finding a common solution of the classical variational inequality problem and a fixed point problem. Research along these lines is performed either by relaxing the assumptions on the mappings in the settings (for instance, commonly seen assumptions for the mapping involved in the fixed point problem are nonexpansive or strictly pseudocontractive) or by adding a general system of variational inequalities into the settings. In this paper, we consider both possible ways in our settings. Specifically, we propose an iterative method for finding a common solution of the classical variational inequality problem, a general system of variational inequalities and a fixed point problem of a uniformly continuous asymptotically strictly pseudocontractive mapping in the intermediate sense. Our iterative method is hybridized by utilizing the well-known extragradient method, the CQ method, the Mann-type iterative method and the viscosity approximation method. The iterates yielded by our method converge strongly to a common solution of these three problems. In addition, we propose a hybridized extragradient-like method to yield iterates converging weakly to a common solution of these three problems. MSC:49J30, 47H09, 47J20.

Highlights

Let H be a real Hilbert space with the inner product ·, · and the norm ·, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C
The main theme of this paper is to study the problem of finding a common element of the solution set of variational inequality problem (VIP) ( . ), the solution set of general system of variational inequalities (GSVI) ( . ) and the fixed point set of a selfmapping S : C → C
We study the problem of finding a common element of the solution set of VIP ( . ), the solution set of GSVI ( . ) and the fixed point set of a self-mapping S : C → C, where the mapping S is assumed to be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence {γn} such that Fix(S) ∩ Ξ ∩ VI(C, A) is nonempty and bounded

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and the norm · , let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C. For S to be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence {γn} such that Fix(S) is nonempty and bounded, Sahu et al [ ] proposed an iterative Mann-type CQ method in which the iterates converge strongly to a fixed point of S.

Results

Conclusion