Convex Minimization with Constraints of Systems of Variational Inequalities, Mixed Equilibrium, Variational Inequality, and Fixed Point Problems

Lu-Chuan Ceng,Chin-Tzong Pang,Ching-Feng Wen,Cheng-Wen Liao

doi:10.1155/2014/105928

Abstract

We introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

Highlights

Let C be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C
We introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Frechet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space
We prove strong convergence theorem for the iterative algorithm under suitable conditions

Summary

Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. The above iterative schemes are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense; see, for example, [11,12,13]. Motivated and inspired by the above facts, we first introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the CMP (13) with constraints of several problems: finitely many GMEPs, finitely many VIPs, the GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. The results obtained in this paper improve and extend the corresponding results announced by many others

Preliminaries

Convex Minimization Problems with Constraints

Fixed Point Problems with Constraints