A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

Siwaporn Saewan,Poom Kumam

doi:10.1155/2010/123027

Abstract

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

Highlights

Let E be a Banach space with norm ·, C a nonempty closed convex subset of E, and let E∗ denote the dual of E
We introduce a new hybrid projection method for finding a common solution of the set of common fixed points of two countable families of relatively quasi nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in a real uniformly smooth and 2-uniformly convex Banach space
By using the ∗ -condition, we prove the new convergence theorems for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of two countable families of relatively quasi-nonexpansive mappings, zeros of maximal monotone operators and the solution set of variational inequalities for an αinverse strongly monotone mapping in a 2-uniformly convex and uniformly smooth Banach space

Summary

Introduction

Let E be a Banach space with norm · , C a nonempty closed convex subset of E, and let E∗ denote the dual of E. We know that if T is maximal monotone, the solution set T −10 {x ∈ D T : 0 ∈ T x} is closed and convex. Let E be a smooth, strictly convex and reflexive Banach space, let C be a nonempty closed convex subset of E and let T : E ⇒ E∗ be a monotone operator satisfying D T ⊂ C ⊂. When T is a maximal monotone operator, a well-know method for solving 1.9 in a Hilbert space H is the proximal point algorithm: x1 x ∈ H and, xn 1 Jrn xn, n 1, 2, 3, . Let E be a real Banach space and let C be a nonempty closed convex subset of E and A : C → E∗ be an operator. Recall that let A : C → E∗ be a mapping

Then A is called i monotone if

Preliminaries

Main Results

Application to Complementarity Problem