Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

Prasit Cholamjiak,Suthep Suantai,Yeol Je Cho

doi:10.1155/2012/308791

Abstract

We introduce a Halpern‐type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.

Highlights

Let E be a real Banach space, C a nonempty, closed, and convex subset of E, and E∗ the dual space of E
The purpose of this paper is to investigate strong convergence of Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces
Let A : C → E∗ be a continuous and monotone mapping, let f be a bifunction from C × C to R satisfying A1 – A4, and let φ be a lower semicontinuous and convex function from C to R

Summary

Introduction

Let E be a real Banach space, C a nonempty, closed, and convex subset of E, and E∗ the dual space of E. One classical way often used to approximate a fixed point of a nonlinear self-mapping T on C was firstly introduced by Halpern 1 which is defined by x1 x ∈ C and xn 1 αnx 1 − αn T xn, ∀n ≥ 1, 1.1 where {αn} is a real sequence in 0, 1. If f ≡ 0, the generalized mixed equilibrium problem 1.2 reduces to the following mixed variational inequality problem: finding x ∈ C such that. The purpose of this paper is to investigate strong convergence of Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Examples and numerical results are given in the last section

Preliminaries and Lemmas

Main Results

12 This shows that

Examples and Numerical Results

Conclusion