A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

Pongsakorn Sunthrayuth,Poom Kumam

doi:10.1155/2013/230392

Pongsakorn Sunthrayuth, Poom Kumam

Open Access

https://doi.org/10.1155/2013/230392

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Abstract

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.

Highlights

Throughout this paper, we denoted by R the set of all real numbers
We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space
We always assume that H is a real Hilbert space with inner product ⟨⋅, ⋅⟩ and induced norm ‖ ⋅ ‖ and C is a nonempty, closed, and convex subset of H

Summary

Introduction

Throughout this paper, we denoted by R the set of all real numbers. We always assume that H is a real Hilbert space with inner product ⟨⋅, ⋅⟩ and induced norm ‖ ⋅ ‖ and C is a nonempty, closed, and convex subset of H. In 2010, Ceng and Yao [12] proposed the following relaxed extragradient-like method for finding a common solution of generalized mixed equilibrium problems, a system of generalized equilibria (9), and a fixed point problem of a kstrictly pseudocontractive self-mapping S on C as follows: zn = Sr(Θn ,φ) (xn − rnΨxn) , yn = SλG11 [SλG22 (zn − λ2A 2zn) − λ1A 1SλG22 (zn − λ2A 2zn)] , xn+1 = αnu + βnxn + γnyn + δnSyn, ∀n ≥ 0, (12). With the general iterative algorithm (14), we introduce a new iterative method for finding a common element of a fixed point problem of a nonexpansive semigroup, the set solutions of a general system of generalized equilibria in a real Hilbert space. The main result extends various results existing in the current literature

Preliminaries

Main Results

Some Applications