New Hybrid Steepest Descent Algorithms for Equilibrium Problem and Infinitely Many Strict Pseudo-Contractions in Hilbert Spaces

Peichao Duan

doi:10.1155/2013/139123

Abstract

We propose an explicit iterative scheme for finding a common element of the set of fixed points of infinitely many strict pseudo-contractive mappings and the set of solutions of an equilibrium problem by the general iterative method, which solves the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems are proved. The results presented in this paper improve and extend the corresponding results reported by some authors recently. Furthermore, two numerical examples are given to demonstrate the effectiveness of our iterative scheme.

Highlights

Let H be a real Hilbert space with inner product ⟨, ⟩ and induced norm ‖ ⋅ ‖
We propose an explicit iterative scheme for finding a common element of the set of fixed points of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative method, which solves the variational inequality
Let A : C → H be a nonlinear mapping; we consider the problem of finding x∗ ∈ C such that

Summary

Introduction

Let H be a real Hilbert space with inner product ⟨ , ⟩ and induced norm ‖ ⋅ ‖. Acedo and Xu [10] projected new iterative methods for finding a fixed point of strict pseudo-contractions. Tian [11] revealed the inner contact of Yamada’s algorithm [12] and viscosity iterative algorithm and introduced a new general iterative algorithm combining a k-Lipschitzian and η-strong monotone operator. On this basis, Wang [13] considered a general composite iterative method for infinitely many strict pseudo-contractions in 2010. We combine the operator Ln and the general iterative algorithm to propose a new explicit iterative scheme involving equilibrium problem (5) and an infinite family of strict pseudo-contractions. Further an example will be given to demonstrate the effectiveness of our iterative scheme and another will be given to compare numerical results and convergence rate of the algorithm in this paper and [15]

Preliminaries

Main Result

Numerical Result