Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings

Do D Thanh

doi:10.1186/1687-1812-2014-200

Abstract

We give new hybrid variants of extragradient methods for finding a common solution of an equilibrium problem and a family of nonexpansive mappings. We present a scheme that combines the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this scheme is modified by projecting on a suitable convex set to get a better convergence property under certain assumptions in a real Hilbert space.

Highlights

1 Introduction In this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the induced norm ·
Under certain conditions onto parameters {λk} and {αk}, he showed that the sequences {xk}, {yk} and {tk} weakly converge to the point x ∈ Fix(T) ∩ Sol(f, C) in a real Hilbert space
In this paper, motivated by Ceng et al [, ], Wang and Guo [ ], Zhou [ ], Nadezhkina and Takahashi [ ], Cho et al [ ], Takahashi and Takahashi [ ], Anh [, ] and Anh et al [, ], we introduce several modified hybrid extragradient schemes to modify the iteration schemes ( . ) and ( . ) to obtain new strong convergence theorems for a family of nonexpansive mappings and the equilibrium problem EP(f, C) in the framework of a real Hilbert space H

Summary

Introduction

We always assume that H is a real Hilbert space with the inner product ·, · and the induced norm ·. Where the function f and the mappings Ti, i ∈ , satisfy the following conditions: (A ) f (x, x) = for all x ∈ C and f is pseudomonotone on C, (A ) f is Lipschitz-type continuous on C with constants c > and c > , (A ) f is upper semicontinuous on C, (A ) For each x ∈ C, f (x, ·) is convex and subdifferentiable on C, (A ) F ∩ Sol(f , C) = ∅.

Results

Conclusion