Wiener-Hopf equation technique for solving equilibrium problems and variational inequalities and fixed points of a nonexpansive mapping

Yuehu Wang,Congjun Zhang

doi:10.1186/1029-242x-2014-286

Abstract

In this paper, we introduce some new iterative schemes based on the Wiener-Hopf equation technique and auxiliary principle for finding common elements of the set of solutions of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality. Several strong convergence results for the sequences generated by these iterative schemes are established in Hilbert spaces. As the generation, we also consider two generalized variational inequalities, and obtain some iterative schemes and the proposed strong convergence theorems for solving these generalized variational inequalities, equilibrium problems, and a nonexpansive mapping. Our results and proof are new, and they extend the corresponding results of Verma (Appl. Math. Lett. 10:107-109, 1997), Wu and Li (4th International Congress on Image and Signal Processing, pp. 2802-2805, 2011), and Noor and Huang (Appl. Math. Comput. 191:504-510, 2007).

Highlights

Let H be a real Hilbert space, whose inner product and norm are denoted by ·, · and·, respectively
We introduce some new iterative schemes based on the Wiener-Hopf equation technique and auxiliary principle for finding common elements of the set of solutions of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality
In Section, we extend the variational inequality problem (VIP) in Section and Section, and we get some iterative schemes and strong convergent theorems for solving GVI(K, T) ∩ equilibrium problem (EP)(f ) and GVI(K, T) ∩ EP(f ) ∩ F(S), respectively

Summary

Introduction

Let H be a real Hilbert space, whose inner product and norm are denoted by ·, · and· , respectively. We introduce some new iterative schemes based on the Wiener-Hopf equation technique and auxiliary principle for finding common elements of the set of solutions of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality. We consider two generalized variational inequalities, and obtain some iterative schemes and the proposed strong convergence theorems for solving these generalized variational inequalities, equilibrium problems, and a nonexpansive mapping.

Results

Conclusion