Implicit and explicit iterative methods for mixed equilibria with constraints of system of generalized equilibria and hierarchical fixed point problem

Lu-Chuan Ceng,Ching-Feng Wen,Chin-Tzong Pang

doi:10.1186/s13660-015-0805-2

Abstract

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP, which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then we derive their strong convergence to the unique common solution of the SGEP and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area.

Highlights

Let H be a real Hilbert space with inner product ·, · and induced norm ·, C be a nonempty, closed, and convex subset of H, and PC be the metric projection of H onto C.Let T : C → C be a self-mapping on C
We introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEP) with the constraints of the system of generalized equilibrium problems (SGEP) ( . ) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space
We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP ( . ), which is the unique solution of the HFPP for a strictly pseudocontractive mapping

Summary

Introduction

Let H be a real Hilbert space with inner product ·, · and induced norm · , C be a nonempty, closed, and convex subset of H, and PC be the metric projection of H onto C. ), respectively, converge strongly to the same point x ∈ Fix(T), which is the unique solution to the VIP His results extend and improve Ceng et al.’s corresponding ones [ ] from the nonexpansive mapping T to the strictly pseudocontractive mapping T and from the contractive mapping f to the Lipschitzian mapping V. We make use of weaker control conditions than the previous ones for the sake of proving strong convergence Utilizing these results, we first propose the composite implicit and explicit relaxed extragradientlike schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and derive their strong convergence to the unique common solution of the SGEP

Preliminaries

Main results

Applications

Concluding remarks