Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems

Zhaoli Ma,Lin Wang

doi:10.1186/1687-1812-2014-104

Abstract

In this paper, the demiclosed principle for a k-asymptotically strictly pseudononspreading mapping is shown. Meanwhile, an iterative scheme is introduced to approximate a common element of the set of common fixed points of k-asymptotically strictly pseudononspreading mappings and the set of solutions of mixed equilibrium problems in Hilbert spaces, and some weak and strong convergence theorems are proved. The results presented in this paper improve and extend some recent corresponding results.

Highlights

Let H be a real Hilbert space with the inner product ·, · and the norm ·
X∗ ∈ EP(F) if and only if x∗ ∈ C is a solution of the variational inequality Tx, y – x ≥ for all y ∈ C, i.e., x∗ is a solution of the variational inequality
Inspired and motivated by the recent works of Zhao and Chang [ ], Quan and Chang [ ], etc., in this paper, we propose an iterative scheme to approximate a common element of the set of solutions of k-asymptotically strictly pseudononspreading mappings and mixed equilibrium problem in infinite-dimensional Hilbert spaces

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and the norm ·. Zhao and Chang [ ] proposed the following algorithm for solving k-strictly pseudononspreading mappings and equilibrium problem in Hilbert spaces. Inspired and motivated by the recent works of Zhao and Chang [ ], Quan and Chang [ ], etc., in this paper, we propose an iterative scheme to approximate a common element of the set of solutions of k-asymptotically strictly pseudononspreading mappings and mixed equilibrium problem in infinite-dimensional Hilbert spaces. Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C → C be a continuous k-asymptotically strictly pseudononspreading mapping. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a k-asymptotically strictly pseudononspreading and uniformly L-Lipschitzian mapping.

Main results

Application to a convex feasibility problem