Strong convergence theorems for quasi-nonexpansive mappings and maximal monotone operators in Hilbert spaces

Huan-Chun Wu,Cao-Zong Cheng,De-Ning Qu

doi:10.1186/1029-242x-2014-318

Abstract

We present the strong convergence theorem for the iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators in Hilbert spaces. Our results extend and improve the recent results of Takahashi et al. (J. Optim. Theory Appl. 147:27-41, 2010) and Takahashi and Takahashi (Nonlinear Anal. 69:1025-1033, 2008).MSC:47H05, 47H09, 47J25.

Highlights

Let H be a real Hilbert space and let K be a nonempty closed convex subset of H
A mapping T : K → K is nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ K
Motivated by the above results, especially by Chuang et al [ ] and Takahashi et al [ ], we obtain the strong convergence theorem for the iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators in Hilbert spaces

Summary

Introduction

Let H be a real Hilbert space and let K be a nonempty closed convex subset of H. It is well known that the fixed-point set of a quasi-nonexpansive mapping is closed and convex (see [ , ]). Lin and Takahashi [ ] introduced an iterative sequence that converges strongly to an element of (A + B)– ∩ F– , where F is another maximal monotone operator.

Results

Conclusion