The strong convergence theorems for split common fixed point problem ofasymptotically nonexpansive mappings in Hilbert spaces

Xin-Fang Zhang,Li Juan Qin,Lin Wang,Zhao Li Ma

doi:10.1186/1029-242x-2015-1

Abstract

In this paper, an iterative algorithm is introduced to solve the split common fixedpoint problem for asymptotically nonexpansive mappings in Hilbert spaces. Theiterative algorithm presented in this paper is shown to possess strong convergencefor the split common fixed point problem of asymptotically nonexpansive mappingsalthough the mappings do not have semi-compactness. Our results improve and developprevious methods for solving the split common fixed point problem. MSC: 47H09, 47J25.

Highlights

Introduction and preliminariesThroughout this paper, let H and H be real Hilbert spaces whose inner product and norm are denoted by ·, · and ·, respectively; let C and Q be nonempty closed convex subsets of H and H, respectively
Some authors proposed some iterative algorithms to approximate a split common fixed point of other nonlinear mappings, such as nonspreading type mappings [ ], asymptotically quasi-nonexpansive mappings [ ], κ-asymptotically strictly pseudononspreading mappings [ ], asymptotically strictly pseudocontraction mappings [ ] etc., but they just obtained weak convergence theorems when those mappings do not have semi-compactness. This naturally brings us to the following question
We introduce the following iterative scheme

Summary

Introduction and preliminaries

Throughout this paper, let H and H be real Hilbert spaces whose inner product and norm are denoted by ·, · and · , respectively; let C and Q be nonempty closed convex subsets of H and H , respectively. Using the iterative algorithm above, in , Moudafi [ ] obtained a weak convergence theorem for the split common fixed point problem of quasi-nonexpansive mappings in Hilbert spaces. Some authors proposed some iterative algorithms to approximate a split common fixed point of other nonlinear mappings, such as nonspreading type mappings [ ], asymptotically quasi-nonexpansive mappings [ ], κ-asymptotically strictly pseudononspreading mappings [ ], asymptotically strictly pseudocontraction mappings [ ] etc., but they just obtained weak convergence theorems when those mappings do not have semi-compactness. This naturally brings us to the following question.

Main results

Applications and examples Application to the equilibrium problem