Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings

Huanhuan Cui,Fenghui Wang,Luchuan Ceng

doi:10.1155/2018/9610257

Abstract

We are concerned with the split common fixed point problem in Hilbert spaces. We propose a new method for solving this problem and establish a weak convergence theorem whenever the involved mappings are demicontractive and Lipschitz continuous. As an application, we also obtain a new method for solving the split equality problem in Hilbert spaces.

Highlights

The split common fixed point problem (SCFP) is an inverse problem that aims to find an element in a fixed point set such that its image under a linear transformation belongs to another fixed point set
If U and T are both metric projections, the SCFP is reduced to the well-known split feasibility problem (SFP)
We studied the split common fixed point problem in Hilbert spaces

Summary

Introduction

The split common fixed point problem (SCFP) is an inverse problem that aims to find an element in a fixed point set such that its image under a linear transformation belongs to another fixed point set. It is shown that if τn is chosen in (0, 2/‖A‖2), the sequence generated by method (2) converges weakly to a solution of problem (P1). This result was extended to quasi-nonexpansive operators [7], demicontractive operators [8, 9], two groups of finitely many firmly quasi-nonexpansive mappings [10, 11], and the more general common null point problem [12]. Wang and Xu [19] recently proposed another choice of the stepsize: ρn They proved that if mappings U and T are nonexpansive, the sequence {xn} generated by (3) and (5)-(6) converges weakly to a solution of problem (P1). As a result, based on our extension, we obtain a new method for solving the split equality problem in Hilbert spaces

Preliminaries

The Case for Demicontractive Continuous Mappings

New Methods for the Split Equality Problem

Conclusions