Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms

• Published in: Optimization Vol. 64; no. 12; pp. 2619 - 2630
• Main Author: Zhao, Jing
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.12.2015; Taylor & Francis LLC
• Subjects: Algorithms; feasibility problem; Hilbert space; Iterative methods; Mathematical problems; Nonlinear equations; quasi-nonexpansive mapping; simultaneous iterative algorithm; split fixed-point problem; Studies; weak convergence
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2014.883515

Let , and be real Hilbert spaces, let and be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem: Section.Display where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that, by taking and , we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi's algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated.