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

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OPTIMIZATION 2015年第12期64卷 2619-2630页

作者： Zhao, Jing Civil Aviat Univ China Coll Sci Tianjin Peoples R China

Let H-1, H-2 and H-3 be real Hilbert spaces, let A : H-1. H-3 and B : H-2. H3 be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem: Find x. F(U), y. F(T) such that Ax = By, (1) where U : H-1. H-1 and T : H2. H-2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U) = {x. H1 : Ux = x} and F(T) = {x. H-2 : T x = x}. Note that, by taking H-2 = H-3 and B = I, 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.

关键词： feasibility problem split fixed-point problem quasi-nonexpansive mapping weak convergence simultaneous iterative algorithm

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