Iterative approximation for split common fixed point problem involving an asymptotically nonexpansive semigroup and a to
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Iterative approximation for split common fixed point problem involving an asymptotically nonexpansive semigroup and a total asymptotically strict pseudocontraction Prasit Cholamjiak1* and Yekini Shehu2 *
Correspondence: [email protected] 1 School of Science, University of Phayao, Phayao 56000, Thailand Full list of author information is available at the end of the article
Abstract In this paper, we prove the strong convergence theorem for split feasibility problem involving a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically strict pseudocontractive mapping in Hilbert spaces. Our main results improve and extend some recent results in the literature. MSC: 47H06; 47H09; 47J05; 47J25 Keywords: total asymptotically strict pseudocontractive mapping; nonexpansive semigroup; split common fixed-point problems; strong convergence; Hilbert spaces
1 Introduction In this paper, we assume that H is a real Hilbert space with the inner product ·, · and the norm · . Let I denote the identity operator on H. Let C and Q be nonempty, closed and convex subsets of real Hilbert spaces H and H , respectively. The split feasibility problem (SFP) is to find a point x ∈ C such that Ax ∈ Q,
(.)
where A : H → H is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. The SFP attracts the attention of many authors due to its application in signal processing. Various algorithms have been invented to solve it (see, for example, [–] and references therein). Note that the split feasibility problem (.) can be formulated as a fixed point equation by using the fact PC I – γ A∗ (I – PQ )A x∗ = x∗ ;
(.)
that is, x∗ solves SFP (.) if and only if x∗ solves fixed point equation (.) (see [] for details). This implies that we can use fixed point algorithms (see [–]) to solve SFP. A pop© 2014 Cholamjiak and Shehu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Cholamjiak and Shehu Fixed Point Theory and Applications 2014, 2014:131 http://www.fixedpointtheoryandapplications.com/content/2014/1/131
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ular algorithm that solves SFP (.) is due to Byrne’s CQ algorithm [] which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne [] applied KM iteration to the CQ algorithm, and Zhao and Yang [] applied KM iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the KM algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces. Now let us recall the definitions of some operators that will be used in this pape
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