Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert
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RESEARCH
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Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces Shih-sen Chang1 , Jong Kyu Kim2* , Yeol Je Cho3 and Jae Yull Sim4 *
Correspondence: [email protected] 2 Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Korea Full list of author information is available at the end of the article
Abstract The purpose of this article is to study the weak- and strong-convergence theorems of solutions to split a feasibility problem for a family of nonspreading-type mapping in Hilbert spaces. The main result presented in this paper improves and extends some recent results of Censor et al., Byrne, Yang, Moudafi, Xu, Censor and Segal, Masad and Reich, and others. As an application, we solve the hierarchical variational inequality problem by using the main theorem. MSC: 47J05; 47H09; 49J25 Keywords: split feasibility problem; convex feasibility problem; k-strictly pseudo-nonspreading mapping; demicloseness; Opial’s condition
1 Introduction Throughout this paper, we assume that H is a real Hilbert space, D is a nonempty and closed convex subset of H. In the sequel, we denote by ‘xn → x’ and ‘xn x’ the strong and weak convergence of {xn }, respectively. Denote by N the set of all positive integers and by F(T) the set of fixed points of a mapping T : D → D. Definition . Let T : D → D be a mapping. () T : D → D is said to be nonexpansive if Tx – Ty ≤ x – y,
∀x, y ∈ D.
() T is said to be quasi-nonexpansive if F(T) is nonempty and Tx – p ≤ x – p,
∀x ∈ D, p ∈ F(T).
(.)
() T is said to be nonspreading if Tx – Ty ≤ Tx – y + Ty – x ,
∀x, y ∈ D.
(.)
It is easy to prove that equation (.) is equivalent to Tx – Ty ≤ x – y + x – Tx, y – Ty,
∀x, y ∈ D.
(.)
© 2014 Chang et al.; 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.
Chang et al. Fixed Point Theory and Applications 2014, 2014:11 http://www.fixedpointtheoryandapplications.com/content/2014/1/11
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() T is said to be k-strictly pseudo-nonspreading [], if there exists a constant k ∈ [, ) such that Tx – Ty ≤ x – y + k x – Tx – (y – Ty) + x – Tx, y – Ty,
∀x, y ∈ D. (.)
Remark . It follows from Definition . that () if T is nonspreading and F(T) = ∅, then T is quasi-nonexpansive; () if T is nonspreading, then it is k-strictly pseudo-nonspreading with k = . But the converse is not true from the following example. Thus, we know that the class of k-strictly pseudo-nonspreading mappings is more general than the class of nonspreading mappings. Example . [] Let R denote the set of real numbers with the usual norm. Let T : R → R be a mapping defined by Tx =
x, x ∈ (–∞, ), –x, x ∈ [, ∞).
(.)
Then T is a k-strictly pseudo-nonspre
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