Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces
© Shih-sen Chang et al. 2010
Received: 20 October 2009
Accepted: 28 December 2009
Published: 12 January 2010
The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi- -asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.
The problem of finding a point in the intersection of closed and convex subsets of a Banach space is a frequently appearing problem in diverse areas of mathematics and physical sciences. This problem is commonly referred to as theconvex feasibility problem (CFP). There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning . The advantage of a Hilbert space is that the projection onto a closed convex subset of is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert space. In 1993, Kitahara and Takahashi  deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in uniformly convex Banach space (see also, O'Hara et al.  and Chang et al. ). It is known that if is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space , then the generalized projection from onto is relatively nonexpansive. In 2005, Matsushita and Takahashi  reformulated the definition of the notion and obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Qin et al. , Zhou and Tan , Wattanawitoon and Kumam , Li and Su , and Takahashi and Zembayashi  extend the notion from relatively nonexpansive mappings or quasi- -nonexpansive mappings to quasi- -asymptotically nonexpansive mappings and also prove some weak and strong convergence theorems to approximate a common fixed point of finite or infinite family of quasi- -nonexpansive mappings or quasi- -asymptotically nonexpansive mappings.
It should be noted that theblock iterative algorithm is a method which often used by many authors to solve the convex feasibility problem (see, e.g., Kikkawa and Takahashi , Aleyner and Reich ). Recently, some authors by using the block iterative scheme to establish strong convergence theorems for a finite family of relativity nonexpansive mappings in Hilbert space or finite-dimensional Banach space (see, e.g., Aleyner and Reich , Plubtieng and Ungchittrakool [13, 14]) or uniformly smooth and uniformly convex Banach spaces (see, e.g., Sahu et al.  and Ceng et al. [16–18]).
Motivated and inspired by these facts, the purpose of this paper is to use the modified block iterative method to propose an iterative algorithm for solvingthe convex feasibility problems for an infinite family of quasi- -asymptotically nonexpansive. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend the corresponding results in Aleyner and Reich , Plubtieng and Ungchittrakool [13, 14], and Chang et al. .
In the sequel, we use to denote the set of fixed points of a mapping and use and to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We also denote by and the strong convergence and weak convergence of a sequence respectively.
Following Alber , the generalized projection is defined by
Lemma 2.2 (see ).
Let be a smooth, strictly convex, and reflexive Banach space, a nonempty closed convex subset of , a mapping, and the set of fixed points of . A point is said to be an asymptotic fixed point of if there exists a sequence such that and We denoted the set of all asymptotic fixed points of by .
The class of quasi- -asymptotically nonexpansive mappings contains properly the class of quasi- -nonexpansive mappings as a subclass and the class of quasi- -nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.
Example 2.7 (see ).
Let be a uniformly smooth and strictly convex Banach space and a maximal monotone mapping such that (the set of zero points of ) is nonempty. Then the mapping is closed and quasi- -asymptotically nonexpansive from onto and
Let be the generalized projection from a smooth, strictly convex and reflexive Banach space onto a nonempty closed convex subset . Then is relative nonexpansive, which in turn is a closed and quasi- -nonexpansive mapping, and so it is a closed and quasi- -asymptotically nonexpansive mapping.
Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and a nonempty closed convex subset of . Let be a closed and quasi- -asymptotically nonexpansive mapping with a sequence . Then is a closed convex subset of .
that is, which implies that . Thus from (2.17) we have Since and has the Kadec-Klee property, we have . Since is uniformly smooth and strictly convex, by Remark 2.1(ii) it yields that is hemi-continuous. Therefore . Again since by using the Kadec-Klee property of , we have . This implies that . Since is closed, we have . This completes the proof of Lemma 2.11.
3. Main Results
In this section, we will use the modified block iterative method to propose an iterative algorithm for solving the convex feasibility problem for an infinite family of quasi asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach spaces with the Kadec-Klee property.
We divide the proof of Theorem 3.2 into five steps.
The following theorem can be obtained from Theorem 3.2 immediately.
Since is an infinite family of closed quasi- -nonexpansive mappings, it is an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings with sequence . Hence . Therefore the conditions appearing in Theorem 3.2: is a bounded subset in " and "for each , is uniformly -Lipschitz continuous" are of no use here. In fact, by the same methods as given in the proofs of (3.13), (3.20) and (3.29), we can prove that , and (as ) for each . By virtue of the closeness of mapping for each , it yields that for each , that is, . Therefore all conditions in Theorem 3.2 are satisfied. The conclusion of Theorem 3.3 is obtained from Theorem 3.2 immediately.
For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property (note that each uniformly convex Banach space must have the Kadec-Klee property).
This work was supported by the Natural Science Foundation of Yibin University (no. 2009Z3) and the Kyungnam University Research Fund 2009.
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