- Research Article
- Open Access
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.
- Banach Space
- Nonexpansive Mapping
- Lower Semicontinuity
- Common Fixed Point
- Nonempty Closed Convex Subset
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.
exists for all . is said to be uniformly smooth if the above limit exists uniformly in .
If is a uniformly smooth Banach space, then is uniformly continuous on each bounded subset of
If is a reflexive and strictly convex Banach space, then is hemicontinuous, that is, is norm- -continuous.
If is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping is single-valued, one-to-one, and onto.
A Banach space is uniformly smooth if and only if is uniformly convex.
Each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence if and then .
Next we assume that is a smooth, strictly convex, and reflexive Banach space and is a nonempty closed convex subset of . In the sequel we always use to denote the Lyapunov functional defined by
It is obvious from the definition of that
Following Alber , the generalized projection is defined by
Lemma 2.2 (see ).
Let be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Then the following conclusions hold:
for , if and only if
If is a real Hilbert space , then and is the metric projection of onto .
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 .
A mapping is said to beclosed if for any sequence with and , then .
From the definition, it is easy to know that each relatively nonexpansive mapping is closed.
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.
Next, we give some examples which are closed and quasi- -asymptotically nonexpansive mappings.
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.
Since is arbitrary, the conclusion of Lemma 2.10 is proved.
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, .
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.
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.
where , is the generalized projection of onto the set and for each , is a sequence in satisfying the following conditions:
(a) for all
(b) for all
Then converges strongly to
We divide the proof of Theorem 3.2 into five steps.
We first prove that and both are closed and convex subset of for all .
This implies that is closed and convex. The desired conclusions are proved. These in turn show that and are well defined.
We prove that is a bounded sequence in .
Next, we prove that for all .
Now, we prove that converges strongly to some point .
This together with (3.13) and (3.29), yields . Hence from (3.29) we have , that is, . In view of (3.29) and the closeness of , it yields that . This implies that .
Finally we prove that .
In view of the definition of , from (3.32) we have . Therefore, . This completes the proof of Theorem 3.2.
The following theorem can be obtained from Theorem 3.2 immediately.
where for each , is a sequence in satisfying the following conditions:
(a) for all ;
(b) for all .
Then converges strongly to .
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).
For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings or quasi- -nonexpansive mapping to an infinite family of quasi- -asymptotically mappings;
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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