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Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces
Journal of Inequalities and Applications volumeÂ 2010, ArticleÂ number:Â 869684 (2010)
Abstract
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 quasiasymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with KadecKlee property. The results presented in the paper improve and extend some recent results.
1. Introduction
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 [1]. 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 [2] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in uniformly convex Banach space (see also, O'Hara et al. [3] and Chang et al. [4]). 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 [5] 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. [6], Zhou and Tan [7], Wattanawitoon and Kumam [8], Li and Su [9], and Takahashi and Zembayashi [10] extend the notion from relatively nonexpansive mappings or quasinonexpansive mappings to quasiasymptotically nonexpansive mappings and also prove some weak and strong convergence theorems to approximate a common fixed point of finite or infinite family of quasinonexpansive mappings or quasiasymptotically 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 [11], Aleyner and Reich [12]). 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 finitedimensional Banach space (see, e.g., Aleyner and Reich [12], Plubtieng and Ungchittrakool [13, 14]) or uniformly smooth and uniformly convex Banach spaces (see, e.g., Sahu et al. [15] and Ceng et al. [16â€“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 quasiasymptotically nonexpansive. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with KadecKlee property. The results presented in the paper improve and extend the corresponding results in Aleyner and Reich [12], Plubtieng and Ungchittrakool [13, 14], and Chang et al. [19].
2. Preliminaries
Throughout this paper we assume that is a real Banach space with the dual and is the normalized duality mapping defined by
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.
A Banach space is said to bestrictly convex if for all with . is said to be uniformly convex if, for each , there exists such that for all with â€‰ is said to be smooth if the limit
exists for all . is said to be uniformly smooth if the above limit exists uniformly in .
Remark 2.1.
The following basic properties can be found in Cioranescu [20].

(i)
If is a uniformly smooth Banach space, then is uniformly continuous on each bounded subset of

(ii)
If is a reflexive and strictly convex Banach space, then is hemicontinuous, that is, is normcontinuous.

(iii)
If is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping is singlevalued, onetoone, and onto.

(iv)
A Banach space is uniformly smooth if and only if is uniformly convex.

(v)
Each uniformly convex Banach space has the KadecKlee 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 [21], the generalized projection is defined by
Lemma 2.2 (see [21]).
Let be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Then the following conclusions hold:
(a) for all and ;

(b)
if and , then
(26)

(c)
for , if and only if
Remark 2.3.
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 .
Definition 2.4.
A mapping is said to berelatively nonexpansive [5] if , and
A mapping is said to beclosed if for any sequence with and , then .
Definition 2.5.
A mapping is said to bequasinonexpansive if and
A mapping is is said to be quasi Ï• asymptotically nonexpansive [7], if and there exists a real sequence with such that
Remark 2.6.
From the definition, it is easy to know that each relatively nonexpansive mapping is closed.
The class of quasiasymptotically nonexpansive mappings contains properly the class of quasinonexpansive mappings as a subclass and the class of quasinonexpansive 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 quasiasymptotically nonexpansive mappings.
Example 2.7 (see [7]).
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 quasiasymptotically nonexpansive from onto and
Example 2.8.
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 quasinonexpansive mapping, and so it is a closed and quasiasymptotically nonexpansive mapping.
Let be a uniformly convex Banach space, be a positive number and be a closed ball of . Then, for any given subset and for any positive numbers with , there exists a continuous, strictly increasing, and convex function with such that, for any with ,
Lemma 2.10.
Let be a uniformly convex Banach space, a positive number and a closed ball of . Then, for any given sequence and for any given sequence of positive numbers with there exists a continuous, strictly increasing, and convex function with such that for any positive integers with
Proof.
Since and for all with , we have
Hence, for any given and any given positive integers with it follows from (2.12) that there exists a positive integer such that Letting , by Lemma 2.9, we have
Since is arbitrary, the conclusion of Lemma 2.10 is proved.
Lemma 2.11.
Let be a real uniformly smooth and strictly convex Banach space with KadecKlee property, and a nonempty closed convex subset of . Let be a closed and quasiasymptotically nonexpansive mapping with a sequence . Then is a closed convex subset of .
Proof.
Letting be a sequence in with (as ), we prove that In fact, from the definition of we have
Therefore we have
that is, .
Next we prove that is convex. For any , putting we prove that Indeed, in view of the definition of we have
Since , we have (as ). From (2.4) we have Consequently This implies that is a bounded sequence. Since is reflexive, is also reflexive. So we can assume that
Again since is reflexive, we have . Therefore there exists such that . By virtue of the weakly lower semicontinuity of norm we have
that is, which implies that . Thus from (2.17) we have Since and has the KadecKlee property, we have . Since is uniformly smooth and strictly convex, by Remark 2.1(ii) it yields that is hemicontinuous. Therefore . Again since by using the KadecKlee 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 quasiasymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach spaces with the KadecKlee property.
Definition 3.1.
Let be a sequence of mappings. is said to bea family of uniformly quasiasymptotically nonexpansive mappings, if and there exists a sequence with such that for each
A mapping is said to be uniformlyLipschitz continuous, if there exists a constant such that
Theorem 3.2.
Let be a uniformly smooth and strictly convex Banach space with KleacKlee property and a nonempty closed convex subsets of . Let be an infinite family of closed and uniformly quasiasymptotically nonexpansive mappings with a sequence and Suppose that for each is uniformly Lipschitz continuous and that is a nonempty and bounded subset in Let be the sequence generated by
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
Proof.
We divide the proof of Theorem 3.2 into five steps.
Step 1.
We first prove that and both are closed and convex subset of for all .
In fact, It follows from Lemma 2.11 that is closed and convex. Therefore is a closed and convex subset in . Furthermore, it is obvious that is closed and convex. Suppose that is closed and convex for some . Since the inequality is equivalent to
therefore, we have
This implies that is closed and convex. The desired conclusions are proved. These in turn show that and are well defined.
Step 2.
We prove that is a bounded sequence in .
By the definition of , we have for all It follows from Lemma 2.2(a) that
This implies that is bounded. By virtue of (2.4), is bounded. Denote
Step 3.
Next, we prove that for all .
It is obvious that Suppose that for some . Since is uniformly smooth, is uniformly convex. For any given and for any positive integer , from Lemma 2.10 we have
Hence and so for all . By the way, from the definition of , (2.4), and (3.7), it is easy to see that
Step 4.
Now, we prove that converges strongly to some point .
In fact, since is bounded in and is reflexive, we may assume that . Again since is closed and convex for each , it is easy to see that for each . Since , from the definition of , we have
Since
we have
This implies that that is, In view of the KadecKlee property of , we obtain that
Now we prove that .In fact, by the construction of we have that and Therefore by Lemma 2.2(a) we have
In view of and note the construction of we obtain that
From (2.4) it yields Since we have
Hence we have
This implies that is bounded in Since is reflexive, and so is reflexive, we can assume that In view of the reflexive of we see that . Hence there exists such that . Since
Taking on the both sides of equality above and in view of the weak lower semicontinuity of norm it yields that
that is, This implies that and so . It follows from (3.17) and the KadecKlee property of that (as ). Note that is hemicontinuous, it yields that It follows from (3.16) and the KadecKlee property of that
From (3.13) and (3.20) we have that
Since is uniformly continuous on any bounded subset of , we have
For any and any , it follows from (3.8), (3.13), and (3.20) that
In view of condition (b) , we see that
It follows from the property of that
Since and is uniformly continuous, it yieads Hence from (3.25) we have
Since is hemicontinuous, it follows that
On the other hand, for each we have
This together with (3.27) shows that
Furthermore, by the assumption that for each , is uniformly Lipschitz continuous, hence we have
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 .
Step 5.
Finally we prove that .
Let . Since and , we have
This implies 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.
Theorem 3.3.
Let be a uniformly smooth and strictly convex Banach space with KadecKlee property , a nonempty closed convex subset of . Let be an infinite family of closed and quasinonexpansive mappings. Suppose that is a nonempty subset in . Let be the sequence generated by
where for each , is a sequence in satisfying the following conditions:
(a) for all ;
(b) for all .
Then converges strongly to .
Proof.
Since is an infinite family of closed quasinonexpansive mappings, it is an infinite family of closed and uniformly quasiasymptotically 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.
Remark 3.4.
Theorems 3.2 and 3.3 improve and extend the corresponding results in Aleyner and Reich [12], Plubtieng and Ungchittrakool [13, 14] and Chang et al. [19] in the following aspects.

(a)
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 KadecKlee property (note that each uniformly convex Banach space must have the KadecKlee property).

(b)
For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings or quasinonexpansive mapping to an infinite family of quasiasymptotically mappings;

(c)
For the algorithms, we propose a new modified block iterative algorithms which are different from ones given in [12â€“14, 19] and others.
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Acknowledgment
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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Chang, Ss., Kim, J. & Wang, X. Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces. J Inequal Appl 2010, 869684 (2010). https://doi.org/10.1155/2010/869684
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DOI: https://doi.org/10.1155/2010/869684
Keywords
 Banach Space
 Nonexpansive Mapping
 Lower Semicontinuity
 Common Fixed Point
 Nonempty Closed Convex Subset