# Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces

- Shih-sen Chang
^{1}, - JongKyu Kim
^{2}Email author and - XiongRui Wang
^{1}

**2010**:869684

https://doi.org/10.1155/2010/869684

© Shih-sen Chang et al. 2010

**Received: **20 October 2009

**Accepted: **28 December 2009

**Published: **12 January 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 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.

## Keywords

## 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 the*convex 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 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 the*block 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 finite-dimensional 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. [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 solving*the 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 [12], Plubtieng and Ungchittrakool [13, 14], and Chang et al. [19].

## 2. Preliminaries

*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.

*strictly convex*if for all with . is said to be

*uniformly convex*if, for each , there exists such that for all with 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.

- (i)
- (ii)
If is a reflexive and strictly convex Banach space, then is hemicontinuous, that is, is norm- -continuous.

- (iii)
If is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping is single-valued, one-to-one, and onto.

- (iv)
- (v)
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 [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:

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 be*closed* if for any sequence
with
and
, then
.

Definition 2.5.

*quasi-*

*ϕ*

*-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 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 [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 quasi- -asymptotically 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 quasi- -nonexpansive mapping, and so it is a closed and quasi- -asymptotically nonexpansive mapping.

Lemma 2.10.

Proof.

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 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 .

Proof.

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*.

Definition 3.1.

*a family of uniformly quasi*

*asymptotically nonexpansive mappings*, if and there exists a sequence with such that for each

Theorem 3.2.

where , is the generalized projection of onto the set and for each , is a sequence in satisfying the following conditions:

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 .

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 .

Step 3.

Step 4.

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 .

Step 5.

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.

where for each , is a sequence in satisfying the following conditions:

Proof.

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.

Remark 3.4.

- (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 Kadec-Klee property (note that each uniformly convex Banach space must have the Kadec-Klee property).

- (b)
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;

- (c)
For the algorithms, we propose a new modified block iterative algorithms which are different from ones given in [12–14, 19] and others.

## Declarations

### Acknowledgment

This work was supported by the Natural Science Foundation of Yibin University (no. 2009Z3) and the Kyungnam University Research Fund 2009.

## Authors’ Affiliations

## References

- Combettes PL: The convex feasibility problem in inage recovery. In
*Advances in Imaging and Electron Physics*.*Volume 95*. Edited by: Hawkes P. Academic Press, New York, NY, USA; 1996:155–270.Google Scholar - Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions.
*Topological Methods in Nonlinear Analysis*1993, 2(2):333–342.MathSciNetMATHGoogle Scholar - O'Hara JG, Pillay P, Xu H-K: Iterative approaches to convex feasibility problems in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(9):2022–2042. 10.1016/j.na.2005.07.036MathSciNetView ArticleMATHGoogle Scholar - Chang S-S, Yao J-C, Kim JK, Yang L: Iterative approximation to convex feasibility problems in Banach space.
*Fixed Point Theory and Applications*2007, 2007:-19.MathSciNetGoogle Scholar - Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space.
*Journal of Approximation Theory*2005, 134(2):257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.
*Journal of Computational and Applied Mathematics*2009, 225(1):20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleMATHGoogle Scholar - Zhou H, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi--asymptotically nonexpansive mappings. Journal of Applied Mathematics and Computing. In pressGoogle Scholar
- Wattanawitoon K, Kumam P: Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings.
*Nonlinear Analysis: Hybrid Systems*2009, 3(1):11–20. 10.1016/j.nahs.2008.10.002MathSciNetMATHGoogle Scholar - Li HY, Su YF: Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 72(2):847–855.MathSciNetGoogle Scholar - Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(1):45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleMATHGoogle Scholar - Kikkawa M, Takahashi W: Approximating fixed points of nonexpansive mappings by the block iterative method in Banach spaces.
*International Journal of Computational and Numerical Analysis and Applications*2004, 5(1):59–66.MathSciNetMATHGoogle Scholar - Aleyner A, Reich S: Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces.
*Journal of Mathematical Analysis and Applications*2008, 343(1):427–435. 10.1016/j.jmaa.2008.01.087MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Ungchittrakool K: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces.
*Fixed Point Theory and Applications*2008, 2008:-19.MathSciNetView ArticleGoogle Scholar - Plubtieng S, Ungchittrakool K: Strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces.
*Journal of Nonlinear and Convex Analysis*2007, 8(3):431–450.MathSciNetMATHGoogle Scholar - Sahu DR, Xu H-K, Yao J-C: Asymptotically strict pseudocontractive mappings in the intermediate sense.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(10):3502–3511. 10.1016/j.na.2008.07.007MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Petrusel A, Yao JC: A hybrid method for Lipschitz continuous monotone mappings and asymptotically strict pseudocontractive mappings in the intermediate sense. to appear in Journal of Nonlinear and Convex AnalysisGoogle Scholar
- Ceng LC, Petrusel A, Yao JC: Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense. to appear in Taiwanese Journal of MathematicsGoogle Scholar
- Ceng LC, Sahu DR, Yao JC: Implicit iterative algorithms for asymptotically nonexpansive mappings nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings. to appear in Journal of Computational and Applied MathematicsGoogle Scholar
- Chang S-S, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(9):3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleMATHGoogle Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications*.*Volume 62*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleGoogle Scholar - Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics*.*Volume 178*. Edited by: Kartosator AG. Dekker, New York, NY, USA; 1996:15–50.Google Scholar - Chang SS: On the generalized mixed equilibrium problem in Banach spaces.
*Journal of Applied Mathematics and Mechanics*2009, 30(9):1105–1112. 10.1007/s10483-009-0904-6View ArticleGoogle Scholar

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