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

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.

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 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 [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 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 [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

(21)

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   is said to be smooth if the limit

(22)

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

• (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

(23)

It is obvious from the definition of that

(24)

Following Alber [21], the generalized projection is defined by

(25)

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

(27)

A mapping is said to beclosed if for any sequence with and , then .

Definition 2.5.

A mapping is said to bequasi--nonexpansive if and

(28)

A mapping is is said to be quasi- ϕ -asymptotically nonexpansive [7], if and there exists a real sequence with such that

(29)

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 2.9 (see [13, 22]).

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 ,

(210)

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

(211)

Proof.

Since and for all with , we have

(212)

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

(213)

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.

Letting be a sequence in with (as ), we prove that In fact, from the definition of we have

(214)

Therefore we have

(215)

that is, .

Next we prove that is convex. For any , putting we prove that Indeed, in view of the definition of we have

(216)

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

(217)

Again since is reflexive, we have . Therefore there exists such that . By virtue of the weakly lower semicontinuity of norm we have

(218)

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 quasiasymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach spaces with the Kadec-Klee 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

(31)

A mapping is said to be uniformly-Lipschitz continuous, if there exists a constant such that

(32)

Theorem 3.2.

Let be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and a nonempty closed convex subsets of . Let be an infinite family of closed and uniformly quasi--asymptotically 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

(33)

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

(34)

therefore, we have

(35)

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

(36)

This implies that is bounded. By virtue of (2.4), is bounded. Denote

(37)

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

(38)

Hence and so for all . By the way, from the definition of , (2.4), and (3.7), it is easy to see that

(39)

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

(310)

Since

(311)

we have

(312)

This implies that that is, In view of the Kadec-Klee property of , we obtain that

(313)

Now we prove that .In fact, by the construction of we have that and Therefore by Lemma 2.2(a) we have

(314)

In view of and note the construction of we obtain that

(315)

From (2.4) it yields Since we have

(316)

Hence we have

(317)

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

(318)

Taking on the both sides of equality above and in view of the weak lower semicontinuity of norm it yields that

(319)

that is, This implies that and so . It follows from (3.17) and the Kadec-Klee property of that (as ). Note that is hemi-continuous, it yields that It follows from (3.16) and the Kadec-Klee property of that

(320)

From (3.13) and (3.20) we have that

(321)

Since is uniformly continuous on any bounded subset of , we have

(322)

For any and any , it follows from (3.8), (3.13), and (3.20) that

(323)

In view of condition (b) , we see that

(324)

It follows from the property of that

(325)

Since and is uniformly continuous, it yieads Hence from (3.25) we have

(326)

Since is hemi-continuous, it follows that

(327)

On the other hand, for each we have

(328)

This together with (3.27) shows that

(329)

Furthermore, by the assumption that for each , is uniformly -Lipschitz continuous, hence we have

(330)

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

(331)

This implies that

(332)

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 Kadec-Klee property , a nonempty closed convex subset of . Let be an infinite family of closed and quasi--nonexpansive mappings. Suppose that is a nonempty subset in . Let be the sequence generated by

(333)

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 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.

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 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.