Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

Abstract

We establish some strong convergence theorems for a common fixed point of a finite family of relatively nonexpansive mappings by using a new hybrid iterative method in mathematical programming and the generalized projection method in a Banach space. Our results improve and extend the corresponding results by many others.

1. Introduction

Let be a smooth Banach space and the dual of . The function is defined by for all , where is the normalized duality mapping from to . Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of (see [1]), if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive, if for all , and relatively nonexpansive (see [2]), if and for all and . The iterative methods for approximation of fixed points of nonexpansive mappings, relatively nonexpansive mappings, and other generational nonexpansive mappings have been studied by many researchers; see [3–13].

Actually, Mann [14] firstly introduced Mann iteration process in 1953, which is defined as follows: It is very useful to approximate a fixed point of a nonexpansive mapping. However, as we all know, it has only weak convergence in a Hilbert space (see [15]). As a matter of fact, the process (3) may fail to converge for a Lipschitz pseudocontractive mapping in a Hilbert space (see [16]). For example, Reich [17] proved that if is a uniformly convex Banach space with Fréchet differentiable norm and if is chosen such that , then the sequence defined by (3) converges weakly to a fixed point of .

Some have made attempts to modify the Mann iteration methods, so that strong convergence is guaranteed. Nakajo and Takahashi [18] proposed the following modification of the Mann iteration method for a single nonexpansive mapping in a Hilbert space : where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one, then defined by (5) converges strongly to .

The ideas to generate the process (5) from Hilbert spaces to Banach spaces have been made. By using the properties available on uniformly convex and uniformly smooth Banach spaces, Matsushita and Takahashi [10] presented their idea of the following method for a single relatively nonexpansive mapping in a Banach space : where is the duality mapping on and is the generalized projection from onto .

In 2007 and 2008, Plubing and Ungchittrakool [19, 20] improved and generalized the process (6) to the new general process of two relatively nonexpansive mappings in a Banach space: They proved that both iterations (7) and (8) converge strongly to a common fixed point of two relatively nonexpansive mappings and provided that the sequences satisfy some appropriate conditions.

Inspired and motivated by these facts, in this paper, we aim to improve and generalize the process (7) and (8) to the new general process of a finite family of relatively nonexpansive mappings in a Banach space. Let be a closed convex subset of a Banach space and let be relatively nonexpansive mappings such that . Define in the following way: where is the generalized projection from onto the intersection set ; are the sequences in with and for all . We prove, under certain appropriate assumptions on the sequences, that defined by (9) converges strongly to , where is the generalized projection from to .

Obviously, the process (9) reduces to become (7) when and become (8) when . So, our results extend and improve the corresponding ones announced by Nakajo and Takahashi [18], Plubtieng and Ungchittrakool [19, 20], Matsushita and Takahashi [10], and Martinez-Yanes and Xu [21].

2. Preliminaries

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section.

Throughout this paper, let be a real Banach space. Let denote the normalized duality mapping from into given by where denotes the dual space of and denotes the generalized duality pairing.

A Banach space is said to be strictly convex if for and . It is also said to be uniformly convex if for any two sequences , in such that and . Let be the unit sphere of , then the Banach space is said to be smooth provided that exists for each . It is also said to be uniformly smooth if the limit is attainted uniformly for each . It is well known that if is smooth, then the duality mapping is single valued. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . Some properties of the duality mapping have been given in [22]. A Banach space is said to have Kadec-Klee property if a sequence of satisfying that and , then . It is known that if is uniformly convex, then has the Kadec-Klee property; see [22] for more details.

Let be a smooth Banach space. The function is defined by for all . It is obvious from the definition of the function that (1),(2),(3),for all ; see [4, 7, 23] for more details.

Lemma 1 (see [4]). If is a strictly convex and smooth Banach space, then for , if and only if .

Lemma 2 (see [23]). Let be a uniformly convex and smooth Banach space and let be two sequences of . If and either or is bounded, then .

Let be a closed convex subset of . Suppose that is reflexive, strictly convex, and smooth. Then, for any , there exists a point such that . The mapping defined by is called the generalized projection (see [4, 7, 23]).

Lemma 3 (see [7]). Let be a closed convex subset of a smooth Banach space and . Then, if and only if

Lemma 4 (see [7]). Let be a reflexive, strictly convex, and smooth Banach space and let be a closed convex subset of and . Then, for all y C.

Lemma 5 (see [24]). Let be a uniformly convex Banach space and a closed ball of . Then, there exists a continuous strictly increasing convex function with such that for all and with .

Lemma 6 (see [19]). Let be a uniformly convex and uniformly smooth Banach space and let be a closed convex subset of . Then, for points and a real number , the set is closed and convex.

3. Main Results

In this section, we will prove the strong convergence theorem for a common fixed point of a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let us prove a proposition first.

Proposition 7. Let be a uniformly convex Banach space and a closed ball of . Then, there exists a continuous strictly increasing convex function with such that for all n and with .

Proof. If , using Lemma 5 and the convexity of , we have If , the last inequality above also holds obviously. By the same argument in the proof above, we obtain for all . Then, So,

Theorem 8. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of and relatively nonexpansive mappings such that . The sequence is given by (9) with the following restrictions:(a) for all ; (b) and ; (c) with , for all ; (d) and ; or () .Then, the sequence converges strongly to , where is the generalized projection from onto .

Proof. We split the proof into seven steps.
Step  1. Show that is well defined for every .
It is easy to know that , are closed convex sets and so is . What is more, is nonempty by our assumption. Therefore, is well defined for every .
Step  2. Show that and are closed and convex for all .
From the definition of , it is obvious is closed and convex for each . By Lemma 6, we also know that is closed and convex for each .
Step  3. Show that for all .
Let and let . Then, by the convexity of , we have and then, Thus, we have . Therefore, we obtain for all .
Next, we prove for all . We prove this by induction. For , we have . Assume that . Since is the projection of onto , by Lemma 3, we have for any . As by the induction assumption, holds, in particular, for all . This together with the definition of implies that . Hence, for all .
Step  4. Show that as .
In view of (19) and Lemma 4, we have , which means that, for any , Since and , we obtain for all . Consequently, exists and is bounded. By using Lemma 4, we have as . By using Lemma 2, we obtain as .
Step  5. Show that as .
From , we have as . By Lemma 2, we also have , and then, as . We observe that So, Since , , , and as , we have as . Using Lemma 2, we obtain as .
Step  6. Show that .
Since is bounded and , where , , we also obtain that are bounded, and hence, there exists such that . Therefore, Proposition 7 can be applied and we observe that where is a continuous strictly increasing convex function with . And as . From the properties of the mapping , we have for all . From the condition , we have immediately, as , ; from the condition (d), we can also have , as . In fact, since , it follows that for all . Next, we note by the convexity of and (9) that as . By Lemma 2, we have and as for all .
Step  7. Show that , as .
From the result of Step 6, we know that if is a subsequence of such that , then . Because is a uniformly convex and uniformly smooth Banach space and is bounded, so we can assume is a subsequence of such that and . For any , from and , we have On the other hand, from weakly lower semicontinuity of the norm, we have From the definition of , we obtain , and hence, . So, we have . Using the Kadec-klee property of , we obtain that converges strongly to . Since is an arbitrary weakly convergent sequence of , we can conclude that converges strongly to .