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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 281383, 21 pages
http://dx.doi.org/10.1155/2012/281383
Research Article

On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

Department of Computational & Theoretical Sciences, Faculty of Sciences, International Islamic University Malaysia, P.O. Box 141, 25710, Kuantan, Malaysia

Received 5 January 2012; Revised 10 March 2012; Accepted 12 March 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Farrukh Mukhamedov and Mansoor Saburov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically I-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

1. Introduction

Let be a nonempty subset of a real normed linear space , and let be a mapping. Denote by the set of fixed points of , that is, . Throughout this paper, we always assume that is a real Banach space and . Now let us recall some known definitions.

Definition 1.1. A mapping is said to be(i)nonexpansive if for all ,(ii)asymptotically nonexpansive if there exists a sequence with such that for all and ,(iii) asymptotically nonexpansive in the intermediate sense, if it is continuous and the following inequality holds:

Remark 1.2. Observe that if we define then as , and (1.1) reduces to

In [1, 2], Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the study of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicians (see for more details [3, 4]).

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed-convex bounded subset of a uniformly convex real Banach space and is an asymptotically nonexpansive self-mapping of , then has a fixed point.

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [6]. It is known [7] that if is a nonempty closed-convex bounded subset of a uniformly convex Banach space and is an asymptotically nonexpansive mapping in the intermediate sense, then has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings (see, e.g., [8]).

The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping, and asymptotically nonexpansive mapping in the intermediate sense were studied extensively in [520].

There are many different types of concepts which generalize a notion of nonexpansive mapping. One of such concepts is a total asymptotically nonexpansive mapping [21], and second one is an asymptotically -nonexpansive mapping [22]. Let us recall some notions.

Definition 1.3. Let be a nonempty closed subset of a real normed linear space is called a total asymptotically nonexpansive mapping if there exist nonnegative real sequence and with as and strictly increasing continuous function with such that for all ,

Remark 1.4. If , then (1.4) reduces to In addition, if for all , then total asymptotical nonexpansive mappings coincide with asymptotically nonexpansive mappings. If and for all , we obtain from (1.5) the class of mappings that includes the class of nonexpansive mappings. If and , where for all , then (1.5) reduces to (1.3) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.

The idea of the definition of a total asymptotically nonexpansive mappings is to unify various definitions of classes of mappings associated with the class of asymptotically nonexpansive mappings and to prove a general convergence theorems applicable to all these classes of nonlinear mappings.

Alber et al. [21] studied methods of approximation of fixed points of total asymptotically nonexpansive mappings. C. E. Chidume and E. U. Ofoedu [23] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces. Recently, C. E. Chidume and E. U. Ofoedu [24] constructed a new iterative sequence much simpler than other types of approximation of common fixed points of finite families of total asymptotically nonexpansive mappings.

On the other hand, in [22] an asymptotically -nonexpansive mapping was introduced.

Definition 1.5. Let , be two mappings of a nonempty subset of a real normed linear space , then is said to be(i)-nonexpansive if for all ,(ii)asymptotically -nonexpansive, if there exists a sequence with such that for all and .

Best approximation properties of -nonexpansive mappings were investigated in [22, 25]. In [26], strong convergence of Mann iterations of -nonexpansive mapping has been proved. In [27], the weak convergence of three-step Noor iterative scheme for an -nonexpansive mapping in a Banach space has been established. In [28], the weakly convergence theorem for asymptotically -nonexpansive mapping defined in Hilbert space was proved. Recently, in [2931], the weak and strong convergence of explicit and implicit iteration process to a common fixed point of a finite family of asymptotically -nonexpansive mappings have been studied.

In this paper, we introduce a new type of concept of a generalization of nonexpansive mapping’s nation, which is a combination of Definitions 1.3 and 1.5.

Definition 1.6. Let , be two mappings of a nonempty subset of a real normed linear space , then is said to be a total asymptotically -nonexpansive mapping if there exist nonnegative real sequences and with as and the strictly increasing continuous function with such that for all ,

Now let us provide an example of a total asymptotically -nonexpansive mapping, which is not asymptotically nonexpansive mapping.

Example 1.7. Let us consider the space , and let . Define a nonlinear operator by Let , then from one gets .
One can find that Hence, From , we have So, it follows from (1.10) and (1.11) that
Now consider a new Banach space with a norm , where and define a new mapping by Let , then it is clear that . One can see that . Therefore, using (1.14), we obtain We let and . It is clear that and is strictly increasing, and moreover, (1.14) implies that is is a totally asymptotically -nonexpansive mapping. Here, is the identity mapping of .
Now we are going to show that is not asymptotically nonexpansive. Namely, we will establish that for any sequence of positive numbers with and any , one can find such that In fact, choose , as follows: where
From (1.10), one finds that The last equalities with (1.18) imply that This yields the required assertion. Note that has infinitely many fixed points in , that is, .

Example 1.8. Let us consider the Banach space defined as before, and let be a mapping of a segment to itself, that is, with and where . Note that such kind of functions do exist. One can take (see for more details [8]) and
Define a new mapping by here is defined as above (see (1.7)). Using the same argument as the above Example 1.7, we can establish that Moreover, such a mapping is not asymptotically nonexpansive. Note that the mapping with the function has a unique fixed point in .

Remark 1.9. To the best our knowledge, we should stress that the constructed examples are currently only unique examples of totaly asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Before, no such examples were known in the literature.

The aim of the present paper is unification of all known iterative methods by introducing a new iterative scheme for approximation of common fixed points of finite families of total asymptotically -nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced in [24] and allows us to construct more simpler methods than [23, 24].

Namely, let be a nonempty closed-convex subset of a real Banach space and : be a finite family of total asymptotically -nonexpansive mappings, that is, and is a finite family of total asymptotically nonexpansive mappings, that is, here are the strictly increasing continuous functions with for all , and are nonnegative real sequences with as for all . Then for given sequences in , where , we will consider the following explicit iterative process: such that and .

C. E. Chidume and E. U. Ofoedu [24] have considered only a particular case of the explicit iterative process (1.27), in which is to be taken as the identity mappings. One of the main results of ([24], see Theorem 3.5, page 11) was correct, while the provided proof of that result was wrong. Since, in their proof, they used Lemma 2.3, but which actually is not applicable in that situation, the sequence tends to 0. As a counterexample, we can consider the following one: let , and let the sequences , , and be defined as follows: It is then clear that However,

In this paper, we shall provide a correct proof of Theorem 3.5 page 11 in [24]. As we already mentioned in Lemma 2.3 is not applicable the main result of [24]. Therefore, we first will generalize Lemma 2.3 to the case of finite number of sequences. Such a generalization gives us a possibility to prove the mentioned result. On other hand, the provided generalization presents an independent interest as well. Moreover, we extend and unify the main result of [24] for a finite family of total asymptotically-nonexpansive mappings . Namely, we shall prove the strong convergence of the explicit iterative process (1.27) to a common fixed point of the finite family of total asymptotically-nonexpansive mappings and the finite family of total asymptotically nonexpansive mappings . Here, we stress that Lemmas 3.1 and 3.2 play a crucial role. All presented results here extend, generalize, unify, and improve the corresponding main results of [21, 24, 2933].

2. Preliminaries

Throughout this paper, we always assume that is a real Banach space. The following lemmas play an important role in proving our main results.

Lemma 2.1 (see [16]). Let , , and be three sequences of nonnegative real numbers with , . If the following condition is satisfied:(i), then the limit exists.

Lemma 2.2 (see [34]). Let be a uniformly convex Banach space and . Suppose that , are two sequences in such that hold some , then .

Lemma 2.3 (see [14]). Let be a uniformly convex Banach space, and let , be two constants with . Suppose that is a sequence in and , are two sequences in such that hold some , then .

3. Main Results

In this section, we shall prove our main results. To formulate ones, we need some auxiliary results.

First we are going to generalize Lemmas 2.2 and 2.3 for number of sequences from the uniformly convex Banach space , where .

Lemma 3.1. Let be a uniformly convex Banach space and any constants with . Suppose that are sequences in such that hold some , then and for any .

Proof. Let us first prove for any . Indeed, it follows from (3.1) that We then get that , which means .
Now we prove the statement by means of mathematical induction with respect to . For , the statement immediately follows from Lemma 2.2. Assume that the statement is true, for . Let us prove for . To do this, denote Since , we get . On the other hand, one has We then obtain which means . In this case, according to the assumption of induction with the sequence , we can conclude that , if .
Since due to Lemma 2.2, one gets If , then the following inequality implies that . This completes the proof.

Lemma 3.2. Let be a uniformly convex Banach space, and let be two constants with . Suppose that , are any sequences with for all . Suppose that , are sequences in such that hold for some , then and for any .

Proof. Analogously as in the proof of Lemma 3.1, it is easy to show that . Therefore, let us prove the statement for any . Suppose to the contrary, that there exist two numbers such that then there exists a subsequence of such that .
Let us consider the subsequences of , here . Since , there exists a subsequence of such that for all . Since , for all , one gets , and , for all . We know that It then follows that . On the other hand, we have Therefore, . Consequently, Lemma 3.1 implies that . However, it contradicts to This completes the proof.

Proposition 3.3. Let be a real Banach space, and let be a nonempty closed-convex subset of . Let : be a finite family of total asymptotically -nonexpansive mappings with sequences , , where , and let : be a finite family of total asymptotically nonexpansive mappings with sequences , , where . Suppose that there exist , such that , for all and for all , where , then the following holds for any and for any :

Proof. Since : are the strictly increasing continuous functions, where , it follows that and whenever and , where . By the hypothesis of Proposition 3.3, for all and , we then get Since , are total asymptotically -nonexpansive and total asymptotically nonexpansive mappings, respectively, from (3.14) and (3.15), one gets

Lemma 3.4. Let be a uniformly convex real Banach space, and let be a nonempty closed-convex subset of . Let : be a finite family of total asymptotically -nonexpansive mappings with sequences , , where , and let : be a finite family of total asymptotically nonexpansive mappings with sequences , , where , such that . Suppose that , , , for all , and there exist , such that , for all and for all , where . If is the explicit iterative sequence defined by (1.27), then for each , the limit exists.

Proof. Since , for any given , it follows from (1.27) and (3.13) that Again from (1.27) and (3.12), we derive that
Then from (3.17) and (3.18), one finds Here Denoting in (3.19), one gets Since and , it follows from Lemma 2.1 the existence of the limit . This means the limit exists, where is a constant. This completes the proof.

Now we prove the following result.

Theorem 3.5. Let be a uniformly convex real Banach space, and let be a nonempty closed-convex subset of . Let be a finite family of total asymptotically -nonexpansive continuous mappings with sequences , where , and let be a finite family of total asymptotically nonexpansive continuous mappings with sequences , , where , such that . Suppose that , , , for all , and there exist such that , for all and for all , where , then the explicit iterative sequence defined by (1.27) converges strongly to a common fixed point in if and only if

Proof. The necessity of condition (3.23) is obvious. Let us prove the sufficiency part of the theorem.
Since are continuous mappings, the sets and are closed. Hence, is a nonempty closed set.
For any given , we have (see (3.19)) Hence, one finds From (3.25) due to Lemma 2.1, we obtain the existence of the limit . By condition (3.23), one gets
Let us prove that the sequence converges strongly to a common fixed point in . We first show that is Cauchy sequence in . In fact, due to for all , and from (3.24), we obtain Thus, for any positive integers , from (3.27) with ,, we find Therefore, we get for all , where . Taking infimum over in (3.29) gives
Since and , given , there exists an integer such that for all , we have and . Consequently, for all integers and and from (3.30), we derive which means that is Cauchy sequence in , and since is complete, there exists such that the sequence converges strongly to .
Now we show that is a common fixed point in . Suppose for contradiction that . Since is closed subset of , we have that . However, for all , we have This implies that so that as we obtain which contradicts . Hence, is a common fixed point in . This proves the required assertion.

To formulate and prove the main result, we need once more an auxiliary result.

Proposition 3.6. Let be a uniformly convex real Banach space, and let be a nonempty closed-convex subset of . Let be a finite family of total asymptotically -nonexpansive continuous mappings with sequences , where , and let : be a finite family of total asymptotically nonexpansive continuous mappings with sequences , , where , such that . Suppose that , ,, for all , and are sequences with and , for all , here , , and there exist , such that , for all and for all , where . then the explicit iterative sequence defined by (1.27) satisfies the following: for all .

Proof. According to Lemma 3.4 for any , we have . It follows from (1.27) that as . By means of , , for all , from (3.18), one yields that and from (3.13), (3.37), we have for all . Now using with (3.38) and applying Lemma 3.2 to (3.36), one finds for all . Now from (1.27) and (3.40), we infer that On the other hand, from (3.13), we have which implies The last inequality with (3.22), (3.40) yields Combining (3.44) with (3.37), we get Again from (1.27), we can see that From (3.12) and (3.22), one finds for all . Now applying Lemma 3.2 to (3.46), we obtain for all . We then have Consider for all . Then from (3.40) and (3.49), we get for all .

Now we are ready to formulate a main result concerning strong convergence of the sequence .

Theorem 3.7. Let be a uniformly convex real Banach space, and let be a nonempty closed-convex subset of . Let be a finite family of total asymptotically -nonexpansive continuous mappings with sequences , where , and let be a finite family of total asymptotically nonexpansive continuous mappings with sequences , , where , such that . Suppose that , , , for all , and are sequences with and , for all , here , , and there exist , such that , for all and for all , where . If at least one mapping of the mappings and is compact, then the explicitly iterative sequence defined by (1.27) converges strongly to a common fixed point of and .

Proof. Without any loss of generality, we may assume that is compact. This means that there exists a subsequence of such that converges strongly to . Then from (3.34), we have that converges strongly to . Also from (3.34), we obtain that converges strongly to , for all . Since are continuous mappings, so converges strongly to , for all . On the other hand, from (3.35) and continuousness of , we obtain that converges strongly to , and converges strongly to , for all . Due to (3.41), converges to 0, as . Then, converges strongly to and moreover, (3.13) and (3.12) imply that and converge to 0, as , for all . From (3.34), (3.35), it yields that and converge to 0 as , for all . Observe that for all . Taking limit as , we have that and , for all , which means . However, by Lemma 3.4, the limit exists, then which means