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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 615107, 17 pages
http://dx.doi.org/10.1155/2009/615107
Research Article

A New Iteration Process for Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings

1The Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy
2Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria

Received 10 July 2009; Accepted 26 October 2009

Academic Editor: Gelu Popescu

Copyright © 2009 C. E. Chidume and E. U. Ofoedu. 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

Let be a real Banach space, and a closed convex nonempty subset of . Let be total asymptotically nonexpansive mappings. A simple iterative sequence is constructed in and necessary and sufficient conditions for this sequence to converge to a common fixed point of are given. Furthermore, in the case that is a uniformly convex real Banach space, strong convergence of the sequence to a common fixed point of the family is proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem.

1. Introduction

Let be a nonempty subset of a normed real linear space A mapping is said to be nonexpansive if for all

The mapping is called asymptotically nonexpansive if there exists a sequence with such that for all

The mapping is called uniformly -Lipschitzian if there exists a constant such that for all

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] 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.

A mapping is said to be asymptotically nonexpansive in the intermediate sense (see, e.g., [2]) if it is continuous and the following inequality holds:

Observe that if we define

then as and (1.3) reduces to

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] that if is a nonempty closed convex bounded subset of a uniformly convex real Banach space and is a self-mapping of which is asymptotically nonexpansive 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., [4]).

Sahu [5], introduced the class of nearly Lipschitzian mappings. Let be a nonempty subset of a normed space and let be a sequence in such that A mapping is called nearly Lipschitzian with respect to if for each there exists such that

Define

Observe that for any sequence satisfying (1.6), and that

is called the nearly Lipschitz constant. A nearly Lipschitzian mapping is said to be

(i)nearly contraction if for all (ii)nearly nonexpansive if for all (iii)nearly asymptotically nonexpansive if for all and (iv)nearly uniform -Lipschitzian if for all (v)nearly uniform -contraction if for all

Example 1.1. Let Define by It is obvious that is not continuous, and thus, not Lipschitz. However, is nearly nonexpansive. In fact, for a real sequence with and as we have This is because

Remark 1.2. If is a bounded domain of an asymptotically nonexpansive mapping then is nearly nonexpansive. In fact, for all and we have Furthermore, we easily observe that every nearly nonexpansive mapping is nearly asymptotically nonexpansive with

Remark 1.3. If is a bounded domain of a nearly asymptotically nonexpansive mapping , then is asymptotically nonexpansive in the intermediate sense. To see this, let be a nearly asymptotically nonexpansive mapping. Then, which implies that Hence,

We observe from Remarks 1.2 and 1.3 that the classes of nearly nonexpansive mappings and nearly asymptotically nonexpansive mappings are intermediate classes between the class of asymptotically nonexpansive mappings and that of asymptotically nonexpansive in the intermediate sense mappings.

The main tool for approximation of fixed points of generalizations of nonexpansive mappings remains iterative technique. Several authors have studied approximation of fixed points of generalizations of nonexpansive mappings using Mann and Ishikawa iterative methods (see, e.g., [619]).

Bose [20] proved that if is a nonempty closed convex bounded subset of a uniformly convex real Banach space satisfying Opial's condition [21] (i.e., for all sequences in such that converges weakly to some the inequality holds for all in ) and is an asymptotically nonexpansive mapping, then the sequence converges weakly to a fixed point of provided that is asymptotically regular at that is, the limit

holds. Passty [13] and also Xu and Noor [22] showed that the requirement of Opial's condition can be replaced by the Fréchet differentiability of the space norm. Furthermore, Tan and Xu [23, 24] established that the asymptotic regularity of at a point can be weakened to the so-called weakly asymptotic regularity of at defined as follows: is weakly asymptotic regular at if

holds, where denotes the weak limit.

In [17, 18], Schu introduced a modified Mann iteration scheme for approximation of fixed points of asymptotically nonexpansive self-mappings defined on a nonempty closed convex and bounded subset of a uniformly convex real Banach space He proved that the iterative sequence generated by

converges weakly to some fixed point of if Opial's condition holds, for all is a real sequence such that for some positive constants and Neither condition (1.15) nor condition (1.16) is required with Schu's scheme. Schu's result, however, does not apply, for instance, to spaces with because none of these spaces satisfies Opial's condition.

Rhoades [15] obtained a strong convergence theorem for asymptotically nonexpansive mappings in uniformly convex real Banach spaces using the modified Ishikawa-type iteration method. Osilike and Aniagbosor proved in [12] that the results of [15, 17, 18] remain true without the boundedness requirement imposed on provided that Tan and Xu [25] extended the theorem of Schu [18] to uniformly convex Banach space with a Fréchet differentiable norm without assuming that the space satisfies Opial's condition. Thus, their result applies to spaces with

Chang et al. [26] established weak convergence theorems for asymptotically nonexpansive mappings in Banach spaces without assuming any of the following conditions: (i) satisfies the Opial's condition; (ii) T is asymptotically regular or weakly asymptotically regular; (iii) is bounded. Their results improve and generalize the corresponding results of Bose [20], Górnicki [27], Passty [13], Schu [18],Tan and Xu [2325], Xu and Noor [22], and many others.

G. E. Kim and T. H. Kim [4] studied the strong convergence of the Mann and Ishikawa-type iteration methods with errors for mappings which are asymptotically nonexpansive in the intermediate sense in real Banach spaces.

In all the above papers, the mapping remains a self-mapping of nonempty closed convex subset of a uniformly convex real Banach space If, however, the domain of is a proper subset of then the Mann and Ishikawa-type iterative processes and Schu's modifications of type (1.17) may fail to be well defined.

Chidume et al. [28] proved convergence theorems for asymptotically nonexpansive nonself-mappings in Banach spaces and extended the corresponding results of [12, 15, 17, 18, 26].

Alber et al. [29] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied methods of approximation of fixed points of mappings belonging to this class.

Definition 1.4. A mapping is said to be total asymptotically nonexpansive if there exist nonnegative real sequences and with as and strictly increasing continuous function with such that for all

Remark 1.5. If then (1.18) reduces to In addition, if for all then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. If and for all we obtain from (1.18) the class of mappings that includes the class of nonexpansive mappings. If and where for all then (1.18) reduces to (1.5) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.

The idea of Definition 1.4 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.

Another class of nonlinear mappings introduced as a further generalization of nonexpansive mappings with nonempty fixed point sets is the class of asymptotically quasi-nonexpansive mappings which properly contains the class of asymptotically nonexpansive operators with nonempty fixed point sets (see, e.g., [8, 16, 3033]).

A mapping is said to be quasi-nonexpansive if and is called asymptotically quasi-nonexpansive if and there exists a sequence with such that for all and

is said to be asymptotically quasi-nonexpansive in intermediate sense if it is continuous and

Remark 1.6. Observe that if we define then as and (1.22) reduces to

Existence theorems for common fixed points of certain families of nonlinear mappings have been established by various authors (see, e.g., [2, 3437]).

Within the past 30 years or so, research on iterative approximation of common fixed points of generalizations of nonlinear nonexpansive mappings surged. Considerable research efforts have been devoted to developing iterative methods for approximating common fixed points (when they exist) of finite families of this class of mappings (see, e.g., [33, 3846]).

In [16], Shahzad and Udomene established necessary and sufficient conditions for convergence of Ishikawa-type iteration sequences involving two asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings in arbitrary real Banach spaces. They also established a sufficient condition for the convergence of the Ishikawa-type iteration sequences involving two uniformly continuous asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings in real uniformly convex Banach spaces.

Recently, Chidume and Ofoedu [47] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces. More precisely, they proved the following theorems.

Theorem CO1
Let be a real Banach space, let be a nonempty closed convex subset of , and be total asymptotically nonexpansive mappings with sequences such that Let be given by Suppose and suppose that there exist such that for all Then the sequence is bounded and exists, . Moreover, the sequence converges strongly to a common fixed point of if and only if where

Theorem CO2
Let be a uniformly convex real Banach space, let be a nonempty closed convex subset of and be uniformly continuous total asymptotically nonexpansive mappings with sequences such that and Let for some From arbitrary define the sequence by (1.25). Suppose that there exist such that whenever and that one of is compact, then converges strongly to some

It is our purpose in this paper to construct a new iterative sequence much simpler than (1.25) for approximation of common fixed points of finite families of total asymptotically nonexpansive mappings and give necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. A sufficient condition for convergence of the iteration process to a common fixed point of mappings under our setting is also established in uniformly convex real Banach spaces. Our theorems unify, extend and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced for the approxima tion of common fixed points of finite families of several classes of nonlinear mappings. Our iteration process is also of independent interest.

2. Preliminary

In the sequel, we shall need the following lemmas.

Lemma 2.1. Let , , and be sequences of nonnegative real numbers such that Suppose that and Then is bounded and exists. Moreover, if in addition, then

Lemma 2.2 (Zeidler [48, pages 484-485]). Let be a uniformly convex real Banach space and . Suppose that and are two sequences of such that hold for some , then .

3. Main Results

Let be a nonempty closed convex subset of a real normed space Let be total asymptotically nonexpansive mappings. We define the iterative sequence by

where are sequences in such that .

We now state and prove our main theorems.

Theorem 3.1. Let be a real Banach space, let be a nonempty closed convex subset of and let be total asymptotically nonexpansive mappings with sequences such that Let be given by (3.1). Suppose and suppose that there exist such that for all Then the sequence is bounded and exists, .

Proof. Let Then we have from (3.1) that Since is an increasing function, it follows that whenever and (by hypothesis) if In either case, we have for some Thus, for some constant Hence, where and Observe that and So, from (3.5) and by Lemma 2.1 we obtain that the sequence is bounded and that exists. This completes the proof.

3.1. Necessary and Sufficient Conditions for Convergence in Real Banach Spaces

Theorem 3.2. Let be a real Banach space, let be a nonempty closed convex subset of and let be continuous total asymptotically nonexpansive mappings with sequences such that Let be given by (3.1). Suppose and suppose that there exist such that for all Then the sequence converges strongly to a common fixed point of if and only if   where

Proof. It suffices to show that implies that converges to a common fixed point of Necessity
Since (3.5) holds for all we obtain from it that Lemma 2.1 then implies that exists. But, Hence,
Sufficiency
Next, we first show that is a Cauchy sequence in For all integer we obtain from inequality (3.5) that so that for all integers and all We therefore have that for some constant Taking infimum over in (3.9) gives Now, since and given there exists an integer such that for all and So for all integers we obtain from (3.10) that
Hence, is a Cauchy sequence in and since is complete there exists such that as We now show that is a common fixed point of that is, we show that Suppose for contradiction that (where denotes the complement of ). Since is a closed subset of (recall each is continuous), we have that But, for all we have This implies so that as we obtain which contradicts Thus, is a common fixed point of This completes the proof.

Remark 3.3. If are asymptotically nonexpansive mappings, then for all and so that the assumption that there exist such that for all in the above theorems is no longer needed.

Thus, we have the following corollary.

Corollary 3.4. Let be a real Banach space, let be a nonempty closed convex subset of and let be continuous asymptotically nonexpansive mappings with sequences such that Let be given by (3.1). Suppose Then the sequence is bounded and exists, Moreover, converges strongly to a common fixed point of if and only if

3.2. Convergence Theorem in Real Uniformly Convex Banach Spaces

Theorem 3.5. Let be a uniformly convex real Banach space, be a nonempty closed convex subset of and be uniformly continuous total asymptotically nonexpansive mappings with sequences such that and From arbitrary define the sequence by (3.1). Suppose that there exist such that whenever Then .

Proof. Let Then, by Theorem 3.1, exists. Let If then by continuity of we are done. Now suppose We show that We observe that so that taking on both sides of this inequality, we obtain Let be such that as and define Then, for some Thus, Furthermore, This implies that But, So, Hence, by Lemma 2.2, we obtain This completes the proof.

Theorem 3.6. Let be a uniformly convex real Banach space, let be a nonempty closed convex subset of and let be uniformly continuous total asymptotically nonexpansive mappings with sequences such that and From arbitrary define the sequence by (1.24). Suppose that there exist such that whenever and that one of is compact, then converges strongly some

Proof. We obtain from Theorem 3.5 that Using the recursion formula (3.1), we observe that It then follows from (3.24) and (3.25) that Without loss of generality, let be compact. Since is continuous and compact, it is completely continuous. Thus, there exists a subsequence of such that as for some Thus as and from (3.24), we have that Also from (3.24) as Thus, as Now, since from (3.26), as it follows that as Next, we show that Observe that Taking limit as and using the fact that are uniformly continuous we have that and so But by Theorem 3.1, exists, Hence, converges strongly to This completes the proof.

In view of Remark , the following corollary is now obvious.

Corollary 3.7. Let be a uniformly convex real Banach space, let be a nonempty closed convex subset of and let be asymptotically nonexpansive mappings with sequences such that ; and that one of is compact. From arbitrary define the sequence by (3.1). Then converges strongly to to some

Remark 3.8. Observe that the theorems of this paper remain true for mappings satisfying (1.5) provided that In this case, the requirement that there exist such that for all is not needed.

Remark 3.9. A prototype for satisfying the conditions of our theorems is Prototypes for the sequences in this paper are the following:

Remark 3.10. Addition of bounded (or the so called mean) error terms to the iteration process studied in this paper leads to no further generalization.

Definition 3.11. A mapping is said to be total asymptotically quasi-nonexpansive if and there exist nonnegative real sequences and with as and strictly increasing continuous function with such that for all

Remark 3.12. If then (3.29) reduces to In addition, if for all then total asymptotically quasi-nonexpansive mappings coincide with asymptotically quasi-nonexpansive mappings studied by various authors. If and for all we obtain from (3.30) the class of quasi-nonexpansive mappings. Observe that the class of total asymptotically nonexpansive mappings with nonempty fixed point sets belongs to the class of total asymptotically quasi-nonexpansive mappings. Moreover, if and then (3.29) reduces to (1.24).

It is trivial to observe that all the theorems of this paper carry over to the class of total asymptotically quasi-nonexpansive mappings with little or no modifications.

A subset of a real normed linear space is said to be a retract of if there exists a continuous map such that for all It is well known (see, e.g., [28]) that every closed convex nonempty subset of a uniformly convex Banach space is a retract. A map is said to be a retraction if It follows that if a map ia a retraction, then for all in the range of The mapping is called a sunny nonexpansive retraction if for all and

Definition 3.13. Let be a nonempty closed and convex subset of Let be the nonexpansive retraction of onto A nonself map is said to be total asymptotically nonexpansive if there exist sequences in with as and a strictly increasing continuous function with such that for all Let be total asymptotically nonexpansive nonself maps; assuming existence of common fixed points of these operators, our theorems and method of proof easily carry over to this class of mappings using the iterative sequence defined by instead of (3.1) provided that the well definedness of as a sunny nonexpansive retraction is guaranteed.

Remark 3.14. It is clear that the recursion formula (3.1) introduced and studied in this paper is much simpler than the recursion formulas (1.25) studied earlier for this problem.

Remark 3.15. Our theorems unify, extend, and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced (see, e.g., [8, 16, 22, 28, 30, 39, 40, 44, 47, 4958]) for the approximation of common fixed points of finite families of several classes of nonlinear mappings.

Acknowledgment

This author's research was supported by the Japanese Mori Fellowship of UNESCO at The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

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