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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 202095, 8 pages
http://dx.doi.org/10.1155/2013/202095
Research Article

Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

1Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
4Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 4 February 2013; Accepted 4 April 2013

Academic Editor: Qamrul Hasan Ansari

Copyright © 2013 D. R. Sahu et al. 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 reflexive Banach space with a weakly continuous duality mapping . Let be a nonempty weakly closed star-shaped (with respect to ) subset of . Let  =  be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of , which is uniformly continuous at zero. We will show that the implicit iteration scheme: , for all , converges strongly to a common fixed point of the semigroup for some suitably chosen parameters and . Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

1. Introduction

Let be a nonempty subset of a (real) Banach space and a mapping. The fixed point set of is defined by . We say that is nonexpansive if for all in and asymptotically nonexpansive if there exists a sequence in with such that for all in and in .

Set . We call a one-parameter family of mappings from into a strongly continuous semigroup of Lipschitzian mappings if(1) for each , there exists a bounded function such that (2)  for all in ,(3)  for all in , (4) for each in , the mapping from into is continuous.

Note that . If for all in (1), then is called a strongly continuous semigroup of uniformly -Lipschitzian mappings. If for all in (1), then is called a strongly continuous semigroup of nonexpansive mappings. If for all and in (1), then is called a strongly continuous semigroup of asymptotically nonexpansive mappings. Moreover, is said to be (right) uniformly continuous if it also holds:(5) For any bounded subset of , we have

We denote by the set of common fixed points of ; that is, .

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point. Several authors have studied the problem of the existence of fixed points of asymptotically nonexpansive mappings in Banach spaces having rich geometric structure; see [2] and the references therein.

Consider a nonempty closed convex subset of a (real) Banach space . A classical method to study nonexpansive mappings is to approximate them by contractions. More precisely, for a fixed element in , define for each in a contraction by Let be the fixed point of ; that is, Browder [3] (Reich [4], resp.) proves that as , the point converges strongly to a fixed point of if is a Hilbert space (uniformly smooth Banach space, resp.). Many authors (see, e.g., [513]) have studied strong convergence of approximates for asymptotically nonexpansive self-mappings in Banach spaces under the additional assumption as . This additional assumption can be removed when is uniformly asymptotically regular.

Suzuki [14] initiated the following implicit iteration process for a semigroup of nonexpansive mappings in a Hilbert space: Xu [15] extended Suzuki’s result to uniformly convex Banach spaces with weakly sequentially continuous duality mappings. Recently, Zegeye and Shahzad [16] extended results of Xu [15] and established the following strong convergence theorem.

Theorem ZS. Let be a nonempty closed convex bounded subset of a real uniformly convex Banach space with a weakly continuous duality mapping with gauge . Let be a strongly continuous asymptotically nonexpansive semigroups with net . Assume that is a sunny nonexpansive retract of with as the sunny nonexpansive retraction. Assume that and such that for all , , and is bounded. Let in be fixed. (1) There exists a sequence in such that which converges strongly to an element of . (2) Every sequence defined iteratively with any in , and converges strongly to an element of , provided that for some .

Problem 1. Is it possible to drop the uniform convexity assumption in Theorem ZS?

Motivated by Schu [13], the purpose of this paper is to further analyze strong convergence of (6) and (7) for strongly continuous semigroups of asymptotically nonexpansive mappings defined on a set which is not necessarily convex. It is important and actually quite surprising that we are able to do so for the class of Banach spaces which are not necessarily uniformly convex. It should be noted that, in this generality, Theorem ZS does not apply. Our results are definitive, settle Problem 1, and also improve results of Suzuki [14], Xu [15], and Zegeye and Shahzad [16].

2. Preliminaries

Let be a nonempty subset of a (real) Banach space with dual space . We call a mapping   weakly contractive if where is a continuous and nondecreasing function such that ,   for , and .

By a gauge we mean a continuous strictly increasing function defined on such that and . Associated with a gauge , the (generally multivalued) duality mapping is defined by Clearly, the (normalized) duality mapping corresponds to the gauge . In general,

Recall that is said to have a weakly (resp, sequentially) continuous duality mapping if there exists a gauge such that the duality mapping is single valued and (resp, sequentially) continuous from with the weak topology to with the weak* topology. Every space has a weakly continuous duality mapping with the gauge (for more details see [17, 18]). We know that if admits a weakly sequentially continuous duality mapping, then satisfies Opial’s condition; that is, if is a sequence weakly convergent to in , then there holds the inequality

Browder [19] initiated the study of certain classes of nonlinear operators by means of a duality mapping . Define Then, it is known that is the subdifferential of the convex function at ; that is, where denotes the subdifferential in the sense of convex analysis. We need the subdifferential inequality For a smooth , we have or considering the normalized duality mapping , we have

Assume that a sequence in converges weakly to a point in . Then the following identity holds;

Remark 2. For any with , we have

We need the following demiclosedness principle for asymptotically nonexpansive mappings in a Banach space.

Lemma 3 (see [17, Corollary  5.6.4], [10]). Let be a Banach space with a weakly continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and an asymptotically nonexpansive mapping. Then, is demiclosed at zero; that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .

Let be a convex subset of a Banach space and a nonempty subset of . Then, a continuous mapping from onto is called a retraction if for all in ; that is, . A retraction is said to be sunny if for each in and with in . If the sunny retraction is also nonexpansive, then is said to be a sunny nonexpansive retract of . The sunny nonexpansive retraction from onto is unique if is smooth.

Lemma 4 (see Goebel and Reich [20, Lemma 13.1]). Let be a convex subset of a smooth Banach space , a nonempty subset of , and a retraction from onto . Then, the following are equivalent. (a)  is sunny and nonexpansive. (b)  for all  .(c)  for all  .

Lemma 5 (see [21]). Let and be two real sequences such that (i) and , (ii).
Let be a sequence of nonnegative numbers which satisfies the inequality Then, .

3. Existence of Common Fixed Points

We begin with the following.

Proposition 6. Let be a nonempty closed subset of a Banach space . Let be a uniformly continuous semigroup of asymptotically nonexpansive mappings from into itself with a net . Let be a sequence in and a sequence in with for all in . Assume that and for all in and in . (a) There exists a sequence in defined by(b) If the sequence described by (20) is bounded and  , then

Proof. (a) Set . Since for all in , it follows that , and hence for all in . For each in , the mapping defined by is a contraction with Lipschitz constant . Therefore, there exists a sequence in described by (20).(b) Suppose that the sequence in described by (20) is bounded and . From (20), we have
Without loss of generality, we may assume that is bounded away from 1. Then, there exists a positive constant such that for all in . Since is bounded, it follows from (23) that is bounded.
Set . For , we have for all in , where . Note that as . Since is uniformly continuous at , it follows that as . Therefore, as .

Remark 7. (a) If is star shaped with respect to in , then the assumption “ for all in and in ” in Proposition 6 is automatically satisfied.
(b) If is a strongly continuous semigroup of nonexpansive mappings with , the sequence in described by (20) is bounded (see [15, Theorem 3.3]).

Theorem 8. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function , and a nonempty weakly closed subset of . Let be a uniformly continuous semigroup of asymptotically nonexpansive mappings from into itself with a net . Let be a sequence in and a sequence in with for all in satisfying the condition Assume that such that the sequence described by (20) is bounded in . Then, (a) . (b)  converges strongly to an element which holds the inequality

Proof. By Proposition 6 (b), we have as for all . Since is bounded, there exists a subsequence of such that .(a) For each , we have that is, each is asymptotically nonexpansive mapping. Since for all , it follows from Lemma 3 that for all . Hence, .(b) Since is bounded, there exists a constant such that for all in and in . For any in , from (15) and Remark 2, we have Since for all , we have Observing that , , and is weakly continuous, we conclude from (29) that as because is closed.
We prove that converges strongly to . Suppose, for contradiction, that is another subsequence of such that with . Since for all , we have . For any in , we have From (20), we have It follows that for all in and in . Since , we obtain from (32) that Addition of (33) yields Hence, , a contradiction. Therefore, converges strongly to .
Finally, from (32), we conclude that satisfies (26).

Let be a strongly continuous semigroup of asymptotically nonexpansive mappings from into itself and . Motivated by Morales and Jung [22, Theorem 1] and Morales [23, Theorem 2], we define

Corollary 9. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function and a nonempty weakly closed subset of . Let be a uniformly continuous semigroup of asymptotically nonexpansive mappings from into itself with a net . Let be a sequence in and a sequence in with for all in satisfying condition (25). Assume that such that is star shaped with respect to , and set defined by (35) is bounded. (a)For each in , there is exactly one point in described by (20).(b) is nonempty. Moreover, converges strongly to an element in which holds (26).

Corollary 9 improves and generalizes several recent results of this nature. Indeed, it extends [13, Theorem 1.7] from the class of asymptotically nonexpansive mappings to a uniformly continuous semigroup of asymptotically nonexpansive mappings without uniformly asymptotic regularity assumption. In particular, Corollary 9 improves Theorem ZS in the following ways. (1)Convexity of is not required. (2)The assumption “uniform convexity” of the underlying space is not required. (3)For convergence of , condition “ is a sunny nonexpansive retract of ” is not assumed.

Next, we show that is a nonempty sunny nonexpansive retract of .

Theorem 10. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function . Let be a nonempty closed convex-bounded subset of . Let be a uniformly continuous semigroup of asymptotically nonexpansive mappings from into itself with net . Then, is a nonempty sunny nonexpansive retract of .

Proof. By Theorem 8 (a), . As in Theorem 8 (b), for each in the sequence converges strongly to an element in . Define a mapping by By (32), we have for all in in , and in , where is the diameter of set . Letting , we obtain that Therefore, by Lemma 4, we conclude that is sunny nonexpansive.

Remark 11. In view of Remark 7 (b), the nonemptهness of the common fixed point set of a uniformly continuous semigroup of nonexpansive mappings implies that the sequence in described by (20) is bounded. So we can drop the boundedness assumption of domain in Theorem 10, provided that .

Corollary 12. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function and a nonempty weakly closed subset of . Let be a uniformly continuous semigroup of nonexpansive mappings from into itself with . Let be a sequence in and a sequence in such that . Assume that and that is star shaped with respect to . (a)For each in , there is exactly one point in described by (20).(b) converges strongly to an element in which holds (26).

Corollary 12 is an improvement of [14, Theorem 3] and [15, Theorem 3.3], where strong convergence theorems were established in Hilbert and uniformly convex Banach spaces, respectively.

4. Approximation of Common Fixed Points

Theorem 13. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function . Let be a nonempty closed convex bounded subset of and a uniformly continuous semigroup of asymptotically nonexpansive mappings from into itself with a net . For given in , let be a sequence in generated by (7), where is a sequence in and is a decreasing sequence in satisfying the following conditions.(C1)  and ,(C2)  or ,(C3) , (C4)  and .
Then,(a) . (b)  converges strongly to , where is a sunny nonexpansive retraction of onto .

Proof. It follows from Theorem 10 that , and there is a sunny nonexpansive retraction of onto . Set , , , and . Since is bounded, there exists a constant such that Hence, for all , we have Hence, by Lemma 5 and assumptions (C1)~(C4), we conclude that as . Fix , we have for all in , which gives that as . We may assume that as . In view of the assumption that the duality mapping is weakly sequentially continuous, it follows from Lemma 3 that . Thus, by the weak continuity of and Lemma 4, we have Define . From (7), we obtain Since for all in , it follows from Remark 2 that Note that and . Using Lemma 5, we obtain that converges strongly to .

One can carry over Theorem 13 to the so-called viscosity approximation technique (see Xu [21]). We derive a more general result in this direction which is an improvement upon several convergence results in the context of viscosity approximation technique.

Theorem 14. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function . Let be a nonempty closed convex-bounded subset of , a weakly contraction with function , and a uniformly continuous semigroup of asymptotically nonexpansive mappings from into itself with a net . For an arbitrary initial value in , let be a sequence in generated by Here, is a sequence in , and is a decreasing sequence in satisfying conditions (C1)~(C4). Then, (a). (b) converges strongly to in , where and is a sunny nonexpansive retraction of onto .

Proof. It follows from Theorem 10 that , and there is a sunny nonexpansive retraction of onto . Since is a weakly contractive mapping from into itself, it follows from [24, Theorem 1] that there exists a unique element in such that . Such in is an element of . Now, we define a sequence in by
By Theorem 13, we have that . By boundedness of and , there exists a constant such that for all . Observe that By [25, Lemma 3.2], we obtain . Therefore, .

Theorem 15. Let be a real reflexive Banach space with a weakly continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and a weakly contraction. Let be a uniformly continuous semigroup of nonexpansive mappings from into itself with . For given in , let be a sequence in generated by (45), where is a sequence in and is a decreasing sequence in satisfying conditions (C1)~(C3). Then, converges strongly to , where and is a sunny nonexpansive retraction of onto .

Remark 16. (a) For a related result concerning the strong convergence of the explicit iteration procedure to some fixed point of an asymptotically nonexpansive mapping on star-shaped domain in a reflexive Banach space with a weakly continuous duality mapping, we refer the reader to Schu [13].
(b) We remark that condition “ for some ” implies that as . Thus, the assumption “ for some ” imposed in Theorem ZS is very strong. In our results, such assumption is avoided. Under a mild assumption, Theorem 13 shows that the sequence generated by (7) converges strongly to a common fixed point of a uniformly continuous semigroup of asymptotically nonexpansive mappings in a real Banach space without uniform convexity. Therefore, Theorem 13 is a significant improvement of a number of known results (e.g., Theorem ZS and [12, Theorem 4.7]) for semigroups of asymptotically nonexpansive mappings. Corollary 9 and Theorem 13 provide an affirmative answer to Problem 1.

Acknowledgments

This research is supported partially by Taiwan NSC Grants 99-2115-M-037-002-MY3, 98-2923-E-037-001-MY3, 99-2115-M-110-007-MY3. The authors would like to thank the referees for their careful reading and helpful comments.

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