- Research Article
- Open access
- Published:
Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups
Fixed Point Theory and Applications volume 2010, Article number: 301868 (2010)
Abstract
We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.
1. Introduction
Throughout this paper, Let be a real Hilbert space with inner product and norm , and we write to indicate that the sequence converges strongly to . Let be a nonempty closed convex subset of , and let be a mapping. Recall that is nonexpansive if , for all . We denote the set of fixed points of by , that is, . A mapping is said to be asymptotically nonexpansive if there exists a sequence with for all , , and
Mann's iterative algorithm was introduced by Mann [1] in 1953. This iteration process is now known as Mann's iteration process, which is defined as
where the initial guess is taken in arbitrarily and the sequence is in the interval .
In 1967, Halpern [2] first introduced the following iteration scheme:
for all , where and is a sequence in . This iteration process is called a Halpern-type iteration.
Recall also that a one-parameter family of self-mappings of a nonempty closed convex subset of a Hilbert space is said to be a (continuous) Lipschitzian semigroup on if the following conditions are satisfied:
(a),
(b), for all , ;
(c)for each , the map is continuous on
(d)there exists a bounded measurable function such that, for each , , for all
A Lipschitzian semigroup is called nonexpansive if for all , and asymptotically nonexpansive if . We denote by the set of fixed points of the semigroup , that is, .
In 2003, Nakajo and Takahashi [3] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space :
where denotes the metric projection from onto a closed convex subset of . They proved that the sequence converges weakly to a fixed point of . Moreover, they introduced and studied an iteration process of a nonexpansive semigroup in a Hilbert space :
In 2006, Kim and Xu [4] adapted iteration (1.4) to an asymptotically nonexpansive mapping in a Hilbert space :
where as . They also proved that if for all and for some , then the sequence converges weakly to a fixed point of . Moreover, they modified an iterative method (1.5) to the case of an asymptotically nonexpansive semigroup in a Hilbert space :
where as .
In 2007, Zegeye and Shahzad [5] developed the iteration process for a finite family of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups with a closed convex bounded subset of a Hilbert space :
where as and
where as , with , for each
Recently, Su and Qin [6] modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems.
In 2008, Takahashi et al. [7] proved strong convergence theorems by the new hybrid methods for a family of nonexpansive mappings and nonexpansive semigroups in Hilbert spaces:
where , and
where , and .
In this paper, motivated and inspired by the above results, we modify iteration process (1.4)–(1.11) by the new hybrid methods for countable families of asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in [3, 5–8] and many authors.
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the next section.
Lemma 2.1.
Here holds the identity in a Hilbert space :
for all and .
Using this Lemma 2.1, we can prove that the set of fixed points of is closed and convex. Let be a nonempty closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by , such that for all , where is called the metric projection of onto . We know that for and , is equivalent to for all . We know that a Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality
hold for every with . We also know that has the Kadec-Klee property, that is, and imply . In fact, from
we get that a Hilbert space has the Kadec-Klee property.
Let be a nonempty closed convex subset of a Hilbert space . Motivated by Nakajo et al. [9], we give the following definitions: Let and be families of nonexpansive mappings of into itself such that , where is the set of all fixed points of and is the set of all common fixed points of . We consider the following conditions of and (see [9]):
(i)NST-condition (I). For each bounded sequence , implies that for all .
(ii)NST-condition (II). For each bounded sequence , implies that for all .
(iii)NST-condition (III). There exists with such that for every bounded subset of , there exists such that holds for all and
Lemma 2.2.
Let be a nonempty closed convex subset of and let be a nonexpansive mapping of into itself with . Then, the following hold:
(i) with and satisfy the condition (I) with .
(ii) with and satisfy the condition (I) with
Lemma 2.3 (Opial [10]).
Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping such that . If is a sequence in such that and , then .
Lemma 2.4 (Lin et al. [11]).
Let be an asymptotically nonexpansive mapping defined on a bounded closed convex subset of a bounded closed convex subset of a Hilbert space . If is a sequence in such that and , then .
Lemma 2.5 (Nakajo and Takahashi [3]).
Let be a real Hilbert space. Given a closed convex subset and points . Given also a real number . The set is convex and closed.
Lemma 2.6 (Kim and Xu [4]).
Let be a nonempty bounded closed convex subset of and be an asymptotically nonexpansive semigroup on . If is a sequence in satisfying the properties
(a);
(b),
then .
Lemma 2.7 (Kim and Xu [4]).
Let be a nonempty bounded closed convex subset of and be an asymtotically nonexpansive semigroup on . Then it holds that
3. Strong Convergence for a Family of Asymptotically Nonexpansive Mappings
Theorem 3.1.
Let be a nonempty bounded closed convex subset of a Hilbert space and let for be a countable family of asymptotically nonexpansive mapping with sequence for , respectively. Assume such that for all and as . Let . Further, suppose that satisfies NST-condition (I) and (III) with T. Define a sequence in by the following algorithm:
where as . Then converges in norm to .
Proof.
We first show that is closed and convex for all . From the Lemma 2.5, it is observed that is closed and convex for each .
Next, we show that for all . Indeed, let , we have
Thus and hence for all . Thus is well defined.
From and , we have
So, for , we have
for all . This implies that
hence
for all Therefore is nondecreasing.
From , we have
Using , we also have
So, for , we have
This implies that
Thus, is bounded. So, exists.
Next, we show that . From (3.3), we have
Since exists, we conclude that .
Since , we have which implies that . Now we claim that as for all . We first show that as . Indeed, by the definition of , we have
for all and it follows that
Since as we obtain
for all .
Let . Now, for we get
from (3.14) and as , yields
for each Let and take with . By NST-condition (III), there exists such that
By (3.16) and , we get
By the assumption of and NST-condition (I), we have
Put . Since for all , is bounded. Let be a subsequence of such that . Since is closed and convex, is weakly closed and hence . From (3.19), we have that . If not, since satisfies Opial's condition, we have
This is a contradiction. So, we have that . Then, we have
and hence . From , we have . This implies that converges weakly to , and we have
and hence . From , we also have . Since satisfies the Kadec-Klee property, it follows that . So, we have
and hence . This completes the proof.
Corollary 3.2.
Let be a nonempty bounded closed convex subset of a Hilbert space and let be an asymptotically nonexpansive mapping with sequence . Assume such that for all and as . Let . Define a sequence in by the following algorithm:
where as . Then converges in norm to .
Proof.
Setting for all from Lemma 2.2(i) and Theorem 3.1, we immediately obtain the corollary.
Since every family's nonexpansive mapping is family's asymptotically nonexpansive mapping we obtain the following result.
Corollary 3.3.
Let be a nonempty bounded closed convex subset of a Hilbert space and let be a family of nonexpansive mappings with sequence . Assume such that for all and as . Let . Further, suppose that satisfies NST-condition (I) with T. Define a sequence in by the following algorithm:
Assume that if for each bounded sequence , for all implies that . Then converges in norm to .
We have the following corollary for nonexpansive mappings by Lemma 2.2(i) and Theorem 3.1.
Corollary 3.4 (Takahashi et al. [7, Theorem ]).
Let be a bounded closed convex subset of a Hilbert space and let be a nonexpansive mapping such that . Assume that for all . Then the sequence generated by
converges in norm to
4. Strong Convergence for a Family of Asymptotically Nonexpansive Semigroups
Theorem 4.1.
Let be a nonempty bounded closed convex subset of a Hilbert space and let , be a countable family of asymptotically nonexpansive semigroups. Assume such that for all and as . Let be a countable positive and divergent real sequence. Let . Further, suppose that satisfies NST-condition (I) with T. Define a sequence in by the following algorithm:
where as with . Then converges in norm to .
Proof.
First observe that for all . Indeed, we have for all
So, . Hence for all . By the same argument as in the proof of Theorem 3.1, is closed and convex, is well defined. Also, similar to the proof of Theorem 3.1
We next claim that Indeed, by definition of and we have
and then
Since , we have
which in turn implies that
It follows from (4.5) that
Let and for each , we get that
By (4.8) and Lemma 2.7, we obtain that
Furthermore, from (4.9) and Lemma 2.6 and the boundedness of we obtain that . By the fact that for any , where and the weak lower semi-continuity of the norm, we have for all . However, since , we must have for all . Thus and then converges weakly to . Moreover, following the method of Theorem 3.1, . This completes the proof.
Corollary 4.2.
Let be a bounded closed convex subset of a Hilbert space and be an asymptotically nonexpansive semigroup on . Assume also that for all and is a positive real divergent sequence. Then, the sequence generated by
converges in norm to , where as .
Proof.
By Theorem 4.1, if the semigroup , then for all and for all . Hence for all and then, (4.1) reduces to (4.11).
Corollary 4.3 (Takahashi et al. [7, Theorem ]).
Let be a nonempty closed convex subset of a Hilbert space and be a nonexpansive semigroup on . Assume that for all and is a positive real divergent sequence. If , then the sequence generated by
converges in norm to
References
Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4
Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):1140–1152. 10.1016/j.na.2005.05.059
Zegeye H, Shahzad N: Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4496–4503. 10.1016/j.na.2007.11.005
Su Y, Qin X: Strong convergence of monotone hybrid method for fixed point iteration processes. Journal of Systems Science & Complexity 2008,21(3):474–482. 10.1007/s11424-008-9129-3
Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062
Su Y, Qin X: Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Fixed Point Theory and Applications 2006, 2006:-11.
Nakajo K, Shimoji K, Takahashi W: Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces. Journal of Nonlinear and Convex Analysis 2007,8(1):11–34.
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
Lin P-K, Tan K-K, Xu HK: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 1995,24(6):929–946. 10.1016/0362-546X(94)00128-5
Acknowledgments
The authors would like to thank professor Somyot Plubtieng for drawing my attention to the subject and for many useful discussions and the referees for helpful suggestions that improved the contents of the paper. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wattanawitoon, K., Kumam, P. Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups. Fixed Point Theory Appl 2010, 301868 (2010). https://doi.org/10.1155/2010/301868
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/301868