- Research
- Open access
- Published:
Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 167 (2012)
Abstract
Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. We obtain a weak convergence theorem of the explicit averaging cyclic algorithm for a finite family of asymptotically strictly pseudocontractive mappings of K under suitable control conditions, and elicit a necessary and sufficient condition that guarantees strong convergence of an explicit averaging cyclic process to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces. The results of this paper are interesting extensions of those known results.
MSC:47H09, 47H10.
1 Introduction
Let E and be a real Banach space and the dual space of E, respectively. Let () denote the generalized duality mapping from E into given by for all , where denotes the generalized duality pairing between E and . In particular, is called the normalized duality mapping and it is usually denoted by J. If E is smooth or is strictly convex, then is single-valued. In the sequel, we will denote the single-valued generalized duality mapping by .
Let K be a nonempty subset of E. A mapping is called asymptotically κ-strictly pseudocontractive with sequence such that (see, e.g., [1–3]) if for all , there exist a constant and such that
If I denotes the identity operator, then (1) can be written in the form
The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, is the identity, and it is shown by Osilike et al. [2] that (1) (and hence (2)) is equivalent to the inequality
where , .
A mapping T with domain and range in E is called strictly pseudocontractive of Browder-Petryshyn type [4] if for all , there exist and such that
If I denotes the identity operator, then (3) can be written in the form
In Hilbert spaces, (3) (and hence (4)) is equivalent to the inequality
It is shown in [5] that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.
A mapping T is said to be uniformly L-Lipschitzian if there exists a constant such that, for all ,
Let be N asymptotically strictly pseudocontractive self-mappings of K, and denote the common fixed points set of by , where . We consider the following explicit averaging cyclic algorithm.
For a given , and a real sequence , the sequence is generated as follows:
The algorithm can be expressed in a compact form as
where with , a positive integer and . The cyclic algorithm was first studied by Acedo and Xu [6] for the iterative approximation of common fixed points of a finite family of strictly pseudocontractive mappings in Hilbert spaces, and it is better than implicit iteration methods.
In [7] Xiaolong Qin et al. proved the following theorem in a Hilbert space.
Theorem QCKS Let K be a closed and convex subset of a Hilbert space H and be an integer. Let, for each , be an asymptotically -strictly pseudo-contractive mapping for some and a sequence such that . Let and . Assume that . For any , let be the sequence generated by the cyclic algorithm (5). Assume that the control sequence is chosen such that for all and a small enough constant . Then converges weakly to a common fixed point of the family .
Osilike and Shehu [8] extended the result of Theorem QCKS from a Hilbert space to 2-uniformly smooth Banach spaces which are also uniformly convex. They proved the following theorem.
Theorem OS Let E be a real 2-uniformly smooth Banach space which is also uniformly convex, and K be a nonempty, closed and convex subset of E. Let be N asymptotically -strictly pseudocontractive self-mappings of K for some with a sequence such that , , and . Let satisfy the conditions
where and is the constant appearing in the inequality (7) with . Let be the sequence generated by the cyclic algorithm (5). Then converges weakly to a common fixed point of the family .
We would like to point out that the condition () in Theorem OS excludes the natural choice for . This is overcome by this paper. Moreover, we improve and extend the result of Theorem OS from 2-uniformly smooth Banach spaces to q-uniformly smooth Banach spaces which are also uniformly convex. We prove that if satisfies the conditions
where , , then the iterative sequence (5) converges weakly to a common fixed point of the family .
Furthermore, we elicit a necessary and sufficient condition that guarantees strong convergence of the iterative sequence (5) to a common fixed point of the family in q-uniformly smooth Banach spaces.
We will use the notation:
-
1.
⇀ for weak convergence.
-
2.
denotes the weak ω-limit set of .
2 Preliminaries
Let E be a real Banach space. The modulus of smoothness of E is the function defined by
E is uniformly smooth if and only if .
Let . E is said to be q-uniformly smooth (or to have a modulus of smoothness of power type ) if there exists a constant such that . Hilbert spaces, (or ) spaces () and the Sobolev spaces () are q-uniformly smooth. Hilbert spaces are 2-uniformly smooth while
Theorem HKX ([[9], p.1130])
Let and let E be a real q-uniformly smooth Banach space. Then there exists a constant such that, for all ,
E is said to have a Fréchet differentiable norm if, for all ,
exists and is attained uniformly in . In this case, there exists an increasing function with such that, for all ,
It is well known (see, for example, [[10], p.107]) that a q-uniformly smooth Banach space has a Fréchet differentiable norm.
Lemma 2.1 ([[5], p.1338])
Let E be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a nonempty, closed and convex subset of E and be an asymptotically κ-strictly pseudocontractive mapping with a nonempty fixed point set. Then is demiclosed at zero, that is, if whenever such that converges weakly to and converges strongly to 0, then .
Lemma 2.2 ([[2], p.80])
Let , , be sequences of nonnegative real numbers satisfying the following inequality:
If and , then exists. If, in addition, has a subsequence which converges strongly to zero, then .
Lemma 2.3 ([[2], p.78])
Let E be a real Banach space, K be a nonempty subset of E and be an asymptotically κ-strictly pseudocontractive mapping. Then T is uniformly L-Lipschitzian.
Lemma 2.4 Let E be a real q-uniformly smooth Banach space which is also uniformly convex, and let K be a nonempty, closed and convex subset of E. Let, for each , be an asymptotically -strictly pseudocontractive mapping with . Let be the sequence satisfying the following conditions:
-
(a)
exists for every ;
-
(b)
, for each ;
-
(c)
exists for all and for all .
Then the sequence converges weakly to a common fixed point of the family .
Proof Since exists, then is bounded. By (b) and Lemma 2.1, we have . Assume that and that and are subsequences of such that and , respectively. Since E is a real q-uniformly smooth Banach space, which is also uniformly convex, then E has a Fréchet differentiable norm. Set , in (8), we obtain
where b is an increasing function. Since , , for some , then
Therefore,
Hence, . Since , then exists. Since , for all . Set . We have , that is, . Hence, is a singleton, so that converges weakly to a common fixed point of the family . □
3 Main results
Theorem 3.1 Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. Let be an integer and . Let, for each , be an asymptotically -strictly pseudocontractive mapping for some with sequences such that , where , and . Let . Let satisfy the conditions (6) and be the sequence generated by the cyclic algorithm (5). Then converges weakly to a common fixed point of the family .
Proof Pick a . We firstly show that exists. To see this, using (2) and (7), we obtain
where . Since for all n, where , we get . Therefore, (9) implies
Let . Since , we have
then (10) implies exists by Lemma 2.2 (and hence the sequence is bounded, that is, there exists a constant such that ).
Then we prove , . In fact, it follows from (9) that
Then
Since , then (11) implies that . Thus .
For all , we have and . By Lemma 2.3, we know that is uniformly -Lipschitzian, then there exists a constant , such that
Thus
Observe that
Consequently,
Observe also that
Hence,
Consequently, for all , we have
Thus,
Now we prove that for all , exists for all . Let . It is obvious that and exist. So, we only need to consider the case of . Define by
Then for all
By the choice of , we have , so it follows that . For the convenience of the following discussion, set , then .
Set , . We have
and
Set . If for some , then for any so that , in fact, converges strongly to . Thus we may assume for any . Let δ denote the modulus of convexity of E. It is well known (see, for example, [[11], p.108]) that
for all and for all such that , . Set
Then and so that it follows from (12) that
Observe that
and
it follows from (13) that
Since E is uniformly convex, then is nondecreasing, and since , hence it follows from (14) that
Since exits and . Also since and exists, then the continuity of δ and yield uniformly for all . Observe that
Hence , this ensures that exists for all .
Now apply Lemma 2.4 to conclude that converges weakly to a common fixed point of the family . □
Theorem 3.2 Let E be a real q-uniformly smooth Banach space, and let K be a nonempty, closed and convex subset of E. Let be an integer and . Let, for each , be an asymptotically -strictly pseudocontractive mapping for some with sequences such that , where , and . Let . Let satisfy the conditions (6) and be the sequence generated by the cyclic algorithm (5). Then converges strongly to a common fixed point of the family if and only if
where .
Proof It follows from (10) that
Thus , and it follows from Lemma 2.2 that exists.
Now if converges strongly to a common fixed point p of the family , then . Since
we have .
Conversely, suppose , then the existence of implies that . Thus, for arbitrary , there exists a positive integer such that for any .
From (10), we have
and for some , . Now, an induction yields
Since , then there exists a positive integer such that , . Choose , then for all and for all , we have
Taking infimum over all , we obtain
Thus is Cauchy. Suppose . Then for all we have
Thus , , and hence . □
References
Osilike MO: Iterative approximations of fixed points of asymptotically demicontractive mappings. Indian J. Pure Appl. Math. 1998, 29(12):1291–1300.
Osilike MO, Aniagbosor SC, Akuchu BG: Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces. Panam. Math. J. 2002, 12(2):77–88.
Qihou L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 1996, 26(11):1835–1842. 10.1016/0362-546X(94)00351-H
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Osilike MO, Udomene A, Igbokwe DI, Akuchu BG: Demiclosedness principle and convergence theorems for κ -strictly asymptotically pseudo-contractive maps. J. Math. Anal. Appl. 2007, 326: 1334–1345. 10.1016/j.jmaa.2005.12.052
Acedo GL, Xu HK: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036
Qin X, Cho YJ, Ku SM, Shang M: A hybrid iterative scheme for asymptotically k -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2009, 70: 1902–1911. 10.1016/j.na.2008.02.090
Osilike MO, Shehu Y: Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive maps in Banach spaces. Comput. Math. Appl. 2009, 57: 1502–1510. 10.1016/j.camwa.2009.01.022
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K
Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama, Yokohama; 2000.
Bruck RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math. 1979, 32(2–3):107–116. 10.1007/BF02764907
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors have read and approved the final manuscript.
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
Zhang, Y., Xie, Z. Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces. Fixed Point Theory Appl 2012, 167 (2012). https://doi.org/10.1186/1687-1812-2012-167
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-167