- Research Article
- Open access
- Published:
Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings
Fixed Point Theory and Applications volume 2008, Article number: 284613 (2007)
Abstract
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.
1. Introduction
In 2005, Shin-ya Matsushita and Wataru Takahashi [1] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space :
They proved the following convergence theorem.
Theorem 1.1 (MT).
Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty, closed, and convex subset of , let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.1), where is the duality mapping on . If the set of fixed points of is nonempty, then converges strongly to , where is the generalized projection from onto .
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S.Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S.Matsushita and W. Takahashi [1], and some others.
2. Preliminaries
Let be a real Banach space with dual . We denote by the normalized duality mapping from to defined by
where denotes the generalized duality pairing. It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . In this case, is singe valued and also one to one.
Recall that if is a nonempty, closed, and convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This is true only when is a real Hilbert space. In this connection, Alber [2] has recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that is a smooth Banach space. Consider the functional defined as [2, 3] by
Observe that, in a Hilbert space , (2.2) reduces to ,
The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional that is, where is the solution to the minimization problem
existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [2–4]). In Hilbert space, It is obvious from the definition of the function that
Remark 2.1.
If is a reflexive strict convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies From the definition of we have , that is, see [5] for more details.
We refer the interested reader to the [6], where additional information on the duality mapping may be found.
Let be a closed convex subset of , and Let be a mapping from into itself. We denote by the set of fixed points of . is called hemi-relatively nonexpansive if for all and .
A point in is said to be an asymptotic fixed point of [7] if contains a sequence which converges weakly to such that the strong The set of asymptotic fixed points of will be denoted by . A hemi-relatively nonexpansive mapping from into itself is called relatively nonexpansive [1, 7, 8] if .
We need the following lemmas for the proof of our main results.
Lemma 2.2 ({Kamimura and Takahashi [4], [1, Proposition 2.1]}).
Let be a uniformly convex and smooth real Banach space and let be two sequences of . If and either or is bounded, then
Lemma 2.3 ({Alber [2], [1, Proposition 2.2]}).
Let be a nonempty closed convex subset of a smooth real Banach space and . Then, if and only if
Lemma 2.4 ({Alber [2], [1, Proposition 2.3]}).
Let be a reflexive, strict convex, and smooth real Banach space, let be a nonempty closed convex subset of and let Then
By using the similar method as [1, Proposition 2.4], the following lemma is not hard to prove.
Lemma 2.5.
Let be a strictly convex and smooth real Banach space, let be a closed convex subset of , and let be a hemi-relatively nonexpansive mapping from into itself. Then is closed and convex.
Recall that an operator in a Banach space is called closed, if , then .
3. Strong Convergence for Hemi-Relatively Nonexpansive Mappings
Theorem 3.1.
Theorem 3.1 Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty closed convex subset of , let be a closed hemi-relatively nonexpansive mapping such that . Assume that is a sequence in such that . Define a sequence in by the following algorithm:
where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .
Proof.
We first show that and are closed and convex for each . From the definition of and , it is obvious that is closed and is closed and convex for each . We show that is convex for any . Since
is equivalent to
it follows that is convex.
Next, we show that for all . Indeed, we have for all that
That is, for all .
Next, we show that for all , we prove this by induction. For we have Assume that Since is the projection of onto , by Lemma 2.3, we have
As by the induction assumptions, the last inequality holds, in particular, for all . This together with the definition of implies that .
Since and for all , we have
for all . Therefore, is nondecreasing. In addition, it follows from the definition of and Lemma 2.3 that . Therefore, by Lemma 2.4, we have
for each for all Therefore, is bounded, this together with (3.6) implies that the limit of exists. Put
From Lemma 2.4, we have, for any positive integer , that
for all Therefore,
We claim that is a Cauchy sequence. If not, there exists a positive real number and subsequence such that
for all .
On the other hand, from (3.8) and (3.9) we have
Because from (3.8) we know that is bounded, this and (2.4) imply that is also bounded, so by Lemma 2.2 we obtain
This is a contradiction, so that is a Cauchy sequence, therefore there exists a point such that converges strongly to .
Since , from the definition of , we have
It follows from (3.10), (3.14) that
By using Lemma 2.2, we have
Since is uniformly norm-to-norm continuous on bounded sets, we have
Noticing that
which implies that
This together with (3.17) and implies that
Since is also uniformly norm-to-norm continuous on any bounded sets, we have
Observe that
It follows from (3.16) and (3.21) that
Since is a closed operator and , then is a fixed point of .
Finally, we prove that . From Lemma 2.4, we have
On the other hand, since and , for all , we get from Lemma 2.4 that
By the definition of , it follows that both and , whence . Therefore, it follows from the uniqueness of that . This completes the proof.
Theorem 3.2.
Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty, closed, and convex subset of , and let be a closed relative nonexpansive mapping such that . Assume that is a sequences in such that . Define a sequence in by the following algorithm:
where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .
Proof.
Since every relatively nonexpansive mapping is a hemi-relatively one, Theorem 3.2 is implied by Theorem 3.1.
Remaek 3.3.
In recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings have been introduced and studied by various authors [1, 8–11]. In fact, all hybrid iteration methods can be replaced (or modified) by monotone hybrid iteration methods, respectively. In addition, by using the monotone hybrid method we can easily show that the iteration sequence is a Cauchy sequence, without the use of the Kadec-Klee property, demiclosedness principle, and Opial's condition or other methods which make use of the weak topology.
References
Matsushita S-Y, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a banach space. Journal of Approximation Theory 2005, 134(2):257–266. 10.1016/j.jat.2005.02.007
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.
Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. PanAmerican Mathematical Journal 1994, 4(2):39–54.
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002, 13(3):938–945. 10.1137/S105262340139611X
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications. Volume 62. Kluwer Academic, Dordrecht, The Netherlands; 1990:xiv+260.
Reich S: Review of geometry of Banach spaces, duality mappings and nonlinear problems by Ioana Cioranescu. Bulletin of the American Mathematical Society 1992, 26: 367–370. 10.1090/S0273-0979-1992-00287-2
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001, 7(2):151–174. 10.1515/JAA.2001.151
Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(11):2400–2411. 10.1016/j.na.2005.08.018
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
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
Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a banach space. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(6):1958–1965. 10.1016/j.na.2006.08.021
Acknowledgments
The authors would like to thank the referee for valuable suggestions which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China under Grant no. 10771050.
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
Su, Y., Wang, D. & Shang, M. Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings. Fixed Point Theory Appl 2008, 284613 (2007). https://doi.org/10.1155/2008/284613
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/284613