Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces
- Jing Quan1Email author,
- Shih-sen Chang2 and
- Xiongrui Wang1
https://doi.org/10.1186/1687-1812-2012-142
© Quan et al.; licensee Springer. 2012
Received: 8 May 2012
Accepted: 22 August 2012
Published: 5 September 2012
Abstract
The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.
MSC:47H05, 47H09, 49M05.
Keywords
1 Introduction
is called the normalized duality mapping. Let be a nonlinear mapping; denotes the set of fixed points of mapping T.
Alber et al.[1] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. They are defined as follows.
Definition 1.1 Let be a mapping. T is said to be total asymptotically nonexpansive if there exist sequences , with as and a strictly increasing continuous function with such that holds for all and all .
T is said to be total asymptotically quasi-nonexpansive if , there exist sequences , with as and a strictly increasing continuous function with such that holds for all , and all .
Chidume and Ofoedu [2] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings and total asymptotically quasi-nonexpansive mappings in Banach spaces. Chidume et al.[3] gave a new iterative sequence and necessary and sufficient conditions for this sequence to converge to a common fixed point of finite total asymptotically nonexpansive mappings. Chang [4] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.
Recently, many researchers have focused on studying the convergence of iterative algorithms for quasi-ϕ-asymptotically nonexpansive (see [5–9]) and total quasi-ϕ-asymptotically nonexpansive (see [10–12]) mappings. Ye et al.[13] used a new hybrid projection algorithm to obtain strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. Kim [14] used hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings to prove the strong convergence theorems. Saewan [15] used the shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings.
A Banach space E is said to be strictly convex if for and ; it is also said to be uniformly convex if for any two sequences , in E such that and . Let be the unit sphere of E, then the Banach space E is said to be smooth provided exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for each . It is well known that if E is reflexive and smooth, then the duality mapping J is single valued. A Banach space E is said to have the Kadec-Klee property if a sequence of E satisfies that and , then . It is known that if E is uniformly convex, then E has the Kadec-Klee property.
The quasi-ϕ-asymptotically nonexpansive and total quasi-ϕ-asymptotically nonexpansive mappings are defined as follows.
holds for all , and all .
holds for all , and all .
In recent years, many researchers have considered the convergence of asymptotically nonexpansive semigroups [17, 18]. The asymptotically nonexpansive semigroups are defined as follows.
Definition 1.3[17]
- (a)
for each ;
- (b)
for any and ;
- (c)
For any , the mapping is continuous;
- (d)There exist sequences with as such that
holds for all , .
We use to denote the common fixed point set of the semigroup T, i.e., .
Chang [19] used the modified Halpern-Mann type iteration algorithm for quasi-ϕ-asymptotically nonexpansive semigroups to prove the strong convergence in the Banach space. The quasi-ϕ-asymptotically nonexpansive semigroups are defined as follows.
Definition 1.4[19]
- (e)For all , , , there exist sequences with as , such that
holds for all .
2 Preliminaries
This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.
- (f)If , there exist sequences , with as and a strictly increasing continuous function with such that
holds for all , and all .
The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-nonexpansive asymptotically semigroups to prove the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of [5, 6, 9–12, 14, 15, 19] and many others.
In order to prove the results of this paper, we shall need the following lemmas:
Lemma 2.1 (See [16])
- (i)
for all , ;
- (ii)
If and , then , ;
- (iii)
For , if and only if .
Lemma 2.2[19]
Let E be a uniformly convex and smooth Banach space and letandbe two sequences of E. Ifand eitheroris bounded, then.
Lemma 2.3[10]
Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and C be a nonempty closed convex subset of E. Letbe a closed and total quasi-ϕ-asymptotically nonexpansive mapping defined by Definition 1.2. If, then the fixed point setof T is a closed and convex subset of C.
3 Main results
where. Ifandis bounded in C, then the iterative sequenceconverges strongly to a common fixed pointin C.
Proof (I) We prove and all are closed and convex subsets in C.
- (II)
We prove that .
- (III)
We prove that is a Cauchy sequence in C.
- (IV)
Now we prove .
- (V)
Finally, we prove .
In view of the definition of , from (8), we have . Therefore, . This completes the proof of Theorem 3.1. □
Declarations
Acknowledgements
This work was supported by National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345 and No. 11ZA172).
Authors’ Affiliations
References
- Alber YI, Chidume CE, Zegeye H: Approximating fixed point of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006., 2006: Article ID 10673Google Scholar
- Chidume CE, Ofoedu EU: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2007, 333(1):128–141. 10.1016/j.jmaa.2006.09.023MathSciNetView ArticleGoogle Scholar
- Chidume CE, Ofoedu EU: A new iteration process for approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. Int. J. Math. Math. Sci. 2009., 2009: Article ID 615107. doi:10.1155/2009/615107Google Scholar
- Chang S, Kim JK, Joseph Lee HW, Chan CK: A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 37. doi:10.1186/1029–242X-2012–37Google Scholar
- Chang S, Chan CK, Joseph Lee HW: Modified block iterative algorithm for Quasi- ϕ -asymptotically nonexpansive mappings and equilibrium problem in Banach spaces. Appl. Math. Comput. 2011, 217: 7520–7530. 10.1016/j.amc.2011.02.060MathSciNetView ArticleGoogle Scholar
- Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215(11):3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleGoogle Scholar
- Saewan S, Kumam P: Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi- ϕ -asymptotically nonexpansive mappings. Abstr. Appl. Anal. 2010., 2010: Article ID 22. doi:10.1155/2010/357120Google Scholar
- Chang S, Wanga L, Tang Y-K, Wanga B, Qin L-J: Strong convergence theorems for a countable family of quasi- ϕ -asymptotically nonexpansive nonself mappings. Appl. Math. Comput. 2012, 218: 7864–7870. 10.1016/j.amc.2012.02.002MathSciNetView ArticleGoogle Scholar
- Qin X, Huang S, Wang T: On the convergence of hybrid projection algorithms for asymptotically quasi- ϕ -nonexpansive mappings. Comput. Math. Appl. 2011, 61(4):851–859. 10.1016/j.camwa.2010.12.033MathSciNetView ArticleGoogle Scholar
- Chang S, Joseph Lee HW, Chan CK, Zhang WB: A modified Halpern type iterative algorithm for total quasi- ϕ -asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2012, 218(11):6489–6497. 10.1016/j.amc.2011.12.019MathSciNetView ArticleGoogle Scholar
- Wang X, Chang S, Wang L, Tang Y-K, Xu YG: Strong convergence theorems for nonlinear operator equations with total quasi- ϕ -asymptotically nonexpansive mappings and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 34. doi:10.1186/1687–1812–2012–34Google Scholar
- Zuo P, Chang S, Liu M: On a hybrid algorithm for a family of total quasi- ϕ -asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 70. doi:10.1186/1687–1812–2012–70Google Scholar
- Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.MathSciNetView ArticleGoogle Scholar
- Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10Google Scholar
- Saewan S, Kumam P: The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 9Google Scholar
- Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartosator AG. Dekker, New York; 1996:15–50.Google Scholar
- Chang SS, Joseph Lee HW, Chan CK: Convergence theorem of common fixed point for asymptotically nonexpansive semigroups in Banach spaces. Appl. Math. Comput. 2009, 212: 60–65. 10.1016/j.amc.2009.01.086MathSciNetView ArticleGoogle Scholar
- Zegeye H, Shahzadb N, Damana OA: Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings. Math. Comput. Model. 2011, 54: 2077–2086. 10.1016/j.mcm.2011.05.016View ArticleGoogle Scholar
- Chang S, Wang L, Tang Y-K, Zao Y-H, Wang B: Strong convergence theorems of quasi- ϕ -asymptotically nonexpansive semi-groups in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 15. doi:10.1186/1687–1812–2012–15Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.