Convergence Theorems of Three-Step Iterative Scheme for a Finite Family of Uniformly Quasi-Lipschitzian Mappings in Convex Metric Spaces
© T. You-xian and Y. Chun-de. 2009
Received: 9 December 2008
Accepted: 25 March 2009
Published: 31 March 2009
We consider a new Noor-type iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite family of uniformly quasi-Lipschitzian mappings. The results presented in this paper extend, improve and unify some main results in previous work.
1. Introduction and Preliminaries
Takahashi  introduced a notion of convex metric spaces and studied the fixed point theory for nonexpansive mappings in such setting. For the convex metric spaces, Kirk  and Goebel and Kirk  used the term "hyperbolic type space" when they studied the iteration processes for nonexpansive mappings in the abstract framework. For the Banach space, Petryshyn and Williamson  proved a sufficient and necessary condition for Picard iterative sequences and Mann iterative sequence to converge to fixed points for quasi-nonexpansive mappings. In 1997, Ghosh and Debnath  extended the results of  and gave the sufficient and necessary condition for Ishikawa iterative sequence to converge to fixed points for quasi-nonexpansive mappings. Liu [6–8] proved some sufficient and necessary conditions for Ishikawa iterative sequence and Ishikawa iterative sequence with errors to converge to fixed point for asymptotically quasi-nonexpansive mappings in Banach space and uniform convex Banach space. Tian  gave some sufficient and necessary conditions for an Ishikawa iteration sequence for an asymptotically quasi-nonexpansive mapping to converge to a fixed point in convex metric spaces. Very recently, Wang and Liu  gave some iteration sequence with errors to approximate a fixed point of two uniformly quasi-Lipschitzian mappings in convex metric spaces. The purpose of this paper is to give some sufficient and necessary conditions for a new Noor-type iterative sequence with errors to approximate a common fixed point for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. The results presented in this paper generalize, improve, and unify some main results of [1–14].
First of all, let us list some definitions and notations.
If is nonempty, then it follows from the above definitions that an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, and an asymptotically quasi-nonexpansive mapping must be a uniformly quasi-Lipschitzian with . However, the inverse is not true in general.
Definition 1.2 (see [ 9]).
In order to prove our main results, the following lemmas will be needed.
This completes the Proof.
Lemma 1.5 (see ).
Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant Let be the iterative sequence with errors defined by (1.10) and , be three bounded sequences in . Let , , , be sequences in [0,1] satisfying the following conditions:
Then the following conclusions hold:
This completes the proof.
2. Main Results
Let be a complete convex metric space and be a nonempty closed convex subset of Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant . Let be the iterative sequence with errors defined by (1.10) and , , be three bounded sequence in and , , , , , , , and be nine sequences in [ 0,1] satisfying the following conditions:
Similarly, we can obtain the following results.
The authors would like to express their thanks to the referees for their helpful comments and suggestions.
- Takahashi W: A convexity in metric space and nonexpansive mappings. I. Kōdai Mathematical Seminar Reports 1970, 22: 142–149. 10.2996/kmj/1138846111View ArticleMathSciNetMATHGoogle Scholar
- Kirk WA: Krasnoselskii's iteration process in hyperbolic space. Numerical Functional Analysis and Optimization 1982,4(4):371–381. 10.1080/01630568208816123MathSciNetView ArticleMATHGoogle Scholar
- Goebel K, Kirk WA: Iteration processes for nonexpansive mappings. In Topological Methods in Nonlinear Functional Analysis (Toronto, Canada, 1982), Contemporary Mathematics. Volume 21. American Mathematical Society, Providence, RI, USA; 1983:115–123.View ArticleGoogle Scholar
- Petryshyn WV, Williamson, TE Jr.: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 1973, 43: 459–497. 10.1016/0022-247X(73)90087-5MathSciNetView ArticleMATHGoogle Scholar
- Ghosh MK, Debnath L: Convergence of Ishikawa iterates of quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 1997,207(1):96–103. 10.1006/jmaa.1997.5268MathSciNetView ArticleMATHGoogle Scholar
- Liu Q: Iterative sequences for asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2001,259(1):1–7. 10.1006/jmaa.2000.6980MathSciNetView ArticleMATHGoogle Scholar
- Liu Q: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member. Journal of Mathematical Analysis and Applications 2001,259(1):18–24. 10.1006/jmaa.2000.7353MathSciNetView ArticleMATHGoogle Scholar
- Liu Q: Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space. Journal of Mathematical Analysis and Applications 2002,266(2):468–471. 10.1006/jmaa.2001.7629MathSciNetView ArticleMATHGoogle Scholar
- Tian Y-X: Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings. Computers & Mathematics with Applications 2005,49(11–12):1905–1912. 10.1016/j.camwa.2004.05.017MathSciNetView ArticleMATHGoogle Scholar
- Wang C, Liu LW: Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):2067–2071. 10.1016/j.na.2008.02.106MathSciNetView ArticleMATHGoogle Scholar
- Cho YJ, Zhou H, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Computers & Mathematics with Applications 2004,47(4–5):707–717. 10.1016/S0898-1221(04)90058-2MathSciNetView ArticleMATHGoogle Scholar
- Fukhar-ud-din H, Khan SH: Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications. Journal of Mathematical Analysis and Applications 2007,328(2):821–829. 10.1016/j.jmaa.2006.05.068MathSciNetView ArticleMATHGoogle Scholar
- Jeong JU, Kim SH: Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappings. Applied Mathematics and Computation 2006,181(2):1394–1401. 10.1016/j.amc.2006.03.008MathSciNetView ArticleMATHGoogle Scholar
- Zhou H, Kang JI, Kang SM, Cho YJ: Convergence theorems for uniformly quasi-Lipschitzian mappings. International Journal of Mathematics and Mathematical Science 2004,2004(15):763–775. 10.1155/S0161171204309269MathSciNetView ArticleMATHGoogle Scholar
- Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. Journal of Mathematical Analysis and Applications 1998,224(1):91–101. 10.1006/jmaa.1998.5987MathSciNetView ArticleMATHGoogle Scholar
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