- Research Article
- Open Access
Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings
© J.-Z. Xiao and X.-H. Zhu. 2008
- Received: 8 August 2007
- Accepted: 26 November 2007
- Published: 12 December 2007
A family of commuting nonexpansive self-mappings, one of which is weakly contractive, are studied. Some convergence theorems are established for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point. The error estimates of these iterations are also given.
- Banach Space
- Error Estimate
- Convex Subset
- Convergence Theorem
- Nonexpansive Mapping
where is continuous and nondecreasing such that is positive on , and .
It is evident that is contractive if it is weakly contractive with , where , and it is nonexpansive if it is weakly contractive.
As an important extension of the class of contractive mappings, the class of weakly contractive mappings was introduced by Alber and Guerre-Delabriere . In Hilbert and Banach spaces, Alber et al. [1–4] and Rhoades  established convergence theorems on iteration of fixed point for weakly contractive single mapping.
We will establish some convergence theorems for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point and to give their error estimates.
Throughout this paper, we assume that is the set of fixed points of a mapping , that is, ; is defined by the antiderivative (indefinite integral) of on , that is, , and is the inverse function of .
We define iterations which will be needed in the sequel.
where the function takes values in .
We will make use of following result in the proof of Theorem 2.4.
Lemma 1.1 (see ).
Suppose that , are two sequences of nonnegative numbers such that , for all . If , then exists.
where is the Gauss integer of .
By (1.1), (1.2), and (2.11), we deduce
From (2.16) and (2.17), we obtain the error estimate (2.1). This completes the proof.
If in Theorem 2.1, where is the identity mapping of , then we conclude that the sequence converges to the unique common fixed point of weakly contractive mapping , with the error estimate , where . Thus, our Theorem 2.1 is a generalization of the corresponding theorem of Rhoades .
From (2.21)–(2.23), we obtain (2.18). This completes the proof.
Hence, the estimate (2.25) holds. This completes the proof.
This work is supported by the National Natural Science Foundation of China (10671094).
- Alber YI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications. Volume 98. Birkhäuser, Basel, Switzerland; 1997:7-22.View ArticleGoogle Scholar
- Alber YI, Chidume CE, Zegeye A: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications 2006, 2006:-20.Google Scholar
- Alber YI, Guerre-Delabriere S: On the projection methods for fixed point problems. Analysis 2001, 21(1):17-39.MATHMathSciNetView ArticleGoogle Scholar
- Alber YI, Guerre-Delabriere S, Zelenko L: The principle of weakly contractive mappings in metric spaces. Communications on Applied Nonlinear Analysis 1998, 5(1):45-68.MATHMathSciNetGoogle Scholar
- Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Analysis: Theory, Methods and Applications 2001, 47(4):2683-2693. 10.1016/S0362-546X(01)00388-1MATHMathSciNetView ArticleGoogle Scholar
- Hussain N, Khan AR: Common fixed-point results in best approximation theory. Applied Mathematics Letters 2003, 16(4):575-580. 10.1016/S0893-9659(03)00039-9MATHMathSciNetView ArticleGoogle Scholar
- Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996, 202(1):150-159. 10.1006/jmaa.1996.0308MATHMathSciNetView ArticleGoogle Scholar
- Borwein J, Reich S, Shafrir I: Krasnoselski-Mann iterations in normed spaces. Canadian Mathematical Bulletin 1992, 35(1):21-28. 10.4153/CMB-1992-003-0MATHMathSciNetView ArticleGoogle Scholar
- Kirk WA: On successive approximations for nonexpansive mappings in Banach spaces. Glasgow Mathematical Journal 1971, 12: 6-9. 10.1017/S0017089500001063MATHMathSciNetView ArticleGoogle Scholar
- Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44(1):147-150. 10.1090/S0002-9939-1974-0336469-5MATHMathSciNetView ArticleGoogle Scholar
- Ishikawa S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 1976, 59(1):65-71. 10.1090/S0002-9939-1976-0412909-XMATHMathSciNetView ArticleGoogle Scholar
- Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993, 178(2):301-308. 10.1006/jmaa.1993.1309MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.