Existence and convergence of fixed points for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbo
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Existence and convergence of fixed points for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbolic spaces Jingxin Zhang1* and Yunan Cui2 * Correspondence: zhjx_19@yahoo. com.cn 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Full list of author information is available at the end of the article
Abstract Uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity are a natural generalization of both uniformly convexnormed spaces and CAT(0) spaces. In this article, we discuss the existence of fixed points and demiclosed principle for mappings of asymptotically non-expansive type in uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity. We also obtain a Δ-convergence theorem of Krasnoselski-Mann iteration for continuous mappings of asymptotically nonexpansive type in CAT(0) spaces. MSC: 47H09; 47H10; 54E40 Keywords: Asymptotically nonexpansive type, Fixed points Δ-convergence, Uniformly convex W-hyperbolic spaces, CAT(0) spaces
1. Introduction In 1974, Kirk [1] introduced the mappings of asymptotically nonexpansive type and proved the existence of fixed points in uniformly convex Banach spaces. In 1993, Bruck et al [2] introduced the notion of mappings which are asymptotically nonexpansive in the intermediate sense (continuous mappings of asymptotically nonexpansive type) and obtained the weak convergence theorems of averaging iteration for mappings of asymptotically nonexpansive in the intermediate sense in uniformly convex Banach space with the Opial property. Since then many authors have studied on the existence and convergence theorems of fixed points for these two classes of mappings in Banach spaces, for example, Xu [3], Kaczor [4,5], Rhoades [6], etc. In this work, we consider to extend some results to uniformly convex W-hyperbolic spaces which are a natural generalization of both uniformly convex normed spaces and CAT(0) spaces. We prove the existence of fixed points and demiclosed principle for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity. In 1976, Lim [7] introduced a concept of convergence in a general metric space setting which he called “Δ-convergence.” In 2008, Kirk and Panyanak [8] specialized Lim’s concept to CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting. Since then the notion of Δ-convergence has been widely studied and a number of articles have appeared (e.g., [9-12]). © 2011 Zhang and Cui; licensee Springer. 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.
Zhang and Cui Fixed Point Theory and Applications 2011, 2011:39 http://www.fixedpointtheoryandapplications.com/content/2011/1/39
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