Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces
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Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces Buthinah A Bin Dehaish* *
Correspondence: [email protected] Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, 21593, Saudi Arabia Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA
Abstract Let (M, d) be a complete 2-uniformly convex metric space, C be a nonempty, bounded, closed and convex subset of M, and T be an asymptotic pointwise nonexpansive self mapping on C. In this paper, we define the modified Ishikawa iteration process in M, i.e., xn+1 = tn T n (sn T n (xn ) ⊕ (1 – sn )(xn )) ⊕ (1 – tn )xn and we investigate when the Ishikawa iteration process converges weakly to a fixed point of T. MSC: Primary 06F30; 46B20; 47E10 Keywords: asymptotically nonexpansive mapping; asymptotic pointwise nonexpansive mapping; fixed point; inequality of Bruhat and Tits; Ishikawa iteration process; uniformly convex metric space; uniformly Lipschitzian mapping
1 Introduction The class of asymptotic nonexpansive mapping have been extensively studied in fixed point theory since the publication of the fundamental paper []. Kirk and Xu [] studied the asymptotic nonexpansive mapping in uniformly convex Banach spaces. Their result has been generalized by Hussain and Khamsi [] to metric spaces. Khamsi and Kozlowski [] extended their result to modular function spaces. In almost all papers, authors do not describe any algorithm for constructing a fixed point for the asymptotic nonexpansive mapping. Ishikawa [] and Mann [] iterations are two of the most popular methods to check that these two iterations were originally developed to provide ways of computing fixed points for which repeated function iteration failed to converge. Espinola et al. [] examined the convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. Kozlowski [] proved convergence to a fixed point of some iterative algorithms applied to asymptotic pointwise mappings in Banach spaces. In [], the authors discussed the convergence of these iterations in modular function spaces. In a recent paper [], the authors investigate the existence of a fixed point of asymptotic pointwise nonexpansive mappings and study the convergence of the modified Mann iteration in hyperbolic metric spaces. It is well known that the iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical meth© 2013 Bin Dehaish; 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.
Bin Dehaish Fixed Point Theory and Applications 2013, 2013:98 http://www.fixedpointtheoryandapplications.com/content/2013/1/98
ods for solving various nonlinear equations and variational probl
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