Strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in Banach spaces

• Main Authors: Klin-Eam C., Suantai S.
• Format: Article
• Language: English
• Published: 2014
• Online Access: http://www.scopus.com/inward/record.url?eid=2-s2.0-77955421790&partnerID=40&md5=27b89f2562754a5c873a69b02a5f3fff; http://cmuir.cmu.ac.th/handle/6653943832/6241
• Institution: Chiang Mai University

In this paper we propose a new modified viscosity approximation method for approximating common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for a countable family nonexpansive mappings in a reflexive Banach space with uniformly Gateaux differentiable norm under some control conditions. These results improve and extend the results of Jong Soo Jung [J.S. Jung, Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces, Fixed Point Theory and Appl. 2008 (2008) 14 pp., Article ID 167535]. Further, we apply our result to the problem of finding a zero of an accretive operator and extend the results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 5160], Ceng, et al. [L.-C. Ceng, A.R. Khan, Q.H. Ansari, J.-C, Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach space, Nonlinear Anal. 70 (2009)1830-1840] and Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation methods for accretive operator in Banach space, Nonlinear Anal. 69 (2008) 1356-1363]. © 2010 Elsevier Ltd. All rights reserved.