Strong convergence theorems for a countable family of nonexpansive mappings in convex metric spaces

• Main Authors: Withun Phuengrattana, Suthep Suantai
• Format: Journal
• Published: 2018
• Subjects: Mathematics
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=81755181005&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/50109
• Institution: Chiang Mai University

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings {Tn} in convex metric spaces. We prove that the sequence {xn} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when {Tn} satisfies the AKTT-condition, and strong convergence theorems of the proposed iteration to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0) spaces are established under AKTT-condition and the SZ-condition. We also generalize the concept of W-mapping for a countable infinite family of nonexpansive mappings from a Banach space setting to a convex metric space and give some properties concerning the common fixed point set of this family in convex metric spaces. Moreover, by using the concept of W-mappings, we give an example of a sequence of nonexpansive mappings defined on a convex metric space which satisfies the AKTT-condition. Our results generalize and refine many known results in the current literature. Copyright © 2011 Withun Phuengrattana and Suthep Suantai.