Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings { 𠑇 𠑛 } in convex metric spaces. We prove that the sequence { 𠑥 𠑛 } generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when { 𠑇 𠑛 } 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.