Strong convergence theorems for a countable family of nonexpansive mappings in convex metric spaces (Q642739)

Summary: We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings \(\{T_n\}\) in convex metric spaces. We prove that the sequence \(\{x_n\}\) generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when \(\{T_n\}\) 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.