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 ofnonexpansive 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 mappingwhen {Tn} satisfies the AKTT-condition, and strong convergence theorems of the proposed iterationto 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 conceptof W-mapping for a countable infinite family of nonexpansive mappings from a Banach spacesetting to a convex metric space and give some properties concerning the common fixed pointset of this family in convex metric spaces. Moreover, by using the concept of W-mappings, wegive an example of a sequence of nonexpansive mappings defined on a convex metric space whichsatisfies the AKTT-condition. Our results generalize and refine many known results in the currentliterature