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Based on an original idea, namely, a specific way of choosing the indexes of involved mappings, we propose a relaxed Halpern-type iterative algorithm for approximating some common fixed point of a kind of nonlinear mappings and obtain a strong convergence theorem under suitable conditions. Since the involved mappings need no assumption of being uniformly totally quasi-ϕ-asymptotically nonexpansive, and there is no need to compute projections onto intersections of countably many closed and convex sets, the results improve those of the authors with related interest.MSC:47H09, 47H10, 47J25.

Relaxed Halpern-type iteration method for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings