Common fixed points for asymptotic pointwise nonexpansive mappings in metric and banach spaces

Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on C is nonempty closed and convex. We also show that, under some suitable conditions, the sequence {x _k} ^∞ _k=1 defined by x _k+1 = (1 - t _mk)x _k ⊕ t _mkT ^nk _my _(m-1)k, y _(m-1)k = (1 - t _(m-1)k)x _k ⊕ t _(m-1)kT ^nk _m-1y _(m-2)k, y _(m-2)k = (1 - t _(m-2)k)x _k ⊕ t _(m-2)kT ^nk _m-2y _(m-3)k,⋯, y2k = (1 - t _2k)x _k ⊕ t _2kT ^nk ₂y1k, y1k = (1 - t _1k)x _k ⊕ t _1kT ^nk ₁ y0k, y0k = x _k, k ∈ ℕ, converges to a common fixed point of T ₁, T ₂,⋯, T _m where they are asymptotic pointwise nonexpansive mappings on C, {tik} ^∞ _k=1 are sequences in [0, 1] for all I = 1, 2,⋯,m, and {nk} is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.