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 {xk}k 1 defined by xk 1 1 − tmk xk ⊕ tmkT nk m y m−1 k, y m−1 k 1 − t m−1 k xk ⊕ t m−1 kT m−1y m−2 k, y m−2 k 1 − t m−2 k xk ⊕ t m−2 kT nk m−2y m−3 k, . . . , y2k 1 − t2k xk ⊕ t2kT nk 2 y1k, y1k 1 − t1k xk ⊕ t1kT nk 1 y0k, y0k xk, k ∈ N, converges to a common fixed point of T1, T2, . . . , Tm 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.