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)(infinity) defined by x(k+1) = (1 - t(mk))x(k) circle plus t(mk)T(m)(nk)y(m-1)k, y(m-1)k = (1 - t((m-1)k))x(k) circle plus t((m-1)k)T(m-1)(2nk)y(m-2)k, y(m-2)k = (1 - t((m-2)k))x(k) circle plus t((m-2)k)T(m-2)(nk) y(m-3)k, ..., y2k = (1 - t(2k))x(k) circle plus t(2k)T(2)(nk)y1k, y1k = (1 - t(1k))x(k) circle plus t(1k)T(1)(nk)y0k, y0k = x(k), k is an element of N, converges to a common fixed point of T-1, T-2, ..., T-m where they are asymptotic pointwise nonexpansive mappings on C, {t(ik)}(k=1)(infinity) are sequences in [0, 1] for all i = 1, 2,..., m, and {n(k)} is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.