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

Abstract

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=1defined by xk+1= (1 - tmk)xk⊕ tmkTnkmy(m-1)k, y(m-1)k= (1 - t(m-1)k)xk⊕ t(m-1)kTnkm-1y(m-2)k, y(m-2)k= (1 - t(m-2)k)xk⊕ t(m-2)kTnkm-2y(m-3)k,⋯, y2k = (1 - t2k)xk⊕ t2kTnk2y1k, y1k = (1 - t1k)xk⊕ t1kTnk1y0k, y0k = xk, k ∈ ℕ, converges to a common fixed point of T1, T2,⋯, Tmwhere they are asymptotic pointwise nonexpansive mappings on C, {tik}∞k=1are 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. Copyright © 2012 P. Pasom and B. Panyanak.