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 $\{x_k{\}}_{k=1}^{\infty }$ defined by $x_{k+1}=(1-t_{mk})x_k\oplus t_{mk}{T}_m^{n_k}{y}_{(m-1)k},\hspace{0.17em}{y}_{(m-1)k}=(1-t_{(m-1)k})x_k\oplus t_{(m-1)k}{T}_{m-1}^{n_k}{y}_{(m-2)k},y_{(m-2)k}=(1-t_{(m-2)k})x_k\oplus t_{(m-2)k}{T}_{m-2}^{n_k}{y}_{(m-3)k},\dots ,y_{2k}=(1-t_{2k})x_k\oplus t_{2k}{T}_2^{n_k}{y}_{1k},y_{1k}=(1-t_{1k})x_k\oplus t_{1k}{T}_1^{n_k}{y}_{0k},y_{0k}=x_k,k\in \mathbb{N}$, converges to a common fixed point of $T_1,T_2,\dots ,T_m$ where they are asymptotic pointwise nonexpansive mappings on $C$, $\{t_{ik}{\}}_{k=1}^{\infty }$ are sequences in $\left[\mathrm{0,1}\right]$ for all $i=\mathrm{1,2},\dots ,\mathrm{m,}$ and $\left\{n_k\right\}$ is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.