△ -Convergence Problems for Asymptotically Nonexpansive Mappings in CAT(0) Spaces

Abstract

New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping $T$ of a closed convex subset $C$ of a CAT(0) space $X$. Consider the iteration process $\{x_n\}$, where $x_0\in C$ is arbitrary and $x_{n+1}={\alpha }_n{x}_n\oplus (1-{\alpha }_n)T^n{y}_n$ or $x_{n+1}={\alpha }_n{T}^n{x}_n\oplus (1-{\alpha }_n)y_n,y_n={\beta }_n{x}_n\oplus (1-{\beta }_n)T^n{x}_n$ for $n\ge 1$, where $\{{\alpha }_n\},\{{\beta }_n\}\subset (\mathrm{0,1})$. It is shown that under certain appropriate conditions on ${\alpha }_n,{\beta }_n,\{x_n\}$ △-converges to a fixed point of $T$.