Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Abstract

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mapping $T$ of a closed convex subset $C$ of a CAT(0) space $X$. Suppose that the set Fix$\left(T\right)$ of fixed points of $T$ is nonempty. For a contraction $f$ on $C$ and $t\in \left(\mathrm{0,1}\right)$, let $x_t\in C$ be the unique fixed point of the contraction $x\mapsto tf\left(x\right)\oplus (1-t)Tx$. We will show that if $X$ is a CAT(0) space satisfying some property, then $\left\{x_t\right\}$ converge strongly to a fixed point of $T$ which solves some variational inequality. Consider also the iteration process $\left\{x_n\right\}$, where $x_0\in C$ is arbitrary and $x_{n+1}={\alpha }_{n}f\left(x_n\right)\oplus (1-{\alpha }_n)T{x}_n$ for $n\ge 1$, where $\left\{{\alpha }_n\right\}\subset \left(\mathrm{0,1}\right)$. It is shown that under certain appropriate conditions on ${\alpha }_n,\left\{x_n\right\}$ converge strongly to a fixed point of $T$ which solves some variational inequality.