Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces

The purpose of this paper is to study the strong convergence theorems of the moudafi's viscosity approximation methods for a nonexpansive mapping $T$ in CAT(0) spaces without the property $\mathcal{P}$. For a contraction $f$ on $C$ and $t \in (0,1)$, let $x_t\in C$ be the unique fixed point of the contraction $x \mapsto tf(x) \oplus (1-t)Tx$; i.e. \begin{equation*} x_t = tf(x_t) \oplus (1-t)Tx_t, \end{equation*} and \begin{eqnarray*} x_{n+1} = \alpha_n f(x_n) \oplus (1-\alpha_n)T x_n,\ \ n\geq0, \end{eqnarray*} where $x_0\in C$ is arbitrary chosen and $\{\alpha_n\}\subset (0,1)$ satisfies certain conditions. We prove the iterative schemes $\{x_t\}$ and $\{x_n\}$ converge strongly to the same point $\tilde{x}$ such that $\tilde{x}=P_{F(T)}f(\tilde{x})$ which is the unique solution of the variational inequality (VIP) : \begin{eqnarray*} \langle\overrightarrow{\tilde{x}f\tilde{x}},\overrightarrow{x\tilde{x}}\rangle \geq0,\ \ \ x\in F(T). \end{eqnarray*} By using the concept of quasilinearization, we remark that the proof given below is different from that of Shi and Chen \cite{Shi-Chen}. In fact, Strong convergence theorems for two given iterative schemes are established in CAT(0) spaces without the property $\mathcal{P}$.