Approximating Fixed Point in CAT(0) Space by s-iteration Process for a Pair of Single Valued and Multivalued Mappings

Abstract

Suppose $K$ is a closed convex subset of a complete CAT(0) space $X$. $T$ is mapping from $K$ to $X$. $F(T)$ is set of fixed point of $T$ which is nonempty. Sequence $\{x_n\}$ is defined by an element $x_1 \in k$ such that
\begin{eqnarray*}
x_{n+1} &=& P((1-\alpha_n) Tx_n \oplus \alpha_n y_n)\\
y_n &=& P((1-\beta_n) x_n \oplus \beta_n T x_n) \ \ \ \forall \geq
1
\end{eqnarray*} where $P$ is the nearest point projection from $X$ onto $k$. $\{\alpha_n\}, \{\beta_n\}$ are real sequences in (0,1)
with the condition
\begin{eqnarray*}
\displaystyle \sum^{\infty}_{n=1}\alpha_n \beta_n (1-\beta_n)
=\infty
\end{eqnarray*} Then $\{x_n\}$ converges to some point $x^*$ in $F(T)$. This result is extension of the result of Abdul Rehman Razani and saeed Shabhani. [Approximating fixed points for nonself mappings in CAT(0) spaces Springer 2011:65]