ITERATIVE APPROXIMATIONS FOR GENERALIZED NONEXPANSIVE MAPPINGS USING K ITERATION PROCESS IN BANACH SPACES

Abstract

Let $H$ be a nonempty subset of a Banach space $X$. A mapping

$T:H\rightarrow H$ is said to be generalized $\alpha$-nonexpansive if there is a real

number $\alpha\in[0,1)$ such that for all $x,y\in H$, we have

\begin{eqnarray*}

\frac{1}{2}||x-Tx||\leq||x-y||

\end{eqnarray*}

\begin{eqnarray*}

||Tx-Ty||\leq\alpha||Tx-Ty||+\alpha||Ty-x||+(1-2\alpha)||x-y||.

\end{eqnarray*}

In this paper, we obtain some weak and strong convergence theorems

for such mappings using K-iteration process in uniformly convex Banach space setting. Our results extend and improve many results in the literature.