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

Source 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 realnumber $\alpha\in[0,1)$ 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 theoremsfor such mappings using K-iteration process in uniformly convex Banach space setting. Our results extend and improve many results in the literature.