Iteration by Cesaro means for quasi-contractive mappings

Abstract

Let $C$ be a nonempty closed convex subset of a Banach space $E$ and  $T$ be a quasi-contractive mapping on $C$. We prove, the sequence $\{x_{n}\}$, iteratively defined by, \[ \left \{ \begin{array}{l} x_1 \in C \\ y_{n}=s_{n}x_{n}+(1-s_{n})T^{n}x_{n}\\ x_{n+1}=t_{n}x_{n}+(1-t_{n})\frac{1}{n+1}\sum_{j=0}^{n}T^{j}y_{n}, \end{array} \right.  \] is weakly convergent to a point of $F(T)$. Moreover, by a numerical example (using Matlab software), the main result and the rate of convergence are illustrated.