H. K. Pathak, D. O'Regan, M. S. Khan, R. P. Agarwal

## Asymptotic Behavior of Generalized Nonexpansive Sequences and Mean Points

Abstract:
Let $E$ be a real Banach space with norm $\|\cdot\|$ and let $\{x_n\}_{n\geq 0}$ be a generalized nonexpansive sequence in $E$ (i.e., $\ {\|x_{i+1}-x_{j+1}\|}^2\leq \|x_i-x_j\|^2+(\varepsilon(i+1,j+1)-\varepsilon(i,j))^2$ for all $i,j\geq 0$, where the series of nonnegative terms
$\sum\limits_{i,j}\varepsilon(i,j)$ is convergent). Let $K=\mathop{\bigcap}\limits_{n =1}^\infty \overline{\co} \left\{\left\{ x_i-x_{i - 1} \right\}_{i \geq n} \right\}.$ We deal with the mean point of $\{\frac{x_n}{n}\}$ concerning a Banach limit $\mu$. If $E$ is reflexive and $d = d(0,K)$, then we show that $d = d\left(0,\ \overline {\co} \left\{ \frac{x_n - x_0} {n} \right\} \right)$ and there exists a point $z_0$ with $\|z_0 \|=d$ such that $z_0\in \overline {\co} \{ \frac {x_n -x_0} {n}\}$. In the sequel, this result is applied to obtain the weak and strong convergence of $\{\frac {x_n}{n}\}.$

Keywords:
Asymptotic behavior, Banach limit, mean point, nonexpansive sequence, generalized nonexpansive sequence.

MSC 2000: 47H09