Asymptotic behavior of nonexpansive sequences and mean points

Abstract:

Let $E$ be a real Banach space with norm $\Vert \cdot \Vert$ and let $\{x_n\}_{n \ge 0}$ be a nonexpansive sequence in $E$ (i.e., $\Vert x_{i + 1} - x_{j + 1}\Vert \le \Vert x_i - x_j\Vert$ for all $i, j \ge 0$). Let $K = \bigcap _{n = 1}^{\infty }\overline {co}\{\{x_i - x_{i - 1}\}_{i \ge n}\}$. We deal with the mean point of $\{\frac {x_n}{n}\}$ concerning a Banach limit. We show that if $E$ is reflexive and $d = d(0,K)$, then $d = d(0,\overline {co}\{\frac {x_n - x_0}{n}\})$ and there exists a unique point $z_0$ with $\Vert z_0\Vert = d$ such that $z_0 \in \overline {co}\{\frac {x_n - x_0}{n}\}$. This result is applied to obtain the weak and strong convergence of $\{\frac {x_n}{n}\}$.