Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces

Abstract:

Let $X$ be a real Banach space, ${({x_n})_{n \geqslant 0}}$ a nonexpansive sequence in $X$ (i.e., $||{x_{i + 1}} - {x_{j + 1}}|| \leqslant ||{x_i} - {x_j}||$ for all $i,\;j \geqslant 0$), and $C$ the closed convex hull of the sequence ${({x_{n + 1}} - {x_n})_{n \geqslant 0}}$. We prove that ${\lim _{n \to + \infty }}||{x_n}/n|| = {\inf _{n \geqslant 1}}||({x_n} - {x_0})/n|| = {\inf _{z \in C}}||z||$ and deduce a simple short proof for the following result, (i) If $X$ is reflexive and strictly convex, then ${x_n}/n$ converges weakly in $X$ to the element of minimum norm ${P_C}0$ in $C$ with \[ ||{P_C}0|| = \inf \limits _{n \geqslant 1} \left \| {\frac {{{x_n} - {x_0}}} {n}} \right \| = \lim \limits _{n \to + \infty } \left \| {\frac {{{x_n}}} {n}} \right \|.\] (ii) If ${X^{\ast }}$ has Fréchet differentiable norm, then ${x_n}/n$ converges strongly to ${P_C}0$. This result contains previous results by Pazy, Kohlberg and Neyman, Plant and Reich, and Reich and is also optimal since the assumptions made on $X$ in (i) or (ii) are also necessary for the respective conclusion to hold.