Proceedings of the American Mathematical Society

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Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces


Author: Behzad Djafari Rouhani
Journal: Proc. Amer. Math. Soc. 117 (1993), 951-956
MSC: Primary 47H10; Secondary 46B15
MathSciNet review: 1120510
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Abstract: Let $ X$ be a real Banach space, $ {({x_n})_{n \geqslant 0}}$ a nonexpansive sequence in $ X$ (i.e., $ \vert\vert{x_{i + 1}} - {x_{j + 1}}\vert\vert \leqslant \vert\vert{x_i} - {x_j}\vert\vert$ 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 }}\vert\vert{x_n}/n\vert\vert = {\inf _{n \geqslant 1}}\vert\vert({x_n} - {x_0})/n\vert\vert = {\inf _{z \in C}}\vert\vert z\vert\vert$ 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

$\displaystyle \vert\vert{P_C}0\vert\vert = \mathop {\inf }\limits_{n \geqslant ... ...p {\lim }\limits_{n \to + \infty } \left\Vert {\frac{{{x_n}}} {n}} \right\Vert.$

(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.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1120510-8
Article copyright: © Copyright 1993 American Mathematical Society