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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Asymptotic behavior of almost-orbits of nonlinear semigroups of non-Lipschitzian mappings in Hilbert spaces


Authors: Kok-Keong Tan and Hong Kun Xu
Journal: Proc. Amer. Math. Soc. 117 (1993), 385-393
MSC: Primary 47H20; Secondary 47A35, 47H10
MathSciNet review: 1111223
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Abstract: Let $ C$ be a nonempty closed convex subset of a Hilbert space $ H$, $ \mathcal{F} = \{ T(t):t \geqslant 0\} $ be a continuous nonlinear asymptotically nonexpansive semigroup acting on $ C$ with a nonempty fixed point set $ F(\mathcal{F})$, and $ u:[0,\infty ) \to C$ be an almost-orbit of $ \mathcal{F}$. Then $ \{ u(t)\} $ almost converges weakly to a fixed point of $ \mathcal{F}$, i.e., there exists an element $ y$ in $ F(\mathcal{F})$ such that

$\displaystyle {\text{weak-}}\lim \frac{1} {t}\int_0^t {u(r + h)dr = y\quad {\text{uniformly for }}h \geqslant 0.} $

This implies that $ \{ u(t)\} $ converges weakly to a fixed point of $ \mathcal{F}$ if and only if $ \{ u(t + h) - u(t)\} $ converges weakly to zero as $ t$ tends to infinity for each $ h \geqslant 0$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1111223-7
PII: S 0002-9939(1993)1111223-7
Keywords: Almost-orbit, asymptotically nonexpansive semigroup, weakly asymptotically regular, nonexpansive retraction, fixed point, metric projection, asymptotic center
Article copyright: © Copyright 1993 American Mathematical Society