Asymptotic behavior of almost-orbits of nonlinear semigroups of non-Lipschitzian mappings in Hilbert spaces
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- by Kok-Keong Tan and Hong Kun Xu PDF
- Proc. Amer. Math. Soc. 117 (1993), 385-393 Request permission
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 \[ {\text {weak-}}\lim \frac {1} {t}\int _0^t {u(r + h)dr = y\quad {\text {uniformly for }}h \geqslant 0.} \] 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 $h \geqslant 0$.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 385-393
- MSC: Primary 47H20; Secondary 47A35, 47H10
- DOI: https://doi.org/10.1090/S0002-9939-1993-1111223-7
- MathSciNet review: 1111223