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| ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUTCONVEXITY |
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Citation： |
Li Gang,Ma Jipu.ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUTCONVEXITY[J].Chinese Annals of Mathematics B,1998,19(2):209~216 |
| Page view： 1011
Net amount： 664 |
Authors： |
Li Gang; Ma Jipu |
Foundation： |
Project supported by the National Natural Science
Foundation of China and the Science Foundation of Jiansu Province. |
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| Abstract： |
Let $G$ be a semitopological semigroup.
Let $C$ be a nonempty subset of a Hilbert space and
$ \Im =\{T_{t}:t\in G \} $ be a representation of $G$ as
asymptotically nonexpansive type mappings of $C$ into itself such that
the common fixed
point set $ F(\Im)$ of $\Im$ in $C$ is nonempty. It is proved that
$ \bigcap \limits_{s\in G} \overline{\text{co}} \{T_{ts}x:t\in G\}\bigcap F(\Im)$ is
nonempty for each $ x \in C $ if and only if there
exists a nonexpansive retraction $P$ of $C$ onto $F(\Im)$ such that $ PT_{s}=T_{s}P
=P $ for all $ s\in G $ and $P(x)$ is in the closed convex hull
of $ \{T_{s}x: s\in G \} $, $ x\in C $. This result shows that many key conditions
in [1--4, 9, 12--15 ] are not necessary. |
Keywords： |
Nonlinear ergodic theorem, Non-lipschitzian
mappings, Semigroup mappings, Semigroup |
Classification： |
47H09, 47H10 |
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