Asymptotic behavior of almost-orbits of reversible semigroups of non-Lipschitzian mappings in Banach spaces.(English)Zbl 0891.47047

Summary: Let $$C$$ be a nonempty closed convex subset of a uniformly convex Banach space $$E$$ with a Fréchet differentiable norm, $$G$$ a right reversible semitopological semigroup, and $${\mathcal S}= \{S(t): t\in G\}$$ a continuous representation of $$G$$ as mappings of asymptotically nonexpansive type of $$C$$ into itself. The weak convergence of an almost-orbit $$\{u(t): t\in G\}$$ of $${\mathcal S}$$ on $$C$$ is established. Furthermore, it is shown that if $$P$$ is the metric projection of $$E$$ onto the set $$F({\mathcal S})$$ of all common fixed points of $${\mathcal S}$$, then the strong limit of the net $$\{Pu(t): t\in G\}$$ exists.

MSC:

 47H20 Semigroups of nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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