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Ergodic retraction theorem and weak convergence theorem for reversible semigroups of non-Lipschitzian mappings. (English) Zbl 1111.47052

Let \(G\) be a semitopological right reversible semigroup and \(C\) be a closed convex subset of a uniformly convex Banach space \(E\) with a Fréchet differentiable norm. Given the continuous representation \(T=\{T_t\); \(t\in G\}\) of \(G\) as nearly asymptotically nonexpansive selfmaps of \(C\) with \(F(T)\neq \emptyset\), the following assertions are valid: (i) for each almost orbit \(u(.)\) of \(T\), \(\bigcap _{s\in G}\overline{\text{co}} \{u(t)\); \(t\geq s\}\cap F(T)\) is at most a singleton, (ii) \(\bigcap _{s\in G} \overline{\text{co}} \{T_t x\); \(t\geq s\}\cap F(T) \neq \emptyset\) for each \(x\in C\) iff there exists a nonlinear ergodic retraction \(P\) of \(C\) onto \(F(T)\) such that \(PT_s=T_sP=P\) for all \(s\in G\) and \(Px\in \overline{\text{co}} \{T_s x\); \(s\in G\}\) for each \(x\in C\). Further applications of these facts are given to the weak convergence of \(\{u(t)\); \(t\in G\}\) towards some point of \(F(T)\).

MSC:

47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
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