Ergodic theorems for non-Lipschitzian semigroups without convexity
Let \(G\) be a semitopological semigroup. Let \(C\) be a nonempty subset of a Hilbert space and \({\mathfrak I}= \{T_t:t\in G\}\) be a representation of \(G\) as asymptotically nonexpansive type mappings of \(C\) into itself such that the common fixed point set \(F({\mathfrak I})\) of \({\mathfrak I}\) in \(C\) is nonempty. It is proved that \(\bigcap_{s\in G}\overline{\text{co}}\{T_{ts}x: t\in G\}\cap F({\mathfrak I})\) is nonempty for each \(x\in C\) if and only if there exists a nonexpansive retraction \(P\) of \(C\) onto \(F({\mathfrak I})\) such that \(PT_s= T_sP= P\) for all \(s\in G\) and \(P(x)\) is in the closed convex hull of \(\{T_sx: s\in G\}\), \(x\in C\).
asymptotically nonexpansive type mappings
scientific article; zbMATH DE number 1159360