Nonlinear ergodic theorems for a semitopological semigroup ofnon-Lipschitzian mappings without convexity

摘要

LetGbe a semitopological semigroup,Ca nonempty subset of areal Hilbert spaceH, and?={Tt:t∈G}a representation ofGas asymptotically nonexpansive type mappings ofCinto itself. LetL(x)={z∈H:infs∈Gsupt∈G‖Tts?x?z‖=inft∈G‖Tt?x?z‖}for eachx∈CandL(?)=∩x∈C?L(x). In this paper, we prove that∩s∈Gconvˉ{Tts?x:t∈G}∩L(?)is nonempty for eachx∈Cif and only if there exists a unique nonexpansive retractionPofCintoL(?)such thatPTs=Pfor alls∈GandP(x)∈convˉ{Ts?x:s∈G}for everyx∈C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.