Nonlinear ergodic theorems for general curves defined on general semigroups

Abstract

Let G be a (algebra) semigroup and let H be a real Hilbert space. Let F( G) be a set of all mappings from G into H. In this paper, we first try to provide a nonlinear ergodic retraction theorem for the mappings from G to H. Moreover, we introduce the notion of asymptotically nonexpansive-type curves and give the ergodic convergence theorems for this kind of curves. Using the main results, we can also prove the ergodic convergence theorem for the almost-orbits of semitopological semigroups of non-Lipschitzian mappings without convexity.