Amenability and nonlinear flows in dual Banach spaces

Abstract

This paper is devoted to the study of nonlinear flows with weak* compact convex phase spaces sitting in conjugate Banach spaces. Nonlinear fixed point properties for the classes of amenable affine and convex semitopological semigroups are established; giving partial answers to question 1 in Lau and Zhang (J Funct Anal 263:2949–2977, 2012); we also prove related results for amenable semitopological semigroups. Furthermore, by almost periodicity techniques, we derive a non-commutative version of Bruck’s result (Pac J Math 53:59–70, 1974) on the existence of a nonexpansive retract; we also provide a characterization of left amenability property for the space of almost periodic functions for semitopological semigroups, and derive a fixed point property in $$\ell ^1$$ .