Amenability and Fixed Point Properties of Semitopological Semigroups in Modular Vector Spaces

In this paper, we initiate the study of fixed point properties of amenable semitopological semigroups in modular spaces. Takahashi's fixed point theorem for amenable semigroups of nonexpansive mappings, and T. Mitchell's fixed point theorem for reversible semigroups of nonexpansive mappings in Banach spaces are extended to the setting of modular spaces. Among other things, we also generalize another classical result due to Mitchell characterizing the left amenability property of the space of left uniformly continuous functions on semitopological semigroups by introducing the notion of a semi-modular space as a generalization of the concept of a locally convex space.