In this paper, we initiate the study of fixed point properties of amenable or reversible 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.

在本文中，我们开始研究模块化空间中可适应或可逆半拓扑半群的不动点性质。Takahashi的适用于非扩张映射的半群的不动点定理，以及T. Mitchell的适用于Banach空间中的非扩张映射的可逆半群的不动点定理，扩展到了模块化空间的设置。除其他事项外，由于米切尔通过引入半模空间的概念作为局部凸空间的概念的概括，因此米歇尔表征了半拓扑半群上左均匀连续函数空间的左可适应性，因此我们也推广了另一种经典结果。