Extension of Some Common Fixed Point Theorems to Mean Non-expansive Mappings

In 1969, DeMarr proved that a commuting family of non-expansive mappings on a nonempty convex compact subset of a Banach space has a fixed point. Takahashi extended DeMarr’s theorem to the case of discrete amenable semi-groups. In recent years, considerable research has been devoted to the theory of fixed points as well as common fixed points. In the case of semi-topological semi-groups (that is, a semi-group with a Hausdorff topology such that the multiplication is separately continuous), Lau and Zhang studied DeMarr’s theorem under more general conditions for example in the case of the amenability of the space of almost periodic functions as well as the space of weakly almost periodic functions. In this paper, we study several fixed point properties of the mean non-expansive semi-topological semi-groups acting on nonempty convex weakly compact subsets of a locally convex space as well as give extensions of the results of Lau and Zhang.