Common fixed point theorems for semigroup actions of Kannan’s type on strictly convex Banach spaces

Abstract

Let K be a weakly compact subset of a strictly convex Banach space X. Let S be a semitopological semigroup which acts on K so that the action is weakly separately continuous of Kannan mappings with some additional conditions such that for every $$f \in C(K)$$ , the functions $$s \in S \longrightarrow f_x(s) = f(s.x)$$ belongs to Z a closed linear subspace of $$l^{\infty } (S)$$ containing constants and invariant under translations for every $$x \in K$$ . We prove that if Z has a left invariant mean then K has a common fixed point for S.