A weak form of amenability of topological semigroups and its applications in ergodic and fixed point theories

In this paper, we introduce a weak form of amenability on topological semigroups that call $$\varphi $$ -amenability, where is character semigroup. Some basic properties new notion are obtained and by giving some examples, show definition weaker than the semigroups. As noticeable result, for semigroup S, it shown if S -amenable, then amenable. Moreover, -ergodicity introduced proved under conditions Banach space X, -amenability any antirepresntation defined right action equivalent. A relation between existence common fixed point investigated relation, Hahn-Banach property in sense studied.