Abstract
In this paper, we first introduce a broad semigroup of not necessarily continuous mappings in a Banach space which contains discrete semigroups generated by generalized nonspreading mappings and semigroups of phi-nonexpansive mappings. Then we establish a weak convergence theorem of Mann's type iteration for the semigroups of mappings in a Banach space. Using the result, we obtain well-known and new theorems which are connected with weak convergence results in Banach spaces.