Abstract
In this paper, we first introduce a broad semigroup of mappings without continuity in Hilbert spaces which contains discrete semigroups generated by generalized hybrid mappings and semigroups of nonexpansive mappings. Then, using the theory of invariant means, we prove a weak convergence theorem of Mann's type iteration for the semigroups. Next, using Halpern's type iteration, we prove a strong convergence theorem for such semigroups. Using these results, we obtain new and well-known results for semigroups of mappings without continuity in Hilbert spaces.