Weak and Strong Convergence Theorems for Semigroups of Not Necessarily Continuous Mappings (Nonlinear Analysis and Convex Analysis)

In this article, using the concept of strongly asymptotically invariant nets, we first introduce a broad semigroup of not necessarily continuous mappings in a Hilbert space.Fur- thermore, we consider such a semigroup in a Banach space which contains discrete semigroups generated by generalized nonspreading mappings [22] and semigroups of $\phi-$ -nonexpansive mappings [40].Then we prove weak convergence theorems of Mann's type iteration and strong convergence theorems of Halpern's type iteration for the semigroups of mappings in a Hilbert space.Furthermore, we obtain a weak convergence theorem of Mann's type iteration in a Ba- nach space.Using these results, we obtain well-known and new theorems which are connected with weak and strong convergence theorems in a Hilbert space and a Banach space.