Taking as starting point the following viscosity implicit Mann-type iteration process
introduced by T. Suzuki [Proc. Am. Math. Soc. 131, 2133–2136 (2002; Zbl 1031.47038)] for a nonexpansive semigroup , the present authors consider two viscosity iteration processes, the first one an implicit iteration process
while the second one is an explicit iteration process
where is a contraction mapping. The main results of the paper are convergence theorems for these iterative processes toward a common fixed point of the semigroup which is also the unique solution of a certain variational inequality.