Viscosity S-iteration algorithm for finding common fixed point of nonexpansive mappings and application to nonexpansive semigroups

. We study the problem of finding the common element of the set of fixed points of two nonexpansive mappings in Hilbert spaces. Some previous attempts in this direction make some restrictive assumptions which may be difficult to check in practice on the generated sequence. In this paper, we introduce a viscosity S-iteration scheme for finding the common fixed point of two nonexpansive mappings. Under some very mild conditions, we obtain a strong convergence theorem for the sequence generated by our algorithm. We apply our main result to approximating common fixed points of nonexpansive semigroups. We also provide numerical examples to support our main results and illustrate the efficiency and effectiveness of our algorithm by comparing with some existing algorithms in literature. This work generalizes and improves some existing works in the literature in this direction.