Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces

In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We introduce and consider three-step Mann iterations for finding a common solution of a general system of variational inequalities (GSVI) and a fixed point problem (FPP) of an infinite family of nonexpansive mappings in X. Here three-step Mann iterations are based on Korpelevich's extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for three-step Mann iterations involving the GSVI and the FPP in a Hilbert space. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature.