Two-step projection methods for a system of variational inequality problems in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let I (C) be a sunny nonexpansive retraction from E onto C. Let the mappings be gamma (1)-strongly accretive, mu (1)-Lipschitz continuous and gamma (2)-strongly accretive, mu (2)-Lipschitz continuous, respectively. For arbitrarily chosen initial point , compute the sequences {x (k) } and {y (k) } such that where {alpha (k) } is a sequence in [0,1] and rho, eta are two positive constants. Under some mild conditions, we prove that the sequences {x (k) } and {y (k) } converge to x* and y*, respectively, where (x*, y*) is a solution of the following system of variational inequality problems in Banach spaces: Our results extend the main results in Verma (Appl Math Lett 18:1286-1292, 2005) from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.