An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces

Let C be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E. Let T: C→ C be relatively nonexpansive mapping and let A_i: C→ E* be L_i-Lipschitz monotone mappings, for i = 1,2. In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a relatively nonexpansive mapping and the solution set of variational inequality problems for A₁ and A₂. Under some mild assumptions, we show that the proposed algorithm converges strongly to a point in F(T) ∩ VI(C, A₁) ∩ VI(C, A₂). Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.