Construction of a common solution of a finite family of variational inequality problems for monotone mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let A_i: C → H, for i = 1, 2; be two L_i-Lipschitz monotone mappings and let f: C → C be a contraction mapping. It is our purpose in this paper to introduce an iterative process for finding a point in V I(C,A₁) ∩ V I(C, A₂) under appropriate conditions. As a consequence, we obtain a convergence theorem for approximating a common solution of a finite family of variational inequality problems for Lipschitz monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.