Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H . Let F :H →H be an operator such that for some constants k,η > 0, F is k-Lipschitzian and η-strongly monotone on C; that is, F satisfies the following inequalities: ‖Fx− Fy‖ ≤ k‖x− y‖ and 〈Fx− Fy,x− y〉 ≥ η‖x− y‖2 for all x, y ∈ C, respectively. Recall that T is nonexpansive if ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈H . We consider the following variational inequality problem: find a point u∗ ∈ C such that