Generalized Partially Relaxed Pseudomonotone Variational Inequalities and General Auxiliary Problem Principle

Let T : K → H be a nonlinear mapping from a nonempty closed invex subset K of an infinite-dimensional Hilbert space H into H . Let f : K → R be proper, invex, and lower semicontinuous on K and let h : K → R be continuously Fréchet-differentiable on K with h′, the gradient of h, (η,α)-strongly monotone, and (η,β)-Lipschitz continuous on K . Suppose that there exist an x∗ ∈ K , and numbers a > 0, r ≥ 0, ρ(a < ρ < α) such that for all t ∈ [0,1] and for all x ∈ K∗, the set S∗ defined by S∗ = {(h,η) : h′(x∗ + t(x − x∗))(x − x∗) ≥ 〈h′(x∗ + tη(x,x∗)),η(x,x∗)〉} is nonempty, where K∗ = {x ∈ K : ‖x− x∗‖ ≤ r} and η : K ×K →H is (λ)-Lipschitz continuous with the following assumptions. (i) η(x, y) + η(y,x) = 0, η(x, y) = η(x,z) + η(z, y), and ‖η(x, y)‖ ≤ r. (ii) For each fixed y ∈ K , map x → η(y,x) is sequentially continuous from the weak topology to the weak topology. If, in addition, h′ is continuous from H equipped with weak topology to H equipped with strong topology, then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x∗ of the variational inequality problem (VIP): 〈T(x∗),η(x,x∗)〉+ f (x)− f (x∗)≥ 0 for all x ∈ K .