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 assump-tions. (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 aux-iliary 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. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1