Proximal point algorithms and generalized nonlinear variational problems

Let T : K → H be a continuous (η) – pseudomonotone and (η, L) – relaxed monotone mapping from a nonempty closed invex subset K of a finite-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. Let x∗ ∈ K be any fixed solution of the variational inequality problem (VIP): find an element x∗ ∈ K such that〈T(x∗),η(x,x∗)〉+f(x)-f(x∗)⩾0for allx∈K.Then the sequence {xk} generated by the proximal point algorithm converges to x∗.