A general framework for the solvability of a class of nonlinear variational inequalities

Based on a general framework for the auxiliary problem principle involving continuously m-Frechet-differentiable (m greater than or equal to 2) mappings, the approximation-solvability of the following class of nonlinear variational inequality problems (NVIP) involving the generalized partially relaxed monotone mappings is presented. Find an element x* is an element of K such that <T(x*), eta(x, x*)> + f(x) - f(x*) greater than or equal to 0 for all x is an element of K, where T : K --> R-n is a mapping from a nonempty closed invex subset K of R-n into R-n, eta : K x K --> R-n is a mapping, and f : K --> R is a continuous invex function on K. The general class of the auxiliary problems principle is described as follows: for a given iterate x(k) is an element of K and for a parameter rho > 0, determine x(k+1) such that <rhoT(x(k)) + h'(x(k+ 1)) - h'(x(k)), eta(x, x(k+1))> + rho[f(x) - f(x(k+1))] greater than or equal to 0 for all x is an element of K, where h : K --> R is continuously Frechet-differentiable on K.