In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of these methods either requires only pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier known results for solving variational inequalities involving the convex sets.

In this paper, we introduce a new class of variational inclusions involving three operator. Using the resolvent operator technique, we establish the equivalence between the general variational inclusions and the resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general variational inclusions. We also consider the criteria of these iterative methods under suitable conditions. Since the general variational inclusions include the variational inequalities and the related optimization problems as special cases, our results continue to hold for these problems.

In this paper, we introduce and consider some new classes of variational inequalities and the Wiener-Hopf equations. Using the projection technique, we establish the equivalence between the general nonconvex variational inequalities and the fixed point problems as well as the Wiener-Hopf equations. This alternative equivalent formulation is used to study the existence of a solution of the general convex variational inequalities. This equivalence is used to suggest and analyzed several projection iterative methods for solving the general nonconvex variational inequalities. Convergence criteria of these new iterative is also discussed under suitable conditions. Our method of proofs is very simple as compared with other techniques.