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In this paper, a new class of generalized convex function is introduced, which is called the strongly α-preinvex function. We study some properties of strongly α-preinvex function. In particular, we establish the equivalence among the strongly α-preinvex functions, strongly α-invex functions and strongly αη-monotonicity under some suitable conditions. As special cases, one can obtain several new and previously known results for α-preinvex (invex) functions.

It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, the auxiliary principle and the penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.

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