Some characterizations and applications on strongly $\alpha$-preinvex and strongly $\alpha$-invex functions

In this paper, a new class of generalized convex functions is introduced, which is called the strongly quasi $\alpha$-preinvex functions. Some properties of strongly $\alpha$-preinvex functions are studied. We establish the relationships among the strongly quasi $\alpha$-preinvex functions, strongly quasi $\alpha$-invex functions and strongly quasi $\alpha\eta$-monotonicity under some suitable conditions. As applications, a class of perturbed variational-like inequality problems is introduced, some relationships between the perturbed variational-like inequality and optimization problems are established under the assumptions of strongly $\alpha$-invex functions.