In this paper, a new class of nonconvex sets and functions called strongly -convex sets and strongly -convex functions are introduced. This class is considered as a natural extension of strongly -convex sets and functions introduced in the literature. Some basic and differentiability properties related to strongly -convex functions are discussed. As an application to optimization problems, some optimality properties of constrained optimization problems are proved. In these optimization problems, either the objective function or the inequality constraints functions are strongly -convex. 

Csaszar introduced the concept of generalized topological space and a new open set in a generalized topological space called -preopen in 2002 and 2005, respectively. Definitions of -preinterior and -preclosuer were given. Successively, several studies have appeared to give many generalizations for an open set. The object of our paper is to give a new type of generalization of an open set in a generalized topological space called -semi-p-open set. We present the definition of this set with its equivalent. We give definitions of -semi-p-interior and -semi-p-closure of a set and discuss their properties. Also the properties of -preinterior and -preclosuer are discussed. In addition, we give a new type of continuous function

  The aim of this paper is to introduce a new type of proper mappings called semi-p-proper mapping by using semi-p-open sets, which is weaker than the proper mapping. Some properties and characterizations of this type of mappings are given.

The purpose of this paper is to introduce a new type of compact spaces, namely semi-p-compact spaces which are stronger than compact spaces; we give properties and characterizations of semi-p-compact spaces.

In this paper, we define a new type of pairwise separation axioms called pairwise semi-p- separation axioms in bitopological spaces, also we study some properties of these spaces and relationships of each one with the ordinary separation axioms in the bitopological spaces.

The purpose of this paper is to study a new types of compactness in the dual bitopological spaces. We shall introduce the concepts of L-pre- compactness and L-semi-P- compactness .

   The purpose of this paper is to study   new  types of open sets in bitopological spaces.    We shall introduce the concepts of  L- pre-open and L-semi-p-open sets

The significance of the work is to introduce the new class of open sets, which is said Ǥ- -open set with some of properties. Then clarify how to calculate the boundary area for these sets using the upper and lower approximation and obtain the best accuracy.

In this paper, we propose new types of non-convex functions called strongly --vex functions and semi strongly --vex functions. We study some properties of these proposed functions. As an application of these functions in optimization problems, we discuss some optimality properties of the generalized nonlinear optimization problem for which we use, as an objective function, strongly --vex function and semi strongly --vex function.