Formulas for calculating of generalized differentials with respect to a set and their applications

This paper provides formulas for calculating of Fr\'{e}chet and limiting normal cones with respect to a set of sets and the limiting coderivative with respect to a set of set-valued mappings. These calculations are obtained under some qualification constraints and are expressed in the similar forms of these ones of Fr\'{e}chet and limiting normal cones and the limiting coderivative. By using these obtained formulas, we state explicit necessary optimality conditions with respect to a set for optimization problems with equilibrium constraints under certain qualification conditions. Some illustrated examples to obtained results are also established.