Some Properties of Mordukhovich Derivatives with Applications to Locally Variational Inequalities in Banach Spaces

In this paper, we will investigate the connection between Frechet derivatives and Mordukhovich derivatives of single-valued mappings in Banach spaces. We will find some properties of Mordukhovich derivatives of set-valued mappings, which will be demonstrated by the set-valued metric projection operator in finite dimensional Banach spaces. We introduce the concept of locally variational inequalities in Banach spaces, which focus on finding solutions within some specific neighborhoods rather than across the entire global domain; and we find the connection between the solutions of locally variational inequality problems and the Mordukhovich derivatives.