Proximal analysis in reflexive smooth Banach spaces

In the present paper, we introduce and study a new proximal normal cone in reflexive Banach spaces in terms of a generalized projection operator. Two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth Banach spaces. The density theorem for both proximal subdifferentials has been proved in pp-uniformly convex and qq-uniformly smooth Banach spaces. Various important properties and applications of our concepts are also proved.