Local $V$-Moreau envelope of nonconvex functions

Recently, the author in 2024 studied the $V$-Moreau envelope $e_{\lambda ,S}^{V}f$ of nonconvex functions $f$ with index $\lambda >0$ and associated with closed sets $S$ in smooth Banach spaces. In this paper, we developed a corresponding local theory for the local $V$-Moreau envelope of $f$ on a neighborhood of a point $\overline{x}$ in the domain of $f$. A local concept of generalized projection is introduced in connection with the local $V$-Moreau envelope. We investigated the Lipschitz continuity, Fréchet differentiability, and $V$-proximal subdifferentiability of the local $V$-Moreau envelope of $f$. Moreover, we established the local Hölder continuity of the local generalized projection for the class of $V$-primal lower nice functions introduced and studied in 2020 by Bounkhel et al. Our results extended several known theorems from Hilbert spaces to a more general framework of smooth Banach spaces.