We continue the study of the properties of the $V$-Moreau envelope and generalized $(f,\lambda )$-projection that we started in ^[5]. In this paper, we study the differentiability and the regularity of the $V$-Moreau envelope and the Hölder continuity of the generalized $(f,\lambda )$-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.

This paper establishes a successive approximation method designed to compute a point belonging simultaneously to the fixed-point set of a given mapping (relatively nonexpansive) and to the solution set of a nonconvex equilibrium problem within a Banach space. Our present findings provide a unifying generalization of numerous previously obtained results, beginning with the transition from Hilbertian structures to the more general Banach framework, and further encompassing the passage from convex analytical settings to their considerably more delicate nonconvex counterparts. We mainly extend the results proved in ^[27] from convex to nonconvex settings.