Continuous Fréchet Differentiability of the Moreau Envelope of Convex Functions on Banach Spaces

It is shown that the Moreau envelope of a convex lower semicontinuous function on a real Banach space with strictly convex dual is Fréchet differentiable at every its minimizer, and continuously Fréchet differentiable at every its non-minimizer satisfying that the dual space is uniformly convex at every norm one element around its normalized gradient vector at those points. As an application, we obtain the continuous Fréchet differentiability of the Moreau envelope functions on Banach spaces with locally uniformly duals and the continuity of the corresponding proximal mappings provided that both primal and dual spaces are locally uniformly convex.