The Moreau envelope function and proximal mapping in the sense of the Bregman distance

In this paper, we explore some properties of the Moreau envelope function eλf(x)eλf(x) of ff and the associated proximal mapping Pλf(x)Pλf(x) in the sense of the Bregman distance induced by a convex function gg. Precisely, we study the continuity, differentiability, and Clarke regularity of the Moreau envelope function and the upper semicontinuity and single-valuedness of the proximal mapping as well as its relation to the convexity of λf+gλf+g, where λλ is a positive parameter.