Gradient-Proximal like Method for minimizing a Sum of two Convex Functions

For solving the convex optimization problem of finding a minimizer of a sum of two proper closed convex functions on a real Hilbert space, we consider a sequence generated by an inexact iterative scheme, based upon the Moreau-Yosida approximation, in which the quadratic kernel is replaced by a pseudo-distance, tied to Bregman functions. Several classical results (obtained for the Moreau-Yosida approximation) are extended to this context.