Subdifferential Calculus Without Qualification Assumptions

in which A is linear continuous, f1, f2 and g are closed proper convex functions finite at x and A(x) respectively, are not valid without additional assumptions, a well known and disappointing fact, especially for beginners in convex analysis. A general formula for ∂ (f1 + f2) (x) which does not require any qualification condition has been proposed in [15] (see also [16]). But it involves the approximate subdifferential ∂εf of convex analysis. Thus this condition is specific to the convex situation. It is our purpose here to present a formula which is valid without any additional condition and which uses only the ordinary subdifferential of convex analysis. Such an aim has already been pursued in [2], [3], [14] and [29]; moreover, in [2], [3] a connection with a notion of variational sum of operators is pointed out. However the conditions we impose on the sequences involved in our formula differ from the conditions of Attouch-Baillon-Théra and Thibault: they are more precise and their interpretation is easier and more amenable to the general nonconvex, nonsmooth case. Since our approach aims at a link with the general situation of nonsmooth analysis, an interest for such conditions seems to be justified. It may have consequences on the evolution of nonsmooth analysis. Moreover, our conditions bear on the coupling functional and it is well known in nonlinear functional analysis and in the study of partial differential equations that delicate limiting procedures are to be found for such terms (see [7], [10], [19] for instance). Thus the analysis of convex functionals which are obtained in terms of integrals or integro-differential terms as in [5], [8], [13], [25], [26] may benefit from a closer analysis of calculus rules.