Favorable Classes of Mappings and Multimappings in Nonlinear Analysis and Optimization

A generalization of the class of lower C k ?functions introduced by R.T. Rockafellar 28] called lower T k ?functions is proposed in the innnite dimensional case. Mappings of class T k are studied for themselves as they seem to deserve some attention. Other classes of functions such as subconvex functions, subsmooth functions, semismooth functions are either introduced or extended to the case of an innnite dimensional Banach space. R.T. Rockafellar has pointed out in 28] some classes of functions on an open subset W of some Euclidean space which are important from the point of view of nonsmooth analysis. It is our purpose here to extend his study to the innnite dimensional situation (see section 3) and to delineate some notions close to the class of semismooth functions introduced by Miiin 20] (see section 4). We also deal with generalizations of the class of submonotone multimappings considered by Spingarn 32] (see section 2). The lack of local compactness of the space leads us to consider directional convergence, as in 8], 20], 32], rather than ordinary convergence. We also consider (in section 1) a class of G^ ateaux diierentiable mappings whose derivatives satisfy a mild continuity property. We call it the class of T k-mappings because its deenition involves the tangent functor of diierential geometry. In nite dimensional spaces it coincides with the class of C k-mappings. It seems to play an important role in nonlinear analysis; in particular it ts well the case of superposition operators (or Nemitskii operators) between L p spaces. This class is used to deene a generalization of the notion of lower-C k function introduced in 28]; we call it the class of lower-T k functions: In turn, we show that any lower-T k function is semismooth for a large class of subdiierentials. For the applicability to algorithms of the notions considered here, we refer the reader to 21] for instance.