The differentiability properties of the metric projection Pc on a closed convex set C in Hilbert space are characterized in terms of the smoothness type of the boundary of C. Our approach is based on using variational type second derivatives as a sufficiently flexible tool to describe the boundary struc ture of the set C with regard to the differentiability of Pc. We extend results by R.B. Holm...

For distributions, we build a theory of higher order pointwise differentiability comprising, for zero, {\L}ojasiewicz's notion point value. Results include Borel regularity differentials, rectifiability the associated jets, Rademacher-Stepanov type theorem, and Lusin approximation. A substantial part this development is new also zeroth order. Moreover, establish Poincar\'e inequality involving ...

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of G~teaux, Fr6chet, and Hadamard are singled out from the general framework of cr-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equ...

Abstract In this paper, the concept of generalized saddle point(GSP) is employed to discuss the optimization problems of a set of convex functions on a normed linear space X , which presents an equivalence under a special condition between GSP and its optimum solution. A study on integrated convex optimization problem by using Gâteaux and Fréchet differentiability respectivly, and the equivalen...

such as was treated in the previous papers [3] and [4]. However, we shall show that we can replace the strong continuous differentiability of A(t)A(s)~ by its Holder continuity by means of a slight change of the proof. It is quite clear that the differentiability of A(t)A(s)~ is not necessary for the construction of the formal fundamental solution U(t, s) of (0. 1'). In the previous papers, how...

The well-known finite-dimensional first-order open mapping theorem says that a continuous map with a finite-dimensional target is open at a point if its differential at that point exists and is surjective. An identical result, due to Graves, is true when the target is infinite-dimensional, if “differentiability” is replaced by “strict differentiability.” We prove general theorems in which the l...