Nonsmooth analysis: differential calculus of nondifferentiable mappings

Abstract

Abstract.A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus.Introduction. The idea to extend the framework of differential calculus so as to cover more general classes of functions and mappings is by no means new. Basically, it was the underlying idea for the differentiation theory connected with the Lebesgue integral and for the theory of distributions. Both theories deal essentially with what could be called nonlocal aspects of the calculus centered around the Newton-Leibniz and integration by parts formulae. The notion of the value of a derivative at a given point makes no sense in either of them.Nonsmooth analysis appeared in the 1970's just to carry out an extension of the local aspect of the calculus connected with the idea of (linear) approximation of a mapping about a given point. Certain separate ideas and results appeared of course much earlier (one could recall the Dini numbers for instance) but a systematic study began during the last decade when the natural development of the optimization theory made the need for such an extension very acute and, as often happens, practical and heuristic computations were initiated before an adequate theory appeared (see [52] and references therein).It is not surprising that the main impulse came from the optimization theory which has natural mechanisms generating nonsmoothness. But as a result, most of the efforts were applied to obtain more and more refined conditions for extrema with less interest in those aspects of analysis that are less immediately connected with this purpose. The only exception was perhaps the generalized gradients of Clarke whose analytical virtues were recognized from the beginning ([8]-[12], [17],

On Stabilized Point Spectra of Multivalued Systems

“…(1.7) Multivalued operators having the structure (1.7) appears also in Ioffe's fan theory [6]. It is clear that F is positively homogenous.…”

On the Hahn-Banach extension property

“…Recently, Ioffe [9] has established the equivalence of (HB) and (LUB) as a consequence of a more general result on linear selections for fans [8,9]. In this note we wish to show that, using the same vector space, same hyperplane and essentially the…”

“…These arguments, while geometric, are lengthy ad involve considrable case analysis. Elster and Nehse [5,6], using the Bonnice-Silverman-To-Yen proof as a basis, have shown the equivalence of a variety of other convex optimization and positive operator theoretic results.Recently, Ioffe [9] has established the equivalence of (HB) and (LUB) as a consequence of a more general result on linear selections for fans [8,9]. In this note we wish to show that, using the same vector space, same hyperplane and essentially the…”

A New Definition for Generalized First Derivative of Nonsmooth Functions

“…Some of known generalized derivatives are the Clarke's Generalized derivative [1], the Mordukhovich's Coderivatives [4][5][6][7][8], Ioffe's Prederivatives [9][10][11][12][13], the Gowda and Ravindran H-Differentials [14], the Clarck-Rockafellar Subdifferential [15], the Michel-Penot Subdifferential [16], the Treiman's Linear Generalized Gradients [17], and the Demyanov-Rubinov Quasidifferentials [2]. In these mention works, for introducing generalized derivative of non-smooth function is a set which may be empty or including several members.…”