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About differentiability and V BG * class

Turowska M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2014

Vol. 19

245--249

Let X be a finite dimensional real Banach space. We show that if the contingent of the curve Γ : (a, b) → X fulfils some conditions then each parametrization of that curve is V BG * . Stanisław Saks proved that each V BG * function is differentiable at a set of full Lebesgue measure. The result of this paper is a partial converse of that theorem.

Towards historical roots of necessary conditions of optimality: Regula of Peano

Dolecki S., Greco G. H.

Control and Cybernetics

2007

Vol. 36, no 3

491-518

At the end of 19th century Peano discerned vector spaces, differentiability, convex sets, limits of families of sets, tangent cones, and many other concepts, in a modern perfect form. He applied these notions to solve numerous problems. The theorem on necessary conditions of optimality (Regula) is one of these. The formal language of logic that he developed, enabled him to perceive mathematics with great precision and depth. Actually he built mathematics axiomatically based exclusively on logical and set-theoretic primitive terms and properties, which was a revolutionary turning point in the development of mathematics.

Inversion of multifunctions and differential inclusions

Nachi K., Penot J.-P.

Control and Cybernetics

2005

Vol. 34, no 3

871-901

We present a new inverse mapping theorem for correspondences. It uses a notion of differentiability for multifunctions which seems to be new. We compare it with previous versions. We provide an application to differential inclusions.

Extremality and differentiability of convex functions

Bourass A., Ferrahi B., Saidou N.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 1

13-29

We study relations between the differentiability and some geometrical properties of convex functions defined on a Banach space. As a consequence, we get a characterization of Radon-Nikodym property in Banach spaces by geometrical properties of convex functions.