It is proved that if 0 < λ ≤ f(x) ≤ Λ for x ∈ Ω, where Ω ⊂ R is a bounded convex domain, and f is L-Dini continuous on Ω, then there exist infinitely many biLipschitz maps F : Ω → R such that detDF (x) = f(x) for a.e. x ∈ Ω. Moreover, these mappings can be chosen to have convex potentials. We relate our result to a classical theorem by H. M. Reimann; however, the emphasis is on the novel use of...

In this work we focus on nondifferentiable Lyapunov functions where we derive conditions which ensure the existence of time-dependent upper bounds for such functions. The aforementioned conditions are based on the notion of upper right-hand Dini derivative of Lyapunov functions. As an application we study the attractivity and the asymptotic stability of the time–varying class of systems ẋ(t) = ...

It is clear that limy↓x f(y) ≤ limy↓x f(y). Analogously we can define limy↑x f(y) and limy↑x f(y) and we also have limy↑x f(y) ≤ limy↑x f(y). One can verify as in the sequential case that e.g. (1) limy↓x f(y) ≤ A if and only if for all > 0 there exists a δ > 0 such that f(y) < A+ for all y such that 0 < y − x < δ. (2) limy↓x f(y) ≤ A if and only if for all > 0 and δ > 0 there exists an y with 0...

We define the notions of unilateral metric derivatives and “metric derived numbers” in analogy with Dini derivatives (also referred to as “derived numbers”) and establish their basic properties. We also prove that the set of points where a path with values in a metric space with continuous metric derivative is not “metrically differentiable” (in a certain strong sense) is σsymmetrically porous ...

Laura Bazzichi ([email protected]) Marco Dini ([email protected]) Alessandra Rossi ([email protected]) Silvia Corbianco ([email protected]) Francesca De Feo ([email protected]) Camillo Giacomelli ([email protected]) Cristina Zirafa ([email protected]) Claudia Ferrari ([email protected]) Bruno Rossi ([email protected]) Stefano Bombardieri (s.bombardie...

The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f : X → R defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E, we draw the possibility to state our results for arbitrary real linear spa...

Given a lower semicontinuous function f : Rn → R ∪ {+∞}, we prove that the set of points of Rn where the lower Dini subdifferential has convex dimension k is countably (n − k)-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.

E. Riccietti a∗ and S. Bellavia and S. Sello Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, viale G.B. Morgagni 67a, 50134 Firenze, Italia.; Dipartimento di Ingegneria Industriale, Università di Firenze, viale G.B. Morgagni 40, 50134 Firenze, Italia, [email protected]; Enel Ingegneria e Ricerca, Via Andrea Pisano 120, 56122 Pisa, Italia, stefano.sello@en...

We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals.

We study Ruelle operators on expanding and mixing dynamical systems with potential function satisfying the Dini condition. We give an estimate for the convergence speed of the iterates of a Ruelle operator. Our proof avoids Markov partitions. This is the second part of our research on Ruelle operators.