Bianca Satco

Measure differential inclusions - between continuous and discrete

The paper is devoted to the study of the measure-driven differential inclusions dx(t)∈G(t,x(t))dμ(t), x(0)=x0 for arbitrary finite Borel measure μ. This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach.MSC:49N25, 34A60, 93C30, 49J53, 37N35, 34A37.

Decomposability and uniform integrability in Pettis integration

In this paper, we give several characterizations of decomposable subsets of the space of Pettis integrable functions (in particular, a characterization similar to that already known in the Bochner integrability setting). We also introduce the notion WPUI of uniform integrability, and obtain some relations between decomposability and PUI, resp. WPUI uniform integrability concepts. As consequence, conditional weak compactness and sequential weak compactness criteria are given under decomposability assumptions. Finally, an application to second order differential inclusions is presented.

Set valued integrability in non separable Fr\'{e}chet spaces and applications

Abstract We focus on measurability and integrability for set valued functions in non-necessarily separable Fréchet spaces. We prove some properties concerning the equivalence between different classes of measurable multifunctions. We also provide useful characterizations of Pettis set-valued integrability in the announced framework. Finally, we indicate applications to Volterra integral inclusions.

Impulsive nonlocal differential equations through differential equations on time scales

Abstract We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations.

Optimal Control Problems Governed by a Second Order Ordinary Differential Equation with m-Point Boundary Condition

Using a new Green type function we present a study of optimal control problem where the dynamic is governed by a second order ordinary differential equa- tion (SODE) with m-point boundary condition.

On Regulated Functions

Abstract In this paper we investigate the space of regulated functions on a compact interval [0,1]. When equipped with the topology of uniform convergence this space is isometrically isomorphic to some space of continuous functions. We study some of its properties, including the characterization of the dual space, weak and strong compactness properties of sets. Finally, we investigate some compact and weakly compact operators on the space of regulated functions. The paper is complemented by an existence result for the Hammerstein-Stieltjes integral equation with regulated solutions.

Urysohn measure driven integral equations in the space of bounded variation functions and applications

Motivated by the fact that bounded variation (often discontinuous) functions frequently appear when studying integral equations that describe physical phenomena, we focus on the existence of bounded variation solutions for Urysohn integral measure driven equations. Due to numerous applications of Urysohn integral equations in various domains, problems of this kind have been extensively studied in literature, under more restrictive assumptions. Our approach concerns the framework of Kurzweil-Stieltjes integration, which allows the occurrence of high oscillatory features on the right hand side of the equation. A discussion about interesting consequences of our main result (given by particular cases of the measure driving the equation) is presented. Finally, we show the generality of our results by investigating two examples of impulsive type problems (from both theoretical and numerical perspective) and giving an application in electronics industry concerning polarization properties of ferroelectric materials.

A Cauchy Problem on Time Scales with Applications

A Cauchy Problem on Time Scales with Applications We obtain the existence of continuous solutions for a nonlocal Cauchy problem on time scales in Banach spaces, considering non-absolutely convergent deltaintegrals. As this kind of problems contains the classical cases of differential and difference equations and not only these ones, our main theorem offers a very general existence result. We then deduce, by an embedding procedure, a result for impulsive differential equations under a new kind of assumptions on the jump functions.