Convergence Theorems for Varying Measures Under Convexity Conditions and Applications

In this paper, convergence theorems involving convex inequalities of Copson’s type (less restrictive than monotonicity assumptions) are given for varying measures, when imposing convexity conditions on the integrable functions or on the measures. Consequently, a continuous dependence result for a wide class of differential equations with many interesting applications, namely measure differential equations (including Stieltjes differential equations, generalized differential problems, impulsive differential equations with finitely or countably many impulses and also dynamic equations on time scales) is provided

A new result on impulsive differential equations involving non-absolutely convergent integrals

AbstractIn this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations

Approximating the solutions of differential inclusions driven by measures

The matter of approximating the solutions of a differential problem driven by a rough measure by solutions of similar problems driven by “smoother” measures is considered under very general assumptions on the multifunction on the right-hand side. The key tool in our investigation is the notion of uniformly bounded ε-variations, which mixes the supremum norm with the uniformly bounded variation condition. Several examples to motivate the generality of our outcomes are included

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. Keywords: Pettis integral, decomposability, set-valued integral, uniform integrability, sequential weak compactness, differential inclusionQuaestiones Mathematicae 29(2006), 39–5

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

We focus on measurability and integrability for set valued functions in non-necessarily separable Fr'{e}chet 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