BIANCA SATCO

Objectives

The aim of the present project is to obtain, in the general setting of Banach spaces, the existence of regulated and bounded variation solutions for different kinds of differential equations and inclusions.

At the first step, we will focus on (single- and set-valued) differential equations driven by regular Borel measures. Our approach will be different from the other papers in literature treating this subject. More precisely, we will use the Stieltjes integration theory, by regarding regular Borel measures as Lebesgue-Stieltjes measures with bounded variation distribution functions. This will make possible the obtaining of very general existence results (under continuity and growth conditions, with no Lipschitz continuity assumptions), that will cover some delicate situations, e.g. the case of a nonatomic singular measure or the case of hybrid systems with Zeno behaviour (i.e. the case where the impulsive moments accumulate in a finite interval).

Since the space of regulated functions is a really natural framework in the Kurzweil-Stieltjes integration, we will also study the existence of regulated and bounded variation solutions under Kurzweil integrability hypothesis. Notice that we thus significantly broaden the spectrum of considered problems since the integrability in this sense is much more general that classical integrability, allowing, in particular, to consider highly oscillatory functions on the right-hand side. By decomposing the Borel measure into atomic and nonatomic part, we will be able to present a new existence results (by imposing different conditions on the function associated to the atomic part and that associated to the nonatomic part).

At the next step, we will be concerned with the study of semilinear evolution equations when the C0 semigroup of operators is not necessarily compact. We will again place ourselves in a more general setting (than the authors of other related papers), by making use of the properties of Kurzweil-Stieltjes integral in order to obtain regulated solutions. We will also generalize several results obtained for linear problems under Kurzweil-type integrability assumptions, in the framework of generalized differential equations.

We will be concerned as well with associated set-valued problems (i.e. measure driven differential inclusions, respectively semilinear evolution inclusions), using through the research in this direction our experience in the field of set-valued theory.

At the same time, as modeling concrete phenomena in physics often leads to solutions of bounded variation, we will investigate the existence of such solutions to our problems, by imposing stronger hypothesis on the right-hand side of the considered measure-driven or semilinear evolution equations or inclusions.

For all previously mentioned problems, we will also focus on some other features, e.g. the continuous dependence with respect to a parameter or the properties of the set of solutions. Let us finally remark that the project director has previously studied hybrid systems but with a completely different approach (that of time scales theory) and was also interested in applications of integration of highly oscillatory functions to differential and integral equations, but not to semilinear evolution equations.