Mieczysław Cichoń

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.

Some Applications of Nonabsolute Integrals in the Theory of Differential Inclusions in Banach Spaces

In this paper we present a brief historical review about multivalued integrals and its relations with differential inclusions. Then a new theorem about existence of solutions (in some weak sense) for differential inclusions in Banach spaces is proved (by using some properties of nonabsolute integrals).

EXISTENCE OF SOLUTIONS OF THE DYNAMIC CAUCHY PROBLEM IN BANACH SPACES

Abstract In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales xΔ(t)=f(t,x(t)),x(0)=x0,t∈Ia,

A note on Peano’s Theorem on time scales

\matrix{{x^\Delta (t) = f(t,x(t)),} \hfill & {} \hfill \cr {x(0) = x_0 ,} \hfill & {t \in I_a ,} \hfill }

On Regulated Functions

where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.

On continuous dependence of solutions of dynamic equations

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

On the existence of solutions for quadratic integral equations in Orlicz spaces

Abstract In this paper we investigate the dynamic Cauchy problem in Banach spaces. We check how dense a time scale must be in such a way that Peano’s Theorem holds and we present a counterexample to Peano’s Theorem on a time scale with only one right dense point.

Weak solutions of differential equations in Banach spaces

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.