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Using the relations between the solutions of generalized ODEs and the solutions of other types of equations, we translate our results to measure differential equations and impulsive differential equations. The main reference for this chapter is [29].
The aim of Chapter 14 is to bring together the theory of semidynamical systems generated by generalized ODEs. We show the existence of a local semidynamical system generated by a nonautonomous generalized ODE of the form
where
Chapter 15 is intended for applications of the theory developed in some of the previous chapters to a class of more general functional differential equations, namely, measure FDE of neutral type. In Section 15.1, some historical notes ranging from the beginnings of the term equation, passing through “functional differential equation,” and reaching functional differential equation of neutral type are put together. Then, we present a correspondence between solutions of a measure FDE of neutral type with finite delays and solutions of a generalized ODEs. Results on existence and uniqueness of a solution as well as continuous dependence of solutions on parameters based on [76] are also explored.
We end this preface by expressing our immense gratitude to professors Jaroslav Kurzweil, Štefan Schwabik (in memorian) and Milan Tvrdý for welcoming several members of our research group at the Institute of Mathematics of the Academy of Sciences of the Czech Republic so many times, for the countably many good advices and talks, and for the corrections of proofs and theorems during all these years.
October 2020
Everaldo M. Bonotto
Márcia Federson
Jaqueline G. Mesquita
São Carlos, SP, Brazil
1 Preliminaries
Everaldo M. Bonotto1, Rodolfo Collegari2, Márcia Federson3, Jaqueline G. Mesquita4, and Eduard Toon5
1Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo, São Carlos, SP, Brazil
2Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, MG, Brazil
3Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo, São Carlos, SP, Brazil
4Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Brasília, DF, Brazil
5Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil
This preliminary chapter is devoted to two pillars of the theory of generalized ordinary differential equations for which we use the short form “generalized ODEs”. One of these pillars concerns the spaces in which the solutions of a generalized ODE are generally placed. The other pillar concerns the theory of nonabsolute integration, due to Jaroslav Kurzweil and Ralph Henstock, for integrands taking values in Banach spaces. As a matter of fact, such integration theory permeates the entire book. It (the theory of non absolute integration) is within the heartwood of the theory of generalized ODEs, appearing (the same theory of nonabsolute integration) in the integral form of a very special case of nonautonomous generalized ODEs, namely (now we mention the name of the special case of generalized ODEs), measure functional differential equations.
The solutions of a Cauchy problem for a generalized ODE, with right‐hand side in a class of functions introduced by J. Kurzweil in [147–149], usually belong to a certain space of functions of bounded variation (see Lemma 4.9). However, since functions of bounded variation are also regulated functions in the sense described by Jean Dieudonné and, more generally, by the group Nicolas Bourbaki, and because the space of regulated functions is more adequate for dealing with discontinuous functions appearing naturally in Stieltjes‐type integrals, it is important to present a substantial content about this space. Thus, the first section of this chapter describes the main properties of the space of regulated functions with the icing of the cake being a characterization of its relatively compact subsets due to D. Franková.
Regarding functions of bounded variation, which are known to be of bounded semivariation and, hence, of bounded
In the third section of this chapter, we describe the second pillar and main background of the theory of generalized ODEs, namely, the framework of vector‐valued nonabsolute integrals of Kurzweil and Henstock. Here, we call the reader's attention to the fact that we refer to Kurzweil vector integrals as Perron–Stieltjes integrals so that, when a more general definition of the Kurzweil integral is presented in Chapter 1, the reader will not be confused. One of the highlights of the third section is, then, the integration by parts formula for Perron–Stieltjes integrals.
An extra section called “Appendix,” which can be skipped in a first reading of the book, concerns other types of gauge‐based integrals which use the interesting idea of Edward James McShane. The well‐known Bochner–Lebesgue integral comes into the scene and an equivalent definition of it as the limit of Riemannian‐type sums comes up.
1.1 Regulated Functions
Regulated functions appear in the works by J. Dieudonné [58, p. 139] and N. Bourbaki [32, p. II.4]. The raison d'être of regulated functions lies on the fact that every regulated function
