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It is well known that the remarkable theory of generalized ordinary differential equations (we write generalized ODEs, for short) was born in Czech Republic in the year 1957 with the brilliant paper [147] by Professor Jaroslav Kurzweil. In Brazil, the theory of generalized ODEs was introduced by Professor Štefan Schwabik during his visit to the Universidade de São Paulo, in the city of São Paulo, in 1989.

Nevertheless, it was only in 2002 that the theory really started to be developed here. The article by Professors Márcia Federson and Plácido Táboas, published in the Journal of Differential Equations in 2003 (see [92]), was the first Brazilian publication on the subject. Now, 18 years later, members of the Brazilian research group on Functional Differential Equations and Nonabsolute Integration decided to gather the results obtained over these years in order to produce a comprehensive literature about our results developed so far, regarding the theory of generalized ODEs in abstract spaces.

Originally, this monograph was thought to be organized by professors Márcia Federson, Everaldo M. Bonotto, and Jaqueline G. Mesquita, with the contribution of the following authors: Suzete M. Afonso, Fernando G. Andrade, Fernanda Andrade da Silva, Marielle Ap. Silva, Rodolfo Collegari, Miguel Frasson, Luciene P. Gimenes, Rogelio Grau, Maria Carolina Mesquita, Patricia H. Tacuri, and Eduard Toon. However, after a while, it became a production of us all, with contributions of everyone to all chapters and to the uniformity, coherence, language, and interrelationship of the results. We, then, present this carefully crafted work to disseminate the theories involved here, especially those on Kurzweil–Henstock nonabsolute integration and on generalized ODEs.

In the introductory chapter, named Preliminaries, we brought together two main issues that permeate this book. The first one concerns the spaces where the functions within the right-hand sides of differential or integral equations live. The other one concerns the theory of nonabsolute vector-valued integrals in the senses of J. Kurzweil and R. Henstock. Sections 1.1 and 1.2 are devoted to properties of the space of regulated functions and the space of functions of bounded -variation. Among the main results of Section 1.1, we mention a characterization, based on [96, 97], of relatively compact sets of the space of regulated functions. Section 1.2 deals with properties of functions of bounded -variation where the Helly's choice principle for abstract spaces is a spotlight. The book [127] is the main reference for this section. The third section is devoted to nonabsolute vector-valued integrals. The basis of this theory is presented here and results specialized for Perron–Stieltjes integrals are included. We highlight substitution formulas and an integration by parts formula coming from [212]. Other important references to this section are [72, 73, 210].

The second chapter is devoted to the integral as defined by Jaroslav Kurzweil in [147]. We compiled some historical data on how the idea of the integral came about. Highlights of this chapter include the Saks–Henstock lemma, the Hake-type theorem, and the change of variables theorem. We end this chapter with a brief history of the Kapitza pendulum equation whose solution is highly oscillating and, therefore, suitable for being treated via Kurzweil–Henstock nonabsolute integration theory. An important reference to Chapter 2 is [209].

Before entering the theory of generalized ODEs, we take a trip through the theory of measure functional differential equations (we write measure FDEs for short). Then, the third chapter appears as an embracing collection of results on measure FDEs for Banach space-valued functions. In particular, we investigate equations of the form

where is a memory function and the integral on the right-hand side is in the sense of Perron–Stieltjes. We show that these equations encompass not only impulsive functional dynamic equations on time scales but also impulsive measure FDEs. Examples illustrating the relations between any two of these equations are also included. References [85, 86] feature as the foundation for this relations. Among other topics covered by Chapter 3, we mention averaging principles, covering the periodic and nonperiodic cases, and results on continuous dependence of solutions on time scales. References [21, 82, 178] are crucial here.

In Chapter 4, we enter the theory of generalized ODE itself. We begin by recalling the concept of a nonautonomous generalized ODE of the form

where takes a pair of a regulated function and a time to a regulated function. The main reference to this chapter is [209]. Measure FDEs in the integral form described above feature in Chapter 4 as supporting actors, because now their solutions can be related to solutions of the generalized ODEs, whose right-hand sides involve functions which look like

This characteristic of generalized ODEs plays an important role in the entire manuscript, since it allows one to translate results from generalized ODEs to measure FDEs.

Chapter 5, based on [78], brings together the foundations of the theory of generalized ODEs. Section 5.1 concerns local existence and uniqueness of a solution of a nonautonomous generalized ODE with applications to measure FDEs and functional dynamic equations on time scales. Second 5.2 is devoted to results on prolongation of solutions of generalized ODEs, measure differential equations, and dynamic equations on time scales.

Chapter 6 deals with a very important class of differential equations, the class of linear generalized ODEs. The origins of linear generalized ODEs goes back to the papers [209–211]. Here, we recall the notion of fundamental operator associated with a linear generalized ODE for Banach space-valued functions and we travel on the same road as the authors of [45] to obtain a variation-of-constants formula for a linear perturbed generalized ODE. Concerning applications, we extend the class of equations to include linear measure FDEs.

After linear generalized ODEs are investigated, we move to results on continuous dependence of solutions on parameters. This is the core of Chapter 7 which is based on [4, 95, 96, 177]. Given a family of generalized ODEs, we present sufficient conditions so that the family of their corresponding solutions converge uniformly, on compact sets, to the solution of the limiting generalized ODE. We also prove that given a generalized ODE and its solution , where is a Banach space, one can obtain a family of generalized ODEs whose solutions converge uniformly to on .

As we mentioned before, many types of differential equations can be regarded as generalized ODEs. This fact allows us to derive stability results for these equations through the relations between the solutions of a certain equations and the solutions of a generalized ODEs. At the present time, the stability theory for generalized ODEs is undergoing a remarkable development. Recent results in this respect are contained in [3, 7, 80, 89, 90] and are gathered in Chapter 8. We also show the effectiveness of Lyapunov's Direct Method to obtain several stability results, in addition to proving converse Lyapunov theorems for some types of stability. The types of stability explore here are variational stability, Lyapunov stability, regular stability, and many relations permeating these concepts.

The existence of periodic solutions to any kind of equation is also of great interest, especially in applications. Chapter 9is devoted to this matter in the framework of generalized ODEs, whose results are also specified to measure differential equations and impulsive differential equations. Section 9.1 brings together a result which provides conditions for the solutions of a linear generalized ODE taking values in to be periodic and a result which relates periodic solutions of linear nonhomogeneous generalized ODEs to periodic solutions of linear homogeneous generalized ODEs. Still in this section, a characterization of the fundamental matrix of periodic linear generalized ODEs is established. This is the analogue of the Floquet theorem for generalized ODEs involving finite dimensional space-valued functions. In Section 9.2, inspired in an approach by Jean Mawhin to treat periodic boundary value problem (we write periodic BVP for short), we introduce the concept of a -periodic solution for a nonlinear homogeneous generalized ODE in Banach spaces, where and . A result that ensures a correspondence between solutions of a -periodic BVP and -periodic solutions of a nonlinear homogeneous generalized ODE is the spotlight here. Then, the existence of a -periodic solution is guaranteed.

Averaging methods are used to investigate the solutions of a nonautonomous differential equations by means of the solutions of an “averaged ” autonomous equation. In Chapter 10, we present a periodic averaging principle as well as a nonperiodic one for generalized ODEs. The main references to this chapter are [83, 178].

Chapter 11 is designed to provide the reader with a systematic account of recent developments in the boundedness theory for generalized ODEs. The results of this chapter were borrowed from the articles [2, 79].

Chapter 12 is devoted to the control theory in the setting of abstract generalized ODEs. In its first section, we introduce concepts of observability, exact controllability, and approximate controllability, and we give necessary and sufficient conditions for a system of generalized ODEs to be exactly controllable, approximately controllable, or observable. In Section 12.2, we apply the results to classical ODEs.

The study of exponential dichotomy for linear generalized ODEs of type

is the heartwood of Chapter 13, where sufficient conditions for the existence of exponential dichotomies are obtained, as well as conditions for the existence of bounded solutions for the nonhomogeneous equation

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 belongs to a compact class of right-hand sides. We construct an impulsive semidynamical system associated with a generalized ODE subject to external impulse effects. For this class of impulsive systems, we present a LaSalle's invariance principle-type result. Still in this chapter, we present some topological properties for impulsive semidynamical systems as minimality and recurrence. The main reference here is [4].

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