their specific applications, and in particular their use in reinterpreting the concept of functional-differential equation. This is the viewpoint adopted in the present substantial monograph, whose red wire is to show that many types of evolution equations in Banach space can be treated in an unified and more general way as a special case of the Kurzweil generalized differential equations.
The general ideas are introduced and motivated through an elegant treatment of the measure functional differential equations, where in the integrated form of the differential equation, the Lebesgue measure is replaced by some Stieltjes one. The approach covers differential equations with impulses and the dynamic equations on time scales. The generalized differential equations in Banach spaces are then introduced and developed in a systematic way. Most classical problems of the theory of ordinary differential equations extending to this new and general setting. This includes the existence, continuation, and continuous dependence of solutions, the linear equations, the Lyapunov stability, the periodic and bounded solutions, the averaging method, some control theory, the dichotomy theory, questions of topological dynamics, and the measure neutral functional differential equations.
It is not surprising that the richness and wideness of the content of this substantial monograph is the result of the intensive team activity of 14 contributors: S.M. Afonso, F. Andrade da Silva, M. Ap. Silva, E.M. Bonotto, R. Collegari, F.G. de Andrade, M. Federson, M. Frasson, L.P. Gimenes, R. Grau, J.G. Mesquita, M.C. Mesquita, P.H. Tacuri, and E. Toon. Most of them have already published contributions to the area and other ones got a PhD recently in this direction. Each chapter is authored by a selected subgroup of the team, but it appears that a joint final reading took place for the final product. Each chapter provides a state-of-the-art of its title, and many classes of readers will find in this book a renewed picture of their favorite area of expertise and an inspiration for further research.
In a world where, even for scientific research, competition is too often praised as a necessary driving force, this monograph shows that the true answer is better found in an open, enthusiastic, and unselfish cooperation. In this way, the book not only magnifies the work of Jaroslav Kurzweil and of Štefan Schwabik but also their wonderful spirit.
October 2020
Jean Mawhin
Louvain-la-Neuve, Belgium
Preface
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
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
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
