List of Contributors
Suzete M. Afonso Departamento de Matemática – Instituto de Geociências e Ciências Exatas Universidade Estadual Paulista “Júlio de Mesquita Filho” (UNESP) Rio Claro–SP, Brazil
Fernando G. Andrade Colégio Técnico de Bom Jesus Universidade Federal do Piauí Bom Jesus–PI, Brazil
Fernanda Andrade da Silva Departamento de Matemática Instituto de Ciências Matemáticas e de Computação (ICMC) Universidade de São Paulo São Carlos–SP, Brazil
Marielle Ap. Silva Departamento de Matemática Instituto de Ciências Matemáticas e de Computação (ICMC) Universidade de São Paulo São Carlos–SP, Brazil
Everaldo M. Bonotto Departamento de Matemática Aplicada e Estatística Instituto de Ciências Matemáticas e de Computação (ICMC) Universidade de São Paulo São Carlos–SP, Brazil
Rodolfo Collegari Faculdade de Matemática Universidade Federal de Uberlândia Uberlândia–MG, Brazil
Márcia Federson Departamento de Matemática Instituto de Ciências Matemáticas e de Computação (ICMC) Universidade de São Paulo São Carlos–SP, Brazil
Miguel V. S. Frasson Departamento de Matemática Aplicada e Estatística Instituto de Ciências Matemáticas e de Computação (ICMC) Universidade de São Paulo São Carlos–SP, Brazil
Luciene P. Gimenes (in memorian) Departamento de Matemática – Centro de Ciências Exatas Universidade Estadual de Maringá Maringá–PR, Brazil
Rogelio Grau Departamento de Matemáticas y Estadística División de Ciencias Básicas Universidad del Norte Barranquilla, Colombia
Jaqueline G. Mesquita Departamento de Matemática Instituto de Ciências Exatas Universidade de Brasília Brasília–DF, Brazil
Maria Carolina Mesquita Departamento de Matemática Instituto de Ciências Matemáticas e de Computação (ICMC) Universidade de São Paulo São Carlos–SP, Brazil
Patricia H. Tacuri Departamento de Matemática – Centro de Ciências Exatas Universidade Estadual de Maringá Maringá–PR, Brazil
Eduard Toon Departamento de Matemática Instituto de Ciências Exatas Universidade Federal de Juiz de Fora Juiz de Fora–MG, Brazil
Foreword
Since the origins of the calculus, the development of ordinary differential equations has always both influenced and followed the development of the integral calculus. Newton's theory of fluxions was instrumental for solving the equations of mechanics, Leibniz and the Bernoulli brothers explored the differential equations solvable by quadrature, Euler developed similar numerical methods for computing definite integrals and approximate solutions of differential equations, that Cauchy converted into existence results for the integral of continuous functions and the solutions of initial value problems.
Following this distinguished line, Jaroslav Kurzweil solved in 1957 the delicate question of finding optimal conditions for the continuous dependence on parameters of solutions of ordinary differential systems. To this aim, he introduced a new concept of integral generalizing Ward's extension of the Perron integral, and showed how to define it through a technically minor but basic modification of the definition of the Riemann integral. It happened that, under this type of convergence, the limit of a sequence of ordinary differential equations could be a more general object, a generalized differential equations, defined in terms of the Kurzweil integral. As it often happens in mathematics, a similar integral was introduced independently in 1961 by Ralph Henstock during his investigations on the Ward integral.
This new theory, besides providing a simple, elegant, and pedagogical approach to the classical concepts of Lebesgue and Perron integrals, shed a new light on many classical questions in differential equations like stability and the averaging method. The theory was actively developed in Praha, around Kurzweil, by Jiří Jarník, Štefan Schwabik, Milán Tvrdy, Ivo Vrkoč, Dana Fraňková, and others. Their results are beautifully described in the monographs Generalized Ordinary Differential Equations of Schwabik and Generalized Ordinary Differential Equations of Kurzweil.
Contributions came also from many other countries. In particular, inspired by the fruitful visits of Márcia Federson in Praha and the enthusiastic lectures of Štefan Schwabik in São Carlos, a school was created and developed at the University of São Paulo and spread into other Brazilian institutions. Those researches gave a great impetus to the theory of generalized ordinary differential equations from the fundamental theory to many new important applications.
Because of its definition through suitable limits of Riemann sums, the Kurzweil–Henstock integral directly applies to functions with values in Banach spaces. This is particularly emphasized by the monograph Topics in Banach Space Integration of Schwabik and Ye GuoJu. It was therefore a natural and fruitful idea to consider generalized ordinary differential equations in Banach spaces, not for just the sake of generality,