Generalized Ordinary Differential Equations in Abstract Spaces and Applications

GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES AND APPLICATIONS

Explore a unified view of differential equations through the use of the generalized ODE from leading academics in mathematics

Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations, including Measure Functional Differential Equations (measure FDEs). It presents a uniform collection of qualitative results of Generalized ODEs and offers readers an introduction to several theories, including ordinary differential equations, impulsive differential equations, functional differential equations, dynamical equations on time scales, and more.

Throughout the book, the focus is on qualitative theory and on corresponding results for other types of differential equations, as well as the connection between Generalized Ordinary Differential Equations and impulsive differential equations, functional differential equations, measure differential equations and dynamic equations on time scales. The book's descriptions will be of use in many mathematical contexts, as well as in the social and natural sciences. Readers will also benefit from the inclusion of:

• A thorough introduction to regulated functions, including their basic properties, equiregulated sets, uniform convergence, and relatively compact sets

• An exploration of the Kurzweil integral, including its definitions and basic properties

• A discussion of measure functional differential equations, including impulsive measure FDEs

• The interrelationship between generalized ODEs and measure FDEs

• A treatment of the basic properties of generalized ODEs, including the existence and uniqueness of solutions, and prolongation and maximal solutions

Perfect for researchers and graduate students in Differential Equations and Dynamical Systems, Generalized Ordinary Differential Equations in Abstract Spaces and App-lications will also earn a place in the libraries of advanced undergraduate students taking courses in the subject and hoping to move onto graduate studies.

Auteur

Everaldo M. Bonotto, PhD, is Associate Professor in the Department of ­Applied Mathematics and Statistics, at ICMC-Universidade de São Paulo, São Carlos, SP, Brazil.

Márcia Federson, PhD, is Full Professor in the Department of Mathematics at ICMC-Universidade de São Paulo, São Carlos, SP, Brazil. Jaqueline G. Mesquita, PhD, is Assistant Professor at Department of Mathematics at the University of Brasília, Brasília, DF, Brazil.

Texte du rabat
Explore a unified view of differential equations through the use of the generalized ODE from leading academics in mathematics

Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations, including Measure Functional Differential Equations (measure FDEs). It presents a uniform collection of qualitative results of Generalized ODEs and offers readers an introduction to several theories, including ordinary differential equations, impulsive differential equations, functional differential equations, dynamical equations on time scales, and more. Throughout the book, the focus is on qualitative theory and on corresponding results for other types of differential equations, as well as the connection between Generalized Ordinary Differential Equations and impulsive differential equations, functional differential equations, measure differential equations and dynamic equations on time scales. The book's descriptions will be of use in many mathematical contexts, as well as in the social and natural sciences. Readers will also benefit from the inclusion of:

• A thorough introduction to regulated functions, including their basic properties, equiregulated sets, uniform convergence, and relatively compact sets
• An exploration of the Kurzweil integral, including its definitions and basic properties
• A discussion of measure functional differential equations, including impulsive measure FDEs
• The interrelationship between generalized ODEs and measure FDEs
• A treatment of the basic properties of generalized ODEs, including the existence and uniqueness of solutions, and prolongation and maximal solutions Perfect for researchers and graduate students in Differential Equations and Dynamical Systems, Generalized Ordinary Differential Equations in Abstract Spaces and App­lications will also earn a place in the libraries of advanced undergraduate students taking courses in the subject and hoping to move onto graduate studies.

Contenu
List of Contributors xi

Foreword xiii

Preface xvii

1 Preliminaries 1
*Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, and Eduard Toon*

1.1 Regulated Functions 2

1.1.1 Basic Properties 2

1.1.2 Equiregulated Sets 7

1.1.3 Uniform Convergence 9

1.1.4 Relatively Compact Sets 11

1.2 Functions of Bounded B-Variation 14

1.3 Kurzweil and Henstock Vector Integrals 19

1.3.1 Definitions 20

1.3.2 Basic Properties 25

1.3.3 Integration by Parts and Substitution Formulas 29

1.3.4 The Fundamental Theorem of Calculus 36

1.3.5 A Convergence Theorem 44

Appendix 1.A: The McShane Integral 44

2 The Kurzweil Integral 53
*Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Jaqueline G. Mesquita*

2.1 The Main Background 54

2.1.1 Definition and Compatibility 54

2.1.2 Special Integrals 56

2.2 Basic Properties 57

2.3 Notes on Kapitza Pendulum 67

3 Measure Functional Differential Equations 71
*Everaldo M. Bonotto, Márcia Federson, Miguel V. S. Frasson, Rogelio Grau, and Jaqueline G. Mesquita*

3.1 Measure FDEs 74

3.2 Impulsive Measure FDEs 76

3.3 Functional Dynamic Equations on Time Scales 86

3.3.1 Fundamentals of Time Scales 87

3.3.2 The Perron -integral 89

3.3.3 Perron -integrals and PerronStieltjes integrals 90

3.3.4 MDEs and Dynamic Equations on Time Scales 98

3.3.5 Relations with Measure FDEs 99

3.3.6 Impulsive Functional Dynamic Equations on Time Scales 104

3.4 Averaging Methods 106

3.4.1 Periodic Averaging 107

3.4.2 Nonperiodic Averaging 118

3.5 Continuous Dependence on Time Scales 135

4 Generalized Ordinary Differential Equations 145
*Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita*

4.1 Fundamental Properties 146

4.2 Relations with Measure Differential Equations 153

4.3 Relations with Measure FDEs 160

5 Basic Properties of Solutions 173
*Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon*

5.1 Local Existence and Uniqueness of Solutions 174

5.1.1 Applications to Other Equations 178

5.2 Prolongation and Maximal Solutions 181

5.2.1 Applications to MDEs 191

5.2.2 Applications to Dynamic Equations on Time Scales 197

6 Linear Generalized Ordinary Differential Equations 205
*Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Miguel V. S. Frasson*

6.1 The Fundamental Operator 207

6.2 A Variation-of-Constants Formula 209

6.3 Linear Measure FDEs 216

6.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs 220

7 Continuous Dependence on Parameters 225
*Suzete M. Afonso, Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita*

7.1 Basic Theory for Generalized ODEs 226

7.2 Applications to Measure FDEs 236

8 StabilityTheory 241
*Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rog…