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.
Table of Contents
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 Δ-integral 89
3.3.3 Perron Δ-integrals and Perron-Stieltjes 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), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon
8.1 Variational Stability for Generalized ODEs 244
8.1.1 Direct Method of Lyapunov 246
8.1.2 Converse Lyapunov Theorems 247
8.2 Lyapunov Stability for Generalized ODEs 256
8.2.1 Direct Method of Lyapunov 257
8.3 Lyapunov Stability for MDEs 261
8.3.1 Direct Method of Lyapunov 263
8.4 Lyapunov Stability for Dynamic Equations on Time Scales 265
8.4.1 Direct Method of Lyapunov 267
8.5 Regular Stability for Generalized ODEs 272
8.5.1 Direct Method of Lyapunov 275
8.5.2 Converse Lyapunov Theorem 282
9 Periodicity 295
Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Maria Carolina Mesquita
9.1 Periodic Solutions and Floquet’s Theorem 297
9.1.1 Linear Differential Systems with Impulses 303
9.2 (θ,T)-Periodic Solutions 307
9.2.1 An Application to MDEs 313
10 Averaging Principles 317
Márcia Federson and Jaqueline G. Mesquita
10.1 Periodic Averaging Principles 320
10.1.1 An Application to IDEs 325
10.2 Nonperiodic Averaging Principles 330
11 Boundedness of Solutions 341
Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 11.1 Bounded Solutions and Lyapunov Functionals 342
11.2 An Application to MDEs 352
11.2.1 An Example 356
12 Control Theory 361
Fernanda Andrade da Silva, Márcia Federson, and Eduard Toon
12.1 Controllability and Observability 362
12.2 Applications to ODEs 365
13 Dichotomies 369
Everaldo M. Bonotto and Márcia Federson
13.1 Basic Theory for Generalized ODEs 370
13.2 Boundedness and Dichotomies 381
13.3 Applications to MDEs 391
13.4 Applications to IDEs 400
14 Topological Dynamics 407
Suzete M. Afonso, Marielle Ap. Silva, Everaldo M. Bonotto, and Márcia Federson
14.1 The Compactness of the Class F0(Ω,h) 408
14.2 Existence of a Local Semidynamical System 411
14.3 Existence of an Impulsive Semidynamical System 418
14.4 LaSalle’s Invariance Principle 423
14.5 Recursive Properties 425
15 Applications to Functional Differential Equations of Neutral Type 429
Fernando G. Andrade, Miguel V. S. Frasson, and Patricia H. Tacuri
15.1 Drops of History 429
15.2 FDEs of Neutral Type with Finite Delay 435
References 455
List of Symbols 471
Index 473
