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 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.
Edited by:
Everaldo M. Bonotto,
Márcia Federson,
Jaqueline G. Mesquita
Imprint: John Wiley & Sons Inc
Country of Publication: United States
Dimensions:
Height: 10mm,
Width: 10mm,
Weight: 454g
ISBN: 9781119654933
ISBN 10: 1119654939
Pages: 512
Publication Date: 24 August 2021
Audience:
Professional and scholarly
,
Undergraduate
Format: Hardback
Publisher's Status: Active
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
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.
