Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Description

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. Cover Title Page Copyright Contents List of Contributors Foreword Preface Chapter 1 Preliminaries 1.1 Regulated Functions 1.1.1 Basic Properties 1.1.2 Equiregulated Sets 1.1.3 Uniform Convergence 1.1.4 Relatively Compact Sets 1.2 Functions of Bounded ℬ‐Variation 1.3 Kurzweil and Henstock Vector Integrals 1.3.1 Definitions 1.3.2 Basic Properties 1.3.3 Integration by Parts and Substitution Formulas 1.3.4 The Fundamental Theorem of Calculus 1.3.5 A Convergence Theorem Appendix 1.A: The McShane Integral Chapter 2 The Kurzweil Integral 2.1 The Main Background 2.1.1 Definition and Compatibility 2.1.2 Special Integrals 2.2 Basic Properties 2.3 Notes on Kapitza Pendulum Chapter 3 Measure Functional Differential Equations 3.1 Measure FDEs 3.2 Impulsive Measure FDEs 3.3 Functional Dynamic Equations on Time Scales 3.3.1 Fundamentals of Time Scales 3.3.2 The Perron Δ‐integral 3.3.3 Perron Δ‐integrals and Perron–Stieltjes integrals 3.3.4 MDEs and Dynamic Equations on Time Scales 3.3.5 Relations with Measure FDEs 3.3.6 Impulsive Functional Dynamic Equations on Time Scales 3.4 Averaging Methods 3.4.1 Periodic Averaging 3.4.2 Nonperiodic Averaging 3.5 Continuous Dependence on Time Scales Chapter 4 Generalized Ordinary Differential Equations 4.1 Fundamental Properties 4.2 Relations with Measure Differential Equations 4.3 Relations with Measure FDEs Chapter 5 Basic Properties of Solutions 5.1 Local Existence and Uniqueness of Solutions 5.1.1 Applications to Other Equations 5.2 Prolongation and Maximal Solutions 5.2.1 Applications to MDEs 5.2.2 Applications to Dynamic Equations on Time Scales Chapter 6 Linear Generalized Ordinary Differential Equations 6.1 The Fundamental Operator 6.2 A Variation‐of‐Constants Formula 6.3 Linear Measure FDEs 6.4 A Nonlinear Variation‐of‐Constants Formula for Measure FDEs Chapter 7 Continuous Dependence on Parameters 7.1 Basic Theory for Generalized ODEs 7.2 Applications to Measure FDEs Chapter 8 Stability Theory 8.1 Variational Stability for Generalized ODEs 8.1.1 Direct Method of Lyapunov 8.1.2 Converse Lyapunov Theorems 8.2 Lyapunov Stability for Generalized ODEs 8.2.1 Direct Method of Lyapunov 8.3 Lyapunov Stability for MDEs 8.3.1 Direct Method of Lyapunov 8.4 Lyapunov Stability for Dynamic Equations on Time Scales 8.4.1 Direct Method of Lyapunov 8.5 Regular Stability for Generalized ODEs 8.5.1 Direct Method of Lyapunov 8.5.2 Converse Lyapunov Theorem Chapter 9 Periodicity 9.1 Periodic Solutions and Floquet's Theorem 9.1.1 Linear Differential Systems with Impulses 9.2 (θ,T)‐Periodic Solutions 9.2.1 An Application to MDEs Chapter 10 Averaging Principles 10.1 Periodic Averaging Principles 10.1.1 An Application to IDEs 10.2 Nonperiodic Averaging Principles Chapter 11 Boundedness of Solutions 11.1 Bounded Solutions and Lyapunov Functionals 11.2 An Application to MDEs 11.2.1 An Example Chapter 12 Control Theory 12.1 Controllability and Observability 12.2 Applications to ODEs Chapter 13 Dichotomies 13.1 Basic Theory for Generalized ODEs 13.2 Boundedness and Dichotomies 13.3 Applications to MDEs 13.4 Applications to IDEs Chapter 14 Topological Dynamics 14.1 The Compactness of the Class ℱ0(Ω,h) 14.2 Existence of a Local Semidynamical System 14.3 Existence of an Impulsive Semidynamical System 14.4 LaSalle's Invariance Principle 14.5 Recursive Properties Chapter 15 Applications to Functional Differential Equations of Neutral Type 15.1 Drops of History 15.2 FDEs of Neutral Type with Finite Delay References List of Symbols Index EULA