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.

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• List of Contributors xiForeword xiiiPreface xvii1 Preliminaries 1Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, and Eduard Toon1.1 Regulated Functions 21.1.1 Basic Properties 21.1.2 Equiregulated Sets 71.1.3 Uniform Convergence 91.1.4 Relatively Compact Sets 111.2 Functions of Bounded B-Variation 141.3 Kurzweil and Henstock Vector Integrals 191.3.1 Definitions 201.3.2 Basic Properties 251.3.3 Integration by Parts and Substitution Formulas 291.3.4 The Fundamental Theorem of Calculus 361.3.5 A Convergence Theorem 44Appendix 1.A: The McShane Integral 442 The Kurzweil Integral 53Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Jaqueline G. Mesquita2.1 The Main Background 542.1.1 Definition and Compatibility 542.1.2 Special Integrals 562.2 Basic Properties 572.3 Notes on Kapitza Pendulum 673 Measure Functional Differential Equations 71Everaldo M. Bonotto, Márcia Federson, Miguel V. S. Frasson, Rogelio Grau, and Jaqueline G. Mesquita3.1 Measure FDEs 743.2 Impulsive Measure FDEs 763.3 Functional Dynamic Equations on Time Scales 863.3.1 Fundamentals of Time Scales 873.3.2 The Perron Δ-integral 893.3.3 Perron Δ-integrals and Perron–Stieltjes integrals 903.3.4 MDEs and Dynamic Equations on Time Scales 983.3.5 Relations with Measure FDEs 993.3.6 Impulsive Functional Dynamic Equations on Time Scales 1043.4 Averaging Methods 1063.4.1 Periodic Averaging 1073.4.2 Nonperiodic Averaging 1183.5 Continuous Dependence on Time Scales 1354 Generalized Ordinary Differential Equations 145Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita4.1 Fundamental Properties 1464.2 Relations with Measure Differential Equations 1534.3 Relations with Measure FDEs 1605 Basic Properties of Solutions 173Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon5.1 Local Existence and Uniqueness of Solutions 1745.1.1 Applications to Other Equations 1785.2 Prolongation and Maximal Solutions 1815.2.1 Applications to MDEs 1915.2.2 Applications to Dynamic Equations on Time Scales 1976 Linear Generalized Ordinary Differential Equations 205Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Miguel V. S. Frasson6.1 The Fundamental Operator 2076.2 A Variation-of-Constants Formula 2096.3 Linear Measure FDEs 2166.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs 2207 Continuous Dependence on Parameters 225Suzete M. Afonso, Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita7.1 Basic Theory for Generalized ODEs 2267.2 Applications to Measure FDEs 2368 StabilityTheory 241Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon8.1 Variational Stability for Generalized ODEs 2448.1.1 Direct Method of Lyapunov 2468.1.2 Converse Lyapunov Theorems 2478.2 Lyapunov Stability for Generalized ODEs 2568.2.1 Direct Method of Lyapunov 2578.3 Lyapunov Stability for MDEs 2618.3.1 Direct Method of Lyapunov 2638.4 Lyapunov Stability for Dynamic Equations on Time Scales 2658.4.1 Direct Method of Lyapunov 2678.5 Regular Stability for Generalized ODEs 2728.5.1 Direct Method of Lyapunov 2758.5.2 Converse Lyapunov Theorem 2829 Periodicity 295Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Maria Carolina Mesquita9.1 Periodic Solutions and Floquet’s Theorem 2979.1.1 Linear Differential Systems with Impulses 3039.2 (θ,T)-Periodic Solutions 3079.2.1 An Application to MDEs 31310 Averaging Principles 317Márcia Federson and Jaqueline G. Mesquita10.1 Periodic Averaging Principles 32010.1.1 An Application to IDEs 32510.2 Nonperiodic Averaging Principles 33011 Boundedness of Solutions 341Suzete 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 34211.2 An Application to MDEs 35211.2.1 An Example 35612 Control Theory 361Fernanda Andrade da Silva, Márcia Federson, and Eduard Toon12.1 Controllability and Observability 36212.2 Applications to ODEs 36513 Dichotomies 369Everaldo M. Bonotto and Márcia Federson13.1 Basic Theory for Generalized ODEs 37013.2 Boundedness and Dichotomies 38113.3 Applications to MDEs 39113.4 Applications to IDEs 40014 Topological Dynamics 407Suzete M. Afonso, Marielle Ap. Silva, Everaldo M. Bonotto, and Márcia Federson14.1 The Compactness of the Class F0(Ω,h) 40814.2 Existence of a Local Semidynamical System 41114.3 Existence of an Impulsive Semidynamical System 41814.4 LaSalle’s Invariance Principle 42314.5 Recursive Properties 42515 Applications to Functional Differential Equations of Neutral Type 429Fernando G. Andrade, Miguel V. S. Frasson, and Patricia H. Tacuri15.1 Drops of History 42915.2 FDEs of Neutral Type with Finite Delay 435References 455List of Symbols 471Index 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.

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• List of Contributors xiForeword xiiiPreface xvii1 Preliminaries 1Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, and Eduard Toon1.1 Regulated Functions 21.1.1 Basic Properties 21.1.2 Equiregulated Sets 71.1.3 Uniform Convergence 91.1.4 Relatively Compact Sets 111.2 Functions of Bounded B-Variation 141.3 Kurzweil and Henstock Vector Integrals 191.3.1 Definitions 201.3.2 Basic Properties 251.3.3 Integration by Parts and Substitution Formulas 291.3.4 The Fundamental Theorem of Calculus 361.3.5 A Convergence Theorem 44Appendix 1.A: The McShane Integral 442 The Kurzweil Integral 53Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Jaqueline G. Mesquita2.1 The Main Background 542.1.1 Definition and Compatibility 542.1.2 Special Integrals 562.2 Basic Properties 572.3 Notes on Kapitza Pendulum 673 Measure Functional Differential Equations 71Everaldo M. Bonotto, Márcia Federson, Miguel V. S. Frasson, Rogelio Grau, and Jaqueline G. Mesquita3.1 Measure FDEs 743.2 Impulsive Measure FDEs 763.3 Functional Dynamic Equations on Time Scales 863.3.1 Fundamentals of Time Scales 873.3.2 The Perron Δ-integral 893.3.3 Perron Δ-integrals and Perron–Stieltjes integrals 903.3.4 MDEs and Dynamic Equations on Time Scales 983.3.5 Relations with Measure FDEs 993.3.6 Impulsive Functional Dynamic Equations on Time Scales 1043.4 Averaging Methods 1063.4.1 Periodic Averaging 1073.4.2 Nonperiodic Averaging 1183.5 Continuous Dependence on Time Scales 1354 Generalized Ordinary Differential Equations 145Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita4.1 Fundamental Properties 1464.2 Relations with Measure Differential Equations 1534.3 Relations with Measure FDEs 1605 Basic Properties of Solutions 173Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon5.1 Local Existence and Uniqueness of Solutions 1745.1.1 Applications to Other Equations 1785.2 Prolongation and Maximal Solutions 1815.2.1 Applications to MDEs 1915.2.2 Applications to Dynamic Equations on Time Scales 1976 Linear Generalized Ordinary Differential Equations 205Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Miguel V. S. Frasson6.1 The Fundamental Operator 2076.2 A Variation-of-Constants Formula 2096.3 Linear Measure FDEs 2166.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs 2207 Continuous Dependence on Parameters 225Suzete M. Afonso, Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita7.1 Basic Theory for Generalized ODEs 2267.2 Applications to Measure FDEs 2368 StabilityTheory 241Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon8.1 Variational Stability for Generalized ODEs 2448.1.1 Direct Method of Lyapunov 2468.1.2 Converse Lyapunov Theorems 2478.2 Lyapunov Stability for Generalized ODEs 2568.2.1 Direct Method of Lyapunov 2578.3 Lyapunov Stability for MDEs 2618.3.1 Direct Method of Lyapunov 2638.4 Lyapunov Stability for Dynamic Equations on Time Scales 2658.4.1 Direct Method of Lyapunov 2678.5 Regular Stability for Generalized ODEs 2728.5.1 Direct Method of Lyapunov 2758.5.2 Converse Lyapunov Theorem 2829 Periodicity 295Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Maria Carolina Mesquita9.1 Periodic Solutions and Floquet’s Theorem 2979.1.1 Linear Differential Systems with Impulses 3039.2 (θ,T)-Periodic Solutions 3079.2.1 An Application to MDEs 31310 Averaging Principles 317Márcia Federson and Jaqueline G. Mesquita10.1 Periodic Averaging Principles 32010.1.1 An Application to IDEs 32510.2 Nonperiodic Averaging Principles 33011 Boundedness of Solutions 341Suzete 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 34211.2 An Application to MDEs 35211.2.1 An Example 35612 Control Theory 361Fernanda Andrade da Silva, Márcia Federson, and Eduard Toon12.1 Controllability and Observability 36212.2 Applications to ODEs 36513 Dichotomies 369Everaldo M. Bonotto and Márcia Federson13.1 Basic Theory for Generalized ODEs 37013.2 Boundedness and Dichotomies 38113.3 Applications to MDEs 39113.4 Applications to IDEs 40014 Topological Dynamics 407Suzete M. Afonso, Marielle Ap. Silva, Everaldo M. Bonotto, and Márcia Federson14.1 The Compactness of the Class F0(Ω,h) 40814.2 Existence of a Local Semidynamical System 41114.3 Existence of an Impulsive Semidynamical System 41814.4 LaSalle’s Invariance Principle 42314.5 Recursive Properties 42515 Applications to Functional Differential Equations of Neutral Type 429Fernando G. Andrade, Miguel V. S. Frasson, and Patricia H. Tacuri15.1 Drops of History 42915.2 FDEs of Neutral Type with Finite Delay 435References 455List of Symbols 471Index 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.

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• List of Contributors xiForeword xiiiPreface xvii1 Preliminaries 1Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, and Eduard Toon1.1 Regulated Functions 21.1.1 Basic Properties 21.1.2 Equiregulated Sets 71.1.3 Uniform Convergence 91.1.4 Relatively Compact Sets 111.2 Functions of Bounded B-Variation 141.3 Kurzweil and Henstock Vector Integrals 191.3.1 Definitions 201.3.2 Basic Properties 251.3.3 Integration by Parts and Substitution Formulas 291.3.4 The Fundamental Theorem of Calculus 361.3.5 A Convergence Theorem 44Appendix 1.A: The McShane Integral 442 The Kurzweil Integral 53Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Jaqueline G. Mesquita2.1 The Main Background 542.1.1 Definition and Compatibility 542.1.2 Special Integrals 562.2 Basic Properties 572.3 Notes on Kapitza Pendulum 673 Measure Functional Differential Equations 71Everaldo M. Bonotto, Márcia Federson, Miguel V. S. Frasson, Rogelio Grau, and Jaqueline G. Mesquita3.1 Measure FDEs 743.2 Impulsive Measure FDEs 763.3 Functional Dynamic Equations on Time Scales 863.3.1 Fundamentals of Time Scales 873.3.2 The Perron Δ-integral 893.3.3 Perron Δ-integrals and Perron–Stieltjes integrals 903.3.4 MDEs and Dynamic Equations on Time Scales 983.3.5 Relations with Measure FDEs 993.3.6 Impulsive Functional Dynamic Equations on Time Scales 1043.4 Averaging Methods 1063.4.1 Periodic Averaging 1073.4.2 Nonperiodic Averaging 1183.5 Continuous Dependence on Time Scales 1354 Generalized Ordinary Differential Equations 145Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita4.1 Fundamental Properties 1464.2 Relations with Measure Differential Equations 1534.3 Relations with Measure FDEs 1605 Basic Properties of Solutions 173Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon5.1 Local Existence and Uniqueness of Solutions 1745.1.1 Applications to Other Equations 1785.2 Prolongation and Maximal Solutions 1815.2.1 Applications to MDEs 1915.2.2 Applications to Dynamic Equations on Time Scales 1976 Linear Generalized Ordinary Differential Equations 205Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Miguel V. S. Frasson6.1 The Fundamental Operator 2076.2 A Variation-of-Constants Formula 2096.3 Linear Measure FDEs 2166.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs 2207 Continuous Dependence on Parameters 225Suzete M. Afonso, Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita7.1 Basic Theory for Generalized ODEs 2267.2 Applications to Measure FDEs 2368 StabilityTheory 241Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon8.1 Variational Stability for Generalized ODEs 2448.1.1 Direct Method of Lyapunov 2468.1.2 Converse Lyapunov Theorems 2478.2 Lyapunov Stability for Generalized ODEs 2568.2.1 Direct Method of Lyapunov 2578.3 Lyapunov Stability for MDEs 2618.3.1 Direct Method of Lyapunov 2638.4 Lyapunov Stability for Dynamic Equations on Time Scales 2658.4.1 Direct Method of Lyapunov 2678.5 Regular Stability for Generalized ODEs 2728.5.1 Direct Method of Lyapunov 2758.5.2 Converse Lyapunov Theorem 2829 Periodicity 295Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Maria Carolina Mesquita9.1 Periodic Solutions and Floquet’s Theorem 2979.1.1 Linear Differential Systems with Impulses 3039.2 (θ,T)-Periodic Solutions 3079.2.1 An Application to MDEs 31310 Averaging Principles 317Márcia Federson and Jaqueline G. Mesquita10.1 Periodic Averaging Principles 32010.1.1 An Application to IDEs 32510.2 Nonperiodic Averaging Principles 33011 Boundedness of Solutions 341Suzete 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 34211.2 An Application to MDEs 35211.2.1 An Example 35612 Control Theory 361Fernanda Andrade da Silva, Márcia Federson, and Eduard Toon12.1 Controllability and Observability 36212.2 Applications to ODEs 36513 Dichotomies 369Everaldo M. Bonotto and Márcia Federson13.1 Basic Theory for Generalized ODEs 37013.2 Boundedness and Dichotomies 38113.3 Applications to MDEs 39113.4 Applications to IDEs 40014 Topological Dynamics 407Suzete M. Afonso, Marielle Ap. Silva, Everaldo M. Bonotto, and Márcia Federson14.1 The Compactness of the Class F0(Ω,h) 40814.2 Existence of a Local Semidynamical System 41114.3 Existence of an Impulsive Semidynamical System 41814.4 LaSalle’s Invariance Principle 42314.5 Recursive Properties 42515 Applications to Functional Differential Equations of Neutral Type 429Fernando G. Andrade, Miguel V. S. Frasson, and Patricia H. Tacuri15.1 Drops of History 42915.2 FDEs of Neutral Type with Finite Delay 435References 455List of Symbols 471Index 473