Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

✍ Scribed by Murat Adıvar, Youssef N. Raffoul

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 426
• Edition: 1st ed. 2020
• Category: Library

✦ Synopsis

Motivated by recent increased activity of research on time scales, the book  provides a systematic approach to the study of the qualitative theory of boundedness, periodicity and stability of Volterra integro-dynamic equations on time scales. Researchers and graduate students who are interested in the method of Lyapunov functions/functionals, in the study of boundedness of solutions, in the stability of the zero solution, or in the existence of periodic solutions should be able to use this book as a primary reference and as a resource of latest findings. This book contains many open problems and should be of great benefit to those who are pursuing research in dynamical systems or in Volterra integro-dynamic equations on time scales with or without delays. Great efforts were made to present rigorous and detailed proofs of theorems. The book should serve as an encyclopedia on the construction of Lyapunov functionals in analyzing solutions of dynamical systems on time scales. The book is suitable for a graduate course in the format of graduate seminars or as special topics course on dynamical systems.

The book should be of interest to investigators in biology, chemistry, economics, engineering, mathematics and physics.

✦ Table of Contents

Preface
Contents
1 Introduction to Stability and Boundedness in Dynamical Systems
1.1 Introduction
1.2 Introduction to Lyapunov Functions
1.3 Delay Dynamic Equations
1.4 Resolvent for Volterra Integral Dynamic Equations
1.4.1 Existence of Resolvent: Lp Case
1.4.2 Shift Operators
1.4.3 Continuity of r(t,s)
1.5 Periodicity on Time Scales
1.6 Open Problems
2 Ordinary Dynamical Systems
2.1 Boundedness
2.1.1 Applications
2.2 Exponential Stability
2.2.1 Applications
2.3 Perturbed Systems
2.4 Delay Dynamical Systems
2.4.1 Delay Dynamical Systems-Linear Case
2.4.2 Delay Dynamical Systems-Nonlinear Case
2.4.3 Applications
2.4.4 Perturbed Vector Nonlinear Delayed Systems
2.5 Open Problems
3 Functional Dynamical Systems
3.1 Uniform Boundedness and Uniform Ultimate Boundedness
3.2 Applications to Volterra Integro-Dynamic Equations
3.3 More on Boundedness
3.3.1 Applications to Nonlinear Volterra Integro-DynamicEquations
3.4 Exponential Stability
3.4.1 Applications to Volterra Integro-Dynamic Equations
3.5 Open Problems
4 Volterra Integro-Dynamic Equations
4.1 Generalized Gronwall's Inequality and Functional Bound
4.2 Principal Matrix Solutions
4.2.1 Existence and Uniqueness
4.2.2 Existence of Principal Matrix
4.2.3 Variation of Parameters
4.3 Necessary and Sufficient Conditions for Stability via Resolvent
4.3.1 Introduction
4.3.2 New Resolvent Equation and Variation of Parameters
4.3.3 Uniform Stability
4.3.4 Applications and Comparison
4.4 Necessary and Sufficient Conditions via Lyapunov Functionals
4.4.1 Applications and Comparison
4.5 Classification of Solutions
4.5.1 Classifications of Positive Solutions
4.5.2 Applications
4.5.3 Classifications of Negative Solutions
4.6 Open Problems
5 Exotic Lyapunov Functionals for Boundedness and Stability
5.1 Various Results Using Lyapunov Functionals
5.1.1 Necessary and Sufficient Conditions: The Scalar Case
5.1.2 Necessary and Sufficient Conditions: The Vector Case
5.2 Shift Operators and Stability in Delay Systems
5.2.1 Generalized Delay Functions
5.2.2 Exponential Stability via Lyapunov Functionals
5.2.3 Instability
5.2.4 Applications and Comparison
5.3 Open Problems
6 Volterra Integral Dynamic Equations
6.1 The Resolvent Method
6.1.1 Existence and Uniqueness
6.2 Continuation of Solutions
6.3 Lyapunov Functionals Method
6.3.1 Applications to Particular Time Scales
6.4 Open Problems
7 Periodic Solutions: The Natural Setup
7.1 Neutral Nonlinear Dynamic Equation
7.1.1 Existence of Periodic Solutions
7.2 Nonlinear Dynamic Equation
7.2.1 Existence of Periodic Solutions
7.2.2 Stability
7.3 Delayed Dynamic Equations
7.3.1 Stability
7.3.2 Periodicity
7.4 Integro-Dynamic Equation
7.4.1 Existence of Periodic Solutions
7.4.2 Applications to Scalar Equations
7.5 Schaefer Theorem and Infinite Delay Volterra Integro-Dynamic Equations
7.5.1 Existence of Periodic Solutions
7.5.2 Application to Infinite Delay System
7.6 Connection Between Boundedness and Periodicity
7.6.1 Existence
7.6.2 Connection Between Boundedness and Periodicity
7.7 Almost Linear Volterra Integro-Dynamic Equations
7.7.1 Periodic Solutions
7.8 Almost Automorphic Solutions of Delayed Neutral DynamicSystems on Time Scales
7.8.1 Almost Automorphy Notion on Time Scales
7.8.2 Exponential Dichotomy and Limiting Results
7.8.3 Existence Results
7.9 Large Contraction and Existence of Periodic Solutions
7.9.1 Existence of Periodic Solution
7.9.2 Classification of Large Contractions
7.10 Open Problem
8 Periodicity Using Shift Periodic Operators
8.1 Periodicity in Shifts
8.1.1 Periodicity in Quantum Calculus
8.1.1.1 Connection Between Two q-Periodicity Notions on qN0
8.2 Floquet Theory on Time Scales Periodic in Shifts δ
8.2.1 Floquet Theory Based on New Periodicity Concept:Homogeneous Case
8.2.2 Floquet Theory Based on New Periodicity Concept:Nonhomogeneous Case
8.2.3 Floquet Multipliers and Floquet Exponents of UnifiedFloquet Systems
8.3 Stability Properties of Unified Floquet Systems
8.4 Existence of Periodic Solutions in Shifts δ for Neutral Nonlinear Dynamic Systems
8.4.1 Existence of Periodic Solutions
8.5 Open Problems
References
Index

✦ Subjects

Mathematics;Calculus; Differential equations