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This book is devoted to the qualitative theory of functional dynamic equations on time scales, providing an overview of recent developments in the field as well as a foundation to time scales, dynamic systems, and functional dynamic equations. It discusses functional dynamic equations in relation to mathematical physics applications and problems, providing useful tools for investigation for oscillations and nonoscillations of the solutions of functional dynamic equations on time scales. Practice problems are presented throughout the book for use as a graduate-level textbook and as a reference book for specialists of several disciplines, such as mathematics, physics, engineering, and biology.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Calculus on Time Scales

Abstract
This chapter is devoted to a brief exposition of the time scale calculus. A detailed discussion of the time scale calculus is beyond the scope of this book; for this reason the author confines to outlining a minimal set of properties needed in the further proceeding.
Svetlin G. Georgiev

Chapter 2. Dynamic Systems

Abstract
Suppose that \(\mathbb {T}\) is a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively.
Svetlin G. Georgiev

Chapter 3. Functional Dynamic Equations. Basic Concepts, Existence, and Uniqueness Theorems

Abstract
Suppose that \(\mathbb {T}\) is a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively, such that \((0, \infty )\subset \mathbb {T}\). Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 4. Linear Functional Dynamic Equations

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 5. Stability for First-Order Functional Dynamic Equations

Abstract
Let \(\mathbb {T}\) be an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let also, \(\beta =\min \{t: t\in \mathbb {T}\}\) and r > 0.
Svetlin G. Georgiev

Chapter 6. Oscillations of First-Order Functional Dynamic Equations

Abstract
Let \(\mathbb {T}\) be a time scale that is unbounded above with forward jump operator and delta differentiation operator σ and Δ, respectively.
Svetlin G. Georgiev

Chapter 7. Oscillations of Second-Order Linear Functional Dynamic Equations with a Single Delay

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively.
Svetlin G. Georgiev

Chapter 8. Nonoscillations of Second-Order Functional Dynamic Equations with Several Delays

Abstract
Suppose that \(\mathbb {T}\) is a time scale that is unbounded above with forward jump operator and delta differentiation operator σ and Δ, respectively.
Svetlin G. Georgiev

Chapter 9. Oscillations of Second-Order Nonlinear Functional Dynamic Equations

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 10. Oscillations of Third-Order Functional Dynamic Equations

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 11. Oscillations of Fourth-Order Functional Dynamic Equations

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 12. Oscillations of Higher-Order Functional Dynamic Equations

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 13. Shift Operators

Abstract
Suppose that \(\mathbb {T}\) is an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let \(t_0\in \mathbb {T}\).
Svetlin G. Georgiev

Chapter 14. Impulsive Functional Dynamic Equations

Abstract
Let \(\mathbb {T}\) be a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively.
Svetlin G. Georgiev

Chapter 15. Linear Impulsive Dynamic Systems

Abstract
Suppose that \(\mathbb {T}\) be an unbounded above time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let also, \(0\in \mathbb {T}\), \(\{t_k\}_{k\in \mathbb {N}}\subset \mathbb {T}\) be such that
$$\displaystyle 0\leq t_0<t_1<\ldots <t_k<\ldots , $$
limk t k = , and t k, \(k\in \mathbb {N}\), are right-dense. Denote J = [t 0, ).
Svetlin G. Georgiev

Backmatter

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