# Book:Martin Bohner/Dynamic Equations on Time Scales

## Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales

### Contents

Preface
Chapter 1. The Time Scales Calculus
1.1. Basic Definitions
page 1
Definition $1.1$
Theorem $1.7$
1.2. Differentiation
Definition $1.10$
Theorem $1.16 (i)$
Theorem $1.16 (ii)$
Theorem $1.16 (iii)$
Theorem $1.16 (iv)$
Theorem $1.20 (i)$
Theorem $1.20 (ii)$
Theorem $1.20 (iii)$ (and Theorem $1.20 (iii)$)
Theorem $1.20 (iv)$
Theorem $1.20 (v)$
Theorem $1.24 (i)$
Theorem $1.24 (ii)$
1.3. Examples and Applications
1.4. Integration
Definition $1.57$
Definition $1.58$
Theorem $1.60(i)$
Theorem $1.60(ii)$
Theorem $1.60(iii)$
Theorem $1.60(iv)$
Theorem $1.60(v)$
Definition $1.62$
Theorem $1.65$
Theorem $1.67$
Corollary $1.68(i)$
Corollary $1.68(ii)$
Corollary $1.68(iii)$
1.5. Chain Rules
1.6. Polynomials
1.7. Further Basic Results
1.8. Notes and References
Chapter 2. First Order Linear Equations
2.1. Hilger's Complex Plane
Hilger complex plane
Hilger real axis
Hilger alternating axis
Hilger circle
Hilger real part
Hilger imaginary part
Hilger pure imaginary
2.2. The Exponential Function
2.3. Examples of Exponential Functions
2.4. Initial Value Problems
2.5. Notes and References
Chapter 3. Second Order Linear Equations
3.1. Wronskians
3.2. Hyperbolic and Trigonometric Functions
3.3. Reduction of Order
3.4. Method of Factoring
3.5. Nonconstant Coefficients
3.6. Hyperbolic and Trigonometric Functions II
3.7. Euler-Cauchy Equations
3.8. Variation of Parameters
3.9. Annihilator Method
3.10. Laplace Transform
3.11. Notes and References
4.1. Preliminaries and Examples
4.2. The Riccati Equation
4.3. Disconjugacy
4.4. Boundary Value Problems and Green's Function
4.5. Eigenvalue Problems
4.6. Notes and References
Chapter 5. Linear Systems and Higher Order Equations
5.1. Regressive Matrices
5.2. Constant Coefficients
5.4. Asymptotic Behavior of Solutions
5.5. Higher Order Linear Dynamic Equations
5.6. Notes and References
Chapter 6. Dynamic Inequalities
6.1. Gronwall's Inequality
6.2. Hölder's and Minkowski's Inequalities
6.3. Jensen's Inequality
6.4. Opial Inequalities
6.5. Lyapunov Inequalities
6.6. Upper and Lower Solutions
6.7. Notes and References
Chapter 7. Linear Symplectic Dynamic Systems
7.1. Symplectic Systems and Special Cases
7.2. Conjoined Bases
7.3. Transformation Theory and Trigonometric Systems
7.4. Notes and References
Chapter 8. Extensions
8.1. Measure Chains
8.2. Nonlinear Theory
8.3. Alpha Derivatives
8.4. Nabla Derivatives
8.5. Notes and References
Solutions to Selected Problems
Bibliography
Index