Difference between revisions of "Book:Martin Bohner/Dynamic Equations on Time Scales"

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Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales

Online versions

Chapters 1-3 hosted by Martin Bohner

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

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

Chapter 4. Self-Adjoint Equations

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

5.3. Self-Adjoint Matrix Equations

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