Dynamic Equations on Time Scales

1. The Time Scales Calculus. - 1. 1. Basic Definitions. - 1. 2. Differentiation. - 1. 3. Examples and Applications. - 1. 4. Integration. - 1. 5. Chain Rules. - 1. 6. Polynomials. - 1. 7. Further Basic Results. - 1. 8. Notes and References. - 2. First Order Linear Equations. - 2. 1. Hilger's Complex Plane. - 2. 2. The Exponential Function. - 2. 3. Examples of Exponential Functions. - 2. 4. Initial Value Problems. - 2. 5. Notes and References. - 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. 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. - 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. - 6. Dynamic Inequalities. - 6. 1. Gronwall's Inequality. - 6. 2. Holder'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. - 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. - 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. Language: English