Focal Boundary Value Problems for Differential and Difference Equations

Abstract

Preface. 1: Continuous Problems. 1.1. Introduction. 1.2. Abel-Gontscharoff Interpolation. 1.3. Solution of Linear Problems. 1.4. Existence and Uniqueness. 1.5. Picard's and Approximate Picard's Methods. 1.6. Quasilinearization and Approximate Quasilinearization. 1.7. Integro-Differential Equations. 1.8. Delay-Differential Equations. 1.9. Necessary and Sufficient Conditions for Right Disfocality. 1.10. Tests for Right and Eventual Disfocalities. 1.11. Green's Functions. 1.12. Monotone Convergence. 1.13. Uniqueness Implies Uniqueness. 1.14. Uniqueness Implies Existence. 1.15. Continuous Dependence and Differentiation with respect to Boundary Values. 1.16. Right Disfocality Implies Right Disfocality. 1.17. Right Disfocality Implies Existence. 1.18. Differential Inequalities Imply Existence. 1.19. Infinite Interval Problems. 1.20. Best Possible Results: Control Theory Methods. 1.21. Converse Theorems. 1.22. Focal Subfunctions. 1.23. Generalized Problem I. 1.24. Generalized Problem II. 1.25. A Singular Problem. 1.26. A Problem with Impulse Effects. Comments and Remarks. References. 2: Discrete Problems. 2.1. Introduction. 2.2. Discrete Abel-Gontscharoff Interpolation. 2.3. Existence and Uniqueness. 2.4. Picard's and Approximate Picard's Methods. 2.5. Quasilinearization and Approximate Linearization. 2.6. Necessary and Sufficient Conditions for Right Disfocality. 2.7. Tests for Right and Eventual Disfocalities. 2.8. Green's Functions. 2.9. Monotone Convergence. 2.10. Continuous Dependence and Differentiation with Respect to Initial and Boundary Values. 2.11. Differences with Respect to Boundary Points. 2.12. Uniqueness Implies Existence. 2.13. Generalized Problems. Comments and Remarks. References. Index.