Sturmian Theory for Ordinary Differential Equations

Inhaltsverzeichnis

• I. Historical Prologue.
• 1. Introduction.
• 2. Methods Based Upon Variational Principles.
• 3. Historical Comments on Terminology.
• II. Sturmian Theory for Real Linear Homogeneous Second Order Ordinary Differential Equations on a Compact Interval.
• 2. Preliminary Properties of Solutions of (1.1).
• 3. The Classical Oscillation and Comparison Theorems of Sturm.
• 4. Related Oscillation and Comparison Theorems.
• 5. Sturmian Differential Systems.
• 6. Polar Coordinate Transformations.
• 7. Transformations for Differential Equations and Systems.
• 8. Variational Properties of Solutions of (1.1).
• 9. Comparison Theorems.
• 10. Morse Fundamental Quadratic Forms for Conjugate and Focal Points.
• 11. Survey of Recent Literature.
• 12. Topics and Exercises.
• III. Self-Adjoint Boundary Problems Associated with Second Order Linear Differential Equations.
• 1. A Canonical Form for Boundary Conditions.
• 2 Extremum Problems for Self-Adjoint Systems.
• 3. Comparison Theorems.
• 4. Comments on Recent Literature.
• 5. Topics and Exercises.
• IV. Oscillation Theory on a Non-Compact Interval.
• 2. Integral Criteria for Oscillation and Non-Oscillation.
• 3. Principal Solutions.
• 4. Theory of Singular Quadratic Functionals.
• 5. Interrelations Between Oscillation Criteria and Boundary Problems.
• 6. Strong and Conditional Oscillation.
• 7. A Class of Sturmian Problems on a Non-Compact Interval.
• 8. Topics and Exercises.
• V. Sturmian Theory for Differential Systems.
• 2. Special Examples.
• 3. Preliminary Properties of Solutions of (2.5).
• 4. Associated Riccati Matrix Differential Equations.
• 5. Normality and Abnormality.
• 6. Variational Properties of Solutions of (3.1).
• 7. Comparison Theorems.
• 8. Morse Fundamental Hermitian Forms.
• 9. Generalized Polar Coordinate Transformations for Matrix Differential Systems.
• 10. Matrix Oscillation Theory.
• 11. Principal Solutions.
• 12. Comments on Systems (3.1) Which are Not Identically Normal.
• 13. Comments on the Literature on Oscillation Theory for Hamiltonian Systems (3.1).
• 14. Higher Order Differential Equations.
• 15. Topics and Exercises.
• VI. Self-Adjoint Boundary Problems.
• 2. Normality and Abnormality of Boundary Problems.
• 3. Self-Adjoint Boundary Problems Associated with (B).
• 4. Comparison Theorems.
• 5. Treatment of Self-Adjoint Boundary Problems by Matrix Oscillation Theory.
• 6. Notes and Comments on the Literature.
• 7. Topics and Exercises.
• VII. A Class of Definite Boundary Problems.
• 2. Definitely Self-Adjoint Boundary Problems.
• 3. Comments on Related Literature.
• 4. Topics and Exercises.
• VIII. Generalizations of Sturmian Theory.
• 2. Integro-Differential Boundary Problems.
• 3. A Class of Generalized Differential Equations.
• 4. Hestenes Quadratic Form Theory in a Hilbert Space.
• 5. The Weinstein Method of Intermediate Problems.
• 6. Oscillation Phenomena for Hamiltonian Systems in a B*-Algebra.
• 7. Topological Interpretations of the Sturmian Theorems.
• Abbreviations for Mathematical Publications Most Frequently Used.
• Special Symbols.
• Author Index.