Reid Sturmian Theory for Ordinary Differential Equations

Erscheinungsjahr 2012
ISBN: 978-1-4612-6110-0
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 31, 560 Seiten, eBook

Reihe: Applied Mathematical Sciences

Zielgruppe

Research

Autoren/Hrsg.

Reid, William T.

Weitere Mitwirkende

Burns, J.

Herdman, T.

Ahlbrandt, C.

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.- 1. Introduction.- 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.- 1. Introduction.- 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.- 1. Introduction.- 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.- 1. Introduction.- 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.- 1. Introduction.- 2. Definitely Self-Adjoint Boundary Problems.- 3. Comments on Related Literature.- 4. Topics and Exercises.- VIII. Generalizations of Sturmian Theory.- 1. Introduction.- 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.