Comparison and Oscillation Theory of Linear Differential Equations

Mathematics in Science and Engineering, Volume 48: Comparison and Oscillation Theory of Linear Differential Equations deals primarily with the zeros of solutions of linear differential equations.

This volume contains five chapters. Chapter 1 focuses on comparison theorems for second order equations, while Chapter 2 treats oscillation and nonoscillation theorems for second order equations. Separation, comparison, and oscillation theorems for fourth order equations are covered in Chapter 3. In Chapter 4, ordinary equations and systems of differential equations are reviewed. The last chapter discusses the result of the first analog of a Sturm-type comparison theorem for an elliptic partial differential equation.

This publication is intended for college seniors or beginning graduate students who are well-acquainted with advanced calculus, complex analysis, linear algebra, and linear differential equations.

Inhalt
Preface
Chapter 1. Sturm-Type Theorems for Second Order Ordinary Equations

1. Comparison Theorems for Self-Adjoint Equations

2. Additional Results of Leighton

3. Extension to General Second Order Equations

4. Comparison Theorems for Singular Equations

5. Comparison Theorems for Eigenfunctions

6. Reid's Comparison Theorems on Focal Points

7. Levin's Comparison Theorems

8. The Order of Zeros

Chapter 2. Oscillation and Nonoscillation Theorems for Second Order Ordinary Equations

1. The Oscillation Criteria of Hille and Nehari

2. Conditionally Oscillatory Equations

3. Nehari's Comparison Theorems

4. The Hille-Wintner Comparison Theorem

5. Hille's Necessary and Sufficient Conditions for Nonoscillatory Equations

6. Leighton's Oscillation Criteria

7. Potter's Oscillation Criteria

8. Hille's Kneser-Type Oscillation Criteria

9. Nonoscillation Theorems of Hartman and Wintner

10. Asymptotic Estimates for the Number of Zeros of a Solution of (1.1) or (2.1)

11. Nonoscillation Criteria for Hill's Equation

12. Nonoscillation Criteria for Complex Equations

Chapter 3. Fourth Order Ordinary Equations

1. Introduction

2. Separation Theorems

3. Comparison Theorems for (3.2) and (3.3)

4. Comparison Theorems for Other Fourth Order Equations

5. Comparison Theorems for Eigenfunctions

6. Nonoscillation Theorems

7. Leighton and Nehari's Sufficient Conditions for Nonoscillatory Equations

8. Comparison Theorems for Nonoscillation

9. Howard's Comparison Theorems for Eigenvalue Problems

Chapter 4. Third Order Ordinary Equations, nth Order Ordinary Equations and Systems

1. Introduction

2. Separation Theorems for Third Order Equations

3. Comparison Theorems for Third Order Equations

4. Oscillation Criteria for Third Order Equations

5. Separation and Comparison Theorems for nth Order Equations

6. General Oscillation Theorems

7. Nonoscillation Theorems for Systems of Differential Equations

8. Whyburn's Second Order System

Chapter 5. Partial Differential Equations

1. Introduction

2. Comparison Theorems for Self-Adjoint Equations in Bounded Domains

3. Comparison Theorems for General Second Order Elliptic Equations

4. Comparison Theorems on Unbounded Domains

5. Extension to Complex-Valued Solutions and Subsolutions

6. Lower Bounds for Eigenvalues

7. Oscillation Theorems

8. Comparison Theorems for Eigenfunctions

Bibliography

Author Index

Subject Index

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