Oscillation Theory of Two-Term Differential Equations

1. Front Matter

   Pages i-vii

   PDF

2. Introduction

   • Uri Elias

   Pages 1-10

3. The Basic Lemma

   • Uri Elias

   Pages 11-26

4. Boundary Value Functions

   • Uri Elias

   Pages 27-41

5. Bases of Solutions

   • Uri Elias

   Pages 42-48

6. Comparison of Boundary Value Problems

   • Uri Elias

   Pages 49-59

7. Comparison Theorems for Two Equations

   • Uri Elias

   Pages 60-66

8. Disfocality and its Characterization

   • Uri Elias

   Pages 67-97

9. Various Types of Disfocality

   • Uri Elias

   Pages 98-110

10. Solutions on an Infinite Interval

    • Uri Elias

    Pages 111-130

11. Disconjugacy and its Characterization

    • Uri Elias

    Pages 131-140

12. Eigenvalue Problems

    • Uri Elias

    Pages 141-160

13. More Extremal Points

    • Uri Elias

    Pages 161-175

14. Minors of the Wronskian

    • Uri Elias

    Pages 176-192

15. The Dominance Property of Solutions

    • Uri Elias

    Pages 193-207

16. References

    • Uri Elias

    Pages 209-216

17. Back Matter

    Pages 217-226

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Oscillation theory was born with Sturm's work in 1836. It has been flourishing for the past fifty years. Nowadays it is a full, self-contained discipline, turning more towards nonlinear and functional differential equations. Oscillation theory flows along two main streams. The first aims to study prop­ erties which are common to all linear differential equations. The other restricts its area of interest to certain families of equations and studies in maximal details phenomena which characterize only those equations. Among them we find third and fourth order equations, self adjoint equations, etc. Our work belongs to the second type and considers two term linear equations modeled after y(n) + p(x)y = O. More generally, we investigate LnY + p(x)y = 0, where Ln is a disconjugate operator and p(x) has a fixed sign. These equations enjoy a very rich structure and are the natural generalization of the Sturm-Liouville operator. Results about such equations are distributed over hundreds of research papers, many of them are reinvented again and again and the same phenomenon is frequently discussed from various points of view and different definitions of the authors. Our aim is to introduce an order into this plenty and arrange it in a unified and self contained way. The results are readapted and presented in a unified approach. In many cases completely new proofs are given and in no case is the original proof copied verbatim. Many new results are included.

• Boundary value problem
• DEX
• Finite
• Lemma
• Natural
• boundary element method
• character
• differential equation
• eigenvalue problem
• equation
• extrema
• function
• functions
• operator
• theorem
• ordinary differential equations

• Department of Mathematics, Technion Israel Institute of Technology, Haifa, Israel

  Uri Elias

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