Oscillation Theory of Two-Term Differential Equations

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.

Klappentext

This volume is about oscillation theory. In particular, it considers the two-term linear differential equations Lny + p(x)y = 0, where Ln is a disconjugate operator of order n and p(x) has a fixed sign. Special attention is paid to the equation y(n) + p(x)y = 0. These equations enjoy a very rich structure and are the natural generalization of the Sturm-Liouville operator. Our aim is to introduce an order among the results which are distributed over hundreds of research papers, and arrange them in a unified and self-contained way. Many new proofs are given and the original proof is never copied verbatim. Numerous new results are included. Among the topics which are discussed are oscillation and nonoscillation, disconjugacy, various types of disfocality, extremal configurations of zeros, comparison theorems, classification of solutions according to their behaviour near infinity and their dominance properties. Audience: This work will be of interest to researchers and graduate students interested in the qualitative theory of differential equations.

Inhalt

0. Introduction.- 1. The Basic Lemma.- 2. Boundary Value Functions.- 3. Bases of Solutions.- 4. Comparison of Boundary Value Problems.- 5. Comparison Theorems for Two Equations.- 6. Disfocality and Its Characterization.- 7. Various Types of Disfocality.- 8. Solutions on an Infinite Interval.- 9. Disconjugacy and Its Characterization.- 10. Eigenvalue Problems.- 11. More Extremal Points.- 12. Minors of the Wronskian.- 13. The Dominance Property of Solutions.- 13. References.- 13. Index.