Oscillation Theory of Two-Term Differential Equations

✍ Scribed by Uri Elias (auth.)

• Publisher: Springer Netherlands
• Year: 1997
• Tongue: English
• Leaves: 227
• Series: Mathematics and Its Applications 396
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-vii
Introduction....Pages 1-10
The Basic Lemma....Pages 11-26
Boundary Value Functions....Pages 27-41
Bases of Solutions....Pages 42-48
Comparison of Boundary Value Problems....Pages 49-59
Comparison Theorems for Two Equations....Pages 60-66
Disfocality and its Characterization....Pages 67-97
Various Types of Disfocality....Pages 98-110
Solutions on an Infinite Interval....Pages 111-130
Disconjugacy and its Characterization....Pages 131-140
Eigenvalue Problems....Pages 141-160
More Extremal Points....Pages 161-175
Minors of the Wronskian....Pages 176-192
The Dominance Property of Solutions....Pages 193-207
References....Pages 209-216
Back Matter....Pages 217-226

✦ Subjects

Ordinary Differential Equations