Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions
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989 Angelo B. Mingarelli
Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Angelo B. Mingarelli Department of Mathematics, University of Ottawa 585 King-Edward Avenue, Ottawa, Ontario, Canada K1N 9B4
AMS Subject Classifications (1980): Primary: 45J 05,45005,47 A 99 Secondary: 34 B 25, 34 C 10, 39 A 10, 39 A 12, 47 B 50 ISBN 3-540-12294-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12294-X Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the materia is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Quest' opera
e
umilmente dedicata
ai miei cari genitori Giosafat Oliviana e al mio fratello Marco A.M.D.G.
e
PREFACE The aim of these notes is to pursue a line of research adopted by many authors (We Feller, M.G. Krein, 1.5.
Kac,
F.V. Atkinson, W.T. Reid, among others) in order to develop a qualitative and spectral theory of Volterra-Stieltjes integral equations with specific applications to real ordinary differential and difference equations of the second order. We begin by an extension of the classical results of Sturm (comparison theorem, separation theorem) to this more general setting.
In chapter 2 we study the oscillation theory
of such equations and, in Chapters 3,4,5, apply some aspects of it to the study of the spectrum of the operators generated by certain generalized ordinary differential expressions associated with the above-mentioned integral equations. In order to make these notes self-contained some appendices have been added which include results fundamental to the main text.
Care has been taken to give due credit to those
researchers who have contributed to the development of the theory presented herein - any omissions or errors are the author's sole responsibility. I am greatly indebted to Professor F.V. Atkinson at whose hands I learned the subject and I also take this opportunity to acknowledge with thanks the assistance of the Natural Sciences and Engineering Research Council of Canada for continued financial support.
My sincere thanks go to Mrs. Frances Mitchell
VI
for her expert typing of the manuscript. Finally, I am deeply grateful to my wife Leslie Jean for her constant encouragement and patience and I also wish to thank Professor A. Dold for the possibility to publish the manuscript in the Lecture Note series.
Angelo B. Mingarelli Ottawa, April 1980.
TABLE OF CONTENTS
x
INTRODUCTION CHAPTER 1 Introduction
1
1.1.
Comparison Theorems for Stieltjes Integ
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