Regular Variation and Differential Equations
This is the first book offering an application of regular variation to the qualitative theory of differential equations. The notion of regular variation, introduced by Karamata (1930), extended by several scientists, most significantly de Haan (1970), is
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Vojislav Maric
Regular Variation and Differential Equations
Springer
Author Vojislav Maric Fruskogorska 47 21000 Novi Sad, Yugoslavia E-mail: [email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Maric, Vojislav: Regular variation and differential equations / Vojislav Maric, - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in mathematics; 1726) ISBN 3-540-67160-9
Mathematics Subject Classification (2000): 34A45, 34ClO, 34E05, 26A12 ISSN 0075- 8434 ISBN 3-540-67160-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a company in the BertelsmannSpringer publishing group. © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10700377 4113143/du
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To the memory of Vojislav G. Avakumovic.
Preface The notion of regular variation was discovered by Jovan Karamata in his famous paper of 1930 "Sur une mode des croissance reguliere des fonctions". Karamata's aim was Tauberian theory, one of the highlights of the epoch marked by the work of eminent analysts, predominantly that of G.H. Hardy, J.1. Littlewood and also of E. Landau, culminating in N. Wiener's general Tauberian theorem in 1932. However, in addition to proving Tauberian theorems first for LaplaceStieltjes and later for the more general integral transforms of convolution type, regular variation was soon applied in Abelian theorems, giving in fact asymptotic behavior of integrals and series, the Fourier ones in particular. Further applications in analysis include Mercerian theorems, analytic number theory, complex analysisentire functions in particular. With W. Feller's well known treatise of 1968, [14]' regular variation was recognized as a major tool in the probability theory and its applications. A new impetus to the subject was provided by the L. de Haan work in 1970, [23], where he introduced a substantial generalization of regular variation, aiming again primarily at the pr
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