Theory of a Higher-Order Sturm-Liouville Equation
This book develops a detailed theory of a generalized Sturm-Liouville Equation, which includes conditions of solvability, classes of uniqueness, positivity properties of solutions and Green's functions, asymptotic properties of solutions at infinity. Of i
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1659
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Vladimir Kozlov Vladimir Maz' ya
Theory of a Higher-Order Sturm-Liouville Equation
Springer
Authors Vladimir Kozlov Department of Mathematics University of Linkoping 58183 Linkoping Sweden e-mail: [email protected] Vladimir Maz'ya Department of Mathematics University of Linkoping 58183 Linkoping Sweden e-mail: [email protected] Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kozlov, Vladimir A.: Theory of a higher order Sturm-Liouville equation / \Z Kozlov and V. Maz'ya. - Berlin; Heidelberg; New York; Barcelona; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo: Springer, 1997 (Lecture notes in mathematics; 1659) ISBN 3-540-63065-1 Mathematics Subject Classification (1991): 34-02, 34EIO, 34005, 35-16 ISSN 0075- 8434 ISBN 3-540-63065-1 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 Berlin Heidelberg 1997 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 SPIN: 10520387 46/3142-543210 - Printed on acid-free paper
Introduction
The present book is devoted to the ordinary differential equation
(at + k+)m+ (-at - L)m-w(t) - w(t)w(t)
= f(t),
(0.1)
where at is the differentiation with respect to the variable t, and k+, k.: are real numbers, k.; > k:, and m+, rn., are positive integers. In the case k.; = " - ik+)m+(>.. - iL)m_'
"J>'={3
e E (L, k+). For t > 0 we have
g(t)
=
m+-l
(' )m+-1-8
ie-k+tim_-m+ "" ----'--z-'t:-_ Z:: (rn., -1- s)! 8=0
X
(-m_)(-m_-1) ... (-m_-s+1)('k 0 k-Fg(t)
= (k+ -
k_)n+j+l-m+-m_e-k+t
xP(m_ - j.m.; - n; (k+ - L)t) and for t (at
(1.17)
0
ek-t(-at-L)m_-I(at+k+)m+V(t)=-ltek-Tf(T)dT+CI' Using (1.28) for j = rn., - 1 and the positivity of
1
00
ek-T f(T)dT
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