Opial Inequalities with Applications in Differential and Difference Equations

406 Pages·1995·8.47 MB·English

by  Ravi P. Agarwal

Opial Inequalities with Applications in
Differential and Difference Equations
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume320
Opial Inequalities
with Applications in
Differential and
Difference Equations
by
Ravi P. Agarwal
and
Peter Y. H. Pang
Department of Mathematics,
National University of Singapore,
Kent Ridge, Singapore
SPRINGER-SCIENCE+BUSINESS MEDIA B.V.
A C.I.P. Catalogue record for this book is ayailable from the Library of Congress
ISBN 978-90-481-4524-9 ISBN 978-94-015-8426-5 (eBook)
DOI 10.1007/978-94-015-8426-5
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Preface
CHAPTER 1
Opial's Inequality
1.1. Introduction 1
1.2. Opial's proof 2
1.3. Olech's proof 5
1.4. Beesack's proof 7
1.5. Levinson's proof 8
1.6. Mallow's proof 9
1.7. Hua and Pederson's proof 9
References 10
CHAPTER 2
Generalizations of Opial's Inequality
2.1. Introduction 11
2.2. Beesack's generalization I 11
2.3. Hua's generalization 14
2.4. He and Wang's generalization 16
2.5. Yang's generalization 17
2.6. Maroni's generalization 22
2.7. Calvert's generalization 24
2.8. Redheffer's generalization 29
2.9. Beesack and Das' generalization 32
2.10. Beesack's generalization II 45
2.11. Shum's generalization I 52
2.12. Lee's generalization 56
2.13. He's generalization 61
2.14. Pachpatte's generalization I 63
vi
2.15. Lin and Yang's generalization 66
2.16. Lin's generalization 71
2.17. Godunova and Levin's generalization 73
2.18. Qi's generalization 76
2.19. Hwang and Yang's generalization 77
2.20. Rozanova's generalization 82
2.21. Pachpatte's generalization II 84
2.22. Mitrinovic and PecariC's generalization 87
2.23. Fagbohun and lmoru's generalization 92
2.24. Hong, Yang and Du's generalization 96
2.25. Boyd and Wong's generalization 100
2.26. G.Milovanovic and I.Milovanovic's generalization 101
2.27. Beesack's generalization III 105
2.28. Shum's generalization II 109
2.29. Sinnamon's generalization 117
2.30. Boyd's generalization 119
2.31. Vranceanu's generalization 121
References 123
CHAPTER 3
Opial Inequalities Involving
Higher Order Derivatives
3.1. Introduction 128
3.2. Willett's extension 128
3.3. Das' extension 129
3.4. Yang's extension 134
3.5. Cheung's extension 135
3.6. Li's extension 137
3.7. Fink's extension 139
3.8. Pachpatte's extension 145
3.9. Agarwal and Pang's extension I 152
3.10. Agarwal and Pang's extension II 165
3.11. Mitrinovic and PecariC's extension 179
3.12. Fagbohun and lmoru's extension 180
3.13. Boyd's extension 182
3.14. FitzGerald's extension 192
3.15. Rozanova's extension 198
3.16. Agarwal's extension 200
References 205
vii
CHAPTER4
Opial Inequalities in Several
Independent Variables
4.1. Introduction 208
4.2. Agarwal and Pang's result I 209
4.3. Agarwal and Pang's result II 220
4.4. Pachpatte's result 236
4.5. Agarwal and Sheng's result 241
4.6. Cheung's result 271
4.7. Necaev's result 273
References 278
CHAPTERS
Discrete Opial Inequalities
5.1. Introduction 283
5.2. Lasota's inequality 284
5.3. Wong's inequality 285
5.4. Lee's inequality 287
5.5. Pachpatte's inequality 291
5.6. G.Milovanovic and I.Milovanovic's inequality 294
5.7. Beesack's inequality 297
5.8. Agarwal and Pang's inequality I 308
5.9. Agarwal and Pang's inequality II 326
5.10. Agarwal and Pang's inequality III 342
References 354
CHAPTER 6
Applications
6.1. Introduction 356
6.2. Uniqueness of initial value problems 357
6.3. Upper bounds of solutions 362
6.4. Uniqueness of boundary value problems 372
References 386
Name Index 390
ix
Preface
Inequalities which involve integrals of functions and their derivatives, whose
study has a history of about one century, are of great importance in mathemat
ics, with far-reaching applications in the theory of differential equations, ap
proximations and probability, among others. This class of inequalities includes
the Wirtinger, Lyapunov, Landau-Kolmogorov, and Hardy types to which an
abundance of literature, including several monographs, have been devoted. Of
these inequalities, the earliest one which appeared in print is believed to be
a Wirtinger type inequality by L. Sheeffer in 1885 (actually before the re
sult by Wirtinger), which found its motivation in the calculus of variations.
Improvements, generalizations, extensions, discretizations, and new applica
tions of these inequalities are constantly being found, making their study an
extremely prolific field. These inequalities and their manifold manifestations
occupy a central position in mathematical analysis and its applications.
In the year 1960, the Polish mathematician Zdzidlaw Opial {1930 - 1974)
published an inequality involving integrals of a function and its derivative,
which now bears his name. While it has been shown that inequalities of this
form can be deduced from those of Wirtinger and Hardy type, the importance
of Opial's result, however, is in the establishment of the best constant. Imme
diately after its publication, several simplifications of the original proof were
offered, and generalizations and discrete analogues were considered. In the
subsequent three decades, the study of Opial type inequalities has grown into
a substantial field, with many important applications.
This is believed to be the first book dedicated to the theory of Opial type
inequalities. In writing this book, we have taken a historical approach. We
offer a complete survey of results in the field, starting with Opial 's landmark
paper, traversing through its generalizations, extensions and discretizations,
almost always with detailed proofs. These results are then compared and
contrasted, mostly in the form of remarks. To present a complete picture in
the historical development, some redundancy of material has been allowed.
Throughout the book, we have arranged the material by the authors' names.
In chapter 1, we present the original proof of Opial and five subsequent
simplified versions, which appeared in the years 1960 - 1965.
In chapter 2, we have collected the various generalizations of Opial's in-
X
equality involving one or more functions, but remaining in the context of first
order derivatives. The arrangement of material is not chronological, but is in
the order of generality or complexity. This is the longest chapter in the book
and we have tried to follow the historical development as faithfully as possible.
It is divided into 31 sections and covers 60 publications.
In chapter 3, we have put together extensions of Opial's inequalities which
involve higher order derivatives of functions. In this and the subsequent chap
ters, we have tried to avoid excessive repetition of material. Usually, the latest
and most general results are stated, and the earlier results are then deduced
from them, mostly by way of remarks. Inevitably, historical chronology is not
as thorough as in chapter 2. This chapter contains 16 sections and covers 32
publications.
In chapter 4, Opial type inequalities in several independent variables are
discussed. These results generalize several inequalities in the previous three
chapters. The subject matter is arranged in 7 sections and 58 publications are
covered.
In chapter 5, we have included discrete Opial type inequalities. Discrete
analogues of continuous results have recently attracted much interest in math
ematical analysis, not just for their importance from the application and com
putation point of view, but as a subject in its own right. Several monographs
devoted exclusively to discrete topics have appeared in the last few years. This
chapter comprises 10 sections and covers 28 publications.
Finally, some of the important applications of Opial type inequalities in
the theory of differential and difference equations, such as uniqueness of initial
value problems, existence and uniqueness of boundary value problems, and
upper bounds of solutions, are discussed in chapter 6. In this chapter, the
power of Opial type inequalities is demonstrated as they are used to obtain
results which are sharper or more general than those known previously. This
chapter contains 4 sections and covers 43 publications.
R.P.Agarwal
P.Y.H.Pang
CHAPTER 1
Opial's Inequality
1.1. Introduction
In the year 1960, Opial (6] established the following interesting integral
inequality :
Theorem 1.1.1. Let x(t) E C(1l[O, h] be such that x(O) = x(h) = 0, and
x(t) > 0 in (0, h). Then, the following inequality holds
fh lx(t)x'(t)l dt ~ !!:. fh (x'(t))2 dt. (1.1.1)
lo 4 lo
In (1.1.1), the constant h/4 is the best possible.
A weaker form of (1.1.1) can be obtained rather easily by combining the
Cauchy-Schwarz inequality and Wirtinger's inequality which states that : for
x(t) E C{ll(O, h] such that x(O) = x(h) = 0,
(1.1.2)
Indeed, we have
< !!:. fh (x'(t))2 dt.
lo
1r