Dichotomies in Stability Theory
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629
w. A. Coppel
Dichotomies in Stability Theory
Springer-Verlag Berlin Heidelberg New York 1978
Author
W.A. Cappel Department of Mathematics Institute of Advanced Studies Australian National University Canberra, ACT, 2600/Australia
AMS Subject Classifications (1970): 34D05, 58F15 ISBN 3-540-08536-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08536-X Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
PREFACE
Several years ago I formed the view that dichotomies, rather than Lyapunov's characteristic exponents, are the key to questions of asymptotic behaviour for nonautonomous differential equations.
I still hold that view, in spite of the fact that
since then there have appeared many more papers and a book on characteristic exponents.
On the other hand, there has recently been an important new development
in the theory of dichotomies.
Thus it seemed to me an appropriate time to give an
accessible account of this attractive theory. The present lecture notes are the basis for a course given at the University of Florence in May, 1977.
I am grateful to Professor R. Conti for the invitation to
visit there and for providing the incentive to put my thoughts in order.
I am also
grateful to Mrs Helen Daish and Mrs Linda Southwell for cheerfully and carefully typing the manuscript.
CONTENTS
STABILITY
Lecture 4.
Lecture 8.
Appendix Notes
1
EXPONENTIAL AND ORDINARY DICHOTOMIES
10
DICHOTOMIES AND FUNCTIONAL ANALYSIS
20
ROUGHNESS
28
DICHOTOMIES AND REDUCIBILITY
38
CRITERIA FOR AN EXPONENTIAL DICHOTOMY
1+7
DICHOTOMIES AND LYAPUNOV FUNCTIONS
59
EQUATIONS ON
67
R AND ALMOST PERIODIC EQUATIONS
DICHOTOMIES AND THE HULL OF AN EQUATION
71+
THE METHOD OF PERRON
87
92 95
Subject Index
98
1.
STABILITY
A dichotomy, exponential or ordinary, is a type of conditional stability.
Let us
begin, then, by recalling some facts about unconditional stability. The classical definitions of stability and asymptotic stability, due to Lyapunov, are well suited for the study of autonomous differential equations.
For non-
autonomous equations, however, the concepts of uniform stability and uniform asymptotic stability are more appropriate. Let
x(t)
be a solution of the vector differential equation
(1)
f( t, x )
which is defined on the halfline
uniformly stable if for each any solution s
0
x(t)
0 S t
0
00
there is a corresponding
0
Ix Cs ) x(t) I < E
Ix(t)
is said to
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