DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I ORDINARY DIFFERENTIAL EQUATIONS
This is Volume 55 in R’IATHEMA’I’ICS I N SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California A complete list of the boolts in this series appears at the end of this volume.
DIFFERENTIAL AND
INTEGRAL INEQUALITIES Theory and Applications Volume I ORDINARY DIFFERENTIAL EQUATIONS
V. LAKSHILIIKANTHAM and S. LEELA {Jnioersity of Rho& Islmiii Kiiqstow, Rliotlc Islaiid
A C A D E RI I C P R E SS
New J’ork and London
1969
COPYRIGHT 0 1969, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W.l
LIBRARY OF CONGRESS CATALOG CARDNUMBER: 68-8425
PRINTED I N THE UNITED STATES OF AMERICA
Preface
This volume constitutes the first part of a monograph on theory and applications of differential and integral inequalities. 'The entire work, as a whole, is intended to be a research monograph, a guide to the literature, and a textbook for advanced courses. T h e unifying theme of this treatment is a systematic development of the theory and applications of differential inequalities as well as Volterra integral inequalities. T h e main tools for applications are the norm and the Lyapunov functions. Familiarity with real and complex analysis, elements of general topology and functional analysis, and differential and integral equations is assumed. T h e theory of differential inequalities depends on integration of differential inequalities or what may be called the general comparison principle. T h e treatment of this theory is not for its own sake. 'The essential unity is achieved by the wealth of its applications to various qualitative problems of a variety of differential systems. T h e material of the present volume is divided into two sections. T h e first section consisting of four chapters deals with ordinary differential equations while the second section is devoted to Volterra integral equations. T h e remaining portion of the monograph, which will appear as a second volume, is concerned with differential equations with time lag, partial differential equations of first order, parabolic and hyperbolic respectively, differential equations in abstract spaces including nonlinear evolution equations and complex differential equations types. T h e vector notation and vectorial inequalities are used freely throughout the book. Also, because of the several allied fields covered, it becomes convenient to use the same letter with different meanings in different situations. This, however, should not cause confusion, since it is spelled out wherever necessary. T h e notes at the end of each chapter indicate the sources which have V
vi
PREFACE
been consulted and those whose ideas are developed. Some sources which are closely related but not included in the book are also given for guidance. We wish to express our warmest thanks to our colleague Professor C. Corduneanu for reading the manuscript and suggesting improvements. Our thanks are also due to Professors J. Hale, N. Onuchic, and C. Olech for their helpful suggestions. We are immensely pleased that our monograph appears in a series inspired and edited by Professor R. Bellman and we wish to express our gratitude and warmest thanks for his interest in this book.
V. LAKSHMIKANTHAM S. LEELA Kingston, Rhode Island December, I968
Contents
PREFACE
V
ORDINARY DIFFERENTIAL EQUATIONS Chapter 1 . 1.0. Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.
Chapter 2. 2.0.
2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 2.13. 2.14. 2.15 2.16.
3
Existence and Continuation of Solutions Scalar Differential Inequalities Maximal and Minimal Solutions Comparison Theorems Finite Systems of Differential Inequalities Minimax Solutions Further Comparison Theorems Infinite Systems of Differential Inequalities Integral Inequalities Reducible to Differential Inequalities Differential Inequalities in the Sense of Caratheodory Notes
3 7 11 15 21 25 27 31 37 41 44
Introduction Global Existence Uniqueness Convergence of Successive Approximations Chaplygin’s Method Dependence on Initial Conditions and Parameters Variation of Constants Upper and Lower Bounds Componentwise Bounds Asymptotic Equilibrium Asynlptotic Equivalence A Topological Principle Applications of Topological Principle Stability Criteria Asymptotic Behavior Periodic and Almost Periodic Systems Notes
45 45 48 60 64 69 76 79 84 88 91 96 100 102 108 120 129
vi i
...
CONTENTS
Vlll
Chapter 3. 3.0. Introduction 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.1. 3.8. 3.9. 3.10. 3.1 I . 3. I ? . 3.13. 3.14. 3.15. 3.16. 3.11. 3.18. 3.19. 3.20. 3.21.
Basic Comparison Theorems Definitions Stability Asymptotic Stability Stability of Perturbed Systems Convcrse Theorems Stability by the First Approximation Total Stability Integral Stability I,”-S ta hi I i ty Partial Stability Stabilit) of Differential Inequalities Boundcdness and Lagrange Stability Eventual Stability Asymptotic Behavior Relative Stability Stability with Respect to a Manifold .-\lmost Periodic Systems Uniqueness and Estimates Continuous Dependence and the Method of Averaging Notes
Chapter 4 . 4.0. Introduction 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10.
Main Comparison Theorem Asymptotic Stability Instability Conditional Stability and Boundedness Converse Theorcms Stability in Tube-like Domain Stability of Asymptotically Self-Inwriant Sets Stability of Conditionally Invariant Sets Existence and Stability of Stationary Points Notes
131 131 135 138 145 155 158 177 186 191 199 205 209 212 222 229 24 1 244 245 254 257 264 267 267 269 273 277 284 293 291 305 308 311
VOLTERRA INTEGRAL EQUATIONS Chapter 5. 5.0. Introduction
Integral Inequaiitics 5.2. Local and Global Existence 5.3. Comparison Theorems 5.4. Approximate Solutions, Bounds, and Uniqueness 5.5. Asymptotic Behavior 5.6. Perturbed Integral Equations 5.7. Admissibility and Asymptotic Behavior 5.8. Integrodifferential Inequalities 5.9. Notes 5.1.
31 5 315 319 322 324 327 333 340 350 354
CONTENTS
ix
Bibliography
355
AUTHORINDEX
385
SUBJECT INDEX
388
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DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I
ORDINARY DIFFERENTIAL EQUATIONS
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ORDINARY DIFFERENTIAL EQUATIONS
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Chapter 1
1 .O. Introduction This chapter is an introduction to the theory of differential inequalities and therefore forms a basis of the remaining chapters. After sketching the preliminary existence and continuation of solutions of an initial value problem for ordinary differential equations, we develop fundamental results involving differential inequalities. Basic comparison theorems that form the core of the monograph are treated in detail. While considering the system of differential inequalities (finite or infinite), we find it convenient to utilize the notion minimax solutions, and consequently our treatment rests on this notion. Certain useful integral inequalities that can be reduced to the theory of differential inequalities are also presented. Some results on differential inequalities of Caratheodory's type are also dealt with.
1.1. Existence and continuation of solutions Let R" denote the real n-dimensional, euclidean space of elements u = ( u l ,u2 ,..., un). Sometimes, we shall denote also the (a + 1)-tuple (t, u l , u, ,..., un) as an element, and Rn+I shall denote the space of elements ( t , u l , uz ,..., un) or ( t , u). Let 11 u 11 be any convenient norm. As usual, we shall use R instead of R'. Let E be an open (t, u)-set in I?"+'. We shall mean by C [ E ,R"] the class of continuous mappings from E into R". Iff is a member of this class, one writes f E C[E, R"]. Let us consider a system of first-order differential equations with an initial condition u' = g(t, u ) , u(t0) = u 0 , (1.1.1) where u' = du/dt,u0 = ( u I 0 ,u,,, ,..., unO),and g E C [ E ,R"]. A solution of the initial value problem (1.1.1) is a differentiable function of t such 3
4
CHAPTER
1
that ~ ( t , , = ) ZL, , ( t , u(t))E E, and u'(t) = g(t, u(t)) for a t-interval J containing t,, . This means that u ( t ) has a continuous derivative. From these requirements on the continuous function u(t), it follows that it satisfies the integral equation
I n order to prove the classical Peano's existence theorem, we have to introduce the notion of an equicontinuous family of functions. 1)EFINITION 1.1.1. A family of functions F = { f ( u ) } defined on some ti-set E C R" is said to be equicontinuous if, for every E > 0, there exists a S = S ( E ) , independent of f E F and also u l , u2 E E, such that 11 f ( u J -f(u2)ll -E, whenever / / u1 - u2 I/ < 6. T h e following theorem shows the fundamental property of such a family of functions, the proof of which will be omitted.
THEOREM 1.1.1. (Ascoli-Arzela). Let F = ( f ] be a sequence of functions defined on a compact u-set E C R", which is equicontinuous and equibounded. Then, there exists a subsequence ( f n } , n = 1, 2, ..., which is uniformly convergent on E. THEOREM 1.1.2. (Peano's Existence Theorem). Let g E CIRo , Rn], whcrc R,, is the set [ ( t ,u ) : to t to a, 11 u - u, /I 61; 11 g(t, u)\l M on R,, . Then, the initial value problem (1.1.1) possesses at least one solution u ( t ) on t, t t, 01, where 01 = min(a, 6 / M ) .
< < +
<
<
< < +
Pyoof. 1,et uo(t) be a continuously differentiable function, on [to 5, t,,], S > 0, such that uo(t,) = u, , 11 u,(t) - u,, /I 6, and 11 u i ( t ) 11 M . For 0 < E 6, we define a function u f ( t )= u,(t) on [to- ~ -S, to]and -
% ( t ) = UO
on [t, , t, and
<
<
<
+ 4, where
+ It g(s, u,(s
-
( 1.1.2)
.)) ds
t0
C Y = ~
min(ol, E ) . Observe that uXt) is differentiable
I/ u d t ) - *o II
+
(1.1.3)
on [to 8, to EJ. If 0 1 ~< 01, we can use (1.1.2) to extend uXt) as a continuously differentiable function over [to- 6, to a 2 ] , a2 = min(iu, 2 ~ )such , that (1.1.3) holds. Continuing in this way, uc(t) can be defined over [to 6, to a ] so that it has a continuous derivative and -
+
~
+
1.1.
5
EXISTENCE AND CONTINUATION OF SOLUTIONS
satisfies (1.1.3) on the same interval. Furthermore, I/ u:(t)ll < M , and therefore { u c ( t ) }forms a family of equicontinuous and uniformly bounded functions. An application of Theorem 1.1.1 shows the existence of a sequence {en> such that > > *.. en+O as n--t a,and u(t) = limn+mu,(t) exists uniformly on [to - 6, t,, a]. Since g is uniformly continuous, we obtain that g(t, u,,(t - E , ) ) tends uniformly to g ( t , u(t)) as n + co,and, hence, term-by-term integration of (1.1.2) with E = E , , a1 = 01 yields
+
This proves that u ( t ) is a solution of (1.1.1). T h e following corollary of Peano's Theorem is useful in applications. COROLLARY 1.1 .I. Let E be an open ( t ,u)-set in Rn+l and E, be a and 11 g(t, u)II M on E. compact subset of E. Suppose that g E C[E, Rrb] Then, there exists an a = a(E, E,, , M ) such that, if (to , u,,) E E, , (1.1.1.) has a solution, and every solution exists on [ t o ,t, a]. I n that case, when g is not bounded on E, we can replace the set E by an open subset El having a compact closure in E and containing E,, .
<
+
T h e next theorem deals with the problem of extending the solutions up to the boundary of E.
THEOREM 1.1.3. Let E be an open ( t ,u)-set in Rn+l,and let g E C[ E ,R"]
< < a,.
and u ( t ) be a solution of (1.1.1) on some interval to t u(t) can be extended as a solution to the boundary of E.
Then
Proof. Let E l , E , ,... be open subsets of E such that E = (J E,; the closures El, E, ,... are compact, and En C It then follows from Corollary 1.1.1 that there exists an E , > 0 such that, if ( t o ,uo)E E m , all solutions of (1.1.1) exist on t,, t to E , . Choose n, so large that (a,, , ~(u,,))E Enl . Then, ~ ( tcan ) be extended E,,], and, if (a, E , ~, u(ao E,,)) E En1, over an interval [a,,, a, u(t) can be further extended over [a,, + E , , , a, + 2 4 . This argument can be repeated until we get the extension of u(t) over the interval t,, t a, , where a, = a, N,E,,, N , is an integer 2 1 , such that
< < + +
+
< <
(a1
, u(a1)) 4 En, .
+
+
Choose n, so large that (al , u(al))E Enz . Arguing as before, we arrive at an integer N 2 3 1 such that u(t) can be extended over t, t a2, 0 2 = a, N2E?t2 7 and ( a , 7 u ( 4 ) 4 Enz . Proceeding in this way, we are led to a sequence of integers
+
< <
6
1
CHAPTER
n, < n2 < ... and numbers a, < a, < a2 < . - -such that u(t) has an extension over [to, a), where a = lim,+m a, and that (a,, u(a,)) $ . Thus, the sequence (a,, ,u(uk)) is either unbounded or has a cluster point on the boundary of E. T o show that u(t) tends to the boundary of E as t ---t a, we must show that no limit point of {tk , u(tk))is an interior point of E as t, 4a. Since this follows from the lemma below, the theorem is proved.
Erik
LEMMA 1. I . I . Let g E C[E,R"], where E is an open (t, u)-set in Rn+l.Let u(t) be a solution of (1.1.1) on an interval to t < a, a < 00. Assume that there exists a sequence {t,) such that to tk+ co as k -+ co and uo = 1imk+%u(tk) exists. If g ( t , u) is bounded on the intersection of E and a neighborhood of ( a , uo),then
<
lim u(t) t-a
=
<
( I . 1.4)
uo.
If, in addition, g(a, uo) is defined such that g(t, u ) is continuous at ( a , uo),then u(t) is continuously differentiable on [ t o ,a] and is a solution of (1.1.1) on [ t o ,a].
Proof. Let E
I/ u
uo /I
<
> 0 be sufficiently small. Consider the set I?: O < a - t < E,
<
E. Let M ( e ) be so large that IIg(t, u)ll M ( E ) for ( t , u ) E E n I?. If, for R sufficiently large, 0 << a - t, e / 2 M ( ~and ) 11 u(t,,) - uo // ~ / 2 then , -
<
<
I1 4 t ) - u(t,c)ll for t ,
M(E)(a- tk) G 4 2
a. If this is not true, there is a t, such that t,<< t,
/ / u(tl) - u(t,,) I( = rcl(c)(a - t k )< ~ / 2 It. therefore follows that :c
(1.1.5)
< a,
< & + / / u(tk) uo / / < for t, < t < t , . This implies 11 u'(t)/I < M ( E )for t,c < t < t, . Consequently, 11 u ( t ) - uo 11
-
E
II 4 t l ) - u(t,Jll G M(E)(t, - tk) < M ( 4 ( a - t k ) . This proves (1.1.5), which, in turn, shows that (1.1.4) holds. T h e last part of the lemma follows from the fact that u'(t) = g(t, u(t))+ g ( a ,
COROLLARY I . 1.2. Let g E
=
E
[ ( t ,u ) : t ,
u")
as
t -+ a.
C[E, R"], where
< t < t" + a ( a < a),u E R"].
1.2.
7
SCALAR DIFFERENTIAL INEQUALITIES
Let u(t) be a solution of (1.1.1). T h e n the largest interval of existence of u(t) is either [ t o ,to + a] or [ t o ,a), 6 to + a and /I u(t)l/ -+ og as t46.
<
1.2, Scalar differential inequalities We adopt the following notation for Dini derivatives: D+u(t) = lim sup h-l[u(t h-O+
+ h)
-
u(t)],
D+u(t) = liin inf h-l[u(t + h ) - u ( t ) ] , h-Oi
D-u(t)
= lim
sup h-'[u(t
D-u(t)
= lim
inf h-l[u(t
+ h)
h-*O-
h-0-
+ h)
-
-
u(t)],
u(t)],
+
where u E C[(to, to a), R].When D+u(t) = D+u(t),the right derivative will be denoted by u;(t). Similarly, uL(t) denotes the left derivative. DEFINITION1.2.1. Let E be an open ( t , u)-set in R2 and g E C[E, R]. Consider the scalar differential equation with an initial condition u' = g(t, u ) ,
u(t,) = ug
.
+
Suppose z, E C[[t,, to a ) , R], z,;(t) exists for t E [to, to (t, v(t))E E. If v(t) satisfies the differential inequality a t )
< g(t>a ( t ) ) ,
t
fE [to
9
to
(1.2.1)
+ a),
and
t- a ) ,
it is said to be an under-function with respect to the initial value problem (1.2.1). On the other hand, if q t )
> g(t, +)),
t
6
[to , to
+ a),
v(t) is said to be an over-function.
A fundamental result on scalar differential inequalities is the following:
THEOREM 1.2.1. Let E be an open ( t , u)-set in R2 and g E C[E, R]. Assume that 21, w E C[[t,, to a), R] and ( t , v(t)), ( t , w ( t ) ) E E , t
E
[to, to
+ + u). Suppose further that .(to)
< w(to),
(1.2.2)
8
1
CHAPTER
and, for t E (to, to
+ a ) , the inequalities (1.2.3) (1.2.4) (1.2.5)
Proof.
If assertion (1.2.5) is false, then the set
z
=
[t E [to, to
is nonempty. Defining t , Furthermore,
=
+ a): w ( t ) < v(t)]
inf 2, it is clear from (1.2.2) that to
< t, . (1.2.6)
v(t1) = 4 t l )
and
4t) < 4 t h
t
E
Using (1.2.6) and (1.2.7), we obtain, for small h 74tl
+ h) h
-
u(t1)
>
+ h)
4tl
(1.2.7)
[to , tl).
h
< 0,
-4
tl)
7
which in its turn implies D-v(tl)
2 D-w(t,).
(1.2.8)
T h e inequalities (1.2.3), (1.2.4), and (1.2.8) together with (1.2.6) lead us to the contradiction At1 7 4 t l ) )
>A t 1
7
4tl)).
Hence 2 is empty, and the statement (1.2.5) follows.
REMARK1.2.1. I t is obvious from the proof that the inequalities (1.2.3) and (1.2.4) can also be replaced by D-4t) < g(t, v(t))t D-w(t) >> g ( t , w ( t ) ) ,
respectively. Note that the proof does not demand the validity of the inequalities (1.2.3) and (1.2.4) for all t E (to, to a). T h e following refinement is a consequence of this observation.
+
1.2.
9
SCALAR DIFFERENTIAL INEQUALITIES
THEOREM 1.2.2. Let the assumptions of Theorem 1.2.1 hold, except that the inequalities (1.2.3) and (1.2.4) are satisfied for t E 2, = [t E (to , t, u ) : v(t) = w(t)]. Then (1.2.5) remains valid. I n fact, Theorem 1.2.1 can be subjected to further refinements. T o this end, we require the following simple lemmas. Although we state them for scalar functions, it is easy to see that they are true for vector functions as well. Unless otherwise specified, let S denote an at-most countable u). subset of [ t o ,t,
+
+
+
LEMMA 1.2.1. (Zygmund). Suppose that u E C[[t,, t, a), R] and the inequality Du(t) 0 for t E [to , to u ) - S, D being a fixed Dini derivative. Then, u(t) is nonincreasing in t on [ t o ,to u).
+
<
+
+
LEMMA 1.2.2. Let n, w E C [ [ t , , t, a), R ] , and for some fixed Dini derivative Dv(t) < w ( t ) for t E [ t o , to u ) - S. Then, D-v(t) < w ( t ) for t E [to, to a).
+
Proof.
Define the function m(t) = v ( t ) -
+
st
w(s) ds.
to
It then follows, from the assumption, that Dm(t) = Dv(t)
~
W(t)
< 0,
2E
[ t o ,to
+ a) s. -
Hence, by Lemma 1.2.1, m(t) is nonincreasing in t on [t, , to Consequently, D-m(t) = D-v(t)
-
<
~ ( t ) 0,
t E [ t o , to
+ u).
+ a),
and the lemma is proved. REMARK1.2.2. I n the light of Lemma 1.2.2, it is clear that Theorem 1.2.1 remains true when the inequalities (1.2.3) and (1.2.4) hold for t E [ t o ,t, u ) - S , D being any fixed Dini derivative. It will now be shown that any solution of the initial value problem (1 2.1) can be bracketed between its under- and over-functions.
+
THEOREM 1.2.3.
Let n(t), w ( t ) be under- and over-functions with respect to the initial value problem (1.2. l), respectively, on [to , to + a). If u(t) is any solution of (1.2.1) existing on [t, , t, + u ) such that ~ ( t ,= ) un = w(t,),
(1.2.9)
10
CHAPTER
1
then v(t) < u(t) < w(t),
E
[t”, t,
+ a).
(1.2.10)
Proof. We shall prove the right half of the inequality (1.2.10). Similar reasoning can bc used for the left half. Let w(t) and u(t) be an overfunction and a solution of (1.2.1), respectively. Lct m(t) = w(t) - u(t). Then, m;(f,,) > 0 because of (1.2-9). I t follows that m ( t ) is increasing to the right of to in a sufficiently small interval to t t, E , which implies that
< < +
u(t,
Furthermore,
+
€)
u’(t)
and for t E [t,,
+
< g(t, 4 t ) ) s(t,749)
w’, (t) E,
to
< w(t, 4-€).
+ a). A direct application of Theorem 1.2.1 yields that u(t)
.= w ( t ) ,
t
t
+ a).
[t,, , t ,
This proves the theorem.
COROLLARY 1.2.1. and
Let E be an open ( t , u)-set in R2,g, ,g, ’?l(t, u)
< gz(t, u),
E
C[E,R],
( t , u ) E E.
Let ul(t),u2(t)be any two solutions of u; = g d t , u),
respectively, existing on [ t o ,to udt) ,%(f), f E [ t o 9 to 4.
+
~
4 = &(t, u),
+ a ) such
that ul(t,)
< uz(t,).
Then
COROLLARY I .2.2. Let E be an open ( t , u, v)-set in R3, and g E C [ E ,R ] , and g ( t , u, v) is nondecreasing in v for each fixed t and u. Let u , v t C[[t,, t, 4- a ) , I?] such that u’(t), v’(t) exist, (t, u(t), ~ ’ ( t ) ) , ( t ,~ ( t ~ ) ,’ ( t E) B ) for t E [to , to u). Assume that the inequalities
+
4 t h u’(t)) < 0, hold for t t [ t o ,to t E [ t o , to a).
+
+ u).
A t , U ( t ) , v’(t)) ‘Then, u(t,)
0
< v(t,)
implies u(t) < u(t) for
1.3.
11
MAXIMAL AND MINIMAL SOLUTIONS
1.3. Maximal and minimal solutions T h e notion of maximal and minimal solutions of (1.2.1) will now be introduced.
DEFINITION 1.3.1. Let r ( t ) be a solution of the scalar differential equation (1.2.1) on [ t o ,t, a). Then r ( t ) is said to be a maximal solution of (1.2.1) if, for every solution u(t) of (1.2.1) existing on [ t o ,t, a), the inequality
+
+
u(t)
< r(t),
t E [ t o to 7
+ a)
(1.3.1)
holds. A minimal solution p ( t ) may be defined similarly by reversing the inequality (1.3.1). We shall now consider the existence of maximal and minimal solutions of ( I .2.1) under the hypothesis of Peano’s existence theorem.
C[R, , R], where R, is the rectangle to t t, a, I u - uo 1 6, and I g(t, u)I M on R, . T h e n there exist a maximal solution and a minimal solution of (1.2.1) on [t,,, to a ] , where 01 = min(a, 6/(2M 6)).
THEOREM 1.3.1.
< < +
Let g
E
<
<
+
+
Proof. We shall prove the existence of the maximal solution only, since the case of the minimal solution is very similar. Let 0 < E b/2. Consider the differential equation with an initial condition
<
+
u‘ = g(t, ).
Observing that gdt,
u(t,) =- uo
E,
4 =g(t,4 +
+
(1.3.2)
€.
E
is defined and continuous on R, : t ,
< t < to + a ,
+
<
1u
- (ug
+ €)I
< b/2,
lg, (t, u)I M (b/2) on R,, we deduce from Theorem 1.1.2 that the initial value problem (1.3.2) has a solution u(t, E ) on the interval [ t o ,to a ] , where 01 = min(a, b/(2M 6)). For 0 < E~ < el E , we have R, C R, and
+
<
.(to
7
€2)
+
< to 4 ,
+
U ’ ( t , €2)
< R(t, u ( t ,
U ’ ( t , €1)
> g (t, u(t, €1)) -t €2
€2))
€2, 7
t E [to , t” 4-.I.
12
1
CHAPTER
We can apply Theorem I .2.1 to get t E [to tn
u ( t , el),
~ ( l€ 2, )
9
i a].
Since the family of functions u(t, E ) is equicontinuous and uniformly bounded on [to, t, 011, it follows by Theorem 1.1.1 that there exists a decreasing sequence { E ~ ]such that eTL-+ 0 as n --t m, and the uniform limit r ( t ) = lim u(t, en)
+
n-*m
+
exists on [ t o ,t, a ] . Clearly, r(t,) = Z L ~. T h e uniform continuity of g implies that g(t, u(t, E , ) ) tends uniformly to g(t, r(t)) as n -+ a, and thus term-by-term integration is applicable to u(t, 4 = uo -1-
'n
+ s( R(&
u(s,
to
4 )ds,
which in turn shows that the limit r ( t ) is a solution of (1.2.1) on [f"
, 2"
+
(YI.
We shall now show that r ( t ) is thc desired maximal solution of (1.2.1) on [t,,, to t] satisfying (1.3.1). Let u(t) be any solution of (1.2.1) existing on [z,, , t, 4-n ] . Then,
+
~ ( t ,= ) ug u'(t)
u'(t, 6 )
for t
E
[to, t,,
+
X]
and
E
U(t)
< ug
+
E
== U ( t n ,
df, 4 t ) ) + ', At, u(t, .)I +
, ,
E),
€7
< h/2. By Remark 1.2.1, we obtain that < u(t, €1,
t
E
[to , to
-t
a].
T h e uniqueness of the maximal solution shows that u(t, e ) tends uni0. This proves the theorem. formly to r ( t ) on [to , to 1.1 as E
+
--f
This cxistence theorem, together with the extension Theorem 1.1.3, implies the following:
TIIEOKEM 1.3.2. Let g E C [ E , R ] , where E is an open ( t , u)-set in R2 and ( t o, u,,) E E. Then (1.2.1) has maximal and minimal solutions that can be extended to the boundary of E. T h e lemmas given below are useful in certain later applications.
I~EMMA 1.3.1. Txt the hypothesis of Theorem 1.3.2 hold, and let [t,,, to 4-u ) be the largest interval of existence of the maximal solution
1.3.
13
MAXIMAL AND MINIMAL SOLUTIONS
+
r ( t ) of (1.2.1). Suppose. [ t o ,t l ] is a compact subinterval of [ t o , to u). T h e n there is an E , > 0 such that, for 0 < E -< c0 , the maximal solution r ( t , E ) of Eq. (1.3.2) exists over [ t o ,t l ] , and lim r ( t , E )
=
c-*O
r(t)
uniformly on [ t o ,t l ] .
Proof. Let En be an open bounded set, I?,) C E, and ( t , r ( t ) ) E E for t E [t,,, t l ] . We can choose a b > 0 such that, for t E [to, t l ] , the rectangle R t : [t,t
is included in Eo for E that
+ 61,
Iu
(r(t)
~
+ .)I
< 6,
< b/2. Let 1 g(t, u)I < M on Eo . T h e n it is evident I g(t, u )
+ I < + 6/2 6
<
on R t , for t E [ t o ,tl] and 0 < E b/2. Consider the rectangle RI, . It follows from Theorem 1.3.1 that the maximal solution r ( t , E ) of (1.3.2) exists on [t, , to 711, 7 = min(b, 26/(2M 6)). Note that 7 does not depend upon E . Furthermore, proceeding as in Theorem 1.3.1, we can conclude, in view of the uniqueness of the maximal solution r ( t ) of (1.2. I), that lim r ( t , G) = ~ ( t )
+
+
€-O
uniformly on [ t o ,to
+ 71. This implies that
+ 7,
l i i Y(t"
= Y(t"
6)
+ 7).
< 6/2 such that, for 0 < <
Consequently, there is an
E
Y(t0
+ 7, < r(t0 + 7) + 6)
E
~
we , have
E.
We can now repeat the foregoing argument with respect to the rectangle Rfo+,, E < , to show that there exists an c2 < such that, for E < E$ , the maximal solution f ( t , E ) of 21'
exists on [to
=g(t, u)
+
E,
+ q, to + 2711, and
+ 7)
U(t,
lim f ( t , E )
+ 7,to + 2711. For r ( t ,E )
= f(t,
€1,
E
t
+ 7) +
E
= r(t)
E-0
uniformly on [to r(t, E ) by defining
= r(t0
E
< E $ , we can extend the function [to
+ 7, t o + 271.
14
1
CIIAPTER
It is clear that r ( t , E ) is the maximal solution of (1.3.2) on [ t o ,to and lim v(t, C ) = r ( t )
+ 271,
E-0
+
uniformly on [ t o ,to 271. By induction, it can be shown that there is an c0 = E , such that [ t o ,tl] C [ t o ,t, nq], that the maximal solution r(t, E ) of (1.3.2) exists on [to, to nq] for 0 -: E < E , , and that
+
+
lim r ( t , C ) C-0
= r(t)
+ nq]. T h e lemma is thus proved. LEMMA1.3.2. Let g E C[[t,, to + u] x R, R] and nondecreasing for each t E [to , to + a]. Assume that uniformly on [ t o ,to
g(t, 0)
I g(tj .)I
<M
0,
on
[to
7
to
in u
(1-3.3)
+ a] x R ,
(1.3.4)
and u ( t ) = 0 is the unique solution of u' = R(4 u ) ,
on [to, t ,
(1.3.5)
u(to) = 0
+ a]. Then, the successive approximations u"(t) = M(t - t o ) ,
As,
%+l(t) = Jf
ds
'0
are well defined; 0 and
< untdt) < % ( t )
lim u J t )
n-m
Moreover, for every n u' = g (t, u )
exists on [ t o , t,
L:
0
on
[to to 9
+ a],
uniformly on [ t o ,to
+ a].
(1.3.7)
3 I , the maximal solution m(t) of -I- k ( t , U n - l ( t ) ) ,
%(to)
=
0,
4-a ] , and lim rn(t) = 0
n-.m
(1.3.6)
uniformly on
[ t o , to
> 0,
+ a].
(1.3.8)
<
Proof. An easy induction proves ( I .3.6). Since, by ( 1 .3.4), I uk(t)l M , using Theorem 1.1.1, we can conclude that limn+mu,(t) I- u(t)uniformly
1.4.
15
COMPARISON THEOREMS
on [to , to + a ] .It is clear that u ( t )satisfies u'(t) = g ( t , u(t))and u(t,) By (1.3.5), it follows that u ( t ) = 0, and (1.3.7) is proved. Given E > 0, there is an n 3 a(.) such that
1 k ( t , un-1(t)I <
t
€9
[t" , t o
=
0.
+ a],
because of (1.3.3) and (1.3.7). Now an argument similar to that of Lemma 1.3.1 proves (1.3.8). 1.4. Comparison theorems
An important technique in the theory of differential equations is concerned with estimating a function satisfying a differential inequality by the extremal solutions, of the corresponding differential equation. One of the results that is widely used is the following comparison theorem:
THEOREM 1.4.1. Let E be an open ( t ,u)-set in R2 and g E C [ E ,R]. Suppose that [ t o ,to a ) is the largest interval in which the maximal a ) , R ] , ( t , m ( t ) ) E E for solution ~ ( tof) (1.2.1) exists. Let m E C[(t,, t, t E [to, to a ) , rn(t,) uo , and for a fixed Dini derivative,
+ <
+
+
Dm(t)
t
E
[to , to
+ a)
-
S. Then, m(t)
Pyoof.
< g(t, m ( t ) ) ,
< +),
t
6 [to
, to
(1.4.1)
+ a).
(1.4.2)
From Lemma 1.2.2, it follows that (1.4.1) can be replaced by D-m(t) ,< 'dt,m ( t ) ) ,
t
+
(t" > to
E
(1.4.3)
a).
Let to < T < to + a. By Lemma 1.3.1, the maximal solutions (1.3.2) exist on [to , T ] for all E > 0 sufficiently small, and O E'
m(t) < r(t, E ) ,
t
of
(1.4.4)
~ ( t= ) lim r ( t , c)
uniformly on [ t o ,TI. Using (1.3.2) and Theorem 1.2.1, we derive that
u ( t , c)
(1.4.3)
[ t o , 71.
and
applying (1.4.5)
T h e last inequality, together with (1.4.4), proves the assertion of the theorem.
16
1
CHAPTER
REMARK1.4.1. If the inequality (1.4.1) is reversed and m(t,) 2 u,, , then we have to replace the conclusion (1.4.2) by m(t) 3 p ( t ) , where p ( t ) is the minimal solution of (1.2. I).
Theorem 1.4.1 can also be proved under a weaker hypothesis.
THEORFM 1.4.2.
2
Let m(t),r ( t ) be as in Theorem 1.4.1, and =
[t t [t,,, t,
+ a) :
Y(t)
< m ( t ) < r ( t ) + €01,
for some c0 > 0. If (1.4.1) is satisfied for t E 2 at-most countable subset of 2,then (1.4.2) holds.
-
9,where 3
(1.4.6)
is an
Proof. I t is enough to prove (1.4.5). As before, Lemma 1.2.2 implies that (1.4.3) is satisfied for t E Z. Proceeding as in the proof of Theorem 1.2.1, we arrive at a t, such that m(td
In view of (1.4.4), there exists an r ( t , 6)
=Y(t, en
9
.).
> 0 such that
< y(t) i€0,
t
E
[to
, 71.
RIoreover, we have r ( t ) -'r(t, , E), and hence there results the inequality r ( t ) < r ( t , €)
t
€
[ t o , 71.
It thereforc follows from (1.4.6) that
which implies that t , E 2. Hence, (1.4.3) is satisfied for such a t , , and this is sufficient to establish the desired result. We now give a modification of Theorem 1.2.1. It is evident that the proof of Theorem 1.2.1 breaks down if we do not assume one of the relations (1.2.3) and (1.2.4) to be a strict inequality. This, however, can be relaxed provided g satisfies a further assumption.
THEOIIEM 1.4.3. Let the hypothesis of Theorem 1.2.1 hold except that the inequalities (1.2.3) and (1.2.4) are replaced by
d d t l U(t))>
[ 1.4.7)
D-w(t> 3 d t , w ( t ) )
(1.4.8)
f)-v(t)
1.4.
17
COMPARISON THEOREMS
+
for t E ( t o ,to u). Assume further that, for each t E [to , TI,g satisfies the condition g( t >~
2
1 - )g ( t > ~ 2 )
where G E C[[t,, to of
-G(T
+
-
T E
t , ~1 - uZ),
( t o ,t,
~1
3~
+ u ) and 2
,
(1.4.9)
+ a) x R, R], and r(t) = 0 is the maximal solution G(t,u),
U' =
~ ( t ,= ) 0.
T h e n (1.2.5) holds. Proof. Proceeding as in the proof of Theorem 1.2.1, there exists a t, E (t, , to a) such that 4 t l ) = W(tl>, (1.4.10)
+
and
~ ( t< ) ~ ( t ) , to
+
< ti.
Define vl(t) = n(t, t, t ) and w l ( t ) = w(t1 view of (1.4.10) and (1.4.11), yields that ~
+ to
(1.4.1 1) -
t). This, in (1.4.12)
ui(to) = Wi(tn):
s(t)< W l ( t ) ,
(1.4.13)
t E [ t o > tll.
Setting m ( t ) = w l ( t ) vl(t), the definitions of n, , w1 and the assumptions (1.4.7) and (1.4.8) imply the inequality ~
D-m(t)
=
D-w,(t)
where gl(t, u ) = -g(tl (1.4.9) to arrive at
D-s(t)
~
+ to
D-m(t)
-
< g1(t, W l ( t ) )
- gl(4
s(t)),
t, u). Since (1.4.13) holds, we can use
-< G(t,m(t)),
t E [to
9
tll.
By Theorem 1.4.1, we have m(t)
< r(t),
t
E
[to , tll,
(1.4.14)
where r ( t )is the maximal solution of u' = G(t,u), such that r(t,) = m(t,). From the definition of m ( t ) and (1.4.12) and (1.4.13), we deduce that m ( t ) 3 0, t E [t, , t l ] , and nz(t,) = 0. Then, the inequality (1.4.14) and the assumption r ( t ) = 0 show that v ( t ) = ~ ( t ) , t E [to , ti],
which, however, is contrary to the assumption (1.4.1 1) and the definition of t , . Hence, the set 2 is empty, and the theorem is proved.
18
CIIAPTER
1
T o give another comparison theorem that, in certain situations, is more useful than Theorem 1.4.1, we require the following result:
+
THEOREM 1.4.4. Let E be the product space [to, to a ) x R2 and g E C [ E , R ] . Assume that g is nondecreasing in v for each t and u. Suppose that ~ ( tis) the maximal solution of the differential equation u' = g(t, u, u ) ,
existing on [to , to
u ( f o )= u,
+ a ) , and
t € [to, t,
r ( t ) 3 0,
(1.4.1 5)
2; 0
(1.4.16)
4-u).
Then, the maximal solution r l ( t ) of u(t,) = uo ;3 0,
u' = gl(t, u ) ,
where g l ( t , 21)
= g(t, u, ~ ( t ) )exists ,
r ( t ) = rdt),
on [to, t, t
E
[t" , to
(1.4.17)
+ u ) and
+ a).
(1.4.18)
Proof. By Theorems 1.3.1 and 1.3.2, the maximal solution r l ( t ) of (1 .4.17) exists on an interval [t,,, to $- (Y], (Y a, which can be extended to the boundary of E. This implies that either r l ( t ) is defined over [ t o ,t, a ) or there exists a t , < t, a such that
+
+
and this yields, from Theorem 1.4.1, that
-
as far as r l ( t ) exists. I t follows from (1.4.16), (1.4.19), and (1.4.20) that Yl(t,)
(1.4.21)
-km
as t,. -+ t l - . We shall show that (1.4.21) cannot be true. For this purpose, consider the maximal solution r ( t , E) of u' = g (t, u, u )
+
E,
u(to)= u g
+
6,
ug
2 0,
(1.4.22)
1.4.
19
COMPARISON THEOREMS
+
+
which, by Lemma 1.3.1, exists on [to, t, v ] , v > 0, and t , v < to for sufficiently small E > 0. Moreover, we have from (1.4.22) that
+ a,
and Hence, one gets, from Theorem 1.2.1, the inequality r(t) < r(t,E),
t
-t .I.
[to , t ,
E
(1.4.24)
Since g is nondecreasing in o, (1.4.23) and (1.4.24) lead to
> g d t , r ( t , .I),
r'(4 .)
t
E
[to , t,
+ .I.
) ri(to, €1. But ri(t) = g(t, ri(t), r(t)) gi(t, ri(t)), t E [to > ti), and ~ i ( t o < An application of Theorem 1.2.1 again shows that ri(t> < r ( t , E ) ,
t
E [to
7
( I -4.25)
ti)-
Since r ( t , €) exists on [to , t , + v], v > 0, (1.4.2t) leads us to a contradiction because of (1.4.25), and this proves the existence of r l ( t ) on [to to 9
+ a).
T o prove (1.4.1 S), we now see that (1.4.20) is true for t Furthermore,
E
[t, , t,
+ a).
r;(t> = A t 7 rdt>>= d t t rl(t>?+>)-
From the monotonic character of g in
ZI
and (1.4.20), one gets
G s(4 r d t ) , ri(t))* Theorem 1.4.1 now gives that ri(t)
G ~ ( t ) , t~
[ t o , to
+ a).
This inequality, together with (1.4.20), proves (1.4.18), as is desired.
THEOREM 1.4.5. Let the hypothesis of Theorem 1.4.4 hold; m E C [ [ t o, to a), R ] such that ( t , m ( t ) , o) E E, t E [to , to a ) , and m(t,) < uo . Assume that for a fixed Dini derivative the inequality
+
+
Dm(t>
is satisfied for t E [to , to we have
< g(t, m(t),4
(1.4.26)
+ a ) - 5'. Then, for all < r ( t ) , t E [to , to + a ) ,
m(t)
ZI
<
Y(t),
t
E
[to, t" -4- a).
(1.4.27)
20
CHAPTER
1
+
<
r ( t ) , t E [to, to a). Then, using the monotonicity of g Proof. Let a in z‘, the inequality (1.4.26) reduces to
where gl(t, m ( t ) ) = g(t, m(t), r ( t ) ) . If r l ( t ) is the maximal solution of (1.4.17), ‘Theorem 1.4.4 shows that r l ( t ) exists on [to, to a ) and (1.4.18) is true. Now a straightforward application of Theorem 1.4.1 assures the inequality (1.4.27).
+
Assume that, for each dition
T
(to , to
E
g ( t , ul) - ~ ( t ,u2) 2: -G(T
whcre G E C[[t,,, to of
+ a ) and t E [to, TI, g satisfies the con+ to
-
t a ) x R , R ] , and U’ =
G(t,u),
t , u1
--
uZ),
u1
3 U2
9
r ( t ) - . 0 is the maximal solution ~ ( t ,: ) 0.
+
Then vz(to)< u,, implies m ( t ) < u(t), t E [ t o ,to a ) , where u ( t ) is any ) u,,, existing on [to , to + a). solution of u’ = g ( t , u), ~ ( t , ,= T h e maximal and minimal solutions may be defined to the left of t o , and their existence may be proved using the previous arguments with necessary modifications. A result parallel to Theorem 1.4.1, concerning the minimal solution to the left, is useful in later applications. TTc shall state this as a theorem, omitting its proof.
THEOREM 1.4.6. Let E be an open ( t ,u)-set in R2 and g E C [ E ,R ] . Suppose that m E C[(t(, a, t o ] ,R ] , ( t , m ( t ) )E E for t E (to- a, to], m(t,,) 2 uO, and for any fixed Dini derivative ~
Dm(t)
< g (f, m(t)),
t
t (to
-
a, to).
Then
f 4 t ) ‘z dt), as far as p ( t ) exists to the left of to , p ( t ) being the left minimal solution of (1.2.1).
I .5.
21
FINITE SYSTEMS OF DIFFERENTIAL INEQUALITIES
1.5. Finite systems of differential inequalities Many of the results considered so far for scalar differential inequalities will now be extended, in the sections that follow, to finite systems of differential inequalities. T o avoid repetition, let u s agree on the following: the subscript i ranges over the integers 1, 2, ..., n; let 0 k n ; the subscriptsp and q range over the integers 1, 2 ,..., k and k 4- 1, k 2 ,..., n, respectively. We shall be using vectorial inequalities freely, with the understanding that the same inequalities hold between their corresponding components. We shall consider the differential system with an initial condition, written in the vectorial form
< < +
u'
= g(t, u),
u(tn) = u o ,
whereg E C[E, R"] and E is an open ( t , u)-set in
(1.5.1)
R"1-1.
+
DEFINITION 1.5.1. Let v E C [ [ t ,, to a), R " ] ; ( t , v ( t ) )E E, and v;(t) exists for t E [tn , to a). T h e function v ( t )is said to be a k under ( n k ) over-function with respect to the initial value problem (1.5.1) if
+
~
%J,+(t)< g,(t, 4 t ) ) >
+
74,d t )
> g,(t, 4 t ) )
hold for t E [t,, , to a). If v(t) satisfies the reversed inequalities, it is said to be a k over ( n - k) under-function. These definitions clearly include the definitions of under- and overfunctions as special cases, viz., K = 0 or k = n. We require that the function g(t, u ) should satisfy certain monotonic properties, which are listed below. DEFINITION 1.5.2. T h e function g(t, u ) is said to possess a mixed quasimonotone property if the following conditions hold:
(i) gp(t,u ) is nondecreasing in u j , j = 1, 2,..., k , j f p , and nonincreasing in uQ. (ii) g,(t, u ) is nonincreasing in up and nondecreasing in u j , j = k + 1 , k + 2 ,..., n, j # q . Evidently, the particular cases k = n and k = 0 in the mixed quasimonotone property correspond to quasi-monotone nondecreasing and quasi-monotone nonincreasing properties of the function g(t, u), respectively. Furthermore, g(t, u ) is said to possess mixed monotone property if, in conditions (i) and (ii), j # p , j f q are not demanded.
22
CHAPTER
1
An extension of Theorem 1.2.1 which plays an equally important role is the following: T H E O R E M 1.5.1. Let (i) g E C[E, R"], where E is an open ( t , u)-set in R"-I'; (ii) zi, w E C[[t,, to a), R"], ( t , v(t)), ( t , w(t)) E E for t E [t,,, t,, a ) ;and (iii)g(t, u ) possess a mixed quasi-monotone property. Assume further that
+
+
(1 5 3 ) (1 5 4 ) (1 5 5 ) (1 S.6)
Proof. Define mlJ(t)= wIJ(t) because of (1.5.2), ~
n J t ) and m J t )
m,(t*) 1 0 ,
=
vq(t) - wq(t). Then,
i = 1, 2)...)n.
(1.5.8)
Suppose that the assertion (1.5.7) is not true. Then, the set
z
(J [ t E [to , to 4-a): m,(t) < 01 11
=
2
1
is nonernpty. Let t , = inf 2. By (1.5-8), it is obvious that t, > t o . Since the set Z is closed, t , E Z , and consequently there exists a j such that m,(tl) = 0.
(1 5 9 )
If (1.5.9) is not true, one would have m,(t,) <<0, which implies m,(t) < 0 in a sufficiently small neighborhood to the left of t, . This contradicts the definition of t , , and therefore (1.5.9) is valid. Moreover, (1.5.10) and (1 5 1 1 )
1.5. FINITE Suppose that 1 < j gives
SYSTEMS OF DIFFERENTIAL INEQUALITIES
< k. Th en (1.5.11), &(tl
9
4tlN
< &(tl
23
along with (1.5.3) and (1.5.5), 9
4tl)).
( 1.5.12)
T h e mixed quasi-monotone property of g ( t , u ) in u, in view of (1.5.9) and (1.5.10), yields (1.5.13) &(tl 4 t l ) ) < &(tl w(t1)). 1
1
T h e inequalities (1.5.12) and (1.5.13) lead us to a contradiction. If, on the other hand, k 1 j \< n, arguing as before, we arrive at the contradiction
+ < &(tl
>
4tlN
> &(tl
7
Ntl)),
using the relations (1.5.4), (1.5.6), (1.5.9), (1.5.10), (1.5.11), and the mixed quasi-monotone property of g(t, u ) in u. Hence the set 2 is empty, and (1.5.7) is proved.
COROLLARY I .5.1. Let conditions (i), (ii), and (iii) of Theorem 1.5.1 be satisfied. Assume that, for t E (to, to a ) , the inequalities
+
DWt)
< g(t, W ) ,
D-m(t)
> g(t, 4))
hold. Then, v(to) < w(to)implies v(t) < 4 t h
t
E
[to , t,
+ a).
REMARK1.5.1. Notice that the proof of Theorem 1.5.1 remains unchanged even when the inequalities (1.5.3)-( 1.5.6) are replaced by
< sdt, f4th D-.,(t) 2 g*(t, 4 t ) ) , D--Wp(t) 2 g p ( t ,w ( t ) ) , D-w*(t) < g,(C 4 t ) ) . D-.,(t)
REMARK1.5.2. One can, in Theorem 1.5.1 and the following corollary, use any fixed Dini derivative D in place of D-, the corresponding inequalities being satisfied only for t E [to , to + a) - S. This follows from Lemmas 1.2.1 and 1.2.2. T h e next theorem is an analog of Theorem 1.2.3.
24
CHAPTER
1
THEOREM 1.5.2. I,et v(t),w ( t ) be k under (n - k) over-, k over ( n - k) under-functions, respectively, for t E [t,,, t, + a ) , with respect to the initial value problem (1.5.1). Assume that g ( t , u ) has mixed quasimonotone property. Let u ( t ) be any solution of (1.5.1) existing on [ t o ,to a ) such that
+
v(t,)
Then
for t
E
[to , to
+ a).
(1.5.14)
= U" = W ( t " ) .
% ( t ) < %(t> < zuv(t),
( 1.5.15)
%(t) > 4
(1.5.16)
4 > %(t)
Pmof. If (1.5.15) and (1.5.16) hold for t, < t < fo , fo sufficiently close to t o , then one can deduce the assertion of the theorem by the application of Theorem I .5. I and the subsequent Remark 1.5.1. Indeed, such a f, exists. For, defining 7n,(t)
=
%(t)
~
m,(q
%(t),
% ( t )- % ( t )
and noting thdt m,(t,,) = 0 because of (1.5.14), it is easy to deduce that mi, +(to)> 0, which implies m,(t) is increasing in a sufficiently small neighborhood of t o , say t, t t , . Similar argument with
< <
m;T(t) = Z U D ( t )
~
mZ(t) - U * ( t )
UD(t),
-
w,(t)
< <
shows that nz:(t) increases in t, t t, , t, being sufficiently close to t, . Now, the desired 2, min(t, , t J . T h e proof is therefore complete. ~
COROLLAKY 1.5.2. Let v ( t ) ,w ( t ) be under- and over-functions, respectively, with respect to the initial value problem (1.5.1) for t E [to, to a ) . .Assume that g(t, u ) is quasi-monotone nondecreasing in 21. Let u ( t ) be any solution of (1.5.1) existing on [t, , t, a ) such that
+
+
v(t,)
= U"
== Z"(t,).
Then, v(f)
< u ( t ) < U..(t),
t 6 [ t o , f"
-I a).
COROLLARY 1.5.3. Let (i)f, g E C [ E , R"], where E is an open (t, u)-set in R"I1; (ii) either f or g possess a mixed quasi-monotone property; g,,(t, u ) , f;,(t, ). >g,(t, u), ( 4 u ) E E ; ( 4 4 t h 44 be any (iii) f,,(t,). two solutions of u' = f ( t , u), v' = g(t, v), existing on [ t o ,to a), respectively, such that u0,],<. Z~,,I , u,,* > o,,*. T h e n
+
%(t)
< %(t)'
4 4 > 4%
t
E
[to , to
+ a).
1.6.
25
MINIMAX SOLUTIONS
1.6. Minimax solutions DEFINITION1.6.1. Let r(t) be a solution of the differential system (1.5.1) existing on [ t o , to a) such that, for every solution u(t) of (1.5.1) on [ t o , to a), the inequalities
+
+
4 4
< rzI(t),
% ( t )2 Y J t ) ,
3 rp(t),
uq(t)
or ~ p ( t )
< rq(t),
+4
(1.6.1)
, to t-a)
(1.6.2)
[to , t o
t
E
t
E [to
are satisfied. I n case of ( I .6. I), r(t) is called a k max ( n k) mini-solution of (1.5.1), whereas, in case of (1.6.2), it is said to be a k mini ( n - k) max-solution. I n either case, r(t) is said to be a minimax solution. A k max ( n - k) mini-solution reduces to a maximal solution when k = n and to a minimal solution when k = 0. Similarly, a k mini ( n - k) max-solution coincides with a minimal solution and a maximal solution when k = n and k = 0, respectively. ~
As minimax solutions include both maximal and minimal solutions as special cases, we consider below the existence problem for minimax solutions only.
THEOREM 1.6.1. Let g E CIRo , R"], where Ro : t o
<
< t < to + a,
/I u
- uo II
< 6,
and 11 g(t, u)II M on R, . Assume further that g(t, u) possesses a mixed quasi-monotone property. Then, there exists a k max ( n - k) miniand a k mini ( n - k) max-solution of (1.5.1) on [ t o , to 71, where 7 = min(a, 6/(2M b)).
+
+
Proof.
Let 0
< E < h/2. Consider the initial value problem
+
4 = g,(C
u) E, u:, = g,(t, u ) - E ,
Up(to) =
uo.?,
uq(to)= U o g
+
6;
- E.
1
Observe that g, E C[R,, R"], where g,(t,
R, : [ ( t ,U ) E Rn+': to
and R , C R, . Also,
4 = g(t, 4 i
< t < to +
U,
6,
/I u
- (uo
<)I1 < 6/21,
(1.6.3)
26
1
CHAPTER
It therefore follows from Peano’s Theorem 1.1.2 that the initial value problem (1.6.3) has a solution u(t, c) on the interval [ t o ,t, 71, 7 = min(a, 6/(2M -1 6)). Let 0 -1 e2 < E . Then, we have
+
<
"Ate , € 2 ) < up(to 4,
uq(tn
7
ux4
< g d t , u(t,
€2)
4(f,€2)
‘I
U X f ? €1) > ‘
-t-
€2))
, €2) > .,(to
, €1);
;
€2
& ( t , u(t, 4) - €2 ;
s,(t,
ZGf,€1) < g,(C
44 €1)) u(t, 4)
+
~
;
€2 €2.
An application of Theorem 1.5.1 yields
+
%At,
€2)
< %(t, 4
udt, €2)
> ua(t,€1).
for t E [t, , t , 771. Since the family of functions u ( t , E ) is equicontinuous and uniformly bounded, one can establish that lim u(t, E , )
=
en 4
+
~(t)
71 and that r ( t ) is a solution of (1.5.1) on uniformly on [to , t, [fn > 4, 171. T o show that r ( t ) is a k max ( n - k) mini-solution of (1.5.1) on [t,,, t, 171, we have to prove that (1.6.1) is satisfied. Let u(t) be any solution of (1.5.1) existing on [ t o ,t, -k 171. Then,
+
+
< U J t n , e), uo(tn) > ug(tn uC(t) < R B ( t , 4 t ) ) + E ;
Z4tn)
gq(t,u ( t ) )
uXt)
~
u x t , €1 2 g,(t, u ( t , 6 ) )
4 ( t , .) < g,(4
for
E
< b/2. By Theorem
u(t,
e;
+
E;
-E
1.5.1, it follows that
4 4 < %(t, E ) , % ( t ) > uo(t, €1
for t E [to, t,
+ 71. Consequently, <
u D ( t ) lim u,,(t, 6 ) = r D ( t ) , <-n
for t
E
[to
+ 771.
u,(t)
> lirn u,(t, c) <-to
=
r,(t)
9
c);
1.7.
FURTHER COMPARISON THEOREMS
27
T h e existence of k mini ( n k) max-solution can be established by changing the signs of E in (1.6.3) and proceeding on similar lines. This proves the theorem. ~
This existence theorem for minimax solutions, together with the extension Theorem 1.1.3, implies the following:
THEOREM 1.6.2. Let hypotheses (i) and (iii) of Theorem 1.5.1 hold. Then, if ( t o ,uo)E E, (1.5.1) has minimax solutions that can be extended
to the boundary of E.
COROLLARY 1.6.1. Let the hypothesis of Theorem 1.6.2 hold. Let [to, t,, u ) be the largest interval of existence of k max ( n - k) minisolution r ( t ) of (1.5.1). Suppose [ t o ,t l ] is a compact subinterval of [ t o ,to u). T h e n there exists an co > 0 such that, for 0 < E < to , the k max ( n - k) mini-solution r ( t , E ) of the system ( I -6.3) exists on [to, t l ] , and lim r ( t , C) = ~ ( t )
+ +
e-,O
uniformly on [to, t l ] .
1.7. Further comparison theorems We shall be concerned, in this section, with comparison theorems for finite systems of differential inequalities. These are, naturally, extensions of some of the results in Sect. 1.4. As will be seen, minimax solutions play an essential role.
THEOREM 1.7.1. Assume that the hypotheses
(i) and (iii) of Theorem 1.5.1 hold. Suppose that [ t o ,to a ) is the largest interval in which the k max ( n - k) mini-solution r ( t ) of (1.5.1) exists. Let m E C"to t o a ) , R"],( t , 4 9 ) E E, t E [ t o , to -1- a),
+
?
+
m,(t,) G
*o,n
>
mc(to) 2 *o,c
?
(1.7.1)
and, for a fixed Dini derivative, the inequalities (1.7.2) hold for t E [to, to
+ u)
-
S. Then,
28
CHAPTER
1
Proof. By Lemma 1.2.2, it follows that (1.7.2) is equivalent to the inequalities (1.7.4)
+
+
for t E ( t o ,t, u). Lct T E ( t o ,to a). Then, the existence of k max - k ) mini-solution r ( t , c) of (1.6.3) on [ t o ,71,for all E > 0 sufficiently small, satisfying r ( t ) = lim r ( t , 6) (I -7.5) <-0 (n
uniformly on [to, TI, is a consequence of Corollary 1.6.1. By Thcorern 1.5.1 and the relations (1.6.3) and (1.7.4), we deduce that (1.7.6)
for t E [ t o ,TI. T h e last inequalities, in view of (1.7.5), prove the conclusion (1.7.3).
REMARK1.7.1. If, in Theorem 1.7.1, the inequalities (1.7.1) and ( I .7.2) are reversed, the assertion (1.7.3) becomes % ( t ) ‘2 P,,(t)>
mdt)
where p ( t ) is the k mini ( n requires obvious changes.
-
< PQ(t),
t E It” > to
+ 4,
k ) max-solution of (1.5.1). T h e proof
T h e following corollary of Theorem 1.7.1 is important in later applications,
COROLLARY 1.7.1. Let condition (i) of Theorem 1.5.1 be satisfied. Suppose that g is quasi-monotone nondecreasing in u. Let [ t o ,to a ) be the largest interval of existence of the maximal solution r ( t ) of (1.5.1). Let m E C [ [ t o to a), and, for a , a),R”],( t , m ( t ) )E E, t E [ t o ,to fixed Dini derivative, the inequality
+
-+
+
Dm(t)
< g(t, m(t))
(1.7.7)
holds for t E [to, to -{- a ) - 5’. Then, 4to)
implies m(t)
G uo t
E
[to , t o
(1.7.8)
+ a).
(1.7.9)
1.7.
29
FURTHER COMPARISON THEOREMS
REMARK1.7.2. If, in Corollary 1.7.1, the inequalities ( I .7.7) and ( I .7.8) are reversed, then the conclusion (1.7.9) is to be replaced by m(t) 3 p(t),
t E [to , f ,
-t a),
where p ( t ) is the minimum solution of (1.5.1). This follows from Remark 1.7.1. T h e next theorem is analogous to Theorem 1.4.2 in this general framework.
THEOREM 1.7.2. Let the hypothesis of Theorem 1.7.1 hold, except that the inequalities (1.7.2) are replaced by ( I .7.10)
where
z, = [t E [ t o , to + a ) : d,(t) < 01, d,(t)
d,(t) = r d t ) - m&),
=
m,(t)
-
r*(t),
and Si is an at-most countable subset of Zi , for each i. Then (1.7.3) remains valid. Proof. T h e proof requires minor changes up to (1.7.5) of the proof of Theorem 1.7. I . Now, proceeding to prove (1.7.6) as in Theorem 15 1 , we arrive at a t , and a j such that 1 j n and
< <
4 t l ) =YAtl
, €1.
(1.7.1 1)
Moreover, it is easy to obtain from Theorem 1.5.1 that YAt)
<
> r,(t, €1
y,(t)
€)>
+
(1.7.12)
for t E [ t o ,71, 7 E ( t o, to a), where r ( t , e ) is the k max ( n - k) minisolution of (1.6.3) which exists on [ t o ,.], by Corollary 1.6.1. I t then follows from (1.7.11) and (1.7.12) that r j ( t l )< m j ( t l )
or
r3(tl) > mj(tl)
if
if
K
1
<j < k
+ 1 < j < n.
This implies, from the definitions of Zi and di(t), that t , E Zj . Hence, the j t h inequality in (1.7.10) is satisfied for such a t , . T h e rest of the proof is identical with the proof of Theorem 1.5.1 in order to arrive at (1.7.6), and this is sufficient to draw the conclusion (1.7.3).
30
CHAPTER
1
THEOREM 1.7.3. Let the hypothesis of Theorem 1.5.1 hold, except that the inequalities D-.*(t) D-wD(t)
+
3 g,(t,
( I .5.4*)
.(t)),
> g,(t, w ( t ) )
(1.5.5*)
are satisfied for t E (to, to a ) , instead of (1.5.4) and (1.5.5). Assume further, for each T E ( t , , t, a), t E [ t o ,71, and for each i, that g satisfies the condition gi(t,U)
-
+
g,(t, U) 3 -G(T
+ to
ui 3 z& , u j = zZi (i # j ) , where G E C[[to, to is the maximal solution of U'
=
G(t,u),
-
t , ui - Ci),
(1.7.13)
+ a ) x R, R], and r(t) = 0
~(t,= ) 0.
Then (1.5.7) holds. Proof. Following the proof of Theorem 1.5.1, we arrive at a t, E ( t o, to u ) and a j ( 1 j TZ) satisfying
+
< <
m,(t1)
=
( 1.7.1 5 ) < t < t, . C ( t ) = v(t, + to t ) , G ( t ) = ~ ( t+, t, t),
rnj(t) 2 0,
<
(I .7.14)
0, to
Let 1 < j k. Define g(t, u) = -g(t, to t, u), and d ( t ) = Gi(t) Then, using (1.5.3) and (1.5.5*), we obtain
+
~
-
~
Ci(t) for t E [to , t J .
This inequality implies, along with (1.7.16) and the assumption (1.7.13), the scalar differential inequality D-d(t)
< G(t,d ( t ) ) ,
t E [to , tll.
Since d(t,) = 0 because of (1.7.14), arguing as in Theorem 1.4.3, one deduces that vj(t)
1~
j ( t ) ,
t E [to tll, 7
1.8.
31
INFINITE SYSTEMS OF DIFFERENTIAL INEQUALITIES
which contradicts the assumption (1.5.2) and the definition of t, E (to , to a). A repetition of the argument to the case Jz 1<j n yields a similar contradiction. This proves that the set 2 is empty, and the desired result (1.5.7) follows.
+
+
<
+
COROLLARY 1.7.2. Let E be the product space [ t o ,to a) x R2n and g E C[E, R"].Assume that g is quasi-monotone nondecreasing in u for each (t, v) and monotone nondecreasing in 21 for each ( t , u). Suppose that r ( t ) is the maximal solution of the differential system u' = g(t, u, u),
u(to)= uo 3 0
+
existing on [to , to a ) , and that r ( t ) 3 0, t maximal solution r l ( t ) of
[to, to
+ u). Then,
the
u(to) = uo 3 0
u' = gl(t, u),
exists on [ t o ,to
E
+ a ) , where
gdt, u ) = g(t, u, r ( t ) )
and r ( t ) = rl(t), t E [to, to
+ a).
COROLLARY 1.7.3. Let the assumptions of Corollary 1.7.1 hold; m E C [ [ t o to , a), R"] such that ( t , m(t), u ) E E, t E [tu , to a), and m(to) uo . Assume that for a fixed Dini derivative the inequality
<
+
+
Dm(t)
+
< g(t, m ( t ) ,v)
is satisfied for t E [to , to a ) -- S. Then, for all 21 we have m(t) r ( t ) , t E [ t o ,to a).
<
+
< r ( t ) , t E [to, t, + a),
1.8. Infinite systems of differential inequalities
A classical result of Perron is that the maximal solution of a scalrr differential equation can be obtained as the least upper bound of the family of functions m(t) that satisfy the inequality
with the same initial condition m(to) = uo . Similar arguments hold for infinite systems of differential inequalities, provided that the maximal solution of a single equation is known. We shall first formulate an abstract
32
CHAPTER
1
version of this method and then apply it to show the existence of minimax solutions for an infinite system of differential equations and also obtain inequalities. Let El ,Fl be two partially ordered sets with the partial ordering We use the same symbol of order relation, namely, for both the sets. Assume that the following conditions hold:
<.
<,
x,y,Z
E
El,
x
x,yeEl,
y
x
f,y, 2 eF1, f
f
<j ,
€F1 implies
f
y < x
9
< f.
< .%
< z;
(1.8.1)
x=y;
(1.8.2)
< 2;
(1.8.3)
imply x imply
imply f
(1.8.4)
Corresponding to the sets E , and Fl , let us consider two partially ordered sets I&, F2 , with the dual order relation, denoted by the symbol 3, satisfying conditions (1.8.1 *)-(1.8.4*) analogous to (1.8.1)-(1.8.4). We shall use u , v, w and U,5,W for elements belonging to E, and F, , respectively. Let the operators P I , P, be defined on E l , E , , taking values in 17, , F2 , respectively. Furthermore, let the functions Q1 ,Q2 be defined on El x El x E, , El x E, x E , , taking values in Fl ,F 2 , respectively. Consider the simultaneous equations (1.8.5)
By a solution of (1.8.5), we shall mean an ordered pair (x,u), x E El , E, such that x, u satisfy Eqs. (1.8.5) simultaneously. X solution Y = ( E , 71) of (1.8.5) is said to be a minimax solution, if for every solution (x, 21) of (1.8.5) the relations
ZL E
x
U > T
are satisfied. T h e functions 0,, Q, are said to possess a mixed quasi-monotone property whenever the following conditions hold: y1 ,y 2 E El , y1 y , imply that
<
and
Q1(x, y1 , 21) QZ(Y1
u1 , uz t E ,
7
-sQ1(x,y z , u ) ,
x € El ,
*>a) ;Q2(Y,, u, a),
u E B, ;
(1.8.6)
u, 21 E E, ;
(1 23.7)
x,Y E E, ;
(1.8.8)
, u1 3 u, imply that L)l(X?
Y , u1)
<
Y , u2),
Ql@,
1.8.
INFINITE SYSTEhlS OF DIFFERENTIAL INEQUALITIES
33
We now define the sets Ul
= [X E
El : P1(x)< Q1(x,X, u),u E E,];
( 1.8.10 )
(1 3 . 1 1) E, : P,(u) 2 Q,(x, U , u),x E El]. T h e following theorem is concerned with the existence of the minimax solution of (1.8.5).
U,
= [U E
THEOREM 1.8.1. Let P, , P, ,Q, ,Q, be as defined previously. Suppose that Q1 , Q, have the mixed quasi-monotone property. Assume further that there exist two functions z,,x, defined on El, E, such that zl(E,) C E l , z,(E,) C E, , satisfying the follwing conditions: ( I .8.12)
and (1.8.1 3)
<
imply that x zl(y) , u >, z2(v).Let the sets U , , U , defined in (1.8.10) and (1.8.11) be nonempty. Then,
Zl(E1)c U, ,
Zz(E2)
c u, -
(1.8.14)
Moreover, the existence of (sup U , , inf U,) implies the existence of (sup zl(U,), inf z2(U,)), and vice versa. Also, sup U, = sup zl(U,), inf U , = inf z,( U,), and r = (sup U , , inf U,) is the minimax solution of ( I .8.5).
Proof.
Let x E E, , u E E,
. Then,
from (1.8.12), ( 1.8.15)
34
CHAPTER
1
which, in view of the definitions of the sets U , , U , , imply that Z1(X) t
u,,
Zz(U) E
u, .
This proves the assertion (1.8.14). We shall show that z1 , are increasing functions. For, let y1 y 2 , zj >, 21, , where y1 ,y, E El and 71, , v2 E E, . Using again (1.8.6) and (1.8.9) in (1.8.12), we get
<
Pl(Zl(Y1))
G Ql(Zl(Y1)I Yz >
PZ(%(.l))
2 Q,(Yl
7
0 2
9
4
7
Z2(.1)),
and this, because of (1.8.13), proves that
d Z1(3’2)>
ZI(Y1)
Suppose now that that
2 .2(.21.
Z Z ( 4
8 = sup U , , 7 = inf U , ; we Zl(S)
< 6,
have, by (1.8.14), (1.8.17)
, 4 1 7 2)17.
<
On the other hand, since x 6,u 3 7 for any x E U , , u E U , , the mixed quasi-monotone property of 0, , Q 2 , together with the definitions of U , , U, , yields Pl(.) < V l ( X , S, T ) , P.,(,) 2 I ) z ( t , 7, U).
I t then follows from (1.8.13) that x f
Zl(S),
u >, 47).
Consequently, f = z,(<), 7 = ~ ~ ( from 7 ) (1.8.17). I t is evident from (1.8.12) that Y = ( f , 7) is a solution of (1 A.5). Let y y * , 71 3 u* for y E zl( U,), v E z2(U J . Then
<
Zdt) f
y*,
Zz(T)
3 .*-
T h e monotonic nature of z1 , z2 yields
zl(x) < ~ ~ ( 6 )for all y E z1(U1),
y
=--
v
= z&)
I t therefore implies that
> z2(r])
8
for all
sup xl( U,), 7
:
uE
=
z2(U2).
inf x,( U,). T h e fact that
.=(5 , 7) is- the minimax solution of (1.8.5) follows from the definitions of U , , r J z .
1.8.
35
INFINITE SYSTEMS O F DIFFERENTIAL INEQUALITIES
If ( = sup zl(U,), 7 = inf z,( U,), using similar arguments, one can prove that ( = sup U , , 7 = inf U , , respectively, and that Y = 7) is the minimax solution of (1.8.5). This completes the proof of the theorem.
(e,
Consider the system u; = f ; ( t , u1 , u2 ,... u, ; v1 , v, )...)v,), )
Ui(t") =
uio,
1
v;
= &(t, u1 , u2 ,... u, )
; v1 , v,
)...,us),
(1.8.18)
V j ( t 0 ) == Zljo,
1
<.I'< q,
whcre p , q are arbitrary (may be infinite). T h e existence of minimax solutions for finite systems of differential equations follows when p , q are both finite. T h e minimax solution for infinite systems is covered by the other choices of p and q (either p or q or both may be infinite). I n case p , q are both infinite, the functions f i and g, in (1.8.18) are to be interpreted as f i (t, u1 , u, ,...; v 1 , v, ,...) and g j ( t , u1 , u2 ,...; vl , v 2 ,...), respectively. Let the functions fi and gj satisfy the following assumptions: and (i) f i ( t , u1 , u2 ,..., ur, ; v1 , v2 ,..., a), gj(t, u1 , u2 ,..., up ; vl , v2 ,..., v,) are defined for [to, to a] and arbitrary u1 , u, ,..., u, and v 1 , v 2 ,..., vq , (ii) There exists constants M iand Nisuch that
+
Ifdt, u1
I g j ( t , ui
9
u2
up
>.--3
; v1 ,
uz )-..,up ; ~1
fort~[t,,t,+a],l
9
,**-,
~2
j...,
< Mi uQ)I< Nj %)I
9
<j
(iii) T h e functions fi and gj are continuous in the sense that, if t'Y-+t,uia+ui,uje+vj(l < i < p , 1 < j < q ) , t h e n fi(ta, &(t", u l a ,
%Iy,...,
u2a
-
upa;vile, v,", ...,%*) -+fi(t,
,...,up";vl", COZ"
,... a*") )
u1
,ug ,..., u, ; 81 ,w 2 ,...,v,),
gj(t, u1 , u2
,..., up; v1 , v2 )...)v,).
(iv) T h e functions f i and g . ossess mixed quasi-monotone property, p i.e., for each i = 1, 2 ,...,p a n d j = 1, 2,..., q, (al) f i ( t , u1 , u, ,..., up ; v 1 , v, ,..., v,) is monotonic nondecreasing in u l , uz ,..., u i p 1 ,ui+l,..., up and monotonic nonincreasing in 'u1 , 'u2 ,...,vuq ; J.
36
CHAPTER
1
,..., ull ; vl , v, ,..., v,) is monotonic nonincreasing ZL, , , til ,..., ti1, and monoconic nondecreasing in vl , v, ,...,v. 1-1 , V ] 11 ,..., vq . (al) g,(t,
in
11,
Xov we have the following:
TIIEOREM 1.8.2. Assume that the functions f i and gj satisfy conditions (i)-(iv). Then there exists a minimax solution ( u t ( t ) ,vT(t))of (1.8.18) on [t,,, f , -1- a ] . Furthermore, if mi(t), nj(t)are continuous functions defined on [t,,, t,, a ] ,satisfying the inequalities
+
nz,(to)
< uin,
nj(to)
>
7170,
ml(t), ~ a z ( t ) , - *m*D,( t ) ; nl(t), nz(t),.-*> nq(t)),
D ; m j ( t )s < f ; ( f ,
D+/L,(t) 1.3g,(t, q ( t ) ,m d t ) , . . . , m&); fil(t), nz(t),..., @)),
and nlL(t)
Pmg. [t,, , I , ,
< &t),
t E [to, to
n,(t) 5 v3*(t),
+ a].
Let (+i(t)),{ $ j ( t ) }be two sequences of continuous functions on
+ a] such that, for t E [ t o , to + a ] ,
Denote
+,(t)
<
$j(t)
::vjo :
ZLj"
+ Mi(t l- N,(t
-
to),
i
=
1, 2,...)p ,
-
to),
j
=
1, 2 ,..., q.
i+i(t)S = (+l(t), + d t ) Y > 4 f l ( t )= ) YY {$j(f))
($l(t),
1
%z(t),..*,
% q ( t ) ) =--V *
I,et E l , E, stand for systems of sequences y = {qh,(t)},ZI = {$j(t)}.If $#p(t)], y, = {$:."(t)} are two sequences such that, if y 1 ,y z E E, , then yl y 2 implies +;"(t) $ y ) ( t ) on [to, to a] and for each i. Similarly for z', , u2 E E, , v 1 3 v, means that $ j l ) ( t )3 $iz)(t) on [to , f , -1- a ] and for each j . Let F , , P, stand for the systems of sequences of continuous functions on [t,,, t , + a] which take values in the real extended line, the order relations in F , , F2 being the same as those in El , E, , respectively. It is easy to verify that conditions (1.8.1)-( I .8.4) and (1.8.1 *)-( 1.8.4") are satisfied. Let us now define the operators P I ,P, and the functions Q1 , 0,as follows:
3'1
~
<
<
+
{wL(q;, P2,(71) = P ' % j ( t ) > > i 01(*~, Y , a) = . f L ( t , bl(t),.*.i (bi-l(t), ~ i ( t )4i+l(t)>..*,+g(t); PdY)
=
O~(JJ> 71, u) = gj(t,+l(t),
+D(t);
$l(t),$At),***, 4q(t)),
%l(t>,.*., %f-l(t),'At), 4j+l(t),***,4Qb(t)).
1.9.
INTEGRAL INEQUALITIES REDUCIBLE T O DIFFERENTIAL INEQUALITIES
37
Clearly the functions PI , Q2 satisfy the mixed quasi-monotone property. For any pair of sequences {$i(t)}, {t,bj(t)},define f i ( t , Y)
=
.fdt,95i(t),.--~$ ’ i - ~ ( ~ yi) ,
$l(t),
$~+i(~),..-,
Zi(t,P ) = gj(t, A(t),...,+,tt); $l(t)>..*,$j-l(t)>
~j
1
&dt),..*,d ’ ~ ( ~ ) ) ;
d’i+l(t),...,d’n(t)>.
Let, for each i, ri(t) be the maximal solution of Y’ =
h(t,Y),
Yi(t0) == uio,
and, for eachj, pj(t) be the minimal solution of P’ = Ej(G P ) ,
Pj(t0)
=
vjo.
Since the functions f i and gj satisfy (ii), the existence of r i ( t ) and pj(t) on [to , to u ] is ensured. Let us now define the functions z, , z2 by
+
%({$At)))= {fj(t)>.
~ , ( ~ g l i ( l ) )= ) {Yi(t)}>
From Theorem 1.4.1, it follows that the functions z, , z2 satisfy (1.8.13). Moreover, the sets U , , U , are nonempty, since uio - Mi(t - to)E U , and zjjO + Ni(t - t o )E U , , because of (ii). Furthermore,
I Yk(t)l
< Mi
9
I P;(t)l
< Nj .
Therefore, the family of functions {ri(t)},{ p j ( t ) } are equicontinuous and uniformly bounded. This proves that sup z1( Ul)=: {sup Y i ( t ) S , inf z,( U,)
+
= {inf p j ( t ) }
are continuous on [t,, t, u ] . T h e assertion of Theorem 1.8.2 now follows from Theorem 1.8.1.
1.9, Integral inequalities reducible to differential inequalities We shall consider, in this section, only those integral inequalities that are reducible to differential inequalities. We begin with one of the simplest and most useful integral inequalities.
THEOREM 1.9.1.
+
Let m, v E C[[t,, t, a), I?+], where R , denotes the nonnegative real line. Suppose further that, for some nonnegative constant C , we have (1.9.1)
38
CHAPTER
1
Then m(t)
< C exp j t
v(s) ds,
t
[to, to
E
t,l
Proof.
If C
+ u).
(1-9.2)
> 0, it follows from (1.9.1) that
which, by integration, yields C
+ J:vv(s)m(s) dsI
~
log C
<
fo
u(s) ds.
This incquality, together with ( I .9. l ) , gives (1.9.2). If C 0, then (1.9.1) holds for every constant C, > 0, and therefore the previous argument gives (1.9.2) with C = C, . Letting C, + 0 implies m ( t ) 7 0. This proves the theorem.
COROLLARY 1.9.1. 1,et m, u E C[[t,, to and satisfy the inequality m(t)
< n(t) -1-
It
+ a ) , R,], n E C [ [ t ,, to + a), R],
v(s)m(s)ds,
to
t
E
[ t o , to
+ u).
Then we have
If, in addition, the derivative n'(t) exists for t
E
[to, to + a ) , then
A generalization of Theorem 1.9.1 is the following analog of Theorem 1.4.1 which, however, requires the monotony of g with respect to u.
THEOREM I .9.2. Let E be an open ( t , u)-set in R2 and g E C[E,R]. Suppose that g(t, u ) is monotonic nondecreasing in u for each t. Let m E C"t0 to a), RI, (4 4 4 ) E E, t f [to , t o a), m(t,) %J , and 9
+
+
<
1.9.
INTEGRAL INEQUALITIES REDUCIBLE TO DIFFERENTIAL INEQUALITIES
Then m(t)
< r(t),
t
6 [to
+ 4,
, t"
39
(1.9.4)
where r ( t ) is the maximal solution of (1.2.1)existing on J.
Proof.
Define
so that (1.9.5)
and Since g is monotonic in u, using (1.9.5), we obtain the differential inequality v'(t)
< g ( t , v(t)),
t
E [to
, to
+ a).
From an application of Theorem 1.4.1, we deduce that v(t)
< r(t),
t E [to ,to
+ a).
T h e assertion (1.9.4) is now immediate because of (1.9.5).
REMARK1.9.1. Notice that one could prove Theorem 1.9.1 using an argument similar to that of Theorem 1.9.2, although we have given the classical proof.
+
COROLLARY 1.9.2. Let m, v E C[[t,, to a ) , R,], g E C[R+, R,], g(u) monotone in u and g(0) = 0. Assume that m(t>
< m, + 1't o v(s)g(m(s)>ds,
t
E
[ t o ,to
+ a).
Then m(t)
< w-l[zo(m,>
+ Ft
to
74s) dsl,
to
< t < t, ,
where
N1.
=
jud7/g(7), UO
uo
> 0;
+
w-'(u) is the inverse function of w(u), and (to, t l ) C [to, to a ) such 1 that w(mo)+ Jt,v(s) ds is in the domain of definition of w-'(u).
40
CIIAPTEK
1
COKOLL~ARY 1.9.3. Let m , p E C [ [ t , , to + a), R] and m(t)
< m ( b ) + J tt o [ K m ( s )+ p(s)l ds,
K > 0.
Then m(t) ' ::m(to)exp[K(t- to)]
+ j'p ( s ) e x p [ K ( t
-
s)] ds,
to
t tz [to , t o
+ a).
COKOI,I.ARY 1.9.4. Let the assumptions of Theorem 1.9.2 be satisfied except that the integral inequality (1.9.3) be replaced by
where
TZ E
C[[t,,, t,
+ a ) , R]. Then (1.9.4) takes the form
4 4 ,< 4 t ) + Y ( t ) ,
t
E:
[to , to
+ a),
where r ( t ) is the maximal solution of u'
= g(t, n(t)
+ u),
u(2,) = 0
existing on [to , t , 4-a ) , It is casy to extend Theorem 1.9.2 to finite systems of integral inequalities. Actually, we prove such a result in a more general form.
THEOREM 1.9.3. Let assumption (i) of Theorem 1.5.1 hold, and suppose that g(t, u ) has the mixed monotone property in u. Let m E C[[k,,, to a ) , R"],( t ,m ( t) )E E , t E [to, to a ) , and the inequalities
+
+
m,(to) G
% (t)<: ;
hold for t E [to, to
for t t [ t o , to (1.5.1).
U0.D
%(to)
+ a). Then % ( t ) e .,(f>,
+ a), where r ( t ) is
%(to)
+ Jt g&,
> U0.Q
3
4 s ) ) ds
to
m,(t) b .dt>
(1.9.6)
the K max (n - k) mini-solution of
1 .lo.
Proof.
41
DIFFERENTIAL INEQUALITIES IN THE SENSE OF CARATHEODORY
Let the vector function v(t) be equal to
so that mP(t)
and
< %(t), v'(4
%(t)
24
(1.9.7)
2 )
= g(t, v ( t ) ) .
T h e mixed monotonic character of g in u shows, in view of the inequalities (1.9.7), that % (t)
< g,(t, 4 t ) ) l
VXt)
>, gdt, n ( t ) ) ,
t E [ t o t" 9
+ 4.
Theorem 1.7.1 is now applicable, and we get %(t)
<
YP(t),
% ( t ) 3 r*(t),
t
E
[to , t o
+
a).
T h e inequalities (1.9.6) result from (1.9.7), and the theorem is proved.
I .lo. Differential inequalities in the sense of C a r a t h e o d o r y Let the function g(t, u ) be defined on an open ( t , u)-set E C R2,taking values in R. g(t, u ) is said to satisfy the Caratheodory condition if (i) g(t, u) is continuous in u for each fixed t and Lebesgue measurable in t for each fixed u ; and (ii) M ( t ) is a summable function on [to , to + u] and I g(t, ).I < M ( t ) , (4 u) E E. By a solution u(t) of the differential equation with an initial condition u'
= g(t, u),
.(to) = uo ,
(1.10.1)
we mean an absolutely continuous function u(t) satisfying (1 .lo. 1) almost everywhere on [ t o ,to u]. By the classical theorem of Caratheodory, there exists a solution of (1.10.1)under the foregoing conditions. Moreover, existence of maximal and minimal solutions and the problem of extension of solutions can be shown in just a similar way as before. T h e following theorem on differential inequalities of Caratheodory type is of interest.
+
42
CHAPTER
1
THEOREM 1.10.1 Let (i) the function g(t, u ) be defined on an open ( t , u)-set E C Rz, taking values in R and satisfying the Caratheodory's condition; and (ii) r ( t ) bc the maximal solution of (1.10.1) existing on [ t o , t,) -1- u ] . Assume that rn E C[[t,, t , -1- a],R] and is of bounded variation on [ t , , t, u ] such that its singular part is a nonincreasing function. Suppose further that
+
d ( t ): . ;
almost everywhere on [ t o ,to
+ a ] . Then 4t")
implies m(t)
Proof.
(1.10.2)
f ( t , m(t))
< r(t),
<
(1.10.3)
Ul)
t € [t" , to
+ u].
(1.10.4)
Define the function f ( 4 ).
=
<
if m ( t ) u, if u 6 m ( t ) ,
dt,U)
lg(t, m ( t ) )
which satisfies the Caratheodory's condition. 1,et r l ( t )denote the maximal solution of u(t,) =: U" . ( I . 10.5) u' : f ( t , u ) , We claim that ~
m(t)
<:
t
r,(t),
€
[ t o , to
+ u].
( I . 10.6)
If this were not true, let, without any loss of generality, (tl , t z ) be an open interval such that
(1.10.7)
4 t l ) = Yl(t1)
and " ( t , -1- 12) >; rl(t,
11 > 0 sufficiently small. For t ( 1.10.5) the inequality m'(t)
--
rXt)
E
+- h),
( I . 10.8)
( t l , t z ) , we obtain from (1.10.2) and
< s(t,4 4 ) - f ( t ,
Vl(t)).
( 1 .10.9)
T h e definition of f ( t , u), together with (1.10.7), (1. IO.8), and (1.10.9), gives "'(t) - r ; ( q 0, which in its turn implies
<
( 1.10.10)
1.10.
DIFFERENTIAL INEQUALITIES I N T H E SENSE OF CARATHEODORY
43
Since m(t) is of bounded variation, we have m(tl
+ h)
m(h)
=
+
tl+h
m’(s) ds
tl
+ B(tl + h),
(1.10.11)
where P ( t ) is the singular part of m(t). The relations (1.10.7), (1.10.10), and (1.10.11) lead to m(t,
-t-
h)
P(tl
~
+ h) <
Yl(t,
+ h).
As P(tl) = 0 and P ( t ) is nonincreasing by assumption, -P(tl and consequently
+ h) <
m(t,
Yl(t1
+ h) 3 0,
+ h),
which contradicts (1.10.8). This proves (1.10.6). T h e definition of f now shows, because of (1.10.6), that r l ( t ) is a solution of (1.10. I), and therefore Yl(t>
e r(t>,
t E [ t o , to
+ .I
This completes the proof.
THEOREM 1.10.2. Let assumptions (i) and (ii) of Theorem 1.10.1 be satisfied. Assume that m ( t ) is absolutely continuous for t E [to, to + a] and satisfies (1.10.2) almost everywhere on [t,, , to a]. Then (1.10.3) implies (1.10.4). If m(t) is absolutely continuous on [ t o ,to a ] , the singular part P ( t ) of m ( t ) is identically zero. This remark shows that Theorem 1.10.2 is a consequence of Theorem 1.10.1.
+
+
THEOREM 1.10.3. Let assumption (i) of Theorem 1.10.1 hold. Assume that g ( t , u ) 2 0; m E C [ [ t o, to a ] , R] and satisfies, for small h > 0,
+
Then (1.10.3) implies (1.10.4). Proof. It follows from the inequality (1.10.12) that m(t) is absolutely continuous over any interval in [ t o ,to a]. Consequently, m’(t) exists almost everywhere on [ t o ,to a]. Moreover, (1.10.12) implies that the derivative m’(t) satisfies the relation
+
I m’(t)l almost everywhere on [ t o ,to from Theorem 1.10.2.
+
< g(t, m(tN
+ a]. T h e assertion (1.10.4) now follows
44
CHAPTER
1
1.11, Notes
Theorems 1.1.2 and 1. I .3 are taken from Hartman [5]. Theorem 1.1.2 is due to Peano [2], and the proof uses a device of Tonelli [l]. For the type o f results on differential inequalities in Sect. 1.2, see Babkin [l], Cafcrio [ I , 21 and Chaplygin [ 11. AIaxim‘il and minimal solutions are considered by Peano [I]. Also see I
, 0. Theorem 1.4.5 is new. ‘I’he results of Sects. 1.5, 1.6, and 1.7 have been adopted from the work o f Rurton and Whyburn [l], who introduced the notion of minimax solutions. See also Kamke [2] and Wazewski [3, 4, 61 for results on e\tremal solutions for systems and corresponding theorems on ditferential inequalities. Section 1.8 contains the work of Lakshmikantham and Leela [4]. For allied results, see Mlak and Olech [l] and Mlak [2]. I o r ‘I‘heorem 1.9.1, see Bellman [3], Giuliano [I], Gronwall [l], and Peano [I]. Corollary 1.9.2 is due to Bihari [I]. Also, see Lakshmikantham [9] and Langcnhop [l]. ‘I’heorem 1.9.2 is due to Viswanatham [2, 41. See also Haiada [ l ] and Cafiero [3]. For a generalization of this result to systems, see 1,akshmikantham [2] and Opial [l]. Theorem 1.9.3 is new. Theorem 1.10.1 is taken from Olech and Opial [l]. ‘l’he result of Theorem 1.10.3 is due to Lakshmikantham [l]. See also Olech and Opial [I]. For Caratheodory’s existence theorem, see McShane [ 11. N e m proofs of ‘l’heorems 1.1.2, 1.3.1, and 1.4.1 are given by Corduncanu [ 171. For differential inequalities with the limiting initial conditions, see hIamedov [I]. For collateral reading on differential and integral inequalities, see Szarski [l] and Walter [3].
Chapter 2
2.0. Introduction T h e most important techniques in the theory of differential equations involve a systematic use of the theory of differential inequalities, or what may be termed as the integration of differential inequalities. We present, in this chapter, a number of varied results depending essentially on this approach. Many of our theorems and their proofs will involve estimates with respect to some convenient norm. T h e choice of a norm as a medium for our arguments is a natural one, although it is seldom the best choice. A better candidate is, of course, the so-called Lyapunov function, which is more flexible. An approach based on Lyapunov functions will be postponed to later chapters.
2.1. Global existence We shall use the Tychonoff's fixed point theorem for locally convex linear spaces to prove the global existence of solutions of the differential system x' = f ( t , x),
x(t,) = xo ,
t,
> 0,
(2.1 .I)
where f~ C [ J x R", R"], J being the half-line [0, 03). Let us state the fixed point theorem of Tychonoff in the following form.
THEOREM 2.1.1. Let B be a complete, locally convex, linear space and B, a closed convex subset of~B. Let the mapping T : B + B be continuous and T(B,) C B, . If T(B,) is compact, then T has a fixed point in B,. T h e main result of this section runs as follows. 45
46
2
CHAPTER
THEOREM 2.1.2.
L c t f E C[J x R”, R”], and, for ( t , x)
llf(t! .x)ll
E
< g(t7 I1 x It),
J x Rn, (2.1.2)
where g E C [ J x R , , R , ] and the function g(t, u ) is monotonic nondecreasing in u for each t E J . Assume that, for every u, > 0, the scalar differential equation u’ = R ( t , u ) ,
u(t,)
ug
,
t,
5 ,
0
(2.1.3)
has a solution u ( t ) = u(t, t o , u,,) existing for t 3 t, . Then, for every x,,E R7’such that 11 xo I/ u, , there exists a solution x ( t ) = x ( t , t, , x,) of (2.1.1) for t 2 t, , satisfying
<
/I x(t)lI
<~ ( t ) ,
t
3 to.
Pyoof. T o apply Theorem 2.1.1, lct us consider the real vector space B of all continuous functions from [ t o ,m) into Rn, the topology on B being that induced by the family of pseudo-norms (p,(x)}:=,, , where for x E B,
II 4t)ll.
pn(.x) = t,”yP,
A fundamental system of neighborhoods is then given by { IjrL}:=i=l ,
where
V,
[x E B : pn(x)
~:
< 11.
Under this topology, B becomes a complete, locally convex, linear space. Let us now define a subset B,, of B as follows: B,
=
[x t B : /I x(t)/l
< u(t),t 3 f,],
(2.1.4)
where u(t) is a solution of (2.1.3) existing for t 3 t, . I t is clear that, in the topology of B, the set B, is closed, convex, and bounded. Consider the integral operator defined by T ( x ) ( t ) = Y,,
+
1:”
f ( ~x(s)) , ds,
(2.1.5)
whose fixed point corresponds to a solution of the system (2.1.1). Evidently, the operator T is compact in the topology of B , and therefore T(B,) is compact in view of the boundedness of B, . T o prove the theorem, it remains to be shown that T(B,) C B, . T o this end, we observe that for any x E B, , (2.1.6)
2.1.
47
GLOBAL EXISTENCE
because of (2.1.5) and (2.1.2). Using the monotonic character of g(t, ZL) in u , the definition of B,, , and the fact that u ( t ) is a solution of (2.1.3) such that 11 x,,I/ u, , it follows from (2.1.6) that
<
II W(t)ll G 49. This implies T(B,,)C B, and completes the proof. COROLLARY 2.1.1.
Let f e C[f x R", Rn]and
Ilf(t, 41 ,< YtMll II)
(2.1.7)
for ( t , x) E J x Rn, where h(t) 3 0 is continuous for t E J; g(u) 3 0 is continuous for u 3 0, g(0) = 0, g(u) > 0, zc > 0, and g(u) is nondecreasing in u. If
1"
(2.1.8)
du/g(u) = a3
un
for u,, > 0, then, for every x,,E Rn, there exists a solution of (2.1.1) for t 3 to . Proof. T h e result follows from Theorem 2.1.2, if we show that the differential equation (2.1.3) has a solution existing for t 3 to . Clearly, the equation u' = A(t)g(u),
U ( t " ) = ug
>0
(2.1.9)
may be solved. For, if we write
it is easily seen that the function G(u) is strictly increasing in u, and so the inverse function exists. I n view of the assumptions concerning g, the domain of the inverse function is [0, a), and therefore the solution u(t) of (2.1.9) is defined for t 3 to .
REMARK2.1.1. It is possible to obtain, from Theorem 2.1.2, some qualitative information of solutions of (2.1. l), knowing the behavior of solutions of the related scalar differential equation (2.1.3). T h e result that follows illustrates this fact.
THEOREM 2.1.3. x'
Consider the system = A(t)x
+f(t,
XI,
x(t,)
= xo ,
to >, 0,
(2. I. 10)
48
CHAPTER
2
where A ( t )is a continuous, n x n matrix for t E J and f E C [ J x R", R"]. Let U ( t )denote the principal matrix solution of the linear system x'
=
A(t)x
where g E C [ J x R, , A!,] g(t, u) is monotone nondecreasing in u for each fixed t , and the scalar differential equation u' = -0u
-1- kg(t,u ) ,
has a solution u ( t ) existing for t
u(t,) = uo
>0
3 to such that
lim u ( t ) = 0. 1-
riLi
Then, there exists a solution x ( t ) of (2.1.10) defined for t
to such that
lim x ( t ) = 0.
t
-0,
2.2, Uniqueness A simple criterion that implies the uniqueness of solutions is the following Perron's condition.
THFORFM 2.2.1. Assume that (i) the function g(t, u)is continuous and nonnegative for t,, t to a, 0 u 2b, and, for every t , , to . t , . t, a, u(t) = 0 is the only differentiable function t , , which satisfies t on to
+
<
< < + u'
for to
< t,
= g ( t , u),
< <
(2.2.1)
u(to) = 0
; (ii) f E C[R, , R"],where
R, : t"
< t < to + a,
11 x - XG 11
< b,
2.2.
49
UNIQUENESS
4,, IIf(t, 4 -.f(t,-Y)lI < g(t, /I x --Y 11).
and for (4 x), ( t ,Y ) E
(2.2.2)
Then, the differential system x' = f ( t , x),
has at most one solution on to
x(t,)
(2.2.3)
= x,
< t < t, + a.
Proof. Suppose that there are two solutions xl(t) and x 2 ( t )of the system (2.2.3) on t, t t, a. Define m ( t ) = 11 x,(t) - x2(t)11. Then,
< < +
DWt)
< I1 4 ( 4 - "G(t)ll = Ilf(t, xdt)) - f ( C < g(t, W ) ,
X7Lt))ll
using (2.2.2). Also, m(t,) = 0. For any t , such that to < t , we obtain from Theorem 1.4.1 the inequality m(t)
< r(t),
t,
< to + a,
< t < t, ,
where r ( t ) is the maximal solution of (2.2.1). T h e assumption (i) now assures that mft) = 0 on to t < t , , proving the theorem.
<
COROLLARY 2.2.1. T h e function g(t, u) = Ku, K > 0, is admissible in Theorem 2.2.1. It is an easy exercise to verify that g(t, u ) satisfies assumption (i) of Theorem 2.2.1. I n this case, the condition (2.2.2) just reduces to the well-known Lipschitz condition. Although Corollary 2.2.1 is a direct consequence of Theorem 2.2.1, we give below a proof that is instructive. Proof of Corollary 2.2.1. Let m ( t ) be the same function defined previously, and let m ( t ) = n(t)eLt, where L > K is a constant. It is enough to show that n ( t ) = 0 on t, t to a. Suppose, on the contrary, that max n(t) > 0, t o < t 4t o f a
< < +
and that the maximum occurs at t = u. We have, at On the other hand, using (2.2.2), we obtain n'(o)eLu
+ Ln(a)eLu= m'(a) < llf(% x d 4 ) - f ( % xz(4)ll < Kn(u)eLu.
u, n'(o) =
0.
50
CHAPTER
< L , that n‘(n) < 0, contradicting
This implies, because of the choice K n’(o) 0. Thus, n(t) 0 on t, t ~
2
< < to + a.
~
T h e next result is known as liamke’s uniqueness theorem, which is, evidently, more general than that of Perron and is sufficient for many practical cases, since it includes as special cases many known criteria.
THFOKEM 2.2.2.
Assume that (i) the function g(t, u)is continuous and nonnegative for t, < t t, -1- a, 0 u 2b, and, for every t, , t, 2, to a, u ( t ) .~ 0 is the only function differentiable on t, t < t , and continuous on t,, t -.t , , for which e .
-_
<
+
< <
<
L ,
u’(4 =
Af,u ( t ) ) ,
u(t,)
u&)
=
=
f,
< t < t, ,
(2.2.4) (2.2.5)
0;
(ii) the hypothesis (ii) of Theorem 2.2.1 is satisfied except that the condition (2.2.2) holds for (t,x), ( t ,y ) E R, , t # 1, . Then, the conclusion of Theorem 2.2.1 is valid. We shall first prove the following: T H F O R ~ h ~2.2.3. I J x t the function g(t, u ) verify hypothesis (i) of Theorem 2.2.2. Assume that the function ~ , ( tu, ) is continuous and nonnegative for t, t t, a, 0 u 2h, gl(t, 0) = 0, and
< < + ~ i ( t u, )
< <
< s(t,u ) ,
+
Then, for every t , , t, < t , t, a, u ( t ) function on t,, t it, , which satisfies
<
for t ,
i ,
u’
---
gl(t, u),
(2.2.6)
t f to. ~
0 is the only differentiable (2.2.7)
u(t,J = 0
. t, .
Proof. Let us show that the maximal solution r ( t )of (2.2.7) is identically zero. Suppose, on the contrary, that there exists a n, t , CT -< t, a, such that ~ ( o> ) 0. Because of the inequality (2.2.6), we have : 4
v’(t)
.< g ( t , r ( t ) ) ,
If p ( t ) is the minimal solution of
t,
< u.
+
2.2.
51
UNIQUENESS
an application of Theorem 1.4.6 shows that P(t)
< +),
(2.2.8)
as far as p ( t ) exists to the left of u. T h e solution p ( t ) can be continued to t = t o . If p(T) = 0, for some T, to < T i u, we can effect the continuation by defining p ( t ) = 0 for to < t < T . Otherwise, (2.2.8) ensures the possibility of continuation. Since r(t,) = 0, liml_,to+ p ( t ) = 0, and we define p(to) = 0. Furthermore, since gl(t, u ) is continuous at ( t o ,0) and gl(to , 0) = 0, r;(to) exists and is equal to zero. This, because of (2.2.8), implies that p;(to) exists and p;(t,) = 0. But we have assumed that g(t, u ) satisfies hypothesis (i) of Theorem 2.2.2. Xence, p ( t ) =- 0. This contradicts the fact that p(o) = r(u) > 0. Therefore, r ( t ) = 0, and the proof is complete.
COROLLARY 2.2.2. T h e function g(t, u ) = A(t)u, where A ( t ) 3 0 is to a, satisfies the requirements of Theorem continuous on t, < t 2.2.3, provided that
< +
+ A(t)]e-”(t) > 0,
lim+sup[l
t-to
where p(t) =
Proof.
J
t0
A ( 4 ds,
(2.2.9)
t f t” .
(2.2.10)
Consider the differential equation (2.2.1 1 )
u’ = A(t)u.
+
T h e solutions u ( t ) 0 of (2.2.11) are nonvanishing constant multiples of the function e - p ‘ l ’ , p ( t ) being given by (2.2.10). T h e derivative of this function is A ( t ) e - P ( [ ) . Since A ( t ) 3 0, it follows from assumption (2.2.9) that every solution u(t) 0 of (2.2.11) violates at least one of the two limiting conditions (2.2.5). Hence, the function g(t, u) = A(t)u satisfies hypothesis (i) of Theorem 2.2.2.
+
COROLLARY 2.2.3. Let the assumptions of Theorem 2.2.3 hold except that the function gl(t, u ) is continuous on to < t t, a, 0 u 26. Then, for every t , , to < t, < t, a, u(t) 5-0 is the only function differentiable on to < t < t, and continuous on to t < t, , for which u;(t,) exists,
< + <
+
u’(t) = gdt, 4
and (2.2.5) holds.
th
to
9
< <
52
CHAPTER
2
+
Pmof. \Ye have to show that all the solutions u(t) 0 are such that they violate the limiting properties (2.2.5). Assuming the contrary and proceeding as in the proof of Theorem 2.2.3, it is easy to prove the stated result. Pmof of Theorem 2.2.2.
Define the function (2.2.12)
+
<
<
for to t < to a, 0 < zi 2b. Since f ( t , x) is continuous on I-?, , g,(t, 7[) is continuous on t, < t t, a, 0 u 26. It is clear that condition (2.2.2) holds for the function gf(t, u ) because of (2.2.12). 31oreol er, gr(t, ).
<
< +
< <
< g(C ).
< <
for t,, . t to + a, 0 zi 2h. Theorem 2.2.3 is now applicable with gl(t, 7 1 ) = g,(t, zi), and therefore g,(t, u ) satisfies the assumptions of Theorem 2.2.1. This establishes Theorem 2.2.2. COROLLARY 2.2.4. T h e conclusion of Thcorem 2.2.2 holds if condition (2.2.2) is replaced by
0, then g(t, u ) = X(t)+(u) is admissible in COROLLARY 2.2.5. If t, Theorem 2.2.2 provided that A ( t ) 3 0 is continuous for 0 < t a; +(u) is continuous for zi 2 0 and +(O) = 0, + (u ) > 0 for u > 0; and
<
:
T h e following is yet another criterion of uniqueness cf solutions which generalizes the earlier ones. T h e statement of the results involves the existence of two controlling functions.
TIIFOR~--LI 2.2.4. Alssuniethat (i) the functions A ( t ) ,B(t)are continuous and nonnegatixe on t,, < t < to a such that A(t,) = B(t,) = 0, K ( t ) ,0, t > , and
+
f,
limAL4(t)/B(t) = 0;
t *fo
(2.2.13)
2.2.
53
UNIQUENESS
(ii) the functions gl(t, u), g 2 ( t ,u) are continuous and nonnegative for t, < t t, + a, 0 u 2b; (iii) all the solutions u(t) of
<
< <
(iv) the only solution v(t) of (2.2.15)
D' = '?At, ).
on t,
< t < to + a such that (2.2.16)
lim+v(t)/B(t) = 0
t-to
is the trivial solution; (v) f E C[R, , R"],and, for ( t ,x), ( t ,y ) E R,, t f to, (2.2.17) Then, the differential system has at most one solution on to
< t < t, + a.
Before proceeding to the proof of Theorem 2.2.4, it is convenient to prove the following:
THEOREM 2.2.5. Let the functions A(t), B(t), gl(t, u), and g z ( t ,u ) fulfill hypotheses (i), (ii), (iii), and (iv) of Theorem 2.2.4. Suppose that the function g ( t , u ) is continuous and nonnegative for t, t to a, 0 u 2b, g(t, 0) e 0, and
< < +
< <
(2.2.18) Then, u ( t ) = 0 is the only differentiable function on t, which satisfies for to
< t < to + a.
u'
= g(t, u ) ,
U(t") =
0
< t < to + a (2.2.19)
Proof. We shall show that the maximal solution r ( t ) of (2.2.19) is identically zero. Assuming, on the contrary, that there exists a 0 such that Y(U) > 0 and proceeding as in the proof of Theorem 2.2.3, making use of relations (2.2.18) and (2.2.19), we obtain PAt)
< r(t),
(2.2.20)
54
CHAPTER
2
as far as p 2 ( t ) exists to the left of u, where p 2 ( t ) is the minimal solution of (2.2.15) such that pL(~) = r(u). As before, we can continue p2(t) up to t, by defining pz(to) : 0. Since p2(t) 0, we have
+
t-tO+ lim
# 0,
pz(t)P(t)
which, in view of (2.2.20), implies that lim+v(t)/B(t) # 0.
t-tto
This, together with assumption (2.2.13), shows that there exists a t, such that r(t1)
>4tJ.
(2.2.21)
Let p l ( t ) be the minimal solution of (2.2.14) such that pl(tl) = r ( t l ) . Then it can be shown, arguing similarly, that p l ( t ) can be continued up to t, , pl(t,) = 0, and
0
t"
(2.2.22)
Since, by hypothesis (iii), all solutions u(t) with u(t,) = 0 of (2.2.14) must obey u ( t ) A(t),to t t, a, we must have
< < -+
<
Pl(4
<4 t h
to
< t < t , + a.
This is absurd because of (2.2.21) and the fact that pl(tl) Hence, pl(to) > 0, which implies, in view of (2.2.22), that 0
contradicting the assumption r(t,)
=
=
r(tl).
< +")> 0.
Pmof qf Theorem 2.2.4. Consider the function g(t, u ) gf(t, u), where gf(t, u ) is the function defined by (2.2.12). By combining the respective -~
arguments in the proofs of Theorems 2.2.2 and 2.2.5, it is easy to show that gf(t, u ) verifies Perron's uniqueness conditions of Theorem 2.2.1, which is sufficient to establish the uniqueness of solutions. REMARK2.2.1. Whenever f ( t , x) is assumed to be continuous on R, , it follows from the foregoing considerations that the uniqueness conditions of Theorems 2.2.2 and 2.2.4 can be reduced to that of Perron's condition. If the pair of functions g,(t, u), g 2 ( t ,u ) satisfies the hypotheses of Theorem 2.2.4, we can also show that there exists a function g(t, u )
2.2.
55
UNIQUENESS
that fulfills the uniqueness criteria of Kamke as given in Theorem 2.2.2. This is the content of the following:
THEOREM 2.2.6. Let the functions A(t), B(t), gl(t, u), and gz(t,u ) satisfy hypotheses (i), (ii), (iii), and (iv) of Theorem 2.2.4. Then, there exists a function g(t, u ) verifying assumption (i) of Theorem 2.2.2.
Proof. Define the function g(t, u ) by g(t,
4 = min[g,(t,
u), gdt,
41.
(2.2.23)
Then g satisfies (2.2.18). T o prove the stated result, it is enough to show that no nontrivial solution of (2.2.4) fulfills the limiting conditions (2.2.5). I n fact, the assumption that there exists a differentiable function u(t) satisfying the differential equation (2.2.4) and the conditions (2.2.5) for which u(a) > 0, to < CT < to a, leads, following the proof of Theorem 2.2.5, to the contradiction that u(t,) > 0.
+
COROLLARY 2.2.6. T h e functions gl(t, u ) = K1ua, gz(t,u ) = K,(u/t) are admissible in Theorem 2.2.4, if 0 < 01 < I , K2(1 - a ) < 1, with A(t) = K,(l - ~ ~ ) t l / ( l and - ~ ) , B(t) = t K 2 . We shall now show that, if certain conditions of Theorem 2.2.2 are violated, Eq. (2.2.3) has nonunique solution. We prove this for the case n = 1 and t, = 0.
<
THEOREM 2.2.7. Let g(t, u ) be continuous on 0 < t a, 0 < u < b, Suppose that, for each t , , g ( t , 0) = 0, and g(t, u ) > 0 for u > 0. 0 < t , < a, u(t) 0 is a differentiable function on 0 < t < t , , and continuous on 0 < t < t, for which uL(0) exists,
+
0 < t < t, ,
u’ = g ( t , a),
and u(0) = u i ( 0 ) = 0.
Let f E C I R o ,R], where R, : 0 ( t ,Y ) E R, t # 0,
< t < a, 1 x I < b,
and, for ( t , x),
7
I f ( t , x) -f(t,Y)I
>g(t,
I x -Y I ) .
(2.2.24)
Then, the scalar differential equation x’ = f ( t , x),
has at least two solutions on 0
x(0)
< t < a.
=0
(2.2.25)
56
CHAPTER
2
Pmf. Let us first suppose that f ( t , 0) = 0, so that, putting y we obtain the inequality
=
0,
lf(t,41 > d t l I x I), because of condition (2.2.24). Sincef(t, x) is continuous and g(t, u) > 0 for u > 0, it follows that eitherf(t, x) < 0 or f ( t , x) > 0, for x .f 0. This implies that either (2.2.26)
or (2.2.27)
By hypothesis, there exists a u, 0 < u < a, such that u(u) > 0. Let y ( t ) be the minimal solution of x’ = f ( t , x), y(a) = u(u). Then, using an argument similar to that in the proof of Theorem 2.2.3 and the inequality (2.2.26), it can be shown that y ( t ) u ( t ) to the left of (T, as far as y ( t ) exists. Moreover, y ( t ) can be continued u p to t = 0 and
<
0
< y ( t ) < u(t),
0
< u.
Since u(0) z u;(O) = 0, we obtain from the foregoing relation y(0) = y’,(O) = 0. This proves that the differential equation (2.2.25) has a solution y ( t ) , not identically zero. On the other hand, since f ( t , 0) 0, (2.2.25) admits the identically zero solution. Hence, we have two different solutions for Eq. (2.2.25). Corresponding to the case (2.2.27), we can employ a similar reasoning to arrive at the same conclusion. We shall now remove the restriction f ( t , 0) = 0. Let x o ( t ) be a t < a. Using the transformation solution of (2.2.25), existing on 0 - x x,,(t), we get
<
v
-
-
2’ =
x’
-
=f ( t ,
x
xi(t)
= f ( t , x) - f ( t , x,(t))
+ x,(t)
- f ( t , xo(t))
(2.2.28)
= F ( t , 2).
Evidently, F ( t , 0) L 0. I t follows, therefore, that x = 0 is one solution of (2.2.28) through (0, 0). But the previous considerations show that (2.2.28) has a solution z ( t ) different from the identically zero solution. This implies that x ( t ) = x ( t ) x,,(t) is not identically equal to x,(t), and the theorem is proved. We have assumed the continuity of f ( t , x) in the uniqueness results
+
2.2.
57
UNIQUENESS
that are discussed so far. We shall consider, in what follows, the differential system x' =f(t,
x),
(2.2.29)
x(0) = 0,
< <
<
t a, 11 x 11 b. A solution of where f ( t , x) is defined on R, : 0 (2.2.29) in the classical sense will mean a function x(t) that is continuous on 0 t a, differentiable on 0 < t < a, and that satisfies (2.2.29) for 0 < t < a. Suppose that x(t), y ( t ) are two solutions of (2.2.29), existing on 0 t a ; then, the requirement
< <
< <
which is satisfied when f ( t , x) is continuous at (0, 0), is a necessary condition for the uniqueness of solutions. This can be generalized by the following: (2.2.30)
<
a, and where the function B(t) is continuous, positive on 0 < t B(O+)= 0. That condition (2.2.30) is not sufficient for uniqueness is seen by the following:
LEMMA 2.2.1. Suppose that the function B(t) is continuous and a such that B(O+)= 0. Then there exists an infinity positive on 0 < t of functionsf(t, x) such that (2.2.29) has more than one solution satisfying the condition (2.2.30).
<
Proof. t
0
We first construct a function A(t)having a nonnegative derivative
< < aand
lim A(t)/B(t)= 0.
t-0+
We proceed as follows. Divide the interval [0, a] into subintervals I, such that z1= (42, a), z, = (a/4,a/2)).... Suppose that 6 , such that
=
inf, B(t). Find a positive linear function I,, on Il L,(a)
=
b, ,
L,(a/2) ,< &(a).
58
CHAPTER
2
Then find L, on I , such that L,(a/2)
= b,/2
< &,(u/2) L 2 ( 4 4 ) e 4L2(u/2). L,(a/2)
if 6,/2
< L,(a/2),
if b,/2
> L,(u/2),
We continue this process and then connect the linear functions near the points a/2% by suitable functions having nonnegative derivatives (for example, by arcs of parabolas). This modification gives us the function A(t) with the required properties. Having constructed the function A(t), it is easy to define f (t, x) by f ( t , x) = x"A'(t)
(0 < a
< 1).
Then, x(t) = 0 and x ( t ) = (1 - c ~ ) l / ( l [- ~A)(t)]/(lpm) are solutions of (2.2.29). It is clear that any two solutions satisfy the condition (2.2.30), and, hence we have the proof. It is also easy to prove the following fact.
LEMMA 2.2.2. Suppose that f ( t , x) is defined on R, and is continuous at (0, 0). T h e n there exists a function B(t) on 0 t 1 such that (2.2.30) is satisfied.
< <
Proof.
Let x(t), y ( t ) be two solutions of (2.2.29) on 0
< t < a. Define
m(t> = /I x(t> - Y(t)Il.
Observe that m(0) = 0, and, because of the assumed continuity of x) at (0, 0), we have lim m ( t ) / t = 0.
f(t,
t-.o+
Setting B(t) = sups
tQ
m(s)/s, it is easily verified that lim m ( t ) / B ( t )= 0. -o+
I
+
This is possible if m(s) 0 in some neighborhood of the origin; otherwise, the existence of B(t) is trivial. We notice that the continuity requirement off(t, x) at (0,O) is stronger than the condition (2.2.30). T o see this, definef(t, x) as follows:
I
1, f ( t , x) = x / t ,
0,
x > t, 0 < x < t, x < 0.
2.2.
59
UNIQUENESS
< <
T h e solutions of (2.2.29) are then given by x(t) == Izt, where 0 Iz 1. Take B(t) = t 1 / 2 . Clearly, the relation (2.2.30) is satisfied even though f ( t , x) is not continuous a t (0,O). These considerations lead to
THEOREM 2.2.8. Suppose that x(t), y ( t )are any two solutions of (2.2.29) a satisfying (2.2.30), where B(t) is positive, continuous on 0 < t with B(O+)= 0. Let the functiong(t, u ) 3 0 be continuous on 0 < t a, 0 u b, and the only solution u(t)of
< <
< <
on 0
u’ = g(t, u )
< t < a such that
lim u ( t ) / B ( t )= 0
t-O+
is the trivial solution. Assume further that the functionf(t, x) is defined on Ro and satisfies I I f ( t 7
< g(t7 I1 x - Y II)
4 - f ( t ,Y)II
for ( t , x), (t,y ) E Ro , t # 0. Then there exists at most one solution of (2.2.29) on 0 t a.
< <
Proof. Define m(t) = (1 x(t) - y ( t ) (I, where x ( t ) , y ( t ) are any two solutions of (2.2.29) existing on 0 t a. Then m(0) = 0, and
< <
o.44 G llf(t, 4 t ) ) -f(t,y(t))lI
< g(t, m(t>).
If we suppose that, for some u, 0 as in Theorem 2.2.3, that p(t)
< u < a,
<4 t h
t
<
m(u) > 0, we can show,
0,
as far as p ( t ) exists, where p ( t ) is the minimal solution of u’ = g(t, u ) ,
u(u) = m(u).
Furthermore, as before, p ( t ) can be continued up to t = 0 and p(t) m(t), 0 t 0. Then, because of the assumed condition (2.2.30), we have
0
<
<
< <
0
< lim p(t)/B(t)< lim m ( t ) / B ( t )= 0, t-O+
t-O+
which, by hypothesis, implies that p ( t ) = 0. This contradicts p(u) m(u) > 0, and hence m(t) = 0, 0 t a. T h e proof is complete.
< <
=
60
CHAPTER
2
2.3. Convergence of successive approximations The answer to the question of whether or not a solution of the system (2.2.3) can always be obtained as a limit of the sequence or a subsequence of the successive approximations is negative. It is not difficult to construct an example such that the solution of (2.2.3) is unique, although no subsequence of the successive approximations converges to that unique solution. It turns out, however, that, with an additional restriction of monotony ofg(t, u ) in u in Theorems 2.2.1,2.2.2, and 2.2.4, convergence of successive approximations to the unique solution follows. Suppose that g ( t , u ) of Theorem 2.2.2 is monotone nondecreasing in zi for fixed t, in addition to the hypotheses of the theorem. Then, defining
< < +
< <
instead of (2.2.12) for t, t to a, 0 u 26, we note that g,(t, u ) is monotone nondecreasing in u for each t. Thus, it follows that
and, by Theorem 2.2.3, g,(t, u) satisfies the hypotheses of Theorem 2.2.1. Similarly, if we assume that gl(t, u ) and g2(t, u ) of Theorem 2.2.4 are monotone nondecreasing in u for each t, then Theorem 2.2.5 shows that g,(t, u ) = g ( t , u ) also verifies the assumptions of Theorem 2.2.1, in view of the fact that
It is therefore enough to prove the convergence of successive approximations for Theorem 2.2.1 with an additional restriction of monotony on g ( t , u).
THEOREM 2.3.1. Let the assumptions of Theorem 2.2.1 hold. Suppose M on further that g ( t , u ) is nondecreasing in u for each t, l l f ( t , x) 11 R, , and = min(a, b / M ) . Then, the successive approximations defined by
<
(Y
(2.3.1)
< <
exist on t, t t, -k 01 as continuous functions and converge uniformly on this interval to the solution x ( t ) of (2.2.3).
2.3.
61
CONVERGENCE OF SUCCESSIVE APPROXIMATIONS
< < to +
Proof. Suppose that xk(t)is defined and continuous on to t and satisfies 11 xk(t) xo 11 b for R = 0, 1, 2 ,..., n. Write -
<
xn+1(t) = xo
+ sIo
f(S, x n ( 4
01
ds.
Then, since f ( t , x,(t)) is defined and continuous on to the same holds for x n t l ( t ) . Moreover, it is also clear that
< t < to +
01,
< Ma < 6. Thus, by induction, the successive approximations are defined and continuous on to t to 01 and
< < + II xn+i(t)
-
xn
I1 < 6,
=
0, 1,2,....
We shall now define the successive approximations for Eq. (2.2.1) as follows: udt) = M(t
-
to),
un+i(t) = fOg(s, un(s)) ds,
tn
< t < to +
01.
Then,
(2.3.2) (L.3.3)
the inequalities (2.2.2), (2.3.3), and the monotonic character of g ( t , u ) in u give
Thus, by induction, the inequality (2.3.4)
62
CHAPTER
2
(2.3.5)
because of the monotonicity of g ( t , u ) in u.An application of Theorem 1.4.1 yields that
II % ( t )
-
Xm(t)ll
< yn(t),
to
< t < to +
01,
where r,(t) is the maximal solution of Y'
=d t , Y )
-i- 2g(t, un-l(t>),
%&(to)
=
0
for each n. Since the conditions of Lemma 1.3.2 are satisfied r,(t) -+ 0 uniformly on to t to 01, as n + CO. This implies that xn(t) converges uniformly to x ( t ) on to t to 01 as n + CO. By Theorem 2.2.1, the solution of (2.2.3) being unique, this x ( t ) is the unique solution of (2.2.3).
< < +
< < +
Another proof of Theorem 2.3.1. It can be easily shown that the sequence of approximations (2.3.1) is uniformly bounded and equicontinuous on to t to 4-N, and therefore there exists uniformly convergent subsequences. Suppose that xn(t) - xnPl(t) + 0 as n + 00; then (2.3.1) implies that the limit of any such subsequence is the unique solution x(t) of (2.2.3). It then follows that a selection of a subsequence is unnecessary and that the full sequence x,(t), xl(t), x2(t),... converges uniformly to x(t).
< <
2.3.
63
CONVERGENCE OF SUCCESSIVE APPROXIMATIONS
Thus, to prove Theorem 2.3.1, it is sufficient to show that m(t) = 0, where m ( t ) = lim sup (1 x n ( t ) - xn-l(t)ll. n-w
(2.3.6)
We shall first show that m ( t ) is continuous for t E [t, , t, Ilf(t, x) 11 M on R, , we see that
<
II 4 t l ) - xn-l(tl)ll
< ll xn(tz) xn-l(tz)ll + 2 M I tl < m(tz) + 2MI tl t , I + -
-
for large n, if
E
-
+ a ] . Since
tz
I
E
> 0. Hence, we have m(t1)
< m(t2) + 2MI tl
As t, , t, can be interchanged and
E
I 4tl) -4tz)I
-
t,
I
+
E.
> 0 is arbitrary, we obtain
< 2 M I tl
t, I 9
~
which proves the continuity of m ( t ) . T h e assumption (2.2.2), together with the relation (2.3. l), yields
< J t g(s, II xn(s) xn-l(s)II) ds. For a fixed t in the interval ( t o , t, + a ] , there is a sequence of integers II xn+dt> - xn(t)ll
n, < n2 < and that
0 . .
-
t0
such that 11 ~ ~ + ~ xn(t) ( t )11 tends to m(t) as n m*(s)
exists uniformly on to
=
lim
n = n,-m
/I xn(s)
-
=
nk 4 co,
X~-~(S)II
< s < to + a. Thus, (2.3.7)
Since g is assumed to be monotone nondecreasing in u and m*(s) we obtain from (2.3.7) the inequality
< m(s),
<
By Theorem 1.9.2, m(t) r(t), where r ( t ) is the maximal solution of (2.2.1). As Eq. (2.2.1) is assumed to possess only identically zero solution, it follows that r(t) = 0, which in turn shows that m(t) = 0 on [ t o ,to a ] . This completes the proof.
+
64
2
CHAPTER
2+4. Chaplygin’s method
We are interested in establishing a method of approximation of the solution of a given differential equation by means of solutions of an associated linear equation and in estimating the difference between them. This is precisely what the Chaplygin’s method accomplishes. For convenience, we shall first consider the case of scalar differential equation.
THEOREM 2.4.1. Let f E C I R o ,R ] , where R, is the rectangle to t 0 in R, . Let to + 01 the functions uo = uo(t),vo = vo(t)be differentiable for to t such that (t, uo(t)),( t , vo(t))E R, and
+
<
<
< <
< f ( t , *o(t)), vat> > f ( t ,vo(t)>7 *Xt)
*,(to) = xo
7
(2.4.1)
vo(t0) = xo
-
(2.4.2)
Then, there exists a Chaplygin sequence {un(t),vn(t)) such that *At) < * n + d t )
< x ( t ) < vn+1(t) < vn(t), %(to)
= xo =
t
E
[ t o I to
+ 4,
%(to),
where x ( t ) is the unique solution of x‘ = f ( t , x),
(2.4.3)
x ( t o ) = xo
+
existing on [to , to a ] . Also, u,(t) and v,(t) tend uniformly to x(t) on [to , to a] as n -+ GO. If, in addition, for a suitable constant ,f3,
+
0 < vo(t) - *,(t)
then
I %(t)
-
v,(t)l
< 2is/22n,
< is, t
E
[to , to
+ .I.
(2.4.4)
Proof. T h e functions uo(t),vo(t), and x(t) satisfy the assumptions of Theorem 1.2.3, and therefore we have %(t)
< x ( t ) < .o(t),
We now define the functions
t
E
( t o , to
+ .I
2.4. Observe that, for t =
65
CHAPLYGIN’S METHOD
to,
fAt0
9
, x; uo
x; uo 7 no) = f&o
Y
.o).
Let ul(t), vl(t)be the solutions of the linear differential equations 4 ( t ) =f1(t, u d t ) ; uo , %I,
%(to) = xo
>
(2.4.7)
q t ) =fdt, s(t); uo %),
Vl(t0) =
xo
1
(2.4.8)
9
which exist on [to, to offi in (2.4.5) result
+ a ] . From the inequality (2.4.1) and the definition uLxt)
< f(4 uo(t)) = fdt, uo(t);*o
>
no),
which, because of Theorem 1.2.1 and the following remark, yields uo(t)
< %l(t),
t
E
(to 9 to
+
. I ] .
(2.4.9)
A similar reasoning with (2.4.2) and (2.4.6) shows that
We shall next show that the functions ul(t), vl(t) also satisfy the differential inequalities (2.4.1) and (2.4.2), respectively. Since fz(t, x) is strictly increasing in x, using (2.4.5), (2.4.7), and the mean value theorem, it is easy to deduce that
On the other hand, 4 ( t ) < f ( t , uo(t)) = f2@,
uo(t>;uo >
Vo),
and consequently, we have, applying Theorem 1.2.1 again,
Furthermore, it is readily seen that (2.4.13)
66
2
CHAPTER
and .f(t>s ( t ) ) = f(tJuo(t)) + f z ( t ,
uo(t)"dt)
+ i f z z ( t , O[vi(t)
-
uo(t)l udt)
- uo(t)12,
< E < ni(t).
(2.4.14)
T h e relations (2.4.8), (2.4.1 I), (2.4.12), (2.4.13), and (2.4.14), together with the repeated applications of mean value theorem and the assumption M t , t) > 0, imply v;(t) = fi(f, v1(t>; uo vo) 1
>f ( t ,s ( t ) ) ,
+ 4.
t E (to to 1
(2.4.15)
Since the functions ul(t), vl(t), and x(t) verify the assumptions of Theorem I .2.3, we obtain %(t) < x ( t ) < s(t), t
E
( t o to 7
+
011,
which, in view of the inequalities (2.4.9) and (2.4.10), gives uo(t) < ul(t) < x ( t ) < vl(t)
< vo(t),
t
E
( t o ,to
+ a].
(2.4.16)
T h e foregoing considerations define a transformation T that assigns to a given couple of functions (uo(t),vo(t))a new couple (ul(t),vl(t))satisfying the same inequalities (2.4.1) and (2.4.2), respectively, such that (2.4.16) holds. This implies that (u1 7 v1) =
mu0
9
.")I.
It therefore follows that we can apply the transformation T to the couple ( u l ,ul) to get (u2, v2). A repeated application of the transformation T provides a well-defined Chaplygin's sequence (%+l
> %+l) =
W
n
>
741
of functions satisfying the following relations:
6) (ii) (iii)
uXt)
4 ( t ) > f ( t , v,(t)), u,(t)
(3 4 + * ( t ) (v)
< .f(t,un(t)),
%(to)
=
xo ;
vn(to) = xn ;
< ~ , + ~ (< t ) x ( t ) < ~ , + ~ (< t )v,(t), = f d t 7
t
E
( t o , to
+ a];
%+At); u,(t>, vn(t));
4 + 1 ( t ) = f d t , 7-5n+1(t);
% ( t ) >vn(t))-
It is clear from (iii) that the sequences {urn},{urn}are monotonic and uniformly bounded on [ t o ,to a ] . Furthermore, they are equicontin-
+
2.4.
67
CHAPLYGIN'S METHOD
uous, in view of the fact that, for each fixed n, u, , v, are solutions of linear equations. Hence, an application of Theorem 1.1.1 proves the uniform convergence of un(t),vn(t)to x ( t ) as n --f 00. Let (2.4.17)
and
<
<
Assume that 0 v,(t) - u,(t) (2Haek@)--l= p. Clearly, (2.4.4) holds for n = 0. Suppose it is true for a certain fixed n, i.e., (2.4.19)
From the definition of ~ % + ~ (~t % ) ,+ ~ (and t ) , the mean value theorem, it follows that
where u,(t)
where u,(t)
we obtain, from (2.4.17), (2.4.18), and (2.4.20),
Furthermore, we also have and
I4
-
I un+1(t)
un(t)I -
< I vTdt>
- Un(t)l
4 1 1 G I vnw - un(t)l-
These estimates, together with (2.4.19) and (2.4.21), lead to the differential inequality 22/32 D+lvn+1(t) - un+1(t)l
< KI %+l(t>
- un+1(t)l
+ H22"+1?
68
2
CHAPTER
which, in view of Theorem 1.4.1, yields
Since
S eK(lPs)ds < meKa, we get 1
to
Thus, by induction, the relation (2.4.4) is true for all n, and consequently we have, by (iii),
I x ( t ) - U,(t)i
and
G 2/3/2'"
1 x ( t ) - v,(t)l ,< 2/3/22". This completes the proof. Let us now consider the differential system x' = f ( t , x),
x(t,) = x,
.
(2.4.22)
I n this case, we shall be able to demonstrate only the lower Chaplygin's sequence {un},under some additional restrictions.
THEOREM 2.4.2.
Let f E C[R,, Rn],where R, is the set,
< t G to + a ,
R, :
<
I1 x
-
xo II
< b.
Let l / f ( t ,x) (1 M on R , . We suppose that f ( t , x) is quasi-monotone nondecreasing in x, for each t E [to , t, + a], and that af ( t , x ) / a x exists and is continuous on R,. Let u,(t) be continuously differentiable on [to , a ] , where 01 = min(a, b / M ) ,( t ,u,(t)) E R,,and ui(t) < f ( t , u,(t)), ",(to) = x, . Furthermore, let
+
f ( t , x) +f,(t,
x)
. ( y - x) < f ( t , y )
if
x
(2-4-23)
Then, there exists a Chaplygin sequence {un(t))such that un(to)= x, , %(t)
< Un+l(t) < 4 t h
t E
[ t o , to
+ 4,
where x(t) is the solution of (2.4.22) existing on [to , to uniformly on [ t o , t,
+ .I].
lim u,(t)
n-m
= x(t)
+ a ] , and
2.5.
69
DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS
Proof. We notice, first of all, that afi/axj >, 0 for i # j . This follows from the quasi-monotonicity of f ( t , x). Moreover, applying Corollary 1.5.1, we obtain uo(t)
< X(t),
t 6 [to ,to
+ .I.
Corresponding to the linear equation (2.4.7),we have now to consider the linear system defined by
=
(2.4.24)
Y(t0) = xo *
uo(t)),
Observe that f ( t , y; u,(t)) possesses the quasi-monotone property in y because afi/axj2 0, i # j . Hence it follows by Corollary 1.5.1 that uo(t)
< Ul(t),
t E [to t o
+ 4,
where ul(t) is the solution of (2.4.24). The assumption (2.4.23) implies that u;(t> = f ( t ,u l w ; u o w
Thus, we have uo(q
< x@),
< ul(4 < 4 t h
t E [to , t o
+ 4.
t E [to 7 t o
+ 4.
As in Theorem 2.4.1, we can define the transformation verifying u d t ) = T[uo(t)l-
T h e rest of the argument is but a repetition of the proof of Theorem 2.4.1 with appropriate changes. This establishes the method of Chaplygin for systems.
2.5. Dependence on initial conditions and parameters We shall consider the problem of continuity and differentiability of solutions x ( t , to , xo) of the differential system x' = f ( t , x),
(2.5.1)
70
CHAPTER
2
with an initial condition (2.5.2) with respect to the initial values (to , xo).
LEMMA 2.5.1.
Let f~ C [ J x R", Rn],and let
Assume that ~ * ( tto, , 0) is the maximal solution of U' =
G ( t ,u),
through ( t o ,0). Let x(t, t o , xo) be any solution of (2.5.1) and (2.5.2). Then,
ll x ( t , t o ,xo) - g o II e Y * ( t ,
THEOREM 2.5.1.
to I O),
t >, t o -
L e t f E C [ J x R", R"],and, for ( t , x), ( t ,y ) E J x R",
where g t C [ J x R, , R,]. Assume that u(t) = 0 is the unique solution of the differential equation U' = g ( t , u )
(2.5.4)
such that @(to) = 0. Then, if the solutions u(t, t o , uo) of (2.5.4) through every point ( t o ,uo) are continuous with respect to initial conditions
2.5.
DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS
71
( t o ,uo), the solutions x(t, t o ,xo) of (2.5.1) and (2.5.2) are unique and continuous with respect to the initial values ( t o ,xo).
Proof. Since the uniqueness of solutions follows from Theorem 2.2.1, we have to prove the continuity part only. T o that end, let x(t, t o ,xo), y ( t , to ,y o ) be the solutions of (2.5.1) through (to, xo), (to,yo), respectively. Defining m ( t ) = I1 x ( t , to 7 xo) - Y ( t , to Yo)ll,
the condition (2.5.3) implies the inequality
e
D f f 4 t > g(t, 4 t h
and, by Theorem 1.4.1, we obtain where r(t, to , 11 xo - y o 11) is the maximal solution of (2.5.4) such that u(to)= 11 xo - yo 11. Since the solutions u(t, to , uo) of (2.5.4) are assumed to be continuous with respect to the initial values, it follows that lim r ( t , to , I/ xo
J(o-yo
-
Yo I l l
= r ( t , to
7
01,
and, by hypothesis, r ( t , t o ,0) = 0. This, in view of the definition of m ( t ) , yields that lim x ( t , to , xo)
xo-Yo
= r(t,t o
,Yo),
which shows the continuity of x ( t , t o ,xo) with respect to x,, . We shall next prove the continuity with respect to initial time t o . If x(t, t o , xo), y ( t , t, ,xo), t , > t o , are the solutions of (2.5.1) through ( t o ,xo), ( t l , xo), respectively, then, as before, we obtain the inequality D+m(t)
where Also,
< gft, W ) ) ,
m(t> = /I x ( t , to > xo) - Y ( t , t , 4 t l ) = ll X ( t ,
?
>
to xo) - xo It. ?
Hence, by Lemma 2.5.1, m(t1)
and, consequently, m(t)
< r*(t,
< r"(t),
x0)Il.
7
to 9 01,
t
> t, ,
12
CHAPTER
2
where f(t) =
q t , tl , Y * ( t ,
7
to ,O))
is the maximal solution of (2.5.4) through (tl , r * ( t l , t o , 0)). Since = 0, we have
r * ( t , , t o , 0)
lim f ( t , t, , ~ * ( t,,t o ,0 ) ) = f ( t , t o ,0 ) ,
tl-fo
and, by hypothesis, T ( t , t o , 0) is identically zero, thus proving the continuity of x ( t , t o , xu) with respect to to .
COROLLARY 2.5.1. T h e function g ( t , u ) in Theorem 2.5.1.
= Lu,
L
> 0,
is admissible
THEOREM 2.5.2. Let f E C[E, Rn],where E is an open ( t , x, p)-set in R"+"L+l,and for p = po , let xo(t) = x(t, to , xo , p,,) be a solution of x' = f ( t , x,Po), existing for t
4 t O ) = xo
7
(2.5.5)
3 to . Assume further that lim f ( t ,x , Y ) = f ( t ,x, Po),
w-wo
(2.5.6)
uniformly in ( t , x), and, for ( t , x1 , p), ( t , x2,p ) E E, I l f ( t 7
x1
9
Y) -f(t,
x2
, P)Il
e At, /I
x1 - xzll),
(2.5.7)
whereg E C[/ x R, , R,]. Suppose that u(t) = 0 is the unique solution of (2.5.4) such that u(tn) = 0. Then, given E > 0, there exists a S ( E ) > 0 such that, for every p, 11 p p,, I/ < S ( E ) , the differential system ~
x' = f ( t , x,P),
x(tn) = xo
(2.5.8)
admits a unique solution x ( t ) = x(t, t o ,x,, , p ) satisfying
I1 ~
( t ) xo(t)ll ~
<
t
€9
> to
Proof. T h e uniqueness of solutions is obvious from Theorem 2.2.1. From the assumption that u(t) = 0 is the only solution of (2.5.4), it
+
follows, by Lemma 1.3.1, that, given any compact interval [to , to U ] contained in J and any E > 0, there exists a positive number 7 = V(E) such that the maximal solution r(t, to , 0, 7) of 24'
= g(t, 24)
+7
2.5.
exists on to
DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS
< t < to + a and satisfies r ( t , to , 079)
< 6,
t
E
[to , to
+ 4.
Furthermore, because of the condition (2.5.6), given 7 a 6 = 6 ( q ) > 0 such that
Ilf ( t , X, P ) - f ( t ,
3 9
PJI
13
> 0, there exists
< 17
provided
I1 P Now, let
E
> 0 be
- Po
II < 8.
given, and define
Nt)= I1
- XO(~)lL
where x(t), x,(t) are the solutions of (2.5.8) and (2.5.5), respectively. Then, using the assumption (2.5.7), we get D+m(t)
< g ( t , 4 t ) ) + llf ( t , xo(%
From this, it turns out that, whenever D+m(t)
11 p
P ) - f(t, xo(t), P0)Il.
- po 11
< 6,
< g(t, 4 t ) ) + ?-
By Theorem 1.4.1, we have
< r(t,t o , O,d,
t
3 to,
II x ( t ) - x,(t)Il < e ,
t
2 to,
m(t)
and hence provided that
/I P
- Po
II < 8.
Clearly, 6 depends on E since q does. T h e proof is complete.
LEMMA 2.5.2. Let f E C [ J x D,R"],where D is an open, convex set in R",and let af /ax exist and be continuous. Then,
74
CHAPTER
2
the convexity of D implies that F ( s ) is defined. Hence, (2.5.9)
SinceF(1) = f ( t , x2) andF(0) (2.5.9) from 0 to I .
=f
( t , xl),the result follows by integrating
T H E O R E M 2.5.3. Assunic that f E C [ J x Rn,Rn] and possesses continuous partial derivatives af /& on J x Rn. Let the solution xO(t)= x(t, t o , x,,)of (2.5.1) exist for t 3 t o , and let
Then
exists and is the solution of Y'
=
H ( t , t o , x0)y
(2.5.10)
such that O(to, to , xo) is the unit matrix; (ii)
ax(t, f, , XI)) at,
exists, is the solution of (2.5.10), and satisfies the relation
First wc shall prove conclusion (i). Let h be a scalar and e, = (eTcl, ..., ek?() be the vector such that ekj = 0 if j f k and ekk = 1. Then, for small h, let
Proof.
x(t, h )
=
x(t, t o ,
x0
+ e&),
which is defined on J , and lim x(t, h ) = xo(t) h-0
uniformly on J . Since
2.5.
DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS
75
applying Lemma 2.5.2 with x2 = x(t, h), x1 = x,(t), we have
If we write
the existence of ax(t, t o ,xo)/2xo is equivalent to the existence of the limit of x h ( t ) as h 3 0, since ~ ( t, ,h ) = xo ekh, xh(t,) = ek . Thus, xh(t) is the solution of the initial value problem
+
y‘ == H ( t , ‘ 0 ,
9
h)y,
Y(tO) = ek
(2.5.12)
7
where
As x(t, h) + x,(t) as h + 0, by the continuity of lim H(t, to , xo , h) h-0
=
af /ax,
it follows that
H(t, t o ,xo)
uniformly on J . Considering (2.5.12) as a family of initial value problems depending on a parameter h, where H(t, to , xo , h) is continuous for t E J , h being small and y arbitrary, and observing that the solutions of (2.5.12) are unique, it is clear that the general solution of (2.5.12) is a continuous function of h. In particular, limb+,, xl,(t) = x ( t ) exists and is the solution of (2.5.10) on J. This implies that ax(t, t o ,xo)/axo exists and is the solution of (2.5.10). T o prove (ii), define
Since (2.5.1) has unique solutions, we have x ( t , to
+ h, xo) = x(t, to , “(to , to + h,
and therefore hGhh(t)= x(t, tn >
-
.To)),
to , tn + h, xo)) - x ( t , to , xo).
Because ax(t, t o ,xo)/axoexists and is continuous and .(to
, to
+ h, xo)
.(to , to , xo) = xo
as
h
-
(2.5.13)
0,
76
CHAPTER
2
it follows from (2.5.13) that hqt)
xO],
=
0. By the mean value theorem, there exists a 0 ash such that --f
where 0
as h
--f
=
0, ,k
=
(2.5.14) 1 , 2,...,n
< B < 1. Notice that, for each k,
0. Thus, (2.5.14) shows that
0, which implies that ax(t, to , xo)/ato = limh+oi r l ( t )exists and satisfies (2.5.1 1). This completes the proof.
as h
--f
2.6. Variation of constants Let us prove some elementary facts about linear differential systems,
x'
=
(2.6.1)
A(t)x,
where A ( t ) is a continuous n x n matrix on J. Let U ( t ) be the n x n matrix whose columns are the n-vector solutions x ( t ) , x(t), being so chosen to satisfy the initial condition U(t,) = unit matrix. Since each column of U ( t )is a solution of (2.6. l ) , it is clear that U satisfies the matrix difierential equation U'
=
U(to)= unit matrix.
d(t)U,
THEOREM 2.6.1.
(2.6.2)
Let A ( t )be a continuous n x n matrix on J. T h e n the fundamental solution U ( t )of (2.6.2) is nonsingular on J , More precisely, det U ( t )
exp
1 :
/ t t0
where tr A ( t ) =
CF=laii(t).
tr A ( s ) ds,
t E J,
2.6. Proof.
VARIATION OF CONSTANTS
77
T h e proof depends on the following two facts:
(i) d(det U ( t ))/d t= sum of the determinants formed by replacing the elements of one row of det U ( t )by their derivatives. (ii)
T h e columns of U ( t )are the solutions of (2.6.1).
Simplifying the determinants obtained in (i) by the use of (ii), we get d dt
- det U ( t ) = tr A(t)det U ( t ) .
T h e result follows, since U(to)= unit matrix.
THEOREM 2.6.2.
Let y ( t ) be a solution of Y’
=
A(t)Y + F ( t , Y ) ,
(2.6.3)
where F E C[J x Rn,R”],such that y(to)= y o . If U ( t ) is the matrix solution of (2.6.2), then y ( t ) satisfies the integral equation
This, because of (2.6.2), yields z’(t) = U-’(t)F(t,Y ( t ) ) ,
whence z ( t ) = Yo
+
Jt
U-l(s)F(s,y(s))ds.
t0
Multiplying this equation by U ( t )gives (2.6.4). COROLLARY 2.6.1. Let A(t) be a continuous n x n matrix on J such that every solution x ( t ) of (2.6.1) is bounded for t 3 to . Let U ( t )be the fundamental matrix of (2.6.1). Then, U-l(t) is bounded if and only if
is bounded from below.
78
2
CHAPTER
We shall now consider the nonlinear differential system (2.5.1). T h e following theorem gives an analog of variation of parameters formula for the solutions y(t, to , x,,)of
Y' = . f ( t , y ) +F(t,Y).
(2.6.5)
THEOREM 2.6.3.
Let ~ , F CE[ J x RrL,R"], and let af /ax exist and be continuous on J x R". If x ( t , t o , x,,) is the solution of (2.5.1) and (2.5.2) existing for t >, t o , any solution y ( t , t o , xo) of (2.6.5), with ~ ( t , ,= ) xo , satisfies the integral equation Y ( t , to .o> 7
=
x ( t , to 7 %>
+ It @ ( t ,
s,
to
Y ( S , to , x,,))F(s,Y ( S , to , x,,)) ds
(2.6.6)
for t 2 t,, , where @(t,t o , x,,) = ax(t, to , xo)/axo. Proof.
Write y ( t ) = y(t, t o , x,,). Then, dx(t, s , y ( s ) ) ds
-
ax(t,s,y(s))
+
q t ,S?Y(S))
aY
as
= @ ( t ?s,Y N ) [ Y ' ( S )
.Y" (2.6.7)
YG))l,
using Theorem 2.5.3. Noting that x(t, t , y ( t , to , x,,)) = y(t, t,, , x,,) and y'(s) - f ( s , y(s)) = F(s, y(s)), by integrating (2.6.7) from to to t, the desired result (2.6.6) follows.
THEOREM 2.6.4. Let f E C [ J x Rn, R"], and aflax exist and be continuous on J x R". Assume that x ( t , t o , x,,) and x(t, t,, ,y o ) are the solutions of (2.5.1) through (to , x,,) and (to ,yo), respectively, existing for t 3 t o , such that x,,, y o belong to a convex subset of R". Then, for t >, t o , x ( t , t" , Y") - x ( t , to xo) = 7
[I;
@(G to , xo
+
S(Y0 - XI)))
4.
(Yo - xo).
(2.6.8)
Proof. Since xo,y o belong to a convex subset of R", x ( t , to,x,, is defined for 0 s I . Thus,
< <
and hence the integration from 0 to 1 yields (2.6.8).
+ s( yo
-
x,,))
2.7.
79
UPPER AND LOWER BOUNDS
2.7. Upper and lower bounds Consider the differential system
and the differential inequality
where f , fl E C [J x R", R"], and 6 E C [J , R,]. DEFINITION 2.7.1.
By a &approximate solution of
on [to, co), we mean a function y ( t ) such that y E C [J , R"],y ' ( t ) exists on J - S , S being an at-most countable subset of J , and satisfies (2.7.2) on J - S.
THEOREM 2.7.1. that
Let g E C [ J x R, , R,] and u, a, 8 E C [ j ,R,] such (2.7.4)
for t
> t o . Let D
=
[x,y
E
Rn : 1) x - y 11
=
u(t)
and
Ij x - y jl
= v(t),t
> to].
Assume that f,fl E C [J x Rn,R"], and
for t > to and x, y E Q. If x ( t ) , y ( t ) be a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , co) such that
(2.7.6)
Proof. If (2.7.6) is not true, the set
80
CHAPTER
2
is nonempty. Arguing as in the proof of Theorem 1.2.1, we arrive at a t , > to such that either or In either case, it follows from the definition of 9 that, at t = t, , x(tl), y ( t l )E $2, and therefore, defining m ( t) = I/ x ( t ) - y ( t ) 11, we get
I m;(tl)l
< I/ x’(t1) -Y‘(tl)ll G
Ilf(t1
x(t1))
-fAh ?Y(tl))ll
+ I/ Y’(t1) -fdG
?Y(tl))ll.
This, together with (2.7.2) and (2.7.5), implies
A repetition of the rest of the proof of Theorem 1.2.1, with appropriate changes, proves (2.7.6).
THEOREM 2.7.2.
Let the assumptions of Theorem 2.7.1 hold except that (2.7.4) and (2.7.5) are replaced, respectively, by (2.7.7)
for t > to , x, y E Rtl. Suppose further that, for each t E [to , 71,g satisfies the condition
1 g(t7 ~
1 - )g ( f ,
uZ)I
< G(7 + t o
-
t , ~1 - uz),
7E
~1
[to , 00) and
2~
2 , (2.7.9)
where G E C [ J x R, , R,] and r ( t ) = 0 is the maximal solution of U’ =
G(t,u),
~ ( t , )= 0.
Then, the inequality (2.7.6) remains valid.
(2.7.10)
2.7.
81
UPPER AND LOWER BOUNDS
Proof. By a repeated application of Theorem 1.4.3, we can prove (2.7.6). For this purpose, it is enough to see that g 6 and -(g 6) satisfy the condition (1.4.9), in view of (2.7.9). Also, (2.7.8) implies that
+
+
for t
> to.
+ S(t)l < m;(t) < g(t7 4 t ) ) + s(t)>
-[g(t, 4 t ) )
THEOREM 2.7.3.
Let g E C [ J x R, , R,], 6 E C [ J ,R,], and r ( t ) , p ( t ) be the maximal and the minimal solutions of
respectively, existing on [to, 00). Let Q = [ x7 Y and
where
c1
E
R" :P ( t ) - €2 II x - Y
r(t)
<
< II x
- y I1 < p(t> II < ~ ( t ) el , t 2 to],
+
, c2 > 0. Assume that, for t > to , x, y
E 9,
IIf(t! x) - f d ~ > Y ) l l G g(t, II x - Y 11).
(2.7.12)
If x(t), y ( t ) be a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , 00) such that
< II -2^o- Y o II < uo ,
no
then p(t)
G II 4 t ) - Y(t)ll
< r(t),
t
(2.7.13)
2 to.
Proof. Let T E [to, 03). By Lemma 1.3.1, the maximal and the minimal solutions r ( t , E ) and p ( t , c) of u' = g(4 u )
v'
=
+ s(t) +
-[g(t, v)
exist for sufficiently small E
4tO)
€7
+ 8 ( t ) + €1,
= UO
v(to)= vo
+
~
E, E
> 0, and r ( t ) = lim r(t, E ) , 6-0
p(t)
=
lim p ( t , t-0
E)
uniformly on [ t o ,TI. In view of this, there exist r ( t , €1 < r ( t ) p(t9
c)
+
€1
> p ( t > - €2
,
, c2 > 0 such that
82
CHAPTER
for t for d
2
E
[t,, , TI. Furthermore, an application of Theorem 1.2.1 yields that,
E
[f,
, 71,
r ( t ) < r(4 € ) r
f(t)
At,
4.
It now follows that, f o r t E [ t o ,TI, (2.7.14)
T o prove (2.7.13), it is enough to show that
Assuming the contrary and arguing as in Theorem 2.7.1, we get either
or These relations show, because of the inequalities (2.7.14), that, in either case, x ( t l ) , y(tJ E 8. By following the rest of the standard argument, it is easy to prove (2.7.15). This completes the proof.
REMARK2.7.1. Evidently, Theorem 2.7.3 holds when the condition (2.7.12) is satisfied for all x,y E Rn instead of 8. Similar comment is valid for Theorem 2.7.1 also. T h e bounds obtained in the foregoing theorems are on a general setup. They include a number of special cases. For instance, if 6(t) I= 0, we get the estimates of the difference of solutions of (2.7.1) and (2.7.3), respectively; whereas if, in addition, f ( t , x) = fl(t, x), the same results yield the growth conditions between any two solutions of the system (2.7.1). On the other hand, if fl(t, x) f ( t , x), error estimates between a solution and a 6-approximate solution of the system (2.7.3) are obtained. Furthermore, if S ( t ) E 0 and fi(t, x) = 0, these results provide the upper and lower bounds of solutions of the system (2.7.1). ~
For future use, the following well-known result is stated as
COROLLARY 2.7.1.
Let f~ C [ J x IIf(f,
).
-
f ( f >Y)II
R",R"],and, for t 3 @,
< L(f>llx
~
Y
I/
1
x, y E R",
2.7. where L
E
83
UPPER AND LOWER BOUNDS
C [ J ,R,]. Then, for t >, to ,
where x ( t ) , y ( t ) are any two solutions of the system (2.7.1), through ( t o xo), ( t o Yo), respectively. I n the foregoing results, the upper bounds obtained are increasing functions of t, since the assumptions demand that g(t, u) 3 0 and 8 ( t ) > 0, and therefore give very little information about the growth of solutions for large time. We give below a different set of assumptions that yield sharper bounds because the function g(t, u ) need not be restricted to be positive. 9
7
THEOREM 2.7.4. Let g E C [ J x R, , R],6 E C [ J ,R,], and r ( t ) be the maximal solution of u' = g(t, u )
+qt),
u(t,) = u g ,
existing on [ t o , 00). Assume that, for t
II X - Y
+ h[f(t,
X) -fi(t,
Y>III
for all sufficiently small h
< /I x
-Y
II
E
J , x, y
+ hg(t,I/ x
E
Rn,
-
y 11)
+ O(h),
(2.7.17)
> 0. Then, II xo - Yo II < uo
implies
I/
-Y(t)ll
< r(t)l
t 3 to
1
(2.7.18)
x(t), y ( t ) being a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, existing on [ t o , 00). Proof.
Consider the function
44 We have, for small h
> 0,
=
II x(t>-Y(t>ll.
84
CHAPTER
where r ( h ) / h+ 0 as h (2.7.17), that
2
It therefore follows, using (2.7..2) and
+ 0.
D+m(t)
< g(t, m(t))+ q t ) ,
which, by Theorem 1.4. I , yields the estimate (2.7.1 8). 2.8. Componentwise bounds
Instead of the differential inequality (2.7.2), we shall be considering a system of differential inequalities given by
I Y’
- f A t , Y)l
< s(4,
(2.8.1)
where 6 E C [ ] ,R,”]. Here and in what follows, we mean by j x 1 a vector whose components are I x1 1, I x2 1, ..., I x, I for any x E R”. Note that 8(t) is a scalar function in (2.7.2), whereas it is a vector in (2.8.1). In this case, the &approximate solution of (2.7.3) must satisfy (2.8.1) in place of (2.7.2).
THEOREM 2.8.1. Let g E C [ ] x R,”, R,”] and possess the quasimonotone nondecreasing property. Let u,v, S E C [ J , R,”] such that, for t > t , , (2.8.2)
Suppose that f,fl E C [ ] x R”, RTL], and, for t
> t, , x,y E Qi, (2.8.3)
If x ( t ) , y ( t ) be a solution and a 8-approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o ,a)such that
(2.8.4)
2.8.
85
COMPONENTWISE BOUNDS
Proof. T h e proof runs parallel to that of Theorem 2.7.1. However, in this situation, the assumption that the set n
2=
(J [t 3 to : q ( t ) < 1 Xi(t) -yi(t)i
< Ui(t)j
i=l
is nonempty leads to the existence of an index j , 1 t , > to such that either
< j < n, and
a
or which shows that x ( t l ) ,y ( t l )E Qj . Consequently, as in Theorem 2.7.1, it is easy to show, using (2.8.3), that - M t l9
+ %(tl)l G 4 A t d < &(tl
"1))
9
m(t1))
+ Utl).
Making use of the quasi-monotone property of g(t, u)and the arguments of Theorem I .5. I , we can prove (2.8.4). The next theorem is analogous to Theorem 2.7.2 for componentwise bounds, the proof of which can be deduced from Theorem 1.7.3, with an observation similar to that of Theorem 2.7.2.
THEOREM 2.8.2. Assume that, in place of (2.8.2) and (2.8.3), we have
for t > t o , x,y E R",other assumptions being the same as in Theorem 2.8.1. Moreover, let, for each T E [to, a),t E [to, T] and for each i - 1 , 2,..., n,
I gdt, u ) u,
>, ii, ,
4
< G(7 + to u, = U, ,
-
t , u,
-
4,
i #j,
where G E C[J x R, , R,], and r ( t ) = 0 is the maximal solution of (2.7.10). Then, the assertion of Theorem 2.8.1 remains true.
86
CHAPTER
2
THEOREM 2.8.3. Let g E C [ J x Rn+,R,"] and possess the quasimonotone nondecreasing property. Assume that r ( t ) ,p(t) are the maximal and the minimal solutions of
and, for each t
> tn , x,y I f i ( t ,x)
E Qi
-fi.i(t?Y)I
Then an
implies P(t)
,
< Ri(G I x -Y
I).
< I xn -yo I < un
< I x ( t ) -Y(t)l < r ( t ) ,
t
3 to
7
x ( t ) , y ( t ) being a solution and a &approximate solution of (2.7.1) and
(2.7.3), respectively, existing on [t, , a). T h e proof of this theorem can be constructed by following the respective arguments of Theorems 2.7.3, 2.7.1, and 1.5.1 with necessary modifications.
THEOREM 2.8.4. I,et S E C [ J ,Rtn], g E C [ / x R+n, R"], and g possess the mixed quasi-monotone property. Suppose that f,fiE C [J x Rn, Rn], a n d , f o r e a c h t > , t , , , p - 1 , 2,..., k , q = k + l , k + 2 ,..., n,
and x ( t ) , y ( t ) are any solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , a). (i) If r ( t ) is the k max(n u'
-
= g(t, ).
k) mini-solution of
+qt),
u(t,)
=
u,
2.8.
then, for t
87
COMPONENTWISE BOUNDS
> t o ,we have (2.8.5)
whenever
for t > t o ,then (2.8.6) implies (2.8.5) provided that, for each T E [ t o ,GO), t E [to,TI, and for each i = 1, 2,...,n,
+
g i ( t , u ) -gg,(t, c)
3 -G(T to - t , ui - $), ui 3 zii , ui = z i i , i #j,
where G E C [ J x R, , R,], (2.7.10).
and r ( t ) = 0 is the maximal solution of
THEOREM 2.8.5. Let g E C [ J x R+",R,"] and possess the quasimonotone nondecreasing property. Let r ( t ) be the maximal solution of u' = g(t, 4
+qt),
existing on [to, co),where S E C [ J ,R,"].
I x -Y
+ h [ f ( t ,x) -fdt,r)ll
4td
=
uo
9
Assume that, for t
< I x -Y
I
E
J, x,y
E
+ k ( t , I x -Y I) + O(h)
R",
88
CHAPTER
for all sufficiently small h
2
> 0. Then,
~ ( t )y,( t ) being a solution and a 6 approximate solution of (2.7.1) and
(2.7.3), respectively, existing on [to, a).
2.9. Asymptotic equilibrium
We shall continue to consider the differential system (2.7.1). DEFINITION 2.9.1. We shall say that the differential system (2.7.1) has asymptotic equilibrium if every solution of the system (2.7.1) tends to a finite limit vector f as t 03 and to every constant vector .$ there is a solution x ( t ) of (2.7.1) on to t < co such that limt+mx(t) = 4. T h e following theorem gives sufficient conditions for the system (2.7.1) to have asymptotic equilibrium: ---f
THEOREM 2.9.1.
<
Let f~ C [ J x R",R"] and
where g E C [ J x R ~ ,,R,] and monotone nondecreasing in u for each t E J . Assume that all solutions u ( t ) of u'
= g ( t , u),
u(t,)
= uo
>0
(2.9.2)
are bounded on [ t o , 03). Then the system (2.7.1) has asymptotic equilibrium. Proof. Let x ( t ) be any solution of (2.7.1). Then, it is easy to deduce from Theorem 2.7.1 that
where r ( t ) is the maximal solution of (2.9.2) such that [I xo 11 = uo . Since, by assumption, every solution of (2.9.2) is bounded on [to , a), it follows from (2.9.3) that every solution x ( t ) of (2.7.1) is bounded on [to a). ?
2.9.
Furthermore, for any t
89
ASYMPTOTIC EQUILIBRIUM
> t , > t o ,we have
II 44
- x(t1)ll
<
<
st
llf(S,
tl
f1&,
e J;,.(s>
+))I1
ds
I1 x(s)ll)
ds
y(s))
ds
(2.9.4)
= r(t) - Y(tl),
using (2.9.1) and (2.9.3) and the monotonic character of g(t, u) in u. Since g is nonnegative, every solution u(t) of (2.9.2) is nondecreasing in t , and hence the boundedness of all solutions of (2.9.2) shows that the maximal solution ~ ( ttends ) to a limit as t + co. This implies that, given an E > 0, we can choose a t, > 0 sufficiently large so that 0 < u(t)
-
r(tl) < 6
for all
t
> t, .
It then follows, because of (2.9.4), that
// x(t)
-
for all
x ( t , ) / /< E
which proves that
t
> t, ,
lim x ( t ) = 5.
(2.9.5)
t-w
T o prove that the system (2.7.1) has asymptotic equilibrium, it remains to be shown that, for every constant vector 5 such that 11 5 1) uo , there exists a solution x ( t ) of (2.7.1) on [ t o ,00) such that (2.9.5) holds. For this purpose, let x,(t) be a solution of (2.7.1) such that
<
xn(t,
+ n) = 5
(n = 1 , 2, ...).
(2.9.6)
+
If r,(t) is the maximal solution of (2.9.2) with Y n ( t , n) = 1) 411, because of the nondecreasing character of every solution of (2.9.2), it follows that We claim that
/I 5 II yn(t)
<
Yn(&
+ < r(t, + n).
< r(t),
It this were not true, let, for some
t D
2 to
+ n.
> to + n,
> r(.).
(2.9.7)
90
CHAPTER
2
Then, by taking the larger of r , ( t ) and r ( t ) , we can construct a solution of 5 11) whose value at is greater than that of the maximal solution r ( t ) ,which is absurd. Hence, (2.9.7) is true. As before, for any t, > t,, and t > t, ,
(2.9.2) through ( t o ,11
(5
I1 , y j L ( t )
xn(h)Il ,<
~
<
Since 11 x j L ( t 11) r?,(t), t tonicity of g, yields that
-I
.1
tl
g(s, I/ .T;~(.~)II) ds.
3 to + n, (2.9.7), together with the mono-
I1 '%(t)
4tl)ll
~
<
Y(tl),
~
which assures that ~ , ~ tends ( t ) to a limit vector x,,(GO) as t + GO uniformly in n. Therefore, Ilf(t, x,,(t))/ / is bounded on every bounded t-interval uniformly in n. Since x X t ) = f ( t !xn(t)),
the family (.x?,(t)f is equicontinuous on every bounded t-interval. Thus, there is a subsequence { x j l k ( t ) )that converges uniformly on every bounded t-interval as k + co. As limt+mx,,(t) + x,(co) for each n, this subsequence may be chosen such that the corresponding sequence of limits {x,,,(co)} also converges as k + co. Now, {xTlb(t)}converges uniformly on [t,,, GO) to a continuous limit function x ( t ) as k + co. Evidently, x ( t ) is a solution of (2.7.1), and therefore lim x ( t ) = ~ ( c o ) .
t-.u
Also, as k -+
GO, SUP
to'
t<m
II x,,(t)
-
%,(.n)
- x(t)ll
-
07
4.n),
and these two facts, together with (2.9.6), imply that x(c0) = (. Now x ( t ) is the desired solution, and this completes the proof of the theorem.
COROLLARY 2.9.1.
If the function g ( t , u ) in (2.9.1) is of the form
' d t ,u> = X(t)&u),
2 0 is continuous for t~ J and +(u) 3 0 is continuous for 3 0, +(0) = 0, +(u) > 0, u > 0, and montonic nondecreasing in u,
where A ( t ) u
and if
(2.9.8)
the conclusion of Theorem 2.9.1 holds.
2.10.
91
ASYMPTOTIC EQUIVALENCE
Theorem 2.9.1 has a corollary for the case that (2.7.1) is replaced by x'
(2.9.9)
A(+ + F ( t , x),
=
where A(t) is a continuous n x n matrix and F Let X ( t ) be a fundamental matrix for
E
C [ J x R", R"].
X ( t o ) = unit matrix,
x' = A(t)x,
(2.9.10)
so that the transformation
x
reduces (2.9.9) to y'
=
X(t)y
(2.9.1 1)
= x-1(t )F(t, X ( t ) Y ) .
Thus, an application of Theorem 2.9.1 to (2.9.11) gives
COROLLARY 2.9.2. Let A(t) be a continuous matrix for t E J and X ( t ) be a fundamental matrix for (2.9.10). Let F E C[J x R", R"], and, for ( t ,Y ) E J x R", /I X-Yt)F(t, X(t)Y)lI < h(t)llY 11, (2.9.12) where h(t) >, 0 is continuous for t
E
J , and
s z X ( s ) ds
< a.
Furthermore, let x ( t ) be a solution of (2.9.9) on some t-interval to the right of to . T h e n x ( t ) exists for all t 2 to , lim X-l(t)x(t) = [,
(2.9.13)
t-m
and, conversely, given a constant vector f , there is a solution x ( t ) of (2.9.9) satisfying (2.9.13). An interesting special case in which the hypotheses of Corollary 2.9.1 are satisfied is that of the linear homogeneous system (2.9.10), where
1; I/
4 s ) l l ds
< a-
I t is enough to take h(t) = 11 A(t)11 and $(u) = u.
2.10, Asymptotic equivalence Suppose we are given the following two differential systems:
x'
=f1(t,
x),
Y'
=f i ( t , Y ) ,
x(h)
=
xo
Y(to) = Yo
1
(2.10.1 ) (2.10.2)
92
CHAPTER
where fi, f 2 valence.
E
2
C [ J x R”, R”]. We shall first define asymptotic equi-
DEFINITION 2.10.1. T h e differential systems (2.10.1) and (2.10.2) are said to be asymptotically equivalent if, for every solution y ( t ) of (2.10.2) [a(t) of (2.lO.l)], there is a solution x ( t ) of (2.10.1) [ y ( t ) of (2.10.2)] such that as
x(t) -y(t)+O
THEOREM 2.10.1
2)
co.
>g(t,u)
for t 3 to such that u ( t ) + 0 as t Suppose further that
+ h(fi(t,
+
Let zc(t) be a positive solution of u’
II X -Y
t
4
< I/ x -Y
-fZ(t,Y))Il
I1
GO,
where g E C [ J x R, , R].
+ hg(t, II x - y 11) + O(h)
(2.10.3)
for all sufficiently small h, t 3 to , and I] x - y 11 = u(t). Then, the systems (2.10.1) and (2.10.2) are asymptotically equivalent. If, in addition, one of the systems has asymptotic equilibrium, then the other system also has asymptotic equilibrium.
Proof. Let us first suppose that y ( t ) is a solution of (2.10.2) defined for t >, t,, . Let x ( t ) be a solution of (2.10.1), defined on some t-interval
to the right oft,, such that
/I 4 t o )
- Y(4J)ll
< .(to>.
Clearly, such a solution exists. Define Then as far as x ( t ) exists. If this assertion is false, let t , be the greatest lower bound of numbers t > t o , for which m ( t ) u(t) does not hold. Since m ( t ) and u(t) are continuous functions, we have, at t = t, ,
<
4 t l ) = 4tl)
and m(t,
+
12)
> U(t,
+ h),
h
> 0.
This implies the inequality (2.10.4)
2.10.
93
ASYMPTOTIC EQUIVALENCE
I n view of the condition (2.10.3), one also gets, at t
=
t, ,
D+m(t,) G d t l 7 m(tl)),
which is a contradiction to (2.10.4). Hence,
/I x(t) -y(t)ll
< u(t)
is true as far as x(t) exists. Now, using Corollary 1 .I .2, it follows that x(t) exists for all t 3 to , since y ( t ) and u ( t ) are assumed to exist for t 3 to . Moreover, as limL+m u(t) = 0, lim I/ x(t) -y(t)ll
t--.m
=
(2.10.5)
0.
On the other hand, if x ( t ) is a solution of (2.10.1) existing on [ t o , a), arguing as before, we can conclude that there exists a solution y ( t ) of (2.10.2) on [ t o , KI) such that (2.10.5) is satisfied. It therefore follows that the systems (2.10.1) and (2.10.2) are asymptotically equivalent. If one of the systems has asymptotic equilibrium, the asymptotic equilibrium of the other system is a consequence of (2.10.5). T h e proof is complete. T h e next theorem gives sufficient conditions for the asymptotic equivalence of the systems (2.9.9) and (2.9.10).
THEOREM 2.10.2. Let A(t) be a continuous matrix for t E J and F E C [J x Rn,R"]. Suppose that
I1F(t>41 < Yt)ll x It. where A ( t )
3 0 is continuous for
t
E
J , such that
s I h ( s ) ds
< 03.
Assume that all the solutions of (2.9.10) are bounded as t lim inf t-m
J:o
(2.10.6)
tr A(s)ds
>
+ CQ
-03.
and (2.10.7)
Then, the systems (2.9.9) and (2.9.10) are asymptotically equivalent. Proof.
Let Y ( t )be a fundamental matrix of (2.9.10). Setting Y(t),(t) = r(t),
94
2
CHAPTER
it is easy to verify that x ( t ) is a solution of (2.9.9) if and only if v ( t ) satisfies a' = Y-'(t)F(t, Y(t),). (2.10.8) Using (2.10.6), (2.10.7), and the assumption that all the solutions of (2.9.10) are bounded, we get
II W f ) F ( t ,y(wll < II W ~ ) l l IY(t)llll l 21 Ilqq
< KII
?J
llA(9,
where K is some constant. Hence, Corollary 2.9.1 implies that (2.10.8) has asymptotic equilibrium. Now, any solution y(t) of (2.9.10) can be written as Y ( t ) = Y(t)S,
4 being a constant column vector. Therefore, 44
-y(t)
=
Y(t)[,(t>- [I,
and the desired result follows, since Y ( t )is bounded on ( t o , a). T h e asymptotic equivalence of the systems (2.10.1) and (2.10.2) can also be considered on the basis of the variation of parameters formula for nonlinear systems developed in Theorem 2.6.3.
THEOREM 2.10.3.
Assume that (i)fi
,.f2
E
C [ J x Rn,Rn],
exists and is continuous on J x R";(ii) dj1(t, t, , x,)(Q2(t, t o ,yo))is the fundamental matrix solution of the variational system
2.10.
ASYMPTOTIC EQUIVALENCE
95
Then, there exists a solution x ( t ) of (2.10.1) [ y ( t ) of (2.10.2)] on [to , a) satisfying the relation lim x ( t ) - y ( t ) = 0. (2.10.10) i-m Pyoof. Let y ( t ) = y ( t , to , y o ) be a given solution of (2.10.2) existing on [ t o , a).Define a function x ( t ) by the relation x(t) =Y(t)
+\
W
' t
@l(hs,y(s))[f,(s,y(s)) -fl(s>Y(S))lds.
(2.10-11)
Since the integral converges by assumption (iii), it follows that x ( t ) is well defined, and, consequently, (2.10.10) is satisfied. It therefore suffices to prove that x(t) is a solution of (2.10.1). For this purpose, we observe, as in Theorem 2.6.3, that
Here use is made of the relation (2.5.11) and the fact that
T h e relations (2.10.1 1) and (2.10.12) yield
Moreover, we have
96
CHAPTER
2
1,et us differentiate (2.10.1 l), recalling that y ( t ) ,Q l ( t , t, , x,) are the solutions of (2.10.2) and (2.10.9), respectively, and using (2.10.12) to obtain
This reduces to, in view of (2.10.14),
T h e relation (2.10.13) implies that x ( t ) is a solution of (2.10.1) with
On the other hand, if x ( t ) is a solution of (2.10.1) existing on [ t o , a), we can show exactly in a similar way that there exists a solution y ( t ) of (2.10.2) on [ t o , m) such that (2.10.10) holds. Thus the theorem is established.
2.1 1 . A topological principle
This topological principle is concerned with the differential system Y'
= f ( t , x),
.(to)
= X"
,
to 2 0,
(2.1 I. 1)
where f E C [ E ,RT1],E being an open ( t , x)-set in Rn+l. Let E, be an open subset of E , r?E, the boundary, and l?, the closure of E, .
DEFINITION 2.1 1.1. A point ( t o , x,) E E n aE, is said to be an egress point of E,) with respect to the system (2.11.1) if, for every solution x ( t )
2.1 1.
97
A TOPOLOGICAL PRINCIPLE
< <
of (2.1 l.l), there is an E > 0 such that ( t , x ( t ) ) E Eo for to E t to . An egress point ( t o ,xo) of Eo is called a strict egress point of E, if to E , for a small E > 0. ( t , x ( t ) ) E, for to < t T h e set of all points of egress (strict egress) is denoted by S(S*). It is clear that S* C S. ~
< +
DEFINITION 2.1 I .2. If A C B are any two sets of a topological space and T : B 4 A is a continuous mapping from B onto A such that W( p ) = p for every p E A , then T is said to be a retraction of B onto A. When there exists a retraction of B onto A , A is called a retract of B. T h e following examples would sufflce to illustrate the concept of retraction.
<
Example 1. Let B = [x E R" : /I x 11 cx] and A = [x E R" : /I x 11 = a ] . T h e n A is not a retract of B. For, if there exists a retraction w : B + A , then there exists a continuous map of 3 into itself, x 4 -n(x), without fixed points. This contradicts the Brouwer's fixed point theorem. Example 2.
Let C = [ ( x , u ) E Rnfm: 11 x
(1
[(x, uo) E Rnim: /I x 11
B
=
A
=B
nC
=
=
01,
<
01,
u
arbitrary],
uo fixed],
[(x, uo) E Rn+" : 11 x I/
=
01,
u,fixed].
From example 1 , it can be seen that A is not a retract of B, whereas it is a retract of C, because we can choose a retraction ~ ( xu, ) = (x,u,).
THEOREM 2.1 1 . 1 . Let f E C [ E , R"], where E is an open ( t , %)-set in Rn+l. Assume that, through every point of E , there passes a unique
solution of the system (2.11.1) and that the solutions depend continuously on initial values. Let Eo be an open subset of E . Suppose that all egress points of E, are strict egress points, i.e., S = S*. Let Z be a nonempty subset of Eo u S such that 2 n S is a retract of S , but is not a retract of Z. Then, there exists at least one point (to , xo) E Z n Eo such that the solution arc ( t , x ( t ) ) of (2.1 1.1) remains in E , on its maximal interval of existence to the right of t o . Proof. Suppose that the conclusion of the theorem is not true. Then, for every ( t o ,xo) E Z - S, there exists a t , = t , ( t o , x,), t , > to such that the solution x ( t ) = x(t, to , xo)of (2.1 1.1) exists on [to,t , ] , ( t , x ( t ) )E E, for [to , t,) and ( t , , x ( t , ) ) E S.
98
CHAPTER
Define a map T , : Z (I)
~ , ) ( ,t x,) ,
-
(ii) ~ , ) ( t,,xt,)
--f
2
S such that
( t , , x(tl))
( t o , x,)
if if
( t o ,x,)
, x,)
(f,
E
EZ
~
S;
Z n S.
We shall show that no is continuous because of the assumption S = S* and the continuous dependence of the solutions on the initial values. Let ( t o , x,))E Z n En and ( t * , x*) be sufficiently near to ( t o ,xo). T h e n the solution x ( t , t*, x*) exists on [t*, t , €1 for some small E > 0 and
+
(i)
( t ,s ( t , t * , x*))
E
E, ,
[ t * , t,
(ii) ( t , r(t, I*, x*)) $ E,, ,
t
=
-
t,
€3;
4-e .
This implies that, at t t t>(t*,x*), ( t t ,x(tf, t", x*)) E S and E , which shows that (tf, x(tF, t*, x*)) is a continuous j tt t, I function of ( t %, x*). T h i s proves that T , is continuous at ( t o ,xo). A similar reasoning holds if (to , x,) E 2 n S. 1,et T be a retraction of S onto 2 n S. I t then follows that the composite map m r , : 2 + 2 n S is a retraction. This contradicts the assumption that X n S is not a retract of 2. T h e theorem is proved. -
T o give an idea of the interplay of the conditions in Theorern 2.1 1.1, let u s consider (2. I l . l ) , where E = J x R,E, = [(t,x) : t E J , 1 x I < 61. T h e boundary aE0 consists of the half-lines x : fb. T h e assumption that f ( t , 6) > 0 , f ( t , -6) 0 guarantees that S = S" = aE0. T h e set Z can be chosen as 2 [ ( t ,x) : t t, , 1 x I < b] and 2 n S as the set of two points (to , i b ) . T h e n 2 n S is a retract of S but not of 2. Theorem 2.1 1 . I now shows that there exists at least one point (to, xo), 1 yo 1 6, such that a solution of (2.1 1.1) exists and satisfies I x ( t ) I < b for t 2 t o . Given a differential system, the choice of the set E,, , for which Theorem 2.11.1 can be applied successfully, may be rather difficult. However, in some cases, it is possible to overcome this difficulty. 1,et ZL E C [ E ,K"] and x ( t ) be a solution of (2.1 1.1). T h e function u ( t , x) is said to possess a trajectory derivative u'(t, x) at the point (t,), xo) along the solution x ( t ) of (2.1 1. l ) if u(t, x ( t ) ) has a derivative at t t, , in which case
-
-
-
:
.'(to
>
=
Luff, x(t))l;pt".
If ~ ( tx), is continuously differentiable with respect to ( t , x), the trajectory derivative d ( t , x) exists and is equal to
2.1 1.
99
A TOPOLOGICAL PRINCIPLE
where the centered dot denotes the usual scalar product of vectors. T h e following theorem provides a suitable choice of E, and the set of egress points in terms of certain functions.
THEOREM 2.11.2. Let f E C [ E ,R"], u E C [ E ,R p ] , and u E C [ E ,R*].Let (i) E,
=
[ ( t ,x) : u j ( t ,x)
< P , 1 < fz < 41; L, = [ ( t ,x) : u,(t, x) = 0 and u3(t,x) < 0, %(t, x! < 0, 1 < j < P , 1 < < sl; M , = [ ( t ,x) : v,(t, x) = 0 and u,(t, x) < 0, %(t, x) < 0, 1 < j < P , 1 < < 41. %(t, ).
(ii)
(iii)
and
<0
< 0, 1 < j
Assume further that the trajectory derivatives ui(t, x), v;(t, x) exist on
L, , M8 and satisfy
u x t , ).
> 0,
( t ,). E L , ,
4 ( t ,).
< 0,
( t ,).
E MI3
(2.11.2) (2.1 1.3)
,
respectively, along all solutions through ( t , x). Then 3)
a
,=l
B=1
UL,-uM,
S=S*=
Proof. We shall first show that S n M , is empty. For, if ( t o ,x,) and x ( t ) is a solution of (2.11. l ) , from (2.11.3) we have vB(t,x ( t ) )
> 0,
for small
[to- E , to)
E
E
M, ,
> 0,
which shows that ( t , x ( t ) )4 E, because of (i). This means that ( t o ,xo) is not a point of egress. Since
it follows that a
21
s*csc(a~,,nE)-( J M , C U L , a=l
a-1
On the other hand, if
u n
( t o > xo) E
m=l
u a
L,
-
B=1
MB
7
a
( J M ~ ~ . (2.11.4)
8-1
100
CHAPTER
2
<
then, from (ii) and (iii), we get uj(to, xo) 0 and vk(t0, x,,) < 0, j I ,..., p , k I ,..., q. T h e assumption (2.11.2) yields that there is an E :> 0 such that 2
~
if ( t o, xo) $ L j ; and + ( t , x(t))
< 0,
[ t o - E , t,
+ €1
for all k.
Hence, ( t o ,xo) E S*, and
4-1
n=l
This, together with (2.11.4), establishes the theorem. 2.12. Applications of topological principle
Considcr the two differential systems x' = f d t , 4,
(2.12.1)
Y' = f i ( t , Y ) ,
(2.12.2)
where fi ,f.L E C [ J x R",R"]. Let g E C [ J x R+ , R] and u(t) be a positivc solution of the differential inequality u'
< g(t, u).
(2.12.3)
Givcn that y ( t ) is a solution of (2.12.2) for t there is a solution x ( t ) of (2.12.1) such that
II x ( t ) - Y(t)ll < 4 t h
t
3 t o , we shall show that 3 to .
(2.12.4)
THI~OREM 2.12.1. Let fi ,f2 E C [ ] x Rn,R"] and g E C [ J x R+ , R]. Assume that the systems (2.12.1) and (2.12.2) possess unique solutions through every point and that the solutions depend continuously on the
2.12.
APPLICATIONS OF TOPOLOGICAL PRINCIPLE
101
initial values. Let y ( t ) be a solution of (2.12.2) and u ( t ) a positive solution of (2.12.3), for t >, t, . Suppose further that
+ h ( f l ( t , 4 -f2(t,Y(t)))ll 2 I/ x Y(t)ll + hg(t, II x Y(t)ll)
II x
-Y(t),
-
(2.12.5)
-
for all sufficiently small h and 11 x - y ( t ) /I = u(t). Then, if T > t, is given, there exists a solution x(t) of (2.12.1) defined for t >, T satisfying (2.12.4) for t 3 T . Proof. We wish to apply Theorems 2.11.1 and 2.11.2 to deduce the result. Defining =
[ ( t ,4 : I/ x -y(t)ll
u(t,
=
II x
u(t, x)
=
to - t ,
4
-y(t)ll
-
< 4% t > t o ] ,
4th
it follows that
< 0, v(t,x ) < 01,
Eo
=
[ ( t ,x ) : u ( t , x )
L
=
[ ( t ,x ) : u ( t , x) = 0,v(t,x )
M
=
[ ( t ,x ) : u(t, X)
< 01,
< 0,~ ( t , = 01. X)
If ~ ( t= ) /I x ( t ) - y ( t ) 11, where x ( t ) is a solution of (2.12.1) such that, for some t = t , >, t, , a ( t l ) = u(tl), the condition (2.12.5) yields a(t1
+ h)
= =
I1 “ ( t ,
+ h)
+ h)ll
-At1
II 4 t l ) + hfl(t1
2 4 t l ) + hg(t1 >
+4 4 4 G ) ) + +), 7
x(t1))
- Y ( t d - hf2(tl
where el(h)/h,e,(h)/h, and e(h)/hall tend to zero as h the inequality m’(t1)
3d t l * 4 t l ) )
=d t l
,4 t l ) ) .
Using the inequalities (2.12.3) and (2.12.6), we obtain u’(t, , XI)
= a’(tl) - U ’ ( t l )
3 g(t1 > 4 t l ) ) 0,
(tl
3
-
u’(t1)
x1) E L ,
where x1 = x(tl). Moreover, v’(t, , xl)
=
-1,
(tl , X1) E M.
7
Y(fl>) - 62(h)ll
+ 0.
This implies (2.12.6)
102
CHAPTER
2
Thus, in view of Theorem 2.1 1.2, we have
,c = s*= L - M. Let
T
> t,, be given. Since AS
=
[ ( t ,x) : t
> t o ,[ ( x -y(t)ll
defining [ ( t ,-y) : t
7,
one sees that Z nS
-
[ ( t ,x) : t
=
11 %.
= T,
- y(t)l/
=
u(t)],
< U(T)],
/ / x - y ( ~ ) l= l .(.)I.
Observing that Z is a closed ball in Rn and 2 n S is the boundary of the ball 2 in R", it is clear that 2 n S is not a retract of 2.However, the mapping
T
:S
S n 2 given by
~ ( tx), x*
-
( t * , x*),
34.1
-~
+
with
t*
= T,
("X - Y ( w 4 T ) / u ( t ) l ,
is a retraction. Consequently, we conclude from Theorem 2.1 1.1 that there exists at least one point ( T , xo) E Z - S such that the solution arc ( t , x ( t , T , xJ) of (2.1 2.1) remains in Eo on its maximal interval of existence. Sincc z r ( t ) and y ( t ) cxist for all t T 3 t o , it follows that the maximal intcrval of existence of x ( t ) is [ T , a). Hence (2.12.4) holds, and the proof is complete.
REMARK 2.12.1. If, for every solution y ( t ) of (2.12.2) [ x ( t )of (2.12.1)], there exists a t,, and g(t, a ) such that (2.12.3) is satisfied by some positive function u ( t ) that tends to zero as t -+ GO and (2.12.5) is satisfied for 11 A y ( t )11 - u(t) (11 ~ ( t ) y 11 u ( t ) ) , then the systems (2.12.1) and (2.12.2) are asymptotically equivalent. :
2.13. Stability criteria Wc consider the differential system x' - f ( t , x),
x(to) = xo ,
where f E C [ / x S , , R"],S, being the set So
--
[x E R" : I/ x / j
< p].
(2.13.1)
2.13.
103
STABILITY CRITERIA
Assume that f ( t , 0) = 0, so that (2.13.1) admits the trivial solution. Let x ( t ) = x(t, t , , x,,) be a solution of (2.13.1) through ( t , , x,,).
DEFINITION 2.13.1. T h e trivial solution of (2,13.1) is said to be (i) stable if, for every E > 0 and to E J , there exists a 6 > 0 such that Ij x, 11 < 6 implies /I x ( t ) 11 < E , t to; (ii) asymptotically stable if it is stable and if there exists a 6, > 0 such that 11 x, 11 < 6, implies x ( t ) + 0 as t -+ co.
>
THEOREM 2.13.1. Let g E C [ J x R , , R] and g(t, 0) the functionf(t, x) satisfies
II x 4-W t , .)I1
< II x I1 -1- Mt,II x Ill-t O(h)
for ( t ,x) E J x S, and for all sufficiently small h or asymptotic stability of the trivial solution of u’
= 0. Assume
= g(t, u),
that
(2.13.2)
> 0. Then, the stability (2.13.3)
u(t,) = u,,
implies likewise the stability or asymptotic stability of the trivial solution of the system (2.13.1). Pyoof. Let the solution u = 0 of (2.13.3) be stable. Then, given 0 << E < p, and to E J , there exists a 6 > 0 with the propcrty that
u,,< 6 implies u(t, t , , u,,) < E , t 3 t o . It is easy to claim that, with these E and 6, the trivial solution of (2.13.1) is stable. If this were false, there would exist a solution x(t) of (2.13.1) and a t , > to such that
I1 x(t,)ll For t
E
=
/I x(t)ll
E,
<
t,
E,
< t < t,
[ t o ,tl], using the condition (2.13.2), it follows that D+m(t>
< s(t,m(t>),
(2.13.4)
where m(t) = /I x ( t ) 11, and hence, by Theorem 1.4.1, choosing 11 x, we obtain
/I 4t)ll
< ~ ( tto,
7
II xn II),
11 = u 0 , (2.13.5)
t E [to
where r(t, t, , 11 x, 11) is the maximal solution of (2.13.3). At t we therefore arrive at the following contradiction: t
=
I/ X(tl)ll
thus justifying our claim.
< f(t,
>
to
7
II Xo lo <
€7
=
t,
,
104
CHAPTER
2
Suppose that the solution u - 0 is asymptotically stable. Since this implies, by definition, stability of u = 0, the stability of the trivial solution of (2.13.1) is a consequence of the foregoing argument. This means that the inequality (2.13.4) holds for all t 3 to , and hence (2.13.5) is valid for t >, t,, . It is now clear, by hypothesis, that, if /I xo I( < S o , lim->, x ( t ) = 0. 'The proof is complete. Let us demonstrate the significance and practicability of the assumptions of Theorem 2.13.1 by an example. Letf(t, x) = Ax,where A is an n x n constant matrix. Since
I1 A! it follows that
+ hAx Ii < I/ I + hA /Ill x II,
'Thus, defining the logarithmic norm p ( A ) == lim
[IlZ -1
h-O+
where e(h)/h-+ 0 as h
-j
h
I1 - 11
(2.13.6)
0. T h e function g(t, u ) is therefore given by g(t, u ) = p(A)u.
Clearly, g(t, 0)
E
0, and the general solution of (2.13.3) is u(t, t , uo) = uo = p w w
-
4~1.
<
Thus, the trivial solution u = 0 of (2.13.3) is stable if p ( A ) 0 and asymptotically stable if p ( A ) < 0. Hence, the stability or asymptotic stability of the trivial solution of (2.13.1) follows from Theorem 2.13.1. From the definition (2.13.6), it is easily seen that p ( u A ) = O(p(A),
O(
3 0,
< II A 4'2 + B) < 44 + P ( m I P(A)I
(2.13.7) (2.13.8)
/I7
(2.13.9)
and, from (2.13.8) and (2.13.9), IPV) -
< /I '4
-
B
ii.
(2.13.10)
2.13.
105
STABILITY CRITERIA
T h e value of p ( A ) depends on the particular norm used for vectors and matrices. For example, if /I x 11 represents the Euclidean norm, p ( A ) is the largest eigenvalue of & [ A A*], A* being the transpose of A, whereas the corresponding matrix norm /I A I/ is the square root of the largest eigenvalue of A*A. On the other hand, if I] x jl = I xi I, and 11 A I/ = sup,, IT=, I aili 1, then
+
r
n
1
We further remark that every eigenvalue of A has real part less than or equal to p ( A ) . For, if h is an eigenvalue of A , and x a corresponding eigenvector of norm 1, then il(I+ h A ) x 11
~
=
jl x 11
< II Z + hA /I
On the other hand, ll(I+hA)x 11
~
<
+ hX 1
11 x 11
11
-
-
1
-
h Re X
1
hp(A)
for
h + Of.
for
h + O+.
Therefore, Re h p(A). Let us now take f ( t , x) = A(t)x, where A(t) is a continuous n x n matrix on J. I n this case, g(t, u) = p [ A ( t ) ] u . We observe that p [ A ( t ) ] is continuous on J , by virtue of the inequality (2.13.10) and the continuity of A(t).T h e general solution of (2.13.3) is of the form u(t, to uo) 7
=
uo exp
and, hence, the trivial solution u lim sup t-m
and is asymptotically stable if lim sup t -rm
Jlo
]lo
=
[Jt
to
P[A(S)I ds],
0 of (2.13.3) is stable if
p[A(s)]ds
< 00
p[A(s)]ds =
03.
Therefore, the corresponding stability properties of the trivial solution of (2.13.1) follow from Theorem 2.13.1.
THEOREM 2.13.2. Assume that (i) f E C[J x S o ,R"],f ( t , 0) E 0, and fz(t, x) exists and is continuous on J x So;(ii) y [ f z ( t ,O)] satisfies 1
t
lim sup -t-m t - t o J t op[fx(s, O)] ds
=
01
< 0.
Then, the trivial solution of (2.13.1)is asymptotically stable.
(2.13.11)
106
2
(:HAPTER
Proof. Since f ( t ,0) = 0, given such that
E
> 0, it
f ( t , x) = J;.(t, 0)x
is possible to find a S ( E ) > 0 (2.13.12)
4-F ( t , x),
wherc
~ l F ( t , x ) l ~ cllx~l if 11 x / j < 6 (2.13.1 3) uniformly in t. Let E '> 0 and t, E be given. By the condition (2.13.1 l ) , it follows that we have, for large t > t,, , =<
i;g, d . f A S , 0)l and, if
E
is small enough, lim cxp[c(t
t .a
~
Cis
< 3(t
=
t"),
~
J" p[.f,(s, 011 ds] =:
to ) -1-
0.
(2.1 3.1 4)
f0
Thus, K
01
max e x p [ c ( t
t0
-
t,'Y>
exists, and we choose K
to)
+ 1' p[fs(.s, o)] is] '0
2 1 and 6, such that KSl
< 6(c).
(2.13.1 5 )
11 x(,11 -- 6, , we claim that jl x ( t ) -:'6, t 3 t o , where ,x(t, t , , x,,) is any solution of (2.13.1). If this were false, there
Then, if
x(t)
~
exists a t ,
> t, with the property that 11 'Y(t,)lJ
=
6,
11 x(t)ll
< 6,
t"
< t < t, .
(2.13.16)
Defining nr(t) = / / x ( t ) 11, we observe that
I! , x ( t ) I- h f ( t , s(t)lI < 11 I
+ 4fz(fv0)Iiil 4t)JI + hll F ( f ,x(t))ll,
for t E [t, , t,], because of (2.13.2), and hence ?at)
.< p[f&
O)l.z(t)
+ 11 F ( t , .2.(t))ll,
which, in view of thc relations (2.13.16) and (2.13.13), yields m; ( t )
,< [ P [ f n . ( f ,0)l
I - EIm(t),
t
E [to
t
tll.
Theorem 1.4.1 then implies, choosing u ~= , m(t,), that, for t m(t)
.< m(to) exp [ ~ (--t to)-1-
it '0
p [ f z ( s , O)] ds],
(2.1 3.17) E
[to , t J ,
(2.13.18)
2.13.
107
STABILITY CRITERIA
and we are led to an absurdity:
because of relations (2.13.15), (2.13.16), and the fact that /I xo 1) < 6,. This proves that, whenever 11 xo ( 1 <’ 6,, we have 11 x ( t ) 11 < 6, t 3 t o , and therefore the inequality (2.13.17) is true for all t 3 to . Consequently, (2.13.18) holds for all t 3 t, . It now follows from (2.13.14) and (2.13.18) that Iim{+, x ( t ) = 0, if E is small enough, which establishes the stated result.
THEOREM 2.13.3. Assume that the hypothesis of Theorem 2.13.1 holds except that the inequality (2.13.2) is replaced by x . f ( t ,x)
< I1 x llR(t, II x II),
( t ,).
E
J x
AT,.
(2.13.19)
Then, the conclusion of Theorem 2.13.1 remains valid. Proof,
We proceed as in Theorem 2.13.1 to get
I/ 4tl)lI
= E,
II 4t)lI
e
Now, using (2.13.19) and setting m ( t ) the inequality m(t)m’(t)
e
E,
=
to
< t < t,
11 x ( t ) (1, we obtain, for t E [ t o ,t l ] , (2.13.20)
g(t, m w .
Choose 11 xo 11 = uo . We wish to prove the relation (2.13.5). For this purpose, it is enough to show that
where u(t, E ) is a solution of
being sufficiently small positive quantity. Assuming the contrary and following the proof of Theorem 1.2.1, we arrive at a t, , to < t, t, such that E
<
m(t2) = U ( t , ,
c),
m’(t,)
2 u’(f,,
€>.
(2.1 3.22)
108
2
CHAPTER
Ry assumption on g(t, u), we see that ~ ( t , c,) > 0, and therefore m(t,) > 0. Thus, the relations (2.13.20), (2.13.21), and (2.13.22) lead to the contradiction g(t2 , u(t2 , .))
proving m(t)
+ eg(t2 , m(t2)), E
u(t, E ) , t E [to , t J , which implies (2.13.5) because of the
,<
fact that lim,,,, u(t, c ) = r ( t , to , 11 xo 11) uniformly on [to, tl]. T h e rest of the proof is the same as in Theorem 2.13.1.
2.14. Asymptotic behavior
We shall present here several results on the asymptotic behavior of solutions of differential systems.
THEOREM 2.14.1. small h > 0,
I1 .t'
~
Assume that F
<
Y I- h(F(t7x) - F(~,Y))II /I x
E
-
C [ J x Rn, R"], and, for sufficiently Y I/
+ hg(f, I/ x -yII) + O(h),
(2-14.1)
where g E C [ J s R, , R]. If every solution u ( t ) of u' = g ( t , u )
+ II F ( t , 0)lL
(2.14.2)
44,) = *o > 0
tends to zero as t + CO, then every solution x(t) of x'
Proof.
= F ( t , x),
Let x(t) be any solution of (2.14.3) such that 11 xo 11 m ( t ) = I1
Th en , for small h
---f
~
<
ZI,
. Define
4t)N
> 0, we have
whcrc eCh)/lz 0 as h 0, we obtain
y
(2.14.3)
%(to)= xo
-
0. Now, using the condition (214.1) with
2.14.
109
ASYMPTOTIC BEHAVIOR
and this gives, by Theorem 1.4.1, the estimate
< r(t),
/I 4t)lI
t
3 t"
(2.14.4)
f
where r ( t ) is the maximal solution of (2.14.2). T h e conclusion follows from the hypothesis and the relation (2.14.4).
THEOREM 2.14.2.
Assume that F E C [ J x R", R"] and (aF/ax)(t, x) exists and is continuous on J x R". Let x . [ H ( t ,x)
+ f f * ( t ,41. < 211 x Ilg(t, I1 x II),
(2.14.5)
where H * is the transpose of H , which is given by H ( t , x)
=
J1
n
(aF/ax)(t,xs) ds,
and g E C [ J x R, , R]. Then every solution x ( t ) of (2.14.3) tends to zero as t -+ 00, if every solution u ( t ) of (2.14.2) tends to zero as t + co. If, in particular, F ( t , 0) = 0, then, the trivial solution of (2.14.3) is asymptotically stable whenever the null solution of (2.14.2) is asymptotically stable.
Proof.
If x ( t ) is any solution of (2.14.3), write m"t)
=
11 X ( t ) l l 2 .
Then 2m(t)m'(t) = 2x(t) - F ( t ,x(t)).
Since F ( t , x) - F ( t , 0) =
J
aF( t ,xs) ~-
ax
0
ds
. X,
using (2.14.5), we get the inequality 2m(t)m'(t) G 211 4t)llllF(t,0)Il
+ 211 x(t)llg(t, 11 4t)Ih
which implies, arguing as in Theorem 2.13.2, that
/I 4t)ll
< r(t),
t
2 to,
r ( t ) being the maximal solution of (2.14.2) such that stated result is clear from this estimate.
THEOREM 2.14.3.
11 x,,/I
< uo . T h e
Let U ( t )be the fundamental matrix solution of x'
=
A(t)x,
(2.14.6)
110
CHAPTER
2
A ( t ) being a continuous n x n matrix and U(t,) F E C[/ x Rn, R r t ]F, ( t , 0) == 0, t E J , and
I1 U-'(t)F(t, r4t)Y)ll
=
unit matrix. Let
< g(4 I/ Y ll),
(2.14.7)
where g E C [ J x R , , K,]. Assume that the solutions u(t) = u(t, t o ,u,,) of
u' = g ( t , u ) ,
(2.14.8)
u(t,) = U"
are bounded for t 2 t o . Then, the stability properties of the linear differential system (2.14.6) imply the corresponding stability properties of the null solution of x'
Proof.
=
- l ( t ) x -tq t , x ) ,
x(t,)
= X"
.
(2.14.9)
T h e linear transformation x =
reduces (2.14.9) into
U(t)y
WW(t, U ( t ) Y ) .
y'
(2.14.10)
<
Let y ( t ) be any solution of (2.14.10) such that y(tJ - x,,and 11 x, 11 uo . Then, if m(t) - /I y ( t ) 11, it is easy to obtain, in view of (2.14.7), the differential incquality D+m(t)
' .<
g ( t , m(t)),
and hence, by Theorem 1.4. I , we arrive at the inequality lIY(t)ll
< r(t)9
t
to
(2.14.11)
?
r ( t ) being thc maximal solution of (2.14.8). If x ( t ) is any solution of (2.14.9), we deduce, from the relation (2.14.1 I ) and the transformation x L'(t)y, that II x(t)ll < Y ( t ) l l q t ) l l , t 2 t" . (2.14.12) ~~
Since all the solutions of (2.14.8) are assumed to be bounded, it follows from (2.14.12) that the stability properties of the null solution of (2.14.9) are implied by the corrcsponding stability propertics of (2.14.6).
THEOREM 2.14.4. Assume that the fundamental matrix solution U ( t ) of the linear system (2.14.6) vcrifies
11
U(t)ji
<M
11 U-l(s)lI
< M,
t,
< s < t.
(2.14.13)
2.14.
111
ASYMPTOTIC BEHAVIOR
(2.14.14)
(2.14.1 5 )
Then, the stability properties of the null solution of (2.14.9) depend on the corresponding stability properties of the linear system (2.14.6). Proof. By Theorem 2.6.2, any solution x ( t ) of (2.14.9) satisfies the integral equation ~ ( t=) U(t)xo -t
SI
U(t)U-l(s)F(s,x(s)) ds.
(2.14.16)
T h e estimate (2.14.14) reduces (2.14.16) to the inequality
It x(t)ll
e II u(t)ll[llxo II +
Jt
I1 U-Ys)lIYs)lI
x(s)ll dsl,
t0
which, by writing m(t) = 11 x(t)li/ll U ( t )11 and using (2.14.13), shows that
for some constant K > 0. T h e conclusion is now immediate.
THEOREM 2.14.5. Let the assumptions of Theorem 2.14.4 hold except that the inequalities (2.14.13) are replaced by
11
u(t)il< M ,
11 u(t)U-l(s)II < N ,
to
< s < t.
(2.14.17)
Then, all solutions x ( t ) of (2.14.9) exist for t 3 to and verify the estimate
/I 4t)Il
< KIIx,
11,
t 3 to
7
(2.14.18)
for some K > 0. If, in addition, y ( t ) is the solution of the linear system (2.14.6) with y(to) = xo such that limt+wy ( t ) = 0, then 1imL.,%x ( t ) = 0.
I12
CHAPTER
2
Pmof. T h e integral equation (2.14.16) gives, using the conditions (2.14.17) and (2.14.14),
I/ ?c(t)Il /<
+ J^'
I1
Mll.~"
tu
which, by Theorem 1.9.1, leads to
/I 4t)ll
t 3 to ,
MA(s)lI x(s)ll ds,
< Mll Xo I1 exp
[I:w 4
< A-11 .L(,/I,
t
U
25 to ,
> 0, proving (2.14.18). If y ( t ) is the solution of (2.14.6) such that Iimt+=y ( t ) = 0, given E > 0, there exists a T ( E )such that 11 y(t)li < E for all t >, T ( E ) .Hence, for t 22 T(c), we h a w , successively, using (2.14.17) and (2.14.14), for some constant K
s(t) =y ( t )
<
/ / .v(t)lj
+
E
Jt
U(t)U-l(s)F(s,x(s)) ds,
'0
exp
MA(s) ds
J^:,,
< RE,
for some constant k > 0, which is independent of that lirnl ,I, x ( t ) = 0.
E
and T . This proves
THEOKIJM 2.14.6. Assume that (i) A is an n x n constant matrix and the characteristic roots of A have negative real parts; (ii) F E C[J x R", Rn], and, given any E > 0, there exist 6(t), T ( E )> 0 such that
I/ F ( t , .)I1
< €11
X
I/
provided 11 x 11 < 6 ( ~ )and t 3 T ( E ) (iii) ; G E C [ J x Rn,R"] and there OL and t E J , exists an .I >- 0 such that, if 11 x /I i
I1 G(t,xlll where y
E
< 74%
C [ J ,R,] and tfl
p(t) =
j t
y(s) cis -+
o
as
t
+
co.
2.14.
113
ASYMPTOTIC BEHAVIOR
Then, there exist To 3 0, 6 > 0 such that, for every to 2 To and I/ xo 11 < 6, any solution x ( t ) = x(t, to , xo) of the differential system X' =
A X + F ( t , X) + G(t,x),
satisfies
lim x ( t )
t t m
=
x(to) = xo
(2.14.19)
0.
If, in particular, (2.14.19) possesses trivial solution, then the trivial solution is asymptotically stable.
Proof. o
By assumption (i), it follows that there exist constants K
> 0 such that
11 eAt Ij
< KcUt,
t
0.
2 1 and (2.14.20)
Choose E so that 0 < E < min(a/K, a). Because of assumption (ii), we can choose T ( E )3 1 and 6 ( ~ ) E . Let To 3 T ( E )be so large that t To implies that
<
1:
exp[-(o
-
&)(t
- s)]y(s) ds
< S(e)/2K = 6, .
(2.14.21)
We shall prove below that such a choice is possible. Observe that
= Jt
Also, for p
~ ( udu )
t0
> 0,
whence e-Bt
J' eBsy(s) ds < e-Bt 1
Applying L'Hospital's rule on
for t
> to > 1.
I14
CHAPTER
2
it can be easily verified that
j t essy(s)ds
lim
t-m
for all /3
=0
1
> 0. T h e validity of (2.14.21) is now clear.
Let to 2 To and 11 x,,// x ( t ) = exp[A(t
-
-26, .
to)]x,
Then, as long as 11 x(t)/(< 6(~), we have
+
/:o
eA(t-s)[F(s,x(s)
+ G(s,x(s))]
ds,
from which, using assumptions (ii), (iii), and the estimate (2.14.20), it follows that
II x(t)lleuf
+
< Kal exp(ot,)
By Corollary 1.9.1, we obtain /I x(t)/jeot < kT6, exp(ut,,) exp[Kc(t
~
so that
I/ x(t)ll
< KS, e x p - ( o
-
Kc)(t - t,,)]
i:
to)]
[ell
x(s)ll
+
1'
+ y(s)]KeuSds.
Keusy(s)exp[Kc(t - s)] ds,
to
+ K It
exp[-(a
-
Kc)(t
~
(2.14.22)
which, using (2.14.21), yields
11 x(t)]l < KSl <<
s)] y(s) ds,
to
KS,
+ K J: exp[-(o
4-$a(€)
=
Kc)(t - s)] y(s) ds
~
S(r).
Thus, the inequality (2.14.23)
II 4t)ll <
holds on [to , a), which implies that (2.14.22) is true for t 3 to . Hence, lim x ( t )
t-m
= 0.
If x = 0 is the solution of (2.14.19), it is clearly asymptotically stable. For, we have immediately from (2.14.23) that (( x(t)\( < E whenever // x,,/ [ < a(<), since we have chosen S ( E ) E . T h e proof is complete.
<
Notice that p ( t ) = J, y(s) ds + 0 as t as t --t n3 or J" y(s) ds < co . 0 This observation suggests the following: 1+1
-+
co holds if either y ( t )
---f
0
2.14. COROLLARY 2.14.1. system
115
ASYMPTOTIC BEHAVIOR
T h e conclusion of Theorem 2.14.6 is valid for the
x' = Ax
+ F ( t , + Gl(t, + G2(t, 2)
x)
2)
if conditions (i) and (ii) of Theorem 2.14.6 hold and, for small 11 x 11, G,(t, x) + 0 as t + m uniformly in x and 11 G,(t, x) I/ h(t)l/x 11, where X E C[J,R,],and 1 : X(s) ds < co.
<
THEOREM 2.14.7.
Assume that f E C [ / x S,, , R"],f ( t , 0) exists and is continuous on J x S , , and, for t E J ,
IIfAt, 4 -fdt)
0)li
< K /I x I/.
= 0, f,(t,
x)
(2.14.24)
Suppose that lim sup ( t t-m
-
j t p [ f z ( s ,O)] ds
t,)-l
= uo
to
< 0.
(2.14.25)
Let y(t) be the solution of the variational system Y'
where x(t)
=
=fz(t, x ( t ) ) y ,
for t
(2.14.26)
x(t,) =
X"
(2.14.27)
3 to . Then, we have the estimate
3 to and lim,,,y(t)
Proof.
= xn
x(t, to , xo) is the solution of x' = f ( t , x),
existing for t
to>
=
0.
Consider the function m ( t ) = IIy(t)ll. Observe that
where e(h)/h-+ 0 as h -+ 0. Furthermore, from the definition of p ( A ) given by (2.13.6), the inequality (2.13.10), and the assumption (2.14.24), we obtain
~ [ f d 4t))l t, Hence, it follows that
< &it7
0)l + Kli 4t)ll.
(2.14.29)
116
CHAPTER
2
-4s thc hypotheses of Theorem 2.13.2 are satisfied, the trivial solution of (2.14.27) is asymptotically stable, and therefore, if 11 x, I/ is small t 2 t, . Consequently, choosing 11 x, I/ sufficiently enough, 11 x(t)ll small, we have, from (2.14.30),
v,
which, by Theorem 1.4. I , leads to the estimate (2.14.28). Moreover, by condition (2.14.25), it results that, if 7 is small,
This, together with (2.14.28), implies that liml+my ( t ) = 0.
REMARK2.14.1. T h e condition (2.14.25) implies that the solutions yo(t)of the variational system
have the property that liml+myo(t)= 0. For, setting m(t) = IIy,(t)ll,
we obtain
D+m(t)
< P [ f Z ( t , O)lm(t),
and hence, by Theorem 1.4.1,
T h e assumption (2.14.25) assures that liml->va yo(t) = 0. Thus, in essence, Theorem 2.14.7 guarantees the asymptotic behavior of the solutions of (2.14.26), whenever there exists a similar behavior for the solutions of (2.14.31). From these considerations, we infer the following lemma, which is interesting in itself. T,EMMA 2.14.1. Let A(t) be a continuous n x n matrix on J . If x(t) is the solution of s' = A ( t ) x ,
we have
x(t,) = .xo ,
2.14.
COROLLARY 2.14.2.
Under the assumptions of Theorem 2.14.7, if ,t
1
then u
117
ASYMPTOTIC BEHAVIOR
(2.14.32)
< 0.
Proof. Since the trivial solution of (2.14.27) is asymptotically stable by Theorem 2.13.2, choosing (1 xo 11 small, we can make 11 x(t)ll < 7, t 3 t o , and hence we have 0
<
00
+ K.I
because of (2.14.29). It therefore follows that, by choosing 7 sufficiently small, u can be made less than zero whenever go < 0.
THEOREM 2.14.8. Assume that f E: C[J x R", R"],and f Jt , x) exists and is continuous on J x R". Suppose further that x ( t ) = x ( t , t o ,xo) is the solution of (2.14.27) with the property lim x ( t )
t-m
= 0.
(2.14.33)
If defined by the relation (2.14.32) is less than zero, then every solution x(t, to ,yo) of (2.14.27) such that xo ,yo belong to a convex subset of Rn satisfies lim x ( t , to ,yo) = 0. (2.14.34) t-m (5
Let x ( t , to , xo), x ( t , to , y o ) be the solutions of (2.14.27) such that xo , y o belong to a convex subset of Rn.Then, by Theorem 2.6.4, we get Proof.
and hence, for t 3 t o ,
II x ( t , t o ro)ll
< I! x ( t , to
> x0)ll
+ I/ yo
-
xo ll exp
[j t I-L[f&, t,
x(4)lds],
by virtue of Lemma 2.14.1. T h e relation (2.14.32) implies that
for sufficiently large t, and therefore the assumption u 0.
This, together with (2.14.33), assures (2.14.34).
< 0 yields
118
CHAPTER
2
COROLLARY 2.14.3. In addition to the hypotheses of Theorem 2.14.8, if we assume that f ( t , 0) = 0, then the trivial solution of (2.14.27) is asymptotically stable.
THEOREM 2.14.9. Let f E C [ J x R",R"], f ( t , 0) = 0, and f 2 ( t , x) exist and be continuous on J x R". Let G E C [ J x R", R"],and, given any E > 0, there exists a 6 = S ( E ) > 0 such that
II G(t,Y)ll
<4 Y
I19
<
provided IIy 11 a(€). Assume that the trivial solution of (2.14.27) is asymptotically stable and that u defined by (2.14.32) is negative. Then, the trivial solution of the perturbed system
is asymptotically stable. Proof. Let x ( t ) = x ( t , to , y o ) ,y ( t ) = y ( t , to , yo) be the solutions of (2.14.27) and (2.14.35), respectively. Then, by Theorem 2.6.3,
Moreover, by Theorem 2.6.4, we infer that
Let
E
> 0 be given. By assumption on G ( t , y ) ,it follows that
2.14.
119
ASYMPTOTIC BEHAVIOR
provided IIy(t)ll < a(€). Theorem 1.9.1 readily gives the estimate
for t 3 to , which implies that 11 y(t)l/ remains less than S ( E ) if 1) yo )I is small enough, because the condition u < 0 shows that, for sufficiently small E > 0,
Thus, (2.14.36) is valid for t 3 t, , and the asymptotic stability of the trivial solution of (2.14.35) follows.
THEOREM 2.14.10. Assume that (i) f E C [ J x R",R"],f ( t , 0)= 0, and f z ( t , x) is continuous on J x Rn;(ii) p [ f z ( t , O)] -u, u > 0, t E J ; (iii) G E C [ J x Rn,R"],G(t,0) = 0, and there exists an a > 0 such that, if 11 x 11 < a, t E J , 11 G(t,x)II y(t), where y E C [ J ,R,] and
<
<
p(t> =
1
t+l
t
y(s) ds -+
o
as
t
-j
co.
Then, the trivial solution of (2.14.35) is asymptotically stable. Proof.
Let E
> 0 be given such that 0 < E < min(u, a ) . Choose To 3 1
so large that, for t
s:
T o ,we have
exp[-(cr
-
c)(t
- s)]y(s)
ds
< YE) - = 6, , 2
(2.14.37)
<
E. This choice is possible, as shown in Theorem 2.14.6. where a(€) It is easy to show that, whenever 11 xg I] < a,, /I x(t)ll < S ( E ) , t >, to . For, otherwise, there would exist a t, > to 3 'Tosuch that
II 4tl)ll Define m ( t ) inequality
=
= 6(E),
ll 4t)ll
< 6(c),
t
E
[to >
4.
11 x ( t ) 11. Then, for t E [ t o ,t l ] , we obtain the differential
m;(t>
< P [ f Z ( t ? O)lm(t) + /I F(t7 4t))ll + It G(t,4t))ll < -(a 4 4 t ) + y(t). -
Here we have used assumptions (ii) and (iii) of the theorem, in addition to the relations (2.13.12) and (2.13.13) and the argument employed in Theorem 2.13.2.
120
CHAPTER
2
An application of Theorem 1.4. I gives
/I 4t)ll
< /I xo I/ exP[-(u
+ for t
E
[t, , t,]. At t S(c)
=
Jl0
exP[-(a
-
.)(t
-
441
- E)(t -
s)1 A s )
(2.14.38)
t , , there arises an absurdity
< 6, -k j:'
< 6,
+ 6,
exp[-(o =
-
c)(tl
-
s)] y(s) ds
S(E),
because of (2.14.37). This proves that, if 11 x,,11 < a,, Ij x(t)ll < a(€), t 2 t, , which, in its turn, implies the inequality (2.14.38) for all t 3 to . Since S(E) E , the stated result follows, as in Theorem 2.14.6.
<
+
COROLLARY 2.14.4. T h e function f ( t , x) = Ax F ( t , x), where A is an n x n constant matrix such that p ( A ) -0 and F ( t , x) satisfies assumption (ii) of Theorem 2.14.6, is admissible in Theorem 2.14.10.
<
2.15. Periodic and almost periodic systems We shall be concerned in this section with the existence of periodic and almost periodic solutions of differential equations. Let us first state the following:
LEMMA 2.15.1. Let E be an inductively ordered set, and let T be a transformation from E into E such that, for any x E E, we have T ( x ) x. Then, there exists at least one point x E E satisfying T ( x ) = x. As an application of this lemma, we prove an existence theorem for periodic solutions. THEOREM 2.15.1. Assume that (i) g E C [ J x R., , R],g(t, u) is nondecrcasing in u for each t E J , periodic in t with a period W , and the differential equation (2.1 5.1) u' = At, u) admits a periodic solution of period w ; (ii) f E C [ J x R",R"],f ( t , x) is periodic in t with a period w , and, for t E J , x,y E R", and sufficiently small h > 0,
2.15.
121
PERIODIC AND ALMOST PERIODIC SYSTEMS
where fi E C [ J x R",R"];(iii) the functions f,fi , and g are smooth enough to assure the existence and uniqueness of solutions, and the system Y' = S d h Y ) (2.1 5.3) has a bounded nondecreasing solution. Then, the differential system x'
=f
admits a periodic solution of period
(2.15.4)
( t , x)
w.
Proof. Let y ( t ) be the bounded monotonic solution of (2.15.3) such that y(to)= y o , to >, 0. Suppose that u(t) is the periodic solution of (2.15.1) of period w . It is possible to choose to and uo such that u(to) = u g >, 0,
>, 0,
u(t) - ug
t
>, t o .
(2.15.5)
Define m ( t ) = 11 x ( t ) - y(t)ll, where x(t) is the solution of (2.15.4) with the property x(to) = y o . Clearly, m(to) = 0. We have, using the condition (2.15.2), the differential inequality D+m(t>
< g ( 4 m(t)>.
Consider the solutions u(t, c) of u' = g(t, u )
for sufficiently small
E
+
u ( t o ,E )
E,
=
uo
+
E
> 0. Setting
because of the nondecreasing nature of g(t, u ) we get
According to Theorem 1.2.1, we infer that m(t) < p(t, E),
t
But limp(t, E ) <-0
= lim E-0
[u(t, E )
-
(uo
2 to .
+
E)]
= u(t)
-
uo
uniformly in t, and therefore (2.15.6)
122
CHAPTER
2
+
Define the point x(t,)by p , . Take the point ~ ( t , w ) on the solution x ( t ) and denote it by p , . Let T be the transformation that takes any point p, t o p , by the foregoing process. Since the functionf(t, x) is periodic in t with a period w , any solution passing through the fixed point, under the preceding transformation, is clearly a periodic solution. Hence, it is enough to show the existence of a fixed point, under the transformation T defined previously. Since u ( t ) is periodic and u(t,) = u, , we have u(t,,
+ nu)
~
U" = 0,
n
= 0,
I , 2 ,....
It therefore follows from (2.15.6) that X(t,
+- nu)
= y(t,
+ nu),
n
= 0,
1 , 2,....
By assumption, y ( t ) is a bounded monotonic solution of (2.15.3). Thus, nw) form a bounded, monotonic, and denumerably infinite set. If their upper bound is also included, the set becomes inductively ordered. Hence, the application of Lemma 2.15. I yields a fixed point, and the theorem is proved.
+
it is evident that the points %(to
COROLLARY 2.15.1.
If, in addition to the hypotheses of Theorem 2.15.1,
fi(t, y ) is periodic in t with a period w , the assertion of Theorem 2.15.1 remains valid. I n particular, fi(t, y ) = 0 is admissible. We remark that the monotony of g(t, u ) can be dispensed with if the
periodic solution u(t) of (2.15.1) has the property that, for some t, 0, u(t,,) = 0 and u ( t ) 3 0 for t 3 0. Another set of sufficient conditions for the existence of a periodic solution is given by
THEOREM 2.15.2. tial equation
Assume that (i) g E C [ J x u' = g ( t , u ) ,
u(0) = uo
R, , R], and the differen20
(2.15.7)
has unique solution u(t, 0, uo) such that lim u ( t , 0, u,) t-m
=
0;
(ii) f E C [ J x R",R"],and f (t,x) is periodic in t with a period w and is smooth enough to guarantee the existence and uniqueness of solutions of the system (2.15.4), and, for t E J , x, y E R?&,
I1 x
~
Y -1- h [ f ( t ,X) - f ( t ,
.Y)III < I/ x - Y I1 + hg(t, II x - y 11)
+ O(h)
(2.15.8)
2.15.
123
PERIODIC AND ALMOST PERIODIC SYSTEMS
for sufficiently small h > 0; (iii) the system (2.15.4) has a bounded solution existing on [0, a). Then, the system (2.15.4) admits a periodic solution of period w . Proof. Let x,(t) = xo(t,0, xo) be the bounded solution of (2.15.4) defined for t >, 0. We shall show that, under the assumptions of the theorem, we have lim xo(t W ) - x o ( t ) = 0. (2.15.9) f-m
+
Let x ( t ) = xo(t
+ w)
-
x o ( t ) . Then,
z'(t>= f ( t
+
W,
=f ( t , x d t )
xo(t
+
W))
- f ( t , xo(t))
+ z(t)>- f ( C
xo(t>),
because of the periodicity of f ( t , x) in t. Hence, setting m ( t ) = I/ x(t)ll, we observe, for small h > 0, that mft
+ h) < II .o(t + -f(C
W)
- xo(t)
xo(t))lll
+ II
+ h [ f ( t ,x , ( t ) + z ( t ) )
+)I19
where c(h)/h-+ 0 as h -+ 0. Using condition (2.15.8), it is easy to get the inequality D+m(t)
< g(t, 49,
which implies, by Theorem 1.4.1, choosing uo = 11 x0(w) - xo 11,
II xo(t
+
W)
%(t)ll < 4 4 0,II X O ( W )
~
-
xo Ill,
t
2 0,
u(t, 0, uo) being the unique solution of (2.15.7). Consequently, in view of assumption (i), the relation (2.15.9) is valid. Since xo(t) is bounded, the sequence {xn} = {xo(nw)) is bounded. Hence, a subsequence {xnJ can be extracted which converges to a point x*. It follows from (2.15.9) that, for any n, lim xo[(n
and thus,
n-m
+l)~]
-
xo(nw) = 0 ,
lim xn,+* = x*.
(2.15.10)
k+m
+
Observing that the functions defined by x(t, 0, x,) = xo(t nw) are the solutions of (2.15.4) through (0, xn), where x, = xo(nw),in view of the periodicity off(t, x) in t , we see that the fact lim xnr
k-m
= x*
124
2
CHAPTER
implies Iim .v(t, 0 , xn,)
k
= x ( t , 0, x * ) ,
t-
and hence, from (2.15.10), we obtain lim x ( t , 0 , x,,,,)
0 x*).
= x(t,
k+n
However, because of the uniqueness of solutions,
Since
lim x ( t , O
k-m
=
.lcn,+,)
=
we have X(t,
0 , x*)
=
lim x ( t
k-*m
x(t
x(t
-1
+
w,
w , 0,
xnk)
0 , x*),
+ w , 0, x*).
This means that the solution that satisfies the initial condition x(0) is periodic with period W. T h e proof is complete.
=
x*
An analogous result holds for almost periodic systems. We shall start with the following
DEFINITION 2.15.1. A functionfe C [(- GO, co) x Rn,R"]is said to be almost periodic in t uniformly with respect to x E S, for any compact subset S C R", if, given any 7 > 0, it is possible to find a l(7) such that, in any interval of length 1(7), there is a T such that the inequality llf(t
is satisfied for t
E
+
). - f ( t , x)ll
7,
< 17
(- co, a), x E S.
THEOREM 2.15.3. L e t f E C[(--00, co) x R", R"],andf(t, x) be almost periodic in t uniformly with respect to x E S, for any compact set S C R", and be smooth enough to ensure the existence and uniqueness of solutions of (2.15.4). Furthermore, suppose that, for sufficiently small h > 0,
II x - y I W t , 4
-f(t,y)lll
< II x - y
11[1
-
ah]
+ O(h),
t
2 0, (2.15.1 1)
2.15.
125
PERIODIC AND ALMOST PERIODIC SYSTEMS
where 01 > 0, and that the almost periodic system (2.15.4) admits a bounded solution x ( t , t o , x,) with a uniform bound B. Then, (2.15.4) admits an almost periodic solution that is uniformly asymptotically stable.
Proof. Let x(t) be the bounded solution of (2.15.4), defined on [ t o ,co) so that I/ x(t)ll B, t 3 t o , where B does not depend on t o . Let rk be a sequence of numbers such that rk -+ co as k -+ co and
<
f(t f
Tk
Y
x, - f ( t , ).
-
uniformly for t E (- co, 00) and x E S, any compact set in R". For any p, let k, = k,(p) be the smallest value of k for which r k , p 3 t o . We have
+
/I x ( t
+
Tk)l/
< B,
t
2 B,
k 3 k,(B).
+
We shall now show that the sequence of functions { x ( t rJr)), k 3 kO(,3) converges to a continuous bounded function w ( t ) defined on [p, m), with a bound independent of /3, and that the convergence is uniform on all compact subsets of [p, CO). Since the boundedness of x ( t ) is uniform with respect to t o , it is sufficient to prove that the sequence ( x ( t rX.)>, k 2 k,(& forms a Cauchy sequence on any compact subset of [/ICO). , Let U be any compact subset of [p, a),and let E > 0 be given. Choose an integer n, = no(€,p) 3 k, so large that, for k, >, n o ,
+
(2.15.12)
m(t)
Setting t ,
=
t
+ r6,, t, = t + rk,, we
have
D+m(t)
where 6 is an E@-translation to 6 3 0, that is,
+
number of f ( t , x) for x E S such that (2.15.13)
126
CHAPTER
2
where x E S, any compact set in R”, and t E (- co, GO). I n view of the condition (2.5.1 l ) , we deduce
which, because of relations (2.15.12) and (2.15.13), yields D%Z(t)
< -am(t) + ( € 4 2 ) ,
t
2 to.
By Theorem 1.4.1, we get
This proves the existence of a function w ( t ) defined on bounded by B. Since /3 is arbitrary, w ( t ) is defined
[p, co) and
This proves the existence of a function w ( t ) defined on [/3, a)and bounded by B. Since /3 is arbitrary, w ( t ) is defined on (- co, co), and we have as K co, v(t 7 k )- w ( t ) + 0
+
---f
uniformly on all compact subsets of (- co, a). Next we show that w ( t ) is differentiable and that it satisfies (2.15.4). Observe that lim, ,, x’(t 4-T / \ ) exists uniformly on all compact subsets of (- co, GO), and, consequently,
1
=
lim lim
=
lim -I [w(t
h-0
~t-o
k-.-o
h
h ~
[x(t
+
+ h)
T~
-
-1h )
~(t)],
2.15.
127
PERIODIC AND ALMOST PERIODIC SYSTEMS
which proves that w'(t) exists. Also, w'(t) = lim x'(t k-m
= =
+
T
~
)
+ lim [ f ( t + x(t + lim f ( t f
Tk
k+m
x(t
7
Tk))
Tic,
k-r,
+ lim [ f ( t + k-m
Tk
Y
Tk))
-f(t
+
Tk
9
w(t))]
w(t))]
=At, w(t)), proving that w ( t ) is the solution of (2.15.4). It remains to be shown that w ( t ) is almost periodic. For this purpose, we need to show that, for any E > 0 and any r for which
Ilf(t + T > x )- f ( t , 411 < E
uniformly on (-
00,
co) and all x E S , we have
II w(t
+
7) - w(t)/l
< EL
uniformly for t E (- co, co), where L is some constant independent of E and 7. Suppose y is such that
< E/~BoI.
e-"Y
Let z ( t ) = w(t
+
T)
- w(t), so
that
+
w(t
z'(t) = f ( t
7,
+
T))
- f ( 4 w(t)).
Let 6 be an <-translation number off(t, x) for R?&,such that t - y 8 3 0, that is,
+
llf(t
+ 8, x) - f ( t , .)I1
< E,
If m ( t ) = I/ x(t)ll, we get, for small h m(t
+ h) < /I w(t + ).
-
w(t)
(2.15.14)
S , any compact set in
xE
x E s,
t
(-a, a).
> 0,
+ h [ f ( t+ 8, w(t +
7))
-f(t
+ 8, w(t))lll
+ hllf(t + 8 , 4 t + - f ( 4 w(t + .)I1 + hllf(t + 6 , 4 t ) ) - f ( C 4t))Il -t hllf(t + r , w ( t + .)) - f ( t , w(t + .)I1 + II +)lL TI)
which, as before, yields the differential inequality D+m(t)
< -OIm(t) + 3t,
128
2
CHAPTER
whence, by Theorem .4.1, we have
I/ 4
t
+
T)
3E +; [I
- e-Ov].
T h c boundedness of w ( t ) and the relation (2.15.14) show that
uniformly in t , where L = 4/a. T h e uniform asymptotic stability of w ( t ) can be easily verified. T h i s coinpletes the proof
COROLLARY 2.15.2. Assume that (i) A ( t ) is a continuous n x n matrix on (- m, m), almost periodic in t , and p [ A ( t ) ]<:
a!
-01,
> 0,
t
2 0;
(ii)fE C [ ( - co, a), R7q,a n d f ( t ) is almost periodic in t and is bounded. Then, the system x’ A ( t ) x - l - f ( t ) (2.15.1 5 ) admits an almost periodic solution that is uniformly asymptotically stable. Proof. It is easy to check that the assumptions of Theorem 2.15.3 are satisfied except the existence of a bounded solution. Hence, it only needs to be verified that (2.15.15) has a bounded solution x(t, t o , xo) with a uniform bound 13 for t 3 t , . I n fact, under the assumptions, it turns out that all the solutions x (t , t , , xo), to E (- m, a), xo E R” are uniformly bounded. Let -m
SUP ,(.
a
llf(t)ll
=
B,
1
and let x ( t ) x ( t , t,, , xo) be any solution of (2.15.15) such that t,, E (- m, co) and /I xo / / R,/a. Then, it can be shown that
<
:
11 ~ ( t ) l< l 5B1/2m = B, If this were not true, there would exist a t ,
I1 r(t,)ll
=
B,
II 4t)Il
< B,
t
2 to.
> to such that t
E
[to
7
tll.
(2.15.16)
2.16.
129
NOTES
Let I9 be a BJ3-translation number of A ( t ) x + f ( t ) for x E S , a compact set in R”,such that t, I9 >, 0, that is,
+
+
where ~ ( h ) /+ h 0 as h --+ 0. Observing that t fl 2 0, assumption (i), together with the relations (2.15.16) and (2.15.17), yields
Hence, by Theorem 1.4.1,
which which leads leads to to an an absurdity, absurdity, using using the the fact fact that that I/xJ I/xJ (2.15.16); (2.15.16);
<
This (2.15.15) and and T h i s proves proves the the uniform uniform boundedness boundedness ofof solutions solutions ofof (2.15.15) establishes establishesthe thecorollary. corollary. 2.16, Notes
T h e results of Sect. 2.1 are due to Stokes [I]. For a more general global existence theorem using Tychonoff’s fixed point theorem, see Corduneanu [I]. Theorem 2.2.1 is given by 0. Perron, (see Kamke (I]). T h e proof of Corollary 2.2.1 is new. T h e general uniqueness theorem 2.2.2 is due to Kamke [l]. T h e proof given in the text is based on that of Olech [4]. For Theorem 2.2.3, see Lakshmikantham [12] and Olech [4]. Corollary 2.2.2 is a result of Wintner [l5]. Corollaries 2.2.4 and 2.2.5 are Nagumo’s and Osgood’s uniqueness criteria, respectively. Theorem 2.2.4 is due to Brauer [2]. Theorem 2.2.5 and the proof of Theorem 2.2.4 are taken from Walter [2]. Corollary 2.2.6 is a result of Krasnosel’skii and Krein [2].
130
CHAPTER
2
See also Brauer [l], Kooi [I], and Luxemburg [l]. Beginning from nonuniqueness Theorem 2.2.7, the remaining results of Sect. 2.2 are due to Lakshmikantham [5, 121. T h e proof of Theorem 2.3.1 is taken from Wazewski [9], whereas the second proof of Theorem 2.3.1 is from Hartman [5]. See Olech and Plis [l] for the proof that the monotony assumption in Theorem 2.3.1 cannot be dropped in general. See also Bihari [I], Cafiero [2, 41, Coddington and Levinson [l], Diaz and Walter [I], Dieudonne [l], LaSalle [l], and Viswanatham [I]. The results of Sect. 2.4 are taken from Chaplygin [l] and Lusin [I]. Lemma 2.5.1 and Theorem 2.5.1 are new: the idea is taken from Turowicz [l]. Theorem 2.5.2 is adopted from Antosiewicz [7]. T h e rest of the results of Sect. 2.5 are taken from Hartman [6]. Theorems 2.6.3 and 2.6.4 are due to Alekseev [I]. Most of the results of Sects. 2.7 and 2.8 are adopted from Lakshmikantham [7] and Walter [3]. See also Bihari [2], Brauer [5], and Langenhop [I]. Section 2.9 contains the work of Brauer [ll]. Theorem 2.10.1 is from 1,akshmikantham and Onuchic [I]. Theorems 2.10.2 and 2.10.3 are adopted from Brauer [4, 121. See also Cesari [I] and Wintner [4, 5, 7-91. Theorems 2.11.1 and 2.11.2 are due to Wazewski [2] and are very useful in the study of differential equations. T h e proofs are taken from Hartman [5]. Theorem 2.12.1 is due to Lakshmikantham and Onuchic [l]. For further results and references on the application of Wazewski’s topological principle, see Cesari [l] and Hartman [5]. Theorem 2.1 3.1 is adopted from Lakshmikantham [7], whereas Theorem 2.13.2 is due to Brauer [lo]. For Theorem 2.13.3, see Conti and Sansone [l]. Theorems 2.14.1 and 2.14.3 are adopted from Lakshmikantham [7]. For Theorem 2.14.2, see Krasovskii [4] and Zubov [I]. See Cesari [I] for Theorems 2.14.4 and 2.14.5. Theorem 2.14.6 is due to Strauss and Yorke [I]. Theorems 2.14.7, 2.14.8, and 2.14.9 are taken from Brauer [lo]. See also Cesari [l], Coppel [l], Hartman [5], and Markus and Yamabe [I]. Theorem 2.15.1 is due to Lakshmikantham [7]. Theorem 2.15.2 is new. T h e proof uses a result from Halanay [2]. Theorem 2.15.3 is new. See also Cartwright [l], Deysach and Sell [l], Massera [2], Miller [3], Seifert [5, 61, and Sell [4].
Chapter 3
3.0. Introduction As is well known, Lyapunov's second method has its origin in three simple theorems that form the core of what he himself called his second method for dealing with questions of stability. It is widely recognized, today, as an indispensable tool not only in the theory of stability but also in studying many other qualitative properties of solutions of differential equations. T h e main characteristic of this method is the introduction of a function, namely the Lyapunov function, which defines a generalized distance from the origin of the motion space. As a result, the concept of Lyapunov function, together with the theory of differential inequalities, furnishes a very general comparison principle under much less restrictive assumptions. We present, in this chapter, a variety of qualitative problems bringing out the real significance of the comparison technique.
3.1. Basic comparison theorems Consider the differential system x' = f ( t , x),
where f~ C [ J x R", R"]. For function
x(t0) = x0
,
to 6
1,
(3.1.1)
V E C [ J x R",R,], we define the
1 D+V(t,x) = lim sup - [V(t ti-.^+ h
+ h, x + hf(t, x))
-
V(t,x)]
(3.1.2)
for ( t , x) E J x Rn. Occasionally, we write D+V(t, to denote that the definition of D+V(t,x) is with respect to the system (3.1.1). We can now formulate the basic comparison theorems. 131
132
CHAPTER
3
THEOREM 3.1.1. Let V E C [ J x RE,R,] and V ( t ,x) be locally 1,ipschitzian in x. Assume that the function D+V(t,x) of (3.1.2) satisfies Z l V ( t ,x)
< g (t, V(t,x)),
( t , x)
E
J x Rn,
(3.1.3)
where g E C [ J x R, , R]. Let r ( t ) = r ( t , to , uo) be the maximal solution of the scalar differential equation u' = g ( t , u),
existing to the right of to . If x(t) existing for t 3 to such that
u(to) = uo =
2 0,
(3.1.4)
x ( t , to , xo) is any solution of (3.1.1)
(3.1.5)
then (3.1.6)
Proof. Let x(t) be any solution of (3.1.1) defined for t 3 to such that (3.1.5) holds. Define m(t) = q t , X ( t ) ) ,
so that m ( t J
< u o . For
sufficiently small h
> 0,
we have
where ~ ( h ) /+ h 0 as h + 0. Since V ( t ,x) is locally Lipschitzian in x, we get, using (3.1.3), the inequality L> ' m ( t ) < s(t,m ( t ) ) ,
t E J.
Applying Theorem 1.4.1, we obtain the desired result (3.1.6). REMARK 3.1.1.
Let S, = [x E Rn : 11 x
I/ < p ] ,
and assume that the condition (3.1.3) holds for ( t , x) E J x S, . If x ( t ) is any solution of (3.1.1) such that 11 xo 11 < p, then (3.1.5) implies (3.1.6) as far as x ( t ) remains in S, to the right of to . COROLLARY 3.1. I . zero, i.e.,
If the function g(t, u ) in Theorem 3.1.1 is identically W V ( t ,x)
< 0,
( t , X) E J x
Rn,
3.1.
BASIC COMPARISON THEOREMS
133
then the function V ( t ,x ( t ) ) is nonincreasing in t , and V ( t ,X(t))
< V(to
9
t
Xo),
3to.
COROLLARY 3.1.2. Assume that the hypotheses of Theorem 3.1.1 hold except that the condition (3.1.3) is to be satisfied only for ( t , x) E J x 52, where SZ = [X E R" : ~ ( t< ) V ( t ,X) < ~ ( t+ ) co , t 3 to], c0 being some positive number. Then, the conclusion of Theorem 3.1.1 is true.
Proof. We choose uo = V ( t , , xo) and proceed as in the proof of Theorem 3.1.1 to obtain Z being the set
D+m(t)
< g(t, m ( t ) ) ,
t
E
z,
z= [t E J : r ( t ) < m ( t ) < Y ( t ) + 61. Theorem 1.4.2 now assures the stated result. Sometimes, the following variants of Theorem 3.1.1 are more useful in applications.
THEOREM 3.1.2. Assume that the hypotheses of Theorem 3.1.1 hold except that the inequality (3.1.3) is replaced by A ( t P + V t X) ,
+ v, X)D+A(t)< g(t, V ( t ,x)A(t)),
(3.1.7)
for ( t , x) E J x R", where the function A(t) > 0 is continuous for t E J , and 1 D+A(t) = lim sup - [A(t h++ h
Then U t o xo)A(to)
imp1ies q t , X(t))A(t)
Proof.
+ h)-
A(t)].
< uo
< r(t),
t
(3.1.8)
2 to -
(3.1.9)
Defining VAt, X)
=
V ( t ,X)A(t),
it is easy to show that Vl(t,x) satisfies the assumptions of Theorem 3.1.1.
134
For, if h
CHAPTER
3
> 0 is sufficiently small,
and therefore, using the assumption (3.1.7), we get
T h e estimate (3.1.9) follows immediately from Theorem 3.1.1.
THEOREM 3.1.3. Let the hypotheses of Theorem 3.1.1 hold except that, instead of the inequality (3.1.3), we now assume
o+v(t, x) + 9(ll
.X
Ill < R(& v(t,x)),
( t , x) E I x I?”,
(3.1.10)
where +(u) 3 0 is continuous for u 3 0, $(O) = 0, and $(u) is strictly increasing in u. Suppose further that g(t, u ) is nondecreasing in u for each t E J . T he n (3.1.5) implies that
Proof.
Consider the function
<
m(t). T h e monotonic character of g(t, u ) in u, together with the condition (3.1.lo), now yields so that V ( t ,x ( t ) )
D+ 4 t )
< R(t, m ( t ) ) ,
and the assertion (3.1.11) follows from Theorem 1.4.1.
DEFINITION 3.1.1. T h e function V ( t ,x) is said to be mildly unbounded if, for every T > 0, V ( t ,x) + as 11 x 11 co uniformly for t E [0, TI. T h e mild unboundedness of V ( t ,x) guarantees that, whenever V ( t ,x ( t ) ) is finite, 11 x(t)ll is also finite. T h e assumption that the solutions x(t) of (3.1.1) exist for all t 3 t o , therefore, becomes superfluous, if V ( t ,x) is further assumed to be mildly unbounded in the foregoing theorem. From this observation stems the following global existence theorem. ---f
3.2.
135
DEFINITIONS
THEOREM 3.1.4. Let V EC [ J x Rn, R,], V(t,x) be mildly unbounded and locally Lipschitzian in x. Suppose that g E C [ J x R, , R] and r ( t ) is the maximal solution of (3.1.4) defined for t 3 t, . Assume that (3.1.3) holds. Then, every solution x(t) of (3.1.1) exists in the future, i.e., for all t 3 to , and (3.1.5) implies (3.1.6). Proof. Suppose that the assertion that every solution x(t) of (3.1.1) exists for all t >, t, is false. Then, by Corollary 1.1.2, there exists a t, > t, such that x(t) cannot be extended to the closed interval t, t t , , which implies that there cannot exist an increasing sequence {t,} + t,- such that 11 x(tn)1l is bounded. This, in its turn, yields that 11 x(tn)lI + 00 as t, + t,- . On the basis of Theorem 3.1.1, it follows that (3.1.5) implies (3.1.6) for to t t, . By the assumption that V ( t ,x) is mildly unbounded, the fact that r ( t ) exists for all t >, t, , and (3.1.6), there arises a contradiction as t, + t,- . Hence, the global existence of solutions x(t) of (3.1.1) is proved, which, in turn, assures the estimate (3.1.6) for t 3 t, whenever (3.1.5) holds. T h e proof is complete.
< <
< <
3.2. Definitions Let x ( t , t o , xo) be any solution of the differential system x' = f ( t ,XI,
.(to)
=
,
to
2 0,
(3.2.1)
where f E C[J x S,, , Rn],S, being the set
s,= [x E Rn : 11 x 11 < p].
(3.2.2)
Assume that f ( t , 0) = 0, t E J , so that x = 0 is a (trivial) solution of (3.2.1) through ( t o ,0). We now list a few definitions concerning the stability of the trivial solution.
DEFINITION 3.2.1.
T h e trivial solution x
=
0 of (3.2.1) is
(S,) equistable if, for each E > 0, t, E J , there exists a positive function 6 = 8(t,, E ) that is continuous in t, for each E such that the inequality IlxnIl
implies
/I x ( t , to .")I1 9
<8
< E,
t
3 to ;
(S,) uniformly stable if the 6 in (S,) is independent of t, ;
136
CHAPTER
3
(S,) quasi-equi asymptotically stable if, for each E > 0, to E J , there exist positive numbers IS,, = &,(to) and T = T ( t o ,C ) such that, for t t,, -I T and 11 x ~/I, -: 8, ,
>
I/ x ( t , t,,
7
x0)Il
< E;
(S,) quasi uniformly asymptotically stable if the numbers 6, and T in (S,) are independent of to ; (S,) eqzti-asymptotically stable if (S,) and (S,) hold simultaneously; (S,) un;formly asymptotically stable if (S,) and (S,) hold together; ( S , ) quasi-equi asymptotically stable if, for each E > 0, 01 > 0, and to E J , there exists a positive number T = T(t, , E , a ) such that // xo (1 01 implies
<
I/ x ( t , t o , xo)ll < E,
t
2 to
+ T;
(S,) quasi uniformly asumptotically stable if the T in (S,) is independent of t" ; (S,) completely stable if (S,) holds and (S,) is verified for all 01, O < Y co; (Slo) uniformly completely stable if (S,) holds and (S,) is verified for all ct, 0 < (Y -< m; unstable if ( S , ) fails to hold. (S'J REMARK3.2.1. Sometimes the notion of quasi-asymptotic stability may be relaxed somewhat as in (S,) and (S8).Clearly the E, 01 given in the preceding definitions must be less than p of (3.2.2), and therefore thc concepts (S,)-(S,) are of local nature. If, on the other hand, p = co, so that S , R",the corresponding concepts of stability would be of global character. These considerations lead to (S,) and (Sl,,).We note further that the definitions (S,)and (S,) may hold even whenf(t, 0) + 0. I n other words, the assumption about the existence of the trivial solution is not necessary. I n characterizing Lyapunov functions, it is convenient to introduce certain classes of monotone functions. DEFINITION 3.2.2. A function + ( r ) is said to belong to the class .X if 4 E C[[O,p), K,],#(O) = 0, and +(r) is strictly monotone increasing in Y . I~EFINITIO 3.2.3. N 4 function ~ ( tis) said to belong to the class 9 if C [ ] ,R,],u(t) is monotone decreasing in t , and ~ ( t+ ) 0 as t -+ co.
uE
3.2.
137
DEFINITIONS
DEFINITION 3.2.4. A function + ( t , r ) is said to belong to the class %A? if 4 E C [ J x [0, p), R+], 4 E S” for each t E J , and 4 is monotone increasing in t for each Y > 0 and +(t, Y ) -+ 00 as t oc) for each r > 0. -j
DEFINITION 3.2.5. A function V ( t ,x) with V ( t , 0) = 0 is said to be positive deJinite (negative deJinite) if there exists a function 4 ( r ) E X such that the relation
DEFINITION 3.2.6. A function V ( t , x) with V ( t , 0) = 0 is said to be strongly positive dejinite if there exists a function + ( t , r ) E XA? such that
q t , ). 3 d ( t , I1 x ll),
( t , x) E
J
x
s, .
DEFINITION 3.2.7. A function V ( t , x) 3 0 is said to be decrescent if a function + ( Y ) E S” exists such that V(t,x) <+(I1
x
(44
II),
E
J x s,.
T o use the second method of Lyapunov, which attempts to make statements about the stability properties directly by using suitable functions, we need to study the scalar differential equation u’
= g(t, u),
u(to) = uo
2 0,
tn
0,
(3.2.3)
where g E C [ J x R, , R].We suppose that g ( t , 0) 3 0 so that u = 0 is a solution of (3.2.3) through ( t o ,0). Furthermore, this assumption also implies that the solutions u ( t ) = u ( t , t o , uo) of (3.2.3) are nonnegative for t 2 to so as to assure that g ( t , u ( t ) ) is defined. Corresponding to the stability definitions ( Sl)-(&), we designate by (Sf)-(S$)the concepts concerning the stability of the solution u = 0 of (3.2.3). DEFINITION 3.2.8. The trivial solution u = 0 of (3.2.3) is said to be if, for each E > 0, to E J , there exists a positive function 6 = 6 ( t o , E) that is continuous in to for each E such that
(Sp) equistable
u(t, to uo)
<
provided uo
€7
< s.
t
3 tn,
138
3
CHAPTER
T h e definitions (S$)-(S$) may be formulated similarly. Notice as before that, for the concepts (ST) and (23:) to hold, the assumption g ( t , 0) F- 0 is not necessary.
3.3, Stability We begin with the following stability criteria, recalling the definition of the function D+V(t, x) given in (3.1.2).
THEOREM 3.3.1. Assume that there exist functions V(t,x) and g ( t , u) satisfying the following conditions:
(i) g E C [ J x R, , R] and g ( t , 0) = 0. (ii) V E C [ J x S o ,R,],V ( t ,0) E 0, and V(t,x) is positive definite and locally Lipschitzian in x. (iii)
for ( t , x)
E
J x So ,
< g(t, q t , x)).
D+V(t,).
Then, the equistability of the trivial solution of (3.2.3) implies the equistability of the trivial solution of the system (3.2.1).
Proof.
By assumption, a function b(r) of class 3? exists such that V ( t ,x) 3 b(ll x \I),
( t , x) E
J x
So.
(3.3.1)
Let 0 < E < p and toE J be given. Since the solution u = 0 is equistable, given b ( ~ > ) 0, to E J , there exists a positive function S = S(t, , E ) that is continuous in to for each E , such that uo 8 implies
<
u ( t , t o , uo)
< b(E),
t
3 to.
(3.3.2)
Choose uo = V ( t o xo). , Since V ( t ,x) is continuous and V(t,0) E 0, it is possible to find a positive function 8, = S,(t, , E ) that is continuous in to for each E , satisfying the inequalities
/I xo /I
<
simultaneously. We claim that, if
I1 x(t7 t o
9
V t o %>
61,
xo)ll
9
(1 xo 11 <
< a,,
€3
t
<6
2 to
(3.3.3)
*
Suppose that this is not true. Then, there would exist a solution x ( t ) x ( t , to , xJ with /I xo I/ 8, and a t , > to such that
<
/I x(t,)ll
= E?
I1 4411 < 6 ,
t
E
[to
f
tll,
=
3.3.
139
STABILITY
(3.3.4)
because of (3.3.1). This means that (1 x(t)ll < p for t E [to , t l ] ,and hence the choice uo = V ( t o xo) , and condition (iii) give, as a consequence of Theorem 3.1.1, the estimate v(t,4 t ) )
< Y(t> to
9
uo),
t
E [to
, tJ,
(3.3.5)
where r ( t , t o , uo) is the maximal solution of (3.2.3). T h e relations (3.3.2), (3.3.4), and (3.3.5) lead to the contradiction b(c)
< V(tl
>
4tl))
<
Y(t1
, t o , uo) < b ( 4 ,
proving (&). T h e proof of the theorem is complete.
COROLLARY 3.3.1. T h e function g ( t , u ) = h(t) +(u), where A ( t ) is continuous on J , +(u) 0 is continuous for u 2 0, +(u) > 0 for u > 0, is admissible in Theorem 3.3.1 provided
>
(3.3.6)
for some uo > 0, every to 2 0 and to
< t < co.
Proof. All that we have to verify is that the solution u is equistable. Define
=
0 of (3.2.3)
s,"
otherwise, J(u) = J" ds/+(s), for sufficiently small E > 0. If ds/+(s) = R co, then J ( u ) s'; a monotone function mapping the interval [0, co) homeomorphically onto the interval [0, R). T h e solution u(t, t o , uo) of (3.2.3) is given by
<
(3.3.7) as long as
or
1 40
CHAPTER
3
Thus, from (3.3.7) and the fact that J is a homeomorphism, it is easy to see that u = 0 of (3.2.3) is equistable when (3.3.6) holds. This proves the corollary. 3.3.2. Under the assumptions of Theorem 3.3.1, the uniform stability of the solution u = 0 of (3.2.3) also implies the cquistability of the trivial solution of (3.2.1). 1'IIEOREM
T h e proof follows from the proof of Theorem 3.3.1. In this case, although 6 is independent of t o , the relation (3.3.3) shows that 6, is not independent of t o . Consequently, one gets only the equistability of the trivial solution of (3.2.1).
I'Toof.
COROLLARY 3.3.2.
Assume that there exists a function V ( t ,x) verifying
the following conditions:
(i) V E C [ J x S, , R,], V ( t ,0) = 0, V ( t ,x) is positive definite and locally Iipschitzian in x. 0, ( t , x) E J x So. (ii) D-I V ( t ,x)
<
Then, the trivial solution of (3.2.1) is equistable. It is important to note that, when (ii) holds, the scalar differential equation (3.2.3) reduces to u' =
and as a result Theorem 3.3.2.
0
u(t,) = u,
7
(S2) is
3 0,
to
2 0,
satisfied. Thus, Corollary 3.3.2 follows from
THEOREM 3.3.3. I n addition to the hypotheses of Theorem 3.3.1, assume that V ( t ,x) is decrescent. Then, the equistability of the null solution of (3.2.3) assures the equistability of the solution x = 0 of (3.2.1). Proof. Since V ( t ,x) is decrescent, there exists a function u(r) E .X such that ( 4 4 E 1 x s,. v t , x) < a(ll x II), We follow the proof of Theorem 3.3.1 except that we choose uo = x,) 11). Hy assumption, (SF) holds, and therefore 6 = 8(t,, E ) depends on t,, . As U ( T ) E X , the existence of a positive function 6, = Sl(to , c) satisfying the inequalities
I1 x" II
< 6,
9
4 xo It)
<
(3.3.8)
simultaneously is clear. T h e rest of the proof is very much the same.
3.3.
141
STABILITY
COROLLARY 3.3.3. T h e function g ( t , u ) tinuous on J and / I A ( s ) ds
=
h(t)u, where h(t) is con-
< 03
for every to 3 0, is admissible in Theorem 3.3.3.
THEOREM 3.3.4. Let the hypotheses of Theorem 3.3.1 hold. Assume further that V ( t ,x) is decrescent. Then, the uniform stability of the solution u = 0 of (3.2.3) guarantees the uniform stability of the trivial solution of (3.2. I). Proof. Following the proof of Theorem 3.3.3, it is easy to see that 6, does not depend on t o . For, by assymption of the uniform stability of the null solution of (3.2.3), 6 is independent of t o ,and (3.3.8) shows that 6, is also independent of t o . COROLLARY 3.3.4. T h e function g ( t , u ) = h(t) +(u) defined in Corollary 3.3.1 is admissible in Theorem 3.3.4 if (3.3.6) holds and if either J.", ds/$(s) < 00 or (i) l.",ds/+(s) = cc and (ii) h(s) ds is bounded above for every t , to t < co uniformly in t o .
l;o
<
COROLLARY 3.3.5. Assume that there exists a function V ( t ,x) fulfilling the following assumptions: (i) V EC [ J x S, , R,], V ( t ,x) is positive definite and decrescent, and V ( t ,x) is locally Lipschitzian in x. (ii)
D+V(t,x)
< 0, ( t ,x) E J
x S, .
Then, the trivial solution of (3.2.1) is uniformly stable. T h e definition of uniform stability of the solution x = 0 given in (S,) can also be formulated by means of a monotone function, as can be seen by the following
THEOREM 3.3.5. T h e trivial solution of (3.2.1) is uniformly stable if and only if there exists a function u(r) E % verifying the estimate
II x(t, to for /I xo II
< P.
I
x0)Il
< 4 xo It),
t
2 to
(3.3.9)
Proof. T h e sufficiency of the condition is immediately clear. T o prove the necessity, consider, for a given E > 0, the least upper bound of all positive functions a(€), and designate it by 8 = 8 ( ~ ) .Then 11 xo I/ 8
<
142
3
CHAPTER
<
implies 11 x ( t , t o , xo)ll E for t >, t o , and, if 6, > 8, there exists at least one xo such that, for /I xo I/ < a,, I( x ( t , t o , xo)lI exceeds the value E at some time t. Clearly, the function S(E) is positive for E > 0; it is nondecreasing and tends to zero as E + 0; and it may be discontinuous. We now choose a continuous, monotonically increasing function S * ( E ) satisfying S*(E) 8(e). Then, the inverse function a(r) = 6*-l(r) satisfies (3.3.9). T h e proof is complete.
<
We shall now prove a result that gives sufficient conditions for unstability of the solution x = 0 of (3.2.1).
THEOREM 3.3.6. Let there exist functions V(t,x) and g ( t , u ) satisfying the following properties:
e,
V ( t ,x) = 0 (i) V E C[G, R,], V ( t ,x) is locally Lipschitzian in x on for all ( t , x) E - G, and V ( t ,x) is positive and bounded on G, where G C J x Sp is some open set such that G has at least one boundary point ( T ,0), T > 0.
e
(ii) g E C[J x R, , R+],and D+V(t,x) >> g ( t , V ( t ,x))
2 0,
( t ,x) E G .
(iii) For to > T , the solutions u(t, t o , uo) of (3.2.3), for arbitrarily small uo > 0, are either unbounded or indeterminate, for t 2 t o . Then the trivial solution of (3.2.1) is unstable. There exists a point ( t o ,x,,)E G, xo f 0, in the vicinity of ( T , 0). Let x(t) = x(t, t o ,x,,) be any solution of (3.2.1). Then, the Lipschitzian nature of V(t,x) and condition (ii) yield Proof.
V ( t ,x ( t ) ) 2 V ( t , , xg)
=
uo
> 0,
(3.3.10)
for all t >, 0, for which ( t , x ( t ) ) E G. Since V ( t ,x) = 0 for all (t,x) E - G, it follows from (3.3.10) that (t, x(t)) E G for t 2 t o . Moreover, we also have
e
which, in view of Remark 1.4.1, implies that V t , 4 t ) ) 2 d t , to
7
U"),
t
2 to ,
(3.3.11 )
where p ( t , t o , uo) is the minimal solution of (3.2.3). Since V ( t ,x) is bounded by assumption, the estimate (3.3.1 1) leads to an absurdity,
3.3.
143
STABILITY
if we assume the trivial solution of (3.2.1) is stable. This proves the theorem.
THEOREM 3.3.7. L e t f E C[(-co, co) x S o ,R"] andf(t, x) be periodic in t with a period w . Then, under the hypotheses of Theorem 3.3.1, the trivial solution of (3.2.1) is equistable for to E (- 00, co). Proof. Let 0 < E < p and to E (- co, co) be given. I t is possible to choose an integer k such that to kw 2 0. Since the solution u = 0 of (3.2.3) is equistable, given b ( ~ > ) 0, t, kw >, 0, there exists a positive function 6 = 6 ( t o , c) such that uo 6 implies
+
u(t
u(t
+
+ kw, to + kw, uo) < We),
<
t
+ kw, to + kw, uo) being any solution of u' = g(t
+ kw, u ) ,
u(to
+ kw)
3to,
(3.3.12)
uo .
(3.3.13)
=
+ kw, xo) and obtain 6 , = S,(t, II xo I1 < 8, , w o + kw, xo) < 8
We choose uo = V(to
, c) satisfying
simultaneously, as in the proof of Theorem 3.3.1. With this 6, , the equistability of the trivial solution of (3.2.1) follows. Supposing this were false, there would exist a t , > to such that
II 4tl)ll
=
<
/I +Ill
E?
t
€9
E
(3.3.14)
[to Y t11
for some solution x ( t ) of (3.2.1) such that 11 xo 11 function m(t) = V ( t kw, , x ( t ) ) ,
< 6,.
Consider the
+
and, for small h m(t
+ h)
-
> 0,
m(t)
< Jwf(t,
-f(t
+ kw, 49 II
+LI1+)I1 + [V(t + kw + h, 4 t ) + hf(t + V ( t + kw, x(t))l,
kw, 4 t ) ) )
-
where L is the Lipschitz constant and r(h)/h + 0 as h + 0. T h e periodicity of f ( t , x) and condition (ii) of Theorem 3.3.1 give the inequality
o w ) < g(t
+ kw, m(t)),
t
6
[to 7
tll,
which implies the estimate, by Theorem 1.4.1,
v(t + kw, x ( t ) ) < r(t + kw, to + kw, uo),
t E [to ti], 9
(3-3.15)
144
r(t
3
CHAPTER
+ kw, t , 4-kw, uo) being the maximal solution of (3.3.13). b ( ~< ) V(tl + kw, < ~ ( tI , k w , t o + kw, uo) < b(c),
Thus,
~ ( t l ) )
using relations (3.3.1), (3.3.12), (3.3.14), and (3.3.15). This contradiction proves the assertion of the theorem.
If, in particular, the function f ( t , x) of system (3.2.1) is known to be periodic in t or autonomous and is smooth enough to assure uniqueness of solutions, it is possible to infer more information about the stability of the null solution of (3.2.1). T o this effect, we have
THEOREM 3.3.8. IJet f E C[(-co, GO) x S , , R 7 L ] , f (xt), be periodic in t with a period w, and the system (3.2.1) admit unique solutions. Then, the stability of the trivial solution of (3.2.1) is necessarily uniform. Proof. By the periodicity of f ( t , x) in t , it follows that, if x(t, t o , xo) is a solution of (3.2.1), then x(t -1 w , t , , x,) is also a solution. Furthermore, the uniqueness of solutions shows that, for any interger k, ~ ( $t k w , t , & kw, x,) = ~ ( tt , , x,). For each fixed to , t , E (- co, a), let
Since S ( t , , E) is continuous in to for each E , if we let 6, it IS clear that 8, > 0. For D 8 < 6 , , we define
=
inf,,< to+
S(t,),
4 s ) = 020 sup ll "(t" + 0,to %)ll. 9
Notice that c ( 6 ) is a monotone increasing function of 6, and hence, if S ( E ) is the inverse function of €(a), we havc
I/ x ( t , t o
1
~o)lI
< E,
for every t, E [O, w], provided /I x, /I we can choose an integer k such that kw
: t, < ( k + I)w
<
t
2 to
< S(E).
or
-(k
7
Let t , be arbitrary. T h e n
+ l ) w < to < -kw.
< t, + ( k + 1)w < w .
<
Hence, either 0 to - kw w or 0 Consequently, if 11 xo /I 6 ( ~ ) either ,
/I ~ or for t
/I
G X ( ~ ,
tn
9
( f t ,o
xo)lI
<
7
xo)ll
I1 x ( t
11 x ( t
-
kw, t o
-
kw, xo)ll
<E
+ ( k II)w, + (k + l)w, xo)ll <
2 t o , and the uniformity of the stability is evident.
E,
3.4.
145
ASYMPTOTIC STABILITY
3.4. Asymptotic stability We present, in this section, a number of results concerning the asymptotic stability of the solution x = 0 of the system (3.2.1).
THEOREM 3.4. I . Let there exist functions V(t,x) and g(t, u ) fulfilling the following assumptions: (i) g E C [J x R+ , R] and g(t, 0) = 0. (ii) V E C [ J x S o ,R,], V ( t ,0) = 0, and V ( t ,x) is positive definite and locally Lipschitzian in x. (iii) D+V(t, x)
< g(t, V(t,x)),
(t,x) E J x S,,.
Then, the equi-asymptotic stability of the solution u = 0 of (3.2.3) assures the equi-asymptotic stability of the trivial solution of (3.2.1). Proof. Let (S:) hold. Then, by definition, (SF) and (S:) are satisfied, and therefore only quasi-equi asymptotic stability needs to be proved, since Theorem 3.3.1 guarantees (Sl). T o prove (S3),let E > 0, to E J be given. It then follows from (S?) that, given b ( ~ > ) 0, t, E J , there exist positive numbers 6, = S,(to) and T = T ( t o ,E ) such that u(t, t o , uo)
t
2 t"
+ T,
(3.4.1)
<
whenever u, 6,. Choose u, = V ( t o x ,,). Since V ( t ,x) is continuous and V ( t ,0) = 0, we can find, as in the proof of Theorem 3.3.1, a positive number 8, = 8,(t,) satisfying the inequalities
I1 Xo I/
< 8,
V(to > xo)
Y
< 8,
(3.4.2)
together. We have also, in view of (SJ, the inequality (3.3.5) holding for t 3 t o . Suppose, now, that there is a sequence {tk),t, 3 to + T and t, + co as k + co such that I/ x(t, , t o ,xo)lI 3 E , where x(t, t o ,x,) is some solution of (3.2.1) starting in I/ x, /I 8,. This leads to the contradiction
<
b(c)
< V(t,
, 4 t k >
to
> Xo))
< Y(tk , to
7
uo)
< Kc),
because of (3.3.1), (3.3.5), and (3.4.1). Thus, (S,) is proved, which implies the equi-asymptotic stability of the solution x = 0 of (3.2.1). REMARK 3.4.1. It is possible that 8, obtained in (3.4.2) may be such that 8, > 8,. where S$ = S ( t , , p), in which case the inequality (3.3.5)
146
CHAPTER
3
need not hold for all t 3 t, whenever 11 xo 11 < 8, . T o avoid this situation, it is necessary to redefine 8, = min[8, , &,3in the foregoing proof. COROLLARY 3.4.1. T h e function g(t, u ) = A ( t ) $(u), where h(t) is continuous on J , +(u) >, 0 is continuous for u >, 0, +(a) > 0 for u > 0, is admissible in Theorem 3.4.1 provided that there exists a T , to T co, verifying the relation
< <
THEOREM 3.4.2. Under the assumptions of Theorem 3.4.1, the uniform asymptotic stability of the trivial solution of (3.2.3) also implies the equi-asymptotic stability of the solution x = 0 of (3.2.1). Proof. T h e proof runs parallel to the proof of Theorem 3.4.1. First of all, following the proof of Theorem 3.3.2, we note that 6, is not independent of to although 6 is. Similarly, 8, occuring in (3.4.2) is also not independent of t, , even though 6, is. COROLLARY 3.4.2. T h e conclusion of Theorem 3.4.2 remains true if the function g(t, a ) : --+(a), where #(a)E .K.
Proof. We shall show that (S:) holds. Consider the scalar differential equation (3.2.3), which now takes the form u'
+(a),
.(to) = ug
3 0,
t,
2 0,
t
2 to,
whose solutions are easily seen to be u(t, t o , 110)
where
=
I--"J(.,)
-
(t
-
~IJl,
(3.4.3)
and J - l is the inverse function of J . Given any E > 0, we first observe that (S,) holds with S ( E ) = E . Furthermore, we can also conclude from (3.4.3) that ( S z ) is satisfied with 6, = p and T ( E )= J ( p ) - I(€),for any E p. Thus, it follows that the solution u = 0 of (3.2.3) is uniformly asymptotically stable.
.:
3.4.3. Let the hypotheses of Theorem 3.4.1 hold. Suppose further that V ( t ,x) is decrescent. Then, if the solution u = 0 of (3.2.3) is equi-asymptotically stable, the trivial solution of (3.2.1) is equiasymptotically stable.
THEOREM
3.4.
147
ASYMPTOTIC STABILITY
Proof. T h e equistability of the solution x = 0 is immediate from Theorem 3.3.3. T o prove ( S 3 ) ,we follow the proof of Theorem 3.4.1 and choose u,, = u(II x, 11). Then, instead of (3.4.2), we have the inequalities (3.4.4) II xn II < s o ~ ( xo I I 11) < so holding simultaneously. T h e rest of the proof runs almost similar. T h e fact that 8, and T are not independent of to shows that (S,) holds. This and the proof is complete. proves (S5), COROLLARY 3.4.3. tinuous on J and
T h e function g ( t , u ) ! ; A @ ) ds
for every to
---f
= A(t)u,
where
A(t)
is con-
--Go
> 0, is admissible in Theorem 3.4.3.
THEOREM 3.4.4. Let the hypotheses of Theorem 3.4.1 hold, and let V ( t ,x) be decrescent. Then, uniform asymptotic stability of the solution u = 0 of (3.2.3) guarantees likewise the uniform asymptotic stability of the null solution of (3.2.1). Proof. Since uniform stability of the solution x = 0 follows from Theorem 3.3.4, it remains to be shown that (S,) holds. T o do this, we follow the proof of Theorem 3.4.3 and observe that, in view of (3.4.4), 8, is independent of t,. Th at the number T ( E )depends only on E follows from the condition (5'2). Hence, (S,) is satisfied, which, in its turn, proves (S6).
THEOREM 3.4.5. Assume that there exist functions V(t,x) and g(t, u ) obeying the following conditions:
(i) g E C [ J x R, ,R] and g(t, 0) = 0, t E J . (ii) V E C [ J x S,, , R,], V ( t ,0) = 0, t E J , and V ( t ,x) is strongly positive definite and locally Lipschitzian in x.
<
D+V(t, x) g(t, V(t,x)), ( t , x) E J x So . Then, if the solution u = 0 of (3.2.3) is equistable, the trivial solution of (3.2.1) is equi-asymptotically stable. (iii)
Proof. By assumption (ii), V(t,x) is strongly positive definite, which implies that there exists a function b(t, u ) E %Zsuch that
q t , x) 3 b(t, II x II),
( t , x) E
J x
s,.
(3.4.5)
148
Define b,(u)
CHAPTER
=
3
b(0, u). Then
p be given and t , E J . Since (ST)holds, given bl(q) > 0, Let 0 7 to E J , there exists a positive function 6 = 6(t, , q), which is continuous in t , for each 7 such that u,, 6 implies I
<
Choosing uo = V ( t , , xu), we can find a positive function 5, as in the proof of Theorem 3.3.1, such that the inequalities
=
& ( t o , q),
hold together. Furthermore, by Theorem 3.3.1, we see that the solution = 0 of (3.2.1) is equistable, using (3.4.6). Let q be fixed, and let 5, denote the number S,(t, , 7 ) . T o prove (&), let 0 < E < q, to E J be given, and let 11 xu 11 6, . Since b(t, u ) E XZ,there exists a T = T(to,6) satisfying the relation
x
<
+
If (tr.) is a sequence such that t , >, to T , t , -+ m as k + 00, such that I( x(t,,. , t o , xo)ll >, E whenever 11 x, 11 < a,, it would follow from (3.4.5), (3.4.7), (3.4.9), and Theorem 3.1.1 that
This is a contradiction, since b(t, , c) + co as t, + co. Thus, (S,) and (S,) hold simultaneously, and the theorem is proved.
THEOREM 3.4.6. Under the assumptions of Theorem 3.4.5, the uniform stability of the solution u = 0 of (3.2.3) also implies the equi-asymptotic stability of the solution x = 0 of (3.2.1). Pmof. By assumption, 6 is independent of to in the foregoing proof. However, 8, 1 6, is not independent of to because of (3.4.8). Moreover, 7' also depends on t o , in view of (3.4.9), and thus (S,) holds.
COROLLARY 3.4.4. Theorem 3.4.6.
The
function
g(t, u)
E
0
is
admissible
in
3.4.
149
ASYMPTOTIC STABILITY
THEOREM 3.4.7. Assume that there exist functions V ( t ,x), g(t, u),and A(t) satisfying the following properties: A(t) > 0 is continuous for t
(i)
E
J and A ( t ) -+ co as t
---f
co.
(ii) g E C [ J x R, , R ] and g(t, 0 ) = 0, t E J. (iii) V EC [ J x S, , R,], V ( t ,0 ) = 0, t E J , and V ( t ,x) is positive definite and locally Lipschitzian in x.
<
A(t)D+V(t,x)+ V ( t ,x ) D + A ( t ) g(t,V(t, x ) A ( t ) ) ,( t , x) E J x S,. Then, if the null solution of (3.2.3) is equistable, the solution x = 0 of (3.2. I ) is equi-asymptotically stable. (iv)
Proof. Let 0 < 7 < p, to E J be given, and let u = min,,,A(t). By assumption (i), u > 0. Define 7” = ob(7), where b(u) E .X is the same function as in (3.3.1), obtained because of the positive definiteness of V ( t ,x). Assume that (Sf) holds. Then, given 7* > 0, to E J , there exists a 6 = 6(t, , 7 ) that is continuous in t, for each 7 such that 4 4 t o uo) < ?I*,
t
3
<
(3.4.1 1)
2 to,
if ua 6. Choose ua = A(t,) V(t,, x,). Arguing, as in Theorem 3.3.1, we conclude the existence of a positive function 8 = 8(t, , 7 ) such that
/I xnI/ < 8,
~ ( t o ) ~ ( txu> o,
<8
(3.4.12)
hold jointly. With this 8, condition (8,)holds. For otherwise we are led to a contradiction, as in the proof of Theorem 3.3.1,
4 ~ < A(ti)v(ti ) ,~
( f 3, to
xo))
<
ti
t o , uo)
< rt*,
because of the definition of 7” = ob(7) and the application of Theorem 3.1.2. For a fixed 7 < p, designate by So = 6,(to) the number 8(to, 7). Let now 0 T * ,
t
2 to + T .
(3.4.13)
T h e choice u,, = A(t,) &‘(to, x,,), the positive definiteness of V(t,x), and Theorem 3.1.2 give A(t)b(II x ( t , t n , xo)II)
for t 11 x, [I
(3.4.14)
where x ( t , to , xo) is any solution of (3.2.1) such that 6,. If there exists a sequence { t k } ,t,b 2 to + T , t, -+ co as
2 t,, i
< A ( t ) V ( t ,x ( t , fo , xg)) < ~ ( t o, , uo)
150
k
4
j / x,, / /
3
CHAPTER
I
such that 11 x ( t , &t,, , , xo)il 3 E for some solution satisfying 8, , we obtain from (3.4.1 1) and (3.4.14) the inequality
< rl*,
.4(t,c)b(t)
which contradicts (3.4.13). This proves that (S,) holds, and consequently the solution x = 0 is equi-asymptotically stable. T h e theorem is proved.
THEOREM 3.4.8.
Under the hypotheses of Theorem 3.4.7, the uniform stability of the solution ZL 0 of (3.2.3) also implies the equi-asymptotic stability of the trivial solution of (3.2.1). ~
Proof. As in the proof of Theorem 3.4.6, it is easy to see that 6, and T depend on t o , and consequently (8,) holds.
THEOREM 3.4.9. Assume that there exists a function V ( t ,x) enjoying the following properties: and V ( t ,x) is positive definite, decrescent, (i) I/ E C [J x S, , R,], and locally Lipschitzian in x. (ii)
D+V(t,x)
< -+(I]
x li), ( t , x) E J
x S,, , where 4 E X".
Then, the trivial solution of (3.2.1) is uniformly asymptotically stable.
Proof. Let 0 . t p, t,, E J be given. Since V ( t ,x) is positive definite and decrescent, there exist functions a, h E 2"such that ~
h(ll ?L'Il)
< L7(t,).
,< .(I1
2
ll),
s,.
( t ,). fzJ x
(3.4.15)
By Corollary 3.3.3, it follows that (S,) holds. Hence, condition (ii) and Theorem 3.1.3 with g(t, u ) I 0 give the inequality
for t
t,. Designate by 6, the number Q),
and choose T ( E )=
<
u(S,,)/+(S(c)), mhcre S ( E ) corresponds t o E in (&). Suppose that )I x,, 11 6, and that we would have / / x ( t , t,, , x,,)ll S(t) for t,, t t,, T(E).
Then, for t E [t,,, t,,
+ 7 ' ( ~ ) ] ,we get, from (3.4.16),
1 I t , 4 t ) ) -2 q t , , , .")
s 44,)
which, for t 0
=
t,,
~
+ T ( t ) , reduces to
< b(qc))
.L. LTt,,
0.
~
4(S(,))(t
.b(S(,))(t
~
~
< < +
t")
t,,),
+ T ( c ) ,$4, + 7'(€)))i.(So)
-
4(8(E))T(E)
151
3 . 4 . ASYMPTOTIC STABILITY
This contradiction proves that there exists a t , E [ t o ,to that 11 x(t, , t o , xo)\\< 6(e). Thus, in any case, we have
+ T ( E ) ]such
<
whenever (1 xo I( S o , proving the uniform asymptotic stability of the solution x = 0 of (3.2.1). Notice that condition (ii), together with (3.4.15), yields that D+V(t,x )
< -+[u-"V(t, .))I
ESE
-+*(V(t, x)),
where $* E .X, and therefore the conclusion of Theorem 3.4.9 follows right away from Corollary 3.4.2. However, the proof given previously is of interest in other situations.
THEOREM 3.4.10. Assume that there exist functions V ( t ,x), g(t, u) satisfying the following properties: (i) V E C [ J x S, , R,], and V ( t ,x) is positive definite, decrescent, and locally Lipschitzian in x. (ii) g E C [ J x R, , R ] , and, for every pair of numbers 01, p such that 0 < 01 p < p, there exist 8 = 8(01,p) 3 0, K = k(01,p) > 0 satisfying
<
g(t,u)
< -A,
01
< u < p,
t
.> 0.
(iii) D+V(t,x) d g(t, V ( t ,x)), ( t , x) E J x S,, . (iv) For any function X E C [ J ,R,],
Then, the trivial solution of (3.2.1) is uniformly asymptotically stable. Proof. Let x ( t ) = x ( t , t o ,xo) be any solution of (3.2.1). Defining m(t) = I/ x(t)ll, we obtain, from (3.4.17), m;(t)
< I1 x'(t)ll < Ilf(4 x(t))ll < Y t ) m ( t ) .
By Theorem 1.4.1, it follows that
152
3
C'IIAPTER
as far as (1 x(t)(\ p. Consider thc interval [ t o ,T] for some p. Then, since which 11 .z(t)l/ <
Ji0
h(s) cls
w e obtain
T
>, to for
-< j'h(s) ds, 0
I1 4f)ll< I1 X" II e N T ,
[to , TI,
t
(3.4.18)
where N N ( T )= supo A(t). Since V ( t ,,x) is positive definite and ctescresccnt, (3.4.15) holds. Let 0 < c c p , toE J be given. Choose 8, S , ( c ) such that 481) < 4 6 ) . -
-
It is clcar that s1 E. Let 8 : B[a(S,), A(€)], tZ = k[a(S,), b ( ~ ) ] and , 8 ( ~= ) 8,ecN0. Wc choosc I/ xo 11 6 so that (3.4.18) assures that 11 x(t)ll < 6 , , to t B, which implies that
< <
s .
.(I1 4f)ll)< a(S,),
C'(t, t ( t ) )
t"
< t < 8.
We now claim that "(t, x(t)) < b(€),
t
2 8.
If this wcrc not truc, thcrc exist t , , t, such that t,
> t, >, 8, satisfying
qf, , x(tl)) = ~(s,), Tt2
and n(S,)
Hence, at t
--
1
r(t2))
-
b(c),
< " ( t , x(f)) < h ( € ) ,
t
E
[tl , t,].
(3.4.19)
t , , there results I ) ' K(tl , x(tl)) > 0.
(3.4.20)
On the other hand, as t , 2 8 and (3.4.19) holds, we obtain, from conditions (ii) and (iii) and the fact that V ( t ,x) is locally 1,ipschitzian in x, the inequality I ) ' l'(f,
>
dtl))
< A t , , Vtl :< -k < 0 ,
7
x(t1)))
which contradicts (3.4.20), thus proving that V ( t ,x ( t ) ) < A(€) for t 3 8. It therefore follows that, if (1 xl, (1 -'8, V ( t ,x ( t ) ) << b(a), t >, t o , and
3.4.
153
ASYMPTOTIC STABILITY
consequently, in view of (3.4.15), the uniform stability of the trivial solution of (3.2.1) is proved. Let us denote by So the number 6 ( p ) obtained by setting E = p, and let 0 < E < p . Let 6 = S ( E ) be the same function as before. Assume 11 x,,11 < 8,. Choose
where
T o prove uniform asymptotic stability, it is enough to show that there exists a t, E [to , to TI satisfying /I x(t,)ll < S ( E ) ; since it would then follow that 11 x(t)ll < c, t 3 to T in any case. Suppose there is no such t , ; then,
+
+
a(€) G I1 4t)ll G P ,
t
E
[t" t o 9
+ TI,
which, because of (3.4.15), shows that
Using assumptions (i) and (iii), we get
This absurdity proves that 11 x(t)iI < S ( E ) < E , t 3 t, and the theorem is completely proved.
11 x,,11 < 8,,
-t T , whenever
We have seen in Theorem 3.3.5 that the uniform stability of the trivial solution of (3.2.1) can be formulated by means of monotone functions. We may likewise state the following theorem with respect to uniform asymptotic stability.
154
3
CHAPTER
THEOREM 3.4.1 1. T h e trivial solution of (3.2.1) is uniformly asymptotically stable if and only if there exist functions a E X , (r E 9such that
I/ . ~ ( tto, , xo)ll for
1) xo I/
< 41xn I l ) ~ ( f
-
t
to),
2 to >
(3.4.21)
-, p.
Pyoof. If (3.4.21) is satisfied, it is easy to verify that (S,) holds, and hence sufficiency of the condition is evident. Assume now that the trivial solution of (3.2.1) is uniformly asymptotically stable, so that (S,) holds. Let { e n } be a positive, monotonic sequence, converging to zero as n + GO. Let T , ( E )= inf T ( C ? ~ + ~for ) en+,
Then,
I/ X" I/ < 6 0 ,
t
I1 .x(t, to , xojll < [E,,+~,
en],
TI(€,),
E
E,
.
> t" iTI(€)
assures that
Define T*(e) linear in T,(E,, , l ) . Note that
< <
<
T*(E,+,)
6.
T l ( ~ 1 1 +and 2 ) , T*(r,)
:
lim ?'](en)
n-r
=
03.
T h e equality T1(~,,+l) = may occur on finite parts of the sequence which can be eliminated. T h e function T*(e) is continuous, monotone decreasing, and lim6-,oT * ( E )= CO. Moreover, t 3 t,, T*(e) implies t
Hence, if
entl
< < E
5
to i- T*(%)= t"
E,,
, we have
+ Tl(%I+l).
+
This shows that the function T(e) occurring in (S,) can be chosen continuous and monotonic. Let q ( T ) be the inverse function of T ( E ) . Then, it follows that
3.5.
155
STABILITY OF PERTURBED SYSTEMS
whenever I/ x, 11 < 6, , which proves that uniform asymptotic stability is equivalent to (3.4.21).
3.5. Stability of perturbed systems We shall consider the perturbed system
where j ; R
E
C [J x S, , Rn].
THEOREM 3.5.1.
Let there exist a function V ( t ,x) satisfying the following conditions:
(i) V E C[J x S, , R,], V(t,x) possesses continuous partial derivatives with respect to t and the components of x, and
for ( t , x) E J x S o . V(t,x) B [I x (la, A , B being positive constants. (iii) A 11 x 114 (iv) w E C [ J x R, , R,], w ( t , 0) = 0, w ( t , u) is monotone nondecreasing in u for each t, and
<
<
11 R(t, x)ll
< w ( t , II x 1 ").
Then, the stability properties of the trivial solution of (3.2.3), with U"
L + All w(t, ~
Urn),
imply the corresponding stability properties of the solution x
(3.5.1). Proof.
Consider the function U ( t ,x)
=
(3.5.2) =
0 of
156
CIIAPTER
3
In view of thc assumptions of the theorem, if we define
we obtain, after some computations, that
< U ( t ,x) < Bl'Q/I x 11,
I/ 2 /I
U ' ( t ,). ,< 'dt, V t , x)),
.>
( t ,x) €
J x so,
(t,
J
'R
>
where ~ ( tu ,) is the same function given by (3.5.2). T h e conclusion of the theorem is then a direct conseqeunce of Theorems 3.3.3, 3.3.4, 3.4.3, and 3.4.4. COROLLARY 3.5.1. Let the assumptions of Theorem 3.5.1 hold with 2, m = 1, and w ( t , u ) = X(t)u, where h(t) 3 0 is continuous on J such that N =
Then, the solution x
=
0 of (3.5.1) is asymptotically stable.
THEOREM 3.5.2. Let there exist functions V(t,x), g l ( t , u), and w(t, u ) fulfilling the following conditions: (i) V t C [ J x SR, R ,], V ( t ,x) is Lipschitzian in x for a function k ( t ) 3 0 continuous on J , and
411x II)
< L'(t, x) < 411x II),
t , x) E J
x s, ,
(3.5.3)
where a, h E f . (ii) g,
E
C [ J x R, , R],g,(t,0)
= 0, and
n + l 7 ( t 4? ( 3 2.1) < gl(t? f7(k x)),
( t ,x) E J x
s,
1
(iii) w E C [ J x R, , R,], w(t, 0) = 0, w ( t , u ) is nondecreasing in u for each t, and I / R(t,.y)II
< w ( t , II x II)?
( t ,).
E
Then, the stability properties of the solution u
At, ).
= Rl(t>).
J x s,. =
0 of (3.2.3) with
4- k ( t ) w ( t ,W U ) ) ,
(3.5.4)
where b ' ( u ) IS the inverse function of b(u), imply the same kind of stability properties of the trivial solution of (3.5.1).
3.5.
157
STABILITY OF PERTURBED SYSTEMS
Proof. Let us define the function D+V(t,x) with respect to the perturbed differential system (3.5.1) as follows:
using assumptions (ii) and (iii). This, together with (3.5.3) and the monotonicity of w ( t , u ) in u, leads to the differential inequality D'
v(t,
X)(:5.5.1)
< g(t, V ( t ,x)),
( t ,).
E
J
x s, ,
where g(t, u ) is given by (3.5.4). Now, it only remains to apply Theorems 3.3.3, 3.3.4, 3.4.3, and 3.4.4 to get the desired result. COROLLARY 3.5.2. T h e functions b(u) = u,gl(t, u ) = -au, a > 0, and w(t, u ) = X(t)u, X ( t ) 3 0 being continuous on J and satisfying
are admissible in Theorem 3.5.2, to guarantee the uniform asymptotic stability of the solution x = 0 of (3.5.1) provided k is the Lipschitz constant for V ( t ,x). COROLLARY 3.5.3. T h e functions w ( t , u ) = h ( t ) $ ( u ) , gl(t, u ) = 0, where X E C [ J ,I?,], 4 E X , are admissible in Theorem 3.5.2 to yield the uniform stability of the trivial solution of (3.5.1)) provided that k is the 1,ipschitz constant for V(t,x) and :J- X(s) ds < GO. COROLLARY 3.5.4. T h e functions w(t, u ) = h(t)$(u), gl(t, u ) = -C(u), where h E C [ J ,I?+], 4,C E .Y, are admissible in Theorem 3.5.2 to assure that the trivial solution of (3.5.1) is uniformly asymptotically stable, provided that Fz is the Lipschitz constant for V ( t ,x), k 4 ( k 1 ( u ) )< ,C(u), for some 01 > 0, and limt+m[-t 01 h(s) ds] = -a for all to 3 0.
+ J-7'
158
CIIAPTER
3
3.6. Converse theorems This section will be devoted to a variety of results concerning the construction of Lyapunov functions. Let us first define the notion of generalizcd exponential asymptotic stability. DEFINITION 3.6.1. T h c trivial solution of (3.2.1) is said to be (,Sll) generalized exponentially asymptotically stable if
where K ( t ) > 0 is continuous for t E J , p E X' for t E J , and p ( t ) + GO as t + co. I n particular, if K ( t ) = K > 0 , p ( t ) = at, a: > 0. We have the exponential asymptotic stability of the trivial solution of (3.2.1)
THEOREM 3.6.1. Assume that the solution x = 0 of (3.2.1) is generalized exponentially asymptotically stable and thatf(t, x) is linear in x. Suppose further that p'(t) exists and is continuous on J . T h e n there exists a function V ( t ,x) satisfying the following properties: (i) V E C [ J x S o , R,], and V ( t ,x) is Lipschitzian in x for the function K(t).
<
<
V ( t , x) K ( t )/I x 11, ( t ,x) E J x S, . (ii) 11 ?L^ 11 D + V ( t , x) -p'(t) V ( t ,x), ( t ,x) E J x S, . (iii) Proof.
<
Define
Then, from (3.6. l ) , it follows that (ii) is satisfied. Let s, y E So . Then,
3.6,
159
CONVERSE THEOREMS
To arrive at this estimate, we have used the facts thatf(t, x) is linear in x and that the solutions x(t, t o , x,,) obey the inequality (3.6.1). We shall now prove that V ( t ,x) is continuous. Let 6 >, 0; then
+
Since V ( t ,x) is Lipschitzian in x and x( t 6, t , x) is continuous in 6, the first two terms on the right-hand side of the preceding inequality are small when /I x - x* I( and S are small. Let us consider the third term. Observe that x(t
+ 6 + u, t + 6, x ( t + 5,t , x))
= x(t
+6 +
t , x).
0,
Hence we have
we notice that a(S) is nondecreasing and tends to a(0) as S + 0, since I\ x(t u, t , x)ii exp[ p ( t D ) - p(t)] is a bounded continuous function for all u 2 0. Thus,
+
+
implies that the third term tends to zero as 6 the continuity of V ( t ,x).
+ 0.
T h u s we have verified
160
CHAPTER
3
Furthermore, using the uniqueness of solutions and the definition (3.6.2), 1
D+T-(t,x ( t ) ) = lim sup - [I'(t -1- h, x ( t ih, t , .)) h-0F h =
lim sup h.0'
=
lim sup A-0'
1 ~
12
1
h
~
[sup / / x ( t -t- h 030
sup
[I/ x(t
U ' h
-4-
u, t
11
x {..p(p(t)
<
-
V ( t ,x)]
+ h, x ( t + h, t , %))I1
+ u, t , x)ii e x p ( p ( t -1-
< lim sup 1 [sup 1 1 .z.(t + u, t , )I./ rO &.O+
~
u)
c s p ( p ( t -t u)
~
~
p(t
+ h))
p(t))
0
-
p ( t -1- h ) )
~
I}]
p ' ( t ) l ' ( t , ,x).
Since, for small h > 0, r-(t 7-h, 2
+ hf(t, x))
~-
+ h, t , x) s - hf(t, .)/I r q t 4-h, s(t + h, t , x)) V ( t ,x),
< K ( t )I/ .x(t
l-(t,Y)
-1-
~
-
it easily follows that fl' l'(t, x)
' \
~
p'(t)V(t,x),
proving (iii). T h e theorem is completely proved.
A similar result is true even when f ( t , x) is nonlinear in x provided it is assumed to satisfy a Lipschitz condition in x for a constant L. T h e next theorem substantiates this remark.
THEORE~I 3.6.2.
1,et the trivial solution of (3.2.1) be generalized cxponcntially asymptotically stable and f ( t , x) satisfy a Lipschitz condition in x for a constant 1, = L ( p ) > 0. Assume that p ' ( t ) exists and is continuous for t E J . Suppose further that K ( t ) is bounded and, for some q, 0 ,. q 1, there cxists a number T > 0 such that <.
K ( t )c x p [ - q ( p ( t I T ) - P ( t ) ) l
< 1,
t
E
1.
(3.6.3)
3.6.
161
CONVERSE THEOREMS
Then, there exists a function V(t,x) fulfilling the following conditions:
(i)
V E C [ ] x SDo, R,], 0 < p0
I U t , 4 - V(t,Y)l (ii) (iii) Proof.
< eLT0 sup
.o-’T
< p, and V ( t ,x) obeys
exp[(l
~
mJ(t
+
0) -
P(t))lll x - Y 11.
< v(t,x) < K ( t ) I! x ll, (t, x) E J x So0. P V ( t , x) < -(1 - q)p’(t) V ( t ,x), ( t , x) E J x !I x II
SDo .
Let q, T be given so that the relation (3.6.3) holds. Define V ( t ,x )
=
sup /I x(t --I- (I, t , x)ll exp[( 1 - p){p(t uao
+
0) -
p(t))]. (3.6.4)
Since K ( t ) is assumed to be bounded and p ( t ) E X , the function V(t,x) is defined for ( t ,x) E J x SD,, where p0 = p / M , M = supLGJ K(t). T o prove that V(t,x) satisfies the Lipschitz condition, we first observe that
and, since (3.6.3) holds, we have
Accordingly, by Corollary 2.7.1, we get
T h e relations (ii) and (iii) can be verified as in the proof of Theorem 3.6.1. T h e proof is complete.
COROLLARY 3.6.1. If the trivial solution x = 0 of (3.2.1) is exponentially asymptotically stable, that is, p ( t ) = at, CY > 0, K ( t ) = K > 0, then T = log Klqct, and M = KL+(-q)a/qa, where n/l is the Lipschitz constant for V ( t ,x), are admissible in Theorem 3.6.2. It is possible to prove the previous theorems, under milder assumptions, in a different way.
162
3
CHAPTER
THEOREM 3.6.3.
Assume that
(i) f E C [ J x S, , R”],f(t,0) = 0, andf(t, x) satisfies
< L ( t ) I/ x
IIf(t, x ) -f(t,Y)II
-y
11,
( t , x ) , ( t ,y ) E
1X
L(t) 0 being continuous on J ; (ii) There exists a p E X’ for t E J , p ( t ) 4 co as t exists, and
11 x ( t , 0, xo)ll
< K II xo II exp[-P(t)l,
t
S,
, (3.6.5)
-+00,
2 t o , K > 0,
p’(t)
(306.6)
where x(t, 0, xo) is the solution of (3.2.1) through (0, xo). Then, there exists a function V ( t ,x) enjoying the following properties: (1) V E C [ J x S o ,R,], and V ( t ,x) is Lipschitzian in x for a continuous function K ( t ) > 0.
<
(2) I1 x I1 G V(t,4 K ( t ) I1 x 11, ( t , x) E J x s, . ( 3 ) Di-V(t, x) -p’(t) V ( t ,x), ( t , x) E J x S , .
<
Pmof. Let us denote x = x(t, 0, xo) so that xo = x(0, t , x), because of the uniqueness of solutions of (3.2.1), which is assured by condition (i). We now define V ( t ,x) = Ke-fi(t)11 x(0, t , %)I\. (3.6.7) I t is then evident that V E C [ J x S, , R,], V ( t ,0) of (3.6.6), we have II x II < q t , x).
= 0,
and, because
Since the solutions of (3.2.1) are unique, it follows that V(t
+ h, x ( t + h, t , x)) = Ke-p(t+h)jl x(0, t + h, x ( t + h, t , .)I1 -
Ke-P(t+h’I/ 40,t , x)IL
and hence we obtain
=
+(t)V(t, x).
If x(t, 0, xo), x(t, 0, yo) are the two solutions of (3.2.1) through (0, xo), (0, yo), respectively, the condition (3.6.5) yields
3.6.
I63
CONVERSE THEOREMS
by virtue of Corollary 2.7.1. Letting x = x(t, 0, xo), y get, from the preceding estimate, the inequality
=
x ( t, 0, yo), we
Consequently,
= K ( t )/I
* - Y /I)
+
if we define K ( t ) = K exp[-p(t) l:L(s) ds]. This proves that V ( t ,x) satisfies a Lipschitz condition in x for a function K ( t ) > 0. T h e upper estimate in (2) follows from the Lipschitz condition by setting y = 0 and observing that V ( t ,0) = 0. As in the proof of Theorem 3.6.1, one can deduce (3) from (3.6.8). T h e theorem is proved.
3.6.2. If in Theorem 3.6.3, the functions L(t) and p ( t ) COROLLARY are such that J:L(s) ds p(t), and V ( t ,x) satisfies the Lipschitz condition in x for a constant K > 0. We have already seen that the concepts of stability and asymptotic stability can be defined by means of simple inequalities involving certain monotone functions. We give below some converse theorems in terms of differential inequalities. As will be seen, the approach depends upon the differentiable properties of solutions with respect to the initial values and yields, in a unified way, a method of constructing Lyapunov functions.
<
THEOREM 3.6.4. Suppose that (i) the function j~ C[J x So,R"],f(t,0) exists and is continuous for ( t , x) E J x So; (ii)
= 0,
PI,p2 E Z such that < I1 x ( t , 0 , < PA1 xo ID,
and af(t, %)/ax
there exist functions
A(11xo 11)
X0)ll
tE
I,
(3.6.9)
where x(t, 0, xo) is the solution of (3.2.1) through (0, xo); (iG) the function g E C [ J x R, , R],g(t, 0) 3 0, and ag(t, a ) / & exists and is continuous for (t, u ) E J x R, ;
164
CHAPTER
3
(iv) the solution ~ ( t0,, uo) of (3.2.3) fulfills the estimate
< 44 0 , U,,) < Y 2 ( U o ) ,
Yl(U,)
t
E
J,
(3.6.10)
where y 1 , y 2 E .T. Then, there exists a function V ( t ,x) with the following properties: ( 1 ) V E C [J x S, , K,], and V ( t ,x) possesses continuous partial derivatives with respect to t and the components of x.
(2) h(lI x 11) (3)
< V ( t ,x) < 411x II), =g(t,
( t , x) E
J
q t , x)), ( t , x) E
x S, , a, b E z-.
J x so.
Proof. I n view of hypothesis (i), the existence and uniqueness of solutions of (3.2.1), as well as their continuous dependence on the initial values, is assured. Also, the solutions x ( t , t, , x,), ( t o ,x,,)E J x S, are differentiable functions with respect t o the initial values, and x = 0 is the trivial solution. Furthermore, on the basis of Theorem 2.5.3, we have and
where @(t,to , x,,) = Bx(t, t o , xo)/axois the fundamental matrix solution of the variational system
such that @(t,,, t, , xo) is the unit matrix. Similar conclusions hold for the solutions of (3.2.3) as (iii) is satisfied. Let now x ( t , 0, xJ, u(t, 0, u,) be the solutions of (3.2.1), (3.2.3) through (0, xo), (0, uo), satisfying the inequalities (3.6.9) and (3.6.10), respectively. Denote x(t, 0, x,,) by x so that x,,= x(0, t , x). T h i s is clear by virtue of the uniqueness of solutions of (3.2.1). Choose any continuous function p(x) possessing continuous partial derivatives ap(x)/ax for x E S, such that rnl(I1
II)
< d x ) < %(/I
x II),
Define the function V ( t ,).
=
u[t, 0 , &(O,
a1
7
t , .))I.
a2 E
x-0
(3.6.12)
3.6.
165
CONVERSE THEOREMS
Because of the continuity of x(0, t , x), p(x), and u(t, 0, u,) with respect to their arguments and the fact that u = 0 is the trivial solution of (3.2.3),
it follows that V E C [ J x Y, ,R,]. Since the functions x(0, t , x), u(t, 0, u,) are differentiable with respect to the initial values and ap(x)/ax is assumed to exist, we see that
ax(o, t , at,
+
Thus, (1) and (3) hold. Since x = x ( t , 0, x,) and x, (3.6.9) that
P;l(l
x
II)
ax(o, t , x) . f ( t , x) ax,
=
= 0.
x(0, t , x), it follows from the inequality
< /I 407 t , 4 < P;l(l
x ll),
(3.6.13)
where Pr', / 3 ~ lare inverse functions of PI , p2 and hence belong to X . Using the inequalities (3.6.10), (3.6.12), and (3.6.13) successively, the definition of V ( t ,x) gives J q t ,x)
=
u[t,0, p(x(0, t , 411
2 Yl(cL(X(0,t , 4)) 2 Y d 4 1 40,t , x)II)) 2 Y d ~ l ( P i l(11 x 11)) = b(11 x 11) and
166
3
CHAPTER
Obviously, the functions a, b E 37,since the functions y1 , 0 1 ~,pz', y z , a 2 , and P;' all belong to class ,X. This proves (2), and the proof is complete. COROLLARY 3.6.3. Assume that f E C[J x S, ,R"],f ( t , 0) = 0, and af(t, x)/ax exists and is continuous for ( t , x) E J x S,. Suppose also that, for any solution x(t, 0, x,,) through the point (0, x,,) of (3.2.1),
Then, there exists a function V ( t ,x) such that: (1)
V E C [ J x S, , R,], V ( t ,0)
0, and V ( t ,x) possesses con-
=
tinuous partial derivatives with respect to t and the components of x;
(2) b(ll x 11) (3)
< V(t,x),(t,). V ( t ),.
E
x So 6 E ,x;
J
1
a v t , ).
= ____
at
= 0,
(b,
+
av(t7 4 . f ( t , ). a x-
x) E J x S " .
T h e following variant of Theorem 3.6.4 is of interest in some situations.
THEOREM 3.6.5.
Let
(i) f~ C [ J x S, , R"],f ( t , 0) Ilf(t!
=
-f(t,y)ll
0, andf(t, x) satisfies
< Ldt) /I x
-
Y II
for ( t , x), ( t , y ) E J x S, , where L,(t) 3 0 is continuous on J ; (ii) there exists a function p2 E .X such that
II x(t, 0, %)I1
< P,(ll
xo
10,
t
2 0,
x(t, 0, xo) being the solution of (3.2.1); (iii) g E C [ J x K, , R],g ( t , 0) 1 0, and g(t, u ) verifies
I At, 4 - g(f7 v)l
< L,(t) I
zL - 7l
I
for t E J , u, ZI >, 0, L,(t) 2 0 being continuous on J ; (iv) the solution u(t, 0, u,,) of (3.2.3) fulfill the estimate n(.o)
< 4 t >0, a,),
t
2 0,
Y1 €37.
3.6.
167
CONVERSE THEOREMS
Then, there exists a function V(t,x) with the following properties: (I) V E C [ j x S, , R,], and V ( t ,x) is positive definite and satisfies a Lipschitz condition for a continuous function K ( t ) 3 0.
<
D + W , x) g(t, V ( t ,XI), ( t ,4 E J x So. Proof. Since the uniqueness of solutions, as well as their continuous dependence on initial values, is guaranteed by (i) on the basis of Corollary 2.5.1, if we let x = x (t, 0, xJ, it follows that xo = x(0, t , x) as previously. We define
(2)
V ( t ,4 = u(t, 0, !I 4 0 , t , x)ll),
where u(t, 0, uo) is the solution of (3.2.3) through (0, u,,). Note that, by assumption (iii), u = 0 is the trivial solution of (3.2.3) and that the solutions u(t, t o , u,,) are unique. Furthermore, we infer from uniqueness of solutions of (3.2.1) that V(t
+ k , x ( t + k , t , x))
=
=
and therefore
+ k , 0, /I 40, t + k , x ( t + h, t , x))ll> u(t + k , 0, /I 40, t , u(t
1 D+V(t,x ( t ) ) = lim sup - [ V ( t &+Of k
X)II)>
+ k , x ( t + h, t , x))
-
V ( t ,x)]
= u’(t,0 , !I 40,t , .)!I) = g(t,
V t , XI).
According to Corollary 2.7.1, if we let x = x(t, 0, x,,), y condition (i) implies, as in Theorem 3.6.3, the inequality
Moreover, assumption (iii) also implies that
Thus, for ( t , x), ( t , y ) E J x S o ,we have
(3.6.14) =
x(t, 0, yo),
168
CHAPTER
3
+
if K ( t ) = exp[Ji (L,(s) L2(s))ds]. This, together with (3.6.14), enables us to deduce, as in Theorem 3.6.1, that
where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20), it follows that
COROLLARY 3.6.4. Theorem 3.6.5.
The
function
g(t, u) = 0
is
admissible
in
T h e next theorem deals with the converse problem for asymptotic stability.
THEOREM 3.6.6.
Let assumptions (i) and (iv) of Theorem 3.6.4 hold. Suppose that the solution x ( t , 0, xo) of (3.2.1) satisfies
< P,(I
II x ( t , 0, .T")ll
t
X" Il)o(t),
where p2 E X , c E 9'. Assume that there exist functions y such that y'(u)
3K
b
0,
and r(u")qt)
t
E
.X, 6 E 9
K, > 0,
8 ( t ) 3 klU(t),
< u ( t , 0, uo),
(3.6.15)
2 0,
3 0.
(3.6.16)
Then, there exists a function V(t, x) satisfying V ( t ,0 ) 0, (1) V E C [ J x s, , R,], V ( t ,x) is positive definite and possesses continuous partial derivatives with respect to t and the components of x; :
(2)
V'(4
x) = g(t,
V t , x)),
(4 x) E
1 x so.
Pmof. Let x ( t , 0, x,,), u ( t , 0, uo) be the solutions of (3.2.1), (3.2.3) obeying the estimates (3.6.15), (3.6.16), respectively. Choose any
a
continuous function p(x) having continuous partial derivatives ap(x)/ax for x E S,, such that y(0) = 0 and
Bz(ll x II)
Defining
V(4 ).
(3.6.17)
W
= u [ t , 0, P ( X ( 0 , t ,
4)1,
it can be readily shown as in Theorem 3.6.4 that V E C [ J x So,R,] and satisfies (2). Moreover, V(t,0) = 0 follows from x(0, t , 0) = 0, R > 0, y(0) = 0, and u(t, 0,O) = 0. From the assumption y’(u) there results Y(u1uz) ku1uz. (3.6.18)
z
Furthermore, by virtue of the fact that x the inequality (3.6.15) yields
=
x(t, 0, xo) and xo = x(0, t, x),
(3.6.19)
where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20),
which implies, on account of the assumption 8 ( t ) is positive definite. T h e proof is complete.
2 Rla(t), that V(t,x)
COROLLARY 3.6.5. If u E 2 is a differentiable function for t E J , then the function g ( t , u ) = [a’(t)/u(t)]uis a candidate in Theorem 3.6.6. REMARK 3.6.1. Notice that, in Theorems 3.6.5 and 3.6.6, we have not assumed that the trivial solution of (3.2.3) is stable and asymptotically stable, respectively, since we do not need, in the proof, such
170
3
CHAPTER
a specific assumption. However, these hypotheses are required to prove
direct theorems. Nevertheless, the lower estimates of the solutions ~ ( t0,, u0) of (3.2.3) are compatible with the corresponding stability requirements. It can be seen from the proof of Theorem 3.6.4 that the lower estimate on x ( t , 0, xo) and the upper estimate on u(t, 0, uo) are useful only to prove the dccrescent nature of V ( t ,x). Observe also that we need only the stability information of solutions starting at to = 0, and this is a definite advantage. Undcr the rather general assumptions of Theorem3.6.6, it is not possible to show that. V ( t ,x) is decrescent. This can, however, be achieved in the following: 'rHE O R EM 3.6.7. Let assumptions (i) and (iv) of Theorem 3.6.4 hold. Suppose that, in place of (3.6.15), we have
Plll %J Y 4 t ) < I1 x ( t , 0, X")ll
< All
where R, p1 , p2 > 0 are constants, and of (3.2.3) allow the estimate 4u"qt)
where A, , A,
p > 0,
G
E
Xo
t >, 0,
Ilao(t),
2. Let the solution u(t, 0, uo)
< 4 t , 0, uo) < huo8(t),
t
z 0,
> 0 are constants, and 6 E 9 such that,
for some constant
syt) = U B ( t ) .
Then, there exists a function V ( t ,x) that is decrescent and that obeys (I), (2) of Theorem 3.6.6.
Proof.
By choosing a continuous function p ( x ) so as to satisfy
kill X /I0 < ("(x)
< k,lI
.^c
k , I R,
I T 3
I
B
0,
and following the proof of Theorem 3.6.6 with necessary modifications, we can easily construct the proof of this theorem.
THEOREM 3.6.8. Let assumption (i) of Theorem 3.6.4 hold, and let there exist functions u l , u2 E 9 such that
PlII xo / / ~ i (-t to)
< /I
4)
~ ( t l
~o)ll
< Bzll
Xo
IluAt
-
to),
t
> to
(3.6.20)
p,, p2 > 0 being constants and x ( t , t o ,XJ being the solution of (32.1). Then, there exists a function V ( t ,x) satisfying the following with
properties:
3.6.
171
CONVERSE THEOREMS
(1) V E C [ J x S, , R,], and V ( t ,x) is positive definite, decrescent, and possesses continuous partial derivatives with respect to t and the components of x. -eV(t, x), N > 0, ( t , X) E J x S o . (2) V'(t,X)
<
Proof.
Define the function, for some fixed T
> 0 that we choose later,
Because of assumption (i), one can argue, as before, to show that V E C [ J x S, , R,] and V ( t ,x) is continuously differentiable. Furthermore, from the upper estimate of (3.6.20), we have
and hence
(3.6.21)
where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20), it follows that
Thus, we have shown that (1) is verified. To prove (2), observe that
O n the other hand, using relation (3.6.1 l), we see that
Consequently,
172
CHAPTER
3
We now fix T by choosing it so large that
This is possible, since upE 2.Evidently, from this choice results the inequality V ' ( t ,x ) < - g 11 x 112, which, in view of (3.6.21), leads to
_-=
setting
&i=
~
d ( t ,x ) ,
1/[2P(T)]. T h e theorem is proved.
COROLLARY 3.6.6. condition
Instead of the lower estimate in (3.6.20), the llf(t,
1)I.
< -%)I1
ZL'
I/,
( t ,x ) E
I
x
s,
is admissible in Theorem 3.6.8.
TIIEOREM 3.6.9. Let the trivial solution of the system (3.2.1) be uniformly asymptotically stable. Suppose that
Ilf(t, XI)
-f(t,
%>ll
< L(t)/l
x1
for ( t ,q),( t , xg)E J x S, , where L(t)
~
xz
I1
0 is continuous on J , and
Then, there exists a function V ( t ,x) with the following properties:
, V ( t ,x) is positive definite, decrescent, (1) V E C [ J x S o ,R ! ] and and satisfies 1
v(t,.TI)
Y t , %)I
< MI1x1
-
% II
for ( t , xi), ( t , EJ x . D + V ( t , x ) C ( V ( t , x ) ) , ( t , x )J~ x S,,CE%. (2) C y J
<
AJls(s,)
Pyoof. Let us choose a function G(r) such that G(0) G ( Y )> 0, G"(r)> 0, and lct cx > 1. Since G (T)=
J: du
G"(v)dv 0
=
0, G'(0)
=
0,
3.6.
173
CONVERSE THEOREMS
and
we have, setting u
w/a,
=
( 1) = - i: dw [w'm G"(v)dv
G-
0
<1 [dw f"G"(v)dv 0 1 0
1
= - G(y). 01
0
(3.6.22)
Define
(3.6.23)
If ~(8) is the inverse function of
a(€),
I1 x(t + 0
and therefore
G(ll x ( t
+
U>
Consequently, observing that ( 1
we have
t,4 11 < 41* 11))
1
t , X)ll)
< G(4l * 11))-
+ ma)/( 1 + a) < a, it follows that
v(t*x) < m G ( 4 x 11)).
Since u
> T ( E )implies 11 x ( t + u,t , *)I1 I1 x (t
if
u
2 T(lj x li/a). Thus, G(ll x(t
which, in turn, leads to G(I/4 t
+
D?
t >.)lo
+
D7
t,
< E,
we get
x>ll < II x llb
+ u>t?x)II) < G(ll x IIbh 1+ I +o< olG *I( 01u
< G(ll x 11)
< V(t,21,
because of relations (3.6.22) and (3.6.23). This shows that
1 74
3
CHAPTER
T h e continuity of the function V ( t ,x) implies that there exists a such that v(t,4 = G(llx(t 0 1 t , "411)1 a(J1.
+
If we let x = x(t, to , xJ, x* of (3.2.1) shows that V(t
Denote cr* 1
1
+h
+ au*
+
+ h, x * )
u*
=
-
-
(5.
(1 (1
1
x(t
=
+
7
+ h, t, x), the uniqueness of solutions
+ + u*, t + h, x*) I )
G(/I~ ( t h
Then
+ au*)(l +
+ U*)(l +
0) -
-
1
(a
(1
-
1 +a,* 1 +u*
+ u* + au + auu* - ah + h
0)
+ au
u1
+ o*)(l + ).
1-
1)h
1 +o
It therefore follows that
< V ( t ,x) [I
-
(a
(1
+ u*)(l -
I)h +am)
1'
using (3.6.23), it is easy to obtain D+V(t,x ( t ) )
=
1 lim sup - [ V ( t h-Q+ h
+ h, x ( t + h, t , x))
-
V ( t ,x)]
(3.6.24)
3.6.
because of the fact that limb+*+ I( x* 11 function. We have seen previously that
where
11 x2 /I < S(p),
175
CONVERSE THEOREMS
=
(1 x (1 and that T ( E is) a decreasing
be such that
so that the solutions x ( t , to , xl), x ( t , to , x2) remain in
Let
s,.
If r2 3 rl , we have G(r,) 3 G(r& and hence
On the other hand, if r2
(3.6.25) Since f(t, x) satisfies the Lipschitz condition, by Corollary 2.7.1, we obtain
Furthermore, choosing
the relation (3.6.25) leads to
using the monotonic decreasing character of T ( E ) It . follows from these relations that
176
CHAPTER
3
and thus that
,,
V ( t ,xl) - aAl/ XI
~
x2 11.
These considerations show that, in all cases, V ( t ,xZ)
-
V ( t ,xl)
2
-
o~i4/lx1 - x2 11.
By interchanging the roles of x l ,x2 , we obtain V ( t ,XI) - V ( t ,x2) 3 --olR/l XI - x2 I/,
and therefore there results (3.6.27)
provided xl,x2 f 0. If xg = 0, (3.6.26) yields 0
< V ( t ,21) < 4 1x1 11,
and hence (3.6.27) is true even when x2 = 0. If x1 = x2 = 0, the relation (3.6.27) is trivially satisfied. As previously, it is now easy to obtain (2) from relations (3.6.24) and (3.6.27) and the descrescent character of V ( t , x). Finally, it remains to prove that we can choose G(r) satisfying the required conditions. For this purpose, we may take G(r) == A
J: exp [-KT(!E1)]
dr.
One can easily verify that G(0)
=
0,
G’((Y)= A exp[--KT(S(r)/a)
> 0,
G’(0) = 0,
0, T(0) = a,G’(r) is monotone increasing, and thus since 6(0) G”(r)exists almost everywhere and is positive. T h e proof is complete. Although we have used Theorem 3.6.9 only t o consider stability properties of perturbed systems, we give below a result that makes such a treatment easier.
THEOREM 3.6.10.
Under the assumptions of Theorem 3.6.9, there exists a function w(t, x) satisfying (1) and D+w(t, x) -w(t, x).
<
3.7.
STABILITY BY THE FIRST APPROXIMATION
177
Proof. By Theorem 3.6.9, there exists a function V ( t ,x) such that (1)
and (2) hold. Without loss of generality, we may assume that C(u) is differentiable, C’(0) = 0, and J ds/C(s) = GO. If C(u) does not have 0 these properties, we can choose such a function C,(u) satisfying C,(u) C(u). Consider the function
<
A(.>
=
exp
[J,
ds
for
o < u < p.
It is clear from the properties of C(u)that X(0) = 0 and X’(u) exists and is continuous on 0 u < p, so that I X’(u)i < K. We now definc the desired function w(t, x) by
<
w ( t , x)
= h(V(t,x )).
It is easy to check that this function verifies the required properties. T h e proof is therefore complete.
3.7. Stability by the first approximation Let x,(t) be a solution of (3.2.1). Set y = x -- x,(t), and obtain the equation y’ = x’ - x’( 0 t ) = f ( C 4 - f ( t , xo(t)) =f -
( 4Y
+ xo(t))- f ( 4
ax
xo(t))
+ O(l] y 1 ).
A natural question is whether we can legitimately neglect the terms of the form O(1l y 11). I n the theory of stability by the first approximation, this procedure is justified. T h e following theorem is to that effect.
THEOREM 3.7.1. Suppose that the trivial solution x = 0 of (3.2.1) is exponentially asymptotically stable and f ( t , x) in (3.2.1) is linear in x. Assume further that the function R(t, x) in (3.5.1) satisfies the relation
<
/I R ( t ?)1.I
< Clt x I/
(3.7.1)
p, C being a sufficiently small constant. Then, the trivial for 11 x /I solution x = 0 of (3.5.1) is exponentially asymptotically stable.
Proof. By Theorem 3.6.1, there exists a function V ( t ,x) having the following properties: (i) V EC[/ x S o ,R,], and V ( t ,x) is Lipschitzian in x for a constant K > 0.
178
CHAPTER
3
D+I/(t, X ) ( d . S . l ) G D 'l'(t> 4 b . 2 . 1 )
+ KIIR(t,
(3.7.2)
X)IL
using the fact that V ( t ,x) is Lipschitzian with a constant K . Define m(t) I.'(t, x ( t ) ) ,where ~ ( tis) any solution of (3.5.1) such that 11 xo 11 < &/K. Because of (ii), whenever /Ix,I/ < i p / K , we have m(to) < &p. We claim that m ( t ) < p for t >, t o . If this is not true, there exist numbers t , and t, such that ~
and Thus, D+m(t,) >, 0. On the other hand, since m ( t ) we have, by (ii), that
< P,
II 4t)Il
t"
< p for to < t < t , ,
< t < tl .
Hence, using condition (iii), together with (3.7.1) and (3.7.2), leads to
DWt,)
< -am(t,) + KCIIx(t,)ll = m(t,)[-a
<
+ KC],
because /I x I/ V ( t ,x). Since C is sufficiently small, there exists a > 0 such that C ( N - y ) / K . This implies that D+m(t,) < 0, and this contradiction provcs that m ( t ) < p for t 3 t o . Consequently, (1 x(t)i/ -1p for t >, t o . Thus, whenever 11 xo (1 < & p / K , we have y
<
D+q t , x ( t ) ) < -rV(t, x(t)), and therefore, by Theorem 1.4.1, V ( t ,. x ( t ) )
< V(to
?
xu) exp[-y(t
-
tdl,
t >, t,
*
It is easy to obtain from this inequality a further inequality
/I x ( t , t" > XdIl
< P expl--r(t
~
to)],
t
2 to
9
which proves the stated result. T h e proof of the theorem is complete. T h e linearity o f f ( t ,x) in x can be dropped, iff(t, x) satisfies a Lipschitz condition in x for a constant L = L ( p ) > 0. T h e next theorem is therefore a generalization of Theorem 3.7.1.
3.7.
179
STABILITY BY THE FIRST APPROXIMATION
THEOREM 3.7.2. Assume that the trivial solution x = 0 of (3.2.1) is exponentially asymptotically stable and f ( t , x) satisfies a Lipschitz condition for a constant L = L ( p ) > 0. Let
where N = N ( p ) is sufficiently small. Then, the trivial solution of (3.5.1) is exponentially asymptotically stable. Proof. By Theorem 3.6.2, there exists a function V ( t ,x) with the following properties:
(i) V E C [ J x S, , R,], and V(t,x) satisfies a Lipschitz condition with M = K [ L + P @ ~ 0l / < ~ ~q , < 1. (ii) (iii)
< K I1 x 11, < -qolV(t, x).
I1 x I/ < V t , 4 D+V(t,x)
so.
( t ,4 E J x
Now, following the proof of Theorem 3.7.1, one can prove the stated result. Here we have to choose y > 0 such that N (q y ) / M .
<
~
THEOREM 3.7.3. Assume that the trivial solution of (3.2.1) is exponentially asymptotically stable and that f ( t , x) is linear in x. Suppose further that (i) F E C [ J x S, , R"],and, given any such that IIF(t,x)ll
<
Ell
x I/,
/I x II
E
> 0, there exist a(€), T ( E )
< 6(€),
t
2 q€);
(ii) R E C [ J x S, , Rn], R(t, 0) = 0, and there exists an q such that, if I/ x 11 q,
<
where y
E
>0
C [ J ,R,] and
jy
y(s) ds + 0
as
t
+
a.
Then, there exists a To 3 0 such that, for to 3 To , the trivial solution of x' = f ( t , ).
is asymptotically stable.
+ F ( t , x) + R ( t ,x)
(3.7.3)
180
3
CHAPTER
Pyoof. Since the solution x = 0 of (3.2.1) is assumed to be exponentially asymptotically stable, there exists, by Theorem 3.6.1, a function V ( t ,X) satisfying (a)
V E C [ j x S,, , K,],and V ( t ,x) is Lipschitzian in x for a constant K > 0;
<
<
(b) !I x I! V ( t ,). K /I x /I, ( t , x) E J x so; (c) D+V(t,~ ) ( 3 . 2 . 1 ) < - w V ( t , x), 01 > 0, ( t , X) E J x S o . Let E be given such that 01 ' large that, for t 2 T , , we have A-
<
j': exp[-(a
-
< min(cli/K, 7). Choose T, 2 1 so
E
K€)(t- s)]y(s) ds
< @(E)
= 8,,
(3.7.4)
where 6 ( ~ ) E . As shown in Theorem 2.14.6, this choice is possible. If 11 x Ij . p, it is easy to obtain
+ K[llF(t,x)ll + II &t,
< 1)' v(t>
D LI.'(t,.Y)(a.7.3)
X)C3.2.1)
4111.
(3.7.5)
Consider the function m ( t ) = V ( t ,x ( t ) ) ,x ( t ) = x(t, t,, , x,,) being any solution of (3.7.3). We maintain that, whenever 11 x,,11 < 6,/K, we have I/ x(t)ll a(<), t 3 t,, . If this were false, there would exist a t , > t, >, T, such that
II 4 t J
= 8(€),
II 4t)ll
< S(E),
t
[to
6
ti].
In view of conditions (i) and (ii) and (3.7.5), there results the differential inequality D+m(t)
< -(a
-
+
K€)W(t) K y ( t ) ,
t
<
E
[ t o ,ti].
<
Here, wc have used that /I x(t)ij V ( t ,x ( t ) ) and D+V(t,x ) ( ~ . ~ . ~ ) - ; y V ( t , x). According to Theorem 1.4.1, we can deduce m(t)
< m(to)e x p - ( a
-
K € ) ( t- t")]
3.7.
STABILITY BY THE FIRST APPROXIMATION
using relation (b). At t
=
t , , we shall then have a contradiction
+ 6,
I1 X(tl)ll < 6,
=
S(E)
181
=
S(6)-
>
Thus, 1) xo I/ < S,/K implies 11 x(t)ll < a(€), t to 2 To . Consequently, t o , and the asymptotic stability of the trivial (3.7.6) is valid for all t solution of (3.7.3) follows, as in Theorem 2.14.4.
>
THEOREM 3.7.4. Assume thatfE C [ J x S, , R n ] , f ( t 0) , s 0, af(t, x)/ax exists and is continuous for ( t , x) E J x S o , and that the trivial solution of the variational system x’ = f z ( t ,
(3.7.7)
0)x
is exponentially asymptotically stable. Suppose that assumption (ii) of Theorem 3.7.3 holds. Then, there exists a To >, 0 such that, for to >, T o , the trivial solution of (3.5. I ) is asymptotically stable. Since f ( t , 0)
Proof. have
= 0 and
af(t, x)/&
f ( 4 ).
f d t , 0)x
=
exists and is continuous, we
+ F ( t , x),
where F ( t , x) satisfies assumption (i) of Theorem 3.7.3. Hence, the differential system (3.5.1) takes the form
.’
= f z ( t , 0)x
+ q t , x) + R ( t , x).
It is therefore clear that the stated result follows by Theorem 3.7.3.
THEOREM 3.7.5. Let us suppose that the trivial solution of (3.2.1) is generalized exponentially asymptotically stable and that f ( t , x) is linear in x. Suppose further that the perturbation R(t, x) verifies the estimate
II R(t, .)I1 where w
u for t
< w ( t ,I1 x Ill,
(t,
4E J
x
s,,
C [ J x R, , R,], w(t, 0) = 0, and w(t, u ) is nondecreasing in J. Then, the stability or asymptotic stability of the trivial
E
E
solution of
u‘
=
-p’(t)u
+ K ( t ) w ( t ,u),
u(to) = uo >, 0,
implies the equistability or equi-asymptotic stability of the trivial solution of the perturbed system (3.5.1). Proof. On the basis of Theorem 3.6.1, there exists a function V ( t ,x) fulfilling the following conditions:
182
CHAPTER
3
(i) V E C [ J x S o ,R,], and V ( t ,x) is Lipschitzian in x for a functiorl K ( t ) 3 0. (ii)
< v(t,x) < K ( t )ll x IN,
( t , x) 6 J x so. ni v(t,x)(3.2.1) G -P'(t) v(t,x), ( t , x) E J x
I/ x II
(iii) Thus, whenever
.
/ / x Ij < p, it can be readily verified that
D t V t , X)(3.6.1) G D+V(t,4 ( 3 . 2 . 1 )
which yields a further inequality I)'
s p
I.'(t, X)(a.s.1)
< -p'(t)
I.'(t,
+ K(t)l/w t , x)l'
4 + K(t)w(t,q t , 4)
because of (ii), (iii), and the monotonic character of w(t, u)in u. We can now apply Theorems 3.3.1, 3.3.2, 3.4.1, and 3.4.2 t o obtain the desired result.
THEOREM 3.7.6. Let f E C [ S p ,R"], f(0)= 0, f(m) = olmf(x), m > 1, and the trivial solution of x' = f ( x ) (3.7.8) be asymptotically stable. Then, the trivial solution of the system
(3.7.9)
is exponentially asymptotically stable.
Proof.
Let y(s, s o , x,,) be a solution of (3.7.9) and
s ( t ) being the inverse function of t(s). Set x( t) = y(s(t),so , xo). Then,
3.7.
STABILITY BY THE FIRST APPROXIMATION
183
Furthermore, x(to) = x o , where to = t(so). Since the solution x = 0 of (3.7.8) is assumed to be asymptotically stable, the functionf(x) being autonomous, we have
/I 4t)ll
< 4 xo Il)o(t
-
t
to),
3 to >
where a E X , (T E 9. It therefore follows that
/I Y ( S , so
x0)ll
< 4 xo Il)u[t(s)
- t(s0)l
< 411xo Il)aCP(s - so)] =
4 xo I l ) 4
-
(3.7.10)
so),
where o1 E 9, using the fact that y(u, so , xo) is bounded, and so
From the evaluation (3.7.1 O), the uniform asymptotic stability of the solution y = 0 of (3.7.9) is evident. Clearly, F( y ) is homogeneous in y of first degree. Hence, because of uniqueness of solutions, it results that Y ( &10 , axe)
=
U Y ( % so xo). 9
Moreover, using (3.7.10), we derive that
which implies that a*(.) is linear in u. One can now conclude, on the basis of Corollary 3.6.6 and the facts that a*(.) is linear in u and F ( y ) is homogeneous in y of first degree, that the trivial solution of (3.7.9) is exponentially asymptotically stable.
THEOREM 3.7.7.
Let f E C [ S ,, R"],f ( 0 ) = 0, d)Lf(x) = f ( a x ) , rn > 1, and the trivial solution of (3.7.8) be asymptotically stable. Assume that R E C [ J x S, , R"] and
II Nt,.)I1
< CII x 1Irn,
(1, x)
E
1 x s, ,
(3.7.1 1)
184
CHAPTER
3
C being a sufficiently small constant. Then, the trivial solution of the system x'
=f(.)
+ R(t,).
(3.7.12)
is uniformly asymptotically stable.
Pyoof.
Let x ( t , t o , xo) be a solution of (3.7.12). Define
and let t ( s ) be the inverse function of s ( t ) . Setting y ( t ) = x(t(s),to , xo), it is easy to check that
verifying that y(s ) satisfies the system +ids
=F(Y)
+ R*(s,y),
T h e conditions of Theorem 3.7.1 being fulfilled, it follows that
/I Y(S)ll G k'll X" I/ exp[--or(s whence
-
dl
/I 4 t ( s ) , to , xo)ll < KII2, /I exp[--n(s
(S"
= s(f,)),
-
dl,
and therefore
Since the solution y(s) is defined for all s 2 so , limt+ms ( t ) = 00. This shows that the integral in (3.7.13) is divergent, proving the exponential asymptotic stability of the trivial solution of (3.7.12). T h e theorem is proved.
3.7.
STABILITY BY THE FIRST APPROXIMATION
I85
T h e next theorem is of less general character, which may prove effective in certain concrete cases. T h e importance of the theorem, however, is that a judicious selection of V ( t ,x), reflecting more closely particular properties of the given system, frequently leads to much more precise results rather than yielding to the temptation of choosing V(t,x) as simple as possible, such as V(t,x) = // x /I.
THEOREM 3.7.8. Let the following assymptions hold: (i) There exists a continuously differentiable matrix G(t), which is self-adjoint and positive, that is, the Hermitian form (Gx, x) is positive definite, and 