Stability of Solutions of Differential Equations in Banach Space

CENTRO DE NV[ST!G,;COl'l Y DE ESTUD'):)

B, I

,-,

D::L

E, C A

c' G

Translations

of Mathematical Monographs Volume 43

Stability of Solutions of Differential Equations in Banach Space by Ju. L. Daleckii and M. G. Krein 1'\£:\..';-1',.')',.... ~fl~ , f\L, !ft':

ES',

.~:-""--'-:-:-'I

'.'

".

..f'JiiA.I4.:;.. ,_,_

.? N,

BIBltOTECA American Mathematical Society Providence, Rhode Island

1974

YCTO MQJ;1BOCTb PEllIEHMM 1\M 8 is actually a complex Banach space. The complex hull ~ of a real Hilbert space ~ is more naturally defined by specifying, instead of a norm, a scalar product in such a way that ~ will be a complex Hilbert space. This can be done by setting

(Xl

2) 3)

+

iX2' Jh

+ iY2)

=

(Xb Yl)

+ (X2' Y2) + i [(X2, Yl)

- (Xb Y2)].

II x II ~ limp-;oo II x II (p) = max (II XIII b II xzllz,"" II x. II.J. A projectionP E [.))1 is called an orthogonal projection if x - Px .1. Px or, equivalently, P = P*. 00

2.

15

FUNCTIONS OF BOUNDED LINEAR OPERATORS

§ 2. Functions of bounded linear operators 1. Spectrum and resolvent. Let lB be a complex Banach space. 4) A point A of the complex plane is called a regular point of an operator A if[lB] contains the operator (the value of the resolvent of A at A)

RA

= (A -

E

[lB]

AI)-I.

The set p(A) of all regular points of an operator A is open. Its complement p(A) is called the spectrum of A. The spectrum a(A) is always nonempty, closed and lies in the disk I A I ~ I A ". More precisely, the spectrum a(A) lies in the disk of radius rA

=!~ ~"An"

(the existence of the limit readily follows from the relation I Am+n I ~ "Am" • I An II). In fact, when I A I > r A the series l:;;" A-(Hll Ak converges absolutely in the metric of [lB] inasmuch as the corresponding series of norms is ultimately (for k sufficiently large) majorized by the geometric progression { (rA + C )k / I A for any c > o. Upon multiplying this series by A/ - A, we obtain l. Thus, when I A I > rA, the resolvent always exists and, moreover,

IHI }

RA

= -

l: A-(k+1)Ak.

(2.1)

k=O

It can be shown that the circle IA I = rA always contains a point of the spectrum ~l4nlr is called the spectral radius of A. As can be directly verified, the resolvent equation

a (A). Therefore the limit limn~oo

(A, fl

E

p(A»)

(2.2)

holds for R A• The resolvent RA is an analytic function of Ain a neighborhood of each regular point fl' the absolutely convergent expansion

being valid for I A - fll < 1/ I RI' II· An expansion of the resolvent in a neighborhood of the point at infinity has just been indicated in (2.1). We end this subsection with the following special rule, notable for its simplicity: if I A " < 1 (more generally, rA < 1), then the operator / - A is invertible and (/ - A)-I

= l:

Ak.

k=O

2. Behavior of the spectrum and resolvent of an operator under small perturbations. 0 can be chosen so small that the resolvents R).(X) and R).(A) will differ arbitrarily little from each other on r. But then the projections

I I

r

PA =

-

1 . .f R). (A) di\ -2 nl r

Px

and

= - -2-~.f R). (X) di\ nl r

will differ arbitrarily little from each other. Thus the following proposition holds.

r

THEOREM 2.2. Suppose A E [)H] and is a contour distinguishing a spectral set l1o(A) corresponding to a spectral projection P A. Then for any c > 0 there exists a a-neighborhood Uo(A) such that for all X E Uo(A) the contour will distinguish a spectral set l1o(X) corresponding to a spectral projection P x in the c-neighborhood of PA: II PA - P x II < c.

r

§ 3. Solution of some linear operator equations 1. General formula for the solutions. In certain problems connected with differential equations it is necessary to investigate linear operator equations similar to those of the form AX + XB = Y. We consider the more general equation n

1:

j,k=O

Cjk AjXBk

= Y,

(3.1)

where the Cjk ~:re complex numbers, X is the desired operator and A, Band Yare given operators. In order to consider equation (3.1) in the most general setting we will assume that B E [)Hd, A E [)Hz], Y E [)HI, )Hz] and the desired operator X E [)HI. )Hz]. Let AI and Br denote the linear operators acting on any operator X E [)HI. )H2] that are induced by multiplying X on the left by A and on the right by B respectively:

B

It is easily seen that the operators AI and Br commute for any A [)HI]' With the use of the polynomial

E

[)H2] and

E

n

P(i\, (1)

= 1:

j,k=O

Cjk i\j(1k

we form the operator P A,B ~ P(AI, Br)

n

= 1:

j,k=O

Cjk A{ B;

(E [)HI. ~2])'

(3.2)

I.

22

BOUNDED OPERATORS IN BANACH SPACES

Equation (3.1) now takes the form PA,B X for the existence of an inverse of P A,B. Suppose Art a(A). Then the operator

Y. We will first determine a condition

=

(AI - AI)-l = (A - AI)I- 1

clearly exists. This means in particular that a(AI)

C

a(A).6) In exactly the same

waya(Br) c a(B).

We can now explicitly express the operator P(AI, Br) in terms of the polynomial

peA, fl). In fact, by substituting expressions obtained from (2.3) for the powers of the operators AI and Br in (3.2), we have P(AI, Br)

1

= - - 42 § § P(A, fl) (AI 7r

rAre

- AI)-l (Br - fll)-l dA dfl·

(3.3)

Formula (3.3) suggests the following generalization of the problem under consideration. Let KA,B be the set of all single valued functions ¢(A, fl) that are defined in a neighborhood of the set a(A) x a(B) and analytic in A (fl) in a neighborhood of each point of a(A) (a(B») for fixed fl E a(B) (A E a(A»). For each function ¢(A, fl) E KA,B we define an operator acting in the space [5lh, 5l3 2]:

Formula (3.4) establishes a correspondence, between the functions of class and a certain commutative set of operators acting in [5l31o 5l3zJ, that has the following properties: a) If ¢(A, fl) == 1 then ¢(AI' Br) = l. b) If ¢(A, fl) = a1¢1(A, fl) + a2¢2(A, fl), where a1 and a2 are constants, then KA,B

¢(AI' Br) = a1¢1(AI, Br)

c) If ¢(A, fl)

=

+ a2¢2(AI, Br).

¢1(A' fl)¢2(A, fl) then, as follows from Lemma 2.1, ¢(AI' Br) = ¢l(AI, Br) ¢2(AI, Br).

d) If limn~c>o ¢nCA, fl) = ¢(fl, A) uniformly in a neighborhood of the Cartesian product a(A) x a(B) then lim ¢n(AI, Br) = ¢(AI' Br). n~oo

We note that if X ¢(AI' Br) X

E

[5l3r, 5l3 2] then

-= - 4~2

§ § ¢(A, fl)(A

- AI)-l X (B - fll)-l ddfl.

FA FH

The above remarks imply the following proposition . •llt can in fact be shown that IJ(A/)

= IJ(Ar) = IJ(A).

23

3. SOLUTION OF LINEAR OPERATOR EQUATIONS

THEOREM 3.1. Suppose afunction 00 1m t +c

(h ;?; he).

For any t ;::.. 0 there exists an integer n ;?; 0 such that t Then In pet) ~ n In p(h) + In per) ~ n In ~(h) + nh + r n +r and hence -1' In pet)

1m

1-->00

t

< In p(h) --

h

nh

c (c =

+ r (0

~

r

~

h).

max per) ), O~r~h

(4.6)

.

Comparing (4.5) with (4.6), we conclude that the limit I>

=

(4.5)

=

liml-->oo (p(t)/t) exists.

4. EXPONENTIAL OPERATOR FUNCTION

27

To prove equality (4.4) we note that it is equivalent to the following assertion: in order for a real number p to have a positive number Np corresponding to it such that

I eAt I

~ Np e pt

(t ~ 0),

(4.7)

it is necessary that Re A ~ p for all A E a(A), and sufficient that Re A < P for all

AE a(A). Suppose estimate (4.7) is satisfied. Then according to Lemma 2.2 the spectrum a(At) ( = t a(A» lies in the halfplane Re A ~ In Np + pt (t ~ 0). Hence for any t > 0 the spectrum a(A) lies in the halfplane Re A ~ In Np / t

+ p.

In view of the arbitrariness of t > 0 this implies the first part of the assertion. For a proof of the second part we write eAt in the form eAt

=

-

_1_. feAt RA dA 211:1 r A

'

rA lies completely inside the halfplane Re A < p. Then I eAt I ~ 2~ FAJIeAt III Rd I dA I ~ Np ePt , maxAEr I RA II, I being the length of the contour rA. The

assuming that the contour

where Np = (//211:) theorem is proved. REMARK 4.1. Estimate (4.7) does not hold in general if the spectrum a(A) lies in the closed halfplane Re A ~ p. In order to see this it suffices to consider a finite-dimensional space)8 (0 < dim)8 < 00) and an operator A in it consisting of a single Jordan cell (see Exercises 4 and 16). An application of equality (4.4) to the operator -A gives A

lim

t-.-oo

~~ = t

COROLLARY 4.1. If the estimate a(A) lies on the imaginary axis.

I eAt I

I

min {Re A AEa (A)}. ~ c holds for all t E ( - 00, 00), the spectrum

The converse of tvis assertion is not true (see Remark 4.1). 3. Estimate connected with the theorem on a boundary point of the spectrum. We will need below another estimate of the behavior of eAt that is connected with the location of the spectrum of A. We first note that if Ais an eigenvalue of A and x is a corresponding normalized eigenvector: (A - AI)x = 0, then, as readily follows from (4.2), for example, eAtx = eAlx, and hence I eAtx I = eiRe A. In the case of an arbitrary point of the spectrum the situation is more complicated, although certain estimates can still be given when ). is a boundary point of a(A). Their derivation is based on the following important proposition.

28

I. BOUNDED OPERATORS IN BANACH SPACES THEOREM ON A BOUNDARY POINT OF THE SPECTRUM. For any boundary point E [58]) and any c > 0 there exists a normalized vector

A of a spectrum a(A) (A x (II x I = 1) such that

I (A

- AI) x

I

< c.

PROOF. Suppose A E a(A) and fl E p(A). Then would otherwise get from the obvious identity (A - Al)Rfl - J

=

(4.8)

I Rfl I

~ 1

II fl

- A I. For we

(fl - A)Rfl

that I (A - AI)Rfl - J I < 1, and this would imply (see the special rule mentioned at the end of § 2.1) that (A - AI)Rfl and hence A - AI are invertible, which is impossible. If A is boundary point of a(A), there exists for any c > 0 a fl E p(A) such that I fl - A I < c 12. Then I Rfl I > 21 c and there accordingly exists an element Y ( I y I = 1) such that I RflY I > 21 dor I fl - A I < c 12. Therefore, setting x = RflY I I RflY II, the above identity implies

I (A

- AI)x

I

=

I (A

- Al)Rfl Y III RflY

II I ~ 1/11RflY I + I fl

- A I < c.

A proposition on the estimates referred to above can be formulated as follows. LEMMA 4.l. Let A be a boundary point of a spectrum a(A). For any 0 > 0 and T > 0 there exists a normalized vector x for which

et ReA (1 - 0) ~

I eAt x I

~ et ReA (1

+ 0)

(0 ~ t ~ T).

PROOF. The relations

(eAt - eAt J) x = e.l.t [e(A-,l.I)t - J] x = eAt

t

I e(A-.l.I)r d7: • (A

- AI)x

o

imply the estimate

I eAt x

- eAtx

I

~ et ReA

t

I I e(A-,l.I)r I d7: • I (A o

- AI)X

II·

Irll

Let M = e(A-.l.I)r I d7:. It but remains to choose a normalized vector x that ensures the fuUilment of (4.8) when c = 0 1M.

4. Expressionfor the solutions of the operator equations in terms of the exponential function. We will make use of the properties of the operator function eAt studied above in order to restate some of the results of § 3 in a new form. We consider the equation

AX+ XB= Y, which is obtained from equation (3.8) by substituting - B for B. Suppose that the spectra a(A) and a(B) lie in the interior of the left halfplane. Then the condition

4.

A + fl- #- 0 for (A, fl-)

29

EXPONENTIAL OPERATOR FUNCTION

E

(l(A) x (l(B) is satisfied, and hence equation (4.9) has the

unique solution

__

X -

.c (A :r :r

~1~.c

4

2

1T:

A1)-1 Y(B - fl-1)-1

A

FA F.

(4.10)

dA dp..

+ fl-

We take advantage of the equality

I e(.l+p)t dt,

1 A + fl- = -

00

which is valid for Re A < 0 and Re fl- < O. Substituting this expression in (4.10) and changing the order of integration, we find that

00

= -

JeAtYeBt dt.

(4.11)

o

It can also be verified directly that the integral (4.11),which exists by virtue of estimate (4.7), satisfies equation (4.9). In fact,

- [A ofeAtYeBtdt + 7eAtYeBt dt· BJ = - 7d(eAtYe 0

Bt )

0

= - eAt Ye Bt I';)' = Y. Analogously, if the sets (l(A) and (l(B) lie in the interior of the right halfplane, the solution of equation (4.9) is given by the formula

X =

Je-AtYe- Bt dt.

(4.12)

o

5. Some important renormings of~. Certain norms defined by the exponential operator function playa critical role in stability questions. a) We first assume that the spectrum (l(A) lies in the interior of the left halfplane. Then for any ).I such that Re A
o. Negative, nonnegative and uniformly negative operators (and the meaning of the relation H «0) are defined analogously. Clearly, in order for a nonnegative operator to be invertible it is necessary and sufficient that it be uniformly positive. It is significant that every uniformly positive operator H permits one to introduce in .p a new scalar product (x, y)H = (Hx, y) with respect to which .p remains a complete Hilbert space. This occurs because the new norm I x IIH = (Hx, x)1/2 is topologically equivalent to the original one by virtue of the estimates Am(H) II x 112 ~ II x 111 ~ II H 1111 x 112 We will say below that two different scalar products defined on one and the same set are topologically equivalent if the norms defined by them are topologically equivalent. It can be asserted that any scalar product (x, y)l that is topologically equivalent to the original one can be obtained by means of a formula of the form

( = (Hx,y»),

(x, y)l = (x, Y)H

where H is a uniformly positive operator. We recall that the real part of an operator A Am == ReA ==

E

[.p] is the Hermitian operator

HA + A*),

while its imaginary part is the Hermitian operator A3 == 1m A == (l/2i)(A - A*), so that A = Re A

+

i 1m A.

THEOREM 5.1 (GENERALIZED LJAPUNOV THEOREM). In order for the spectrum of an operator A to lie in the interior of the left halfplane it is necessary and sufficient that there exist a uniformly positive operator W such that

Re(WA) is said to be W-positive if (Wx, x) > 0, W-negative if (Wx, x) 1-

T*)-l H (eiif>I - T)-1 d¢.

0

We recall (see § 2.4) that this means that .p decomposes into a direct sum of invariant (under = .p+ .p_, the spectra IJ± = IJ(A j.p±) of the restrictions of A to.p± lying respectively in the interiors of the right and left halfplanes. 11)

A) subspaces: .p

+

I.

40

BOUNDED OPERATORS IN BANACH SPACES

In view of the fact that all of the points of the unit circle are regular points of T and T* while H»O, the integrand here is uniformly positive.1 2) It follows from (7.6') that (Wx, x) ~ e I x 112 for any x E .p-. The uniform W-negativeness of .p+ is established analogously. We show, finally, that relation (7.4') automatically implies the invertibility of W. In fact,

I

X

112

~

Am tH) (Hx, x)

=

Am tH)

I (Wx, Ax) + (Ax,

Wx)

I~

~~ tH~-

I

Wx

IIII x

II,

and thus an inequality of type I Wx I ~ ell x I is valid. The sufficiency of the condition of the theorem is proved. NECESSITY. Suppose now that the spectrum of A does not intersect the imaginary axis and .p decomposes into a direct sum .p = .p+ .p_ of invariant (under A) subspaces so that the set a+ = a(A l.p+) lies in the interior of the right, while a- = a(A l.p-) lies in the interior of the left, halfplane. Starting from any operator H»O, we form the operator P'S,HP + acting from .p+ into the subspace .p'S, that is invariant under A* and corresponds to its spectral set a'S, = iJ +. According to the results in § 4.4 the operator

+

co

X = -

Se-A*t P'S, HP + e- At dt o

satisfies the equation A* X for fE.p+

+

= -

XA

Pt HP +.

In addition, X

= P'S,XP+ and

00

(XI, f) = -

S(HP + rAt I, P + e- At f) dt < 0. o

Analogously, the operator co

y

=

SeA*tp ~ HP _ eAt dt o

satisfies the equation A* Y + Y A and (YJ, f) > for f E .p_. We put W = X + Y.

°

=

-

P~HP _

and the conditions Y

= P~ YP-

12) We have made use of the following easily proved fact: if F(¢» is a continuous and continuously invertible operator function on [a, b] while H» 0, then f! F*(¢»HF(¢» d¢> O.

»

7. HILBERT SPACE WITH INDEFINITE METRIC

°

Clearly (WI, g) = for fE .fl+, g W-dissipative inasmuch as

.fl-. In addition, the operator A is uniformly

E

A*W+ WA where Hl = P'+ HP +

41

+ P':... HP _ » 0,

=

-

HI>

since

+ (HP _ x, P - x) P + x liz + II P _ x liZ}

(Hl x, x) = (HP + x, P + x)

~ Am (H){ II

~ 1Am (H) {II P+ x II

+

II P- x 11}2 ~ 1Am (H)II

X

liz.

The theorem is proved. In connection with Theorem 7.1 it is useful to have in mind the following general fact. THEOREM 7.2. Suppose W is an invertible Hermitian operator and the space .fl decomposes into a direct sum .fll .flz of W-orthogonal subspaces. If the subspace .fll is W-positive (negative), it is uniformly W-positive (negative).

+

PROOF. Let P 1 denote the projection onto .fll corresponding to the given direct decomposition. Suppose f E .fll and g = W-lj. Then, inasmuch as the form (Wx, y) is positive in .flI> we have Ilf liz = (Wg,f) = (WP1g

+

WPzg,f) = (WP1g,f)

~ ,J (WP1g, P1g)(Wf,f) ~ ,JfWl1. Ilp1W-111·llfll· ,J(Wf,f), which implies (WI, f) ~ m II f liz. 2. u-dichotomic operators and uniform W-contractions. We say that an operator Tis u-dichotomic if its spectrum does not intersect the unit circle and contains both a spectral set (Ji(T) lying in the interior of the unit circle and a spectral set (J.(T) lying in its exterior: (J(T) = (Ji(T) U(J.(T). To this decomposition there corresponds a direct decomposition of the space .fl = .fli .fle into invariant subs paces of the operator T so that the spectra of the restrictions of this operator are

+

-.

(Ji(T)

=

(J(T l.fl,)

and

(J.(T)

=

(J(T l.fle).

In proving Theorem 7.1 we essentially used the properties of an operator T satisfying condition (7.2). It is easily seen that Theorem 7.1 is essentially equivalent to the following proposition on the operators T. THEOREM 7.1'. Suppose TE [.fl] and (J(T) does not cover the unit circle. In order for T to be u-dichotomic it is necessary and sufficient that it be a uniform W-contraction:

42

I.

BOUNDED OPERATORS IN BANACH SPACES

T* WT« W

(7.2)

for some indefinite W ( :;= W* E [.pD. Any operator W (= W*) satisfying condition (7.2) is invertible and such that the invariant subspace .pi of T is W-positive while the invariant subspace .pe of T is Wnegative. An operator W can be chosen so that these subspaces are W-orthogonal. In fact, by hypothesis, the operator T has at least one regular point, on the unit circle. It can be assumed without loss of generality that' = 1 (otherwise we could consider the operator T in place of T). There then exists the operator

,-1 A

= (T + J)(T - 1)-1 (E [.p]),

condition (7.2) for Tbeingequivalent to condition (7.4) for A. 7.1. A uniform W-contraction is either I) u-dichotomic or 2) has a spectrum covering the whole unit circle. COROLLARY

Case 1) is obviously realized. An example will be given below in which case 2) is realized. We first establish a number of propositions. LEMMA 7.3. Let T be a uniform W-contraction. Then all of the points of the unit circle are points of regular typefor T, i.e.for any point, of the unit circle = 1) the operator T is an isomorphism of.p onto a closed subspace.

(1'1

,J

PROOF.

Condition (7.2) means that

(Wx, x) - (WTx, Tx) ~ m II x 112

(x

E

.p),

(7.7)

where m = Am (W - T* WT) > O. We show that this implies the existence of a constant c; > 0 such that (x

E

.p).

(7.8)

In fact, there would otherwise exist a sequence of unit vectors {Xnlll Xn II = I} for which Yn = (T - 'J)xn --+ 0 as n --+ 00. Putting x = Xn, TXn = 'xn + Yn in (7.7), we would then get

mil Xn 112 ~ -- ,(WXn,Yn) - ,(WYn, Xn) - (WYn, Yn) which is impossible. Inequality (7.8) implies that the mapping T closed, B; = (T - 'J).p is also closed.

,J

--+

0,

is an isomorphism. Since

.p is

7. HILBERT SPACE WITH INDEFINITE METRIC

43

REMARK 7.1 t. A comparison of Lemma 7.3 with the theorem on a boundary point of the spectrum (see § 4.3) shows that a point of the unit circle can only belong to a(T) as an interior point. This in turn implies that the unit circle belongs to a(T) if and only if it is contained in an annulus belonging to a( T). THEOREM 7.3. Let T be a uniform W-contraction with Wan invertible operator (W-l E [~)]). In order for T to be u-dichotomic it is necessary and sufficient that T* be a uniform W-l -contraction. PROOF. NECESSITY. If 1 ¢: a(T), the operator A = (T + I)(T - 1)-1 exists. Then A - I = 2(T - 1)-1, i.e. T = (A + I)(A - 1)-1. Substituting the obtained expression for Tin (7.2), we find that WA + A* W «0. Multiplying this inequality from the left and from the right by W-1, we get Re(W-IA*)«O. On the basis of Lemma 7.2 we conclude that (A* + I)(A* - 1)-1 = T* is a uniform W-l -contraction. SUFFICIENCY. If T* is a uniform W-l -contraction, the equality (T* - I)x = 0 implies by Lemma 7.3 that x = O. This in turn implies that ~1 = (T - 1)f) = f), since (T* - I)x = 0 for every vector x .1 ~1' Thus the point A = 1 and hence every other point of the unit circle are regular points of T. In certain cases Theorem 7.3 permits one to verify directly whether the spectrum of an operator T covers the unit circle or does not intersect it. 3. Example of a uniform W-contraction that is not u-dichotomic. Let us construct an example of a uniform W-contraction whose spectrum covers the unit circle. We construct an operator T whose spectrum covers the unit circle and to which at the same time there corresponds even an invertible Hermitian operator W converting it into a uniform W-contraction. Let {ek} ~co be an orthonormal basis in f), f)+ = V ~ ek, f)- = V'(' e_k,13) P + and P _ be the orthogonal projections on .p+ and f)- respectively, and let W = P + P _ ( = W-l). We choose a pair of complex numbers q+ and q-, with 1 q+ 1 < 1 and 1 q- 1 > 1, and define the operator T: n = 1,2, "', n = 0, - 1, - 2, ....

For any x

= L; ~oo

Ck ek E f) we have Tx =

L; q+Ckek-1 k=1 00

(Wx, x)

=

-

(WTx, Tx) =

so that for H 13)

=

L; 1 Ck 12 k=O

+

o L; q- ckek-1,

-1

L; 1 Ck

12 ,

-00

00

L; 1q+ 121 Ck 12 k=1

0

L; 1q- 12 ICk

12,

-00

W - T* WT

The symbol VkEK ek denotes the closed linear span of the system of vectors {ek} kEK.

44

I. BOUNDED OPERATORS IN BANACH SPACES

+ ( I q- 12 -

-1

1) 1:; -00

I

00

Ck

12 ~ 0 1:; -00

I

Ck

12 = O(X, X),

where 0 = min {I - I q+ 12, I q- 12 - I}. Thus H»O, i.e. T is a uniform W-contraction. It is easily seen that T*e n = tj _ e n+1 (n = - 1, - 2, ... ). Therefore (WT*e-1' T*e-1) =

I q- 12 (WeD, eo) = I q- 12 > 0.

On the other hand, (We-h e-1) = - 1 < 0, so that T* is not even an ordinary W-1-contraction (W-1 = W). By Theorem 7.3 (and Corollary 7.1) the spectrum of the constructed operator T covers the whole unit circle. REMARK 7.1. The spectrum of the constructed operator T contains the annulus q+ ~ A ~ q-I (cf. Remark 7.1'). In fact, for any r E q+ q-I) the operator r- 1 T is of the same type as T and hence its spectrum covers the unit circle. This means that the spectrum (J(T) covers the circle I A I = r. We note that the interior points of the considered annulus are points of regular type for Tfor which the co dimension of the space (T - A/)Sj is equal to one. 4. u-dichotomicity conditionsfor uniform W-contractions. The following important assertion is almost trivial.

I I II I

(I I, I

THEOREM 7.4. Suppose W = W* E [Sj] is an invertible operator. Then every uniform (u-dichotomic uniform) W-contraction T E [.p] has a neighborhood in [Sj] consisting of uniform (u-dichotomic uniform) W-contractions. The assertion follows from the fact that the operators W - T*WT and W-1 TW-1 T* are continuous functions of T. THEOREM 7.5. Suppose a uniform W-contraction T is the limit of a sequence {Tn} of u-dichotomic uniform W-contractions. Then T is also a u-dichotomic uniform W-contraction. Proof. Assum~ the contrary. Then the point A = J is a point of the spectrum of T. But according to Lemma 7.3 the operator T - / maps Sj isomorphicallyonto a closed subspace 2 = (T - /)Sj =1= Sj. Therefore by Banach's theorem there exists an operator R in [2, Sj] such that (T - /)R = R(T - /) = /. We choose an n such that T - Tn < 1/ R II· Let us ShOW 14 ) that (Tn - /)Sj =1= Sj.

I

I

I

14) This assertion is a corollary of the following more general proposition of Krein, Krasnosel' ski! and Mil'man on the stability of the deficiency of an operator A EO [lBl> lB 2] injecting lB j onto a closed subspace AlB j • Each operator X in a sufficiently small neighborhood of such an operator A

8.

We have Tn - I

=

45

STABLE W-UNITARY OPERATORS

[I - (T - Tn)R] (T - I), so that (Tn - I) 10

=

(I - Q)B,

where Q = (T - Tn)R. We note that I Q I ~ I T - Tn IIII R I < l. We extend the operator Q onto all of 3) by putting Qx = QP"x, where P" is the orthogonal projection on B. Clearly, I Q I = I Q I < 1. Therefore the operator I - Q is invertible and consequently is a one-to-one mapping of all of 10 onto 10; hence (I - Q)B = (I - Q)B =1= .p. On the other hand, since the operator Tn is a u-dichotomic uniform W-contraction, it follows that 1 1= (J(Tn) and hence (Tn - 1)10 = .p. We have arrived at a contradiction. The theorem is proved. COROLLARY 7.2. Let T(t) (a ~ t ~ b) be a continuous one parameter family of uniform W-contractions. If T(a) is a u-dichotomic operator, so will be T(t) for all t E [a, b].

For suppose there exists a nonempty set of points in [a, b] at which T(t) is not u-dichotomic, and let to denote the infimum of this set. By Theorem 7.5, T(to) is u-dichotomic. But according to Theorem 7.4 there would then exist a neighborhood of to in which T(t) is u-dichotomic, which is impossible.

§ 8. Stable W-unitaryoperators 1. Stability criterion for W-unitary operators. Let W by an invertible Hermitian operator. An operator A will be said to be W-Hermitian if (Ax, Y)w = (x, Ay)w. This condition is equivalent to the equality WA = A*W or WAW-l

=

A*.

(8.1)

An operator U is said to be W-unitary if (Ux, Y)w equivalent to the equality WUW-I

=

=

(x, U-Iy)W, which is

(U-I)*.

(8.2)

We note that the Cayley transform 15) . U

=

(A - iaI) (A

~

+

ial)-l ~.

takes a W-Hermitian operator into a W-unitary one with regular point A = 1. In fact, it is easily seen that U - 1= - 2ia (A

+ il)-I;

(U*)-l - 1= - 2ia (A*

+ il)-I,

also injects ~l onto a closed subspace X~l with codim X~l = codim A~b where codim X~l = dim ~zI X~l. A still more general proposition can be found, for example, in the article [3] of Gohberg and Krein. 15) The real number a*O is chosen sothatthe operator (A + ial)-l will be bounded ( -ia tE o-(A».

I.

46

BOUNDED OPERATORS IN BANACH SPACES

and therefore WUW-I

= 1- 2iWa(A + iI)-I W-I = 1- 2ia(WAW-I + iJ)-I = 1- 2ia(A* + iI)-I = (U*)-I.

In addition, the operator (U - I)-I = (ij2a) (A + iJ) is bounded. One can easily show that the inverse transform A = ai(U + 1) (I - U)-l takes each W-unitary operator with regular point A = 1 into a W-Hermitian operator. We discuss some of the properties of W-Hermitian and W-unitary operators. 1) The spectrum of a W-Hermitian (W-unitary) operator is symmetric with respect to the real axis (unit circle).l6l In fact, the relation WA W-I = A* implies O"(A*) = O"(A). On the other hand, A: E O"(A*) if A E O"(A). The analogous assertion for a W-unitary operator follows in the same way from the similarity of the operators U and (U-I)* (see (8.2»). 2) Let Al and Az be two parts of the spectrum of a W-Hermitian (W-unitary) operator A. We assume that the symmetric image Al of Al with respect to the real axis (unit circle) and Az are separated from each other by nonintersecting contours f'I and r z· Then the invariant subspaces .pI and .pz of A corresponding to the parts Al and Az of its spectrum are W-orthogonal. For a proof it suffices to show that Pi WP z = 0, where I

PI

= - 2. f (AI - A)-I dA

Pz

= -2~ f (AI - A)-I dA

nl r,

and

are the projections on

.pI

nl r,

and

.pz

respectively. Making use of the relation

(AI - A*)-I W = W(Al - A)-I, which is easily deduced from the definition of a W- Hermitian operator, we get

Pi

WP z

= 4~z ="

f f 01 -

A*)-I W(ftl - A)-I dA: dft

FI F2

4~ f f (AI -

A)-I (ftI - A)-I dA: dft

FI F'l.

W f (AI - A)-I (ftl - A)-I dA dft 4n r, r, 1 1 = - -2 f f - - [(AI - A)-I - (ftl - A)-I] dA dft = 0, 4n !" r, A - ft

= - -z f

l')We recall that complex numbers spect to the unit circle if ZI=2 = 1.

ZI

and

Z2

correspond to points that are symmetric with re-

47

8. STABLE W-UNITARY OPERATORS

which follows from Cauchy's theorem (cf. the proof of Lemma 2.1). The analogous assertion for a W-unitary operator can be deduced in the same way (or obtained from what has already been proved by applying the Cayley transform if Whas at least one regular point on the unit circle). The general stability criterion (Theorem 6.1) admits the following sharpening in the case of W-unitary operators. In order to simplify the formulation let us agree to say that a subspace ~ c {> is W"definite if it is either W-positive or W-negative. THEOREM 8.1. In order for a W-unitary operator U to be stable it is necessary and sufficient that {> decompose into a direct sum of two W-orthogonal W-definite subspaces {>1 and {>2 that are invariant under U. PROOF. SUFFICIENCY. By Theorem 7.2., if (8.3)

where {>1 and .fh are W-orthogonal and, for the sake of definiteness, {>1 is Wpositive while {>2 is W-negative, the space {>1 ({>2) is uniformly W-positive (Wnegative). Thus there exists a constant f-l > 0 such that

I(x, x)w I ~ f-l (x, x) The space {>1 «(>z) remains a Hilbert space while U becomes a unitary operator if the scalar product (x, y) in it is replaced by (x, y)w ( - (x, y)w). Thus

which implies II Un I {>k II ~ II W I /f-l (k = 1,2). Inasmuch as the projections P k (k = 1,2) on the {>k are bounded: (k = 1,2), we have

II unll ~ II unP1 11 + II

unP2

11

=

I

Pk

I

~ c

II unl{>lll + I Un 1{>211 ~ 2cll W II /f-l (n = 0, ± 1, ± 2,.··).

NECESSITY. By Theorem 6.1, if U is a stable operator, there exists an invertible operator Stransf6rming Uinto a unitary operator: V = S-l US (V*V = VV* = I). The fact that U is a W-unitary operator implies

W

= U*WU = S*-lV*S*WSVS-1.

Putting G = S*WS, we find that G = V*GV, so that

VG = GV. The operator G is Hermitian, has an inverse and

(8.4)

I. BOUNDED OPERATORS IN BANACH SPACES

48

(Gx, X)

=

(WSx, Sx).

(8.5)

Therefore the real spectrum /Y(G) = /Y+(G) U /Y-(G). Let .p = .p+ EB.p- be the orthogonal decomposition of.p into the corresponding invariant subspaces of G. The invariant (under G) subspaces .p+ and .'0- are G-orthogonal and G-definite (Gx, x) > 0 for x E .'0+ (x =I- 0) and (Gx, x) < 0 for x E.p_ (x =I- 0)). By virtue of (8.4) the subs paces .p± are also invariant under the unitary operator V, so that V.p± = .p±. We now put S.p+ = .p1 and S.p_ = .p2' Then U.pk =.pk (k = 1,2) and according to (8.5) the subspaces .'01 and .p2 will have all of the required properties. 2. Normally W-decomposable operators. Let us agree to say that an operator A is normally W-decomposable if its spectrum /Y(A) can be divided into nonintersecting spectral sets: /Y(A) = /Y1(A) U /Y2(A), so that in the corresponding decomposition .'0

=

.p1

+ .p2

(8.6)

the invariant (under A) subspaces ~.~h and .p2 are uniformly W-definite. Let us agree to use the indexing under which the subspace .p1 will be uniformly W-positive while .p2 will be uniformly W-negative. We note that if A is a W-Hermitian operator (or a W-unitary operator), the subspaces .p1 and .p2 in (8.6) will be W-orthogonal, and by Theorem 7.2 they will be uniformly W-definite whenever they are W-definite. We note some of the properties of normally W-decomposable operators. 1) If a W-unitary operator is normally W-decomposable, it is a stable operator. This assertion follows directly from Theorem 8.1. 2) If an operator A is normally W-decomposable, so will be an operator ¢(A) where the scalar function ¢ E KA is such that ¢(/Y1) ¢(/Yz) = 0. Clearly, the latter condition is always satisfied if the function ¢o.) is one-to-one on the spectrum /Y(A) and, in particular, if ¢ is a linear fractional function. 3) The spectrum of a W-Hermitian normally W-decomposable operator is real. In fact, the Cayley transform of such an operator is a W-unitary operator and hence has a spectrum lying on the unit circle (see Theorem 6.1). 0 (or H» 0), the operator A = WH is normally W-decomposable. 4) If H In fact, assuming, for example, that H 0 (we would otherwise consider the operator -A), we get AW + WA* = 2WHW« O. Applying Theorem 7.1, we obtain the required assertion. We note that the operator A = WH is a W-LHermitian operator.

n

«

«

THEOREM 8.2. The set of all normally W-decomposable operators is open in the Banach space [.p],

In other words, to every normally W-decomposable operator A there corre-

49

8. STABLE W-UNITARY OPERATORS

a

sponds a neighborhood II X - A II < in [.p] consisting of normally W-decomposable operators. Thus the property of being normally W-decomposable is stable. PROOF. Let l and rz be smooth closed contours separating the parts O"l(A) and O"z(A) of the spectrum of A from each other. From Theorem 2.2 it follows that for a sufficiently small > 0 the spectrum of an operator X satisfying the condition II X - A II < also decomposes into parts O"l(X) and O"z(X) separated by the same contours while the spectral projections P{ = Pl(X) and Pf = Pz(X) differ sufficiently little from the corresponding spectral projections PI = Pl(A) and P z = P 2(A). Consequently, for any e > 0 there exists a > 0 such that the inequalities

r

a

a

a

II Pi PI - P{* P{ II < e, II P~ Pz - Pf* Pf II < e, II Pi WPI - P{' WP{ II < e, II n WPz - Pf* WPf II < e are satisfied for I X - A II < (WP{ x, P{x)

=

a.

(WPlx, PI x)

But then

+

([P{* WP{ - Pi WP l ] x, x)

~ m (PIX, PI x) - e

+ m ([Pi PI

II x 112 ~ m (P{x, PIX)

- P{*Pl] x, x) - ell x 112 ~ m IIp{x liz -2ellxll z

and we get (Wx, x) ~ (m - 2e) II x liz when x = PIX. The case when x = Pfx is treated analogously. 3. Strongly stable W-unitary operators. A stable W-unitary operator U is said to be strongly stable if all of the W-unitary operators U' in some a-neighborhood II U ' - U II < of it are stable.

a

THEOREM 8.3. In order for a W-unitary operator U to be strongly stable it is necessary and sufficient that it be normally W-decomposable.· The sufficiency of the condition follows from the preceding theorem and Theorem 8.1. Let us prove its necessity. By Theorem 8.1, if the operator U is strongly stable (and hence stable), the decomposition (8.3) with the known properties will hold. The theorem will be proved if we prove that the strong stability of U implies O"(U

l.pl)

n O"(U I·Pz) =

0·

Let PI and P z denote the projections corresponding to the decomposition (8.3). We introduce in .p the new scalar product (x, y)l

=

(WXl' Yl) - (Wxz, Y2)

where Xk = Pkx, Yk = Pky (k = 1, 2). The new scalar product is definite:

(x, y E

.p),

50

1. BOUNDED OPERATORS IN BANACH SPACES (x =I 0),

and, moreover, as can easily be seen, the norm equivalent to the old one. We put WI = Pi WPI

-

P~

II xiiI

=

(x, x)Vz is topologically

(8.7)

WPz ;

this permits us to write the new scalar product in the form (x, Y

E

oP)·

Since the operator U commutes with the projections Pk (k = 1,2), it is a unitary operator with respect to the new scalar product and its invariant subspaces oPl and oPz are orthogonal to each other. In other words, U is also a Wrunitary operator while the subspaces oPl and oP2 are WI-orthogonal. eO, dE;. (Eo = 0; E;. = E;.-o for 0 < A < 2n) be the spectral resoluLet U = tion of U. In this resolution E;. (0 ~ A ~ 2n) is a Wrorthogonal projection function, i.e. E;. Ep. = Emin(J.,p.) and WlE;. = E1 WI (0 ~ A ~ 2n). We form the operator H = ME;.. Clearly it is WrHermitian, i.e.

J5"

n"

(8.8)

WlH= H*Wl' We show that it is also W-Hermitian, i.e.

WH= H*W.

(8.9)

In fact, since the operators P k (k = 1,2) commute with U, they commute with the projection function E;. and H, and therefore the projections Pt commute with E1 and H*. It therefore follows from (8.8), after mUltiplying it by Pt from the left and by P k from the right, that

Pk WlPkH = H* Pt WI Pk. Nothing that W = Pi WP I + P~ WP 2, we arrive at (8.9). The fact that the operators P k commute with H means that the subspaces oPk are invariant under H. Consequently, to the decomposition (8.3) there corresponds the following representation of H in matrix form:

H = (Hll

o

0 ).

H zz

Then U can be represented in the form U = (exp (iHll )

0 ). exp (iHzz ) ,

a(exp(iHn));

a(U loP2)

o

a(U loPI)

=

= a(exp (iHzz )).

8.

51

STABLE W-UNITARY OPERATORS

The theorem will be proved (by contradiction) if we show that when the spectra (J(H I ~\) and (J( U 1-P2) have a common point i\ there exists in any neighborhood of HaW-Hermitian operator H' whose spectrum contains nonreal points. In fact, this will imply that there exists in any neighborhood of U a W-unitary operator U ' = exp (iH') whose spectrum contains nonunitary points (points not lying on the unit circle) and which are thus not stable. It can be assumed without loss of generality that the common point i\ '# 1, since we could otherwise achieve this by replacing U by a W-unitary operator p U where I p I = 1 and p '# 1. If i\ = exp (iw) (0 < w < 2n) is a common point of the spectra (J(U I-PI) and (J(U 1-P2) then w will be a common point of the spectra (J(Hu) and (J(H22). We first consider the simpler case when w is a common eigenvalue of the operators Hu and H 22 · Let (/JI and CP2 denote corresponding unit eigenvectors: CPk E -Pk, II CPk 111 = 1, Hkk CPk = WCPk (k = 1, 2). For an arbitrary c > 0 we form the operator

_ (Hll H22) _ Gi G, '

H, -

where G, is the operator acting from -Pz into -PI according to the formula and Gi is the WI-adjoint of G, acting from -PI into -Pz according to the formula

Gi x

h(x, CPl)I CP2

= -

= -

ic(Wx, CPI) CP2

(x

E

-PI).

It is easily seen that together with H the operator H, (c > 0) will be W-Hermitian. A simple calculation shows that

H, (CPI

+ CP2)

= (w

+ ic) (CPI + cpz).

Thus the W-Hermitian operator H, (c>O) always has a nonreal eigenvalue. On the other hand, H, will be arbitrarily close in norm to H for sufficiently small c > o. The case when w E (J(Hu) (J(Hzz ) but is not necessarily an eigenvalue for each of the operators Hll and H zz reduces to the preceding one. In fact, using the spectral resolutions of the operators H kk :

n

we form the operators H~%) .

=

(

z". )

w-a

J+ J 0

w+a

a

AdEk ).

+ wi\,

where f\ = J::!:~ dEkJ. (k = 1, 2; 0 < < w). Since, by assumption, w E (J(Hu) (k = 1,2), it follows that Fk '# 0 and w will be an eigenvalue of H j!) with eigenspace Fk-P.

52

I.

BOUNDED OPERATORS IN BANACH SPACES

We have IIHkk -

Hk~)

I

=

/1:1: (A -

w) dEkA II

< 0,

and hence I H - Hij II < 0, where Hij is the W-Hermitian operator defined by the equality Hij = (

~if) H~g)

)

By what has been proved above there exists in any neighborhood of Hij a WHermitian operator with nonreal spectrum; therefore such an operator also exists in any neighborhood of H. The theorem is proved. § 9. Elements of nonlinear analysis

m

1. Contraction principle. In what follows we will assume that is a closed subset of a Banach space ~, although the basic Theorem 9.1 can be formulated for any complete metric space. Let S be a (not necessarily linear) operator mapping m into itself. We call it a contraction if there exists a constant q < 1 such that (9.1)

for any x, y

E

m.

THEOREM 9.1. Suppose the operator Sv is a contraction for some natural number )) ;:::; 1. Then there exists in m one and only one fixed point Xq, of S:

SXq, = Xq,. This point can be obtained from any point Xo Xq,

(9.2) E

mas the following limit:

= lim Snxo. n~oo

PROOF.

Suppose first))

=

1. Then for the sequence

we have according to (9.1)

I Xk+l - Xk II = II SXk - SXk-1 II ~ q II Xk - Xk-l II (k = 1,2, ... ), so that II Xk+l - Xk II ~ qkll Xl - Xo II. Therefore

(9.3)

9.

53

ELEMENTS OF NONLINEAR ANALYSIS

i.e. {x n } is a Cauchy sequence. The fact that 9]( is closed implies the existence of the limit limn~co Xn = Xrp. This limit is a fixed point of S since

I Xrp

- SXrp

I

: 1 and consider the contraction S = S". By what has been proved above S has one and only one fixed point Xrp. On the other hand, since S and S commute, S(SXrp) = S(Sxrp) = SXrp,

i.e. SXrp is also a fixed point of S. Hence by virtue of the uniqueness (9.2) holds. Taking into account that every fixed point of S is a fixed point of S, we conclude that S has no other fixed points. It remains to prove that equality (9.3) holds for any Xo E 9](. To this end we note that every n can be represented in the form n = ml.i + r, where 0 ;;:; r ;;:; l.i - 1. Then Sn Xo = Sr Sm Xo and, since limm~oo Sr Sm Xo = Srxrp = Xrp for fixed r, equality (9.3) is proved. REMARK 9.1. If S itself is a contraction, the difference I Xrp - Snxo I monotonically decreases at the rate of O(qn). We leave it to the reader to formulate what happens when l.i > 1. 2. An application of the contraction principle. Let A E [)B] and let rA = limn~co I An Ill/n be the spectral radius of A (see § 2.1). We consider the equation

+ f, Ax + f, we will have

x = Ax

where f

E )B

is given. If we set Sx

=

(9.4)

n~l

Snx = Anx

+ L:

Akf,

k=O

which implies

I snX2 ..::..

SnXl

I

When r A < 1 and l.i is sufficiently large\ we have I A" I < 1, and hence S" is a contraction. Thus, when rA < 1, equation (9.4) has for any f E )B exactly one solution, which is given by the formula x =

L: A'1 (= lim Snxo)· k=O

n~co

(9.5)

I.

54

BOUNDED OPERATORS IN BANACH SPACES

3. Differentiation offunctions with a vector argument. Let y = f(x) be a function defined in a neighborhood U(xo) of a point Xo of a Banach space j(3o and taking values in a Banach space j(31' The functionf(x) is said to be differentiable (according to Frechet) at Xo if in the neighborhood U(xo) it can be represented in the form f(x)

= f(xo) +

A(x - Xo)

+ 7}1(X, Xo) I x

- Xo II,

(9.6)

where A E [j(3o, j(31] and the function 7}1(X, xo) satisfies the condition lim 7}1(X, xo) = O. X-Xo

The operator A is called the (Frechet) derivative of f(x) at xo, and one writes A

= f'(xo).

As usual, we say that f(x) is differentiable on an open set G if it is differentiable at each point of this set. We will say thatf(x) is continuously differentiable in G if its derivative f'(x) is a continuous function of x with values in [j(3o, j(3d. All of the usual formal rules of differential calculus, which we will not cite here, are retained in the case under consideration. We formulate only the simplest variant of the rule for differentiating a composite function, which will be needed by us in the sequel. If x(t) is a differentiable function of a scalar argument t whose values lie in a domain G of the Banach space j(3o in whichf(x) is differentiable, then (d/dt)f(x(t))

In particular, for x

E

= f'(x(t))

dx/dt.

G and sufficiently small t

+

(d/dt)f(x

ty) = f'(x

+

(y E j(3).

ty)y

Integrating the latter equality from 0 to 1 and changing the notation, we obtain the equality I

f(X2) - f(XI) = Sf' (Xl

+

t(X2 - Xl)) (X2 - Xl) dt,

(9.7)

°

which is valid when the interval Xl + t(xz - Xl) (0 ;::;:; t ;::;:; 1) lies entirely in the set G. REMARK 9.2. It follows from formula (9.7) that if f'(x) is bounded on a convex set G: III' (x) 1J ;: ;:; M (x E G), then Ilf(xz) - f(XI)

I ;: ;:; Mil Xz

- Xl

I

(Xl> Xz E G).

Iff(x) is differentiable in a neighborhood of Xo then the derivativef'(x) itself is a function in this neighborhood with values in the Banach space [j(3o, j(31]. If it too is differentiable at xo, its derivative is denoted by the symbol f"(xo) and called the second derivative of f(x) at Xo. By definition, f"(xo) E [j(3o, [j(3o, j(31]]' In other words, it can be said thatf"(xo) is a bilinear operator taking each pair (hh h z) of elements of j(3o into an element f"(xo)(hh h z) E j(31'

9. ELEMENTS OF NONLINEAR ANALYSIS

55

It is not difficult to obtain the representation f(x) = f(xo)

+ f'(xo)(x

+7jz(x, xo)

Ix

- xo),

- Xo

+ ~

f"(xo)(x - Xo, x - xo)

liz,

(9.8)

where limx~x, 1)z (x, xo) = O. The derivatives of higher order are defined analogously. 4. Inverse function theorem. We now use the contraction principle to obtain an important result of analysis. THEOREM 9.2. Suppose a function y = f(x) is defined and continuously differentiable in a neighborhood U(xo) of a point Xo of a Banach space lBo and takes values in a Banach space lB 1. If the operator T = f'(xo) E [lBo, lB 1] has a bounded inverse T-1 E [lB1, lBo], the function f(x) effects a homeomorphic mapping of a neighborhood V(xo) c U(xo) onto a neighborhood of the point Yo = f(xo). The inverse mapping x = g(y) has in this neighborhood the continuous derivative

g'(y)

= [f'{g (y)]-l.

(9.9)

Iff(x) has continuous derivatives up to order n in a neighborhood ofxo, so will g(y) in a neighborhood of Yo = f(xo). PROOF. We can assume without loss of generality that Xo = Yo = O. The equation f(x) = y is equivalent to the equation

x = x

+

T-1 [y - f(x)] i;! (/) (x).

(9.10)

For fixed y we have

(/)'(x)

= I - T-1f'(x) = T-1 [I' (0) - f' (x)].

Therefore by virtue of the continuity off'(x) there exists a 0 > 0 such that in the closure of the neighborhood Sa,fIl, = {x III x I < o} we have the estimate I (/)'(x) I ~ q < 1, and hence (see Remark 9.2) also the estimate

I (/)(xz)

- (/)(X1)

I

~ q

I Xz

- Xl

I

(Xk E

Sa,fIl,; k = 1,2).

Suppose now

I y I < c=

(1 - q)o /

I

T-1 II·

(9.11)

Then the estimate

I (/) (x) I ~ I (/) (x) - (/) (0) I + II(/) (0) I ~ q I x I + I T-t[[ I y I implies I (/)(x) I ~ 0 provided I x I ~ o. Thus under condition (9.11) the

(9.12)

mapping (/)(x) is a contraction in the closed ball Sa,fIl,. From Theorem 9.1 we deduce that for each y E S"fIl, = {y E lB1 III y I < c} there exists a unique solution x E Sa,fIl, of the equation x = (/)(x) and hence of the equationf(x) = y.

I.

56

BOUNDED OPERATORS IN BANACH SPACES

The inverse function x = g(y) is defined in Se'~l and, since (] can be chosen arbitrarily small, is continuous at y = O. In particular, by taking y sufficiently small, we can obtain an arbitrarily small x = g(y) and hence an operator f'ex) that is close enough to 1'(0) = T to have a bounded inverse. Thus all of the above arguments can still be applied if the pair (xo, Yo) is replaced by the pair (g(y), y), and hence g(y) is also continuous at this point y. In exactly the same way it also suffices to establish the differentiability of g(y) only at y = O. We have y = f(x) = Tx + [f(x) - Tx], where, by virtue of (9.6), f(x) - Tx = r;llxll and r; = 0(1) for x ---+ O. Hence

x = T-l Y - T-l

r; I x I

= T-l y - T-l

r; I g(y)

II·

(9.13)

We now note that (9.12) implies the estimate

I g (y) I

I 1J(g(y) I

=

~ q

I g(y) I + I

T-l

IIII y

II,

from which we get

I g (y) I This shows that for y

---+

ij

~

I T-q I y I / (1

- q).

0

= T-lr;

I g(y) I / I y I

= 0(1).

In this way we obtain the representation (see (9.13) g(y)

= T-l Y

- ij

Iy

II,

showing that g(y) is differentiable at y = 0 and g'(O) = T-l Y = [1'(0)]-1. This proves equality (9.9). The proof of the last assertion of the theorem can be obtained by induction, for which one successively differentiates (9.9). 5. Cone inequality theorem. In this subsection we present a simple assertion, which will be systematically used in more special cases in subsequent chapters. Its proof will be developed for linear operators, although the assertion admits certain generalizations to the nonlinear case. We first intrqduce the following notion. A closed subset Sl: of a Banach space lB is called a cone if it has the following properties: a) Xo E Sl: implies AXo E Sl: for A ~ 0; b) xl. X2 E Sl: implies Xl + X2 E Sl:; c) ± Xo E Sl: implies Xo = O. Suppose a cone Sl: is given in a Banach space lB. We will write

x

rB

then

+ (Ar.-I B + A,,-2 BC ... + Bcr,-I) X2

*0

is a root vector of A of index p - rB. 13. Suppose A and B are Hermitian operators in a Hilbert space ~ and let AM(A)

=

sup (Ax, x),

/)xll=1

A", (A)

=l!xl1=1 inf (Ax, x).

Obtain the estimate (0.9)

Hint. Differentiate the expression ¢(t) = l!eCA+iB)t X\\2 and obtain an estimate of the logarithmic derivative of ¢(t). 14. Suppose ~ is a Hilbert space and A E [~]. Show that if the spectrum a(Re A), where Re A = (A + A*)/2, lies in the closed halfplane Re A~ a, then the spectrum a(A) satisfies the same condition.

61

EXERCISES

Hint. Make use of inequalities (0.9) and Theorem 5.1. 15. Prove the relation e A+B = lim [eAlneB'n]",

(0.10)

n->=

where A, B EO [~] and the limit is taken in the sense of uniform convergence. Hint. Make use of the relation IlecA+B)t - eAt eBtl1 ~ ct 2 (c = const), which is easily deduced from Taylor's formula. Obtain from (0.10) the upper estimate of (0.9). 16. Show that the following estimate is valid for any operator A acting in en. (t ;:::; 0),

(0.11)

where" = max (Re AI A EO a(A)} (see B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemyckii [1], page 131). Hint. The following equality holds for any analytic function f(A) in a domain containing the spectrum a(A) = {Aj} : f(A) = al

+ a2 (A

- AlI)

+ a3(A

- Atf) (A - Azl)

+ ... + an(A

- All) (A - A2l)-··(A - An-II),

where

Hence IaH II complex plane.

~ (Ilk!) max~ I jh+O h Prove that the right derivative of the norm Ilx(t)11 exists at to and, in addition, lim Jx(to ± h)II-=I~(to)1L oullx(to)ll. h+O h 19. According to Exercise 17 the following limit exists for any A A(A) = lim III + hAl-=l. h.O h

EO [~]:

r~-'~ I

62

I. BOUNDED OPERATORS IN BANACH SPACES

Clearly, A(aA) A(A

= aA(A),

if a;:::;O; I A(A) I ~ IIAII;

+ B) ~ A(A) + A (B);

[A(A) - A(B)] ~

IIA - Ell.

At the same time the "measure" A is not a norm since it can take negative values. But one always has A(A)

+ A( -

A) ;:::; 0 (= A(O)).

a) Show that, if m= ,p,

and thus - A ( - A) = Am(AiJl). Let mbe a Banach space of sequences of complex numbers: x Em¢=:> x = (~j) I, the ~j=~j (x) (j = 1, 2,.··) being continuous functionals of x E m. Then to each operator A E [m] there corresponds a matrix lIa jk lll such that the equality x' = Ax is equivalent to the equalities (j = 1,2,.··).

Show that b) If m= c (c is the space of all convergent sequences with norm IIxll A(A) = sup (Reajj j

whereas IIAII = c) If then

m= II

SUPj

L:k

= sup

I ~j I),

+k*j L:·lajkl),

I ajkl.

(II is the space of all absolutely summable sequences with norm IIxll A(A)

= sup (Reajj + L: j k*j

= L:j~11 ~

j

I),

lajkl),

whereas IiAII = SUPk L:jlajkl. 20. Extend the result of Exercise 13 by proving that for any A

E [m]

e- AC - A ) ~ lIe A Ii ~ eACAl.

(0.12)

Estimate (0.12) implies the following estimate for a strip containing the spectrum o(A) (see (2.5) and Lemma 2.2): - A( - A) ~ Re A ~ A(A)

for all A E o(A).

(0.13)

Hint. Using the result of Exercise 18, estimate the derivative with respect to t of the norm lIexp (At)xoll for any Xo Em. Estimate (0.9) i; a special case of a more general estimate of Wintner (see Corollary 111.4.3), while estimate (0.12) is a special case of one of the estimates of S. M. Lozinskii referred to in Exercise 111.19. The notion of the functional A(A) as well as the other results indicated in these exercises are due to S. M. Lozinskii [1] (see also the paper of G. Dahlquist [1]; in the papers of these authors mis a finite-dimensional space). S. M. Lozinski! called the measure A(A) the logarithmic norm of A. 21. Estimate (0.13) implies that the spectrum of an operator A lies in the closed convex domain KA of the complex Cplane obtained by taking the intersection of the halfplanes (0

~

e ~ 2n)

63

EXERCISES

A further restriction of this convex domain can be obtained by applying the formulated result to all possible operators of the form Aa = A - aI, where a is a complex number. For ~ = .p the assertion concerning KA yields the well-known Hausdorff theorem KA = {(Ax, x)lx E.p, Ilxll ~ 1).

When applied to operators acting in en with this or that norm the assertion on KA yields one of the well-known theorems of A. Hirsch (see, for example, Marcus and Mine [1], page 140). 22. An entire function x(Q of a complex argument ~ with values in ~ is called an entire function of exponential type 0 (0 ~ 0 ex)) (or, according to S. N. Bernstein, offinite degree 0-) if


00

(J.

I~ I

Let x(~) = Xo

~ ~2 ~n + lTXI + 2T X2 + ... + II! Xn + ....

Verify that, as in the scalar case (see B. Ja. Levin [1]), a function will be an entire function of exponential type (J if lim SUPn->oo I Xn II lin = (J. 23. According to a theorem of S. N. Bernstein, if an entire scalar function x(~) (~ = Cl) of exponential type is bounded on the real axis, then Ix'(QI

~

(J

sup

~oo

18) A Volterra operator is a completely continuous operator with spectrum concentrated at the origin.

64

I. BOUNDED OPERATORS IN BANACH SPACES

Hint. Assume the contrary and make use of the fact that for a boundary point ;I. of IY(A) not lying on the real axis there exists a sequence of fn E Sj (lIfnll = 1) such that AI. - ;l.fn -> 0 for n -> 00. b) Suppose A E [Sj] and AW» 0 (in this case the operator A is W-l nonnegative). Prove that Sj decomposes into a W-orthogonal sum Sj = Sj+ Sj_ of subspaces invariant under A such that the restrictions A+ = A ISj + and A_ = A ISj_ are similar to a uniformly positive and a uniformly negative operator respectively. Hint. Make use of Theorem 7.1. 27. Suppose A is a W-Hermitian operator with positive spectrum. Show that there exists a unique operator B with positive spectrum such that B2 = A. This operator is W-Hermitian (V. P. Potapov [1]) and Ju. P. Ginzburg [1]). Hint. To construct the operator B make use of the Poincare-Riesz formula (2.3). 28. An operator Y is called a W-nonenlargement if

+

(WY/' Yf)

~

(W/'f),

«

or, in other words, if y* WY w. Show that if Y is a W-nonenlargement having at least one regular point on the unit circle, its adjoint y* is a W-l-nonenlargement (see, for example, V. P. Potapov [1] and Ju. P. Ginzburg [1]). Hint. Apply the Cayley transform to reduce the problem to one of considering W-dissipative operators. 29. Let U be a W-unitary u-dichotomic operator (the spectrum IY(U) does not intersect the unit circle). Show that Sj decomposes into a direct sum of two W-neutral invariant (under U) subspaces in which the spectrum IY(U) is found respectively inside and outside the unit circle (1. S. Iohvidov and M. G. Krein [1]). Hint. Make use of the arguments employed in the proof of Theorem 7.1. The results noted in the following exercises have been taken from an article of M. G. Krein [11]; they will be used in Chapter V. 30. One calls 0 (0 ~ 0 ~ n:) the angle between nonzero vectors x, y E Sj, and writes 0 = 1:::(x, y) (= 1::: (y, x)) or 0(.0) (0 (y-;:i)), if cosO = Re(x,y)/llxIIIIYII.

An equivalent and perhaps more natural definition of the angle 0 = equality

2 sin (012) = Ile x

-

1::: (x, y) is given by the

eyll,

where ex = xlllxll, ey = ylll yll are the unit vectors in the directions of the vectors x, y. Prove that for any Xj E Sj (Xj *- 0; j = 1,2, 3)

1::: (x" X3)

~

1::: (x" X2) + 1::: (X2, X3),

(0.14)

the equality sign homing if and only if the vector X2 lies "between" the vectors Xl and X3, i.e. X2 = ClXl + C3X3 (c" C3> 0). Hint. Using the unit vectors ej = xj/llxjH and arbitrary real numbers t;; (j = 1,2,3), construct the nonnegative form

Calculate the determinant of this form and explain why it is nonnegative. 31. a) Let e(t) (a ~ t ~ b) be a smooth (i.e. continuously differentiable) arc on the unit sphere S of H. Prove that

65

EXERCISES

£II ~

I dt ~ 1: (e(b), e(a»,

(0.15)

the equality sign holding if and only if e(t) lies between e(a) and e(b) for all t E [a, b]. If the unit sphere of Rn (n-dimensional Euclidean space) is taken as S, inequality (0.15) expresses the fact that the geodesics on S are great circle arcs. Hint. Make use of the result of the preceding exercise. b) Generalize (0.15) to the case when the smoothness condition for e(t) is replaced by the requirement that it be absolutely continuous. Hint. Approximate the spherical curve e(t) by the curves eh (t)

=

*:1:

e(7:)d7:/

I {-:z:

e(7:)d7:

I

(a;;;'t;;;'b),


b. 32. Suppose A E [.p] has an inverse A-I E [.pl. The quantity dev(A) ~ sup (1:(Ax, x) Ix E is called the angular deviation of A. Thus 0 ;;;, dev (A) ;;;,

.p, X 7r

*'

o}

and

. f { _R~iAx~1 -.L 0 } cos d ev (A) -- In IIAxl1 Ilxll x E fJ., x -r.

>

Clearly, dev (cA) = dev (A) for c o. a) Verify that dev (A) is a continuous function of A E [.p]. Suppose A, B E [.p] are invertible operators. Since 1: (ABx, x) ;;;, 1: (ABx, Bx) we have dev (AB) ;;;, dev (A)


0, Ilxnll = 1. Consider lim 1:(Ax,,, x n). 33. Prove that for a uniformly positive operator H

<
0 and minus hyperboloids when c < o. It is not difficult to verify that the uniform W-dissipativeness of A implies the relation

d(wx~l; x(t) = 2Re(WAx(t), x(t) < 0

(x(t) "# 0),

which shows that the indefinite form (Wx, x) decreases, for any solution x(t) of equation (1. 2).

3. BOUNDED SOLUTIONS: HOMOGENEOUS CASE

75

If the point Xo is on a minus hyperboloid, the subsequent diminution of (Wx, x) with increasing t means that the trajectory x(t) intersects all of the hyperboloids with larger-in-absolute-value negative values of c and hence goes to infinity. When Xo is on a plus hyperboloid in .f>-, the diminution of (Wx, x), which on .f>- is equivalent to an ordinary norm, leads to an arbitrarily close approach of the trajectory to the center of the hyperboloid. But if Xo is on a plus hyperboloid outside .f>-, the trajectory, after intersecting the hyperboloids (Wx, x) = c with diminishing values of c, enters the cone(Wx, x) = 0 and goes over to a system of minus hyperboloids, receding then to infinity. This follows from the fact that any solution for which P +Xo i= 0 increases unboundedly in norm. We now note that by choosing for the e-dichotomic operator A a Hermitian operator W from the relation 2Re(WA) = WA

+ A*W = -

(P'i,HP+

+ P'!.HP_),

where the P ± are the spectral projections of A corresponding to the invariant subspaces .f>±, we arrive, as follows from the formulas in § 1.4.4, at the form co

(Wx, x) =

J {II HeAtp_x liz - I HrAtP+x IIZ}dt.

o

This form is analogous (when H = J) to the indefinite norm (2.3) in a Banach space 18. It is easily seen that analogous geometric arguments can be developed in a Banach space by considering in place of the hyperboloids the surfaces I x IIA = c. § 3. Boundedness of the solutions of the homogeneous equation 1. First order equation. We wish to determine conditions under which the solutions of the homogeneous equation dx / dt = Ax are bounded on the real line. Since the solutions of this equation are described by the formula x(t) = eAtxo (xo = x(O)), the condition that the solutions be bounded implies the estimate

I eAt Xo I

~ cx,

(- 00 < t < 00),

where the constant Cx , depends only on Xo. Thus the set 01 operators eAt (- 00 < t < 00) is bounded at each element Xo E 18. By the uniform boundedness principle (Theorem 1.1.2) these operators are uniformly bounded:

I eAt I

~

c

(- 00 < t < 00).

(3.1)

The latter estimate, as was shown in Chapter I (Corollary 1.4.1), implies that the spectrum of A lies on the imaginary axis. A more exact result can be obtained if the phase space 18 is a Hilbert space .f>.

76

II.

LINEAR EQUATION WITH CONSTANT OPERATOR

In fact, by virtue of Theorem 1.6.3 condition (3.1) is satisfied if and only if the operator A is similar to a Hermitian operator multiplied by i (a skew-Hermitian operator): A = S-I(iB)S (B* = B). Thus the following result holds. THEOREM 3.1. If each solution of the equation dx / dt = Ax is bounded on the real line, the spectrum a(A) lies on the imaginary axis. If the phase space}B is a Hilbert space, all of the solutions are bounded if and only if the operator A is similar to a skew-Hermitian operator.

2. Second order equation in a Banach space. We consider the second order dif-

ferential equation dZy dt Z

+

_ Ty - 0,

(3.2)

where T E [}B]. Its investigation can be reduced to the investigation of a first order equation in the doubled phase space }B(Z) = }B + }B whose elements are the pairs x = (Xl, Xz) (Xl, Xz E}B) and whose norm is given by the formula X II~ = Xl + Setting y = Xl and dy / dt = Xz, we replace equation (3. 2) by the system

I

dXI - X . dtz,

I liz

I xzllz.

dxz = _ TXl dt '

or by the equivalent equation for a vector X = (XI. xz) in }B(Z) dx/dt

where the operator d

E

= dx,

(3.3)

[}B(Z)] is defined by the operator matrix

It is not difficult to calculate that dZk

The

operato~

= (_ l)k (~k ~k); dZk+1 = (_ l)k( _ ~k+1 ~k).

(3.4)

function edt defining the solutions of equation (3.3) takes the form

co dn co dZk edt - 1: t n - - - 1: t Zk - - n=O n! - k=O (2k)!

dZk+1 tZk+l--k=O (2k + I)!' co

+ 1:

(3.5)

Making use of the correspondence between scalar and operator functions, we put co

cos Tl/Zt

= ~/ -

TktZk l)k (2k)! ;

co

TktZk+1

T-l/Zsin TI/Zt = k'fo(- l)k (2k

In this notation it follows from (3. 4) and (3. 5) that

+

I)!'

3.

77

BOUNDED SOLUTIONS: HOMOGENEOUS CASE

e$t = (

COS P/2t T-l/Z sin P/2t) - T-lIz sin Tl/Z t cos Tl/Z t .

The formula x(t) = e$t Xo, where Xo = (Yo, Yo), now leads us to a representation of the solution of equation (3. 2) satisfying the conditions yeo) = Yo,

(3.6)

y'(O) = Yo,

in the form yet) = (cos P/2t) Yo

+ (T-l/2sin Tl/2t) Yo.

(3.7)

It can be verified by direct substitution that the vector function (3. 7) satisfies equation (3. 2) and the conditions (3. 6). From formula (3. 7) it follows that the boundedness for t E ( - 00, (0) of each solution of equation (3. 2) is equivalent to the boundedness of the operator functions cos Tl/2 t and T-l/Z sin P/z t. We now show that it suffices to require the boundedness of the operator function T-l/2 sin Tl/2t (- 00 < t < (0). We consider the vector function yet) = (T-l/2 sin T1Izt)yo.

Its derivatives are the vector functions y'(t)

= (cos

p/2t)yo

and

y"(t)

= -

(T1I2 sin TlIzt)yo

= -

Ty(t).

By hypothesis, for each fixed Yo E ~ the vector function yet), and hence also = - Ty(t), is bounded. If we prove that y'(t) is also bounded, we can conclude that the set of operators cos p/2 t ( - 00 < t < (0) is bounded at each element Yo E ~, and hence is bounded in norm by virtue of the uniform boundedness principle (Theorem 1.1.2). Thus it remains to show that the boundedness of yet) and y"(t) implies the boundedness of y'(t). We put y"(t)

y" (t) - yet) = J(t).

(3.8)

The function J(t) is bounded on the real line'together with yet) and y"(t). If we consider (3. 8) as a differential equation, we can express the solution of it that is bounded on the real line by means of the easily verified (and well known) formula 3) yet) =

1

2

J e-II-sIJ(s)ds. 00

-00

By differentiating this expression with respect to t, it is easily shown that 3)This formula is a consequence of a general formula involving the Green function of a stationary linear equation (see § 4).

78

II. LINEAR EQUATION WITH CONSTANT OPERATOR sup II y'(t) II ~ SUp Ilf(t) II < I

00.

I

Thus we get that the boundedness of each solution of the second order equation (3.2) on the real line is equivalent to the boundedness of the operator function T-l/2 sin Tl/2 t. We have simultaneously proved the following assertion: in order for all of the solutions of equation (3.2) to be bounded it is necessary and sufficient that the solutions satisfying the condition yeO) = 0 be bounded. 3. Second order equation in Hilbert space. A more precise investigation can be carried out when the space )8 is a Hilbert space .)). THEOREM 3.2. In order for each solution of the equation

-ddt2y2 +

Ty

=

(3.2)

(-oo(t),

(4.26)

where rf>(t)=f(t)- I:kEKA'TJi,e-2biIT is a function whose Fourier coefficients with index in KA;T are equal to zero: ' (4.27) It remains to note that a solution of equation (4.26) under condition (4.27) can be obtained by making use of formula (4.24), since the terms in (4.21) corresponding to the indices k EO KA-T drop out thanks to (4.27). ' If we introduce the "incomplete" Green function FT(t)= -

--L I:

T kEKA;T

(A _ }k1ri I)-le(2kniIT)t

(4.28)

T

we can write an arbitrary T-periodic solution of equation (4.1) in the form T

x(t)= f fT(t-s)f(s)ds+ o

where the

x~ (k EO

I:

kEKA;T

x2e(2bi!T)t,

(4.29)

KA;T) are solutions of equations (4.16).

5. Almost periodic solutions. In this subsection we consider the differential equation dx/dt

=

Ax

+ J(t)

( - 00

< t < (0),

(4.1)

under the assumption that the function J(t) is continuous and almost periodic. We recall that a continuous functionJ(t) on the real line with values in a Banach space ~ is said to be almost periodic (according to Bohr) if for each c > 0 there exists an Le > 0 such that each interval of the real line of length not less than Le contains a point r = -c(c) (an c-translation number) for which

Ilf(t) - J(t + -c) I < c

( - 00

< t
0, t

~

to).

(2.11)

to

It immediately reduces to the previous inequality if p(t) is a bounded function on [to, (0). We will need an estimate for <jJ(t), however, under the more general assumption that

sup I

1 I+T o S p(7:)d7:

7:0

I

= MTo
=

lim In IIx(t) I t

t~co

of each solution x(t) of equation (4.1) satisfies the condition I> ~ M. It is not at all necessary that condition (4.2) be fulfilled in order for all of the solutions of equation (4.1) to have finite Ljapunov exponents (or even to be bounded) (see Exercise 19). Without assuming the fulfilment of condition (4.2), we consider the set:2 of not necessarily finite Ljapunov exponents of all of the possible solutions of equation (4.1). This set is called the (upper) Ljapunov spectrum of equation (4.1). We leave it to the reader to prove that the Ljapunov spectrum Z of the homogeneous equation dx/dt = Ax(A = const) in a finite-dimensional space j8 consists of the real parts of the eigenvalues of A.

By the (upper) Ljapunov exponent of equation (4.1) is meant the quantity sup 1>.

1>1. =

<EI

THEOREM 4.1. The Ljapunov exponent of equation (4.1) coincides with the (upper) Ljapunov exponent of the operator function Vet) = Vet, 0): 1>1. =

lim lnIlV(t)lI. t~co

(4.3)

t

PROOF. Let /l denote the right side of formula (4.3). From the equality x(t) V(t)xo it follows that In IIx(t) I ~ lnIlV(t)1I

+

=

lnllxoll

which implies lim lnllx(t)1I ~ lim In II V(t) II , t~oo

t

-

t~co

t

i. e. I> ~ /l for I> E ~, and hence 1>1. ~ /l. It remains to prove the converse inequality for the case 1>1. < x(t) = V(t)xo of equation (4.1)

I V(t)xo II

00.

For any solution

~ Ne, x,e(KL+e)t,

where c is an arbitrary positive number and Ne,x, is a constant generally depending on c andxo. The latter inequality can be written in the form

II V(t)xoe-(KL+e)tll

~ N"x,.

III.

118

NONSTATIONARY LINEAR EQUATION

The family of operators U(t)e~(KL+e)t (0 ~ t < (0) is bounded at each element Xo E )8. It follows from the uniform boundedness principle that (t

I

~

0).

I

This means that U(t) ~ Nee(KL+e)l, which implies fJ. ~ KL· The theorem is proved. REMARK 4.1. One could analogously introduce the lower Ljapunov exponent K' = lim In lilt of a solution x(t) and then the lower Ljapunov exponent K[ of an equation as the greatest lower bound of lower Ljapunov exponents of its solutions. In this connection K[ = - limHo In U~l(t) It. 2. Bohl exponents. Here we introduce another characteristic of the behavior of the solutions of equation (4.1) which, as will be seen, has more natural properties. Let x(t) = U(t)xo be a solution of this equation. By the (upper) Bohl exponent KB(XO) of this solution is meant the greatest lower bound of all those numbers p for which there exist numbers Np such that

Ilx(t)

I

I

(4.4) for any =

+

'r, t E [0, 00) such that 'r ~

t. If such numbers p do not exist, we put KB(XO)

00.

In exactly the same way the lower Bohl exponent Ks(XO) of a solution x(t) is the least upper bound of those numbers p' for which there exists a number N~ > 0 such that

Ilx(t)

I

~

Np,ep'(H)

Ilx('r) I

(0

~

'r ~

< 00).

t

(4.4')

If K(XO) is a Ljapunov exponent of a solution, clearly, -

00 ~

KB(xo)

~

K(Xo)

~ KB(XO) ~

+

00.

The interval [Ks(Xo), KB(XO)] is called the Bohl interval of the solution in question. The following formulas are easily verified:

KS(XO)

=

lim - 00) to be satisfied it is sufficient and,for the projections P that are uniformly conjugated by U(t), also necessary that

Kp

=

(Kp =

sup

II U(t)PU-l(-r) I
0 and n = [t - r] we put rk

=

r

+k

(k =

n+l U(t)PU-l(r) = II U(rk)PU-l(rk-l), k=l

we get

I U(t)PU-l(r) I .

n+l ~ II I U(rk)PU-l(rk-l) I k=l ~

and, consequently, for

Xo E

Kj,+l

~

Kpen InKp

~ Kpe(t-~)In

Kp,

l8p

IIU(t)xoll = IIU(t)PU-l(r)U(r)xoll ~ Kpe(t-~)lnKpIIU(r)xoll,

i. e. K(P) ~ In Kp. The sufficiency of the condition Kp < ro can be shown analogously. REMARK 4.2. We have incidentally obtained the estimates - InKp

~

KP,(P)

~

KB(P)

~

In Kp.

In particular, the Bohl exponents of equation (4.1) (P inf In I U(:, r) II

= -

satisfy the estimates

O~~~~~l In I U(t, r) II

~ KP, ~ KB ~ REMARK

= /)

sup

InIIU(t, r)ll.

O~t-T~l

4.3. Since [U(t)PU-l(S)][U(S)PU-l(r)] = U(t)PU-l(r),

the first of the conditions (4.6') is equivalent to the condition sup

O~t-r~T

II U(t)PU-l(r) I
O. An analogous remark can be made concerning the second condition of (4.6'). As a corollary of Theorem 4.2 we obtain the following result. THEOREM 4.3. If the operator function A(t) is integrally bounded, the Bohl exponents of equation (4.1) are finite. The proof immediately follows from Lemma 2.4 and formulas (4.7) and (4.9). We now find expressions for the Bohl exponents corresponding to a projection P. THEOREM 4.4. If condition (4.7) is satisfied and the Bohl exponents are finite, they are representable by the formulas KB(P)

=

_ KS(P) =

lim InIIU(z-

+ S)PU-l(Z-) II

l

;

r'l:~oo InIIU(z-)P~-l(z- + s)11 ; J S

T, $--+00

in particular,

KB = lim InIIU(z-

,

- KB =

(4.10)

.

l

+ s)ll:

J

+ s, z-)II

r';i: InIIU(/z-

(4.11)

s

T,S-+OO

PROOF. Suppose p > KB(P). Then inequality (4.8) is satisfied. It follows that Inll U(z-

+ s)PU-l(~llL s

~]~Npmp s

+p '

and, further, 1 A

= II'm-- 1nll U(z- + S)PU-l(Z-) II S

T,S--+OO

~

p,

which implies). ~ KB(P) < roo Suppose now p > ).. By the definition of the upper limit there exists a number T = Tp such that for z-, s ~ Tp lnll U(z-

+ S)PU-l(Z-) II

---'----~"-- ~

s

p,

i. e. II U(t)PU-l(Z-) II ~ ep(t-r) for z-, t - z- ~ Tp. By Remark 4.3 the quantity KT = sUPO~t-r~T II U(t)PU-l(Z-) II is finite. Hence for all t ~ z- ~ T II U(t)PU-l(Z-) II ~ Nep(t-r) ,

III. NONSTATIONARY LINEAR EQUATION

122

where N = max(l, K T e 1pl 1). This same estimate is also valid for 0 ~ 'r ~ t ~ Y. Finally, when 0 ~ 'r ~ T < t, we have

II U(t)PU-l('r) II Setting Np

=

~

II U(t)PU-l(T) II II U(T)PU-l(r) II

~

NeP(t-T) NeP(T-T) = N 2ep(t-T).

max(N, N2), we obtain the inequality

II U(t)PU-l('r) II

~ Npep(t-T)

(t ~ 'r ~ 0),

which implies for x('r)

= U('r)Pxo and x(t) = U(t)PU-l('r)X('r) = U(t)Pxo

the estimate showing that KB(P) ~ p. Thus KB ~ }. and the first formula is proved. The second is established analogously. 3. Stationary equation case. In this case we have THEOREM 4.5. The upper (lower) Ljapunov and upper (lower) Bohl exponents of the homogeneous stationary equation dxfdt = Cx (C = const) coincide with each other and are equal to the supremum (infimum) of the real parts of the numbers }. E a(C). PROOF. For the stationary equation U(t)

=

eCI, U(t, 'r)

=

eC(t-T)

and according to Theorem 1.4.1 KL

= lim

I~co

In IleGl II t

= lim

T~CO

t-T_OO

In IleC(I-T) II t - 'r

=

KB

= max{Re}.l}. E a(C)}.

The assertion in parentheses is proved analogously. DEFINITION. By the strict Ljapunov exponent of equation (4.1) is meant the limit lim In II U(t) " t

I~co

if it exists. Analogously, by the strict Bohl exponent of equation (4.1) is meant In II U('r . 11m

+ s, 'r)" s

,

if this limit exists. We have just convinced ourselves that a stationary equation has strict exponents.

4.

123

LJAPUNOV AND BOHL EXPONENTS

4. Perron's example. We will show by an example that the case ble. We consider the scalar Perron equation

"L < "B is possi-

(4.12) dx/dt = (sin In t + cos In t)x. Here ~ is a one-dimensional space and A(t) is the operator of multiplication by sin (In t + n/4). the function The solution of the equation has the form x(t) = etsinlntx(O), which implies U(t) = etsinlnt. We calculate the upper Ljapunov exponent of the equation:

.v2

"r

..

Let us now show that "B

"B = lim

= I~ 1m InIU(t)1 = I~· 1m sIn In t = 1. t t_oo t-HX)

=

.v2. We have

In IIU(t)U-l(r) II = lim t sin In t t - 1: T~OO t -

T~OO t-r-KX)

1: sin 1:

In 1:

t-f-HX>

=

lim (cos In tav

+ sin In tav),

T~OO

t~T-OO

where tav is a point lying between 1: and t (according to the mean value theorem). We now choose a sequence of pairs of points 1:n, tn as follows: In tn = 2nn

+ n/4 + e;

+ n /4. + n /4 + 'fJn, where

In 1:n = 2nn

Then for 1:n < in < tn we have In in = 2nn Under this choice of 1:n, tn for n --+ 00 we have 1:n --+

tn -

1:n

= e2mr +7r/4(e
0 the upper Ljapunov and upper Bohl exponents of the equation dx/dt

=

A(t

+ l)x

(0

~

t '(t)/if>(t) ~ l/k. Integrating this relation from 1 to t, we get if>(t)

~

if>(l)e(t-l)!k

and hence _1_. X(t)

= rI,'(t) ~ if>(t) > if>(1) e(t-l)!k 'f'

-

k

=

k

.

The latter relation means that (t

~

1),

where)) = 1 /k and Nl = ke1!k /if>(l). . Setting, finally; N = max (Nt. maXO~t~levt I U(t) II), we obtain the required inequality (5.3). The second assertion of the theorem follows upon noting that an arbitrary solution of equation (5.1) differs from the solution distinguished by condition (5.2) by a term U(t)xo: x(t)

=

U(t)xo

+

t

J U(t, T)/(T)dT, o

the boundedness of which follows from estimate (5.3).

5. BOUNDED NESS OF SOLUTIONS

129

The theorem is proved. From Theorem 5.2 below and the fact that there are examples of equations for which A(t) is integrally bounded and KL < < KB (see § 4.5) it follows that the converse of the first assertion of Theorem 5.1 does not hold, i.e. the presence of estimate (5.3) does not guarantee the boundedness of the solution of problem (5.1) -(5.2) for any bounded continuous vector functionf(t) on [0, (0). 2. Role of negativeness of the Bohl exponent in the case of an integrally bounded A(t). A more complete result is obtained under the additional condition of integral boundedness on [0, (0) of the operator function A(t):

°

1+1

J IIA(r) Ildz- ~

M1

(t E [0, 00)).

(5.9)

1

THEOREM 5.2. Suppose condition (5.9) is satisfied. In order for the Cauchy problem (5.1)-(5.2) to have a bounded solution on [0, (0) for every bounded continuousfunction f( t) on [0,(0) it is necessary and sufficient that the Bohl exponent of equation (5.1) be negative. PROOF. The condition is sufficient since the estimate

I U(t, z-) II

~ Ne-v(H)

(v > 0, t

~

z-)

implies

II o U(t, z-)f(z-)dz-II1 ~ Nillfill

J

J0 e-v(/-T)dz- ~ ~ Illflll· v

To prove the necessity of the condition we first show that the boundedness of the solution of the Cauchy problem (5.1)-(5.2) implies the boundedness of the solution of the Cauchy problem { dx(t)/dt = A(t)x x(to) = 0.

+ f(t)

(5.10)

The solution of problem (5.10) is given by the formula 1

x(t) =

JU(t, z-)f(z-)dz-. I,

If the function./(t) were not necessarily continuous, the solution of problem (5.10) could be regarded as the solution of problem (5.1) - (5.2) with free term

°

~ t < to, to.

t ~

In order to avoid working with discontinuous functions we consider the following problem of type (5.1)-(5.2): { dxeCt)/dt = A(t)x,(t) x.(o) = 0,

+ Ie(t),

III.

130

NONSTATIONARY LINEAR EQUATION

where 0

fc(t)

1

{- /(to)(t

=

- to

+ c)

/(t)

t < to - c,

~

for

0

for

to - c

for

t ~

~t
,

where N is a constant satisfying the single condition N

~ max { ~(~;, 02!:~1 ev(t-t,)IIU(t, to) II}, 1 ).!

= k' (1) =

t,+1 d1: X(1:)·

f.

It still remains to show that N can be chosen independently of to. To this end we note that

X(t)

=

I U(t, to) I

~ en,IIA(t) Iidt ~ eM,

(to ~

t ~

to

+

1).

But then in the first place (1) ~

t,+l Je- M,d1:

=

e- M ,

t,

and in the second place max O~t-t,~l

I

{ev(t-t,) U(t, to)

II} ~

max

ev(t-t,)+M, = ev+M,.

O~t-t,~l

Thus we can put N = e1/HM'max{1,k}.

5.

131

BOUNDEDNESS OF SOLUTIONS

The theorem is proved. 3. Essentiality of the condition of integral boundedness. Here we give an example showing that condition (5.9) in Theorem 5.2 cannot be dropped. To this end we consider a continuously differentiable positive scalar function u(t) having the following properties: t

Su(s)ds
nu(n)

b)

u(t),

(n = 1,2,.··; an

+ r(t),

An example of such a function is u(t) = et

r(t) =

o {

nensin2 {2: n (t - n

+ 2an)}

-+

0).

(5.11)

where ~ t

for

n - 1

for

n - 2a n

< n - 2an

~t
0 (depending only on Ao) and co > 0 such that the Bohl exponent of the equation





0 such that the upper Bohl exponent of equation (0.15a) is negative for 0 ~ c co and o


0) if its coefficients are stationary at infinity and the spectrum of its coefficient matrix lies in the interior of the left halfplane. All of these results are actually contained in the above mentioned paper of P. Boh!. Moreover, this paper also essentially contains (for finite-dimensional systems) the test for negativeness of the Bohl exponent formulated in Theorem 6.3, where the property of being stationary at infinity has been replaced by the condition S,.L for sufficiently small (but not arbitrarily small) c 0 and sufO. This result was later rediscovered by N. ficiently large (but not arbitrarily large) L Ljascenko [1,2] and M. A. Rutman [1]. The latter obtained this result for the infinite-dimensional case, generalizing the investigations of M; G. Krein, who had generalized and sharpened Persidskii's results. The following proposition is an elementary corollary of a basic result (concerning nonlinear equations) in the paper of Boh!. In order for an equation

>

>

dx/dt = A(t)x

(0 ~

t< 00)

with bounded continuous coefficients to have the property that sup , j y(t) - x(t)j Sn(~I. ~z). There then exists a pair of unit vectors Xk E ~k (k = 1, 2) for which II Xl + Xz II < o. We put Xl + Xz = x. Then Xk = Pkx and

1 = Ilxkll ~ Ilpkllllxll < Ilpkll 0,

whence l/llpkll < 0 and consequently l/llpk II ~ Sn(~I. ~z)· On the other hand, for aribtrary X E ~

and therefore' .

Sn(~l'~Z) ~ 2x~t

Ilxll _ _2_ Ilplxl - Ilplll'

Analogously, Sn(~-b ~z) ~ 2/llpzll.The lemma is proved. COROLLARY 1.1. The boundedness from above ofa set g> = {P} of projections in a Banach space ~ is equivalent to the boundedness from below of the set

2.

157

KINEMATICALLY SIMILAR EQUATIONS

{Sn (PfB, (/ - p)fB)lp E &} of angular distances between the subspaces PfB and their complements (/ - P)fB. Let P be a projection in a Banach space fB and operator such that U(';o) = I. We consider the projection function (~

U(~) (~ E

B) be an invertible

(1.18)

EB)

and the corresponding subspaces fB(~) = U(~)PfB. If the projection function (1.18) is bounded, we will say that the operator

U(~)

uniformly conjugates the projection P on B.

The significance of this definition is to be found in the fact that, when its condition is satisfied, the set {Sn(P(~)fB, (I - P(~»)fB)l~ E B} of angular distances is bounded from below. § 2. Kinematically similar equations 1. Definition and tests of kinematic similarity of equations. We consider the pair of equations

(t

E

J, k = 1,2).

(2.1)

We say that these equations are kinematically similar on the interval J if it is possible to establish between the totalities of all solutions of these equations a oneto-one correspondence Xz(t) = Q(t)Xl(t)

(t

E

(2.2)

J),

where Q(t) is a bounded linear operator function with a bounded inverse: 1IQ(t) II

~

qb

I Q-l(t) I ~

qz'

(t

E

(2.3)

.1').

Letting Uk(t, -r) denote the evolution operators of equations (2.1), we obtain from (2.2) the relation U 2(t, -r)X2(-r) = Q(t)U1(t, -r)Xl(-r),

which by virtue of the arbitrariness of Xl(-r) and the formula xzC-r) reduces to the equality U2(t, -r)Q(-r) = Q(t)U1(t,-r)

and, in particular, when -r

=

=

Q(-r)xl(r) (2.4)

0, to the equality Uz(t)CU 11 (t) = Q(t),

(2.5)

where C = Q(O). Since the operators 02(t) = U2(t)C and Ol(t) = U1(t) are fundamental operators of equations (2.1), the operator Q(t) can be written in the form ofa ratio offundamental operators of these equations:

158

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

Dz(t)fJ;I(t)

=

Q(t).

(2.6)

Equality (2.6) implies the differentiability of Q(t) and the equality Q'(t)

Az(t)Q(t) - Q(t)AI(t).

=

(2.7)

Suppose on the other hand that Q(t) is the solution of equation (2.7) satisfying the condition Q(O) = C, where C is an invertible operator. This solution, as follows from (111.1.25), has the form Q(t)

=

Uz(t)Q(0)U1I(t),

and hence the operator Q(t) is invertible for each t E f. Since (2.5) clearly implies (2.2) for xz(O) = CXI(O), all five of the relations (2.2), (2.4)-(2.7) are equivalent. This brings us to the following assertion. 2.1. In order for equations (2.1) to be kinematically similar on an interval it is necessary and sufficient that a bounded and boundedly invertible operator function Q(t) on f satisfy one of the folio wing conditions: a) The evolution operators of the equations are connected by the relation LEMMA

f

(2.4) b) For some invertible operator C connected by the relation

E

[58] the Cauchy operators of the equations are

(2.5)

c) There exist fundamental operators Dk(t) (k = 1,2) of equations (2.I)for which Dz(t)D1I(t)

=. Q(t).

(2.6)

d) The operator Q(t) satisfies the differential equation Q'(t)

= Az(t)Q(t) - Q(t)Al(t).

(2.7)

We note that two equations are kinematically similar on a semi-infinite interval if their coefficients coincide outside of a finite interval. For suppose for the sake of definiteness that f = [0, 00) and AI(t) = Az(t) when t ~ to > O. Then when t, 1: ~ to the evolution operators also coincide: UI(t, 1:) = Uz(t, 1:), and~hence

Consequently, for C

where

~

UI(t)

=

UzCt)

= Uz(t, to)U2(to) = UI(t, to)U2(to).

UI(t, to)UI(tO);

UZI(tO)UI(to) we have

159

2. KINEMATICALLY SIMILAR EQUATIONS when when

0 ~ t ~to, t ~ to,

i.e. condition (2.5) is satisfied. THEOREM 2.1. Equations kinematically similar on a halfline have the same upper Ljapunov and the same Bohl exponents. PROOF. From formulas (2.3) and (2.4) it follows that

(l/qlqz)IIU1(t + 'C, t)11 ~ Iluz(t + 'C, t)11

= QlqzllU1(t + 'C, t)ll,

which implies

InlIUz(t + 'C, t)11 _ In(QlQZ) ~ InlIUz(t + 'C, t)11 'C 'C 'C < In II U1(t + 'C, t) II In(QlQz) + -'C 'C . Passing to the upper limit as -

KB (U) Z -

'C

--+ 00

and t

--+ 00,

we obtain the required relation

-I' InIIUz(t+'C,t)11 -- -I' InIIU1(t+'C,t)11 -1m 1m t,

T--HX)

'r

t,

T-KJ()

KB

(U)

l'

'r

The coincidence of the lower Bohl and upper Ljapunov exponents is proved analogously. 2. Reducible equations. The equation dx /dt = A(t)x is said to be reducible according to Ljapunov if it is kinematically similar to a stationary equation dy /dt = By (B = const). Theorems 111.4.5 and 2.1 directly imply the following result. THEOREM 2.2. The upper Ljapunov and upper Bohl exponents of a reducible equation coincide. REMARK 2.1. It is not difficult to verify that the upper Ljapunov and upper Bohl exponents in the case in question are strict. By definition, the reducibility of an equation is equivalent to the relation

U(t)

= Q(t)eBt ,

(2.8)

where Q(t) and Q-l(t) are bounded continuous operator functions on J (one can easily see that it is always possible in this case to take C = J). We note that in the present case Q(t) satisfies the equation

Q'

= A(t)Q- QB.

(2.9)

3. Reduction to an equation with a Hermitian coefficient. Suppose now that the space ~ is a Hilbert space ~. We represent the operator A(t) by the sum of its real

160

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

and imaginary parts: A(t) = Aff/(t) x = V(t)y in the equation

+

iA;:lt), and make the change of variables

dx = A(t)x dt '

~

(2.10)

where V(t) is the unitary operator function satisfying the relations

liJ: = iA;:s(t) V,

V(O) = I.

Here y(t) satisfies the equation

iA;}Vy

+

Vy' = Aff/Vy

+

iA;}Vy

or, finally,

y' = V-I(t)Aff/(t)V(t)y.

(2.11)

Since the operator V(t) is unitary for each t, we obtain the following assertion. LEMMA 2.2. In a Hilbert space equation (2.10) is kinematically similar to equation (2.11), the coefficient of which is unitarily equivalent to the real part of the coefficient of the original equation.

4. Theorem on the decomposition of an equation in Hilbert space. We consider a direct decomposition of the space /p:

/P = /PI

+ /Pz + ... + /Pm

(2.12)

and let PI,,··, Pn be the corresponding projections. Applying the Cauchy operator U(t) of equation (2.10) to the decomposition (2.12), we obtain the new decomposition

/P = /PI(t)

+ /Pz(t) + ... + /Pit),

(2.13)

where /Pk(t) = U(t)/Pk (k = 1,.··, n). The projections corresponding to the decomposition (2.13) are similar to the projections P k and are expressed by means of the formulas

(t

E

f).

(2.14)

THEOREM 2.3: Suppose the projections P k are uniformly conjugated by the operator U(t) on an interval f. Then equation (2.10) is kinematically similar to an equatiqn

dy fdt

=

B(t)y,

(2.15)

the coefficient of which commutes with each of the projections P k (k = 1,.··, n) and which therefore decomposes into a system of independent equations in the phase spaces /Pk of the decomposition (2.12). The kinematic similarity transformation can be chosen so that the following estimate is satisfiedfor some constant c depending only on the projections P k:

2.

161

KINEMATICALLY SIMILAR EQUATIONS

IIB(t)II

ell A(t)11

~

(t E J).

(2.16)

PROOF. The operators (2.14) satisfy the conditions of Theorem 1.2 on J. There therefore exists a bounded invertible operator function S(t) = U(t)R-l(t) satisfying the conditions

(k

= 1,2,.··,n).

We will assume that an equivalent renorming of the space to make the operators P k Hermitian (see the proof of Theorem 1.2) has already been carried out. It then follows from (1.15) that n

R2(t)

= 1:; PkU*(t)U(t)Pk.

(2.17)

k=l

Equation (2.10) is kinematically similar to the equation (B(t) = R'(t)R-l(t)),

dy /dt = B(t)y

(2.18)

since R(t) is clearly a fundamental operator of the latter equation and the ratio U(t)R-l(t) = S(t) satisfies the kinematic siimlarity condition (2.6). Formula (2.17) shows that R(t) and hence B(t) commute with each of the Pk. We must still prove the last assertion of the theorem. Differentiating (2.17), we obtain the relation R(t)R'(t) + R'(t)R(t)

n

= 1:; PkU*(t)[A*(t) + A(t)]U(t)Pk. k=l

By virtue of the boundedness of the Hermitian operator A*(t) exist real constants a and f3 such that a(f,f)

~

([A*(t)

+ A(t)]f,f)

~

f3(f,f)

+ A(t)

there

(f E 4)).

Therefore the relation ([RR'

+ R'R]f,f)

n

=

1:; ([A* k=l

+ A]U(t)Pkf, U(t)Pd)

implies the estimate a 1:; (U(t)Pkf, U(t)Pd)

~

([RR'

+

R'R]f,f)

k

• which can be rewritten in the form a(R2f,f)

~

([RR'

+ R'R]f,f)

~

f3 (R2f,f),

or, settingf = R-lg,

allgl1 2~ ([R'R-l + R-IR']g, g) ~ f3llgI12. Thus we see· that the operator

(2.1 9)

162

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

t (R'R-l + R-lR') = t (E* + E) = Effl

II·

is bounded, it following from (2.19) that !!EffI(t)!! ~ IIA(t) It remains for us to make use of Lemma 2.2 and pass from equation (2.18) to the kinematically similar equation whosecoefficientB(t) = V(t)EffI(t) V-let) is unitarily equivalent to Effl • It is not difficult to see that commutability with the projections Pk is not violated here since the unitary operator Vet) also commutes with them. The operator B(t) which is unitarily equivalent to EffI(t) satisfies the same estimate IIB(t)1I ~ IIA(t)lI· The constant c in (2.16) arises in connection with the renorming of the space. REMARK 2.2: The Cauchy operator of equation (2.15) clearly also commutes with with each of the projections Pk (k = 1,···, n).

§ 3. Exponentially dichotomic equations 1. Basic definitions. In the preceding chapter we were primarily concerned with the equation dxfdt

= A(t)x

(3.1)

when its upper Bohl exponent is negative and hence when all of its solutions decrease exponentially as t ---> 00. We recall that such a situation arises for an equation with a constant coefficient A(t)=A when the spectrum a(A) lies in the interior of the left halfplane. A more complicated situation arises when the spectrum a(A) also contains a component lying in the interior of the right halfplane: (3.2) We recall that the operator A was then said to be exponentially dichotomic. It was shown in § II.2.1 that in this case the space decomposes into a direct sum ~

=

~+

+ Q:L

(3.3)

of subspaces such that the solutions initially in ~+ exponentially increase (decrease) as t---> + 00 (t---> - 00) whereas the solutions initially in ~_, as before, exponentially decrease (increase) as t---> + 00 (t---> - 00). We note that these solutions remain in the fixed subspaces ~+ and ~_ respectively. If P + and P _ are the spectral projections commuting with A that correspond to the decompositions (3.2) and (3.3), there exist positive constants Nand ).i for which

_II

~ Ne-v(t-s)

(t

II eACH) P + I

~ Ne- vCs - t )

(s ~ t).

lIeA(t-S) P

~

s),

(3.4)

We will see at once that an analogous description is possible for an equation with a variable coefficient A(t). And what is more, this description permits us to generalize a number of the results of the preceding chapter.

163

3. EXPONENTIALLY DICHOTOMIC EQUATIONS

DEFINITION 3.1. We will say that a regular ll exponential dichotomy holds on.Jf for the solutions of equation (3.1) (more briefly, the equation is e-dichotomic) if for some to E .Jf the space ~ decomposes into a direct sum (3.5) of closed subspaces such that the following conditions are satisfied: a) The solutions Xl(t) = U(t, to)x~ of equation (3.1) in the subspace ~l(tO) at t = to (x~ E ~1(tO) are subject to the estimate

Il xl(t)II

~ N 1e- v,(t-s)

Il xl(S)II

(t ~

s; t, sE.Jf)

(3.6a)

with some exponent VI > O. b) The solutions xz(t) = U(t, to)x~ of equation (3.1) in the subspace ~z(to) at t = to (x~ E ~z(to)) are subject to the estimate Ilxz(t)

I

~ N ze- V2 (s-t)

IIXz(s)II

(t ~

s; t, sE.Jf)

(3.6b)

with some exponent Vz > o. c) The angular distance between the subspaces ~1(t) = U(t,tO)~I(tO) and ~z(t) = U(t, to)~z(to) cannot become arbitrarily small under a variation of t; more precisely, there exists a constant T > 0 such that (t E .Jf).

(3.7)

Let us make some remarks concerning the above definition. REMARK 3.1. The choice of to is not important. In fact, a change in the value of to in formulas (3.5) - (3.7) will affect only the constants Nk (k = 1, 2). For this reason, in the sequel we will always put to = 0 (assuming that the interval .Jf contains this point) and use the more concise notation ~k(O) = ~k (k = 1, 2). The projections corresponding to the direct decomposition ~ = ~1 ~2 will be denoted by PI and P z. REMARK 3.2. Let Pk(t) (k = 1,2) be the mutually complementary projections on the subspaces ~it) (t E .Jf). Then

+

Pk(t)

=

U(t)Pk U-l(t)

(t E .Jf).

(3.8)

As follows from Lemma 1.1, condition c) in Definition 3.1 is equivalent to the uniform boundedness of the projections Pk(t): (k

= 1, 2; t E .Jf),

(3.9)

i.e. to the uniform conjugatability of the projections P k • REMARK 3.3. Suppose for the sake of definiteness that .Jf contains the halfline 1) For the sake of simplicity we consider a less general situation than the one described in the book of J. Massera and J. Schaffer [1], and for this reason we speak of a regular exponential dichotomy. The word regular will be dropped below.

IV.

164

EXPONENTIAL SPLITTING OF SOLUTIONS

[0, + (0). Then the solutions of equation (3.1) initially in )81 remain bounded on [0, + (0) while the solutions initially in )82 increase unboundedly as t---+ + 00. It turns out, moreover, that all of the solutions x(t) = U(t)xo for which Xo E )81 increase unboundedly as t---+ + 00. For suppose P2XO "# 0. We consider the solution

X2(t)

=

U(t)P2Xo

=

P2(t)U(t)xo

=

P2(t)x(t),

corresponding to an initial value P2Xo E )82' From (3.6b) and (3.9) we obtain the inequality

Ilx(s) I

1

;;;; Ilpz(s)II

Il x 2(S) I

;;;;

1

MN2 e V2 (s-t)

( = JN2 e (s-t) I U(t)P2xoll )

Il x2(t)II

(s;;;; t),

V2

(3.10)

the right side of which unboundedly increases as s---+ 00. Thus the subspace )81 consists of precisely the initial values x(o) of those solutions x(t) of equation (3.1) which remain bounded on [0, + (0). An analogous situation in which )81 is replaced by )82 holds on the left halfline. REMARK 3.4. Suppose J = [0, (0). As was indicated in the preceding remark, the subspace)81 is then uniquely determined by equation (3.1). On the other hand, the subspace )82 can always be replaced by another subspace ~2 that preserves the direct decomposition: )8 = )81 + ~2' It turns out that if condition c) of Definition 3.1 is satisfied, this direct decomposition also determines an exponential dichotomy of the solutions of equation (3.1) in which the constants ].Jk remain the same [it will be shown below that condition c) is a consequence of conditions a) and b) if A(t) is an integrally bounded operator function]. We prove this assertion by verifying condition (3.6b) for the solutions initially in ~2' Suppose Xo E ~2 and x(t) = U(t)xo. Then by virtue of (3.6a) and (3.6b)

Ilx(t)II

~

I U(t)P1x oll + I U(t)P2x oll

~ Nd p 1Xoil + I U(t)P2x oll ~ (Z~ 111;:::\\

+

1) I U(t)P2 x oll·

Combining this inequality with (3.10), we obtain the estimate

Ilx(t)11 ~ M(N1 IIp1X oil + N 2)e. IIp2X oil

V2

(S-t)llx(s)ll,

and ~t only remains to show that the quantity IIp1Xo 11/ IIp2Xo I is bounded for Xo E )82. The latter fact follows directly from Banach's theorem. For the operator P 2 is

165

3. EXPONENTIALLY DICHOTOMIC EQUATIONS

bounded and, as is easily seen, is a one-to-one mapping of)[32 onto ~2. Therefore the inverse mapping is continuous:

and, finally,

We now cite another, in certain cases more convenient, e-dichotomicity condition.

)..11

LEMMA 3.1. In order for equation (3.1) to be e-dichotomic on f with exponents > 0 and)..l2 > 0 it is necessary and sufficient that the conditions

I U(t)P1U-1(S) I I U(t)P2U-1(S) I with certain constants N k (k

~ N1e-v1(t-s)

(t ~ s),

(3.11a)

~ N 2e- Vl (s-t)

(s

~

(3.11b)

t)

= 1, 2) be satisfied on this interval.

PROOF. Inequalities (3.6) immediately follow from (3.11). For, when x(O) = E ~1 and t ~ s, we have

P1X(O)

Ilx(t)11

=

I U(t)P1X(O) I

=

I U(t)P U-1(S)X(S) I 1

~ N1e-v1(t-s)

Ilx(s)ll·

In addition, (3.11) also implies estimate (3.9). Inequality (3.6b) is proved analogously. Conversely, if in (3.6a) we consider the solution X1(S) with initial value x~ = P 1U-1(S)X (x E ~) and take into account (3.9), we obtain the following estimate fort~s:

IIU(t)P1U-1(s)xll

= IIU(t)x~11 ~ N1e-vl(t-S)IIU(s)x~11

I

~ N1e-v,(t-s) U(s)P1U-1(s)xll ~ N1Me-v1(t-s)

Ilxll·

The estimate for t ~ s is obtained in exactly the same way. We can simplify the e-dichotomicity conditions by making the additional assumption that the operator function A(t) is integrally bounded: t+1

J IIA(-c) Ild-c ~

M1

(t

E

f),

(3.12)

t

LEMMA 3.2. If the function A(t) is integrally bounded, condition c) of Definition 3.1 is a consequence of conditions a) and b). PROOF. Estimate (3.12) implies by virtue of Lemma 111.2.4 that I U(t emM1 for integral m.

+ m, t) I

~

166

IV. EXPONENTIAL SPLITTING OF SOLUTIONS

We consider, for some fixed t, a pair of unit vectors Xk(t) E ~h(t) (k = 1,2) and put xi.) = U(., t)Xk(t) (... E f). From (3.6a) and (3.6b) we obtain the estimates !!Xl(t I\xz(t

+ m)1\ + m)1\

~ N 1e- v,ml\xl(t)1\ = N 1e- p ,m, ~ N 2 1e v,mllxz(t)1I

=

N 2 1e v,m,

which show that II XI(t)

+

xz(t) II ~ e- mM, \I U(t

~

+ m, t)Xl(t) + U(t + m, t)X2(t) II e- mM'(l\xz(t + m)1I - IIxl(t + m)l\)

It follows from (1.16) that Sn(5~Mt),

7B z(t»)

~ 1m.

Since the constant 1m > 0 for sufficiently large m, the assertion is proved. 2. Preservation of e-dichotomicity under a kinematic similarity transformation. It is quite easy to establish the following assertion. THEOREM 3.1. If the equations dx fdt

=

Ak(t)X

(k

= 1,2; t Ef)

(3.13)

are kinematically similar and one of them is e-dichotomic on the interval f, this property (with the same exponentS))1 and ))z) is also possessed by the other equation.

PROOF. The Cauchy operators U1(t) and Uz(t) of the kinematically similar equations are connected by a relation of the form (see (2.5») U2(t)

=

Q(t)U1(t)C,

where Q(t) is a bounded operator function with a bounded inverse and C is a bounded invertible operator. Since UZ(t)Pk U 21(S)

=

Q(t)Ul(t)PkU~I(S)Q-l(S),

where Pk = CPkC-l, the assertion of the theorem immediately follows from Lemma 3.1. 3. Green functions. Bounded solutions of the inhomogeneous equation. We now show that an e-dichotomy plays the same role in an investigation of the boundedness of the solutions of the inhomogeneous equation dx fdt = A(t)x

+ f(t)

(t E f)

(3.14)

as condition (3.2) plays for an equation with a constant operator A. To this end we introduce a Green function analogous to the one considered in § IIA

3.

167

EXPONENTIALLY DICHOTOMIC EQUATIONS

Let PI and P z be a pair of mutually complementary projections: PI If U(t) is the Cauchy operator of equation (3.14). we put

G(t -r) ,

U(t)P1 U-l(-r) - U(t)PZU-l(-r) (t, -r E ~).

= {

+ Pz =

t>-r, t < -r

for for

I.

(3.15)

It immediately follows from the definition that the Green function G(t, -r) satisfies the following differential equations on ~ for t i= -r: oG(t, -r)

=

A(t)G(t, -r)

oG~; -r)

=

_

at

(3.16a)

and

G(t, -r)A(-r).

(3.16b)

At t = -r it has a discontinuity such that

G(-r

+ 0, -r) =

- G(-r - 0, -r)

=

+ P Z)U-l(-r) =

U(-r)(P1

+

U(-r)P 1 U-l(-r)

U(-r)PZ U-l(-r)

U(-r)U-l(-r)

= I,

(3.17a)

and analogously

G(t, t

+ 0)

- G(t, t - 0) = - I.

Suppose the functionf(t) is continuous on

g(t)

=

~.

(3.17b)

We consider the integral

S G(t, -r)f(-r)d-r

(3.18)

under the assumption that it exists and permits differentiation under the integral sign. Ifwe set ~ = [a, b] and write t

g(t)

=

b

S G(t, -r)f(-r)d-r + S G(t, -r)f(-r)d-r, t

a

we get

g'(t)

=

G(t, t - O)f(t) - G(t, t

+ O)f(t)

tab a

+ S at G(t, -r)f(-r)d-r + S at G(t, -r)f(-r)d-r a

t b

=

f(t)

+ S A(t)G(t, -r)f(-r)d-r

=

A(t)g(t)

+ f(t).

a

Thus formula (3.18) provides a solution of the inhomogeneous equation (3.14) on ~ as long as the above calculations are permissible. This is always the case when ~ is a finite interval; but special estimates, which can be carried out if equation (3.1) is e-dichotomic, are needed for an infinite or semi-infinite interval.

168

IV. EXPONENTIAL SPLITTING OF SOLUTIONS

Suppose p] and P 2 are the projections corresponding to a direct decomposition = ~] + ~2 for which equation (3.1) is e-dichotomic. The Green function obtained when these projections are substituted in (3.15) is called a principal Green function of equation (3.14). This is the Green function that we will normally make use of in the sequel; for this reason we will usually drop the term principal. Using estimates (3.11), we easily obtain the following estimate for a principal Green function: ~

(3.19) We recall that C(~) denotes the Banach space of bounded continuous functions on J with values in ~ and norm = supJ f(t) II· Estimate (3.19) immediately implies the following result.

I I fill

I

THEOREM 3.2. If equation (3.1) is e-dichotomic on J, the inhomogeneous equation (3.14) has at least one solution x(t) E C(~) for each function f(t) E C(~). This solution is given by the formula x(t) =

S G(t,7:)f(r)dr,

(3.20)

J

where G(t, 7:) is a principal Green function of equation (3.1).

PROOF. Estimate (3.19) implies that the integral (3.20) converges and (3.21) It is easily seen that differentiation under the integral sign is valid in the present case and that therefore, as was shown above, the function (3.20) satisfies equation (3.14) REMARK 3.5. Suppose J = [0, (0). In this case the bounded solution 00

x(t) =

S G(t,7:)f(7:)d7: o

of equation (3.14) on J has the initial value 00

x(O) =

S G(O, 7:)f(7:)d7: = -

o

00

P2

S U-l(7:)f(7:)d7:, 0

belonging to ~2' We obtain the general form of the bounded solutions on [0, (0) of equation (3.14) by adding to the solution already obtained an arbitrary bounded solution of the homogeneous equation (3.1). These are precisely the solutions that are initially in ~1'

3.

EXPONENTIALLY DICHOTOMIC EOUATIONS

169

Thus all of the bounded solutions on [0, 00) of equation (3.14) are represented by the formula x(t) = U(t)y

+ f

(3.22)

G(t, -r)f(r)dr,

o

where y = P1x(0) is an arbitrary element of ~l' REMARK 3.6. The solution (3.20) remains bounded when the boundedness condition for f(t) is replaced by the more general integral boundedness condition t+1

f I f(-r) Ild-r ~ MI'

t

For in this case we have the estimate

Ilx(t) I

~

f I G(t, -r) IIII f(-r) Ild-r J

=

f IIG(t, -r)llllf(-r)lld-r + f IIG(t, -r)llllf(-r)lld-r t?:1: N2 f e-v,(r-t) I f(-r) Ild-r + NI f e-v,(H) Ilf(-r)lld-r

t~t:

~

t~1:

~ N2

f s~o

f~T

e-v,Sllf(t

+ s)llds + Nl f

ev,sllf(t

+ s)llds

s~o

4. e-dichotomicity on a halftine. As was just shown, the e-dichotomicity of equation (3.1) is a sufficient condition for the existence of bounded solutions of the inhomogeneous equation with a bounded free term. In order to explain the extent to which this condition is necessary we must introduce some additional assumptions. We consider the equation on the halfline ~ = [0, 00). The linear manifold ~I consisting of the initial values Xo of the solutions of equation (3.1) that are bounded on [0, 00) is called the (right) !/'-set of this equation. We will assume that ~l is a complemented subspace, i.e. that it is closed and has a direct complement: ~ = ~l + ~2' In the finite-dimensional case this condition is automatically satisfied. In a Hilbert space ,p th~ second part of the condition is superfluous since an orthogonal complement always exists in ,p. We note that this condition is essentially contained in the definition of e-dichotomicity of an equation. It is trivially satisfied for equations with a constant operator whose spectrum does not intersect the imaginary axis. It is easily seen that the closedness and complementedness properties of the !/'-set are not violated under a kinematic similarity transformation. We will subsequently see that they are also preserved under "small" perturbations of e-dicho-

170

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

tomic equations. All of these transformations permit one to widen the class of equations for which the Sf'-set has the indicated properties. If an equation is also defined on the left halfline, the left Sf'-set (to which all of the remarks made above for the right Sf'-set apply) is introduced analogously. LEMMA 3.3. Suppose that equation (3.14) has for each function f(t) least one solution x that is bounded on [0, (0):

Illxlll

=

sup

o::;::t KB(P1), - ).)3 < KB(P3), V3 > KB(P3) and).)2 < KB(P2) ().)1 > ).)3 > 0, ).)2 > V3 > 0), there exist positive numbers Nk (k = 1,3) such that the following estimates are satisfied on J for any x E ~ and t ~ s:

II U(t)P1xll ~ N1e-v,(t-s) II U(s)P1x ll; I U(t)P2x ll ~ N 1ev,(t-s) II U(s)P2x ll; N 31e- v3 (t-s) I U(s)P3x ll ~ II U(t)P3x ll ~ N 3eV3 (t-s) I U(s)P3x ll· .c

z

I

(4.3)

Let us determine some conditions under which in this case the equation

dxfdt = A(t)x has bounded solutions on J. We introduce a Green function by setting

+ /(t)

(4.4)

176

IV. G (t, s)

EXPONENTIAL SPLITTING OF SOLUTIONS

=

s < t, t < s.

{U(t)P1U-l(S) for _ U(t)(P2 + P3)U-l(S) for

(4.5)

Then at least for bounded functionsf(t) with compact support the formula x(t)

=

f G(t, s)f(s)ds

(4.6)

provides certain solutions of equation (4.4). It can be written more explicitly in the form x(t)

f U(t)P1U-l(S)f(s)ds - f U(t)P2 U-l(S)f(s)ds - f =

s 0 and a sufficiently small 0 > o. We wish to determine the nature of the response of the Bohl exponents of an equation to a perturbation. Inasmuch as this has already been done for the upper and lower exponents (see Theorem 111.4.6), we need only examine the behavior of the ends of each gap in the Bohl spectrum. By transforming equation (5.1) with the use of the substitution x = yeAt, we can always shift the point].l = 0 to the center of some gap. The equation then becomes e-dichotomic, i.e. its solutions will satisfy estimates (3.6) and (3.11), in which].ll = ].12 (and it can be assumed that Nl = N 2). We see that it suffices to show that the exponents ].11 and ].iz of the equation are stable in this case. 2. Bounded solutions of the perturbed equation. We first consider the case when oF = [0, (0). Suppose an e-dichotomy of the solutions of equation (4.1) exists for a direct

5.

STABILITY OF BOHL EXPONENTS

179

decomposition ~ = ~1 + ~z with corresponding projections PI and P z• We must first establish the existence of a similar decomposition for the perturbed equation. Let x(t) be a bounded solution on [0, (0) of equation (5.2). It can be regarded as a bounded solution of the equation dx/dt

=

A(t)x

+

f(t),

corresponding to the integrally bounded functionf(t) = B(t)x(t). Formula (3.22) permits one to represent this solution in the form 00

x(t) = U(t)y

+

JG(t, 1:)fC'r)dr

(y

E ~1)

o

and thus to obtain the integral equation 00

x(t) = U(t)y

+ JG(t, 1:)B(1:)x(1:)d1:.

(5.4)

o

Here U(t) and G(t, s) are the Cauchy operator and Green function of the unperturbed equation, and y is a vector of the subspace ~1. It is not difficult to verify that one also has the converse assertion: every bounded solution of the integral equation (5.4) also satisfies the differential equation (5.2).

°

LEMMA 5.1. There exists a 0> such that the satisfaction of condition (5.3) implies the existence of a unique bounded solution on [0, (0) of equation (5.4). This solution is representable in the form

x(t) = F(t)y

(5.5)

where F(t) is a bounded operator function on [0, (0). PROOF. We consider in the space C(~) of bounded continuous vector functions on [0, (0) the linear operator Sx(t) = G(t, 1:)B(1:)x(1:)d1:. It maps C(~) into itself, as follows from Remark 3.6. Using formulas (3.11) for Nl = N z = Nand ))1 = ))z = )), we estimate its norm. We have for n = [t /1:0]

J;'

IllSxll1

00

~ N ..

~

N

J0 e~"lt~TIIIB(1:)lld1:IIIXIII

IIIXIII1:00 t~1 e~"(n~k)To + 1 + k l L r"(k~n~Z)To }

~ NIIIXIII1:00 { 1 + Thus under the condition

1_

~~"TO}.

(5.6)

IV.

180

a
0, i.e. it can be replaced by a condition of type (111.4.16). REMARK 5.3. An analogous result holds when equation (5.1) admits an exponential splitting on the real line. As before, it suffices to consider the case of an e-dichotomy. We can consider equation (5.2) on each halfline (right and left) separately. By virtue of Theorem 5.1 it can be asserted that for sufficiently small ;; > equation (5.2) is e-dichotomic on each of the halflines, the dichotomy on the right (left) halfline being connected with the subspaces ~I and lB2 (lB l and ~2)' where ~I = (/ - P 2RPI )lB l and ~2 = (/ - P I R'P2)lB 2 • Also the operator

°

S

= (/ -

P 2RPI )PI

+ (I -

P I R'P2)P2

=/-

(P2RPI

+ PI R'P2)

is bounded for a sufficiently small (possibly smaller) ;; > 0, invertible (since the operators Rand R' turn out to be small) and such that ~I = SlB l and ~2 = SlB2. Therefore lB = ~I + ~2' It follows from Remark 3.4 that the pair ~b ~2 is connected with an e-dichotomy of the solutions of(5.I) on each of the halflines and hence on the whole real line. 4. Stably decomposing equations. In this subsection we consider equation (5.1) under the assumption that it admits an exponential splitting on an infinite or semi-infinite interval and that the operator A(t) commutes with the projections PI,,", Pn inducing this splitting: A(t)Pk = PkA(t) (k = 1,.··, n). Such an equation decomposes into a system of independent equations in the phase spaces lBk = PklB:

(k

=

1,2,.··,n),

and the Bohl intervals [Vk, Vk] of these equations are pairwise disjoint. We now consider the perturbed equation (5.2) and show (Theorem 5.2) that if the integral boundedness condition (5.3) is satisfied with a sufficiently small ;; > 0, equation (5.2) is kinematically similar to an equation of the form dx jdt

=

[A(t)

+

C(t)]x,

(5.15)

where the operator C(t) is sufficiently small and commutes with the projections Pk •

From this result it will follow independently of Theorem 5.1 that the exponential splitting property is stable for an equation of the described special (quasidiagonal) form. We note that this permits us to eliminate the requirement of integral boundedness of A(t). It will be shown in the next section that any equation of form (5.1) admitting an

184

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

exponential splitting in a Hilbert space is kinematicaHy similar to a system of independent equations of the described type. We first note that if equations (5.2) and (5.15) are kinematically similar the, transforming operator Q(t) = U(t)V-l(t), where U(t) and Vet) are fundamental operators of these equations, must satisfy the equation Q'(t)

=

[A(t)

+

B(t)]Q(t) - Q(t)[A(t)

+

C(t)]

(5.16)

= A(t)Q(t) - Q(t)A(t) + B(t)Q(t) - Q(t)C(t).

We need only select the operator C(t) so that equation (5.16) has a solution Q(t) that is bounded together with the operator function Q-l(t). It is necessary in this connection that C(t) be of quasidiagonal form: (5.17) We will seek a solution in the form Q(t) = 1 + Set), where the operator Set) is small enough to guarantee the invertibility of the operator Q(t). For Set) we have from (5.16) the equation S'(t)

=

A(t)S(t) - S(t)A(t) + B(t)[1 + Set)] - [I + S(t)]C(t).

(5.18)

A solution Set) of equation (5.18) is a solution of the equation S'(t)

(5.19)

A(t)S(t) - S(t)A(t) +F(t)

=

where F(t)

= B(t)[1 + Set)] - [I + S(t)]C(t).

(5.20)

Equation (5.19) has been considered in the preceding section. The formula Set)

=

L:

Pj

J U(t, s)F(s)U(s, t)dsPk

s~t

jk

Pj

(5.21)

JU(t, s)F(s)U(t, s)dsPk s;;;;t

represents the bounded solutions of equation (5.19) satisfying the condition (m = 1,2,.··,n),

(5.22)

provided F(t) satisfies the same condition: PmF(t)Pm = 0 (m = 1,···,n). The latter condition, as follows from (5.17), (5.20) and (5.22), reduces to the relation '" PmB(t)[1

+

S(t)]Pm = Pm[I

+

S(t)](f PkC(t)Pk)pm

Pm[1

+

S(t)]PmC(t)Pm.

=

It is satisfied if one puts C(t) = B(t)[1 + Set)] =

L: PkB(t)[1 +

S(t)]Pk.

k

Taking into account relation (5.22), we obtain ~

..

5.

F(t)

= =

B(t)[I + Set)] - [I + S(t)](f PkB(t)[I + S(t)]Pk )

I: PjB(t)[I + S(t)]Pk - Set) I: PkB(t)[I + S(t)]Pk

k#

=

185

STABILITY OF BOHL EXPONENTS

(5.23)

k

I: Pj[I - S(t)]B(t)[I + S(t)]Pk. k*j

The problem consequently reduces to the determination of a solution S(t) (III Sill

< 1) of the integral equation obtained from (5.21) by substituting for F the transform F[S] defined by the right side of expression (5.23). Denoting the transform defined by the right side of (5.21) by L(F), we write this equation in the form

S

=

(5.24)

L(F[SD.

LEMMA 5.2. For any 0 < q < 1 there exists a number 0 > 0 such that under condition (5.3) equation (5.24) has a unique solution subject to the estimate

(t

IIS(t)II ;;i; q

E

J).

(5.25)

This solution satisfies condition (5.22). PROOF. We will prove the lemma for n = 2. Since the reduction of an equation to a quasidiagonal form can be carried out in successive steps in each of which the space is decomposed into a direct sum of two summands, the general case will then follow. In this simplest case equations (5.21) can be written in the form

Set)

=

PI

J U(t, s)[I -

S(t)]B(t)[I

+

S(t)]U(s, t)dsPz

S(t)]B(t)[I

+

S(t)]U(s, t)dSPl~Ll[S].

(5.21')

s~t

- Pz

J U(t, s)[I -

s?;;t

We recall that, inasmuch as equation (5.1) admits an exponential splitting, we have the estimates II PI U(t, s)11 = II U(t)P 1 U-l(S) II ;;i; N 1ev,(t-s) IlpzU(t, s) II = II U(t)PZU-l(S) II ;;i; N 2e- v,Ct-s)

(t ~ s), } (t ;;i; s),

(5.26)

in which ).)1 < ).)z· We consider th~ metric space Kq of operator functions Set) satisfying condition (5.25) with metric IIIS2 - sllll = supIIS2(t) - SI(t)II· 5

From (5.26) and (5.3) we obtain the estimate IIIL1[S] III ;;i; N 1N 2(1

+

IIISIII)2

J e- to, the spectra (J(A(t») lie in the interior of the union of these rectangles while their resolvents are uniformly bounded on the contours r k of these rectangles, the corresponding constants depending only on the quantity $ and the w-limit operators of A(t). Since Az(t) = A(t) - Al(t), it follows from the first estimate of (6.8) for a sufficiently small > 0 that these same properties will also be possessed by the operator function Az(t). By varying A(t) on a finite interval (and hence changing the equation to a kinematically similar one), we can establish that all of these properties as well as estimates (6.8) will he fulfilled on the whole interval [0, (0). In view of Theorem 6.2 it suffices to prove the theorem for the equation

a

dx Idt = Az(t)x.

(6.9)

We introduce the projections (6.10) They are continuous and differentiable together with the operator AzCt); in fact,

190

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

(6.11)

where

= 21k

Cl

'IT:

sup ).E1,

I [Az(t) -

AI]-I\\

tE [0,00)

(lk

is the length of the contour

rk).

With respect to the projections (6.10) it is possible to construct a conjugation operator O(t) solving the Cauchy problem O'(t) =

n ~

Pk(t)Pk(t)O(t),

0(0)

= I,

(6.12)

k=1

and having the property (6.13)

Since the operator functions Pk(t) (k = 1,,,,, n) are bounded, O(t) uniformly conjugates the projections Pk(O). From Theorem 2.3 it follows that equation (6.12) is kinematically similar to an equation dy fdt = B(t)y the coefficient B(t) and Cauchy operator W(t) of which commute with the projections Pk(O). Let Q(t) be the operator effecting the similarity transformation. Then O(t) = Q(t)W(t)

(6.14)

and from (6.13) it follows that Pk(t) = Q(t)W(t)Pk(O)W-l(t)Q-l(t) = Q(t)Pk(O)Q-l(t).

(6.15)

We note that by the definition of kinematic similarity there exists a constant q > 0 such that

IIQ(t)11 ;£ q, IIQ-l(t)ll;£ q. We now carry out the transformation x into the form ~

tJ:

=

(6.16)

Q(t)y in equation (6.9). It goes over

= [Q-l(t)Az(t)Q(t) - Q-l(t)Q'(t)]y.

(6.17)

The operator Az(t) = Q-l(t)A z(t)Q(t), as follows from (6.15), commutes with the projections Pk(O) , and the spectrum of its restriction to a subspace !Ok = PlO)!O coincides with the part of the spectrum li(Az(t») lying in the interior of

ne,h k

•

It is not difficult to get, using (6.14), that

191

EXERCISES Q'(t)Q~l(t)

=

O'(t)O~l(t)

- Q(t) W'(t) W~l(t)Q~l(t)

n

= I: P£(t)Pk(t) k=l

-

Q(t)B(t)Q~l(t).

From Theorem 2.3 it follows that for some C2 > 0 IIB(t) II

~ c211 ~1 P£(t)Plt) II·

V sing estimates (6.10) and (6.15), we now get that there exists a constant for which

II Q'(t)Q~l(t) I

C3

> 0

~ C30.

We can now estimate the derivative of the operator function A2(t): A2(t) = Q~l(t)A2(t)Q(t) - Q~1(t)Q'(t)Q~1(t)A2(t)Q(t) -

=

Q~1(t)A2(t)Q'(t)

Q~l(t)A2(t)Q(t)

-

-

Q~1(t)Q'(t)Q~l(t)A2(t)Q(t)

Q~l(t)Az(t)Q'(t)Q~l(t)Q(t),

and hence where C4 is a positive constant. With the above estimates we can complete the proof. The equation dy /dt = A2(t)y is equivalent to the system of independent equations (k = 1,2,. ··,n)

in the subspaces {lk = Pk(O){l. We can apply Theorem II1.6.6 to each of these equations and conclude that for a sufficiently small 0 > 0 their Bohl intervals lie within the intervals ().Jk - e /2, ).Jk + e /2). It remains to apply Theorem 5.2 to equation (6.17), which by virtue of (6.15) differs sufficiently little from (6.16). It is not difficult to verify that all of the constants contained in the estimates depend only on e and on estimates of A(t) and its resolvent for sufficiently large t, i.e. in the final analysis on the number e and the w-limit operators of A(t). The theorem is prov€d. EXERCISES 1. Prove the existence of the conjugation operator Q(t, r) described in Theorem 1.1 by assuming (in place of differentiability) that the Pk(t) are continuous functions of bounded variation (Ju. L. Daleckii [3]). Hint. Replace the differential equation (1.10) by an integral equation and make use of the multiplicative Stieitjes integral. 2. Show that the bounded variation condition in Exercise 1 is essential even in a two-dimensional space (L. A. Ivanov [1]).

192

IV.

EXPONENTIAL SPLITTING OF SOLUTIONS

Hint. Consider the operator pet) of orthogonal projection on the straight line forming an angle

oct) with the axis of abscissas. Show that the product P(tn)P(tn-I)···P(tI) has a limit if and only if OCt) is a function of bounded variation. 3. Let P(t) (a ~ t ~ b) be a continuous projection function in a Hilbert space .p. Construct a unitary operator Q q(t) having the property .p(t) = Q It)-p(a), where .p(t) = pet})). Hint. Make use of Theorem 1.2 and the following result (I. C. Gohberg and M. G. Krein [3]): the norm of the difference of two projections on a pair of subs paces achieves a minimum when these projections are Hermitian. 4. Let A(z) be a matrix function of order n that analytically depends on z in some simply connected domain G of the complex plane. Suppose that for each z E G the spectrum O"(A(z» decomposes into spectral sets: O"(z)

== O"(A(z»

= O"I(Z) U O"zez) U ... U O"p(z),

each of which continuously depends on z (in the sense that a contour r k separating 0" iz) from the rest of O"(z) for some z E G will continue to do so under a small variation of z). Show that there exists a matrix U(z) analytically depending on z in G that reduces the matrix A(z) to the quasi diagonal form

Al (z) !

••••• , •• , ••• "', •• ,•• 03 •••••••••••• ",.,, •• ·_

~ A2 (z) ~

"""""""""""~

U-I(z)A(z)U(z) =

where Ak(z) is a matrix function of order nk (k = 1,.··, p) not depending on z (nk= dim ~iz), where ~k(Z) is the invariant subspace of A(z) corresponding to the spectral set O"iz». Hint. Consider the differential conjugation equation (1.10) in a domain of the complex variable z. The matrix U(z) can be expressed in terms of its solution, which is analytic in z. The result noted in this exercise is essentially contained in a paper of Ju. L. Daleckii and S. G. Krein [2]. It was later published by Y. Sibuya (see W. Wasow II]). 5. Show that if the equation x = A(t)x with an integrally bounded coefficient admits an exponential splitting and SO' II B(t) Iidt 00, the equation




a>

+

193

EXERCISES

7. Let A be an operator of simplest type (see Exercise I.1) and let R(t) be an operator function satisfying the condition

f II R(t) II dt o

< 00.

(0.1)

a) Show that the equations dx/dt

= Ax

(0.2)

and dx/dt = Ax

+ R(t)x

(0.3)

are kinematically similar (this means that equation (0.3) is reducible to the form (0.2 ). Hint. By representing the operator Q(t) effecting the kinematic similarity transformation in the form Q(t) = I + Set), obtain for Set) the equation S'(t) = [A

+ R(t)]S(t) -

S(t)A

+ R(t).

(0.4)


0 for 1>0 by virtue of condition 1». Let .9'(/) (- 00 1 00) be a locally integrable operator function with values in [.p]. Exercises 12-14 below will be formulated for the following equation in .p:


o: 1) Re.9'(t)« - al (- 00 1< (0). Then for any(- 00 0

A(t + T) = A(t)

(0

~ 1

< 00).

The Cauchy operator U(I) of equation (1.1) solves the Cauchy problem 198

(1.2)

199

1. MONODROMY OPERATOR

I

d~?)

= A(t)U(t),

(1.3)

U(O) = 1.

It is easily seen that the same problem is also solved by the operator

U1(t) = U(t

+

T)U-1(T).

By virtue of the uniqueness of the solution of problem (1.3) we have U1(t) == U(t), which implies U(t + T) = U(t)U(T). The operator U(T) is called the monodromy operator of equation (1.1). We recall that the spectral radius of U(T) is the radius r of the minimal circle containing its spectrum. This radius is given by the formula r = limk~oo {I II Uk(T) II. (see Chapter I, § 2). We now prove the following assertion. THEOREM 1.1. The periodic equation (1.1) has strict upper Ljapunov and strict upper Bohl exponents which coincide with the logarithm of the spectral radius of its monodromy operator divided by the period: IrL

=

IrB

= T-1 In r.

PROOF. By virtue of the continuous dependence on t of the operators U and U-1 there exists a q such that II U(1:) II ~ q, II U-1(1:) II ~ q for 0 ~ 'C ~ T. Suppose t = nT + 'C and t' = mT + 'C' ('C, 'C' E [0, T]). Then U(t) = U('C)Un(T) and U(t') = U('C')Um(T), which implies U(t, t') = U(t)U-1(t') = U('C)Un-m(T)U-1('C'), and hence (l/q2) II Un-m(T) II ~ II U(t, t') II ~ q211 Un-m(T) II. Taking the logarithm of this system of inequalities and dividing each term by

t - t' (> 0), we get In q2

-~ +

In II Un-m(T) II (n - m)T + ('C -'C')

_ < InIIU(t,t')11
0 (0 < q < 1) for which (n = 1,2,.· .).

(2.3)

To prove the required assertion it suffices to show that the upper Bohl exponent "B(P1) is negative while the lower Bohl exponent "f,,(P2) is positive. We make use of formulas (111.4.10). Let t1 = nIT + Z"1 and t2 = n2T + Z"2 (Z"h 'Z"2 E [0, T]). Then

U(t 2)Pk U-1(t 1)

=

U('Z"2)[Un'(T)Pk U-n,(T)]U('Z"1)

=

U('Z"2)[Un,-n,Pk ]U('Z"1),

and conseque1!tly II U(t2)Pk U-1(t 1) II ;£ M21IUn,-n'lllB., where M = sUPo~ 0; in addition, according to the preceding theorem the operator U(T; A)

2.

EXPONENTIAL DICHOTOMY OF SOLUTIONS

207

is a u-dichotomic W-contraction for sufficiently small A > O. But then it is a u-dichotomic W-contraction for all A > 0 (see Corollary l,7.2). Thus the assertions of Theorem 2.2 and Remark 2.1 hold for the operator U(T;A) for all A > 0 and, in particular, for A = 1. The theorem is proved. 3. The periodic equation of second order. We consider in .p the equation

+ .9(t)y =

0

with a periodic operator function .9(t) (= .9(t Setting

+

d 2y /dt 2

z =

:'i:, x = (~) (=

(2.9) T»).

EB z),

Y

we reduce equation (2.9) to the following equation of first order in the phase space .p(2)

=

.p EB .p:

dx /dt

(2.10)

d(t)x,

=

where sl(t) = (_

~(t) ~)

(I = /.5)'

(2.10')

The monodromy operator U(T) of equation (2.10) will also be called the monodromy operator of equation (2.9). It is easily seen that it has the form U(T)

=

(T»)

IfJ 1fJ'(T) ,

C/J( T) ( C/J'(T)

(2.11)

where C/J(t) and lfJ(t) are the solutions of the equation in [.p] d 2 Y/dt 2

+ .9(t)Y =

0

distinguished by the initial conditions C/J(O) 1fJ(0)

= =

I, C/J'(O) 0, 1fJ'(0)

=

0;

= I.

We introduce an indefinite operator which belongs to [.))(2)] and whose spectrum consists of the points ± 1 : W= (

0

-I

- oI) .

(2.12)

We have

-

Wd

+ d* W

=

(.9 +0 .9* _ 210) .

This relation together with Theorem 2.3 permits us to prove the following proposition.

V. EQUATION WITH PERIODIC OPERATOR FUNCTION

208

THEOREM 2.4. Suppose Re&P(t)

«0

uniformly in t

E ( - 00,

(0).

(2.13)

Then the phase space .p(2) decomposes into a direct sum of two subspaces ~± that are invariant under the monodromy operator U(T) of equation (2.9) and such that the spectrum of U(T)I~+ (U(T)I~_) lies in the interior (exterior) of the unit disk. Each of the subspaces ~± can be given by the equation z = ± K ±y,2l where the operators K± (E [.pD are invertible uniformly dissipative operators.

The operator K+ (K_) will be called the plus (minus) equation operator of the periodic equation (2.9). PROOF. The fulfilment of condition (2.13) implies the fulfilment of the condition of Theorem 2.3 for the corresponding equation (2.10). Therefore the first assertion of Theorem 2.4 is a corollary of Theorem 2.3. Moreover, according to Theorem 2.3 it can be asserted that the subspace ~+ (~_) in the decomposition .p(2) = ~+ + ~_ is uniformly W-positive (W-negative). Thus in regard to ~+, for example, we can assert the existence of a constant m (> 0) such that (Wx, x) ~ m(x, x) for x E ~-'-. Since W has the form (2.12), this property means that - (y, z) - (z, y) ~ m(IYI2

+

(x = y EB

Iz12)

Z E ~+)

or, equivalently Ily - Zll2 =

Ily

+ Zll2

~ 2m (lIyll2

m(lIy - Zll2

+

lIy

+

+

Z1l2)

I z 1l2) (x = y EB ZE~+).

(2.14)

Let S± denote the operators mapping ~+ (c .p(2)) into .p according to the rule S±x = y Z (x E ~+). If S+x = y - Z = 0 for some x E ~+, it will also be true by virtue of (2.14) that y + Z = 0, i.e. x = y = Z = O. Thus the operator S+ is a one-to-one mapping of ~+ onto ilR = S+~+, i.e. to any u E ilR there corresponds a unique pair y, Z E.p such that u = y - Z, y EB Z E ~+. We put +u = y + Z for any u E ilR. Then inequality (2.14) can be written In

+

.Y'

the form lIull2 -

II.Y'+u1l 2 ~ m (lIull2 + II.Y' +uIl 2),

from which if follows that m < 1 and (u

Thus the linear operator Since it follows from y -

E

ilR).

.Y' + acting from ilR into .p is a uniform contraction. Z = u and y + Z = .Y' +u that

2) We say that a subspace E c .pI EB .p2 has the equation X2 = KXb where K E [.ph .p2], if E = [x EB Kx Ix E .pI}' The operator K is called the equation operator for E (more precisely, the equa-

tion operator for E with respect to (.ph .p2»'

2.

209

EXPONENTIAL DICHOTOMY OF SOLUTIONS

21

y =

(u

+ ff +u),

Z

=

21 (ff +u

- u),

we have (2.15)

Let us show that iJR = .p. Suppose iJR is a proper part of .p. Then ff + has an extension ffE[.p] with norm Ilffll = Ilff+II(~[(I-m)/(l +m)]1/2). Such an extension can be obtained, for example, by defining ff on the closure m as the closure of ff + by continuity and setting ff equal to zero on .p iJR. After this, by setting

e

2

=

{(u

+

ffu) EB (ffu - u)lu

E

N,

we obtain a subspace 2 which contains 2+ as a proper part (the operators / ± ff are continuous one-to-one mappings of ,'0 onto itself) and, in addition, is uniformly W-positive (also inequality (2.14) will be satisfied with the same m for all x E 2). But if 2 is an extension of 2+, it follows from 2+ + 2- = .p(2) that the intersection of 2 with 2- contains nonzero elements. Since 2- is a W-negative subspace, we have arrived at a contradiction. Thus iJR = .p, and it therefore follows from (2.15) that 2+

=

{y EB K+yly

E

''o},

(2.15')

«

where K+ = (ff + - /) (/ + ff +)-1. Since Ilff + II < 1, we get Re K+ 0 (see § 1.7). Thus the assertion of the theorem concerning the subspace 2+ is proved. The assertion concerning the subspace 2- can be proved analogously. The theorem is proved. 4. Equation operators as fixed points. We consider a vector x = y the monodromy operator (2.11) into the vector U(T)x

=

Since this vector also lies in

(I/!(T)y

EB K+y E

.I.? + . It

is mapped by

+ IJf(T)K+y) EB (I/!'(T)y + 1Jf'(T)K+y).

.I.? +,

(2.16) In view of its arbitrariness the vector y in this equation can be dropped. We put ¢(Z)

= (I/!'(T) + IF'(T)Z)(I/!(T) + IJf(T)Z)-1

for those Z for which the right side is meaningful. It can be shown (see Exercise 1.41) that under condition (2.13) of Theorem 2.4 the linear fractional function ¢(Z) is meaningful for any Z in the interior 09(f,) = {ZIZE [{ij, ReZ«Oj

of the left operator halfplane and maps it into itself. Relation (2.16) shows that K+ (E 09) is a fixed point of the mapping ¢: ¢(K+) = K+. It can be shown that K+ is the only fixed point of ¢ in 09. Moreover (see Exercises IV.13 and 1.41), for any

v.

210

EQUATION WITH PERIODIC OPERATOR FUNCTION

ZoEm

= n->oo lim

K+

Z.,

where Zn

= ¢(Zn-l)

(n

= 1,2,···).

(2.17)

We consider the linear fractional transformation ¢-I(Z)

=-

(W'(T) - ZW(T»-l((j)'(T) - Z(j)(T»,

= ¢-I(¢(Z» = Z. It maps the interior n~(i) = (Z IZ E [i)], Re Z » 0)

which is the inverse of ¢: ¢(¢-I(Z»

of the right operator halfplane into itself and its only fixed point in n~ is - K_, for which there exists a formula analogous to (2.17).

5. Splitting of the solutions by means of the initial conditions. Let r ± denote the spectral radii of the restrictions of the operators U(T) and U~1(T) to 53+ and 53~ respectively. Then the quantities "B(P +) = In r + « 0) and "B(P~) = - In r ~ (> 0), where P ± are the spectral projections of U(T) corresponding to the subspaces 53±, will be Bohl exponents of equation (2.10) in the phase space .p(2). Thus for any c > 0 there exists an N, > 0 such that Ilx(t) I ~ N,eCKBCP+) +e) (t~s) Ilx(s) I

whenever x(t) is a solution of equation (2.10) with x(O) E 53+ (x(O) E 53~) and - 00 < s < t < 00. The second assertion of Theorem 2.4 permits us to reformulate these relations in terms of the solutions of the original equation (2.9) as follows: suppose y(t) is a solution of equation (2.9) and - 00 < s < t < 00; then if y'(O) = K+y(O), (II y(t) 112

+ I y'(t) 112)112

~ N,eC 0 and a corresponding Ne> 0 estimate (2.18) holds for all solutions y(t) of equation (2.9) with y'(O) = K+y(O) is equivalent to the assertion for any c > 0 and a corresponding M, > 0 the following estimate holds for these solutions:

2. EXPONENTIAL DICHOTOMY OF SOLUTIONS

I yet) I ;:;; Mee(KB(P+) +e) (I-s) II y(s) II

( - 00

211

< s < t < (0).

It is now not difficult to conclude that the least upper bound of Bohl exponents of all solutions yet) of equation (2.9) with y'(O) = K+y(O) coincides with In r +. An analogous assertion can be formulated in regard to the solutions y(t) of equation (2.9) with y'(O) = - K_y(O). 6. The equation of second order with a Hermitian coefficient &P(t). When &P(t) = &P*(t) (- 00 < t < (0), Theorem 2.4 can be sharpened. For suppose &P(t) = &P*(t) (- 00 < t < (0). Then for d(t) of the form (2.10')

we have

where (=

,,*).

It accordingly follows from Theorem 1.3 that U(T) is a ,,-unitary operator:

U* "U

= ,,; U = "-lU*-',,,

and hence that its spectrum (J(U(T» is situated symmetrically with respect to the unit circle. This implies the latter assertion of the following proposition. THEOREM 2.5. Suppose in equation (2.9) &P(t)

= &P*(t)

«0

uniformly in t

E ( - 00,

(0).

(2.20)

Then the equation operators K± of equation (2.9) are uniformly negative Hermitian operators, i.e. K± = K'± O. The spectra of the restrictions of the monodromy operator U(T) to 2+ and 2- are specularly situated with respect to the unit circle.

«

PROOF. Since the fulfilment of condition (2.20) implies the fulfilment of the original condition of Theorem 2.4, the existence of uniformly dissipative operators K± such that z = ± K±y are the equations for the invariant subspaces 2± (c 0\5(2) of U(T) is guaranteed by Theorem 2.4. Since the spectium (J+ = (J(U(T)12+) ((J- = (J(U(T)12-») lies entirely in the interior (exterior) of the unit disk and U(T) is a ,,-unitary operator, it follows from this fact that 2+ and 2_ are ,,-neutral subspaces (see Exercise 1.29). Let Xl = Yl EEl K+Y1 and X2 = Y2 EEl K+Y2 be any two vectors of 2+. The relation ("Xl, X2) = 0 implies (K+Yb Y2) - (Yb K+Y2)

= O.

In view of the arbitrariness of Yb Y2 E~) it follows that K+ ness of K_ can be proved analogously.

= K't-. The Hermitian-

v. EQUATION WITHPERIODIC OPERATOR FUNCTION

212

We give another proof of the Hermitianness of the K± which does not make use of the result of Exercise I.29. Let P + and P _ denote the complementary spectral projections of U(T) corresponding to the decomposition (1(U(T») = (1+ U (1- of its spectrum, so that P ±.p(Z) = £±. Clearly, P"± will be the spectral projections of U*(T) corresponding to the decomposition (1(U*(T») = 0-+ U 0-- in which the spectral sets o-± are the specular images with respect to the real axis of the sets (1±. The corresponding invariant subspaces £1: of U*(T) can be found by means of the formulas £1: = P f.p(Z). It is easily seen that these formulas have the following geometric meaning: £-'i'

= .p(Z)

e £_;

£."'.

= .p(Z)

e £+.

(2.21)

Since, on the other hand, U(T) = /(U*(T»)-l/-l, we have /£."'. = £+ and /£-'i' = £-. In fact, the subspaces /£± are invariant under the operator /(U*(T»)-l/, the restrictions of which to these subspaces have the spectra (o-±)-l = (1+. According to (2.21) a vector Y EB Z (E .p(Z» belongs to £."'. if and only if it is orthogonal to all of the vectors Yl EB K+Yl (Yl E .p), i.e.

From this result we obtain the equation Y = - K-'i'z (y E .p) for the subspace £."'..3) But then, as is easily seen, £+ (= /£."'.) has the equation z = K-'i'y, whereas we already know that it has the equation z = K+y. Therefore K-'i' = K+. Analogously, from the relations £- = /£.+- and £-'i' = .p £_ we can get that K."'. =K_. The theorem is proved. 7. The equation of second order with a parameter. Since for d(t) = (-~(t) ~) and W = (~I we have

e

-J)

JRe(Wd(t»)dt o

= ([ RePJ'(t)dt

oj,

l ° -I

the following proposition can be deduced from Theorem 2.2 by means of the same arguments that were used to obtain Theorems 2.4 and 2.5 (from Theorem 2.3). THEOREM

2.6.

If T

SRePJ'(t)dt» 0,

(2.22)

o

there exists an c >

3)

°such that the assertions of Theorem

I.e. Bot: = (- K:;"z EB z Iz E .p) .

2.4 and a/so, if P(t) is a

3.

LOCALIZATION OF SPECTRUM

213

Hermitian operator function, the assertions of Theorem 2.5 will hold for the periodic equation d 2y/dt 2 when -

E

+ A&>(t) Y

=

(&>(t

0

+

T) = &>(t))

(2.23)

< A < O.

Thus the monodromy operator U(T; A) of the equation will be unstable when A is sufficiently small in absolute value and negative. It will be shown below ( § 6; see also Exercise 8) for a Hermitian operator function satisfying condition (2.22) that the monodromy operator U(T; A) will be strongly stable when A is sufficiently small and positive.

§ 3. Localization theorems on the spectrum of the monodromy operator (>8 = .))) 1. Annular localization. According to Wintner's estimate (lII.4.l9), if >8 = .)), the Cauchy operator of equation (1.1) satisfies the inequalities

II U(T) I II U-l(T) I

G

~ exp A~Am(t))dt ), ~ exp ( -

fAm(Am(t))dt).

This immediately implies the following assertion. 3.1. The spectrum of the monodromy operator U(T) lies in the annulus KA = {(IrA ~ 1'1 ~ R A}, where THEOREM

T

rA

= exp JAm(Am(t))dt, o

T

RA

= exp

JAM(Am(t))dt. 0

Despite the apparent roughness of this localization theorem it has some interesting applications (see Exercise 5 and the commentary on it). 2. Angular localization. If the operator function A(t) is sufficiently "small in the large" on the segment (0, T) then U(t) must be sufficiently close to the identity operator and hence the spectrum of the monodromy operator must lie in the intersection of the ab~ve annulus KA and some sector larg ~ a (a < n). We will cite below two theorems which permit one to distinguish a sector (or a rotated sector) of this type. To deduce these theorems we must make use of the properties of the following functionals, which were introduced in the exercises of Chapter I:

pi

dey U def sup {~(Ux, x)lx ¥- 0, x am U def min {dev(,U)Ii'1

= 1}.

E .))},

v.

214

EQUATION WITH PERIODIC OPERATOR FUNCTION

We recall that the spectrum of U lies in the angle larg pi ~ dev U. Let Ar(Am) = I min (0, - Am(Am), AM(Am» I· THEOREM

3.2. Thefollowing estimate holds: dev (U(T») ~

T

S .v IIA(t) liz -

o

Pr(Am(t»)]2dt (= .9""A).

(3.1)

Thus if.9""A < n, the spectrum of the monodromy operator of equation (1.1) lies in the sector

I arg p I ~ .9""A·

(3.2)

In particular, it lies in the sector T

I arg p I ~ oS IIA(t)lldt.

(3.3)

In proving this and the following theorem (and the intermediate lemmas) we will assume that the operator function A(t) = A(t + T) is continuous. Once the theorems have been proved under this assumption, they can be obtained in the general case by applying them to the approximating equations

and passing to the limit for h ! 0 in the obtained estimates. PROOF. Any solution x(t) = U(t)xo of equation (l.l) clearly satisfies the inequality Ildx/dtll ~

IIA(t)llllx(t)ll·

In addition,4) (3.4)

since by virtue of Wintner's estimates

+ h) I !Am(Am(s))ds ~ In Ilx(tIlx(t)11

t+h

t+h

~! AM(Am(s»)ds .

.,

If we pute(t) = Ilx(t)II-1x(t) or x(t) = Ilx(t)lle(t), we will have

~~

=

:r Ilxll·e + Ilxll ~~.

Since (e, e) = 1, it follows that 4) Whenever x(t) is a continuously differentiable function, IIx(t}11 has a left and right derivative (see Exercise I. 18). It can be assumed for the sake of definiteness that the right derivative of In IIx(t)1I has been taken in (3.4).

3.

215

LOCALIZATION OF SPECTRUM

(de/dt, e)

+ (e, de/dt) =

0,

and consequently 1\

~~ r= (~y+ Ilx 112 ( 4ft Y,

w-

I ~ I = JII%r· ~

v' IIA 112

(~ 1nllx(t)IIY

- [Ar(Affi(t»)p.

Thus (see Exercise 1.31)

1::: (e(T), e(O»)

~ !II ~~

II dt

~ f7 A·

Since 1::: (e(T), e(O») = 1::: (U(T)xo, xo) for an arbitrary nonzero vector Xo E .p, the above estimate implies (3.1) and hence the assertions of the theorem. 3. Angular localization with regard for amplitude. As we know (see § 111.2), equation (1.1) is kinematically similar to the equation

dy/dt = Q-l(t)Affi(t)Q(t)y,

(3.5)

where Q(t) is the unitary operator function solving the Cauchy problem

Q(O) = I.

dQ/dt = iA:iJCt)Q,

Under the substitution y = Qx equation (1.1) goes over into equation (3.5), for which Wet) = Q-l(t)U(t) is a fundamental operator function. The equality U(T) = Q(T) WeT) implies that am U(T)

~

+ am

am Q(T)

WeT)

~

am Q(T)

+ dev

WeT).

We can obtain simple estimates for am Q(T) and dev WeT). The spectral length of a Hermitian operator H, i.e. the length of the shortest segment containing its spectrum (J(H), will be denoted by I(H):

I(H) LEMMA

3.1. Suppose H(t) (0

~

def

t

AM(H) - Am(H). ~

T) is a Hermitian operator function and T

WeT) =

J eH(t)dt.

(3.6)

o

Then dev Wet)

~ ~

JI(H(t»)dt.

(3.7)

°

It can be assumed without loss of generality that H(t) is a continuous operator function. For sufficiently large natural n the product PROOF.

v.

216

EQUATION WITH PERIODIC OPERATOR FUNCTION

n

fi:

(tk = kT/n, Lltk = T/n; k = 1,2,.··,n)

exp (H(tk)Lltk)

1

will be arbitrarily close to WeT). The deviation of this product (see Exercise 1.32) will not be larger than n

I; dey exp (H(t k) Lltk).

(3.8)

k=l

By virtue of the result of L. V. Kantorovic (see Exercise 1.33), for arbitrary G

=

G*

cos (dev (exp G») and consequently sin (dev (exp G»)

=

1 cosh (I(G)/2) ,

tanh (I(G)/2), so that

=

dey (exp G) ~ tan (dev (exp G») = sinh (I(G)/2) = I(G)/2

+

O(/3(G»).

Therefore the sum (3.8) does not exceed

But this implies (3.7). LEMMA

3.2. Suppose H(t) (0

~

t

~

T) is a Hermitian operator function and T

Q(T) =

JeiH(t)dt.

(3.9)

o

Then

dev(~I?Q(T») ~ ~

fI(H(t»)dt,

(3.10)

o

where (0

~

t ~ T).

PROOF. Let Q, ( = O(H») denote the right side of (3.10), which is a trivial estimate unless 0 < 1C. As in the preceding proof, it can be assumed that H( t) is a continuous operator function, so that for sufficiently large n

Qn =

iJ eiH(t.)Jt. == UnUn-l"··U n

1

n;

(tkk = T n' Lltk = T k = 1,2,.··,n)

will differ arbitrarily little from Q(T). The spectrum of Uk = exp (iH(tk) Lltk) lies on the arc

(3.11)

3. LOCALIZATION OF SPECTRUM

217

TR = {e i.l[Am{H(tk»)L1tk ~ A ~ AM{H(tk»)L1tk}' The sum of the lengths of these arcs is equal to I:~=l I{H(tk»)L1tk (= 2wn), which for sufficiently large n will be arbitrarily close to {) and hence less than n when {) < n. Therefore (see Exercise 1.34) the spectrum of the product (3.11) will lie on the product of the arcs Tk (k = 1"",n), i.e. on the arc oflength 2wn with center

'n

=

exp i { ~

iE

+

[AM{H(tk»)

Am{H(tk»)]L1tk }.

Passing to the limit for n -+ 00, we conclude that the spectrum of Q(t) will lie on the arc of length 2{)(H) with center But this is the fact that inequality (3.10) expresses.

'H'

THEOREM 3.3. Suppose the following condition is satisfiedfor equation (1.1): (w

clef)

+J

[/(Am)

o

+

I(A,,)]dt < n.

(3.12)

Then the spectrum of the monodromy operator of equation (1.1) lies in the sector of the complex p-plane

Iarg p where a

E [-

a

I ~ w,

n, n] is determined from the congruence 1

a == -2

T

J[AM(A,,) + Am(A,,)]dt (mod 2n). o

PROOF. We return to the relation U(T) = Q(T) WeT), where Q(t) has the form (3.9) with H(t) = A,,(t) while Wet) has the form (3.6) with H(t) = Q-l(t)Am(t)Q(t). In view of the fact that Q is unitary we have according to Lemma 3.1 that dey WeT) ~

1

2

T I T

JI(Q-IAmQ) dt = 2 JI(Am)dt. o

0

According to Lemma 3.2

dev{e-ir 0 such that every equation

whose Hermitian satisfies the inequality T

J11Jf'(t) o

Jf'l(t)lldt < 0

(4.6)

is stable. LEMMA 4.1. In order for equation (4.2) to be strongly stable it is necessary and sufficient that its monodromy operator U(T) be a strongly stable operator. PROOF. To prove the sufficiency we note that to the Hermitian Jf'(t) there correspond two constants CI > 0 and Cz > 0 such that

(see Lemma III.2.3). This shows that the mapping Jf'(t) ~ U(T) that takes each Hermitian into its monodromy operator is continuous. Thus under a small variation of the Hermitian we obtain a monodromy operator UI(T) lying in a small

4.

221

CANONICAL DIFFERENTIAL EQUATIONS

neighborhood of V(T). From the strong stability of the latter we obtain the stability of Vl(T). To prove the necessity we must show that any f-unitatyoperator V1(T) lying in a small neighborhood of V(T) is the monodromy operator of an equation with a Hermitian J~''t(t) differing little from £,(t). In other words, we must show that the mapping described above is open.We omit a proof of this fact (see Exercise 4). Lemma 4.1 and Theorem 1.8.3 imply the following fact. THEOREM 4.2. In order for a canonical equation to be strongly stable it is necessary and sufficient that its monodromy operator V(T) be normally f-decomposable, i.e. that under a decomposition6 ) :0 = :01 :02 of:o into uniformly f-definite invariant (under V(T)) subspaces the spectra 17(V(T)I£'1) and 17(V(T)I£'2) do not intersect.

+

3. Central stability band of the canonical equation with a parameter. We now consider the canonical equation with a real parameter A: dxjdt = iAf£'(t)x.

(4.7)

The stability (strong stability) points of this equation are those values of A for which it is stable (strongly stable), i.e. for which its monodromy operator V(T; A) is stable (strongly stable). The set of strong stability points of equation (4.7) is obviously open and therefore, if it is not empty, decomposes into a system of open intervals-the stability intervals of equation (4.7). We note that the following representation is valid: T

V(T; A) = 1+ iAf

J£'(s)ds + O(A2)

(A

-+

0).

(4.8)

o

For if we integrate each side of the equality dVjdt = iA.J'£'(t) Vet; A), we get t

U(t; A) = 1+ iA

Jf£'(s)V(s; A)ds.

o

By iterating this equality and setting t = T, we obtain (4.8) (cf. (111.1.11)). THEOREM

4.3.

If the operator £'av =

1 T

T

J£'(s)ds o

is uniformly positive, there exists a maximal interval (A-1' AJ) (30) (the central stability band of the equation) all of whose nonzero points A are strong stability points of equation (4.7). 6)

See Theorem 4.1.

222

V.

EQUATION WITH PERIODIC OPERATOR FUNCTION

PROOF. It suffices to show that A is a strong stability point if it is different from zero and sufficiently small in absolute value. From (4.8) we obtain the equality

U(T; A) = 1+ iA[T ,1Yfav

+

O(A)].

From the properties of normally W-decomposable operators cited in § 1.8 it follows that under the conditions of the theorem the operator T,I Yfav is normally ,I-decomposable. Since the property of normal ,I-decomposability is stable (Theorem 1.8.2), it is not affected by adding a small operator O(A) to T ,IYfav . It is also obviously preserved under multiplication by a scalar iA i= 0 and the subsequent addition of tlie identity operator. Thus the ,I-unitary operator U(T; A) is normally ,I-decomposable. By Theorem I.8.3 it is strongly stable. The theorem is proved. The purpose of the following arguments is an estimation of the central stability band of a canonical equation whose Hermitian satisfies the conditions Yf(t)

>0

(0 ~ t ~ T);

Yfav

I T

=

~T

SYf(s)ds» O. o

Let us agree to call such an operator function Yf(t), a Hermitian of positive type, and the canonical equation corresponding to it, a canonical equation of positive type. LEMMA 4.2. Let [a, b] be a closed interval containing zero at each point A of which there exists the operator

[U(T; A)

+ 1]-1

(E[.p]).

(4.9)

If the Hermitian Yt'(t) in equation (4.7) is of positive type, every nonzero point of [a, b] is a strong stability point of this equation. PROOF.

We consider the operator V(A) = i[1 - U(T; Ami

+

U(T; A)]-l.

We shall prove that V(A) can be represented in the form V(A)

=

,IN(A)

(A

E

[a, b]),

(4.10)

where N(A) is a uniformly positive operator. It will then follow from the properties of normally decomposable operators (see § I.8.2) that V(A) is normallY,l-decom. posable. Since U(T; A) can be expressed in terms of V(A) in the form of a linear fractional transformation, it too is normally ,I-decomposable. In addition, this operator is ,I-unitary. By virtue of Theorem 1.8.3 this operator will be strongly stable. To obtain formula (4.10) we consider the derivative

4. V(A)

223

CANONICAL DIFFERENTIAL EQUATIONS

= ~~ =

i

~

{2[/ + U(T; A)]-1 - I} (4.11)

= - 2i[/ + U(T; A)]-1 dU~; A) [/ + U(T; A)]-I. Let us calculate U(T; A) = dUCT; A)/dA. Differentiating the equation dUet; A)/dt

= iA,IYt'(t)U(t; A)

and the equality U(O; A) = / with respect to A, we get

dU~/}l =

iA,IYt'(t)U(t; A)

+

i,lYt'(t)U(t; A),

U(O; A)

=

o.

We have thus obtained an operator equation of the form of equation (111.1.1), = i,lYt'(t)U(t; A). Its solution is given by formula (111.1.19):

wheref(t)

t

=

U(t; A)

JU(t; A)U-l(S; A),IYt'(S)U(s; A)ds o

t

= iU(t; A),I JU*(s; A)Yt'(S)U(s; A)ds. o

Introducing the notation T

R(A)

=

J U*(s; A)Yt'(S)U(s; A)ds,

(4.12)

o

we get dUCT; A)/dA

= WeT; A),IR(A).

We substitute this expression in (4.11) and make use of the relation

(/ +

U)-IU,I

= (U-l + /),1 = (,IU*,I + I)-I,1 = ,I(U* + I)-I.

Then V(A)

= - 2i[/ + U(T; A)]-I·W(T; A),IR(A)[/ + U(T; A)]-l = 2,1[U*(T; A) + 1]-lR(A)[T; A) + 1]-1.

Since V(O) = O· it follows that A

V(A)

=

2,1

J [U*(T; p) + 1]-IR(p)[U(T; p) + 1]-ldp,

o

and we obtain relation (4.10) with the operator A

N(A)

= 2 J[U*(T; p) + 1]-IR(p)[U(T; p) + 1]-ldp.

(4.13) o Finally, the operator R(p) in the integrand is positive and, for sufficiently small

224

v.

EQUATION WITH PERIODIC OPERATOR FUNCTION

f1 > 0, uniformly positive. For it follows from (4.12) that R{f1) > 0; and if f1 is sufficiently small, the operator U(1:; f1) is close to I, so that R{f1) is close to Sl£{s) ds, and therefore R{f1)>> O. But this means that N(}..)>> 0 (0 < A < b). We note that the operator N(A) receives a positive increase from an increase in the value of it An analogous examination can be made of the case a < A < o. The lemma is proved. We now establish a connection between the existence of the operator (4.9) and the solvability of a special boundary problem. We consider the boundary problem dxfdt = iAf£(t)x,

x{O)

+ x(T)

=

f

(4.14)

A number A is called a regular point ofproblem (4.14) iffor any f E 4' there exists a unique solution x(t) (O ~ t ~ T) depending continuously on f in the sense of the norm SUPO~t~T Ilx{t) II· The complement of the set of regular points of problem (4.14) is called its spectrum. LEMMA 4.3. In order for a point A to be a regular point of problem (4.14) it is necessary and sufficient that the operator [U(T; A) + 1]-1 (E [4'D exist. PROOF. We write the solution of equation (4.14) in the form x(t) = U(t; A)Xo. Substituting this expression in the boundary condition, we obtain the relation [U(T; A) + I]xo = J, which proves the lemma. REMARK 4.2. Since the operator U(T; A) depends continuously on A, the set of points A for which the operator [U(T; A) + 1]-1 is bounded is open. Consequently, the spectrum of problem (4.14) is closed. Let A-I and Al denote respectively the maximal negative and minimal positive points of the spectrum of problem (4.14).7) The operator [U(T; A) + 1]-1 IS bounded in the interval (A-I> AI). Using Lemma 4.2, we obtain the final result. THEOREM 4.4. Suppose £(t) is a Hermitian of positive type. Every nonzero point of the interval (Ll> AI) of regularity ofproblem (4.14) is a strong stability point of the canonical equation (4.7), i.e.

A-I

~

A-I < 0 < Al

~

AI·

4. Characteristic multipliers of the finite-dimensional canonical equation. We consider the canonical equation under the assumption that the space 4' is finite dimensional. The eigenvalues p of the monodromy operator U(T; A) are called the characteristic multipliers of equation (4.7). When the equation is stable, the characteristic mUltipliers p lie on the unit circle. 7) We will not dwell on a proof of the existence of the spectrum of the problem on each of the halflines. In the finite-dimensional case this fact follows from formula (0.21) of Exercise 12; if, in addition, the equation is a Hamiltonian equation, it can be shown that A±1 = A±I.

5.

225

SECOND ORDER EQUATIONS

Of interest is their behavior under a variation of ALet us discuss the behavior of the characteristic multipliers when A is in the interval (A~h AI)' We consider the operator V(A)

= i[I - V(T; AmI + V(T;

A)]~I.

In the proof of Lemma 4.2 it was shown that V(A) = ,IN(A), where N(A) is a positive operator having a positive increase under an increase in the value of A. The equation for the eigenvalues of yeA) can be written in the form N(A)x = fJ-,1x. As is well known, the eigenvalues fJ- can be found with the use of the so-called minimax relations 1 _. (,Ix, x) mm max (N(A)X, x) ,

-/i- -

where the maximum is taken over all vectors satisfying additional linear relations and the minimum, over all such relations. Since the denominator increases under an increase in the value of A, we obtain the following result: the eigenvalues of V(A) corresponding to ,I-positive (,I-negative) eigenvectors increase (decrease) under an increase in the value of A. It is easily seen that the characteristic multipliers p are connected with the eigenvalues fJ- of yeA) by the formula I-p fJ- = i - - l+p

or

p

_i-II r - i+fJ-'

This leads to the following rule: under an increase in the value of A (A E (A~h AI») the characteristic multipliers of thefirst kind (corresponding to ,I-positive eigenvectors of the monodromy operator) undergo a counterclockwise movement while the characteristic multipliers of the second kind (corresponding to ,I-negative eigenvectors of the monodromy operator) undergo a clockwise movement. 8 )

§ 5. Second order equations (infinite-dimensional analog of Hill's equation) 1. Central stability band. We consider in a Hilbert space ,p the differential equation of second ord~r with a real parameter fJd 2y/dt 2

+

fJ-pjJ(t)y = 0,

(5.1)

where pjJ(t) is a periodic Hermitian valued operator function: pjJ(t + T) = pjJ(t) = pjJ*(t}. As in the case of a first order equation, we say that this equation is stable for a given fJ- if all of its solutions are bounded on the real line, and strongly stable if it 8) Additional rules for the movement of characteristic multipliers in more complicated situations can be found in the papers of M. G. Krein [6] and M. G. Krein and G. Ja. Ljubarskii [1].

226

v. EQUATION WITH PERIODIC OPERATOR FUNCTION

remains stable under a small variation of the operator function g>(t).9) In accordance with this we will refer to the values of fl. as stability or strong stability points. If fl. #- 0, we can use the substitution). = .v~, Yl = ).y, Y2 = dyjdt, x = G~) to reduce equation (5.1) to a first order equation in the space .p(2) = .p EEl.p:

~~- = ). (+g>(~) ~)x, which can also be written in the Hamilton canonical form if g>(t) is real:

dx dt

=

'1(0il

~) x

III

i).fYfx.

=

(5.2)

°

Here the upper sign is taken when fl. = ).2 > while the lower is taken when - ).2 < 0. It is obvious that the solutions of equation (5.1) are bounded if the solutions of equation (5.2) are bounded, and hence, if ). is a strong stability point for equation (5.2), fl. = ±).2 will be a strong stability point for equation (5.1). With the use of this fact we can make a number of deductions concerning second order equations. Theorem 4.3 immediately implies the following result. fl. =

5.1. Ifg>(t) is an operator function with a uniformly positive mean value > such that all of the points of the open interval (0, fl.l) are strong stability points of equation (5.1 ).lOJ THEOREM

g>av» 0, there exists a number fl.l

° > °

We will further assume that fl. and g>(t) is an operator function of positive type.llJ In this case we will also call equation (5.1) an equation of positive type. Its Hamiltonian

will also be of positive type. In order to characterize the number fl.1 we must in accordance with Lemmas 4.2 and 4.3 consider the boundary problem

-~~

=

i)./Yf(t)x,

x(o)

+ x(T)

=

f,

where

f =

(1)

(h,J2 E .p),

- if)

°'

In the metric d(!?l'j, !?l'z) = Sifll!?l'j(t) - !?l'2(t)lldt. 10) A generalization and sharpening of this theorem as well as a mechanical interpretation of it are given in Exercise 8. 11) I.e. !?l'(t) > 0 and !?l'av» O. 9)

5.

and find its regular points A. Recalling that x = (Yl) yz and if£' = ( form

~l yICO)

227

SECOND ORDER EQUATIONS

!!1t~ =

= AYz;

+ Yl(T)

=

,2

h;

0/ ), we reduce this problem to the

;;r

AY'(t)yr.

-

+ yz(T)

Yz(O)

=

fz,

or, finally, by setting y = YI/A and y' = Yz, to the form dZy dt Z

+

_ f-LfjJ(t)y - 0,

(5.3)

+ y(T) = gr. y'(O) + y'(T) where in the present case gl = hlA, gz = fz, A = ..;-;;. y(O)

= gz,

In accordance with the definition adopted earlier it is natural to regard a given value of /1 as a regular point of problem (5.3) if there exists a unique solution y(t) for any gl, gz E.p that together with its derivative y'(t) depends continuously on gl and g2' On the basis of the above arguments we immediately deduce the following result from Theorem 4.4. THEOREM 5.2. Suppose fjJ(t) is an operator function of positive type. If /11 is the minimal point of the spectrum of problem (5.3), every point of the open interval (0, /11) is a strong stability point of the equation

dZxldt Z + f-LfjJ(t)x

= 0.

2. Generalization of Ljapunov's test. In this subsection we derive an effective estimate for the central stability band of equation (5.1). To this end we consider in more detail the boundary problem (5.3). By means of the substitution y(t) = z(t)

+ ¢(t),

where ¢(t) = (h12A - (TJ4)fz) + t tJz, we reduce it to the problem of finding a function z(t) satisfying the equation

_c!Z~_ + rI/fjJ(t)z dt 2

,ufjJ(t)"'(t)

= -

'f'

(5.4)

and the homogeneous boundary conditions z(O)

+ z(T) =

0,

z'(O)

+

z'(T)

= 0.

(5.5)

We now take advantage of the fact that the solution of the differential equation - (d Zzldt 2)

=

h(t)

with conditions (5.5) can be written in the form

(5.6)

v. EQUATION WITH PERIODIC OPERATOR

228

FUNCTION

T

Jg(t -

Z(t) =

s)h(s)ds,

o

where g(t) = T/4 - It 1/2 is the Green function of problem (5.6), (5.5) (see, for example, R. Courant and D. Hilbert [1]). It is easily verified that the Green function g(t) has the series representation 00

L:

g(t) =

(5.7)

Ckei(2k+1)t,

k=-oo

where

Ck

= (T/n 2)(2k

+

1)2. We note that 00

L:

Ck

= g(O) = T/4.

k=-oo

Setting h(t) = f.lf!lJ(t)(z + ¢), we conclude that the boundary problem (5.4), (5.5) is equivalent to the integral equation T

z(t) - p

T

Jg(t -

s)f!lJ(s)z(s)ds

o We consider the Hilbert space in.p and scalar square

=

f-t

Jg(t -

i>

(5.8)

of functions z(t) (0 ~ t ~ T) with values

T

«Z))2

s)f!lJ(s)¢(s)ds.

0

T

J(f!lJ(s)z(s), z(s»)ds = J 11f!lJ1I2(s)z(s)11 2ds.

= into lB.

r

g(t - s)(f!lJ(s)z(s), cp) ds l

~ i~ T2

{I

2

11f!lJ1/2(S)Z(s)II·IIf!lJ1/2(S)cpllds

T

T

r

~

-16 0J11f!lJ1/2(s)z(s)11 2ds. 0J11f!lJ1/2(s)cpI12ds

~

16 Ilcp 112 J11f!lJ(s) lids. «Z))2,

T2

T

o

(5.9)

5.

229

SECOND ORDER EQUATIONS

which implies IllKzl1i =

TJT!

O~~ET IIKzl1 ;;;; "4

11&J(s)llds

((z».

(5.10)

It is now obvious how to prove the continuity of (Kz)(t). Let C denote the norm of the operator K acting from ,f> into lB: C = sUPzEvIIIKzIII/((z», and let CI = 11K II denote the norm of this operator in the space,f>. Equation (5.8) can be written in the form

z - fJXz = fJXifJ·

(5.11)

(0

= Xl> = T Xl> = T = T T

= T J(£37l(s)£37l av fi,fi)ds

T211£37lavfi112

=

o

and T

<x, x> ~ <Xl> Xl>

=



= J(£37l(s)fi,fi)ds = T(£37l av fi,fi)· o

Therefore IIKjl1

=

~

sup

"

<x, x>

sup

"I

<Xl> Xl>

= Tsup II

II£37lav fil12

(£37l avfi, fi)

=

TII£37lav ll.

From formula (5.15) we obtain the estimate

which proves the theorem.

§ 6. Expansion of the logarithm of the monodromy operator in powers of a small parameter

1. The equation with a periodic operator function analytically depending on a parameter. We consider in a Banach phase space ~ the differential equation

dxfdt

= A(t; il)x

(6.1)

with a periodic operator function

A(t

+

T; il)

= A(t; il)

( - 00

< t
0). From the expansion (111.1.11) and the corresponding estimates it readily follows that the Cauchy operator U(t; A) (0 ~ t ~ T) of equation (6.1) will be a holomorphic vector function in the disk IAI < R with values in the space CT[lB] of continuous functions F(t) (0 ~ t ~ T) with values in [lB] and norm IIIFllle

= O 0). In general, RI < R. If the space lB is a Hilbert space.p, the localization theorems of § 3 can be used to estimate the radius R j from below (see Remark 3.1). It is not difficult to see that the estimate Rj

~ Ro = sup {p

I sup f .J IIA(t;).)112 Ill::;:p 0

Pr(AiJl(t; ).»j2dt




can be satisfied for any e 0 by choosing 0 sufficiently small. Hint. Make use of the Floquet representation and the considerations of Corollary 4.1. 5. Show that the spectrum of the monodromy operator of the equation d 2yJdt 2

+ fl'(t)y =

0

(0.3)

241

EXERCISES

in

.p with periodic coefficient £JI'(t)

(= £JI'(t

+ T»

lies in the annulus l/r

T

r = exp ( T { 1I£J1'(t)lldt

)112

~

P ~ r, where (0.4)

•

>

The estimate is exact, for example, when £JI'(t) = - pI (p 0).12) Hint. By setting dy/dt = lz (l> 0) and x = yEJjz, transform equation (0.3) into equation (0.1) in the phase space .pEJj~\ where s/(t)

=

Oil) (l-I£JI'(t) 0 .

The norm I V(T)II of the monodromy operator of equation (0.1) can be taken as r. It is estimated by means of Wintner's inequality (III.4.19), after which the estimate is minimized with respect to l. Suppose £JI'(t) 0 (- (X) t (X). It is not known in this case whether the assertion remains in force if (0.4) is replaced by the equality


O. From these same formulas it directly follows that estimate (0.9) holds for complexp(t) as long as it holds for nonnegative pet). 14) Note added/or the translation. The conjecture of A. M. Ljapunov was successfully proved by M. G. Krein [13] at the end of 1971.

243

EXERCISES

&'(t) '"

f

e2


= 0,

where if> is the angular displacement of the pendulum from its initial vertical position while th( or - sign is taken before gjl, depending on whether the pendulum is under or over its poinl

245

EXERCISES

suspension. Thus the "linear theory" permits one to assert that an co> 0 can always be found such that for c co the pendulum will perform stable oscillations independently of whether it was originally under or over its point of suspension, provided the vibration frequency OJ of its point of suspension is sufficiently great (OJ> Q(c)). This paradoxical deduction of the linear theory is well known. P. L. Kapica [1, 2] confirmed it experimentally and proposed a nonlinear theory for the phenomenon. An outline of a general nonlinear theory of motion in a high frequency field is given by L. D. Landau and E. M. Lirsic in [I] (but their arguments have remained unintelligible to the authors of this book). 11. Consider the equation


0 if (0.19)

when A2 and /12 independently range over the spectrum 1l([PO)' In particular, the equation is strongly stable for sufficiently large OJ (OJ> 2v' AM([PO)). In the finite-dimensional case, when the spectrum 1l([Po) consists of the squares of numbers Ph"', Pm (>0), a violation of the strong stability of equation (0.18) can occur only for the following isolated values of the frequency: (j, k

= 1,2, .. ·,m; N = 1,2, .. ·).

Hint. Reduce the equation ji + [P oY = 0 to an equation of first order in the doubled space ,p(2). If it is considered as a periodic equation with period T = 2n/OJ, the spectrum of its monodromy operator will consist of the numbers e±iAT (A2 EO 1l([PO), A> 0) and consequently of the two sets (e iAT I A2 EO 1l([Po)} and (e- aT IA2 EO 1l([PO)}' which are the symmetric images of each other with respect to the real axis. To these sets there correspond j'r-definite invariant subspaces of the monodromy operator in question. One should then make use of the results of § 1.8, according to which this operator turns out to be strongly stable if the above two sets do not intersect. For a further development of this theme see the paper of V. A. lakubovic [6] and the survey of M. G. Krein and V. A. lakubovic [I]. In the following problems the space ,p is assumed to be finite dimensional (,p = Cn or Rn). The assertions of these exercises can be extended to the infinite-dimensional case only under special conditions requiring that the values of the operator coefficients be completely continuous operators of this or that class. 12. Consider the skewperiodic boundary problem {dx/dt x(O)

= OJ' yt'(t )x,

+ x(T) = f

(0.20)

in the phase space Cn with a Hermitian :Yt'(t) of positive type. It is elementary to"prove that all of the eigenvalues of this boundary problerri are real (the problem is selfadjoint). Show that the following two equalities hold for the complete sequence (Ai) of eigenvalues of the problem, taken with regard for their multiplicities (M. G. Krein [6, 7]):

7 1] =

~2

lim 1:: r-O IAjl co). Hint. Estimate the left side of (0.24) for "0 = 1/2 by means of Cauchy's inequality and make use of the fact that T2 tr (/Yf av)2

= 1: (Wj)2 + 1: (Wj)2.

16. Suppose that in aT-periodic m-dimensional Hill equation Y + .9(t)y = 0 the coefficient .9(t) = .9*(t) is an odd function: .9(- t) = - .9(t), and Q = f.9 dt, Qav = O. Then (M. G. Krein [7]) this equation will be strongly stable whenever (0.26) For the scalar case (m = 1) this proposition was obtained (in a completely different way, of course) by A. M. Ljapunov [1] back in 1899 (see the reference in Exercise 8). Hint. Transform the Hill equation into a canonical equation with an even Hermitian .p(t) and make use of the second assertion of Exercise 13. 17. Show that the test (0.26) for strong stability of the Hill equation is exact (M. G. Krein [7]). Hint. Consider the scalar Hill equation Y + APa(t)y = 0 with a periodic generalized function Pa(t) defined in the interval (- T12, T12) by the equality Pa(t)

= oCt + a) -

oct - a)

(- Tl2

< t < Tl2),

where a EO (0, TI2). Verify that the skewperiodic boundary problem 2 { d 2Yl dt + APa(t)y = 0, yeO) + yeT) = y'(O) + y'(T)

has a total of two eigenvalues A+(a)

(> 0) and L(a) =

=0

- A+(a), and

where q = Po, qav = O. The equation Y + APa(t)y = 0 with real A will be strongly stable (e-dichotomic) for 0 A+(a) (I AI A+(a)).




< IAI

248

v. EQUATION WITH PERIODIC OPERATOR FUNCTION

REMARK. If the equation ji + AP.(t)y = 0 is transformed into a Hamiltonian equation with an even Hamiltonian (see Exercise 7), the strong stability test indicated in Exercise 13, when applied to the latter equation, becomes a necessary and sufficient condition for A 0 to be a strong stability point. 18. Suppose that in aT-periodic m-dimensional Hill equation ji + f!/J(t)y = 0 the coefficient f!/J(t) = f!/J*(t) has a mean value f!/J.v = 0 while Q(t) is defined as in the preceding exercise. Let q~ (j = 1,.··, m; ql ~ ..• ~ qm) denote the successive eigenvalues of the operator J'2(Q2).v. Show that the Hill equation is strongly stable whenever

*

I: -/b-, Xo o. It will then foIlow that W - U*(T)WU(T»> 0 (the relations :Yt' 0 and:Yt'» 0 are equivalent in a finite-dimensional .f'». 20. Let aT(.f'» denote the Banach space of T-periodic Hermitians :Yt'(t) (= .1t'*(t) = :Yt'(t + T)) with values in [.f'>] and norm

>

T

1l1:Yt'11i

=f

1I:Yt'(t)lIdt.

° (E [.f'>]) denote a signature operator: f =

Let f f* = f -I, and let U(f) denote the group of all f-unitary operators U E [.f'>]. The operator f induces a mapping !T: aT(.f'» ....... U(f) which takes a Hermitian :Yt'(t) into the monodromy operator of the corresponding equation x = jf:Yt'(t)x. Let ST(.f'>; f) denote the set of all:Yt' E aT(.f'» whose images under!T are strongly stable f-unitary operators U E U($). According to Lemma 4.1, ST(.f'>; f) is an open set in aT(.f'» and therefore divides into domains (connected components) called stability domains. When .f'> is finite dimensional, these domains form a countable set. For real .f'> = R}m and f = Jr , a complete description of the stability domains for m = 1 has been given by V. A. Jakubovic [7], and for any natural m, by I. M. Gel' fand and V. B. Lidskii in their important paper [1]. The results of tHe latter were supplemented and then generalized by V. A. Jakubovic [8, 9] to the general case of a complex finite-dimensional .f'> and any signature operator f (in connection with these papers we note the paper of V. B. Lidskii and P. A. Frolov [1], in which related questions are solved for the T-periodic equation

dx = (:Yt'(t) _ dt

.l dQ )x 2 dt

(Q*(t) = - Q(t))

in a finite-dimensional .f'> and which contains some corrections to the paper [9] of V. A. Jakubovic). See also the papers of W. A. Coppel and A. Howe [1] and S. Diliberto [1.] Among the stability domains in ST(.f'>; f) there is always (independently of the dimension dim.f'> ~ 00) the domain Qt(.f'>; f) which contains all of the Hermitians :Yt'(t) of positive type for

NOTES

249

>

which Aj(J'f) 1 (see Theorem 4.4 on the central stability band). The domain Qt(s.J; ,I) has the obvious property that, if it contains a Hermitian of positive type J'f(t), it also contains the halfopen "segment" qJ'f (0 q :::;; 1). In addition, if J'f belongs to Qi, then so does every Hermitian Yf j of positive type such that J'fj(t) :::;; J'f(t) (this assertion is offered as an exercise). All of the tests for strong stability of a canonical equation of positive type indicated in the preceding exercises are tests of whether or not Aj(J'f) 1, i.e. tests of whether or not a Hermitian Yf belongs to Qi(f»; ,I). These tests are called




Z-tests.

There exist techniques permitting one to obtain from the Z-tests tests of whether or not a Hermitian J'f belongs to the other domains of ST(s.J; ,I). In regard to these techniques and other methods of obtaining tests different from the Z-tests for the strong stability of a canonical equation see the paper of M. G. Krein [7] and the survey paper of M. G. Krein and v. A. Jakubovic [1] (see also the survey article of V. M. Stadinskii [1]). NOTES Equations with periodic coefficients were first investigated in a finite-dimensional space by G. Floquet [1], who essentially introduced the monodromy matrix and obtained formula (1.5). A. M. Ljapunov and H. Poincare discovered that in the case of a Hamiltonian equation the monodromy matrix is a Jr-unitary matrix, and it is from this fact that we obtain the symmetry properties of the spectrum of the monodromy matrix. The e-dichotomicity criterion (Theorem 2.1) is taken from the book of J. Massera and J. Schaffer [1]. The subsequent results of § 2, which make use of the properties of operators in Hilbert space with an indefinite metric, are due to M. G. Krein. For the elementary aspects of Theorem 2.2 see his paper [5]. (See also the commentary to Exercise IV.14.) The localization theorems of § 3 are due to M. G. Krein [11]. Theorem 3.2 was obtained by a sharpening and development of the arguments of J. Massera and J. Schaffer [1] showing that under condition (3.15) the spectrum IJ(U(T» does not intersect the negative half of the real axis. Massera and Schaffer managed to find an analogous test for Banach spaces (see their book [1]). The theory of stability bands for the scalar Hill equation with a parameter was constructed by A. M. Ljapunov for both the cases of a coefficient of constant sign and one of variable sign. He was also the first to obtain existence theorems and various estimates for the central stability band. M. G. Krein [3, 6] constructed the theory of characteristic multipliers of the first and second kind, which permitted him to generalize the results of A. M. Ljapunov and obtain a number of new results for finite-dimensional canonical equations. At the present time there is apparently no stability test for the scalar Hill equation which does not have analogs (often several of them) in the multidimensional, and frequently also in the infinite-dimensional, case. Moreover, the methods developed for systems of equations have in many cases provided tests that are also new for the scalar case (see M. G. Krein and V. A. Jakubovic [1]). The important paper of I. M. Gel'fand and V. B. Lidskii [1] is devoted to the classification and calculation of the homotopy invariants of stable Hamiltonian systems (see also the Notes to Chapter I and Exercise 20 ab·ove). The papers [1, 8, 9] of V. A. Jakubovic are devoted to the further development of the theory of periodic canonical equations in this direction. The first investigations of canonical systems in infinite-dimensional Hilbert space were undertaken by V. I. Derguzov [1, 2], who investigated specific classes of canonical equations with an unbounded selfadjoint Hamiltonian J'f(t). It is to canonical systems of this type that concrete problems in the study of parametric resonance of elastic systems with an infinite number of degrees of freedom reduce. The papers [1-5] of V. N. Fomin are devoted to the further investigation of canonical systems of this type.

250

V.

EQUATION WITH PERIODIC OPERATOR FUNCTION

The papers of V. I. Derguzov moved M. G. Krein to investigate the possibility of adapting the methods at his disposal to the infinite-dimensional case. The results obtained were partially published in [9, 10]. These results of M. G. Krein are presented with revisions and additions in § § 4 and 5 and the corresponding exercises. The method in § 6.2 of calculating the logarithm of the monodromy operator for the elementary equation x = AA(t)x was pointed out by N. P. Erugin [1]. A similar method for equations with almost periodic coefficients of the form x = [Ao + AA(t)]x was applied still earlier by I. Z. Stokalo [1] to obtain asymptotic expansions of the solutions. A simpler presentation of these results has been given by N. P. Erugin [2]. The general case considered here for equations with a periodic coefficient A(t; A) was considered by M. G. Krein in a seminar lecture in 1948, but these results were never published. It was in connection with this work that formulas (1.3.7) were first obtained. In 1959 there appeared an article by V. A. Jakubovic [2] that is also devoted to the application of the small parameter method to canonical systems. His results were generalized to the infinitedimensional case by I. I. Kovtun in [1], where, in particular, he proposed a method for calculating coefficients that is based on the use of formula (1.3.17). For the further development of asymptotic methods in application to canonical equations and the relevant literature see the survey of M. G. Krein and V. A. Jakubovic [1].

CHAPTER

VI

LINEAR DIFFERENTIAL EQUATIONS IN THE COMPLEX PLANE In this chapter we consider a linear differential equation the operator coefficient A(z) of which is an analytic function whose only singularity is a pole of first order at the origin. In § 1 we consider the simplest case, when the spectrum of the principal part Ao of A(z) does not contain pairs of points with integral differences. In § 2 we weaken this condition by assuming only that integral differences correspond to isolated points of the spectrum. In § 3 we show how to calculate in the finitedimensional case the coefficients of a series representing a solution by using the Jordan form of the matrix Ao. In § 4 we consider a method of solution which under certain conditions permits one to also obtain a solution of the equation in the case when there are integral differences of points of the continuous spectrum.

§ 1. The equation with a regular singularity. Simplest case 1. The equation with a regular coefficient. In this chapter we consider some questions concerning the Cauchy problem

dx/dz

=

A(z)x, x(zo)

=

Xo

(1.1)

in a complex domain of the z-plane. The case when the operator function A(z) is regular in a simply connected domain G containing the point Zo can be investigated by means of considerations similar to those at the beginning of Chapter III. The solution of problem (1.1) can be represented in the form x(z) = U(z, zo)xo,

(1.2)

where the operator U(z, zo) satisfies the condition U(zo, zo)

=

I

(1.3)

and the equation dUjdz

=

A(z)U

(1.4)

in the phase space [)8]. This operator has the series expansion (1.5)

251

252

VI.

LINEAR EQUATIONS IN THE COMPLEX PLANE

where Wo(z)

= I,

Wk(z)

=

J• A(z) Wk-I(Z)dz,

'0

the integrals being independent of the path of integration in the domain G. The series (1.5) converges uniformly in every closed part F c G whose points can be reached from Zo along paths of bounded length. The solution (1.2) of problem (1.1) is unique. The general solution of equation (1.4) is V(z) = U(z, zo)C, where C (= V(zo» is an arbitrary bounded operator. 2. Simplest equation with a singularity. Of greater interest are equations with a nonregular coefficient A(z). We confine ourselves to a study of the behavior of a solution of equation (1.1) in a neighborhood of a simple pole Zo of the operator function A(z). It can be assumed without loss of generality that Zo = 0 and hence that A(z) has the form A(z)

= (lIz) Ao + Al + Azz + ... ,

(1.6)

Izl

the series (1.6) converging in a domain 0 < < p. It can easily be shown by the usual methods that if a power series with operator coefficients converges in the interior of a disk, it converges there absolutely, i.e. I: IIAklllzlk < 00. Under the above assumptions it is customary in the analytic theory of differential equations to call equation (1.1) an equation with a regular singularity. We first consider the simplest equation dU

dz

= Ao U

(1.7)

z·

It is not difficult to verify by a direct substitution that the operator function U(z)

=

eAo1uC

=

zA,C,

(1.8)

where C is an arbitrary operator, is a solution of (1.7) for z =F O. This solution is not single valued; it is multiplied by the operator eZ"jA, when one makes a complete circuit about the point Zo = 0 in the positive direction. Any solution of equation (1.7) that is single valued in a simply connected domain not containing the point Zo = 0 in its interior can be represented in the form (1.8) under a suitable choice of C and a single valued branch of the logarithm. For this reason the function zA, and, more generally, any solution of equation (1.4) having the analogous property will be called afundamental solution. 3. Case when the spectrum of the principal part does not contain pairs of points with integral differences. We pass to the consideration of the equation with operator (1.6) zdU -- = (

dz

I: AkZk 00

k=O

)

U,

(1.9)

1.

253

EQUATION WITH A REGULAR SINGULARITY

Izl

assuming that the series in the right side converges in a disk 0 ~ < p. We first suppose that the operator Ao satisfies the following condition: there does not exist a pair of points in the spectrum of Ao whose difference is equal to a natural number:

(n

A-W:f=n

1,2,3,.··; A, ,u E I1(Ao))·

=

(1.10)

By analogy with formula (l.8) we will seek a solution of equation (1.9) in the form (1.11) Formally substituting the expansion (1.11) into equation (1.9) and equating the coefficients of like powers of z, we obtain the system of recursion relations

AoUo - UoAo

=

0,

m

mUm - (AoUm - UmAo)

=

L:

AkUm- k

(m = 1,2,.··).

(1.12)

k=1

We put Uo = 1. To successively determine the operators Um (m > 0) we apply formula (1.3.10). We write (1.12) in the form (1.13)

(m = 1,2,.··),

where Fm = L:;=I AkUm- k. From condition (1.10) it follows that the spectra of the operators Ao and Ao -mI (m = 1,2,.··) do not intersect. Therefore the solution of equation (1.13) has the form (see (1.3.10))

Um = 41 2 n: = __ ~

4n:

§ § (Ao - mI - AI)-IFm(Ao F

,

A_

FCm) 0

§ § F, F,

,uI)-1 dA d

/I

r

l4o=---AI)-1 Fm (Ao - ,uI)-1 dA d,u, A + m - ,u

ro

,u (1.14)

where and rcim) are smooth contours surrounding the spectra of the operators Ao and Ao - mI respectively, the contour rcim) being a parallel displacement of the contour In order to prove the convergence of the series in (1.11) we estimate the norms of the operators Urn. Formula (1.14) readily implies that the following estimate is valid for sufficiently large m:

roo

We note in addition that, under our assumptions, for any PI < P lim IIAkllp~ = 0, k~oo

254

VI. LINEAR EQUATIONS IN THE COMPLEX PLANE

and hence

Since

~

max l"::s"::m-l

we have, beginning with some m

1

{II Usllpn c • -m

m

L; IIAkllp~, k=1

= mo,

I Urn I p;" ~ max {II Usllpn and consequently the sequence I Urn lip;" is bounded for any PI < p. Using this fact for some P2 (PI < P2 < p), we see that in each disk Iz I ~ PI « p) the series in l~s~m-l

(1.11) is majorized by a convergent geometric series and is therefore absolutely convergent. Thus the following theorem is valid.

THEOREM 1.1. If the operator Ao satisfies condition (1.10) and the series L;;;" AkZk converges in a disk < p, equation (1.9) has the solution

Izl

U(z) =

Cfo UkZk ) ZAo = B(z)zAo

(Uo = B(O) = I),

(1.11)

the series L;;;"UkZk converging in the same disk.

REMARK 1.1. Let G p be an open disk of radius p cut along some radius. By choosing a branch of ZA o that is single valued in Gp , we can obtain from (1.11) a single valued solution of equation (1.9). We assume that the boundary of Gp contains the origin. Then any solution that is single valued in this domain can be represented in the form V(z) = U(z)C, where C is a constant operator, i.e. U(z) is a fundamental solution. In fact, since B(O) = I, for sufficiently small Zo the operator B(zo) has a bounded inverse, and therefore the operator U-l (zo)

=

ZOAo B-1 (zo)

exists. In the simply connected domain Gp the operator U(z, zo)= U(Z)U-l(ZO) satisfies the condition U(zo, zo) = I and the differential equation (1.9), which does not have any singularities in this domain. Therefore V(z) = U(z, zo)V(zo) = U(Z)U-l(ZO)V(zo) = U(z)C.

2.

INTEGRAL DIFFERENCES OF EIGENVALUES

255

§ 2. Case of integral differences of eigenvalues 1. Solutions of minimal type of the inhomogeneous equation with a constant coefficient. In this section we consider equation (1.9) without the assumption (1.10) on the spectrum of Ao. We first collect some auxiliary facts. We consider the equation

+ fez),

dx jdz = Ax

(2.1)

under the assumption that A is an invertible operator. Suppose efk

E

'13; k

=

O,I,.··,n)

(2.2)

is a polynomial of degree n with vector coefficients. In this case there exists a solution n

x(z) =

L:

Xk Zk

(Xk

'13; k

E

k=O

=

O,I,.··,n)

(2.3)

of the same form. The coefficients Xk are most readily found by the method of undetermined coefficients. Substituting expressions (2.2) and (2.3) into (2.1) and equating coefficients oflike powers of z, we obtain the system of recursion relations AXn

+ In

=

0,

AXk-1 = kXk -

ik-l

(k = n, n - 1,.··,1),

which implies the equalities Xn= -A-lin, Xn-l = - A-I In-I

-

nA-2 fn.

(2.4) Xj

1

= - -.-, i·

. AJ-I

n

L:

s!A-s Is.

s=j

We recall that an entire function fez) is called a function of mininal type (more precisely, of minimal exponential type) if for any e > there exists a constant Ne such that the estimate Ilf(z) II ~ Neeel"1 holds throughout the complex plane. Suppose

°

00

fez)

= L: fk zk .

(2.5)

k=O

It is known (see Exercise 1.22) that this function is of minimal type if and only if lim (k! II fk 1I)Ilk

=

0.

(2.6)

k~oo

The above result concerning polynomial solutions admits the following generalization.

256

VI. LINEAR EQUATIONS IN THE COMPLEX PLANE

LEMMA 2.1. Equation (2.1) has a unique solution in the class of solutions of minimal type if the operator A has a bounded inverse and the vector functionf(z) is of minimal type. PROOF. We put in analogy with (2.4)

Xj

1

.

00

= - -.-, AJ-l 1:; s!A-SIs J.

(j = 0,1,. .. ).

(2.7)

s=j

For any s > 0 there exists by virtue of (2.6) an index jo such that s! Ills II ~ sS when s > jo. Therefore j!llXjl1

~

I;s! Ills II • IIA-llls-U-ll

S=J

whenj > jo, and hence

~im U!IIXjll}l/j = O. J~OO

Thus the function x(z) = 1:;;;" Xjzj is of minimal type. It follows from (2.7) that

(k = 1,2,.··).

(2.8)

Multiplying (2.8) by Zk-l and summing, we get that x(z) is a solution of equation

(2.1). For a proof of uniqueness it suffices to show that when A is invertible the homogeneous equation dx /dz = Ax cannot have nonzero solutions of minimal type. Suppose x(z) = 1:;k XkZk is a nonzero solution. Then Xk = Akxo/k! and Ilxoll = IIA-kAkxoll ~ IIA-lilk. Ilxkll • k! ~ skllA-lllk for arbitrarily large k and sufficiently small s, which implies Xo = O. LEMMA 2.2. Equation (2.1) has a solution of minimal type if the spectrum of A has an isolated poin(at A = 0 andf(z) is of minimal type. Two such solutions differ by a function whose values belong to the invariant subspace 58 0 of A corresponding to the point A = O. PROOF, Let Po be the projection corresponding to the point A = 0 of a(A). Equation (2.1) decomposes into the pair of equations

dYo/dz dYl/dz

= =

+ go(z), AlYl + gl(Z), AoYo

(2.9) (2.10)

2. INTEGRAL DIFFERENCES OF EIGENVALUES

257

where Yo = Pox, Yl = (/ - PO)X, go = Pof, gl = (/ - PO)f, Ao = APo, Al = A(/ - Po).

In the subspace 581 = (/ - Po)58 equation (2.10) satisfies the conditions of Lemma 2.1 and consequently has a solution Yl(Z) of minimal type. We consider equation (2.9) in the subspace 580 = Po58. In this equation the operator Ao has a spectrum consisting of only the point). = 0, while the function go(z) is of minimal type. Let us show that all of the solutions of this equation are also of minimal type. To this end we estimate the values of the derivatives of a solution of equation (2.9) at the origin. It is easily seen from (2.9) that

We note that under the above assumptions there exists for any c > 0 a constant N, such that

In fact the second estimate follows from (2.5) since go(z) is of minimal type. The first estimate can be deduced from the fact that the operator function (/ _ZAO)-1 is an entire function and the series ~;;ozkA~ representing it converges for any z. Further,

~ N,IIYo(O) II

• ck+l + N;(k + l)ek.

From this inequality one easily deduces the estimate lim {I II y&k+l) (0) II ~ c, k-->oo

which by virtue of the arbitrariness of c implies the equality lim { II ydk+l) (0) I }1/(k+1) = 0, k-->oo

proving the lemma. REMARK 2.1. It is not difficult to see that equation (2.1) has a polynomial solution x(z) if fez) is a polynomial and the dimension of the invariant subspace of Ao corresponding to the point). = 0 is finite. 2. Case when the integral differences correspond to isolated points of the spectrum of Ao. We consider the equation dU z -d Z

=

(00 ) ~ Akzk. U, k=O

(2.11)

VI.

258

LINEAR EQUATIONS IN THE COMPLEX PLANE

without assuming that condition (1.10) is satisfied. We note that an integer n for which there exist A, p

E

(j(Ao) such that

A-p=n

(2.12)

is a point of the spectrum of the transform (2.13)

WX= AoX - XAo

acting in the Banach space [f8] (p(W) does not contain differences of the points of (j(Ao); see § 3 below). We relax condition (1.10) and replace it by the following condition (a): each of the integers nl < nz < ... < np (p < (0) that are representable in the form nk = A - p for A, p E (j(Ao) is an isolated point of the spectrum of W. A solution of equation (2.11) will be sought in the form U(z) =

C~ Uiln

(2.14)

Z)Zk) ZAo

This expression is analogous to (1.11), although the coefficients here are no longer constants but functions of ~ = In z. Substituting the expansion (2.14) into equation (2.11) and equating coefficients of like powers of z, we obtain the system of relations Uk(lnz)(Ao

+ kI) +

Uk(inz) =

k

2: Ak-sUs(lnz)

s=o

(k

= 0,

1,.· .),

(2.15)

which is more conveniently written in the form Uk(~)

=

AOUk(~)

-

Uk(~)(Ao

+ kI) + Fk(~)'

where ~ = In z, Fk(~) = 2:~:::6 Ak-sUs(~). The relation Uo(~)

=

AoUo(~)

-

Uo(~)Ao

corresponding to k = 0 is satisfied if one takes Uo(~) = l. For 0 < k < nl we can take Uk(~) = Uk, i.e. we can assume that not depend on ~. Equation (2.15) then takes the form - AOUk + Uk(Ao

+ kJ)

Uk(~)

does

k-l

=

2: Ak-sUs

s=o

and can be investigated in exactly the same way as under condition (1.10). We next consider equation (2.15) for k = nl' Going over to the space [f8], it can be written in the form (setting X(~) = UnJ~)) X'(~) =

(W - nlI)X(~)

where Wis the transform (2.13) and Fn ,

E

[f8].

+ Fn "

2.

259

INTEGRAL DIFFERENCES OF EIGENVALUES

This equation has a solution X(~) which is a function of minimal type, i.e. an entire function satisfying the condition (2.16) for every c > O. For the sake of definiteness (although not out of necessity) we will assume that it satisfies the condition Pn,X(O) = 0,

(2.17)

where P n , is the spectral projection on the invariant subspace of ~ corresponding to the point nl of its spectrum. Under this condition a solution in the class of functions of minimal type is unique. For larger k we obtain an equation of the form (2.18) where the function Fk(O is of minimal type. The operator ~ - kI here has at the origin either a regular point (this is always true for k > np) or an isolated point of its spectrum. In either case equation (2.18) has solutions of minimal type. We can make the process of choosing such solutions completely determinate if we require that conditions of type (2.17) be satisfied. We have thus indicated a method for successively constructing the coefficients of the series (2.14). We will consider this series in a disk Go of radius R < P cut, for example, along the radius (- R, 0). We choose the branch of the logarithm that varies for z E Go in the halfstrip Do

=

glRe~

< InR,

-

7r < In~ 0 the condition (2.20)

If the integral points of the spectrum of the transform (2. I 3) are poles of the resolvent R ..(2t'), the functions Uk(,) are polynomials. The validity of the last assertion of the theorem readily follows from the arguments presented in the proof of the theorem if one takes into account Remark 2.1. REMARK 2.2. Condition (a) means that there exist isolated points Akj and }kj of O"(Ao) whose difference is an integer:

and that there are no other points in O"(Ao) with integral differences. REMARK 2.3. Condition (2.20) implies that limz--+oB(z) = I. It can be deduced from this relation in the same way as in Remark 1.1 that any solution V(z) of equation (2.11) that is single valued in a disk /z/ < P cut along some radius can be represented in the form V(z) = U(z)C, where U(z) is a solution obtained from (2.14) by choosing a single valued branch of the logarithm, i.e. U(z) is a fundamental solution of the equation.

§ 3. Finite-dimensional case (dim )8 < (0) In this section we discuss the calculations needed to determine a solution of equation (2.11) in the finite-dimensional case. We represent the operator Ao in the form of a Jordan decomposition (see Exercise 1.3): n

Ao =

.E (AjPj + Qj).

(3.1)

j=l

Here the Aj (j = 1,.··,n) are the eigenvalues of Ao, the Pj are the corresponding spectral projections and the Qj are the corresponding nilpotent operators satisfying the relations (j

= 1,.··,n).

(3.2)

262

VI.

LINEAR EQUATIONS IN THE COMPLEX PLANE

Everywhere below we will make use of not the finite dimensionality of Q3 but relations (3.1) and (3.2), and thus our arguments remain valid in the infinitedimensional case whenever Ao is an operator of algebraic type (Exercise 1.9). We first consider the case when condition (1.10) is satisfied, i.e. when Aj - Ak "# n ("# 0) (Aj, Ak E a(Ao» for any integer n. In this case a solution is representable in the form of the series (1.11). To successively calculate the coefficients we must solve the equations

UmAo - AoUm + mUm

=

Fm.

(1.13)

The solution of this equation can be represented in the form

Um

(~ -

= -

ml)-IFm,

where ~ is the transform (2.13) acting in the space [Q3]. Let ifhm(A, fJ.) = (A - fJ. - m)-I. Then it is not difficult to conclude (see § I.3) that (~

- m I)-I = (t) of equation (1.1) can always be reduced to an investigation of the stability of the zero solution of an auxiliary equation. In fact, setting yet) = x(t) - 1>(t), we obtain for yet) the equation (1.8)

dYldt = 1/f(t, y),

where 1/f(t, y) = /(t, Y + 1>(t)) - /(t, 1>(t)). We note that 1/f(t, 0) = O. Here the solution x = 1>(t) of equation (1.1) goes over into the zero solution y == 0 of equation (1.8) while any other solution x = ¢(t) goes over into the difference y = 1>(t) - ¢(t). Equation (1.8) can be reduced to a more convenient form for purposes of study if the function/(t, x) is continuously differentiable with respect to x in a neighborhood of the point x - 1>(t). One can then write

+ 1>(t))

1/f(t, y) = /(t, Y

- /(t, 1>(t)) = A(t)y

+ F(t, y),

where A(t) = /~(t, 1>(t)) and IIF(t, y)11 ~ et(t)llyll, with limy~oet(Y) = O. It sometimes turns out that this convergence is uniform in t, i.e. lim e(y) = O.

(1.9)

y~o

Such a situation holds, for example, if the function/(t, x) has a bounded second derivative or, more generally, if 11/;(t,1>(t)

+ h)

- /~(t, 1>(t)) I ~ e Ilh II·

This follows from the obvious representation (see (1.9.7)) /(t, y

+ 1>(t))

- /(t, 1>(t)) - /;(t, 1>(t))y

1

=

J[/;(t, 1>(t) + ~y)

o

- /;(t, 1>(t))]d~ • y.

Thus equation (1.8) can be written in the form dYldt

=

A(t)y

+ F(t,y).

(1.10)

Such an equation, when condition (1.9) is satisfied, is sometimes said to be quasilinear.

The linear equation dYldt

=

A(t)y

(A(t) = /~(t, 1>(t))

(1.11)

1. EXISTENCE OF SOLUTIONS

281

which plays an especially important role in the investigation of equation (1.10), is called the variational equation of equation (1.1) corresponding to the solution x = ifJ(t). We note that if condition (1.9) is satisfied then for any q > 0 there exists a p > 0 such that

iBlllxll

IIF(t, y)11 ~ q11Y11 ~ p}.

(1.12)

in the ball Bp = {x E We will sometimes proceed directly from equation (1.10) under the assumption that condition (1.12) is satisfied with a sufficiently small fixed q > 0 (this condition is weaker than (1.9»). We consider some important special cases that are frequently encountered in the applications. An equation

dx/dt

(1.13)

f(x),

=

whose right side does not depend explicitly on t, is said to be autonomous. Clearly, if x = ifJ(t) is a solution of equation (1.13), any shifted function x = ifJ(t - to) is also a solution of it. Let Xo be a critical point off(x):f(xo) = O. In this case equation (1.13) has the stationary solution x(t) == xo, while the variational equation (1.11) turns out to be a stationary linear equation dx/dt = Ax with the operator

A = f'(xo).

(1.14)

We obtain another important special case when the function f(t, x) is periodic in t for all values of x: f(t + T, x) = f(t, x). Suppose x = ifJ(t) is a periodic solution of such an equation (with the same period): ifJ(t + T) = ifJ(t). In this case the operator function A(t) = f~(t, ifJ(t») is obviously also T-periodic. In particular, if an autonomous equation has a periodic solution y = ifJ(t), its derivative dy/dt = ifJ'(t) satisfies the variational equation. In fact, differentiating the identity difJ(t) /dt = f(ifJ(t»), we obtain the relation

d~?) = f'(ifJ(t»)ifJ'(t) =

A(t)ifJ'(t).

3. General theorem on the stability of the Bohl exponent. Inasmuch as it was shown above that it suffices to carry out an investigation of stability for the zero solution, we will frequently consider the equation dx/dt

=

f(t, x)

(t

~

0)

(1.1)

under the additional condition

f(t, 0)

=

o.

(1.15)

282

VII. NONLINEAR EQUATIONS

We will say that an equation (1.1) satisfying condition (1.15) has property N, p) (- 00 < )) < + 00, N > 0, p > 0) if every solution x(t) of it for which Ilx(to) II ;;:;; p at some moment to satisfies the estimate

~()),

(1.16) for all t > 'r ~ to for which the solution x(t) is defined. We generalize a definition of § III.4.2 by calling the greatest lower bound of numbers A = -)) for which there exist numbers pv and Nv such that equation (1.1) has property ~()), N v, pv) the (upper) Bohl exponent at zero of this equation and denote it by KB. REMARK 1.2. If the conditions of the local existence theorem are satisfied in a neighborhood of zero and the Bohl exponent at zero of the equation is negative, then the solutions whose initial values lie in a sufficiently small neighborhood of zero can be extended indefinitely (see Remark 1.1). Since, furthermore, these solutions tend to zero as t --+ 00, the zero solution in the case in question is uniformly and asymptotically stable. Because of this fact we will speak below without further ado of the negativeness of the Bohl exponent implying stability. By virtue of the above definition, for each)) > KB the solutions of equation (I. 1) that are contained in the ball B p, = < pv} at some moment of time to satisfy the condition

{xlllxli

(1.17)

It is not difficult to see that for a linear equation the Bohl exponent at zero coincides with the upper Bohl exponent of the equation. We now see that, just as in the linear case, the Bohl exponent is stable with respect to small perturbations of the equation. This fact is implied by the following result.

THEOREM 1.3. Suppose that a function f(t, x) for t E [0, 00) and I x I ;;:;; r is continuous in t and satisfies condition (1.15) and the Lipschitz condition

1[J(t, X2)

- f(t, Xl) II ;;:;; LllX2 -

xtll·

Suppose further ,,that equation (1.1) has property ~()), N, p) for some)) > 0 and p < r. Then for any N1 > Nand))l < )) there exists a number q > 0 such that if a function g(t, x) satisfies the inequality

(t

E

[0,

oo);llxll ; :; r),

(1.18)

the equation dx/dt

=

f(t, x)

has property .?l())t> Nt> P1) for some P1 > O.

+ g(t, x)

(1.19)

1.

283

EXISTENCE OF SOLUTIONS

PROOF. It suffices to show that for some h > 0 every solution of equation (1.19) that is contained in a ball Eo at some moment to satisfies the estimates

Ilx(z- + s)11 ~ N1e-v,s Ilx(z-)II Ilx(z- + h) I ~ e-v,h Ilx(z-) I

(0

~

s

~

h),

(1.20)

for z- ~ to (provided this solution is defined at the corresponding points). For in this case the inequality

Ilx(z- + nh)11 is valid for any z-

~

~ e-v,nhllx(z-)

I

to and any integer n and, finally,

Ilx(t)II

Ilx(z- + nh + s)11 ~ N1e-v,Sllx(z- + nh)11 ~ N1e-v, (S+nh) Ilx(z-) I = N1e-v, Ilx(z-) I (t = z- + nh + s). =

(H)

for any t > z- ~ to We consider a solution yet) of equation (1.1) satisfying the condition Ily(to) I ~ 00, where

00 = min {p /N, p}.

(1.21)

Since it is subject to inequality (1.16), in particular,

I y(t) I

~

Nlly(to)11


0 so that (1.22) and consider on the interval [to, to + h] a solution x(t) of equation (Ll9) satisfying the condition x(to) = Xo E Eo (for those values of t at which this solution is defined). From the relation x(t)

t

t

to

to

= x(to) + Jf(s, x(s»)ds + Jg(s, x(s»)ds

(1.23)

we obtain by virtue of (Ll8) the estimate Ilx(t) I ~

t

IIxoll + (L + q) Jt, IIx(s)llds,

which implies by virtue of Corollary III.2.1 that IIx(t)1I ~ Ilxolle(L+q)h.

(1.24)

Suppose now yet) is a solution of equation (Ll) satisfying the same condition y(to) = Xo. From (1.23) and the equation t

yet) = Xo

+ J(s, y(s»)ds t,

VII. NONLINEAR EQUATIONS

284

we deduce, using (1.24), the estimate Ilx(t) - yet) I ~ L

t

J Ilx(s) -

yes) lids

+ qh Ilxo Ile(L+q)h,

to

which implies by virtue of Corollary III.2.1 again that Ilx(t) - y(t) I ~ qh llxo\\e(2L+q)h ~ qhe(2L+q+v,)he-v,(t-to) Ilxoll· Thus for those values of t have the estimate

E

Ilx(t) I ~ ~

[to, to

+ h) at which the solution x(t) is defined

I

I

I

yet) + IIx(t) - yet) [N + qhe(2L+q+v,)h] e-v,(t-to)

we

Ilxoll.

We now subject the number q (> 0) to the condition qhe(2L+q+v,)h

= min(NI

-

N, 'rj).

(1.25)

Then (to

~

t < to

+ h).

(1.26)

On the other hand, taking into account (1.22) and (1.16), we get IIx(to

+ h) I

~

I y(to + h) I + Ilx(to + h) -

~ [1 - 'rj

y(to + h) I + qhe(2L+q+v,)h]e- v,h IIx(to) II,

which implies by virtue of (1.25) that IIx(to

+ h) I

~ e-v,hllx(to)ll·

(1.27)

Since all of the arguments can be repeated with the point to replaced by the points to + h, to + 2h,.··, we can assert that estimate (1.26) is satisfied for all t ~ to at which the solution x(t) is defined. We now choose 0 = min (ooINI> (0). It then follows from (1.26) for Ilx(to) I < 0 that IIx(?:") I ~ Nlo ~ 00 ('L" ~ to) and hence that all of the above estimates can be constructed with to replaced by 'L". In this connection (1.26) and (1.27) go over into estimates (1.20). Thus the theorem is proved, it following from (1.21) that 0= min {p, pIN, pI NNd,

and the number q being determined from (1.25) and (1.22). COROLLARY 1.1. If the Bohl exponent at zero of equation (1.1) is negative, then the Bohl exponent of the perturbed equation (1.19) for sufficiently small q > 0 is also negative. REMARK 1.3. Suppose the function get, x) satisfies the condition

2.

STABILITY: STATIONARY PRINCIPAL PART

lim Ilg(t, x) ,,~o

Ilxll

I

285

=0

uniformly in t. In this case we will briefly write g(t, x) = o(llxll), bearing in mind that the convergence is uniform in t. Then for any q > 0, no matter how small, we can find a neighborhood of zero in which condition (1.18) is satisfied. Consequently, for a sufficiently small Pl > 0 equation (1.19) will have property &6'(Vl' Nl , Pl) with Vl arbitrarily close to v. Hence in this case the Bohl exponents at zero of equations (1.1) and (1. 19) coincide.

§ 2. Stability and instability of the solutions of the nonlinear equation with a stationary principal part 1. Stability. Suppose the stationary linear equation dx/dt

= Ax

(2.1)

has a negative Bohl exponent /CB and the function F(t, x) is subject to the estimate IIF(t, x)11 ~ qllxll

(t ~ 0;

Ilxll

~ p).

(2.2)

Then from Theorem 1.3 it follows that the equation dx /dt

= Ax + F(t, x)

(2.3)

has a negative Bohl exponent at zero whenever q > 0 is sufficiently small. We will independently obtain this same result in order to illustrate a method (a natural generalization of the so-called second method of Ljapunov) which will be used later to prove an instability theorem. The idea underlying this method was presented in Chapter II. We recall some of the geometric observations made there. If the spectrum a(A) lies in the interior of the left halfplane, the vector field Ax (of tangent vectors to the integral curves of equation (2.1)) at each point x E}B after an appropriate renorming of the space }B turns out to be directed toward the interior of the sphere centered at the origin which passes through the point x. But if a(A) contains a component in the right halfplane there exist points x E }B at which the vector Ax is directed toward the exterior of this sphere. It is obvious that a small perturbation of the equation cannot produce a large change in the ch~racter of the integral curves. We can therefore anticipate that under certain conditions the question of stability of the zero solution of equation (2.3) under a "not too large" perturbation FCt, x) can be resolved by the spectrum of the linear approximation of this equation. These general observations receive a rigorous formulation below. THEOREM 2.1. Suppose the spectrum of the operator A lies in the interior of the left halfplane, so that IleAl ~ Noe-v,t, and suppose condition (2.2) is satisfied for q < vo/No.

I

VII.

286

NONLINEAR EQUATIONS

Then equation (2.3) has a negative Bohl exponent at zero.

I

Consider a solution x(t) of equation (2.3) satisfying a condition Ilx(to) P (the number Po will be chosen below) and let to ~ t < T be an interval in which Ilx(t)11 < p. We introduce in the space )8 a new norm PROOF.

< po

~

co

I x llA

As was shown in Theorem 1.4.2, M = No/vo. We put yet) = rAtx(t). Then

=

J I eAtxlldt. o

mllxll

~ IIxllA ~ Mllxll, where m

dt/dy = rAt (dx/dt - Ax) =

e~AtF(t,

=

l/IIAII and

x)

and co

co

o

t

Ilx(t)IIA = J lI eA (r+t)y(t)lI dZ' = J lI eAry(t)lI dZ'. Let us calculate the divided difference I x(t

+ h) IIA

Ilx(t) IIA

-

h

7 lIeAry(t + h)hI -

= _ ~ ttlleAry(t)lIdZ' + h t

lI eAry(t)1I d'r.

t+h

Denoting the first summand of the right side by .P1(h) and the second by .P2 (h), we estimate them, assuming that to ~ t < T. We first note that

In addition, setting Z' .P2(h) ~

=

s

+ t + h in the integral below, we get

7 lIeAr[y(t + hh) -

r

y(t)] I dZ'

Hh

~

I eAseA(t+h) yet

+ hh -

yet) II ds

=

II eA(t+h) y(t

+ hh -

yet)

t

~ Mil eA(t+h) y(t + hh - yet) II ~ Mil eAh II II eAt yet + hh - y(t)~. The last estimate implies the inequality lim .P2(h) ~ MlleAtY'(t) II h~O

=

M IIF(l, x) I ~qMllxll.

(2.5)

2.

STABILITY: STATIONARY PRINCIPAL PART

287

I

Ik Ilx(t + h) IIA - Ilx(t) IIA ,

Thus for the right first derivative of the norm x(t)

d+ Ilx(t) IIA = lim ~

h

h-+O

which exists, as was shown in Chapter I (see Exercises 1.17 and 1.18), we obtain from (2.4) and (2.5) the estimate

d+llx(t)IIA/dt ~ - vllx(t)IIA' qM)/m > 0 since qM < (vo/No)(No/vo)

where v = (1 From inequality (2.6) we obtain the estimate

(2.6)

1.

=

(to ~ .. ~ t < T),

and, finally, after returning to the original norm, we get

Ilx(t) I M /m.

~

Ne-"(I-T) Ilx(..)I

(to

~

..

~ t

< T),

where N = We put po = min(p, p /N). Then Tcan be taken arbitrarily large. For suppose Ilx(T) = p. Since we can pass to the limit in (2.7) for obtain the contradictory inequality

I

p

= Ilx(T) I

~

Ne-"(T-t,) Ilx(to) I
0;

/lxll

;£p; t ~to).

(2.21)

In fact, estimate (2.21) in a sufficiently small ball implies estimate (2.2) with a sufficiently small q.

2.

291

STABILITY: STATIONARY PRINCIPAL PART

It turns out that we can deduce from (2.21) a result that in a certain sense is more general, the assumption that O"(A) does not intersect the imaginary axis having been dropped. THEOREM 2.3. If the spectrum O"(A) of A contains points lying in the interior of the right halfplane and condition (2.21) is satisfied, the zero solution of equation (2.3) is unstable. PROOF.

Re A (A

Let A = a + (3i be a point of O"(A) having a maximal real part: a (A)). Then for any 1) > 0

~

EO"

(1)

> 0; t

~

z-).

Moreover, as follows from Lemma 1.4.1, for each T > 0 and a sufficiently small 0 there exists a vector ~ = ~iJ,T (II~II = 1) such that

o>

ea(t-t')(l - 0) ~

I eM-I,); I

~ ea(t-to)(l

+ 0)

(0 ~ t - to ~ T).

Let x(t) (t ~ to) be the solution of equation (2.3) satisfying the condition x(to) = Xo. We write it in the form x(t) = X1(t) + xz(t), where I

X1(t)

= eA(I-I,)xo; xz(t) = JeA(T-I,)F(z-, x(z-))dr:. I,

Let Xo = e~iJ,T and let R be a number satisfying the condition 1 + 0 < R < min (2, 1 + 0 + 01), where the number 01 > 0 will be chosen later on. By virtue of the continuity of x(t) there exists an interval (0 ~ t - to ~ T1) on which

I x(t) I

~ eRea(t-t,) .

(2.22)

We estimate both parts of the solution on this interval, assuming that 1)

< ap, eReaT,

~

p.

Under these conditions we have Ilxz(t)

I

I

~

J IleA(I-T) 1IIIF(z-, x(z-)) IldzI, I

~ N~q

J e(a+~)(I-T) Ilx(z-)II1+PdzI, I

~ (eR)1+PN~q

J e(a+~)(t-T)+a(P+1)(T-I,) dz10

and

(2.23)

292

VII.

NONLINEAR EQUATIONS

Il x1(t) II ~ (1

Therefore, provided T1

~

+ o)cea(t-t,).

T,

Ilx(t) II ~ Il x 1(t) II

+ Ilxz(t) II ~ [ 1 + 0 + NrR

(0 < t - to

cPR1+Peap (t-t,) ] celt-to)

ap - r;

~ T1).

If it turns out in this connection that

1+

() +

N~q

ap - r;

cPR1+PeapT, < R,

then the continuity of x(t) implies that condition (2.22) is also satisfied on a larger interval than [to, to + T1]. Therefore inequality (2.22) is always satisfied on the interval [to, to + T1], where T1 satisfies the equation 1

+0+

N~q

ap - r;

cPR1+PeapT,

=

R,

which can be rewritten in the form cea T, = _1_ [(R - 1 - (})(ap - r;) J1IP. R N~qR

(2.24)

For sufficiently small c this equation always has a unique solution T 1(c) > 0, which when 0, Rand r; are given depends only on c and not on T. Thus T can be chosen arbitrarily and we put T = T1(C). . We now note that (2.24) implies cReaT,

=

(ap -

r; )l1P (R _ 1 _ o)llP

N~qR

::S;

-

ollP 1

(ap -

r; )lIP.

N~q

Setting 01 ~ N~qpP I(ap - r;), we ensure the fulfilment of inequality (2.23) and thereby the validity of all of the arguments constructed above. Thus inequality (2.22) is satisfied on the interval 0 ~ t - to ~ Tb where T1(c) is a solution of equation (2.24). But then Ilx(to + T1) II ~ II x 1(to + T1) II .

~ (1 - o)ceaT, =

[1 - 0 -

- Ilxz(to + T1) II N~q

ap - r;

N~q

ap - r;

cPR1+PeapT'ceaT,

R1+PcPeaPT, ] ceaT,

_ _ [(R - 1 - (})(ap - r;)J 1IP _ - (2 R) N~qR1+P -

C

>

o.

293

3. NONSTATIONARY PRINCIPAL PART

Ilxoll

This estimate implies the instability of x(t), since E = can be taken arbitrarily small and yet there still exists a value T1(E) for which Ilx(to + T1) ~ c. 4. Stability of a stationary solution of the autonomous equation. We consider the autonomous nonlinear equation dx

dt = f(x)

(to

~

I

(2.25)

t < 00).

If Xo is a critical point:f(xo) = 0, in a neighborhood of which the functionf(x) is continuously differentiable, the difference y(t) == x(t) - Xo will satisfy the quasi linear equation dyJdt = Ay

+ F(y).

(2.26)

On the basis of what has been said above we can conclude that the behavior of a small solution of this equation is determined by the spectrum of the operator A = f'(xo). If the spectrum (J(A) lies in the interior of the left halfplane, the zero solution of equation (2.26), and hence also the stationary solution x == Xo of equation (2.25), is uniformly and asymptotically stable. But if the points of (J(A) also lie in the interior of the right halfplane, we can assert that this solution is unstable when either (i) (J(A) does not intersect the imaginary axis or (ii) a condition of type (2.21) is satisfied (it is satisfied, for example, if the function f(x) is twice differentiable).

§ 3. The nonlinear equation with a nonstationary principal part 1. Another variation of the theorem on the stability of the Bohl exponent. In this section we consider the question of stability of the solutions of the nonlinear equation dx

dt = A(t)x + F(t, x),

(3.1)

without assuming that its principal part is stationary. As indicated in § 1, it suffices to study conditions for the stability of the zero solution of equation (3.1). Thus we can assume that F(t,O)

= O.

(3.2)

From Corollary 1.1 to Theorem 1.3 it follows that a small addition F(t, x) does not violate the property of negativeness of the Bohl exponent of the equation. We now give an independent proof of this fact, following the same idea as in the proof of Theorem III.4.6, in order to obtain more exact values of the constants entering into the estimates. Also we replace condition (1.18) by the analogous integral condition

IIF(t, x) I

~ r;(t)llxll

(t ~ to;

Ilxll

~ p),

(3.3)

VII. NONLINEAR EQUATIONS

294 where

1

t-+

Co

-7:0 Jt 7J(7:)d7: ;£ q.

(3.4)

An analogous generalization could have been made in Theorem 1.3 (see Exercise 5). THEOREM 3.1. Suppose the equation

dxldt = A(t)x

(to ;£ t < <Xl)

(3.5)

has property ~(J.!, N) (J.! > 0) while the function F(t, x) satisfies condition (3.3), (3.4) for a fixed 7:0' Then when q < J.! IN equation (3.1) has property ~(J.!t, Nt, Po) for some po > 0, Nl = Ne Nco andJ.!l = J.! - Nq > O.

{xlllxli

PROOF. Suppose po < min (pINl , p), Xc E Bpo = ;£ Po} and x(t) is the solution of equation (3.1) satisfying the condition x(7:) = XC' We can always find aT> 0 such that < p ;£ t < + In the indicated interval x(t) can be written in the form

Ilx(t)11

(7:

7:

T).

t

x(t) = U(t,7:)xc

(7: ;£ t < 7: + T).

+ J U(t, s)F(s, x(s»ds

The evolution operator U(t, estimate

(3.6)

7:) of equation (3.5) is by assumption subject to the (t

~

7:).

Using this inequality and inequality (3.3), we obtain from (3.7) the integral inequality

Ilx(t)11

;£ N~lxclle-v(t-c)

t

+ J Ne-v(t-s)r;(s)llx(s)llds c

(7: ;£ t < 7: +

T),

which implies (see Corollary 111.2.2) that on the interval 0 ;£ t -

Ilx(t)II

;£ Nlllxclle-(v-Nq)(t-c)

From estimate (3.7) and the condition

Ilx(t) I

;£ pe-(v-Nq) (t-c)

Ilxcll

7: < T

(Nl = Ne Nco ).

(3.7)

;£ po we obtain the inequality

(0 ;£ t -

7:
0 such that + = p, which contradicts inequality (3.7). Thus inequality (3.7) is satisfied for any t and (t ~ which proves the theorem.

Ilx(7: T)II

7:

7:),

3.

295

NONSTATIONARY PRINCIPAL PART

COROLLARY. For sufficiently small q > 0 the zero solution of equation (3.1) is uniformly and asymptotically stable if equation (3.5) has a negative Bohl exponent.

(Here it is assumed that the perturbed equation satisfies the conditions of the local existence theorem (see Remark 1.2).) 2. Stability of the zero solution of the quasilinear equation. If equation (3.1) is quasilinear, i.e. IIF(t, x)11 = o(llxll), condition (3.4) is satisfied for arbitrarily small q in a sufficiently small neighborhood of x = O. Thus the quasilinear equation (3.1) has property £6>(Vh Nh P1) with V1 arbitrarily close to v; in particular, its zero solution is uniformly stable. It turns out that this result has a converse. THEOREM 3.2. In order for the zero solution of equation (3.1) to be uniformly stable for any perturbation F(t, x) satisfying the quasilinearity condition it is necessary and sufficient that the corresponding unperturbed equation have a negative Bohl exponent.

The sufficiency of the assertion follows directly from the above arguments. We prove the necessity by considering only one perturbed equation (3.1) with F(t, x) = xllxllp for some fixed p > 1. Let Bp = {xlllxli < p} be a ball having the property that every solution x(t) of the equation under consideration with Xo = x(to) E Bp is indefinitely extendable for t > to. Each such solution can be represented in the form x(t) = U(t, to)y(t), where U(t, to) is the evolution operator of equation (3.5). From (3.1) we obtain the following equation for y(t); PROOF.

y'(t) = ¢(t)y(t)

(¢(t) =

It follows from (3.8) that yet)

¢(t) = exp

~ to).

(3.8)

(> 1 for t > to),

(3.9)

I U(t, to)y(t) lip; t

= ¢(t)xo, where

(1. ¢(s)ds)

and therefore by virtue of (3.8) ¢'(t)

=

¢P+1(t) I U(t, to)xo lip·

(3.10)

From this result, taking into account that ¢(to) = 1, we get ¢'(7:) 1 f. II U(7:, to)xo lip d7: = f ¢P+1(7:) d7: ?:. p t

t

(xo E £6> p).

(3.11)

But this estimate implies by virtue of Theorem III.6.2 and Remark III.6.3 that the Bohl exponent ICB of equation (3.5) is negative under the additional assumption of integral boundedness of the coefficient A(t). In the general case, according to the same Theorem III.6.3, it remains for us to show that ICB < 00. To this end we make use of the fact that uniform stability must

296

VII.

NONLINEAR EQUATIONS

hold for the perturbed equation under consideration. We choose an c > 0 and find a 0 such that Ilx(to) < 0 implies < c independently of the choice of to· Then for < min (0, p) we will have

Ilxoll

I

Ilx(t) I

Ilx(t) I

= ¢(t) I U(t, to)xo I
0 there exist a constant M > 0 depending only on A and p and a constant q > 0 depending only on A such that if F(t, x) E (M, q, p), equation (4.4) has one and only one solution x(t) that remains for all t in the ball Bp:

sup

Ilx(t)

-00 I the first term in (4.12) becomes sufficiently smaIl and the trajectory of the solution finds itself in BpI. Let XI(t) and xz(t) be a pair of solutions. From (4.12) we deduce the inequality 00

Ilxz(t) - XI(t) I ~

Ilyzo -

YlOIINe-v(t-t,)

+ S Ne-vlt-slqllxz(s)

- xI(s)llds,

t,

which implies by virtue of Corollary III .2.3 that

Ilxz(t) -

xI(t)!1

~

¢(t)llyzo - YlOll,

where

¢(t) =

J)

+

vv22NJ)_ 2qNJ)

e-(t-t,).Jv'-ZqNv,

and this implies the proof of the second part of the theorem. REMARK 4.5. To each vector Yo E ~L with a sufficiently small norm there corresponds a unique bounded trajectory x(t) (t ~ to) such that Yo = P _x(to). Thus the initial points x(to) of the trajectories that are bounded for t -> + 00 form in B p/ ZN a manifold IJJI+ of dimension equal to the dimension of ~L. In fact, to each point Yo E 58_ Bp/ZN there corresponds one and only one point x(to) such that Yo = P _x(to) and from which there emanates a trajectory that is bounded for

n

t -> 00.

The manifold IJJI+ is conveniently imagined as a "surface" in the Cartesian product of the subspaces P +58 and P -58. The "equation" of this surface, i.e. the relation between Yo = P -x(to) and P +x(to), is implicit in (4.12). If x(to) E Bp/ZN\IJJI+, the solution x(t) leaves any ball BpI (PI < p) for sufficiently large t. When the spectral set(i +(A) is empty, the space 58 coincides with 58- and therefore all of the solutions emanating from Bp/ZN terminate in BpI for t -> 00. By virtue of (4.13) each of them is uniformly and asymptoticaIly stable. 2)

Hence f1. can be made arbitrarily close to the distance from a(A) to the imaginary axis.

VII. NONLINEAR EQUATIONS

302

On the other hand, if (J' _(A) is empty, the manifold 9](+ becomes a single point x, for which P -x = O. In this case there is only one trajectory that is bounded for t > to. It is obtained when Yo = o. Analogous results hold for a left halfline, the role of 58_ being played by 58+. The initial values x(to) of the solutions that remain in Bp as t --+ - 00 fill out a manifold 9](- situated in B p/ 2N• Each point x of this manifold is uniquely determined by its projection y = P +x, and therefore the "dimension" of 9](- coincides with the dimension of 58+. We again consider equation (4.4) on the real line. From Theorems 4.1 and 4.3 and the observations made in Remark 4.5 we obtain the following result. THEOREM 4.4. Suppose the spectrum (J'(A) does not intersect the imaginary axis. Then for any p > 0 there exist M and q depending only on A and p such that if F(t, x) E (M, q, p), a sufficiently small neighborhood of the zero element of 58 will contain manifolds 9](- and 9](+ with the following properties. a) The manifold IJJ(± is homeomorphic to a neighborhood of zero of the subspace 58±. b) The manifolds 9](+ and 9](- have exactly one point z in common. c) The solution xo(t) of equation (4.4) satisfying the condition xo(to) = z is bounded on the realline. d) The solutions x(t) of equation (4.4) satisfying the condition x(to) E 9](+ (9](-) exponentially approach xo(t) as t --+ + 00 (t --+ - (0) and exponentially recede from it as t tends indefinitely far in the opposite direction. REMARK 4.6. If one of the manifolds 9](± reduces to a single point belonging to xo(t), this solution is uniformly and asymptotically stable over the corresponding halfline. But if the dimensions of both 9](+ and 9](- are not equal to zero, we say that this solution is conditionally stable. REMARK 4.7. The above results extend without difficulty to equation (4.1) in the case when its nonstationary principal part admits an e-dichotomy of the solutions. 4. Orbital stability of a periodic solutions of the autonomous equation. We consider the autonomous equation (4.14)

dx/dt = f(x)

with a continuously differentiable function f(x) and having a periodic solution ¢(t) = ¢(t + T) (- 00 < t < 00). We have already indicated that in this case the variational equation dx/dt

= A(t)x

(A(t)

= !'(¢(t»)

(4.15)

has the periodic solution x(t)

= ¢'(t)

(4.16)

and hence (we assume that ¢(t) ¢ const) the spectrum of the monodromy operator U(T) of this equation contains the point A = 1.

4.

SPECTRUM NOT INTERSECTING THE IMAGINARY AXIS

303

We assume that this point is an isolated simple eigenvalue while the rest of the spectrum of U(T) lies in the interior of the unit disk and does not surround the origin. In this case there exists a periodic operator function Q(t) that reduces (4.15) to the form dx /dt = Ax, where A is a constant operator with zero as an isolated simple eigenvalue and the rest of its spectrum (1 -(A) lying in the interior of the left halfplane. We take to as the initial point and assume that Q(to) = I. The change of variable

+

x(t) = rp(t)

Q(t)z(t)

(4.17)

takes (4.14) into the quasilinear equation dz/dt

= Az + F(t, z)

(F(t, 0) = 0),

(4.18)

the function F(t, z) satisfying in a sufficiently small neighborhood of zero the Lipschitz condition IIF(t, zz) - F(t,

Zl)

I

~ q

Ilzz - ztll

with a sufficiently small constant q > O. We denote by Po and P _ the spectral projections of A corresponding to the parts (10 = {O} and (1- of its spectrum. Let - )) = SUP'<EIJ_ Re A. The substitution

z = e- (4z) (I-I,) z converts equation (4.18) into the equation

~:

=

(A + ~ I) z + F(t, z)

(4.19)

where the function F(t, z) = e(v/Z) (I-I,) F{t, e- (v/Z) (I-I,) z) has the same properties as F(t, z). But the principal part of this equation is e-dichotomic and we can therefore apply the results of the preceding subsections to it. Since the only solution of equation (4.19) that is bounded on the real line is the zero solution, the solutions issuing from the manifold 9)1+ (described in Remark 4.5) tend to zero as t -+00. Returning to equation (4.18), we obtain the following estimate for any solution of it satisfying the condition z{to) E 9)1+ : (4.20) where f1. < )) but can be made arbitrarily close to )) by requiring that z{to) lie in a sufficiently small neighborhood of zero. These solutionsz(t) are determined from an equation of type (4.12), which, after one returns from (4.19) to (4.18), takes the form co

z(t)

=

eA(I-I,) P _z(to)

+ S Git I,

where

s)F(s, z(s)ds,

(4.21)

304

VII.

NONLINEAR EQUATIONS

G (){ eAtp_ At _ e-Atpo

=

Po

_

fort> 0, for t < O.

We note that although IIGit - s)11 = Ilpoll (s> t) does not decrease as S-Hy), expression (4.21) is applicable to the solutions under consideration, since by virtue of estimate (4.20) IIF(s, z(s))

I

;:£ q Ilz(s)

I

;:£ cl'qe-l'(s-to) lip _z(to) II·

After this, from (4.21) we obtain the estimate Ilz(to) - P-z(fo) -z(to)

lip

I

I


0 I ReAl> a I ReAl < a

for A E O'z(A), for A E O'I(A).

Thus in our case the equation dx /dt = Ax admits an exponential splitting of its solutions of generally third order. We will say in this connection that the spectrum O'(A) admits an a-separation. As usual, Ph P z, )81 and )8z will denote the spectral projections and invariant subspaces of A corresponding to the spectral sets O'I(A) and O'z(A).

VII.

306

NONLINEAR EQUATIONS

We will assume that F(t, x) is of class (M, q), i.e. it is continuous and subject to the estimates IIF(t, x)11 ~ M,

(5.2)

IIF(t, X(2)) - F(t, X(l)) II ~ qllx(2) - x(1) II

(5.3)

. on the product space (- 00, (0) x )8. The purpose of the following discussion is the construction of certain special integral manifolds for equation (5.1). When the spec.trum of A does not intersect the imaginary axis, these manifolds convert into the individual trajectories studied in the preceding section. In order to explain the essence of the matter we consider as an elementary example the system of differential equations

I

dXl/dt = 0, dx1fdt = Xl, dX2/dt = - X2,

or in vector form, setting X

=

(xl, xl', X2),

d~=

d

(0 ° 0) 10

Ox

° °-1

(= Ax).

°

The spectrum (l(A) in this case consists of the two points ).0 = and ).1 = - 1, and the three-dimensional space )8 decomposes into the direct sum )8 = )81 )82 of the two-dimensional space )81 consisting of the vectors (xl, xl', 0) and the onedimensional space )82 consisting of the vectors (0, 0, X2). The solution of the system has, as can easily be seen, the form

+

It readily follows from this formula that the zero solution of the system, as well as any other solution initially in )81, is unstable. But the whole set of solutions that are initially in )81 (and, incidentally, remain in )81) is stable in the sense that the distance between any other solution and )81 tends to zero as t --+ + 00. We can say that this set of solutions forms a stable integral manifold )81. An analogous situation exists, as we will see below, for more general nonlinear equations. The individual solutions of such equations will not as a rule be stable or even conditionally stable because of the presence of a "critical" part of the spectrum lying on the imaginary axis. At the same time certain sets of solutions, which in general are no longer linear, can turn out to be stable (or conditionally stable) in the sense that near solutions will be "attracted" to them. The "dimension" of such a stable manifold of solutions always coincides with the multiplicity of the critical

5.

STABLE INTEGRAL MANIFOLDS

307

part of the spectrum of the coefficient of the principal linear part of the equation (and if the critical part is absent, the manifold reduces to a single trajectory). Another example of this type of stable integral manifold is the orbit of a periodic solution of the autonomous equation considered in the preceding section. In this section we deduce a functional equation for integral manifolds of a specific form and prove their existence, while in the next section we establish the stability of such a manifold under certain assumptions on the equation. We note that after finding such a manifold, in order to determine the solutions composing it, we must consider, as will be seen, a differential equation in a space of dimension equal to the multiplicity of the critical part of the spectrum, i.e. less than the dimension of the initial phase space. In particular, this dimension can be finite. We proceed to more precise definitions. By an integral manifold of equation (5.1) we will mean a set WI composed of the trajectories of this equation in the product space (- 00, (0) x Q3. Strictly speaking, we depart here from the usual meaning of the term "trajectory" as the set of values of a solution x(t) (- 00 < t < (0) in the phase space Q3 and think of it rather as the set of points (t, x(t») of the space (- 00, (0) x Q3. The component x E Q3 of an element (t, x) will be called its spatial projection. From this point of view a trajectory in the usual sense of the word is the spatial projection of the trajectory being considered by us. Thus, by definition, through each point of an integral manifold WI there passes a trajectory x(t), and each trajectory having at least one point in WI has all of its points in WI. We will consider the integral manifolds of equation (5.1) that can be described by equations of the form

(t E (-00, (0); Xl E Q31) (5.4) = ¢(t, Xl) in the "coordinates" t, Xl = PIX and Xz = Pzx. Here we will assume that the function ¢ is continuous in t and satisfies a LipXz

schitz condition in Xl with a constant not depending on t. The set of all such functions is a linear space, which we denote by L. Let L1J denote the subset of L consisting of the functions satisfying the condition ., !I¢(t, x?) - ¢(t, x?) (x?l, x?) E Q31;

II

~ 7j I x?) - x?) II

7j = const > 0)

(5.5)

(the functions of class L1J)' The integral manifold WI described by equation (5.4) is called a (p,7j)-manifold if ¢ E L1J and satisfies the condition

(t

E (-00,

(0);

Xl E Q31)'

(5.6)

Thus the spatial projection of a (p, 7j)-manifold is completely contained in the

VII.

308

NONLINEAR EQUATIONS

cylinder Up of 1S whose base is the ball Bp(1S z) in 1Sz and whose "axis" is the subspace 1S1. If a trajectory (t, x(t) lies on the considered manifold and cf;(t) = PIX(t) is its projection onto the subspace 1S1o the whole trajectory can be expressed in terms of the function cf;(t) by means of the formula X(t)

= cf;(t) + ¢(t, cf;(t).

(5.7)

For constructing the manifolds of equation (5.1) that are of interest to us it is convenient to write this equation in the form of a system of two equations in the subspaces 1S1 and 1S z. Multiplying (5.1) from the left by the projections PI and P z respectively, we obtain the required system dXI = f dt

l dt d.x2

=

AIXI

+ FI (t, Xl + X2),

A ZX2 + F2(t,

Xl

(5.8a)

+ X2),

(5.8b)

where Xk = Pkx, Fk = PkF, Ak = PkA (k = 1, 2). Let us derive an equation which the function ¢(t, Xl) defining a (p, 1])-manifold 9J1 must satisfy. If X = x(t) is a trajectory lying on this manifold, the functions Xl = cf;(t) = PIX(t) and X2 = ¢(t, cf;(t) must satisfy system (5.8). Here equation (5.8a) takes the form dcf;/dt

=

Alcf;

+ FI(t, cf; + ¢(t, cf;)).

(5.9)

Conditions (5.3) and (5.5) imply the condition II FI(t, cf;"

+ ¢(t, cf;")

- FI(t, cf;'

+ ¢(t, cf;')

II ~ q(1

Therefore equation (5.9) satisfies for each given ¢ E 1S1 a unique solution cf;(t)

1.2 and has for each XlO XlO at t = -c:

E

=

+ 1])IIcf;"

- cf;'II·

(5.10)

L the conditions of Theorem Wet, -c, xlOl¢) that is equal to

(5.11) We note tha! when t, -c and XlO are fixed the function Wet, -c, xlOl¢) is with respect to ¢ a nonlinear operator defined in the space L and taking its values in 1S1. On the other hand the function X2(t) = ¢(t, cf;(t)) is a bounded solution on the real line of the equation dXz/dt

= A2X2 + F2(t, cj;(t) + X2)

(5.12)

and must therefore satisfy the integral equation 00

xz(-c)

=

J G2(-c -

t)F2(t, cf;(t)

+ xz(t)dt,

(5.13)

5.

309

STABLE INTEGRAL MANIFOLDS

where Gz(t) is a principal Green function of the operator A z in the subspace ~z. We now choose for a fixed 'C the solution ¢(t) of equation (5.9) that is equal to a fixed element Xl at t = 'C: ¢(t) = Wet, 'C, XI!¢). From (5.13) we then obtain the integrofunctional equation 00

¢('C, Xl) =

S Gz('C

- t)Fz(t, Wet,

'C,

XI!¢)

+ ¢(t, Wet, 'C, xd¢)))

dt

(5.14) ('C E ( - 00,

<Xl); Xl

E ~l)

for the function ¢(t, Xl). We emphasize that this equation is not simply an integral equation, since ¢ enters into it not only as a function but as an element of the space L. We have thus shown that the function ¢(t, Xl) defining a (p, 7J)-manifold of equation (5.1) must satisfy equation (5.14). On the other hand, a solution ¢(t, Xl) of equation (5.14) satisfying conditions (5.5) and (5.6) defines a (p, 7J)-manifold of equation (5.1). In fact, for any point (XlO, Xzo, to) lying on 9)(, i.e. satisfying the relation Xzo = ¢(to, XlO), equation (5.9) has a solution Xl =

¢(t) = W(t, to, XlO!¢)

with the property Xl (to) = XlO. It accordingly follows from (5.14) and the obvious relation Wet,

'C,

W('C, to, XlO!¢)!¢) = Wet, to, XlO!¢)

that the function Xz = ¢(t, ¢(t)) is a solution of equation (5.12). This result can be formulated in the following manner. LEMMA 5.1. A (p, 7J)-manifold of the differential equation (5.1) is described by a solution ¢ of the integrofunctional equation (5.14) that is of class L~ and satisfies condition (5.6). The trajectories lying on the manifold are determined by the solutions of the differential equation (5.9).

It will be proved below that for given p > 0 and 7J > 0, when M and q are sufficiently small, equation (5.14) has one and only one solution possessing the necessary properties, and that the integral manifold corresponding to it consists of all of the trajectories of equation (5.1) that lie in the cylinder Ilxzll ;£ p. 2. Proof of the existence of (p, 7J)-manifolds. Let us first prove an auxiliary proposition. We note that if the operator A admits an a-separation, there exists a > 0 for which the spectral set O"I(A) lies in the strip IRe AI < a and hence

a

a

IleM11 ;£

Nae(a-alltl,

(5.15)

where Al = PIA. On the other hand, it can be assumed that the spectral set O"z(A) lies in the domain Re A! > a + and hence that the estimate

!

a

310

VII. NONLINEAR EQUATIONS ~ Noe- (aH) III

II G2(t) II

(5.16)

is valid for the Green function G 2(t). LEMMA 5.2. The operator //f(t, -r, xllqS) representing a solution of equation (5.9) satisfies the conditions

1) II//f(t, -r, xi 2)lqS(2)) - //f(t, -r, X?) IqS(l)) I

~ No {lIxiZ)

- XiI) lIe(a-o+~.) II-TI

(5.17)

i

+ q I e(a-0+~')II-Sls~pIW2)(S, Xl) (qS(k) where (3q

-->

0 as q

-->

E

- qS(l)(s, Xl) lids

L7J;

X?)

E )81;

I} k = 1,2),

0, and

PROOF. Let xil)(t)

=

//f(t, -r, x?) WI)), Xl(t)

=

x?)(t)

=

//f(t, -r, X?) W2)),

//f(t, -r, xiI) W2)).

These functions satisfy the integral equations 1

eA,(H)x?)

+ S eA,(I-S)Fl(s, x?)(s) + qS(l)(S, x?)(s)) ds,

x?)(t) = eA,(H)xi2)

+ S eA,(I-S)Fl(s, xi 2)(s) + qS(2)(S, x{Z)(s))) ds,

xiI)(t)

=

(5.19)

1

(5.20)

T

1

Xl(t)

=

eA,(H)xi l )

+ S eA,(I-S)Fl(s, Xl(S) + qS(2)(S, Xl(S))) ds.

(5.21)

T

From (5.19) and (5.21), using estimate (5.15) and the Lipschitz conditions (5.3) and (5.10) for the functions F and qS(l), qS(2), we obtain the estimate iixil)(t) - xl(t)1I ~

1

S IIeA,(I-s) II{ IIFl(s, x?)(s) + qS(l)(S, x?)(s)) +

T

IIFl(s, Xl(S)

+ qS(l)(s, Xl(S)))

- Fl(s, Xl(S) - Fl(s, Xl(S)

1

~ Noq(1

+ r;) Se(a-o) (I-s) IIxiI)(s)

- Xl(S) lids

T

1

+ Noq Se(a-o) (I-S) T

sup IWl)(s, Xl) - qS(2)(S, Xl) lids. x,

+ qS(1)(s, Xl(S)) I + qS(2)(S, Xl(S)) II} ds

S.

311

STABLE INTEGRAL MANIFOLDS ~ 'C

It can now be deduced from Remark III.2.1 that when t

Ilxil)(t) -

XI(t)

I

~ u (t),

where u(t) is a solution of the integral equation

=

u(t)

Naq(1

+ r;)

t

S e(a-a) (t-s)u(s)ds T

t

+ NaqS e(a-a)(t-s)supll¢(l)(s, Xl)

-

¢(Z)(s,

Xl) I ds,

(t-S)supll¢(1)(s, Xl) -

¢(Z)(s,

Xl) I ds.

T

X,

which,as can easily be verified, has the form u(t)

=

t

Naq S e[N,q(1+~)+a-aJ T

X,

Thus

t

~ Naq S e(a- H fl,)(t-S)supll¢(1)(s,

Xl) -

¢(Z)(s,

where the constant f3q = Naq(1 + r;) tends to zero as q Further, (S.20) and (S.21) imply

IlxI(t) -

xiZ)(t)

I

Xl) lids,

(S.22)

x,

T

~

o.

I I XiI) - xiZ) I t + S IleA,(t-S) I I FI(s, XI(S) + ¢(Z)(s, XI(S»))

~ IleA,(t-T)

T

- FI(s, xiZ)(s) ~ Noe(a-o) (t-T)

+ Noq

Ilx?) -

xi Z)

I

t

(1

+ ¢(Z)(s, x?)(s») lids

+ r;) S e(a-a) (t-s) IlxI(s) -

I

x{2)(s) ds.

T

Using Lemma II1.2.1 once again, we readily deduce from this inequality the estimate

IIW(t, 'C, x?) I¢(t, XI. y)

satisfying the integral equation

1>(1:, XI. y) = eA(~-lo)y +

(6.1) 00

J Gz('1: -

t) F2(t, Wet,

The function Xl

= W(t,

'1:,

xll1» + 1>(t, Wet, '1:, Xl 11», y))dt (to

10

'1:,

xlOl1»

~ '1:

to)

(6.5)

and with the norm

We will investigate in this space the operator S: ¢

-t

~ defined by the equality

(6.6) co

= eACr-to)y + LEMMA

).i
0 is sufficiently small and i is sufficiently large

III~III

=

_sup

0 is sufficiently small and I> 0 is sufficiently large.

Let us show that it is a contraction in this space. Using Lemma 5.2, we get that

OJ

;;:; No

S e-(a+O)i T- ti IIF1(t, W(t, ZO,X1!¢C2») + ¢(2)(t, W(t, zo, x11¢(2»), y))

(6.8)

t,

OJ

;;:; Noq Se- (a+O) iT- t i(R1 + R 2)dt, t,

where R1

W(t, zo, X1!¢C1») I e(a-o+(J,) it-Ti ;;:; q a + (3q ~~~ 11¢(2)(s, XI. y) - ¢(1)(s, Xl, y) II, =

IIW(t, zo, X1!¢C2») -

a

R2

y)11 ¢(1)(t, W(t, zo, X11¢(2»), y)11

=

1I¢(2)(t, W(t, zo, X1!¢C2»), y) - ¢(1)(t, W(t, zo, X11¢(1»),

~

IlqI(2)(t, W(t, zo, X11¢(2»), y) +

;;:;

11¢(1)(t, Wet, zo, X1!¢C2»), y) - ¢(1)(t, W(t, zo,x11¢(1)), y)11 e(a-o+fJ,) it-Ti } SUpl _ !¢C2)(S, XI. y) - ¢(l)(s, XI. y) I { 1 + qr; a - + (3q .

a

Substituting these estimates into (6.8) and integrating, we obtain the required result for sufficiently small q > 0 in exactly the same way as in the proof of Theorem 5.1.

7.

319

AVERAGING PRINCIPLE

The proof of the last assertion of the theorem is carried out in exactly the same way as in Theorem 5.1. § 7. Averaging principle 1. A veraging principle for a finite interval. We consider in a Banach space )8 the differential equation

(7.1)

dx/dt = cf(t, x)

containing a small parameter c. We assume that on the set [0, (0) x B, where B is a ball in )8, the functionf(t, x) is bounded, continuous in t and satisfies the Lipschitz condition (XI. X2 E

B)

(7.2)

with a constant not depending on t. We assume further that the mean value off(t, x) with respect to t: fo(x) = lim T-'too

1

T

T

S f(t, x)dt

(7.3)

0

exists for each x E B. Under this assumption it is expedient to consider along with equation (7.1) the equation (7.4)

dx/dt = cfo(x).

It follows from (7.3) that the functionfo(x) also satisfies condition (7.2) in B.

Let y(t) (0;:;:; t ;:;:; To) be a vector function taking values in B that satisfies the equation dy

dt =fo(y)

(7.5)

Then, as can easily be seen, the function xo(t; c) = y(ct) will be a solution of equation (7.4) on the interval t ;:;:; To/c. We will establish below a connection between xo(t; c) and the solution x(t; c) of equation (7.1) that satisfies the condition

°;:;:;

x(o; c) = yeO) (= xo(O; c)).

(7.6)

More precisely, we will prove the following assertion. THEOREM 7.1. Suppose that on the set [0, (0) x B the function f(t, x) is bounded, continuous in t and satisfies the Lipschitz condition (7.2). Suppose further that its mean value (7.3) exists for each x E B. If yet) is a solution of equation (7.5) that takes the interval [0, To] into B, there exists for any r; > an co > such that when < c < co the solution x(t; c) of equation (7.1) satisfying condition (7.6) is subject to the estimate

°

°

°

320

VII.

NONLINEAR EQUATIONS

Ilx(t; e) - y(et) II ~

(0

1)

~

t

~

Tole).

Theorem 7.1 is obtained as a corollary of a general theorem on the continuous dependence of solutions of differential equations on a parameter. To this end we consider the more general (than (7.1») equation dx d7: = /(7:, x; e)

(0

~

7:

~

To; 0

~

e

~

eo)

(7.7)

with a function/(7:, x; e) satisfying the Lipschitz condition

xI/!

II/(7:,X2; e) - /(7:, Xl; e) II ~ CllX2 -

(7.8)

and the condition '!O

To

0

0

lim J/(7:, x; e)d7: = c~o

J/(7:, x; O)d7:

(7.9)

which is called the condition of integral continuity at e = O. Equation (7.7) can be converted into (7.1) by putting /(7:, x; e) = {ffc«7: 1)0$, x) e ~ 00, ox, e- ,

and making the change of variable 7: = et. Condition (7.9) is satisfied in this case since it takes the form lim

) J/ (7:-, X d7: = 7:ofo(x) e . T,

c~o 0

or, equivalently, lim -

e

c~o 7:0

T,1e

J0 /(t, x)dt = fo(x),

which is equivalent to (7.3). Let us prove an auxiliary assertion. LEMMA 7.1. Suppose x(7:) is a continuous/unction on [0, Tol. Then (7.8) and (7.9) imply the relation

PROOF.

T,

T,

lim

J /(7:, x(7:); e)d7: = J /(7:, x(7:); O)d7:.

e~O

0

0

From (7.9) it follows that for any 7:10 7:2 TZ

Tz

"'1

orr

lim e~O

and hence that for any 0

~

E

(7.10)

[0, Tol

J/(7:, x; e)d7: = J/(7:, x; O)d7:, 7:1 < ...
0 there exists an co (0 < co < c1) such that every solution x(7:; c) of equation (7.11) for 0 < c < co that is defined on the interval [0, To] and satisfies the condition x(O; c) = x(O; 0) (= xo) is subject to the estimate

VII.

322

NONLINEAR EQUATIONS

(7.12)

Ilx(-r; c) - x(r; 0) II < r; PROOF.

The functions x(r; c) and x(r; 0) satisfy the integral equations T

x(r; c) = Xo

+ Jf(s,

x(r; 0) = Xo

+ Jf(s, x(s; 0); O)ds.

o

x(s; c); c)ds

and
0)

o

is the Laplace transform of the function V(t, z)

= f(t, Xo + z) - fo(xo + z).

(7.28)

We will use the following well-known proposition to show that the transformation (7.26) has an inverse. 7.2. Suppose a function ifJ(s; J.) (0 ~ s < Cf) taking values in a Banach space and depending on an abstract parameter J. E A satisfies the condition

LEMMA ~

(M =) sup{11

+1

ifJ(s; J.)ds

I I t ~ 0; J.

and, uniformly in J. E A, the condition

lim ~

T~oo T

Then the relation

JifJ(s; J.)ds = O.

0

E

A}
0 proves the required assertion. LEMMA

7.3. The following relations hold uniformly in 1 E ( - 00,

00)

and Z E Bp:

lim eU(t, z; e) = 0,

(7.30)

lim eu;(t, z; e) = O.

(7.31)

dO dO

For by virtue of (7.16) (7.17) and (7.28) the bounded functions V(s + I, z) and V;(s + t, z) satisfy the conditions of Lemma 7.2, the role of the parameter). being played by the pair (I, z). From (7.31) it follows that when 0 is sufficiently small the operator 1 -eu;(/, z; e) (t E ( - 00, (0), Z E Bp) has a bounded inverse, and hence the mapping (7.26) is invertible. In addition, from (7.30) it follows that for any preas0 and an r; 0 such that < Po when (0 O.

e>

e eo

Ilzll

eo>

>

Ilhll

326

VII.

NONLINEAR EQUATIONS

Representing the function u in the form 00

u(t, z; c)

=

eel

JV(s, z)e.-esds,

I

we find that

uf(t, z; c) = cu(t, z; c) - V(t, z),

(7.32)

and consequently

dh

dt

=

dz

,dz

z

V

dt - cU. dt - c u + c .

(7.33)

Using relations (7.26), (7.28) and (7.33), we reduce equation (7.25) to the form

(/ - cu;)dzJdt = c[f(t, Xo + h; c) - f(t, Xo + z; c)]

+ c/o(xo + z) + czu(t, z; c) = c(Az + S(z») + cZu(t, z; c) + c[f(t, Xo + z - cU, c) - f(t, Xo + z; c)] or, after multiplying by the bounded operator (/ - cU;)-l and introducing the "slow" time 7: = ct as the independent variable, to the form

dzJdt

= Az + F(z, 7:; c),

(7.34)

where, as is easily seen, F(z, 7: ; c) has a bounded first derivative F: (z, 7: ; c) and satisfies the relation

F(z, 7:; c) - S(z)

=

O(c).

uniformly in 7: E ( - 00, (0) and z E Bp. Hence for any M > 0 and f.J. > 0 there exist an co and a c < co and Ilzll < /30 we have

IIF(z, 7:; c) II ~ M

/30 such that

when (0 0 an co > 0 such that when (0 0 there exists an ea> 0 such that when (0 0 (t

>

>


0 there exists an N> 0 for which the equation dx/dt

= A(t)x + F(t, x)

(0.12)

has property 88(vo ~ e, N). If p = 1, the assertion is true for e = O. 11. Prove the following theorem (D. L. Kucer [2]). Suppose the equation condition sup

O~'t'~t

x=

A(t)x satisfies the

00

and IIF(t, x)11 ~f(t)llxll (11xll ~ r; t ~ 0),

where

fo fP(t)dt


Then the solutions of equation (0.12) are subject to the estimate IIx(t) I ~ Nllx(to)11 for some O. 12. Prove the following theorem (D. L. Kucer [2]). Let m 1 and p 1. In order for the zero solution of equation (0.12) to be uniformly stable for any F(t, x) satisfying the condition

>

IIF(t, x)11

~f(t)llxlim

(11xll

~

r; t

>

~

0),


O. It can be assumed here that conditions (0.16) are satisfied only in some ball. " An analogous assertion holds with respect to Theorem 6.1. c) Extend this result to the case when 00

f tnM(t)dt2, when the spectrum of the monodromy operator for the variational equation contains points lying in both the interior and exterior of the unit disk, some trajectories will approach a periodic motion for 1->00 and some, for 1->-00. In this connection there can arise a complicated phenomenon discovered by H. Poincare, the so-called homoclinic motion: a trajectory that is asymptotic to a periodic motion for both 1->00 and 1->-00. A deep study of this phenomenon in the multidimensional case has recently been made by Ju. I. Neimark in the paper [1] (which contains a bibliography of the literature on this question). NOTES The existence and uniqueness theorems of § 1 are contained in various textbooks and have been well known for a long time. The result of Theorem 1.3, viz. the stability of the Bohl exponent of a nonlinear equation under nonlinear perturbations, is essentially contained in the works of Bohl (see the Notes to Chapter III and Exercise 9). The proof given by us is based on a method applied by B. A. Scerbakov [1] in a more general situation (see Exercise 5 and the comments to it). Theorems 2.1 and 3.1 (see M. G. Krein [2] and the "Lectures") constitute more special (but somewhat more exact) results. The method employed in Theorem 3.1 makes use of integral inequalities (a similar method was essentially used by Bohl; see the Notes to Chapter III). The method of proof of Theorem 2.1 is a generalization of a method of Ljapunov, as is also the method of proof of Theorem 2.2 (see the "Lectures"), which generalizes another well-known theorem of Ljapunov, viz. his instability theorem. In the theorems of §§2 and 3 mentioned above the perturbing term has first order smallness. The stability of the solutions of nonlinear equations under such an assumption on the perturbations was apparently first studied by O. Perron. The other instability theorem cited by us (Theorem 2.3) was proved by M. A. Rutman [1] in a more complicated way and under more complicated assumptions. Our proof, which is based on the same idea, was obtained by Ju. L. Daleckii after analyzing the original proof. Nonlinear perturbations oflinear e-dichotomic equations were first considered by P. Bohl (for the case when the original equation is stationary) and later by o. Perron (see the Notes to Chapter IV). Almost periodic solutions have been studied in the finite-dimensional case by many authors (G. I. Birjuk [1], N: N. Bogoljubov [1], N. N. Bogoljubov and Ju. A. Mitropol'skii [1] and B. P. Demidovic [2]). In this connection we have waived the usually imposed requirement that the almost periodicity of F(/, x) in 1 be uniform in x. Theorem 4.5 was obtained in the two-dimensional case by A. A. Andronov and A. A. Vitt [1]; a more exact finite-dimensional version due to E. A. Coddington and N. Levinson [1] (see also the book by B. P. Demidovic (3]) has been carried over in the present book to the infinite-dimensional case. The method of investigating multidimensional integral manifolds and their application to the substantiation and development of the averaging principle were proposed by N. N. Bogoljubov in his deep investigation [1]. The ideas of this investigation were used in all of his subsequent works.

NOTES

333

Various versions of results in the finite-dimensional case that are similar to those presented in §§ 5 and 6 have been given in the works of N. N. Bogoljubov and Ju. A. Mitropol'skii (N. N.

Bogoljubov [1] and N. N. Bogoljubov and Ju. A. Mitropol'skii [1]). Ju. A. Mitropol'skii and o. B. Lykova (Ju. A. Mitropol'skii [2] and Ju. A. Mitropol'skii and o. B. Lykova [1]) have carried over some of these results to the case of Hilbert space. The present account is of a more general character and in a number of places contains simpler proofs. It was obtained as a result of analyzing the above mentioned works and is due to Ju. L. Daleckii [6]. The material presented in Chapter IV has permitted us to carry over many of these results to the nonstationary case. The results presented in § 7 are a development of the well-known method of van der Pol. A theorem of the type of Theorem 7.1 was proved for systems with periodic coefficients by L. I. Mandel'stamm and N. D. Papaleksi [1], and for arbitrary systems of finite order by N. M. Krylov and N. N. Bogoljubov [1]. I. I. Gihman [1] noted that this theorem is a corollary of a special theorem on the continuous dependence of the solutions of a differential equation on a parameter. M. A. Krasnosel'skit and S. G. Krein [1] generalized this result and carried it over to the case of an equation in Hilbert space. There exists a number of other generalizations due to J. Kurzweil and Z. Vorel [1], H. A. Antosiewicz [1] and others. We have cited only the simplest version in order to avoid widening the circle of ideas considered in this book, and our presentation follows that of M. A. Krasnosel'skii and S. G. Krein. The results presented in § 7.2 concerning the estimation of the behavior of a solution over an infinite interval of time were obtained in the finite-dimensional case by N. N. Bogoljubov [1]. They were carried over to the case of Hilbert space by Z. Sircenko [1]. We have given a somewhat simpler proof. An extensive survey of the results concerning integral manifolds up to 1961 inclusively is contained in the report of N. N. Bogoljubov and Ju. A. Mitropol'skit [2], while a survey of more recent works can be found in the book of Ju. A. Mitropol'skii and O. B. Lykova [2]. For a survey of the results connected with the method of averaging see the book of Ju. A. Mitropol'skit [3].

CHAPTER

VIII

ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS OF A LINEAR DIFFERENTIAL EQUATION WITH A LARGE PARAMETER In this chapter we find an asymptotic expansion of the solutions of a linear differential equation in the inverse powers of a large parameter appearing in the equation. In the simplest case, when the principal part of the coefficient of a finite-dimensional system has a spectrum consisting of eigenvalues whose multiplicity does not change over the whole interval of time, such expansions have been the object of extensive studies, beginning with the works of Birkhoff and Tamarkin. Usually these expansions are used to prove the completeness of the eigenfunctions of selfadjoint boundary problems. We do not cite such applications since they lie to the side of our main theme and, furthermore, have appeared in textbooks for some time now (see, for example, M. A. Naimark [1]).

Of greater relevance for us is the use of asymptotic methods for the approximate calculation of the monodromy operator of a periodic equation and for the determination of estimates for stability bands. Applications of this kind are partially touched upon in the examples of § 2 and in the Exercises at the end of the chapter. On the other hand, equations with a large parameter can be interpreted in an equivalent form as equations with slowly varying coefficients. In § 1 we present a method of asymptotically decomposing an equation with a large parameter in accordance with a decomposition of the spectrum of the principal part of its coefficient into nonintersecting spectral sets. In § 2, under the assumption that the spectrum of the principal part of the coefficient lies in the left halfplane, we estimate the error sustained in replacing the solution of an equation by its nth approximation. In § 3 we consider the more complicated case when, after going over to slow time, the coefficients of the equation can rapidly oscillate against a background of slowly varying amplitudes. In this case there can arise distinctive resonance phenomena analogous to parametric resonance.

§ I. Approximate decomposition of the equation I. Statement of the problem. We consider in a phase space ~ the differential equation

~~

=

AB(7:; A)¢

(0 ;2;

7:

;2; T),

(1.1)

in which A is a sufficiently large parameter. More precisely, we will assume that A varies in a domain llAo of the complex plane which lies outside the disk of radius Ao (IAI ~ Ao) and contains the ray A ~ Ao. We assume that the following representation holds:

B(7:; A)

= Bo(7:) + (l/A)B(1)(7:; A) 334

(A

E

ll)o)'

(1.2)

1.

APPROXIMATE DECOMPOSITION OF THE EQUATION

335

where the operators Bo('r) and B(l)('r; A) are continuous in '0 on the interval [0, T] and, in addition, ('0 E

[0, T]; A E il).,).

In the sequel some differentiability requirements will also be imposed on these operator functions. It will be assumed everywhere in this chapter that the spectrum O'(Bo('r)) decomposes into several spectral sets: n

O'(Bo('r))

=

U O'k('r), k=l

(1.3)

which do not intersect for any '0 E [0, T] and continuously depend on '0. The latter means that a smooth contour Fk separating a spectral set O'k('rO) from its complement in the spectrum O'(Bo('ro)) for some '00 E [0, T] has the same property for any other sufficiently close values of '0 E [0, T]. We recall that according to the formula

Pk('r) = - - 12. § (Bo('r) - p,l)-ldp, 71:1 r

(k

= 1,.· ·,n)

the spectral projections Pk('r) corresponding to the decomposition (1.3) are continuously differentiable in '0 as many times as the operator Bo('r). Equation (1.1) can be written in a different form if one introduces the new variable t = A'r, viz.

d¢/dt = [Bo('r)

+ £B(1)('rie)]¢

(£ = I/A).

(1.4)

In the case when the variable t has the meaning of time, the variable '0 = £t (for real £ > 0) is called the "slow time". The equations ofform (1.4) are called equations with slowly varying coefficients. When £ = equation (1.4) degenerates into the equation d¢ /dt = Bo(O)¢, whose evolution operator is easily calculated:

°

n

L; Pk(O)eB,(O) p. (0) (t-s). (1.5) k=l It is natural to expect that the basic structure of this expression is preserved for sufficiently small values of £ "# with the constant operators being replaced by slowly varying operators plus terms of order 0(£). We will therefore seek an approximate expression for the evolution operator of equation (1.4) in the form

U(t, s)

=

eB,(O)(t-s)

=

°

_

U(t,to;£)

n

=

L; [Pk('r) k=l

+ £Vk('r;£)]Ylt,to;£),

where the operator Yk(t, to; £) satisfies the equation

dYk(~/O; £)

=

[Pk('r)Bo('r)

+ £Ok('r; £)]Yk(t, to; £)

VIII.

336

EQUATION WITH A LARGE PARAMETER

and the condition (1.6) Returning to the original notation and using (1.6), we rewrite the indicated relations in the form D(7:,

7:0;

dYk('d:O; A)

A) =

~JI +

l

Vk(7:; A)] Yk(7:,

= A[Pk(7:)Bo(7:) + (7:,7:0 E

-t

7:0;

A),

Qk(7:; A)] Yk(7:,

7:0;

(1.7) A)

(1.8)

[0, T]; A Ell).,).

The operators Vk and Qk (k = 1,2,.··, n) are to be found. We will try to choose them so that condition (1.6) is satisfied, i.e. so that the range of Y k lies in the subspace ~h(7:) = Pk (7:)'iS (k = 1,.··, n). We assume that the operator B(l)(7:; A) has an asymptotic expansion in inverse powers of A, i.e. (1.9) where IIB(P+l)(7:; A) II ~ c.

(1.10)

We will seek the operator functions Vk and Qk in the form 1

T Vi7:; A) 1

P

= E/-s VkS(7:),

T Qk(7:; A) =

P

S~l A- sQks(7:)

(1.11 ) (1.12)

and attempt to choose the coefficients of these expansions so that the operator function (1.7) satisfies the equation dD(7:, , (7:,. A')U-( 7:,7:0,. A') d7:7:0; A) -_ AB

1 '" ( 7:,. A, ') + »'Vp

(1.13)

which differs from the equation. dUjdt

= AB(7:; A)U

for the desired evolution operator by a term of order O(A-P). Under certain conditions, which will be described below, a solution of equation (1.13) differs from the evolution operator of equation (1.1) by a quantity of order O(A-P).

Once we have indicated a method of determining the coefficients of the expansions (1.11) and (1.12), the approximate solution of the Cauchy problem for

1.

APPROXIMATE DECOMPOSITION OF THE EQUATION

.),)1

equation (1.1) will be reduced to the solution of equations (1.8) in the phase spaces ~M't"o). Such a reduction is called an approximate (asymptotic for . :l. -+ 00) decomposition of equation (1.1). 2. Some auxiliary propositions. LEMMA 1.1. Let Q('t", 't"o) be a differentiable operator with respect to '0 that "tracks" the subspaces IBi'r):

Pk('r)Q('r, Q('ro,

'00)

=

'00)

= l.

Q('r, 'ro)Pk('ro)

(k

= I, .. ,n),}

(1.14)

If the relation Qk('r; ).) holds identically in

'0,

=

Pk('t")Qk('r; ..:l.)Pk('r) + Q'('t", 'rO)Q-l('r, - Pk('r)Q'('r, 'rO)Q-l('t", 't"O)Pk('r)

'00)

(1.15)

the relation

Pi'ro)Yk('ro, 'ro;..:l.) = Yk('ro,

'00;

..:l.)Pl't"o) = Yl'ro, 't"o;..:l.)

(1.16)

implies the relation ('0 E [O,T]). (1.17) = Yk('r, '00; ).)Pk('ro) = Yk('r, 'ro;..:l.) PROOF. The substitution Yk('r, '00; ).) = Q('r, 't"o)Xk('r, '00;).) takes equation (1.8)

Pk('r)Yk('r,

'00;).)

into the equation dXk/d't" = Q-l('r, 't"o){..:l.Pk('r)Bo('r)

+ Qk('r,..:l.)

- Q'('r, 'rO)Q-l('r, 'ro)}Q('r, 'ro)Xk,

which by virtue of conditions (1.15) and (1.14) reduces to the form dXk/d't" = Pk('t"O)Q-l('t", 't"o){ ..:l.Pk('r)Bo('r)

+ Qk('r; ..:l.)

- Q'('r, 'rO)Q-l('r, 'ro)}Q('r, 't"O)Pk('t"O)Xk. It is not difficult to see that this equation is satisfied by not only Xk('r, '00; ).) but also the operators Xk('t", '00; ..:l.)Pk('ro) and Pk('rO)Xk('t", '00; )'). From (1.16) it follows that all three operators coincide for '0 = '00 and hence for all '0 E [0, T]. Inasmuch as .,.

Pk('r)Yk('r,

'00;).) =

Pk('r)Q('r, 'ro)Xk('r, 'ro;..:l.)

= Q('t", 'ro)Pk('rO)Xk('t", '00; ..:l.) and Yk('r,

'00;

equality (1.17) is proved.

..:l.)Pi't"o) = Q('r, 'ro)Xk('t", 't"o; ..:l.)Pk('ro),

338

VIII. EQUATION

WITH

A LARGE PARAMETER

REMARK 1.1. Condition (1.15) is obviously satisfied for an operator Ok(r; A) of form (1.12) if

Okl(r)

= Pk(r)Okl(r)Pk(r) + Q'(r, -ro)Q-l(r, -ro) - Pi-r)Q'(-r, -rO)Q-l(-r, 7:o)Pk(-r),

Oks(-r) = Pk(-r)Oks(-r)Pk(-r)

(s

~

2).

(1.18) (1.19)

REMARK 1.2. In the sequel it will be convenient to take as Q(-r, -ro) the conjugation operator constructed in § 111.1. It satisfies the differential equation

In this case equality (1.18) takes the form

Okl(-r)

= Pk(-r)Okl(-r)Pk(-r) +

n

~

P;(-r)Pj{-r).

(1.20)

j=l

Condition (1.16) will be satisfied, for example, if Yk(-ro, -ro; A) = Pk(-ro). From Lemma 1.1 and Remarks 1.1 and 1.2 it follows that if the operators Oks(-r) are selected so that they satisfy conditions (1.19) and (1.20), the operators Yk(-r, -ro; A) determined from equations (1.8) will satisfy condition (1.6) for all -r if they satisfy it for -r = -roo 3. Construction of the approximate solution. We proceed to calculate the coefficients of expansions (1.11) and (1.12). Differentiating (1.7) and using these expansions and (1.8), we find

1

=

n

p

T k'f1 s'f1 A- sV;sC-r) Yk(-r, -ro; A)

+ k~l

[I + S~l A-SVkS(-r)] [Pk(-r)Bo(-r) + S~l A-SOkS(-r)] Yi-r, -ro; A) (1.21)

On the other hand, the right side of equation (1.13), after dividing through by

A and using (1.9), (1.7) and (1.11), takes the form

1.

B(-I:; ).)0('1:, '1:0;).)

+ ).-(P+1)f/JP{'1:;).)

= [sto ),-sBs('1:) x

=

339

APPROXIMATE DECOMPOSITION OF THE EQUATION

+

EJ1 + ~1

).-(P+1)B(P+1)('1:; ).)]

).-SVkS('1:)] Yk('1:, '1:0;).)

+

).-(P+1)f/Jp('1:;).)

k~l {BO('1:) + St1 ).-s[Bs('1:) + BO('1:)Vks('1:)]

+ +

(l.22)

).-(P+l)B(P+1)('1:; ),)[1 + st1 ).-sVkS ('1:)]} Yi'1:, '1:0;).)

).-(P+1)f/Jp('1:; )').

The right sides of equalities (l.21) and (l.22) must coincide. By equating the coefficients of like powers of )., making use of relations (1.17) and discarding the common factor Yk('1:, '1:0; ).), we obtain the system of relations Vk1 ('1:)Bo('1:)Pk('1:)

+ Ok1(r')Pi'1:)=

Vk.('1:)Bo('1:)Pk('1:)

+

V';'S-l('1:)Pk('1:)

= B.('1:)Pk('1:)

+

BO('1:)Vk.('1:)Pk('1:)

B1('1:)Pk('1:)

+ Bo('1:)Vk1 ('1:)Pk('1:); s-l

+ Oks('1:)Pi'1:) + r;

j=l

Vk,s-l('1:)Okj{'1:)Pk('1:)

s-l

+ r;

(2~

B s-i'1:)Vkj{'1:)Pi'1:) j=l which is more suitably written in the form

s

~

BO('1:)Vk1 ('1:)Pi'1:) - Vk1 ('1:)Pk('1:)Bo('1:)

(1.23)

= Ok1('1:)Pi'1:) - B1('1:)Pk('1:); BO('1:)Vks ('1:)Pk('1:) - V ks ('1:)Pi'1:)Bo('1:) = OkS('1:)Pk('1:) - Tk.('1:)Pk('1:)

p),

(2

~

s

~

p),

(1.24)

where Tks('1:)

s-l

s-l

;=1

j=l

= r; Bs-i'1:) Vkj{'1:) + Bs('1:) - v,{, S-l('1:) - r; Vk. s-i'1:)Okj{'1:).

(1.25)

Moreover, a comparison of the terms of order O().-(P+1» in (1.21) and (1.22) leads to the equality f/Jp('1:, ).) n {

= k"f1 V£p{'1:)

zp

P

+ S=f+1 ).-(s-P-l) j=~_P [Vk,s-j{'1:)Ok;('1:) - B(P+1)('1:,

- Bs-j('1:)Vk;('1:)] (1.26)

).(1 + st1 ).-sVkS('1:)]} Yk('1:,

'1:0, ).).

VIII.

340

EQUATION WITH A LARGE PARAMETER

The system of equations (1.23)-(1.24) is recursive in nature inasmuch as the expression (1.25) for Tks(-r) contains only those operators Vkj{-r) and Qkj{-r) with subscripts j = 1,.··, s - 1 and the derivative V':'s-l(-r). We will now show how to successively determine all of the operators Qkj and Vkj (j = 1,.··,p; k = 1,.··,n) from these equations with the use offormula (1.3.14). It will be assumed in this connection that the operator function Bj{-r) (j = 1,. .. ,p) has strongly continuous derivatives up to order p + 1 - j inclusively. We begin by considering the equation of first approximation (1.23). Multiplying it from the left by the projection Pi-r), we obtain for the operator XR)(-r)

= Pj{-r)Vkl(-r)Pk(-r)

the equation BO(-r)Xj~l)(-r) - XN)(-r)Bo(-r)

(j = Pj(-r)[Qkl(-r) - B1(-r)]Pk(-r) This equation is satisfied for j = k if one puts

(1.27) =

1,2,. ··,n).

(1.28)

Xkr(-r) = 0

and Pk(-r)Qkl(-r)Pk(-r)

= Pl-r)B1(-r)Pk(-r).

(1.29)

Equalities (1.20) and (1.29) permit us to determine the operator Qk1(-r): n

Qkl(-r) = Pk(-r)B1(-r)Pk(-r)

+ L: P;(-r)P,(-r). ,=1

(1.30)

When j =f. k we have Y(-r) = Pi-r)Y(-r)Pk(-r), where Y(-r) denotes the right side of (1.27) and the projections Pi-r) and Pk(-r) correspond to disjoint spectral sets of Bo(-r). We can therefore apply formula (1.3.14), which gives us

XJ.Nr) _

_1_

§

§

- - 4,,2 F.«) F,«) ___1_

-

§

(Bo(-r) - vI)-IP/-r)[Qkl(-r) - B1(-r)]Pl-r)(Bo(-r) - f.JI)-1 dv df.J v - f.J

§

(1.31)

(Bo(-r) - vI)-IP/-r)[Pf(-r) - B1(-r)]Pk(-r)(Bo(-r) - f.JI)-1 dv df.J. 4,,2 F.«) F,«) v - f.J

Finally, we find the operators Vk1 (-r) by putting (1.32) As has already been mentioned the projections Pk(-r) have as many continuous derivatives as the operator Bo(-r). Therefore, as can be seen from formulas (1.30) and (1.31), the operators Qkl(-r) and Vk1 (-r) have continuous derivatives of order p. We consider the equations for second and higher approximations in a completely analogous manner. Multiplying (1.24) from the left by Pj{-r), we obtain for the

2.

ESTIMATE OF THE ERROR

341

operator the equation Bo('C)Pj{'C)Xj~)(-1:)

- xjt/('C)BO('C)Pk('C) = Pj('C) [OksC'C) - Tks('C)]Pk('C),

(1.33)

which is investigated in exactly the same way as (1.27). Whenj = k we put (1.34) and and get, using (1.19),

OksC'C) = Pi'C)TksC'C)Pk('C)

(s = 2,.··, p).

(1.35)

After this we find

XjZl('C)

_

f f

4n2 r.Cr) rjCr) and put, as before,

(Bo('C) - vI)-lPj{'C)Tk,('C)Pi'C)(Bo('C) - fLI)-l dv dfL (1.36) v - fL Vk,('C) = L; xj%)('C).

(1.37)

#k

We note that the operator function

Tk2('C)

=

B1('C)Vk1('C)

+ Bz('C)

- Vk1('C) - Vk1('C)Okl('C)

is continuously differentiable p - 1 times, and hence so also are the operators Vk1('C) and Ok2('C). It is not difficult to show by induction that the operators Vkr('C) and Okr('C) are continuously differentiable p + 1 - r times. For if this is true when r = s - 1, the operator Tks('C) is according to formula (1.25) continuously differentiable p - s + 1 times, and it follows from formulas (1.35) and (1.37) that this property is also possessed by the operators Vkr('C) and OkrC'C). In particular, th~, operator Vkp{'C) has a continuous derivative. Thus all of the coefficients of expansions (1.11) and (1.12) have been determined so that the operator function (1. 7) satisfies equation (1.13). In this connection, as follows from formula (1.26), the operator function Wp{'C; i\) is continuous in 'C for 'C E [0, T] and i\ E U".

§ 2. Estimate of the error 1. Basic lemmas. We proceed to estimate the error sustained in using the above method for constructing the evolution operator of equation (1.1).

342

VIII.

EQUATION WITH A LARGE PARAMETER

This estimate is obtained under certain additional assumptions connected with the location of the spectrum of the operator Boer). One of these additional assumptions is formulated as follows. (C~) The spectrum a(Bo(-r») of Bo(-r) lies in the interior of the left halfplane for all values of -r E [0, T]. In certain important cases the spectrum a(Bo(-r») can fall on the imaginary axis.

Such a situation can be analyzed under certain additional assumptions described by the following condition.

(C v) The phase space 58 is a Hilbert space ~ and there exists a uniformly positive continuously differentiable operator W(-r) (0 ~ -r ~ T) such that the operator Re(W(-r)Bo(-r») is nonpositive. We note that it automatically follows from condition (C v) that the spectrum

°

a(Bo(-r») lies in the closure of the left halfplane. In fact, for any iJ > the operator Bo(-r) - iJI is obviously uniformly W(-r)-dissipative and hence its spectrum lies in the interior of the left halfplane (see Theorem 1.5.1). An example of an operator satisfying condition (C v) is an operator of the form Bo(-r) = S(-r)H(-r), where H(-r) is a strongly continuously differentiable uniformly positive operator and the operator S(-r) is skew Hermitian: S*(-r) = - S(-r). In fact, in this case it is possible to put W(-r) = H(-r). Then Re(W(-r)Bo(-r») = H(-r)S(-r)H(-r) - H(-r)S(-r)H(-r) =

°

and thus condition (C v) is satisfied. This case includes, in particular, the canonical equations (S = if). Our subsequent analysis will be based on two auxiliary propositions. LEMMA 2.1. Suppose the continuous operator function Bo(-r) with values in [58] satisfies condition (C~) while the operator function B1(-r; A) is uniformly bounded in the domain ilAo = Uio,e o = P = peielp ~ Ao;lol< Oo}:

!!B1(-r; A)II ~ c

(A E Ui"e,; -r E [0, T)].

Then the evolution operator U( -r, -ro; A) of the equation (A

~

Ao; -r

E

[0, T])

(2.1)

is uniformly bounded in Ui"e,/or sufficiently small 00 > 0:

IIU(-r,-ro;A)11 ~

Cl

(AEUi"e,;-rE[O, T]).

(2.2)

If the number AO is sufficiently large, the stronger condition II U(-r, -ro; A) I! ~ Ne-IAlvo(~-")

where N, PROOF.

))0

(A E Ui"e,; -r E [0, T]),

(2.3)

> 0, is satisfied.

The totality of values of the continuous operator function eieBo(-r)

2.

343

ESTIMATE OF THE ERROR

(r E [0, T]; 181 < ( 0) is a precompact set of operators in [~] the spectra of which for sufficiently small 80 lie in the interior of the left halfplane and hence to the left of a straight line Re fJ- = - v (v > 0). It follows from Lemma III.6.2 in this case that there exists a constant N > for which

°

II e XBo «)s II = Ilee Bo«)sIXIII i8

We consider a number (} > ('Z', 'Z'o E [0, T])

~ Ne-IXlvs

(A

E

llIo,oo; 'Z' E [0, T]; s ~ 0).

° having the property that when I'Z' - 'Z'ol

IIBo('Z') - Bo('Z'o) II ~ v/2N. Writing the equation for the evolution operator in the form

(2.4) ~

(}

(2.5)

dU('Z', 'Z'o; A) = ABo('Z'o)U('Z', 'Z'o; A) d'Z'

+ A[Bo('Z')

- Bo('Z'o)]U('Z', 'Z'o; A)

+ B1('Z'; A)U('Z', 'Z'o; A),

we obtain the representation U('Z', 'Z'o; A) = eAB,«o)«-

AM(H) = ~~ '(H)

"m

(Hx, x)

IfXlf2

. f (Hx, x) = ~~\i IfXlf2

P-projection belonging to [lB]: p2 = P R"-n dimensional Euclidean space u(A)-spectrum of an operator A u(A) = u +(A) U u _(A)-decomposition of a spectrum into spectral sets lying respectively in the interiors of the right and left halfplanes u(A) = u.{A) U u .(A)-decomposition of a spectrum into spectral sets lying respectively in the interior and exterior of the unit disk b

Ja -sign for a multiplicative integral with the indicated order of elementary factors .

dx

x=Tt=x

,

377

AUTHOR INDEX Abraham, R., 365 Adamjan, V.M., 143, 365 Agmon, S., 4, 365 Andronov, AA., 332, 365 Anosov, D.V., 195-197, 365 Antosiewicz, H.A, 333, 365

Frolov, P.A., 248, 373 Fuchs, L., 275, 368

Barbasin, E.A., 3, 143, 144, 365 Belkin, M.K., 363, 365 Bellman, R.E., 144, 148, 365 Bernstein, S.N., 63 Birjuk, G.I., 332, 365 Birkhoff, G.D., 334, 363, 365 Bogoljubov, N.N., 3, 332, 333, 363, 365, 372 Bohl, P., 3, 93, 123, 146-148, 196, 197, 329, 332,366 Brodskii, M.L., 68, 148, 366 Brouwer, L.E.J., 146 Bylov, B.F., 61, 146, 331, 366 Cesari, L., 366 Coddington, E.A., 141, 239, 275, 332, 366 Coppel, W.A., 3, 197,248,366 Courant, R., 89, 228, 366 Dahlquist, G.G., 62, 366 Daleckii, Ju. L., 2, 68, 141, 191, 192, 197,275, 332, 333, 361, 362, 363, 365, 366, 367 Demidovic, B.P., 3, 330, 332, 367 Derguzov, V.I., 4, 68, 249, 250, 367 Dieudonne, J., 3, 367 Diliberto, S.P., 248, 367 Domslak, Ju. I., 4, 367 Duffin, R.J., 92, 368 Eidel'man, S.D., 63, 89 Erugin, N.P., 197, 250, 368 Favard, J., 192, 368 Fescenko, S.F., 361, 363, 368 Floquet, G., 201, 218, 239, 240, 249, 368 Fomin, V.N., 4, 249, 368 Frobenius, G., 275, 368

Gantmaher, F.R., 275, 368 Gavrilov, N.I., 148, 368 Gel'fand, I.M., 61, 68, 248, 249,368,369 Germaidze, V.E., 328, 330, 369 Gihman, 1.1., 333, 369 Ginzburg, Ju. P., 64, 67, 68, 139, 369 Gohberg, I.e., 45, 192,246, 352, 353, 363, 369 Grobman, D.M., 61, 146, 331, 366 Hadamard, J., 196, 369 Halilov, Z.I., 4, 369 Halmos, P.R., 240, 370 Hartman, P., 3, 197, 370 Hilbert, D., 89, 228, 366 Hille, E., 274, 275, 370 Hirsch, A, 63 Howe, A., 248, 366 lohvidov, I.S., 64, 68, 369, 370 Ivanov, L.A, 191, 370 Jakubovic, V.A., 2, 142, 146, 197, 242, 245, 248-250, 370, 372 Jonas, P., 68, 370 Kantorovic, L.V., 65, 370 Kapica, P.L., 245, 370 Karaseva, T.M., 244, 370 Kato, T., 197, 371 Korobkova, I.K., 275, 367 Kovtun, 1.1., 68, 250, 371 Krasnosel'skii, M.A., 44, 277, 327, 333, 371 Krasovskii, N.N., 3, 330, 369, 371 Krein, M.G., 1-3, 35, 44, 45, 64, 67, 68, 92, 93, 143, 147, 148, 192, 196, 225, 242-247, 249, 250, 332, 352, 353, 363, 367, 370--372 Krein, S.G., 4, 63, 89, 192, 197, 277, 327, 333, 361, 362, 363, 367, 371, 372 Krylov, N.M., 333, 363, 372 Kucer, D.L., 2, 144, 148, 330, 372

378

AUTHOR INDEX

KUhne, R., 92, 372 Kurzweil, J., 3, 333, 373 Landau, L.D., 245, 373 Langer, H., 68, 92, 372, 373 Levin, B. Ja., 63, 373 Levinson, N., 141, 239, 275, 332, 366 Levitan, B.M., 373 Lidskii, V.B., 68, 242, 248, 249, 369, 373, 375 Lifsic, E.M., 245, 373 Liouville, J., 363 Ljapunov, A.M., 2, 32, 68, 93, 146, 197, 198, 201, 241-243, 247, 249, 279, 285, 287, 332, 373 Ljascenko, N.Ja., 147, 197, 373 Ljubarskii, G.Ja., 225,372 Ljubic, Ju.I., 92, 373 Lozinskii, S.M., 62, 146, 373 Lumer, G., 240, 370 Lykova, O.B., 3, 333, 374 Maizel', A.D., 194, 196, 373 Malkin, I.G., 145, 147, 196, 330, 373 Mandel'stamm, L.I., 333, 374 Marcus, M., 63, 374 Markov, A.A., 35, 374 Markus, L., 197, 374 Massera, J.L., 3, 6, 148, 163, 196, 197, 239, 240, 249, 374 Melik-Adamjan, F.E., 143, 372 Millionscikov, V.M., 148, 374 Mil'man, D.P., 44 Mine, H., 63, 374 Mirocnik, L. Ja., 241, 242, 374 Mitropol'skii, Ju.A., 3, 332, 333, 366, 374 Myskis, A.D., 146, 374 Naimark, M.A., 334, 375 Neigauz, M.G., 242, 375 Neimark, Ju.l., 332, 375 Nemyckii, V., 61, 146, 331, 366 Nikolenko, L.D., 361, 363, 368 Nirenberg, L., 4, 365 Ostrowski, A.M., 68, ·375

379

Papaleksi, N.D., 333, 374 Perov, A.I., 92 Perron, 0., 123, 145, 147, 148, 196, 329, 332, 375 Persidskii, K.P., 147, 148, 375 Phillips, R.S., 67, 68, 375 Pliss, V.A., 375 Poincare, H., 249,331,332 van der Pol, B., 333 Potapov, V.P., 64, 139, 375 Povzner, A.Ja., 328, 375 Pugaeev, V.S., 363, 375 Rabinovic, I.M., 146, 374 Rapoport, I.M., 193, 363, 375 Robbin, J., 365 Rosenblum, M., 68, 375 Rutman, M.A., 2, 35, 147, 332, 372, 376 Scerbakov, B.A., 329, 332, 376 Schaffer, J., 3, 6, 148, 163, 196, 197, 239, 240, 249,374,376 Schneider, H., 68, 375 Sibuya, Y., 192 Silov, G.E., 61, 369 Sinai, Ja.G., 195-197,365 Sircenko, Z.S., 333, 376 Skil', N.I., 361, 363, 368 SmuI' jan, Ju.L., 68, 372 Starzinskii, V.M., 249, 376 Stokalo, I.Z., 250, 376 Svarcman, P.A., 275, 376 Sz.-Nagy, B. von, 34, 68, 376 Tamarkin, J.D., 334, 363, 376 Taussky, 0., 68, 376 Trjitzinsky, W., 363, 376 Vinograd, R.E., 61, 146, 331, 366 Vitt, A.A., 332, 365 Vorel, Z., 333, 373 Wasow, W., 192, 376 Weyl, H., 4, 197, 376 Wintner, A., 62, 146, 148, 197,376

SUBJECT INDEX A-operator, 143 Adjoint, 31 Adjoint equation, 100 Algebraic multiplicity, 59 Almost periodic function, 87 Fourier coefficient of, 92 mean value of, 92 module of, 192 spectrum of, 92 Alternating multiplicative integral, 140 Amplitude, angular, 65 Angle, 64 Angular amplitude, 65 deviation, 65 distance, 156 operator, 67 Approximate decomposition, 337 Asymptotic equivalence operator, 143 phase, 305 Asymptotically comparable equations, 125 equivalent equations, 115 stable solution, 279 Attraction correspondence, 115 Autonomous equation, 281 Averaging principle for a finite interval, 319 for the real line, 327 Balanced operator, 66 Banach space, 10 Banach spaces direct sum of, 13 isometric isomorphism between, 10 isomorphic, 10 isomorphism between, 10 Banach's theorem, 11 Bernstein's theorem, 63 Bistable equation, 113 Bochner integrable function, 95

integral, 95 Bochner's theorem, 87 Bohl exponent lower, 118, 119, 127 upper, 118, 119, 127 strict, 122, 127 at zero, 282 interval, 118, 119 spectrum, 174 Boundary problem regular point of, 224, 227, 362 selfadjoint, 352 spectrum of, 224, 362 Bounded operator, 10 Canonical equation, 142, 218, 225 of positive type, 222, 226 stable, 219, 225 strongly stable, 220, 225 Cauchy operator, 101 sequence, 10 Cayley transform, 45 Central stability band, 221 Characteristic multiplier, 224 of the first kind, 225 of the second kind, 225 Coalescing shift, 265 Compact set, 134 Comparable spectral sets, 273 Complementary projections, 12 Complemented subspace, 12 Complex hull, 14 Condition (C j ), (C 2), (C 3), 323 (C'll), (Cj;), 342 S"L, 136 Cone, 56 Cone inequality theorem, 57 Conjugation equation, 152 Continuously differentiable function, 54 Contraction, 52

380

SUBJECT INDEX

uniform W-,38 with coefficient q, 66 Contraction principle, 52 Convergence property, 330 Countably valued function, 94 Decomposable operator, normally W-, 48 Decomposition of an equation, approximate, 337 of the identity, 13 of a spectrum, normal, 265 stably normal, 265 Definite subspace, W-,47 Deviation, angular, 65 Dichotomic equation, e-, 73, 162, 195 operator, e-, 20 u-,41 Dichotomy, regular exponential, 163 Differentiable function, 54, 96 continuously, 54 Direct complement, 12 sum of Banach spaces, 13 of Hilbert spaces, 14 of subspaces, 12 Disjoint projections, 13 subspaces, 155, 156 Dissipative operator, 33, 34 uniformly, 34 uniformly, W-, 34 Domain, stability, 248 e-dichotomic, equation, 73, 162, 163, 195 operator, 20 Eigenvalue algebraic multiplicity of, 59 index of, 59 normal, 59 mot manifold of, 59 root space of, 59 Element negative, W-, 37 neutral, W-, 37 nonnegative, W-, 37 nonpositive, W-, 37 positive, W-, 37 Elementary shift, 265 Entire function of exponential type, 63 of minimal type, 255

Equation adjoint, 100 autonomous, 281 bistable, 113 canonical, 142,218,226 conjugation, 152 e-dichotomic, 73, 162, 163, 195 Hamiltonian, 142,218 of positive type, 222, 226 quasilinear, 280 reducible, 159 with a regular singularity, 252 resolvent, 15 Riccati operator, 194 with slowly varying coefficients, 335 stable left, 113 right, 112, 219, 225 strongly, 220, 225 uniformly left, 113 uniformly right, 112 unlimitedly, 146 variational, 281 Equation operator, 67, 208 minus, 208 plus, 208 Equations asymptotically comparable, 125 asymptotically equivalent, 115 integrally comparable, 114 kinematically similar, 157 L-diagonal systems of, 193 Evolution operator, 100 Expansion, uniform W-, 206 Exponential dichotomy, regular, 163 operator function, 25 splitting, 174 Exponentially dichotomic operator, 20 Favard's theorem, 192 Floquet representation, 201 of second order, 239 Force gyroscopic, 90 Rayleigh, 90 resistance, 90 Fourier coefficient, 92 Fn!chet derivative, 54 of second and higher orders, 54, 55 Function almost periodic, 87 Bochner integrable, 95 of class K A ,17

KA.B,22

381

382 L~,

SUBJECT INDEX

307

(M,q), (M,q)+, (M,q,p), (M,q,p)+, 297

countably valued, 94 differentiable, 54, 96 entire of exponential type, 63 of minimal type, 255 Green, 80, 167 incomplete, 87, 176 periodic, T-, 85 piecewise analytic, 17 principal, 81, 168 strongly integrable, 95 strongly measurable, 95 Fundamental operator, 101 solution, 252

-

Gaps, 174 Generalized Ljapunov theorem, 32 Green function, 80, 167 incomplete, 87, 176 periodic, T-, 85 principal, 81, 168 Gyroscopic force, 90 Hamiltonian, 218 equation, 142 Hamilton's equation, 218 Hausdorff's theorem, 63 Hermitian, 218 of positive type, 222 Hermitian operator, 31 indefinite, 37 skew-,36 spectral length of, 215 W-,45 Hilbert spaces, orthogonal sum of, 14 Homoclinic motion, 332 Hyperboloids minus, 74 plus, 74 Identification operator, 66 Imaginary part, 32 Incomplete Green function, 87, 176 Indefinite Hermitian operator, 37 scalar product, 37 Index, 59 Integral Bochner, 95 continuity, 320 manifold, 307 multiplicative, 102

alternating, 140 Stieltjes, 103, 140 Riesz,20 shift, 265 Integrally bounded operator function, 116 comparable equations, 114 . Intertwining property, 152 Invariant subspace, 20 Inverse, 11 Inverse function theorem, 55 Isometric isomorphism, 10 Isomorphic Banach spaces, 10 Isomorphism, 10 Kinematically similar equations, 157 Kreln-Krasnosel'skil-Mil'man theorem, 44, 45 L-diagonal systems, 193 Law of inertia, 66 Left 9'-set, 170 stable equation, 113 uniformly, 113 Linear operator, 10 Ljapunov exponent lower, 118, 127 strict, 26 strict upper, 122, 127 upper, 26, 90,117,127 spectrum, upper, 117 theorem, generalized, 32 Logarithmic norm, 62 Lower Bohl exponent, 118, 119, 127 Ljapunovexponent, 118, 127 Manifold integral, 307 (p,7j )-, 307 (p,7j,+)-,(p, 7j,-)-, 315

root, 59 . Markov's theorem, 35 Maximal "-nonnegative subspace, 67 norm, 13 Mean value, 92 Minus equation operator, 208 hyperboloids, 74 Module, 192 Monodromyoperator, 199,207 Multiplicative integral, 102 alternating, 140

SUBJECT INDEX

Stieltjes, 103, 140 Multiplier, characteristic, 224 of the first kind, 225 of the second kind, 225 Negative element, W-, 37 operator, 32 uniformly, 32 subspace, uniformly W-, 38 W-,38 Neutral element, W-, 37 Nonenlargement operator, W-, 64 Nonnegative element, W-, 37 operator, 31 W-,63 subspace maximal fl' -, 67 W-,37 Nonpositive element, W-, 37 operator, 32 subspace, W-, 38 Norm of an element, 10 maximal,13 Norm of an operator, 10 logarithmic, 62 Normal decomposition, 265 stably, 265 eigenvalue, 59 operator, 58 set, 265 Normally W-decomposable operator, 48 Normed space, 10 Norms, topologically equivalent, 10 w-limit operator, 135 Operator, A-, 143 adjoint of, 31 of algebraic type, 59 angular, 67 amplitude of, 65 deviation of, 65-asymptotic equivalence, 143 balanced, 66 bounded, 10 Cauchy, 101 contraction, 52 uniform W-, 38 with coefficient q, 66 decomposable, normally W-, 48 dichotomic e-,20

u-,41 equation, 67, 208 minus, 208 plus, 208 evolution, 100 expansion, uniform W-, 206 exponentially dichotomic, 20 fundamental,101 Hermitian, 31 indefinite, 37 skew-,36 W-,45 identification, 66 imaginary part of, 32 inverse of, 11 linear, 10 logarithmic norm of, 62 monodromy, 199,207 negative, 32 uniformly, 32 nonenlargement, W-, 64 nonnegative, 31 W-,63 non positive, 32 norm of, 10 logarithmic, 62 normal,58 w-limit, 135 positive, 31 uniformly, 32 quasinilpotent, 63 real part of, 32 resolvent of, 15 right spectral bound of, 139 signature, 248 of simplest type, 57 solving, 100 spectral radius of, 199 spectrum of, 15 stable, 34 unitary, 34 strongly stable W-, 49 W-,45 Volterra, 63 Operator function exponential, 25 integrally bounded, 116 of positive type, 226 precompactly valued, 135 stationary at infinity, 136 strongly continuous, 96 Operator halfplane, right, 67 Operators, product of, 11 Orbital asymptotic stability, 305

383

384

SUBJECT INDEX

Orthogonal elements, W-,37 projection, 14 sets, W-,37 sum, 14 vectors, W-, 37 Phase space, 69 Piecewise analytic function, 17 Plus equation operator, 208 hyperboloids, 74 Point of regular type, 42 Positive element, W-,37 operator, 31 uniformly, 32 subspace, W- 37 uniformly W-, 37 Potential, 142 Precompact set, 135 Precompactiy valued operator function 135 Principle ' averaging, for a finite interval, 319 contraction, 52 for the real line, 327 uniform boundedness, 11 Product of operators, 11 Projection, 12 orthogonal, 14 spatial, 307 spectral, 19 uniformly conjugated, 157 Projections mutually complementary, 12 pairwise disjoint; 13 Property ~(J), N), 122 ~(J), N, p), 282 convergence, 330 intertwining, 152 DiN, if», 329 Quasilinear equation, 280 Quasinilpotent op;rator, 63 Rayleigh force, 90 Real part, 32 Reducible equation, 159 Regular exponential dichotomy, 163 point of a boundary problem, 224, 227, 362 of an operator, 15 Resistance force, 90 Resolvent, 15

Resolvent equation, 15 (p, 7])-manifold, 307 (p,7], + )-manifold, 315 (p, 7], - )-manifold, 315 Riccati operator equation 194 ' Riesz integral, 20 Right operator halfplane, 67 9'-set, 169 spectral bound, 139 stable equation, 112,219,225 uniformly, 112 Root manifold, 59 space, 59 9'-set left, 170 right, 169 Scalar product, indefinite, 37 products, topologically equivalent 32 Selfadj~int boundary problem, 352 ' SeparatIOn, a-, 305 Set compact, 134 normal,265 precompact, 135 9'-,169 Sets, orthogonal, W-, 37 Shift coalescing, 265 elementary, 265 integral, 265 Signature operator, 248 Skew-Hermitian operator, 36 Slow time, 326, 335 Solution asymptotic phase of, 305 fundamental, 252 stable, 279 asymptotically, 279 uniformly, 279 Solving operator, 100 Space Banach,10 complex hull of, 14 normed,1O phase, 69 root, 59 Spaces B~nach, direct sum of, 13 Hilbert, orthogonal sum of, 14 isometric isomorphism between 10 isomorphic, 10 '

SUBJECT INDEX

isomorphism between, 10 Spatial projection, 307 Spectral bound, right, 139 length, 215 mapping theorem, 19 projection, 20 radius, 199 set, 19 sets, comparable, 273 Spectrum a-separation of, 305 Bohl,174 A, 90

Ljapunov, upper, 117 normal decomposition of, 265 stably, 265 of a boundary problem, 224, 362 of an almost periodic function, 92 of an operator, 15 Stability band, central, 221 domain, 248 point, 221, 226 strong, 221, 226 stability, orbital asymptotic, 305 Z-tests for, 249 Stable canonical equation, 219, 225 strongly, 220, 225 equation left, 113 uniformly, 113 right, 112 uniformly, 112 unlimitedly, 146 operator, 34 solution. 279 asymptotically, 279 uniformly, 279 W-unitary operator, strongly, 49 Stably normal decomposition, 265 Stationary at infinity operator function, 136 Stieltjes multiplicative integral, 103, 140 Stone's theorem, 37 Strict Ljapunov exponent, 26 Strict upper Bohl exponent, 122 Ljapunovexponent,122 Strongly continuous operator function, 96 integrable function, 95 measurable function, 95 stable canonical equation, 220, 225 stable W-unitary operator, 49

385

Subspace, 12 complemented, 12 definite W-,47 direct complement of, 12 invariant, 20 negative, uniformly W-, 38 W-,38 nonnegative, maximal f, 67 W-,37 nonpositive, W-, 38 positive, uniformly W-, 37 W-,37 Subspaces direct sum of, 12 disjoint, angular distance between, 156 Sz.-Nagy's theorem, 34 Theorem Banach, 11 Bernstein, 63 Bochner, 87 on a boundary point of the spectrum, 28 cone inequality, 57 Favard,192 generalized Ljapunov, 32 Hausdorff, 63 inverse function, 55 Krein-Krasnosel'skii-Mil'man, 44, 45 Markov, 35 spectral mapping, 19 Stone, 37 Sz.-Nagy, 34 Topologically equivalent norms, 10 scalar product, 32 Transform, Cayley, 45 Transformer, 99 u-dichotomic operator, 41 Uniform boundedness principle, 11 W-contraction, 38 with coefficient q, 66 W-expansion, 206 Uniformly conjugated projection, 157 dissipative operator, 34 left stable equation, 113 negative operator, 32 positive operator, 32 right stable equation, 112 stable solution, 279 W-dissipative operator, 34 W-negative subspace, 38 W-positive subspace, 37

386 Unitary operator, 34 strongly stable W-, 49 W-,45 Unlimitedly stable equation, 146 Upper Bohl exponent, 118, 119, 127 strict, 122, 127 at zero, 282 Ljapunovexponent, 26, 90,117,127 strict, 122, 127

SUBJECT INDEX

Ljapunov spectrum, 117 Variational equation, 281 Vectors angle between, 64 orthogonal, W-, 37 Volterra operator, 63 Z-tests, 249