0387986995Basic TheoryC

x

CONTENTS

Chapter VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order

VI- 1. Zeros of solutions

VI- 2. Sturm-Liouville problems VI- 3. Eigenvalue problems

VI- 4. Eigenfunction expansions

VI- 5. Jost solutions

VI- 6. Scattering data

VI- 7. Reflectionless potentials

VI- 8. Construction of a potential for given data

VI- 9. Differential equations satisfied by reflectionless potentials VI-10. Periodic potentials

Exercises VI

Chapter VII. Asymptotic Behavior of Solutions of Linear Systems

VII-1. Liapounoff's type numbers

VII-2. Liapounoff's type numbers of a homogeneous linear system VII-3. Calculation of Liapounoff's type numbers of solutions VII-4. A diagonalization theorem

VII-5. Systems with asymptotically constant coefficients

VII-6. An application of the Floquet theorem

Exercises VII

Chapter VIII. Stability

VIII- 1. Basic definitions

VIII- 2. A sufficient condition for asymptotic stability

VIII- 3. Stable manifolds

VIII- 4. Analytic structure of stable manifolds

VIII- 5. Two-dimensional linear systems with constant coefficients

VIII- 6. Analytic systems in R2

VIII- 7. Perturbations of an improper node and a saddle point VIII- 8. Perturbations of a proper node

VIII- 9. Perturbation of a spiral point VIII-10. Perturbation of a center

Exercises VIII

Chapter IX. Autonomous Systems

IX-1. Limit-invariant sets

IX-2. Liapounoff's direct method

IX-3. Orbital stability

IX-4. The Poincar6-Bendixson theorem

IX-5. Indices of Jordan curves

Exercises IX

CONTENTS	xi
Chapter X. The Second-Order Differential Equation
-t2 + h(x) dt + g(x) = 0	304
d
X-1. Two-point boundary-value problems	305
X-2. Applications of the Liapounoff functions	309
X-3. Existence and uniqueness of periodic orbits	313
X-4. Multipliers of the periodic orbit of the van der Pol equation	318
X-5. The van der Pol equation for a small e > 0	319
X-6. The van der Pol equation for a large parameter	322
X-7. A theorem due to M. Nagumo	327
X-8. A singular perturbation problem	330
Exercises X	334
Chapter XI. Asymptotic Expansions	342
XI-1. Asymptotic expansions in the sense of Poincare	342
XI-2. Gevrey asymptotics	353
XI-3. Flat functions in the Gevrey asymptotics	357
XI-4. Basic properties of Gevrey asymptotic expansions	360
XI-5. Proof of Lemma XI-2-6	363
Exercises XI	365
Chapter XII. Asymptotic Expansions in a Parameter	372
XII-1. An existence theorem	372
XII-2. Basic estimates	374
XII-3. Proof of Theorem XII-1-2	378
XII-4. A block-diagonalization theorem	380
XII-5. Gevrey asymptotic solutions in a parameter	385
XII-6. Analytic simplification in a parameter	390
Exercises XII	395
Chapter XIII. Singularities of the Second Kind	403
XIII-1. An existence theorem	403
XIII-2. Basic estimates	406
XIII-3. Proof of Theorem XIII-1-2	417
XIII-4. A block-diagonalization theorem	420
XIII-5. Cyclic vectors (A lemma of P. Deligne)	424
XIII-6. The Hukuhara-T rrittin theorem	428
XIII-7. An n-th-order linear differential equation at a singular point
of the second kind	436
XIII-8. Gevrey property of asymptotic solutions at an irregular
singular point	441
Exercises XIII	443
References	453

Index

462

CHAPTER I

FUNDAMENTAL THEOREMS OF

ORDINARY DIFFERENTIAL EQUATIONS

In this chapter, we explain the fundamental problems of the existence and unique- ness of the initial-value problem

dy	=	y(to) _ 4o
dt
(P)

in the case when the entries of f (t, y) are real-valued and continuous in the variable

(t, y), where t is a real independent variable and y is an unknown quantity in Rn. Here, R is the real line and R" is the set of all n-column vectors with real entries.

In §I-1, we treat the problem when f (t, y) satisfies the Lipschitz condition in W. The main tools are successive approximations and Gronwall's inequality (Lemma

In §1-2, we treat the problem without the Lipschitz condition. In this case, approximating f (t, y) by smooth functions, f-approximate solutions are constructed. In order to find a convergent sequence of approximate solutions, we use Arzeli -

Ascoli's lemma concerning a bounded and equicontinuous set of functions (Lemma

1-2-3). The existence Theorem 1-2-5 is due to A. L. Cauchy and G. Peano [Peal] and the existence and uniqueness Theorem 1-1-4 is due to E. Picard [Pi1 and E.

Lindelof [Lindl, Lind2J. The extension of these local solutions to a larger interval is explained in §1-3, assuming some basic requirement.,, for such an extension. In §I-4, using successive approximations, we explain the power series expansion of a solution in the case when f (t, y) is analytic in (t, y). lit each section, examples and remarks are given for the benefit of the reader. In particular, remarks concerning other methods of proving these fundamental theorems are given at the end of §1-2.

I-1. Existence and uniqueness with the Lipschitz condition

We shall use the following notations throughout this book:

Y1,	f,
y2	f2
y=	!y = max{jy,j 1Y21-.. ItYn11.
yn	fn

In §§I-1, 1-2, and 1-3 we consider problem (P) under the following assumption.

Assumption 1. The entries of f (t, y) are real-valued and continuous on a rectan- gular region:

R = R((to,6),a,b) = {(t,yl: It - tot <a, ly - e< b},

where a and b are two positive numbers.

I

it - tol <

2	I. FUNDAMENTAL THEOREMS OF ODES

Since f (t, yam) is continuous on R, I f (t, yul is bounded. Let us denote by M the maximum value of I f (t, y-) I on R, i.e., M = mac I f (t, y) I. We define a positive

number a by	if M = 0,
a
a	if M > 0.
min Ca, Ml

To locate a solution in a neighborhood of its initial point, the number or plays an important role as shown in the following lemma.

Lemma I-1-1. If ff = ¢(t) is a solution of problem (P) in an interval

a < a, then I'(t) - 41 < b in It - tol < a, i.e., (t,5(t)) E R((to,co),a,b) for. It - tol <a.

Proof

Assume that Lemma 1-1-1 is not true. Then, by the continuity of fi(t), there exists a positive number $ such that

(i)	< a,
(ii)
	1¢(t) - 41 < b	for	It - tol <'6,
	I4(to+0) -QoI =b	or	10(to-fl)-qol

The assumption a < a < a implies (3 < a. Hence, (t,j(t)) E R for It - tot < p.

Therefore, I f (t, (t))I 5 M for it - to! < 0. From 4'(t) = f (t, ¢(t)) and 4(to) = 4o, it follows that

(1.1.2)		fi(t) = co +	/ t f(s, 0(s))ds	for It -tot < fi.
			io
Hence,
	I4(t) - 4)1 = I1		f(s, i(s))ds < MIt - tol		for	It - tot < P.
		r
This implies that
(1.1.4)	1	(t)-ZbI<-MQ<Ma<Ma<MM=b			for	It - tol<f.
In particular,		0) - cot < b. This contradicts assumption (ii).

Similarly, we obtain the following result.

Lemma 1-1-2. If y' = 4(t) is a solution of problem (P) in an interval it - tol < a < a, then I¢(t,) - (t2)I < Mlt1 - t2l whenever t1 and t2 are in the interval it - tot < a.

{1 - exp(-L(t - to)]).

1. EXISTENCE AND UNIQUENESS WITH THE LIPSCHITZ CONDITION 3

Remark 1-1-3. Using Lemma 1-1-2, it is concluded that if y = j(t) is a solution of problem (P) on an interval It - to I < it < a, then (to + & - 0) and (to - ar + 0) exist.

Now, consider problem (P) with Assumption 1 and the following assumption.

Assumption 2. The function f satisfies a Lipschitz condition on R: i.e., them exists a positive constant L /such that

If(t, 91) - f (t, W2)1:5 LIlh - 1121

whenever (t,yi) and (t, y2) are on the region R (cf. [Lip]).

Note that L is dependent on the region R. The constant L is called the Lipschitz constant of f with respect to R.

The following theorem is the main conclusion of this section.

Theorem 1-1-4. Under Assumptions I and 2, them exists a unique solution of problem (P) on the interval It - to] < a.

In order to prove this theorem, the following lemma is useful.

Lemma 1-1-5 (T. H. Gronwall [Cr]). If

(i)g(t) is continuous on to < t < ti,

(ii)g(t) satisfies the inequality

(1.1.5)	0 < g(t) < K + L g(s)ds on to < t < ti,
	to
then
(I.1.6)	0 < g(t) < Kexp[L(t - to))

onto<t< ti.

Proof.

Set v(t) = Jio g(s)ds. Then, by (1.1.5), ddtt) < K+Lv(t) and v(to) = 0. Hence,

(1.1.7)

{exp[-L(t - to)]v(t)} < Kexp[-L(t - to)]

and, consequently,

exp[-L(t - to)]v(t) <

This, in turn, implies that Lv(t) < K{exp[L(t - to)] - 1). Therefore, g(t) <

K + Lv(t) < K exp[L(t - to)]. 0

Proof of Theorem 1-1-4.

We shall prove this theorem in six steps by using successive approximations which are defined as follows:

Wi(t) _ 4o,

	(1.1.9)		k=0,1,2,...
	t $k+1(t) = Cq +	:o f(s,Ok(s))ds,

4	I. FUNDAMENTAL THEOREMS OF ODES

Step 1. Each function 0k is well defined and (t,45k(t)) E R for It - tol < a

(k = 0,1,2,...).

Proof.

We use mathematical induction in Steps 1 and 2. The statement above is evident for k = 0. Assume that (t, Ok(t)) E R on It - tol < a. Then, I f (t,Ok(t))I < M on it - toI < a. Hence,

t	f (s, Qsk(s))dsl < MIt - tol < Ma < MM = b.
IOk+I(t) -Q =
4	1	-

Thus, (t.0k+1(t)) E R for It - tol < a.

Step 2. The successive approximations given by (1.1.9) satisfy the estimates

MLk	for It - tol < a, k =0,1,2,... .
Ilk+1(t) - mk(t)I < (k 1)1It - toIk+1

Proof.

For k = 0, we have 1m1(t) - y5o(t)I = I J r f(s,co)ds` < MIt - tol. Assume that

I&(t)(t)I <	ML	co	I
	k1-1 It - tolk for it - tol < a. Then,
14k+I (t) - Qk(t)l = Jo		`f (s, k(s)) - f (s	k-1(s))} ds
	< L I J r
		to
	AkL	t	k
		I r It - toIkdsl = (ML1)r It - tolk+1.

+oo

Step 3. The series E (&(t) - 4k-1(t)) is uniformly convergent on the interval

k=1

It - tol<a.

Proof.

By virtue of the result of Step 2, for a given c > 0, there exists a positive integer

N such that

°°	M °o		(LIt - tol)k	M °O (La)k			<
/	L			< L		k!
E I&(t) - 0k-I(t)I <		k=N	k1		k=N
k=N

1. EXISTENCE AND UNIQUENESS WITH THE LIPSCHITZ CONDITION 5

Step 4. The sequence {k(t) : k = 0, 1, 2.... } converges to

do(t) + F, I(&(t) - k-1(t))

k=1

uniformly on it - tol < a as k -+ +oo.

Proof.

Use the identity ¢N (t) = do (t) + E ($k (t) - $k_1(t)) and the result of Step 3.

k=1

Step 5. fi(t) given by (I.1.10) is a solution of problem (P) on It - toI < a.

Proof.

The definition

&+1(t) = e + rt AS, ('1k(s))ds,

Jca

the continuity of f and the results of Steps 3 and 4 imply that

fi(t) = 6o + f f(s, (s))ds. a

Step 6. fi(t) is the unique solution of problem (P) on It - tol < a.

Proof.

Suppose that y"= fi(t) is another solution of problem (P) on an interval it -toI <

& < a for some &. Lemma I-1-1 and Remark 1-1-3 imply that (t, ti(t)) E R on It - tol :5&. Note that

fi(t)

+

f (s, i1(s))ds

on It - tol < &. Hence,

At) - lL(t)I = f' {f(s,.(s)) - f(s,iG(S))} dsl <- L I fo Id(s) - i (s)Idsl

on it - tol < a. Now, by applying Lemma 1-1-5 with K = 0 to the inequality

L f Id(s) - +L(s)Ids

o

on to < t < to+&, we conclude that I¢(t) -tG(t)I = 0 on the interval to < t < to+&. Similarly, the same conclusion is obtained on the interval to - & < t < to. Steps 1 through 6 complete the proof of Theorem 1-1-4. 0

The following lemma gives a sufficient condition that f (t, y") satisfies Assumption

2 (i.e., the Lipschitz condition).

1=1

... , Yh.n are entries of yh (h = 1,2). Hence,

n

44do _ 1: (yia - y2 j)'

Proof.

Fix t E 1, and yl and #2 in V. Then, By, + (1 - O)y"2 E D for O < 6 < 1 since is convex. Setting g(9) = f (t, 09, + (1 - O)y2), we derive

I f(t, y,) - f(t, y2)1 < nLollit

interval, D is a region, and Lo is a constant, and if V is a convex open set, then for t E z, gi E D, and y2 E D.

9(O) = At, 92),

- y"2I

D

for (t, y) E I x D, where I is an

1

6 I. FUNDAMENTAL THEOREMS OF ODES

Lemma I-1-6. If Of (t' y-) (j = 1, 2, ... , n) exist, are continuous, and satisfy the

8yj

condition: I of (t, y j < Lo (j = 1, 2, .. - , n)

9(1) = f(t,91)-

(t, 8y"1 + (1 - B)y'2),

where yh,1, Yh,2,

If(t,91)-f(t,y2)1 =II @(O)dO <nLolii-u21-

This is true for all t E Z, yl E D, and y2 E D. Thus, the lemma is proved.

The following two simple problems illustrate Theorem 1-1-4.

Problem 1. Using Theorem 1-1-4, show that on the interval It - 11 < V"2-, the initial-value problem

dy -	1	y(1) - 5
dt	(3 - (t - 1)2)(9 - (y -

has one and only one solution.

Answer.

The standard strategy is to find a suitable region

R = {(t,y): It-11 < a, ly-51 < b}

and find the maximum value M of if (t, y) I in R. Then, a = min (a, M)- If

a > f, the problem is solved. In other words, the main idea is to trap the solution curve in a suitable region R. To do this, for example, use the fact that the function

f ,y (3 - (t - 1)2)(9 - (y - 5)2) is continuous and satisfies the Lipschitz condition

	If(t,Yi) - f(t,y2)1 <	Iy' -Y21
		5

1. EXISTENCE AND UNIQUENESS WITH THE LIPSCHITZ CONDITION 7

on the region R= {(t, y) : It-11 < f, Iy-51 < 2} (cf. Exercise I-2). Furthermore,

If(t,y)I <-	1	= 1
	(3_(/)2)(9_22)	5
Set a = V25, b = 2, and M = 1 . Then,
	5
a = min ( a,	) =min f, 2	min(v ,10) = f.

Therefore, the given initial-value problem has one and only one solution on the interval It - 11 < f.

Problem 2. Using Theorem I-1-4, show that the initial-value problem

(P)	dy =	1	y()4 = 3
	dt	(1 + (t - 4)2)(5 + (y - 3)2)'

has one and only one solution on the interval -oo < t < +oc.

Answer.

The function

1

,y

(1 + (t - 4)2)(5 + (y - 3)2)

is continuous and satisfies the Lipschitz condition everywhere in the (t, y)-plane.

Also, If (t, y)I < everywhere. This implies that a = min(a, 5b), if Problem (P) is considered in a region

R = {(t, y) : It - 41 < a, ly - 31 < b}.

Hence, if a < 5b, Problem (P) has one and only one solution on the interval it - 41 < a. Now, this solution is independent of a due to the uniqueness. Therefore, we can conclude that (P) has one and only one solution on the interval -oo < t < +oo.

Remark 1-1-7. It is usually claimed that the general solution of the system

dt = f (t, y-) depends on n independent arbitrary constants. Theorem 1-1-4 verifies

this claim if we define the general solution very carefully, since the initial data (to, Cl represent n arbitrary constants for a fixed to. However, if the reader looks into this definition a little deeper, he will find various complicated situations. It is important for the reader to know that the number of independent arbitrary constants contained in the general solution depends strongly on the space of functions to which the general solution should belong. For example, if the differential equation

(E)

tae = 1