0387986995Basic TheoryC

for (x, e) E A(S2) x Se (r2, a2), K and c are suitable positive numbers, t(x, e) is a fundamental matrix solution of the homogeneous system

and I(e) is a C"-valued function of a independent of x.

Assuming that Ao(0) has distinct eigenvalues A1,.12, ... , Al with multiplicities nl, n2i ... , ne, respectively (n1 + n2 + + ne = n) and that AO(O) is in a blockdiagonal form (XII.4.6) with (XII.4.7), where N, (j = 1, 2,... , l) are n, x nj nilpotent matrices, respectively, apply Theorem XII-4-1 to block-diagonalize (XII.5.27). More precisely speaking, there exist three positive numbers 53, r3, 03, and an n x n matrix P(x, e) which is holomorphic in (x, e) E A(53) x So (r3, a3) such that

(i)P(x, e) admits an asymptotic expansion (XII.4.9) in the sense of Poincart as e -+ 0 in S(r3,a3) uniformly in x E A(53),

(ii)the transformation

with an n1 x n, matrix E J (x, c) for each j = 1, 2, ... , t, and E(x, e) admits

an asymptotic expansion in the sense of Poincar6 as t -+ 0 in

SB(rs,a3) uniformly in x E A(63), where Eo(0) = Ao(0).

Let us construct a rhombus 7)(64) with vertices x{'), x(2), xi3i, and xt4i as it is defined in §XII-2, in such a way that

(a)7)(54) C a(53),

(b)it holds that

on the domain V(54) x So(r4,Q4), where 64i r4, and a4 are suitable positive numbers.

El

ztJ

can be written in the form

In this way, we can prove that t 5(x, e) - w'(x, e) is flat of Gevrey order 1 as e - O in

S e, (r1, a i) n Sez (ri , a 1) uniformly for x E A(S) if a positive number 6 is sufficiently small (cf. [Si6; Proof of Lemma 1 on pp. 377-379]). Therefore, by using Theorem X1-2-3, we can complete the proof of Theorem XII-5-2.

Materials of this section are also found in JSi22J.

XII-6. Analytic simplification in a parameter

For a system of linear differential equations of the form

where k is an integer, y E C', and A(t) is an n x n matrix whose entries are holomorphic in a domain

G. D. Birkhoff proved the following theorem.

Theorem XII-6-1 ((Bil). There exists an n x n matrix P(t) such that the entries of P(t) and P(t)-' are holomorphic in domain (D) and that the transformation

(T) y = PMg

reduces system (E) to a system of the form

where B(t) is an n x n matrix whose entries are polynomials in t. Moreover, the matrix P(t) can be chosen so that x = 0 is, at worst, a regular singular point of (E).

Observe that if we set t = x-3, (E) becomes

where Bk(x) (k = 1, 2, ... , or - 1) are n x n matrices whose entries are holomorphic for x E D1.

Remark XII-6-3. In case when a = 1, (XII.6.5) becomes

k=0

In particular, if a = I and m = 0, (XII.6.5) has the form

e dil = {Ao(x) + eB0(x)} u.

It should be noticed that in Theorem XII-6-2, if we put m = 0, then the right-hand side of (XII.6.5) is a polynomial in a of degree at most a without any restrictions on A(x, e). We prove Theorem XII-6-2 by a direct method based on the theory of ordinary differential equations in a Banach space. (See also [Si8j.)

Proof of Theorem XII-6-2.

The proof is given in three steps.

Step 1. Put

From (XII.6.1), (XII.6.5), and (XII.6.6), it follows that the matrices P(x,e) and

B(x, e) must satisfy the equation

From (XII.6.3), (XII.6.7), (XII.6.8), and (XII.6.9), we obtain

k

Am+1+k(x) - Bm+1+k(x)+ E{Ak-h(x)Ph(x) - Ph(x)Bk-h(x)}

(XII.6.10)

h=O

(k=0,1,...,a-1)

Solve this differential equation in a suitable Banach space with the initial condition

P(0) = 0. Actually, we solve the integral equation

If P is determined, then the matrices Bk are determined by (XII.6.13) and (XI1.6.14).

Step 2. We still assume that A(x, c) is holomorphic in domain (XII.6.2). Denote by B the set of all infinite-dimensional vectors P = {Pk : k = 0, 1, 2.... } such that

(i) Pk are n x n matrices of complex entries,

00

(ii) j:boilPkll < oo, where IlPkll is the sum of the absolute values of entries

k=0

of Pk.

For each P, define a norm 11P11 by

Then, we can regard B as a Banach space over the field of complex numbers. Now, we can establish the following lemma.

Lemma XII-6-4. Let F(x; P) be the infinite-dimensional vector whose entries fk(x,P) are given by (XII.6.16). Then, for each positive numbers R, there exist two positive numbers G(R) and K(R) such that

In order to prove this lemma, consider a formal power series of e which is defined

by

Hence, Lemma XII-6-4 follows immediately.

Step 3. We construct the matrix P(x, e), solving the integral equation (XII.6.18) by the method of successive approximations similar to that given in Chapter I. By virtue of Lemma XII-6-4, we can construct a solution P(x) in a subdomain Dl of Do containing x = 0 in its interior. Since (XII.6.18) is equivalent to differential equation (XII.6.15) with the initial condition P(0) = 0, the solution P(x) gives the desired P(x, e). The matrix B(x, c) is given by (XII.6.13) and (XII.6.14). 0

EXERCISES XII

00

XII-1. Find a formal power series solution y = F, e'd,,,(x) of the system of

m=0

differential equations

CO

LY = Ay" + > Embm(x),

m=0

where y E C", A is an invertible constant n x n matrix, and 5,n (x) and bm(x) are

C"-valued functions whose entries are holomorphic in x in a neighborhood of x = 0.

XII-2. Using Theorem MI-4-1, diagonalize the system

XII-3. Using Theorem XII-4.1, find two linearly independent formal solutions of each of the following two differential equations which do not involve any fractional powers of e.

where q(x) is holomorphic in x for small jxj.

Hint. If we set '62 = e, differential equation (2) has two linearly independent solutions e4=/00(x, 0) and a-1/190(x, -/3). The two solutions

do not involve any fractional powers of e.

XII-4. Let x be a complex independent variable, y" E C", z" E C, e be a complex parameter, A(x, y, z, e) be an n x n matrix whose entries are holomorphic with respect to (x, y, z, e) in a domain Do = {(x, y, X, e) : jxj < ro, jyi < Pi, Izl < p2, 0 < jej < ao, j arg ej < flo}, f (x, y, z, e) be a C-valued function whose entries are holomorphic with respect to (x, y, 1, e) in Do, and #(x, ,F, e) be a C"-valued function whose entries are holomorphic with respect to (x, z, e) in the domain Uo = ((x, zl, e) :

jxj < ro, jzj < p2,0 < jej < ao, j argej < /3o}. Assume that the entries of the matrix

+00

with coefficients p"l(x), which are holomorphic in A(50),

c=1

(2)for any real number 0, there exist three positive numbers b, r, and a such that

(S)has an actual solution ¢(x, e), which is holomorphic in (x, E) E 0(b) x

S(r, a, 0), and that ¢(x, c) has the formal solution p(x, e) as its asymptotic

expansion of Gevrey order 1 as c -. 0 in S(r, a, 0) uniformly in x E 0(b)

or

Hint. This is a special case of Theorem XII-5-2.

+M

XII-9. Assume that p(x,E) _ E Emp',,,(x) is a formal solution of a system (S)

M=1

dy

ca dx = f (x, 17, f), where a is a positive integer, y" E C, Ax, y", e) is a C"-valued

function whose entries are holomorphic in a neighborhood of (x, y, f) = (0, 0, 0), and the entries of pm(x) (m = 0.1, ...) are holomorphic in a disk Ixi < ro, where ro is a positive ember. Assume also that p(x, E) E {G[[E]]3}" uniformly for Ixi < ro

and that 0 < s < 1. Show that if S is an open sector in the E-plane with vertex at e = 0 and whose opening is smaller than sir, there exists two positive numbers ri and r2 and a solution $(x, e) of (S) such that the entries of q,(x, e) are holomorphic in (x, E) for Ixt < r1, IEI < r2,( E S, and that (x, E) admits the formal solution p(x, c) as the asymptotic expansion of Gevrey order s as e -+ 0 in S n {I-El < r2} uniformly for IxI < r1.

Note. No additional conditions on the linear part of Ax, y", c) are assumed.

+00

XJI-10. Assume that p(x, c) = F E"`p""m(x) is a formal solution of a system

m--O

e = f (z, y, E), where y E C", f (x, y, e) is a C"-valued function whose entries are

holomorphic in a neighborhood of (x, y, c) = (0, 6, 0), p".. E C[(x]]" (m = 0,1,... ), p"o(0) = 6, and p(O, e) is convergent. Show that p(x, e) is convegent for ]x) < r and IEI < p for some positive numbers r and p.

Hint. Regard p(ET, e) as the solution of the initial-value problem dy = f (.ET, y, E),

00) = P(0. E.

XII-11. Let t and e be complex variables, y" E G", and the entries of a G"- valued function f(t, y, e) be holomorphic with respect to (t, y. E) in a domain 1)o =

E) : ICI < do, -oo < tt < oo, Jyj < po, 0 < IEI < ro, I arg E) < ao). As-

00

some that f(t, y, E) E E'f,,,(t, y-) as e -+ 0 in the sector So= {E : 0 < IEI <

m=0

ro, I arg EI < ao } uniformly with respect to (t, y-) in the domain Do = {(t, y) : J 3'tI <

do, -oo < Iftt < +oc, Iyl < po}, where the coefficients fm(t,y-) are holomorphic in