We shall also de~ne a norm for Yectors, namely Ilxll = + I x~l. CAI~IBRIDGE~ LINEAR DIFFERENTIAL EOUATIONS WITH ALMOST PERIODIC COEFFICIENTS.

PERIODIC COEFFICIENTS.

BY

ROBERT H. CAMERON. 1

i n CAI~IBRIDGE~ MASS.

w i. Introduction.

I . I . W e shall deal w i t h the system of differential equations d~,(t)

d t -- ajl(t)~l(t) + "" + al,~(t)~,~(t) + fl~(t)

d ~ ( t ) a , l ( t ) ~ ( t ) + ... + a~,~(t)~,(t) + fl~(t);

d t

in which t h e f u n c t i o n s a,, (t) and fir are real or complex a.p.~ functions of t h e real variable t, a n d the fit(t) m a y or m a y n o t be identically zero. W e shall seek to determine conditions u n d e r which t h e solutions of (I. II) are of a r a t h e r general t y p e involving a . p . functions. Before characterizing this type of solu- tion more explicitly, we shall introduce a shorter vector terminology.

I . z . T r o u g h o u t this paper we shall use the letters x, y, z, a n d b to denote n-dimensional vectors (or matrices of n rows a n d one column) h a v i n g the compo-

d t .,

n e n t s ~1 . . . . , ~,~; V~, . . . , Vn; ~,, . . . , ~ ; a n d i l l , . . . , fl~. The vector ~-t~L(),..

d--t~(t) d will be denoted by D Ix]; the n-by-n m a t r i x whose elements are a , , will be d e n o t e d by A, a n d the m a t r i x product of A and x will be denoted by A . x . H e n c e in this terminology (I. I I) becomes

(I. 2I) D[x(t)] = A ( t ) . x(t) + b(t).

We shall also de~ne a norm for Yectors, namely Ilxll =

[ ~ 1 + "

+ I x~l.

I . I I )

1 T h i s p a p e r w a s w r i t t e n w h i l e t h e a u t h o r w a s a N a t i o n a l R e s e a r c h F e l l o w . 2 a . p . = a l m o s t p e r i o d i c (in B o h r ' s sense).

I. 3. I f we consider the k n o w n facts in the case where A(t) and b(t) are actually periodic with a c o m m o n period P, we find t h a t all of the solutions of ( I . 2 I ) are of the form

p

(I. 3 I) x(t)----

~_~e~,t'~y(~l(t),

where p is some positive integer, the 2~ are complex numbers, the r, are non- negative integers, and the

yC~l(t)

are a.p. vector functions. 1 I t therefore seems very n a t u r a l to ask whether in the general case (when A (t) ^and b (t) are a.p.) the solutions will all be of the form (I, 3I) with the

y(')(t)

a.p. instead of peri- odic. U n f o r t u n a t e l y such is not the case, as can be readily shown by examples.

H o w e v e r , the analogy with the p e r i o d i c case makes (I. 3 I) seem a n a t u r a l t y p e of solution, and we are therefore going to seek for conditions u n d e r which the solutions will all be of this type. I t is clear t h a t in the general a . p . case we can w i t h o u t loss of generality assume the ~ to be real, for if they have an imaginary part the exponential breaks up into two factors of which the one with the imaginary part can be absorbed into the a . p . vector functions yr

1.4.

Definition.

A vector function

x(t)

will be said to be of the a. p.

t y p e if there exist a positive integer p, real n u m b e r s ~1, . . . , ~v, non-negative integers rl, . . . , rp, and a . p . vector functions y(1)(t), . . . ,

y(")(t)

such t h a t (1.3 I) holds identically in t. The least common module of y(1)(t), . . . ,

y(n)(t)

will be called the module of

x(t).

I. 5. I t is the aim of this paper to obtain necessary and sufficient condi- tions t h a t all of the solutions of (i. 2I) be of the a.p. type and to obtain suffi- cient conditions t h a t a particular solution be of the a . p . type.

i. 6. The Vector b (t) may w i t h o u t loss of generality be t a k e n to be iden- tically zero and equation (I. 2I) replaced by

(1.6I) D [x (t)] --- A (t). x (t).

F o r consider the system

1,

.D [x (t)] = A (t). (t) + (t) b (t)

(i.

⁶²⁾

/ d t ~ + l (t) = o;

Abstrakte fastperiodisehe Funktionen, Acta mathematiea, vol. 61 (I933),

1 ~. BOCUNER~

I49--I84.

where ~,~+~(t) is a scalar function; and let

x*(t)

be the n + I dimensional vector consisting of

x(t)

a n d ~n+~(t); SO t h a t its components are ~,(t), ~(t), . . . ,

~n+i(t).

I f

x(t)is

a solution of ( I . 2 I ) , t h e n for every c o n s t a n t C,

x*(t)~-{Cx(t), C}

is a solution of (I. 52); and every solution

x*(t)

can be w r i t t e n in this form f o r some

x(t)

if .~,~+~(t)~=o. B u t (I.62) is homogeneous in the ~,(t) a n d is really in ~the form (I. 6I). Thus an equation of the form (I. 21) can be reduced to one of the form (I. 6I), a n d theorems proved concerning the homogeneous equation can readily be re-phrased s o as to apply to the non-homogeneous one.

w 2. Decomposition of a Solution of the a.p. Type.

2. I. I n our search for necessary a n d sufficient conditions t h a t all t h e solutions be of the a . p . type, we shall first study the properties of solutions which are of the a . p . t y p e and t h u s obtain necessary conditions. Consequently it will usually be assumed in sections 2 a n d 3 n o t only t h a t A(t) is a . p . but also t h a t one or more solutions of (I. 6I) is of the a . p . type.

I n particular we shall be interested in t h e assymptotic behavior of solutions as t--* + ~ or as t - * - ~ , a n d therefore introduce the following notation. I f there exists a Positive c o n s t a n t C such t h a t [[x(t)[[ <

Cf(t)

for all sufficiently great t, we say t h a t

x ( t ) = 0 [ f ( t ) ] at + oo;

a n d if there exist two positive constants Q and C~ such t h a t

Clf(t)<

[Ix(t)[[ <

< C~f(t)

for all sufficiently g r e a t t, we say t h a t

x(t)--~ O*

If(t)] at + ~ .

The m e a n i n g to be a t t a c h e d to the s t a t e m e n t t h a t x (t) equals 0 [f(t)] or O* [f(t)]

a t - - ~ or at +__ ~ is obvious.

I t is obvious t h a t if a solution

x(t)

of ( I . 6 I ) is of the a . p . type a n d is expressed in the form (I. 3I) with real ~ , a . p .

y(~)(t),

a n d non-negative integers r~, t h e n

(2. II)

~7(t)= O[e~tt r]

at + ~ ,

where ~ i s the g r e a t e s t value of Z~ occuring in its expression of the form (I. 3I), and r is t h e greatest value of r~ for values of ~ such t h a t Z ~ Z . I f Z is the least of t h e ~ instead of t h e greatest, (2.1 I) holds a t - - ~ , a n d if Z, is a con- s t a n t i n d e p e n d e n t of *, (2. II) holds at + ~ . This suggests the

Definition. L e t A (t) be a n y continuous m a t r i x function. Then a solution x(t) of ( t . 6 I ) will be called a primary solution of order 2 and degree r if x ( t ) - ~ O[e ~tt ~] at + ~ .

I n this section it will be shown t h a t if A (t) is a . p . , a n y solution x(t) of t h e a . p . t y p e can be decomposed into t h e sum of a finite n u m b e r of primary solutions.

2.2. L e t A (t) be a n a . p . m a t r i x function, a n d let x(t) be a solution (I. 6I) of t h e a . p . type. T h e n x (t) can be w r i t t e n in the f o r m

( 2 . 2 I ) x(t)=--e~t[yl, o(t) + tyl, l(t) + .." + tr'yl, r~(t)] -t- ""

+ eap[yp, o ( t ) + t y p , ~ ( t ) + ... + t~pyp,~(t)],

where Zl > Jt~ > .-- > Zp, the y~,, (t) are a . p . , a n d none of the functions y~, ~, (t), . . . , yp.~p(t) is identically zero. W e shall first show t h a t when x(t) is w r i t t e n in this way each of the terms in the above expression (i. e., an entire bracketed expression with its exponential multiplier) is itself a solution. To simplify the notation, we shall drop the subscripts f r o m 21 a n d r 1 a n d let

(2. 22) ylz(t) : (r--/z)[ yl, r--,~ (t) (/z ~- O, I, . . . , r)

te

( 2 . 2 3 ) x , (t) = t" y , - , (t) ( , = o, : , . . . ,

(The reason for i n t r o d u c i n g t h e factorials will appear shortly.) H e r e xr(t) is identical w i t h the first t e r m of (2.21), a n d we shall let x*(t) denote the sum of all the other terms. I t follows t h a t x(t)---xr(t)+ x*(t); a n d t h a t for a n y s,

(2.24) lira e -at t'~ x * (t) = o.

t ~ Q o

2.3. W e shall now try, by using (z. 24), to perform a t r a n s f o r m a t i o n on x (t) which will still leave it a solution of (I. 6I) b u t will get rid of x* (t). To do this, we find a sequence of positive numbers h 1, h~, . . . whose l i m i t s ' i s in- finity a n d for which lira A (t + h~) = A (t) u n i f o r m l y in t a n d lim y , (t + h~) = Yt~ (t) u n i f o r m l y in t for each /~ ~ r. Such a sequence exists since for each i we can choose hi so as to be g r e a t e r t h a n i a n d at the same time a I/i-translation n u m b e r for A(t) a n d every y~(t). H a v i n g chosen the sequence, a n d remembering (z. 23) a n d (z. 24), we have

r ! lim

[e-~hT~ x(t + hi)] -~ eXtyo(t)= Xo(t).

i ~ oo

B u t r!

e-XhihT~x(t + hi)

is a solution of

(2.3I)

1) (t)] = A (t + h , ) (t),

a n d t h e r e f o r e t h e limit x 0 (t) of t h e sequence is a solution of t h e l i m i t i n g system (I. 6I). H e n c e o u r m e t h o d is p a r t i a l l y successful, as we have t r a n s f o r m e d

x(t)

in such a way t h a t all t h e terms of ( 2 . 2 i ) e x c e p t t h e first have vanished. How- ever, t h e first t e r m

xr(t)

has become

Xo(t),

a n d we m u s t t h e r e f o r e find some way of g e t t i n g b a c k to

x~(t).

2.4- I f o u r new solution

xo(t)~ e~tyo(t )

h a d c o n t a i n e d an e x t r a f a c t o r t ~, we would have been able to s u b t r a c t o u t the last t e r m of

x~(t)

a n d o b t a i n a new solution to which we could have applied t h e same process. Thus we would have o b t a i n e d t h e r e s u l t t h a t each t e r m of xr (t) was a solution, a n d h e n c e t h a t

x~(t)

was a solution. H o w e v e r , since t h e f a c t o r t r is missing, t h e last t e r m of x~(t) is n o t a solution unless r = o. As a m a t t e r o f f a c t n o n e of t h e t e r m s of

x~(t)

is a solution w h e n r > o, a n d h e n c e o u r process m u s t be modified; a n d we will find t h a t as we r e p e a t t h e modified process we get successively x~(t), x~(t), . . . a n d finally

x~(t)

i n s t e a d of o b t a i n i n g m o n o m i n a l t e r m s f r o m

x~(t).

B e f o r e passing to t h e c o m p l e t e i n d u c t i o n p r o o f t h a t all of t h e x~(t) are solu- tions, we will c a r r y t h r o u g h one m o r e step of the process in o r d e r to see w h a t k i n d of modifications enter.

2.5- W e a g a i n deal with

x(t + h~),

a n d a g a i n t h e only significant t e r m is xr (t + hi), which m a y be w r i t t e n in t h e e x p a n d e d f o r m

9 t ' - e hQ/

I e~(t+hi) - ~ - I - hi) t g o (t -~ hi)] h ~ -1

+ . . . ( t e r m s in hr-2i , h ir-~, etc.)

h r h~. (t + hi)

i hi), a n d

x (t + hi) ~ Xo

B u t t h e first t e r m of this expression is

~Xo(t+

a solution of (2.3 I). T h u s

~:--37534. Aeta mathematica. 69. Iraprim~ lo 1 septembre 1937.

is

x (t + hi) -- ~ x0 (t + h;)

h~

x, (t) = e ~' [y, (t) + ty o

(t)] = (r - - I)! l i r a _{h i r - - 1}

i~| e h i

is a solution of (i. 61).

2.6. F r o m the way in which t h e second step was carried out, we see t h a t t h e subtraction of a multiple of t h e new solution is carried o u t a f t e r t h e argu- m e n t of the solution has been translated, a n d t h a t the multiplier to be used is a f u n c t i o n of the hi. I n succeeding steps a f u r t h e r complication arises because a combination of all the solutions, already obtained has to be subtracted, a n d f u r t h e r summations have to be used. W e now pass to the generM case.

Since we shall need to pass f r o m t h e y~ (t) to t h e

x~(t)

as well as f r o m the

xt~(t )

to the

yt~(t),

we first note t h a t the r equations (2.23) have a unique solu- tion if we regard t h e n functions

y~(t)

as u n k n o w n s ; a n d it can readily be verified t h a t t h e expressions

( - - t ) "

(~-. 6~) y~(t) = e - ~ - W - ~ . . . ( t ) ( ~ = o , . . . , ,-)

~ 0

satisfy (2.23), a n d hence t h a t (2.6I) holds.

2. 7. W e now assume t h a t

x~(t)

is a solution of

(I.

6I) for all tt less t h a n a certain integer q which does n o t exceed r. As before, we consider

x,(t + hi)

a n d obtain on i n t e r c h a n g i n g the order of s u m m a t i o n in (2.5 I),

where (z. 7I)

xr (t+ hi)= e~( t +hi)~ h~ ~ t,-e ~ h~ ~cr_e

(t; he),

~=o ~ ~=~ (" - e)! Y"--~ (t + h,) = ~=o

~(t; h)

-= ~ ( , + ~

V. ~._.(t + h), (~=o,

. . . , ,-).

S u b s t i t u t i n g f r o m (2.6I) in (2.7I), we o b t a i n

~ ( t ;

h) = ~ ~t" ~-'~_~ (--t--h)Ox._.._e(t + h) ~ x.-o(t + h) ~_j t" (--t--h)~

9 ~ 0 Q ~ O o ' ~ 0 ~ = 0

a ~ 0

a n d it follows t h a t if # < q, ~ , ( t , h~) is a solution of (2.31). Thus

r

h~.

x(t, h,) --- ~(t + h,) - 2] ~. *,-e (t; h,)

~ = r - - q + 1 ^"

~ ' - q h~.

~-- x* (t + h~) + e~( t + h i ) ~ z t"-r

~o ~! ,= (,_ r + h;)

is also a solution of (2.3I), and

9 ~ t'--r+q q)! yr--, (t)

lim ( r - - q)I X ( t , h,) = e~ t = xq(t)

~ ~ e ~ h'[--q (v -- r +

~,=r--q

is a solution of ( I . 6 I ) . W e now k n o w that x , ( t) satisfies ( I . 6 ] ) f o r all t t < q + I , and the induction is complete.

2.8. Finally, since x, (t) is a solution, so is x* (t); and the same a r g u m e n t can be applied to it. Thus each group of terms in x(t) having the same expo- nential factors forms a solution. Moreover the expansion (2.2i) of x(t) is unique, for Xo(t), . . . , x,(t) have been given in terms of x(t) by a uniquely defined process, the yl,0(t), . . . , y l r,(t) are uniquely defined in terms of the x,(t), and the y2,0(t), . . . , y2,,~(t), etc. are uniquely defined afterwards in turn. These r e s u l t s will be s u m m e d u p in (2.9).

2.9- I n order to state the results of ( 2 . 2 - - 2 . 8 ) more concisely, we first introduce by means of the following definition a terminology for some of the concepts which have arisen.

Definition. - - L e t A (t) be any continuous square matrix function and let X o ( t ) ~ o , x,(t), x~(t), . . . , x,(t) be solutions of ([. 6[) and ~ a real n u m b e r such t h a t for all tt ~ r,

(2.9~) y~(t) ~ e - " ~ ( - t ) " (t)

~ = 0

is b o u n d e d for all t. Then y,(t) will be called a pseudo-solution of (~. 6I) of order 2 and degree r. M o r e o v e r f o r t * ~ r, x ~ , ( t ) a n d y ~ ( t ) w i l l be called re- spectively its g e n e r a t o r and its m i n o r of degree tt, and xr(t) will be called its leader. Finally, a solution x(t) of (I. 6I) will be called a satisfactory solution of order X and degree r if it is the leader of a pseudo-solution y (t) of order ~ and degree r, and y (t) will be called its associated pseudo-solution.

I f we express the q u a n t i t y e~t~.j t~ ~.y~__~(t) in t e r m s of X o ( t ) , . . . , xt~(t) by means of (2.9I), we obtain x~(t) as a result. Thus we have

L e m m a 1. L e t A (t) be a n y c o n t i n u o u s s q u a r e m a t r i x f u n c t i o n . Ther/ if y~ (t) is a pseudo-solution of o r d e r )~ a n d degree r a n d has t h e m i n o r s

yo(t),..., y,.(t)

a n d g e n e r a t o r s x o(t), . . . , xr(t), it follows t h a t f o r each # ~ r,

x, (t)

^{- -}

Z

~ 0

H e n c e e v e r y s a t i s f a c t o r y solution is a p r i m a r y solution.

Definition. L e t

A(t)

be a n y c o n t i n u o u s square m a t r i x f u n c t i o n . T h e n a solution

x(t)

which is t h e sum of a finite n u m b e r of s a t i s f a c t o r y solutions x(1)(t), . . . ,

x(P)(t)

h a v i n g distinct orders ~ 1 , . . . , ~p is called a d e c o m p o s a b l e solution. I f Z is t h e g r e a t e s t o f ~ , . . . , Zp,

x(t)

is called a d e c o m p o s a b l e solu- t i o n of o r d e r Z.

5Tow in t e r m s of t h e above definitions, we sum up t h e results of ( 2 . 2 - - 2 . 8 ) . T h e o r e m I. L e t A (t) be an a . p . square m a t r i x f u n c t i o n a n d let x (t) be a solution of ( t . 6 I ) of t h e a . p . type. T h e n

x(t)

can be expressed in one a n d only one way in t h e f o r m

p ra t ~

93) y,

o ' = 1 ' v = O

where t h e Z~ are r e a l a n d distinct, t h e y~, ~ (t) are a . p . , a n d n o n e of t h e y~, o (t) are identically zero. ]Yioreover f o r each a =< p, ya, ~(t) is a pseudo-solution of (I. 6I) of o r d e r Z~ a n d degree r~ h a v i n g t h e m i n o r s

y~,o(t), . . . , y~,~a(t).

Finally, x (t) is t h e sum of t h e leaders of yl, ~, (t) . . . . , yp, rp (t), and h e n c e x (t) is decomposable.

w 3. The Necessity of Condition I.

3. I. I n this section we shall show t h a t if A(t) is a . p . and all t h e solu- tions of (I, 6I) are of t h e a . p . type, t h e n f o r each non-trivial solution

x(t)there

exist a real n u m b e r )~ a n d a n o n - n e g a t i v e i n t e g e r r such t h a t

x ( t ) = O* (eZtt r) at

+ r I n fact, we shall show t h a t a similar s t a t e m e n t holds f o r c e r t a i n com- binations of solutions, so t h a t t h e f o l l o w i n g c o n d i t i o n holds.

0 0 n d i t i o n I. T h e system (I. 6I) will be said to satisfy C o n d i t i o n I if to every finite set of solutions

x(~

x(1)(t) . . . . ,

x(P)(t)

(not all identically zero) t h e r e c o r r e s p o n d a real n u m b e r Z a n d a n o n - n e g a t i v e i n t e g e r r such t h a t

P

F, t x (t) = o* t')

~ ' ~ 0

at + ~ .

Since Theorem I enables us (under the above hypothesis) to express solu- tions in terms of pseudo-solutions, we will determine the asymptotic behavior of pseudo-solutions as a preliminary to d e t e r m i n i n g the asymptotic behavior of solutions a n d combinations of solutions. W e therefore develop in this section certain properties of pseudo-solutions - - first some general properties a n d later asymptotic properties.

3.2. W e begin with a general uniqueness lemma.

L e m m a 2. L e t A ( t ) be a n y continuous m a t r i x f u n c t i o n . T h e n every pseudo-solution y ( t ) of (t.51) has a unique order a n d degree a n d a unique set of minors a n d generators.

F o r suppose t h a t y (t) has the set of generators xo (t) . . . . , xr (t) and the corresponding order ~ a n d degree r, and at the same time y(t) has t h e set of g e n e r a t o r s X'o(t ) . . . . , x'r,(t) a n d the corresponding order ~' a n d degree r'. Then if t o is a n y c o n s t a n t value of t, the two solutions

,, ( _ t o ) , , (-- t~ (t) a n d e - X ' t ~ ~ x ~ , _ , ( t ) e--~t~ E v ! Xr--~

'v~O r

are both equal to y (to) when t = t 0, a n d hence are equal for all t. P u t t i n g t = o a n d v a r y i n g to, we obtain a linear relation between the f u n c t i o n s

e -~t~ e -xt~ to, 9 9 e -~t~ t~ ; e-Zto, 9 e -~'to t o . . . . , e -~'t~ t~

with c o n s t a n t vector coefficients n o t all of which are zero (since Xo(t ) a n d x' o(t) are non-trivial). Thus these functions are n o t linearly independent, and ~ = )/.

B u t since I, to, t~o .. . . are linearly i n d e p e n d e n t it follows t h a r r = r' a n d x ~ ( t ) - - x ' ~ ( t ) ~ o for v = o , . . . , r .

Moreover (2.9I) determines the minors in terms of the generators, a n d hence t h e y are also unique.

3.3. N e x t we notice t h a t t h e property of being a pseudo-solution of a given order is i n v a r i a n t u n d e r addition and u n d e r multiplication by a constant.

This f a c t can easily be verified, so no proof will be given.

Lemma 3. Let A (t) be a continuous square matrix function, 2 a real num- ber, and r a non-negative integer (or the symbol + ~ ) . Then the set of pseudo- solutions of (I. 6I) of order ~. and degree ~ r form with the trivial solution a linear manifold.

Definition. I f A (t) is a continuous square matrix function, the manifold consisting of the trivial solution together with set of pseudo-solutions of (I. 61) of order 2 and degree ~ r will be called the pseudo-solution manifold of (I. 6I) of order 2 and degree r. I f r is not specified, it will be understood to be the symbol + ~ .

3.4. Another lemma t h a t holds under fairly general conditions is the following.

L e m m a 4. Let A(t) be any continuous n-by-n matrix function and let yr(t) be a pseudo-solution of (I. 6I) of order ~ and degree r having the genera- tors Xo(t), . . . , xr(l) and minors yo(t), . . . , y~(t). Then if Y0(t) is bounded away from zero for sufficiently great positive values of t, we have for tt =< r

(3.4I) x , ( t ) ~ - O * [ e 2 t t ~] at + oo.

Moreover for each t the vectors Xo (t), . . . , x,(t) are linearly independent, so t h a t r < n and y~ (t) is different from zero for every value of tt and t.

For (3.4I) is an obvious consequence of (2.92), and the linear independence of the vector functions x,(t) follows from (3.4I). But if there is a linear relation between the x , (t) at a certain point t = to, the same relation holds for all t, since a solution which vanishes at one point vanishes identically. Thus the linear independence of the vector functions implies the linear independence of the vectors obtained by giving t a particular value.

3.5. Returning now to the case in which A (t) is a.p. we shall show t h a t under reasonable conditions the property of being a pseudo-solution is not altered by a limiting translation of t.

L e m m a 5. Let A (t) be an a . p . matrix function, and let y~(t) be a pseudo- solution of (I. 6I) of order ~ and degree r whose minors and generators are yo(t), . . . . y~(t) and Xo(t ) . . . . , xr(t) respectively. Also let hi, he, . . . be a sequence such t h a t 2[ ( t ) ~ lira A (t+h~) exists uniformly in t and lira y , (h;) exists for each tt and does not vanish when # = o . Then ~ ( t ) ~ l i m yr(t+h~) exists for all t

i ~ o v

and is a pseudo-solution of order 2 and degree r of

(3" 5 I) D =

Moreover its m i n o r of degree tt is

(3.5 z)

ff~(t)

= lim

y,(t + hi)

( / t - = o , . . , r)

i ~ o o

and its generator of degree tt is

" ( - hi)"

(3.53) :~,, (t) = lim e - X h ~

;[. x , - , ( t + h , ) ( t t = o , . . . , r ) .

~2~0

(3.54) where

To obtain this result, we first express the f u n c t i o n y , ( t + h i ) in terms of the

x~(t+hi)

by means of (2.9I), and then expand the binomials on the right, and interchange the order of summation.

W e find t h a t

y~(t + h,) = e -~t ~ (-- t)~ x*

(t; hi),

#

x; (t; hi) = e- iy, Q- (t +

q~=0

Since x't, (o;

hi) -= y~(h~),

lira x ; ( o , hi) exists for each it; and as i --* or the sequence of systems D Ix (t)] ---- A ( t + hi)" x (t) approach a limiting system uniformly and its sequence of solutions x~ (t; hi) approaches a limit at one point, t---= o.

Thus the sequence of solutions approaches a limit for all t, and this limit is a solution of the limiting system; or in other words, for each re, xt~(t) exists for all t as defined by (3.53), and is a solution of (3.5I). H e n c e we can take limits as i ----> ~ in (3.54), and we find t h a t for each re, fft~ (t) exists for all t as defined

tt

by (3.52), and 77, (t) ---=

e-~t~_j ( - t)" ~

v ! ~ x , - ~ (t). The latter equation t a k e n in con-

V~0

j u n c t i o n with t h e fact t h a t ~ 0 ( o ) = lim

yo(hi)~

o, shows t h a t ~,(t) is a pseudo- solution of order k and degree r of (3.5I), having the Y:t~(t) as generators a n d the ff~(t) as minors.

3.6. W e now show in L e m m a s 6 and 7 t h a t if A(t) is a . p . and all t h e solutions of (t. 6I) axe of the a.p. type, t h e n every pseudo-solution of (I. 6I) is b o u n d e d away from zero.

Lemma 6. L e t Z be a real n u m b e r a n d A (t) a n a . p . m a t r i x f u n c t i o n such t h a t e v e r y pseudo-solution of (I. 6I) of o r d e r Z and degree zero is b o u n d e d away f r o m zero f o r positive t. T h e n all of t h e pseudo-solutions of (I. 6I) of o r d e r Z are b o u n d e d away f r o m zero f o r all positive t.

F o r suppose t h a t t h e r e is a pseudo-solution yr(t) of o r d e r Z a n d some de- gree r w h i c h comes a r b i t r a r i l y close to zero f o r positive t. L e t its m i n o r s be yo(t), . . . ,

yr(t)

a n d let hi, h2, . . . be a sequence of positive n u m b e r s such t h a t lira

yr(hi)=

o. Since each of t h e

y~(t)

is b o u n d e d a n d because of B o c h n e r ' s t h e o r e m on n o r m a l f u n c t i o n s 1 we can choose a subsequence

h'~, h'~

. . . of hi, h~, . . . such t h a t lim

yt,(h'i)

exists f o r each tt and such t h a t A ( t ) ~ lira

A(t+h'~)

exists u n i f o r m l y in t. By hypothesis, lim

yo(h'i)~

o, a n d h e n c e by L e m m a 5,

i ~ a o

?Tt~(t)~ lim yt,(t+h'i )

exists f o r all tt a n d t a n d ff~(t) is a pseudo solution of

i ~ oo

D [x (t)] = ,4 (t).

x(t)

of o r d e r Z a n d d e g r e e r h a v i n g t h e m i n o r s Yo (t), . . . , ?7~(t).

B u t y0(t) is obviously b o u n d e d away f r o m zero, a n d h e n c e L e m m a 4 applies a n d shows t h a t ff~(o)~ o, in spite of o u r a s s u m p t i o n t h a t lira y~ ( h i ) = o. T h u s o u r

i ~ a o

l e m m a holds.

I t is of course clear t h a t if t h e h y p o t h e s i s h a d been t h a t t h e pseudo- solutions of o r d e r Z and d e g r e e zero were b o u n d e d away f r o m zero f o r n e g a t i v e t (or f o r all t) the c o r r e s p o n d i n g conclusion would hold f o r n e g a t i v e t (or f o r all t).

3.7- L e m m a 7. L e t A(t) be an a . p . m a t r i x f u n c t i o n . T h e n all of t h e a . p . pseudo-solutions of (I. 6I) of d e g r e e zero" are b o u n d e d away f r o m zero.

F o r suppose t h a t t h e r e exists an a . p . pseudo-solution y0(t) of d e g r e e zero a n d some o r d e r Z which comes a r b i t r a r i l y close to zero. L e t

Xo(t)~ e~tyo(t )

be t h e g e n e r a t o r of Yo (t) a n d h~, h~ . . . . a sequence such t h a t lim Y0 ( h i ) = o. L e t h ' , h'~, . . . be a subsequenee of hi, h~, . . . such t h a t 2~ (t) --- lira A (t + h'i) a n d ff0(t)~-lim

yo(t+h'~)

exist u n i f o r m l y in t f o r all t. T h e n if we let

e~t~o(t)=--hCo(t )

we have

lim

e-m, Xo(t + h'i)~. ~o(t),

i ~ oD

x Fastperiodische Funktionen I, Muthematische Annalen, vol. 96, pp. II9--I47 , esp. p. I43.

so that x0(t) is a solution of D [ ~ ( t ) ] = A ( t ) . ~ ( t ) . But x ( o ) = ~ 0 ( o ) = o ; so

~o(t) - = o and ~o(t) ~ o; and because of the uniformity of the limits yo(t) -~

~- lira tTo (t - - h'i) -~ o. Thus Xo (t) is identically zero, which is contrary to the

i ~ o o

assumption that it is the generator of yo(t).

3.8. Having proved all the necessary preliminary lemmas, we state our main conclusion for this section.

Theorem II. Let A ( t ) be an a.p. matrix function. Then a necessary condition that all the solutions of (I. 6I) be of the a.p. type is that the system (I. 6I) satisfy Condition I.

To prove this theorem, let x (~ (t) . . . x (p) (t) be any set of solutions of (I. 6I) (not all identically zero), and let

P

(3.81) z ( t ) - - ~ t~ x(')(t).

~ 0

Clearly by the argument used in Lemma I, z (t) ~ o, for if it were we should

P

have ~ to x (') (t)=--o identically in t and t o, and all the x (') (t) would be trivial.

~ 0

Moreover by Theorem I, (3.82)

g~

x(') (t) = ~ e',,~t~,, oy,,,(t)

( ' = ~ . . . . , I~),

where the ~,,~ are real, the e,.~ are non.negative integers, and the

y~,~(t)

are pseudo-solutions of (I. 6I) of order ,~,,~. B u t from Lemmas 6 and 7, for all and a, y , , ~ ( t ) = O*(l) at + ~ , and hence

(3.83) x(') (t) -- o* (e~,' ~,) at + o~,

where ~, is the greatest of ~,,x, ~,,2, . . . , ~,,g,, and q, is the greatest of the e-.~

for which ~,, ~ = ~,. But if we substRue (3.82) in (3.81) we obtain an expression of the same form. For when terms having the same exponential and power of t are combined, pseudo-solutions having the same orders (though not the same degrees) will be combined, and the result will thus still be a pseudo-solution (or else zero, in which the term drops out). Thus the same reasoning that was used in establishing (3.83) holds for z(t), and it follows that the system satisfies

Condition I.

5--37534. A c t a mathematica. 69. I m p r i m 6 le 1 septembre 1937.

3.9. I t is quite clear t h a t if in the statement of Condition I we replace + ~ by - - ~ , Theorem I I will still hold. Of course, we can not replace + by +_ ~ , because the 2 and r occurring in our asymptotic equation will not usually be the same at + ~ and - - ~ . We could on the other hand require the asymptotic equation to hold both a t - - ~ and at + ~ for separate pairs 2, r; and Theorem I I would be somewhat stronger since it would prove the necessity of a stronger condition. However in the future Condition I will usually appear in the hypothesis of our theorems r a t h e r than in the conclusions and therefore we will restrict it to be a one-sided condition as stated. Our theorems will be much stronger in consequence.

w 4. Decomposable Solutions.

4. I. In this section we shall show the connection between Condition I and systems of the form (I. 6I) all of whose solutions are decomposable.

I n the first place we note t h a t the proof of Theorem I I could be carried through just as well if the hypothesis t h a t all the solutions are of the a.p.

type were replaced by the hypothesis t h a t all the solutions are decomposable, provided t h a t we know t h a t the hypothesis of Lemma 6 is satisfied for every ~.

Thus we have

Theorem III. Let A (t) be an a.p. matrix function. Then if all the solu- tions of (I. 6I) are decomposable and all the pseudo-solutions of (I. 6I) of order zero are bounded away from zero for positive t, it follows t h a t Condition I is satisfied.

4.2. We shall show in this section (w 4) t h a t the converse of Theorem I I I is also true. The proof of this fact is rather long, and before carrying it through we shall give an outline of it using for convience the following terminology.

Definition. I f

x(t)-~ O* (e~tt r)

at + ~ , where t. is real and r is a non- negative integer, let A + (x) denote ~ and Z + (x) denote r. Similarly if

x(t)=- -~ O*(eZtt ~)

at - - ~ or at + ~ , let A--(x) and ~ - ( x ) or A(x) and ~(x) denote Z and r respectively.

The converse of Theorem I I I of course has as its hypothesis the assump- tion the

A(t)

is a.p. and Condition I is satisfied. Since Condition I holds, -//+ (x) exists for every non-trivial solution x, and as x ranges over all solutions it takes on only a finite number of values, say ~1, . . - , ~p, where p does not

exceed the order of A (t). These numbers it, will play a prominent role in w h a t follows, for our conclusion will be proved by m a t h e m a t i c a l induction on the it,.

To carry t h r o u g h the induction we shall show the following three things.

(a). There are no pseudo-solutions of order < iti and degree zero, and every pseudo-solution of order ~1 is b o u n d e d away from zero for positive t.

(b). I f every pseudo-solution of order ~ it, and degree zero is b o u n d e d away from zero for positive t, and every solution

x(t)

such t h a t ` 4 + ( x ) < 2~ is decomposable of order < its, then every solution x* (t) such t h a t

`4+(x*)-= it~

is decomposable of order ~ .

(c). I f every solution

x(t)

such t h a t `4+ (x)_--< it, is decomposable of order it~ and every pseudo-solution of order ~ ~ and degree zero is bounded a w a y from zero for positive t, then every pseudo-solution of order ~ it,+i and degree zero is b o u n d e d a w a y f r o m zero for positive t.

I t is obvious t h a t these three s t a t e m e n t s imply t h a t all of the solutiens are decomposable and all pseudo-solutions of zero degree and order _--< itp are bounded away from zero for positive t. Moreover the restriction on the order of the pseudo-solutions can easily be removed, and we have the desired conclu- sion. The p r o o f of (a) is almost obvious, but for the sake of completeness is given in L e m m a 8. The proofs of (b) and (c) are given in L e m m a s II and I e, following the preliminary L e m m a s 9 and IO.

4-3- L e m m a 8. L e t A(t) be an a.p. matrix function such t h a t ( I . 6 0 satisfies Condition I. T h e n if it is the least value which ,4 + ( x ) a s s u m e s for any solution

x(t)

of (I. 6 0 , there is no pseudo-solution of order less than it; and every pseudo-solution of order it is b o u n d e d a w a y from zero for all t.

F o r if there were a pseudo-solution of order less t h a n it, say /z, its m i n o r

yo(t)

of degree zero would also be of order ~t, and

e~tyo(t)

w o u l d be a solution

Xo(t)

of (I.6I). But Xo(t)=

O(e ttt)

at -1- oo, so that _//+(Xo) ~ [ t contrary to the assumption t h a t it is the least value of `4+(x). Again, if

yo(t)is

a pseudo- solution of order it and degree zero,

Xo(t ) = e~tyo(t )

is a solution, and

Xo(t)=O(e ~t)

at + ~ . Thus `4+ (Xo) N it; and this implies t h a t `4+ (Xo) = it and ~ + (x0) = o, so t h a t

Xo(t)-~O*(e zt)

at +oo and

yo(t)-~O*(i)

at + ~ .

4 . 4 . As a preliminary to the proofs of (b) and (c) of (4. z) we give two lemmas concerning transformations on pseudo-solutions.

Lemma 9. L e t it be a real n u m b e r and A (t) an a . p .

n-by-n

matrix func- tion such t h a t (I. 6I) has no pseudo-solution of order 2 and degree zero which

comes arbitrarily close to zero for positive t. L e t h , h~, . . . be a sequence of positive n u m b e r s such t h a t A ( t ) ~ lim A (t + h~) exists uniformly for all t. T h e n

i ~ oo

there exists a subsequence h'l, h'2, . . . of hi, h2, . . . such t h a t for each p s e u d o - solution

y(t)

of (I, 6I) of order ~,

T[y] (t)~--lim y(t + h'i)

exists for all tl More-

t ~ o o

over if such a subsequence is chosen and

y ( t ) a n d y*(t)are

distinct pseudo- solutions of (I. 6I) of order ~, then

T[y] ( t ) ~

T[y*] (t).

F o r in the first place the set of pseudo-solutions of order ). forms with the trivial solution a linear manifold, and this manifold m u s t be of finite dimension- ality since by L e m m a 4 no pseudo-solution of order ~ is of degree ~ n. L e t

y(1)(t) . . . . , y(P)(t)

be a basis for it, and let h'l, h'u, . . . be a subsequence of hi, h~ . . . . such t h a t lim

y(')(h'~)

exists, ( v = I, . . . , p). Then if

y(t)

is any pseudo-solution of (I. 5I) of order ~, lim

y(h'i)

exists, and the same applies to the minors of

y(t)

which are also pseudo solutions. Moreover the corresponding limit is not zero for the minor of zero order. Thus by L e m m a 5,

T[y](t)exists

for all t.

N o w suppose

y(t)

and

y*(t)

are two distinct pseudo-solutions of ( I . 6 I ) of order ~. Then

y(t)--y* (t)

is also a pseudo-solution of order ~, and by L e m m a 5,

T[y--y*](t) = T[y] (t) -- T[y*] (t) is

a pseudo-solution of

D[x(t)]

= A ( t ) - x ( t ) of order ~ and hence is not identically zero.

Lemma 10. L e t ~ be a real n u m b e r and A (t) an a . p .

n-by-n

matrix func- tion such t h a t (I. 6I) has no p s e u d o s o l u t i o n of order )~ and degree zero which comes arbitrarily close to zero for positive t. Then every pseudo-solution of ( I . 6 i ) of order ~ is bounded away from zero for all t. Moreover if hll h~ . . . . is a sequence such t h a t T [y] (t) --~ lim y ( t + hi) exists for each pseudo-solution

y(t)

of ( I . 6 i ) of order )~ and such t h a t uniformly in t, lim

A ( t + h i ) = A ( t ) ,

then the t r a n s f o r m a t i o n T has a single-valued inverse T - ~ defined over the entire pseudo- solution manifold S of (I. 6I) of order ~.

The second s t a t e m e n t follows from the f a c t t h a t according to L e m m a 5, T takes S into a sub-manifold S ' of itself, which is of the same dimensionality as S since T takes a non-trivial element into a non-trivial element. Thus S ' - ~ S, and T - 1 is defined and single valued over the whole of S.

The first s t a t e m e n t follows from the second s t a t e m e n t and L e m m a s 4 and 9.

F o r t h e r e exists a sequence h't, h'~, . . . which approaches + r and for w h i c h

lim

A (t+h'~)= A (t)

uniformly in t; and this sequence has a subsequence h'~', h~', . . .

~ Q o

h t t \

such t h a t lim

y(t§ l)

exists for all t and each pseudo-solution

y(t)

of order it.

Then by w h a t we have just proved, to every pseudo-solution 9(t) of order it there corresponds a pseudo-solution of order it such t h a t lira

y (t+ h~')~-?7(t)

for

i ~ c

all t. B u t y (t) is b o u n d e d a w a y from z e r o for positive t, and ~ (t) is b o u n d e d a w a y f r o m zero for all t.

4. 5. W e now proceed to the proof of proposition ( b ) o f (4. 2), whose conclusion is t h a t each solution

x(t)

for which 1/+ (x)----it is decomposable of order it. To obtain this result we shall have to actually carry out the decompo:

sition of x(t); and this process will be similar to b u t more involved than the decomposition of a solution of the a . p . t y p e given in ( 2 . 2 - - 2 . 8 ) . As in the simpler case, we decompose

x(t)

step by step by mathematical induction. B u t now each step will involve defining and showing the existence of the new q u a n : tities involved, as well as showing t h a t t h e y are solutions or pseudo-solutions of (I. 6I). Moreover each step will involve t w o new solutions and two new pseudo- solutions, the pseudo-solutions being related by a t r a a s f o r m a t i o n of the t y p e defined in L e m m a Io. Finally, each step will depend on a formula obtained in the preceding step, and we shall therefore give these formulas in the s t a t e m e n t of our l e m m a in order to carry on an induction proof based on them.

Lemma 11. L e t

A(t)

be an a . p . n:by-n matrix function such t h a t (I. 6I) satisfies Condition I. Let it be a n u m b e r such t h a t every solution

x'(t)Of

(I. 6I) for which .//+ (x') < it is decomposable of order < it and such t h a t every pseudo:

solution of order it and degree zero is bounded a w a y from zero for positive t.

L e t

x(t)

be a solution of (I. 6I) for which _//+ (x) : it, and let ~ + (x)--~ r. U n d e r these conditions, x (t) is decomposable of order it.

Moreover corresponding to each non-negative integer s ~ r there exist a sequence h~, h~, . . . ---> + ~o and a pseudo-solution

ys(t)

of (I. 6I) having the following properties :

(1) lira

A(t§

for all t.

l ~ ao

(2) F o r every pseudo-solution

y(t)

of order it, lim

y(t§

exists for all t.

i ~ o o

(3) I f

Yo(t), . . . , ys(t)

are the minors of

ys(t),

then for all t and each non- negative /~ ~ s,

x(t+h,)e_a(t+h,) - ~ (t+h,)e _{# !} yr--Q (t + h~)

(4- 5 I) lim e=~--t'

i~ .o (t + h~-)'-t' = o.

To begin the i n d u c t i o n proof of the lemma, we show t h a t the second state- m e n t holds if s = o. I t is obvious t h a t a sequence h'[, h~', . . . can be f o u n d which approaches + ~ a n d has property (]). Moreover it follows from L e m m a 9 t h a t this sequence has a subsequence h'~, h'2, . . . which satisfies (2). B u t since x (t) -~ O* (e ~t t"), x (t) e -~t t - ~ is bounded a n d b o u n d e d away f r o m zero for positive t.

T h u s h'~, h'~, . . . has a subsequence h~, h~ . . . . such t h a t lim x(hi)e--~th~ -" exists.

i ~ a a

H e n c e ~ o ( t ) - r! lim x(tH-h~)e-~h~h-~ -r exists for each t a n d is a non-trivial solu- tion of (I. 6]). Moreover if go(t)----e-~t~o(t),

(4. 52)

go(t) = r! lira x ( t + h,)e--~(t+hi)(t + hi)-"

f o r all t; a n d since x ( t ) e - X t t - " = O*(I) at + ~ , go(t) is b o u n d e d for all t. Thus go(t) is a pseudo-solution of (I. 6I) of order X a n d degree zero a n d has the gene- r a t o r Ec0(t ). By L e m m a Io, there is a unique pseudo-solution yo(t) of order ;~

a n d degree zero such t h a t lim yo(t + h~)~Yo(t). Moreover if we substitute lim yo(t+hi) for g 0 ( t ) i n (4. 52) we find t h a t (4. 5I) is satisfied for s = o . I t e n c e

t ~ Qo

h,, h . ~ , . . . a n d yo(t) satisfy (I), (2), (3); a n d our s t a t e m e n t holds w h e n s = o . C o n t i n u i n g the induction, we assume t h a t the second s t a t e m e n t of the l e m m a is true when s = p - I < r. L e t us designate the corresponding sequence by h'l, h'~ . . . . , t h e corresponding pseudo-solution by yp--1 (t); and the minors a n d generators of y~--x (t) by Yo (t), . . . , yv--1 (t) and Xo (t), . . . , xp-~ (t) respectively.

T h e n (4. 5I) holds for the sequence h'~, h'~ . . . . w i t h tt = s = p - - I , a n d if

r

(4. S3) z ( t ) - x ( t ) - e "

Q=r.'--p + 1

it follows t h a t

(4. 54) lira i n f II

(t)

t-c,-,+,)II = o.

B u {

" ,-e(_ t)"

x(t)- "y, "?.

p ~ r ' - - p + 1 ~=0

a n d h e n c e by C o n d i t i o n I e i t h e r

z ( t ) ~ o

or A + ( z ) a n d ~ + ( z ) b o t h exist. More- over it follows f r o m (4. 54) t h a t e i t h e r z (t)=- o, or _//+ (z) < ~, or A + (z) - - ~ a n d

~ + ( t ) < r - - p

+ I; so in a n y case z ( t ) =

O(e~tV -p) at + ~ .

H e n c e t h e r e exists a s u b s e q u e n c e h , h~, . . . of h'~, h'2, . . . such t h a t

limz(hi)e-~h~h~ -(~--v)

^exists.

i ~ co

A n d since h ' , h'~, . . . has p r o p e r t y (2),

T[y](t)=limy(t+h~)

exists f o r a n r a n d

i ~ Q o

e a c h pseudo-solution

y(t)

of o r d e r ~. T h u s if

T[y~--~] (t)has

t h e m i n o r s Yo (t) . . . . ,

~ - - l ( t ) a n d g e n e r a t o r s ~o(t) . . . . , ~v--~(t), it follows f r o m L e m m a 5 t h a t f o r tt < p -- I, T [Yt~] (t) ---= !7~ (t).

L e t

r h r - - ~

- - x~_e---~ (t + h);

+1

so t h a t f o r each

h, X(t, h)

is a s o l u t i o n of D Ix(t)]-=

A (t + h). x(t)

a n d X ( o , h)----

= z(h).

T h e n lim X ( o ,

h~)e-~h~h~('-v)

exists, a n d

~cp (t) --- (r - - p)! l i m X (t, h~.) e-~h~ ' hT(~-P) exists f o r ull t a n d is a s o l u t i o n of (I. 6I). M o r e o v e r

[z(t + h ) - X(t,

h)] e -~(t+~

Q h . t,0-o

= - y._~ (t+h) y, ~! (~-~)!

(~r---p + 1 a~0

+

he r-e(__ h)" "--~" (t + h) ~

+ ~' ~ Z ~ a! y.--Q--~--.(t+h)

~ = r - - p + 1 ~,=0 a=0

r-p h. tQ_ ~

= - y~-~(t+ h) F, ~! (~-~)!'

Q=r---p4 1 a=O

so t h a t f o r all t

( r - - p ) ! lira [z (t + hi)-- X (t, h~)]

e-X( t+ hi) h-[-(~p)

i o a o

T h u s

(4. 55)

0=r---p + 1

~ - - r + p ~ P tO

T

[Y~-e] (t) (t).

!Tp (t)----(r--/o)! lim

z(t + h~)e-~(t+h~)(t + h~)-('-v)

i---~ oo

exists for all t, and

v te v--~ - - ( t ) " P

~ ( t ) = - ~ ~ e - ~ ~ ~ , . ~__~o(t) + ~ - ~ , ~ , ( t ) = ~ - ~ ~.t" x~" (t).

0 ~ 1 t r = 0 ~ 0

Since z ( t ) = O ( d t t ~p) at + 0r it follows from (4. 55) t h a t ~ p ( t ) i s b o u n d e d for all t, and hence is a pseudo-solution of (I. 6I) of order 4 and degree p having

g o ( t ) . . . . , h ( t ) as minors and X o ( t ) , . . . , ~Cp(t) as generators.

L e t yv(t)=--T-l[~jv](t), and let y:(t), . . . , y~-~(t), yp(t) be its minors. By L e m m a 5, the minors of ~p (t) = T [yp] (t) are I' [y:] (t), . . . , T [y~--~] (t), I ' [yv] (t), and hence by L e m m a 2, T[y$](t)----~j,(t)= T[yg](t) for tt < p - - I . Thus by L e m m a Io, y~,(t)~--yt,(t ) for /~ < p - I, and yp(t) has the minors g o ( t ) , . . . , yv(t).

Moreover we have from (4. 55) and (4. 53)

lim yp (t + h;) ( r - - p ) ! lim x ( t + hi) e-i(t+hi) ^--

_ E (t + h~)~yr_~( t + h~) (t + h~)--r

e=r_~p +i Q!

so t h a t (4. 5I) holds for the pseudo-solution yp(t) and the sequence hi, h2, . . . Thus the induction is complete, and the second s t a t e m e n t of this lemma holds for a l i ' s < r.

Finally, let s ~ r, and let x~(t) be the leader of the pseudo-solution y~(t).

Then by (4. 51),

lim [x (t + h,) -- x, (t + h,)] e - l ( t + h,) _= o,

so t h a t if x* (t) ~ x (t) -- xr (t), lim inf II ^{x ~} (t)II ^{e - ~ t = o .} H e n c e either x* (t) ---= o or .4 + [x*] < X; and in either case x(t) is decomposable of order i.

4-6. P r o p o s i t i o n (c) of (4. 2) is established in the following 1emma.

L e m m a 12. L e t A(t) be an a . p . square matrix function such t h a t (I. 6I) satisfies Condition I. L e t ), be a n u m b e r such t h a t every solution x ( t ) o f (I. 61) for which .4 + (x)=< ~ is decomposible of order < ~, and sueh t h a t every pseudo- solution of order < 4 and degree zero is bounded away from zero for positive t.

L e t 4' be the b a s t n u m b e r greater t h a n i such t h a t there exists a solution x'(t) for which .4+ (x') = i'. Then every pseudo-solution of order < F and degree zero is b o u n d e d away from zero for positive t.

F o r suppose t h a t there exists a pseudo-solution Y0 (t) of order

~,"<= );

a n d degree zero which comes arbitrarily close to zero for positive t and let Xo ( t ) =

e~"tyo(t)

be its generator. T h e n ~ " > 4; a n d _//+ (xo) is less t h a n X' and hence less t h a n or equal to 2. I t follows t h a t x 0 (t) is decomposable of order < ,~, so t h a t

~ t ~

Xo(t)~-- e;'at Z v!ya, ra--~(t),

c i ~ l ~ = 0

where ~ ~ Z 1 > X~ > .-. > Zp a n d the y~,~(t) are pseudo-solutions of order 4o. B u t by L e m m a I o, every

y~.~(t)

is b o u n d e d away from zero for all t, a n d hence

Xo(t)--O*(eZptt~p)

a t - - : r Thus

yo(t)= O*(dzp-z')tt~) at --or

which is impos- sible since Yo (t) is bounded. I t follows t h a t all pseudo-solutions of order ~ Z' and degree zero are bounded away f r o m zero for positive t.

4. 7. W e now state the m a i n theorem of this section, which includes t h e converse of Theorem I I I .

Theorem IV. L e t

A(t)

be an a . p . square m a t r i x f u n c t i o n such t h a t (I. 61) satisfies Condition I. Then every solution of (I. 6I) can be expressed in one a n d only one way as a sum of satisfactory solutions h a v i n g distinct orders. Moreover every pseudo-solution of (I. 6I) is bounded away from zero for all t.

A l m o s t all of the proof of this t h e o r e m has already been given, since the three proposition (a), (b), (c), have been proved in L e m m a s 8, I I , I2. Thus t h e i n d u c t i o n outlined in (4. 2) is complete; and it follows t h a t all solutions of (I. 6I) are decomposable a n d all pseudo-solutions of order n o t greater t h a n the Zp of (4. 2) a n d degree zero are bounded a w a y f r o m zero for positive t. B u t there are no solutions of order greater t h a n ~ a n d degree zero, for if there were a pseudo-solution

yo(t)

of order 4 ' > Zp a n d degree zero,

xo(t)~-eZ'tyo(t)

^{would be}

decomposable, _//-(x0) would exist a n d be less t h a n or equal to ~p, a n d

yo(t)

would be u n b o u n d e d for negative t. Thus it follows from L e m m a IO t h a t all pseudo-solutions are bounded away f r o m zero for all t.

Moreover a solution can be decomposed in only one way, for otherwise the trivial solution would be decomposable a n d ~/+ (o) would exist.

C o r o l l a r y 1. I f

A(t)

is an a . p . square m a t r i x f u n c t i o n such t h a t (I. 6I) satisfies Condition I, it follows t h a t (I. 6I) satisfies t h e condition obtained by replacing + cr by - - ~ in the s t a t e m e n t of Conditiou I.

6--37534. A c t a m a t h e m a t i c a . 69. Imprim6 le 2 septembre 1937.

I t is of course also clear from s y m m e t r y t h a t this negative reflection of Condition I implies C o n d i t i o n I.

C o r o l l a r y 2. I f A(t) is a n a . p . square m a t r i x f u n c t i o n such t h a t (L 6:) satisfies Condition I, it follows t h a t every primary solution of (I. 6I) of order X and degree r is a satisfactory solution of order X a n d degree ~ r.

F o r every solution x(t) is t h e sum of satisfactory solutions of distinct orders, a n d it is clear t h a t if two or more are present a n d x (t) ~ 0 (e ~, t ~') at + a n d x ( t ) ~ 0 ( e z~t *~) at - - ~ , t h e n X,>X.~ a n d the solution in question is n o t primary.

w 5. The Invariance of Condition I.

W e will now show t h a t Condition I is i n v a r i a n t u n d e r limiting translations.

Theorem V. L e t A (t) be an a . p . n-by-n m a t r i x f u n c t i o n such (I. 6 I ) s a t i s - fies Condition I. T h e n if hi, h2, . . . is a sequence of real numbers such t h a t

(t) -- lira A (t + h~)

exists u n i f o r m l y in t, it follows t h a t the t r a n s f o r m e d system

(5. I) D [x (t)] = A (t). x (t)

also satisfies Condition I.

To establish this theorem, let X 1 < X 2 < .-- < Xp be the set of values 1/* (x) can t a k e on as x(t) ranges over all solutions of (I. 5I), a n d for each v ~ p let L , be the linear m a n i f o l d consisting of all satisfactory solutions of order - ~ t o g e t h e r with the trivial solution. Since every solution is decomposable into a sum of satisfactory solutions, the sum of the dimensionalities nl, . . . , ~p of L:, . . . , Lp is n. Moreover for each v ~ p the pseudo-solution manifold L'~ of ( I . 6 : ) of order X, has n, dimensions, for each non-trivial element of L , is a l e a d e r of an element of L'~ a n d t h e correspondence is l - - I since by L e m m a 4 and Theorem I V no pseudo-solution has the trivial solution as a leader.

:Now let h'~, h'~, . . . be a subsequence of h:, h~, , , . such t h a t f o r each

~ p and each element y (t) of L',, T [y] (t) - - lira y (t + h~) exists f o r all t. T h a t such a subsequence can be chosen follows from L e m m a 9 a n d Theorem IV, as does also the fact t h a t T sets up a : - - I correspondence between Z ' , a n d the

m a n i f o l d L:* into which it takes L',. Thus L:* has n, dimensions, and by L e m m a 5 its n o n 4 r i v i a l elements are pseudo-solutions of (5. I) of order it. Since the non- trivial elements of L ' , are bounded away f r o m zero, so are the non-trivial elements of

L',*.

F o r each ~ ~ p , let L~* be the linear m a n i f o l d consisting of the trivial solution t o g e t h e r w i t h the leaders of the non-trivial elements of L~*. T h e n it follows f r o m L e m m a s 2 a n d 4 t h a t t h e correspondence between L:* a n d L* is I - I, so t h a t L~ has n, dimensions. Moreover it follows f r o m L e m m a 4 t h a t if ~(t) is a non-trivial element of L*, _4 + ( ~ ) ~ , ; a n d hence the manifolds L~

a n d L* have only the trivial element in common if /, ~ v. Thus the m a n i f o l d L* + -.. + L~ has n dimensions and is therefore the entire m a n i f o l d of solutions of (5-I); a n d every solution of (5. I) is decomposable.

Finally, let fro(t) be any pseudo-solution of (5. I) of degree zero, a n d let it be its order. Then

e~t~o(t)

is a non-trivial solution of (5, I) a n d is decomposable into the sum x(*,)(t) + ... + x('g)(t); where each

x(~e)(t)

is t h e leader of a non- '*" and vl < "'" < * g . By L e m m a 4, trivial element

y(*e)(t)

of the m a n i f o l d L~Q,

_//(x('e))=it, for each # ~ g , and hence -//+(ffo)-~it~g--it and _ d - ( f f o ) - - i t ~ - - L B u t since if0 (t) is bounded, it,, = it,# = it, g ~- I, a n d if0 (t)--= y(*,)(t). Thus every pseudo-solution of (5. I) of order zero is bounded away f r o m zero. Thus it follows from Theorem I I I t h a t (5. I) satisfies Condition I.

w 6. Stationary Pseudo-solutions.

I n this section we shall obtain sufficient conditions t h a t a pseudo-solution of (I. 6I) be a . p . W e shall assume a t the outset t h a t A (t) is a. p. with a module contained in M and t h a t (I. 6I) satisfies Condition I; a n d we shall show t h a t a pseudo-solution y (t) is a . p . w i t h a "module contained in M if' a n d only if y (t)is w h a t we shall call a positive s t a t i o n a r y f u n c t i o n with respect to M.

Definition. A vector f u n c t i o n

f(t)

will be called (positive) s t a t i o n a r y w i t h respect to a module M if for each real t

(6. 1 I) lim

f ( t + hi)

= f ( t )

whenever hi, h~ . . . . is a sequence of (positive) real numbers such .that for each element q) of M,

lim 9 h~ - o (mod 2 z~).

i ~ o o

W e call particular a t t e n t i o n to the fact t h a t no u n i f o r m i t y is postulated in connection with (6. II). As a m a t t e r of fact, if (6. II) were assumed to hold u n i f o r m l y for all t, a f u n c t i o n which is s t a t i o n a r y with respect to a module M would be a . p . w i t h the module M. However, as the definition stands, a sta- t i o n a r y f u n c t i o n need n o t be a . p . at all, nor even u n i f o r m l y continuous. Thus t h e theorem quoted above as the subject of this section is by no means a mere triviality; and as a m a t t e r of fact it f o r m s the basis for all our later theorems.

The theorem of this section will be proved by the m e t h o d of Favard, which is based on Bochner's definition of a normal function. As this process involves certain iterated limits, we shall first prove in L e m m a I3, t h a t a f u n c t i o n which is s t a t i o n a r y w i t h respect to a module M has a similar property involving iterated limits. As a n o t h e r primilary to the m a i n theorem, we shall show in L e m m a I4 t h a t a part of t h e hypothesis of L e m m a 5 m a y b e replaced by the hypothesis t h a t (I. 61) satisfies Condition I.

6 . 2 . L e r n m a 13. L e t f ( t ) be positive s t a t i o n a r y with respect to a module M, a n d let hi, h,, . . . and k~, k 2 .. . . be sequences of positive numbers such t h a t for each element ~ of M

lim q) (hi + kj) = o (mod 2 re).

i , j ~

T h e n if t o is a real n u m b e r such t h a t g ~- lim lim f ( t o + h~ + kj) exists, it follows t h a t g = f ( t o ) .

F o r corresponding to each positive i n t e g e r n there exists an index j,~ > n such t h a t [g -- l i m f ( t o + hi + kj~)[ <= ~ a n d there exists an index iN > n such t h a t

If(to

+ hi m q- kjn ) - - lim f ( t o + hi + kj~)l <= I .

Then lim f ( t o + hi,, + kj,) = g, a n d for each element 9 of M lim 9 (h~.~ + kj~) = o (rood 2 z). I t follows f r o m the definition of a positive s t a t i o n a r y f u n c t i o n t h a t

lim f ( t o + bin + kj~)=f(to), so t h a t t h e l e m m a is proved.

6. 3. L e m m a 14. L e t A(t) be an a . p . m a t r i x function such t h a t ( I . 6 I ) satisfies Condition I, let y(t) be a n y pseudo-solution of (I. 6I), and let r a n d be the degree a n d order of y(t). T h e n if hi, h~, . . . is a sequence such t h a t

lira y(hl) exists a n d A (t) ----lira A ( t + hi) exists u n i f o r m l y in t, it follows t h a t (t) = lira y (t + hi) exists for all t a n d is a pseudo-solution of order ~ and degree

i ~ oo

r of D [~ (t)] = .~ (t). ~ (t). Moreover if Yt, (t) is the m i n o r of y (t) of degree re,

~ (t) ---- lira yu(t + hi) is the minor of ~(t) of degree ft.

i ~ oo

F o r suppose t h a t there exists a m i n o r yp(t) of y(t) such t h a t lim yp(hi)

i ~ o o

does n o t exist. T h e n since y~(t) is bounded for ft ~ r, there exist two sub-

9 t t t~ !

sequences h'l, h~, . . a n d h~, h~, . . . . . of hi, h2, . . such t h a t yu=* limy~(hi) and y~* = lira y~ (h~') exist for each ft _--< r a n d y~ ~ y~*. I t follows f r o m Theo-

i ~ a o

rem IV t h a t y~ ~ o a n d Y*o* ~ o, a n d f r o m L e m m a 5 t h a t y* (t) = lim y (t + h~) a n d

i ~ a o

y * * ( l ) = l i m y ( t + h ; ' ) are pseudo-solutions of D [ x ( t ) ] - - . 4 ( t ) . ~ r ( t ) o f order ~ a n d

l ~ o o

degree r. B u t y* (o) = y** (o); so by L e m m a 4, Y* (t) --- y** (t), a n d by L e m m a 2 the minors of degree p, lim y~ (t § h~.) a n d lira yp (t + h~') are identical. Thus yp = y p , a n d this contradiction shows t h a t lira y , (hi) m u s t exist for each ft.

H e n c e t h e l e m m a follows from L e m m a 5- 6.4. W e can now prove

T h e o r e m VI. L e t A(t) be a . p . with a module contained in a certain m o d u l e M, a n d let (I. 6I) satisfy Condition I. Then a pseudo-solution y(t) of ( I . 6 I ) is a . p . w i t h a module contained in M if a n d only if y ( t ) i s positive s t a t i o n a r y with respect to M.

I t is obvious t h a t if y (t) is a . p . with a module c o n t a i n e d in ] I , it m u s t be positive s t a t i o n a r y with respect to /I/. W e therefore assume t h a t it is positive s t a t i o n a r y w i t h respect to M a n d seek to show t h a t it m u s t be a . p . w i t h a module contained in M.

Procceeding along the general lines of F a v a r d ' s m e t h o d of proof, we assume t h a t there exists a sequence of positive numbers hi, h~, . . . such t h a t for each (P in M, lira (~ h,. ~- o (rood 2 z), a n d such t h a t lim y (t + hi) does n o t converge to y(t) u n i f o r m l y in t f o r all positive t. T h e n there exist a n u m b e r e > o, a subsequence h'l, h'~, . . . of hi, hl, . . . a n d a sequence of positive numbers tl, t ~ , . . . such t h a t for each index i,

(6.4I) IlY (ti + h~) -- y (ti)II > ~.