1. Introduction and r6sum6 of results LAWRENCE MARKUS and RICHARD A. MOORE OSCILLATION AND DISCONJUGACY FOR LINEAR DIFFERENTIAL EQUATIONS WITH ALMOST PERIODIC COEFFICIENTS 1

OSCILLATION AND DISCONJUGACY FOR LINEAR DIFFERENTIAL EQUATIONS WITH ALMOST PERIODIC COEFFICIENTS 1

BY

LAWRENCE MARKUS and RICHARD A. MOORE Yale University

1. I n t r o d u c t i o n and r6sum6 o f results E q u a t i o n s of t h e form

( r ( x ) y ' ) ' + K ( x ) y = 0, (1)

where r (x) > 0 a n d K (x) are real c o n t i n u o u s f u n c t i o n s on - c~ < x < o0, are classified, b y t h e b e h a v i o r of their real solutions, as ( + ) - o s c i l l a t o r y or n o n - o s c i l l a t o r y . I n t h e first i n s t a n c e one n o n - t r i v i a l (not i d e n t i c a l l y zero), a n d t h e r e b y every, s o l u t i o n vanishes a t a r b i t r a r i l y large abscissas; in t h e second i n s t a n c e e v e r y n o n - t r i v i a l solution is n o n - v a n i s h i n g for sufficiently large abscissas. A special i n s t a n c e of n o n - o s c i l l a t i o n is t h e d i s c o n j u g a t e case i n which e v e r y (non-trivial) solution has a t m o s t one zero on - c~ < x < co. I t is k n o w n t h a t a n e q u a t i o n of t h e form (1) is d i s c o n j u g a t e if a n d o n l y if there is a solution which is everywhere positive.

Our p r i n c i p a l i n t e r e s t concerns t h e s i t u a t i o n where r ( x ) = 1 a n d K (x) = - a + b p ( x ) . Here (a,b) are real p a r a m e t e r s a n d p (x) is a real a l m o s t periodic f u n c t i o n . W e shall note, i n this case, t h a t n o n - o s c i l l a t i o n a n d d i s c o n j u g a c y are coincident. Also we shall f i n d t h a t t h e d o m a i n D i n t h e ( a , b ) - p a r a m e t e r plane, for which t h e c o r r e s p o n d i n g e q u a t i o n s are d i s c o n j u g a t e , is closed a n d convex.

W e generalize t h e t h e o r y of Hill's e q u a t i o n (in which p ( x ) is periodic) b u t , of course, w i t h o u t u s i n g t h e F l o q n e t r e p r e s e n t a t i o n , which is n o t applicable here. F o r example, i n t e r i o r to t h e d i s c o n j u g a c y d o m a i n D there is a basis of solutions each of which has a n a l m o s t periodic l o g a r i t h m i c d e r i v a t i v e . F o r t h e b o u n d a r y of D t h e analysis is more corn- 1 This research was supported by the Office of Scientific Research of the United States Air Force.

7 - 563802. Acta mathematica. 96. Imprim6 le 22 octobre 1956.

100 L A W R E N C E M A R K U S A N D R I C H A R D A . M O O R E

p l i c a t e since h e r e we can d i s p l a y a n e x a m p l e of a n a p p r o p r i a t e e q u a t i o n h a v i n g no (non- t r i v i a l ) b o u n d e d s o l u t i o n f r o m t h e a n a l o g y w i t h H i l l ' s e q u a t i o n one m i g h t e x p e c t a n al- m o s t p e r i o d i c s o l u t i o n in such a case.

F i n a l l y we i n v e s t i g a t e t h e effect on D of p e r t u r b a t i o n s in t h e f u n c t i o n p (x).

2. T h e d i s c o n j u g a c y d o m a i n for e q u a t i o n s w i t h a l m o s t periodic coefficients

D E F I N I T I O N . The diseonjugacy domain D o/

(r(x)y')' + ( - a + b p ( x ) ) y = 0, (2) r (x) > 0 and p (x) being real continuous/unctions on - c~ < x < c~, is the subset o / t h e real (a, b)-pIane wherein the corresponding equations are discon~ugate.

W e shall p i c t u r e t h e b-axis h o r i z o n t a l l y a n d t h e a - a x i s v e r t i c a l l y . Of course, D d e p e n d s on t h e p a r t i c u l a r f u n c t i o n s r (x) > 0 a n d p (x), b u t D a l w a y s c o n t a i n s t h e h a l f - a x i s a >_ 0, b = 0. T h e set of p o i n t s (a, b) for which t h e c o r r e s p o n d i n g e q u a t i o n s (2) a r e o s c i l l a t o r y is t h e o s c i l l a t i o n d o m a i n 0 a n d , w h e n r (x) - 1, t h i s a l w a y s c o n t a i n s t h e h a l f - a x i s a < 0, b = 0.

W e shall be i n t e r e s t e d in t h e d i s c o n j u g a c y d o m a i n D, p a r t i c u l a r l y for r (x) - 1, a n d for v a r i o u s f u n c t i o n s p (x). Below are s e v e r a l e x a m p l e s i n d i c a t i n g s o m e of t h e p o s s i b i l i t i e s for t h e f o r m of D.

E x a m p l e 1. lim p ( x ) = + o % lim p ( x ) = - o % e.g. p ( x ) = x . H e r e D is t h e half-

x - ~ o o x--~ oo

a x i s a > _ 0 , b ~ 0 .

E x a m p l e 2. p (x) -~ O. H e r e D is t h e h a l f - p l a n e a _> 0.

E x a m p l e 3. p (x) u n b o u n d e d a b o v e (below). H e r e D c o n t a i n s no r a y in b < 0 (in b > 0).

H o w e v e r D can c o n t a i n a n i n t e r i o r , e.g., for p ( x ) = x sin (x2), D c o n t a i n s t h e p a r a b o l i c r e g i o n a >_ b~/ 4.

E x a m p l e 4. sup p (x) = M > 0, inf p (x) = m < 0. H e r e t h e l a r g e s t s e c t o r

- o 0 < x < o ~ - o r < x < ~

c o n t a i n e d in D is b o u n d e d b y t h e r a y s a = M b , a > 0 a n d a = rob, a ~ 0. If, say, M > 0 a n d m > 0, t h e n t h e l a r g e s t s e c t o r c o n t a i n e d in D i s b o u n d e d b y t h e r a y s a = M b >_ 0 a n d a = m b ~ O .

E x a m p l e 5. p ( x ) = sin log+ Ix ]. H e r e D is e x a c t l y t h e s e c t o r in w h i c h I bl _< a > 0.

I n s t u d y i n g t h e l i n e a r e q u a t i o n (1) one o f t e n utilizes t h e a s s o c i a t e d R i c c a t i e q u a t i o n

u' + u2/r (x) + K (x) = 0. (3)

F o r a s o l u t i o n y(x) of (1), n o n - v a n i s h i n g on some i n t e r v a l I , u ( x ) = y ' ( x ) r ( x ) / y ( x ) is a s o l u t i o n of (3) on I . Moreover, e v e r y sSlution of (3) can b e so o b t a i n e d . W e shall b e pri-

OSCILLATION A N D D I S C O N J U G A C Y F O R L I N E A R D I F F E R E N T I A L E Q U A T I O N S 1 0 1

m a r i l y i n t e r e s t e d i n t h e case w h e r e r(x) ~- 1 a n d l K ( x ) I < M2, for s o m e r e a l b o u n d M 2.

T h e n a s o l u t i o n u (x) of t h e R i c c a t i e q u a t i o n c a n b e d e f i n e d for all x if a n d o n l y if I u (x) [ < M e v e r y w h e r e .

Also, i n case r (x) - 1 a n d I K ( x ) I < M2, t h e b o u n d e d s o l u t i o n s of t h e R i c c a t i e q u a t i o n fill a closed b a n d w h i c h is e i t h e r e m p t y , a s i n g l e s o l u t i o n c u r v e , or a h o m e o m o r p h of 0 _< y <_ 1, - ~ < x < c~ i n t h e p l a n e . T h i s b a n d is n o n - e m p t y if a n d o n l y if t h e corre- s p o n d i n g l i n e a r e q u a t i o n (1) is d i s c o n j u g a t e . T h e b a n d is closed, t h a t is t h e e x t r e m a l u p p e r a n d l o w e r s o l u t i o n s a r e e a s i l y s e e n t o b e b o u n d e d since o t h e r w i s e e a c h n e a r b y s o l u t i o n u (x) w o u l d s o m e w h e r e s a t i s f y I u (x0) [ > M . B u t t h e n I u (x) l w o u l d g r o w m o r e r a p i d l y to i n f i n i t y t h e n a n u n b o u n d e d s o l u t i o n of u ' = - u 2 + M 2 a n d so u ( x ) w o u l d n o t e x i s t for all r e a l x.

L~M~aA 1. Let r(x) > 0 and K (x) be real almost periodic/unctions. I / t h e equation

(r(x)y')' + K ( x ) y = 0 (1)

is not disconjugate, then it is oscillatory at both +_ ~ .

Proo/. S u p p o s e a n o n - t r i v i a l s o l u t i o n y(x) of (1) v a n i s h e s a t t w o d i s t i n c t p o i n t s x = ~ a n d x =/~. F o r e a c h ~ > 0 t h e r e a r e a r b i t r a r i l y l a r g e ~ - a l m o s t periods, s a y Tn, of r (x) a n d K (x). C o n s i d e r t h e t r a r / s l a t e d e q u a t i o n s

(r(x +T~)y')' + K ( x + T ~ ) y = 0 (ln)

w i t h s o l u t i o n s Yn (x) w h i c h a s s u m e t h e s a m e i n i t i a l v a l u e s a t x = :~ as does y (x). Also t h e r e is a s o l u t i o n Yn(x) of (1) for w h i c h Y ~ ( x + r ~ ) = y~(x).

F o r e a c h ~ > 0 t h e r e exists a n e > 0 s u c h t h a t y~(x) v a n i s h e s o n fl - ~ < x < f i + $.

T h e n Y~(x) v a n i s h e s a t x = ~ + v ~ a n d also n e a r fl + r n . H e n c e e v e r y s o l u t i o n of (1) m u s t v a n i s h o n ~ + ~ n < - x _<fl +T~ + ~. S i n c e t h e t r a n s l a t i o n n u m b e r s rn a r e a r b i t r a r i l y l a r g e (or s m a l l ) , (1) is o s c i l l a t o r y . Q . E . D .

L E M M A 2. Let the real continuous /unctions Kn(x), r ~ ( x ) > 0 , n = l , 2 . . . on - c~ < x < co, converge uni/ormly on each compact interval to K (x) and r (x) > O, respec- tively. I / e a c h equation

(rn(x)y')' + K n ( x ) y = 0 (In)

is disconjugate, then so is

(r(x)y')' + K ( x ) y = 0. (1)

Proo/. S u p p o s e t h a t a ( n o n - t r i v i a l ) s o l u t i o n y ( x ) of (1) h a s t w o d i s t i n c t zeros, x 0 a n d x 1. C o n s i d e r t h e s o l u t i o n s yn(x) of (1,) w i t h i n i t i a l d a t a y , (x0) = 0, Y'n (xo) = Y' (x~).

L e t L be a c o m p a c t i n t e r v a l c o n t a i n i n g x 0 a n d x 1 i n its i n t e r i o r . F o r s u f f i c i e n t l y l a r g e n, [ r~ (x) - r ( x ) I, IK~(x) - K ( x ) l, a n d I Y~(X) - y(x) l a r e s m a l l e r t h a n a n y p r e s c r i b e d

102 L A W R E N C E M A R K U S A N D R I C H A R D A . M O O R E

e > 0 for x E L . T h e r e f o r e y~ (x) v a n i s h e s n e a r x 1. B u t t h i s c o n t r a d i c t s t h e h y p o t h e s i s t h a t (ln) is d i s c o n j u g a t e . T h u s (1) is n e c e s s a r i l y d i s c o n j u g a t e . Q . E . D .

L E M M A 3. L e t r~(x) > 0, K~(x) be real c o n t i n u o u s / u n c t i o n s on - c ~ < x < c ~ and such that

(r~(x)y')' + K ~ ( x ) y = 0 (i = 1, 2) (1~) are non-oscillatory at x = + oo. T h e n [or each t on 0 < t < 1 the equation

[ ( t r l ( x ) § (1 - t)r~(x))y']' + [ t K l ( x ) + (1 - t ) K 2 ( x ) ] y = 0 (lt) is also non-oscillatory at x - + c<).

Proo/. L e t ui (x) = r i (x) y; ( x ) / y i (x), w h e r e y~ (x) is a s o l u t i o n of (li), p o s i t i v e for x > x 0.

T h e n

u~ + u~/r~

(x) + K~(x) = 0 (i = 1, 2) (3~) for x > x 0. C o n s i d e r u t ( x ) = t u l ( x ) + (1 - t ) u~(x). T h e n we c o m p u t e

- t(1 - t) (r2u I - rlu2) 2 "

u~ + u ~ / [ t r 1 + (1 - t)r2] + [ t K 1 § (1 - t)K2]

r l r 2 [tr 1 + (1 - t)re]

T h u s t h e r e is a s o l u t i o n , c o n t i n u o u s for x > x , , of

u' + u 2 / [ t r l + ( i - t)r2] + [ t K 1 + (1 - t ) g 2 ] = O.

T h e r e f o r e (It) is n o n - o s c i l l a t o r y a t x = + c~. Q . E . D .

O n e is o f t e n i n t e r e s t e d i n t h e d i s c o n j u g a c y d o m a i n of t h e t r a n s l a t e s of e q u a t i o n (1) or e v e n of l i m i t t r a n s l a t e s . I f K ( x ) is a l m o s t p e r i o d i c a n d x~ a r e real n u m b e r s s u c h t h a t l i m K ( x + x ~ ) = K * ( x ) u n i f o r m l y o n - c ~ < x < c ~ , t h e n o n e s t a t e s t h a t K * ( x ) is i n

n-§

t h e h u l l H { K ( x ) } g e n e r a t e d b y K ( x ) . I t is k n o w n t h a t K * (x) is a l m o s t p e r i o d i c a n d K ( x ) C H { K * ( x ) } , cf. [5, p. 73].

T H E O a E M 1. L e t r ( x ) > 0 and p ( x ) be real almost p e r i o d i c / u n c t i o n s . T h e n the dis- conjugaey d o m a i n D o[

( r ( x ) y ' ) ' + ( - - a + b p ( x ) ) y = 0 (2) is a closed convex subset o[ the (a,b)-plane. F u r t h e r m o r e , i [ / o r real translates Xn, n - 1, 2, . . . . l i m r (x + x~) - r* (x) > 0 and l i m p (x + x~) = p* (x) u n i / o r m l y on - oo < x < oo, then

n - ~ - o o n - - > r

D* = D, where D* is the discon]ugacy d o m a i n o/

(r* ( x ) y ' ) ' + ( - a + bp* ( x ) ) y = 0. (2*) Proo/. B y t h e a b o v e t h r e e l e m m a s , D is closed, c o n v e x a n d i t s c o m p l e m e n t is t h e o s c i l l a t i o n d o m a i n O. T o s h o w D* = D we n e e d o n l y s h o w t h a t D ~ D* a n d t h e n t h e c o n c l u s i o n follows f r o m s y m m e t r y .

O S C I L L A T I O N A N D D I S C O N J U G A C u F O R LINEAI~ D I F F E R E N T I A L E Q U A T I O N S I 0 3

For a fixed (a,b) E D each solution of (2) and of

(r(x § Xn)y')' § ( - a § bp(x § xn))y

= 0 (2,) is d i s c o n j u g a t e . S u p p o s e t h e r e is a n o n - t r i v i a l o s c i l l a t o r y s o l u t i o n y* (x) of (2*). T h e n , for s u f f i c i e n t l y large i n t e g e r s n, t h e s o l u t i o n of (2~) h a v i n g t h e s a m e i n i t i a l v a l u e s as

y* (x)

m u s t h a v e m o r e t h a n one zero a n d t h u s be o s c i l l a t o r y . B u t t h i s c o n t r a d i c t s t h e d i s c o n j u - g a c y of (2~). T h e r e f o r e

D c D*.

Q . E . D .

COROLLARY.

The oscillation domain 0 o/

(2)

is open, connected and its complement in the (a, b)-plane is D.

Proo/.

Since D a n d 0 are c o m p l e m e n t a r y , O is open. I f

(ao, bo)E O,

t h e n , b y t h e S t u r m - P i c o n e c o m p a r i s o n t h e o r e m , so is (a 0 - $, b0) E O for e~ch ~ > 0. Moreover, 0 con- r a i n s t h e s e c t o r a < - I b ] sup p(x). T h e r e f o r e 0 is c o n n e c t e d . Q . E . D .

W e n e x t p r o c e e d t o a d e t a i l e d s t u d y of t h e f o r m of D. To s i m p l i f y t h e a n a l y s i s we t r e a t o n l y t h e case r(x) ~ 1 a n d we often m a k e t h e c o n v e n t i o n t h a t p ( x ) h a s m e a n zero.

F o r t h e e q u a t i o n (1) L e i g h t o n [9] gives t h e f o l l o w i n g c r i t e r i o n for oscillation:

dx/r(x)

= o o a n d

f K ( x ) d x =oo.

Xo x o

F r o m t h i s , if r (x) - 1 a n d p (x) is r e a l a l m o s t p e r i o d i c w i t h zero m e a n , i t follows t h a t D lies in t h e h a l f - p l a n e a > O, for t h e l i n e a r e q u a t i o n

y" § ( - a + b p ( x ) ) y - O . (L)

B y a r e f i n e m e n t of L e i g h t o n ' s t e s t one c a n show t h a t D, e x c e p t i n g t h e o r i g i n , lies in a > 0.

T H E O R E M 2.

Let p(x) ~ 0 be a real almost periodic/unction o/ mean zero. Then/or y" + ( - a + b p ( x ) ) y

= 0 , (L)

D, excepting the origin, lies in a > 0

Proo/.

W e n e e d o n l y show t h a t for a = 0, b ~: 0 t h e c o r r e s p o n d i n g e q u a t i o n (L) is o s c i l l a t o r y . F o r t h e first case a s s u m e

f p(s)ds

is u n b o u n d e d . T h e n one can r e p l a c e p ( x )

0

b y a f u n c t i o n in

H { p ( x ) ) ,

which we shall still d e n o t e as

p(x),

for which e i t h e r

g~ x

l i m sup

b f p(s)ds

= c~ or lim s u p

b f p (s) ds

= ~ , [4, p. 48]. T h e n f r o m a k n o w n r e s u l t

X ~ 0 X - ~ - - ~ 0

[ l l , p. 138], (L) is o s c i l l a t o r y .

x

F o r t h e s e c o n d case a s s u m e t h a t

f p(s)ds

is a l m o s t periodic. L e t

v(x)

be t h e a l m o s t

0

1 0 4 L A W R E N C E M A R K U S A N D R I C H A R D A . M O O R E

periodic f u n c t i o n with m e a n zero a n d such t h a t v' (x) - p ( x ) . T h e n there exists a n u m b e r

9J

x 0 for which v (x0) = 0 a n d v(x) = f p ( s ) d s . Suppose t h a t t h e R i c c a t i e q u a t i o n

2"11

u' + u 2 + b p ( x ) - O , b~=O (R)

has a b o u n d e d s o l u t i o n . N o w if for e v e r y p * ( x ) 6 H { p ( x ) } , t h e c o r r e s p o n d i n g R i c c a t i e q u a t i o n (R*) h a d a u n i q u e b o u n d e d solution, t h e n it is easy to show (cf. T h e o r e m 9 a n d r e m a r k after T h e o r e m 16) t h a t this solution u ( x ) is a l m o s t periodic. B u t this c o n t r a d i c t s t h e fact t h a t t h e m e a n of u(x) 2 is clearly zero. T h u s we replace p(x) b y some f u n c t i o n i n H p { (x)}, still called p ( x ) , for which there are two b o u n d e d solutions, u 1 (x) a n d u 2 ( x ).

X 3:

N o w ui (x) = u, (Xo) - f ui (s)2ds - b f p (s)ds.

~t'o 2"0

x

L e t lim [ui(x~) - f u~(s)2ds] = w~

2"--> -[- oo 2"0

X

a n d lim [u~(x0) - fu~(s)2ds] = ~i, i = 1, 2.

2 " ~ - - ~r :i,. o

I t c a n be shown, cf. T h e o r e m 14, t h a t l i m i n f l u l ( x ) - u 2(x) l = 0 a n d so ~ 1 = ~ 2 = : r

~o 1 = (0 2 - co. E i t h e r ~ > 0 or eo < 0. S a y to < 0 a n d t h e other case is similar. T h e n u~ (x) <

x

0)/2 - b f p (s) d s = z (x) for large x > k. B u t z(x) is a n a l m o s t periodic f u n c t i o n with n e g a t i v e

2"0

m e a n a n d so 0 < exp f u~(s)ds < exp f z ( s ) d s < K , for some b o u n d K , w h e n x > k. There-

k k

fore t h e linear e q u a t i o n (L) has a basis of b o u n d e d solutions on a half-axis a n d t h u s (L) is oscillatory. B u t this c o n t r a d i c t s t h e a s s u m p t i o n t h a t (R) has a b o u n d e d solution a n d so (L) is oscillatory. Q . E . D .

T H E 0 R E M 3. T h e disconjugacy d o m a i n D o/

y " + ( - a + b p ( x ) ) y - 0, (L)

where p (x) I 0 is real almost periodic w i t h m e a n zero, contains the sector bounded by the rays a = b sup p (x) > 0 a n d a = b i n f p (x) > 0. T h i s is the largest sector belonging to D i n

- - o r - - o r ~

that no other rays are in D. T h e boundary o / D is a continuous curve a(b) which is strictly monotone decreasing on b < 0 a n d increasing on b ~ O.

O S C I L L A T I O N A N D D I S C O N J U G A C Y F O R L I N E A R D I F F E R E N T I A L E Q U A T I O N S 105 Proo/. Consider a r a y a = kb > 0. T h e n for such p a r a m e t e r values (L) becomes

y " + b ( - k + p ( x ) ) y = O. (Lk) T h u s i n b > 0 t h e r a y s for which - k + sup p (x) < 0 yield d i s c o n j u g a t e e q u a t i o n s . I n b < 0, t h e rays which lie i n D correspond to - k § inf p (x) _> 0. F o r a n y other ray, i n a > 0 , b ( - k § becomes positive for x o n some i n t e r v a l . T h u s for large [b[, t h e corresponding e q u a t i o n is oscillatory a n d such r a y s do n o t belong to D.

T h a t t h e b o u n d a r y of D is a m o n o t o n e c o n t i n u o u s curve a(b) follows from t h e fact t h a t D is convex a n d c o n t a i n s a sector. Q . E . D .

COROLLARY. a ( b ) ~ M b /or b--> + c~2; a(b),,~ mb /or b - + - cx3.

If p (x) assumes its s u p r e m u m M or its i n f i m u m m a t some p o i n t , t h e n one can f u r t h e r e s t i m a t e a (b).

THEOREM 4. Let p(x) ~:0 be a real almost periodic /unction o/ mean zero and p ( x ) E C (2). I / p(Xo) = M = sup p(x) (or i/ p(x~) = m = inf p(x)) then the domain

- 0 r < x < o r - r162 < x < o o

D /or the equation (L) has a boundary

a ( b ) = M b - O ( b ~) for b > 0 (or a ( b ) = m b - O ( b 8 9 for b < 0 ) .

Proo/. L e t p(xo) = M . T h e n for each h < p " (x0) t h e r e is a n e > 0 such t h a t p(x) > M

§ 89 (x - x0) 2 on I x - x 01 < e. T a k e e small a n d define k b y 88 (M - k ) / h = - e2. Consider t h e r a y s a = kb > 0 for k j u s t tess t h a n M. A l o n g such rays we h a v e

y " + b ( - k + p ( x ) ) y = O , b > 0 . (L~) F o r these e q u a t i o n s , w h e n e v e r Ix - x 01 <- ~

b ( - k + p(x)) > b ( - k + M + ~ (X - X o ) 2) >7~b8 (M - k ).

N o w if (7 b / 8 ) ( M - k) > - hzr2/(M - k) on a n i n t e r v a l of l e n g t h [ - ( M - k)/h] "~, our e q u a t i o n (L) is oscillatory. T h u s t h e c u r v e

7b ( M - k) = - h z ~ / ( M - k), a = kb

lies i n t h e oscillatory region of (L). T h u s a (b) lies a b o v e M b - 2zeV - 2 h b / 7 for sufficiently large b > 0. A similar c a l c u l a t i o n holds for b < 0. Q.E.D.

I f one assumes t h a t p (x) has a higher order of " f l a t n e s s " a t its m a x i m u m t h e n still s h a r p e r a s y m p t o t i c e s t i m a t e s for a(b) are possible. W e i n d i c a t e t h e results o n l y i n t h e e x t r e m e case where p(x) = M (or p(x) = m) on a n i n t e r v a l

on O < x < o o in case w ( x ) =

0

our a b o v e e q u a t i o n

1 0 6 L A W R E N C E M A R K U S A N D R I C H A R D A . M O O R E

THEOREM 5. Let p(x) 2z 0 be real almost periodic o / m e a n zero. Assume p(x) = M = sup p ( x ) ( o r p ( x ) = m = inf p ( x ) ) / o r x o n a n i n t e r v a l . T h e n , / o r ( L ) , a ( b ) = M b - O ( 1 )

oo < x < ~ - o o < x < ~

/or b > 0 (or a(b) - m b - 0 ( 1 ) / o r b < 0). Thus the boundary curve a(b) is asymptotic to a line o/slope M (or slope m).

Pro@ A s s u m e p ( x ) - M on IX-Xo]<_s. On t h e r a y a =kb, b > 0 , we h a v e

y" + b ( - k + p ( x ) ) y - 0 . (Lk)

B u t b ( - k + p (x)) = b ( - k + 3:1) on Ix - x 01 < e. If b ( - k + M ) k ~2/4c2, (L) is o s c i l l a t o r y . T h u s for a = M b - J r 2 / 4 e 2, large b > 0 , we h a v e oscillation. B e c a u s e of its c o n v e x i t y , a(b) is a s y m p t o t i c t o a line of slope M b e t w e e n a - M b a n d a = M b - ~ 2 / 4 s 2. The case for p(x) = m is similar. Q . E . D .

W e n o w show t h a t , in m o s t i m p o r t a n t cases, t h e b o u n d a r y of D is t a n g e n t t o t h e b-axis a t t h e origin. T h e r e b y D c o n t a i n s t h e m a x i m a l s e c t o r p r o p e r l y in its interior.

T H E O R E M 6. Let p(x) be real almost periodic and f p(s)ds also almost periodic. Then

0

the domain D o/equation (L) has a boundary a(b) which is tangent to the b-axis at the origin.

Pro@ L e t y =exp(~ax)z, where y(x) is a s o l u t i o n of

y " + ( - a + b p ( x ) ) y = O , a > 0 . (L) T h e n one c o m p u t e s

(exp (2 ~'ax) z')' + (exp (2 Vax) bp(x))z = O.

N o w a n o n - o s c i l l a t i o n t e s t of Moore [12] s t a t e s t h a t e q u a t i o n (1) is n o n - o s c i l l a t o r y K(t) dt has a n oscillation < 1 on O < x < o o . F o r

t

x

bf

w(x) = 2 ~ a p(t)dt.

0

L e t w = oscillation f p(t)dt. T h e n t h e p a r a b o l a ( b / 2 ~ / a ) w - 1 or a = (w2/l)b 2 lies in D.

0

T h e r e f o r e a'(b) exists a t b = 0 a n d t h e r e e q u a l s zero. Q . E . D .

COROLLARY. I n case f p(s) is almost periodic, the domain D o/ (L) contains the

0

maximal sector (except /or the origin) in its interior. Also on the boundary curve a(b) o/ D, ( - a + bp(x)) is somewhere positive.

O S C I L L A T I O N A N D D I s c o N J U G A C Y :FOR L I : N E A R DIF:FEtCESITIAL E Q U A T I O : N S 1 0 7

Proo/. S i n c e a ' (0) = 0, t h e c o n v e x d o m a i n D c o n t a i n s t h e r a y s a = m b , a = M b, for b > 0, i n its i n t e r i o r . T h u s D c o n t a i n s t h e r a y s a = M b - ~, a = r o b - s for s o m e s > 0 w h e n I bl is large. S i n c e ( - a + b p ( x ) ) is a r b i t r a r i l y n e a r to zero a t p o i n t s of t h e e x t r e m a l r a y s , ( - a + b p ( x ) ) b e c o m e s p o s i t i v e i n D. Q . E . D .

F i n a l l y we s h o w t h a t i n m a n y i m p o r t a n t cases D is s y m m e t r i c i n t h e a-axis.

T H E O R E M 7. Let p ( x ) be real almost periodic with F o u r i e r /requencies {~n}. I / the { ~ } are rationally independent, then /or the b o u n d a r y curve a(b) o/ D /or the equation (L), we have the s y m m e t r y

a (b) = a ( - b).

Proo/. L e t p ( x ) h a v e F o u r i e r series ~ r162 ix=x w h e r e ~n = $ - ~ . W i t h i n t h e h u l l n = ar

H { p (x)}, c o n v e r g e n c e of t h e F o u r i e r coefficients of t r a n s l a t e s p (x + h~) i m p l i e s u n i f o r m c o n v e r g e n c e o n - c~ < x < c~. N o w o n e c a n select a t r a n s l a t e h~ so t h a t I ~t~h= - ,n I < (89 ( m o d 2 Jr) for n = 1,2 . . . m. T h i s is p o s s i b l e s i n c e t h e f r e q u e n c i e s a r e l i n e a r l y i n d e p e n d e n t o v e r t h e r a t i o n a l s . B u t t h e n we c a n d e f i n e a s e q u e n c e of t r a n s l a t e s p (x + h~) w h o s e F o u r i e r coefficients c o n v e r g e t o t h o s e of - p ( x ) . T h u s - p ( x ) E H { p ( x ) } a n d D is s y m m e t r i c i n t h e a-axis. Q . E . D .

3. Interior of the disconjugacy domain

W e s h a l l n o w c o n s i d e r t h e d i f f e r e n t i a l e q u a t i o n s

y " + ( - a + b p ( x ) ) y - O (L)

for (a, b) i n t h e i n t e r i o r of t h e d i s c o n j u g a c y d o m a i n D. T h e n t h e a s s o c i a t e d R i c c a t i e q u a t i o n

u ' + u 2 + ( - a + b p ( x ) ) - 0 (R)

h a s s o l u t i o n s w h i c h a r e d e f i n e d a n d b o u n d e d o n - c~ < x < c~.

T H E O R E M 8. L e t

u ' + u 2 + K (x) + ~ - 0 (R~)

have bounded solutions on - c~ ~ x <: c~ /or all s m a l l ~, e > ~ ~ O, where K (x) is a real con- t i n u o u s bounded / u n c t i o n on - c~ < x < c~, I K (x) I < M2" T h e n /or each such e there are i n / i n i t e l y m a n y bounded solutions o/ R~ and these /ill a closed bounded d o m a i n Be i n the ( x , u ) - p l a n e . A l s o B~.~ lies interior to B~, i/ Q < ~2. T h e u p p e r and lower bounded solutions uu (x) and uL (x), respectively, o / R 0 are separated, that is,

e 2

i n f I uu(x) - uL (x) I >.~V(522~i ~ + 8ei > O.

1 0 8 L A ~ , V R E : N C E M A R K U S A N D R I C H A R D A . M O O R E

Furthermore/or any two bounded solutions u 1 (x), u 2 (x) o / R o which are not both the extreme solutions, we have

lim [ u l ( x ) - u 2 ( x ) [ = 0 as x---> + c ~ or as x - - - ~ - c ~ .

Proo/. I n t h e (x, u ) - p l a n e each R i c c a t i e q u a t i o n RE defines a slope field. I f ~ > el > 0 t h e n t h e slope of t h e s o l u t i o n c u r v e of RE. is m o r e n e g a t i v e t h a n t h a t of t h e s o l u t i o n c u r v e of R ~ , t h r o u g h e a c h p o i n t (x,u). I f Re, h a d j u s t one b o u n d e d s o l u t i o n on - ~ < x < c~, t h e n i t is e a s i l y seen t h a t Re2 w o u l d h a v e no such b o u n d e d solutions. T h u s e a c h RE has a b a n d BE of b o u n d e d solutions. E a c h b a n d Be is closed ( b y t h e a r g u m e n t j u s t p r e c e d i n g L e m m a 1 of S e c t i o n 2) a n d so h a s a n u p p e r a n d a lower s o l u t i o n c u r v e for t h e c o r r e s p o n d - ing e q u a t i o n Re.

I f e~ > Q > 0, t h e n each s o l u t i o n curve of RE~ w h i c h i n t e r s e c t s e i t h e r e x t r e m a l s o l u t i o n of Rel m u s t b e c o m e u n b o u n d e d . T h u s B~ 2 lies i n t e r i o r t o BE,.

L e t ue(x) be t h e s o l u t i o n of Re, ~ > 0 , t h r o u g h (x0, uu(xo) ) for s o m e x 0. T h e n on x > xo uu(x) > ue(x) a n d , since ue(x) lies a b o v e t h e lower edge of BE, us(x) > UL(X). N o w

U u ( X ) - - u e ( x ) = f [--u~(t) 2 § ~ §

XO

a n d u~(x) - u E ( x ) <_ ( 2 M 2 § for x _ > x o.

B u t t h e n u~(x) - ue(x) = e(x - xo) §

z

w h e r e Q ( x ) = J [ u E ( t ) - u~(t)] [ u e ( t ) § u~(t)]dt.

xo

x

H e n c e I Q (x) l < (2 M 2 + ~) 2 M f (t - xo) d t

~0

o~ IQ

(~)1 <-

M ( 2 M ~ + ~) (x - xo) ~.

A n e l e m e n t a r y c a l c u l a t i o n shows t h a t

u~(xo + h ) - u e ( x o + h) >_~ > 0 ,

where h - 2 M ( 2 M ~ § ) a n d ~ = h v - M ( 2 M 2 § 2.

T h u s u ~ ( x 0 § 0 § B u t x o is a r b i t r a r y a n d so u ~ ( x ) - u L ( x ) > ~ ] on

- c~ < x < c~. I n t h e s t a t e m e n t of t h e t h e o r e m we t a k e e = e / 2 .

L e t u l(x) a n d u s(x) be t w o b o u n d e d s o l u t i o n s of R 0 a n d s a y t h a t u l(x) < u 2(x) a n d t h a t u 1 (x) is n o t t h e lowest b o u n d e d s o l u t i o n of R0 (the o t h e r cases are similar). W e shall

O S C I L L A T I O N A b I D D I S C O I ~ J U G A C Y F O R L I N E A P ~ D I F F E R E N T I A L E Q U A T I O N S 1 0 9

show t h a t

lira l Uu (x) - u 1 (x)l = O.

L e t u ~ (x) be a s o l u t i o n of R o l y i n g below t h e b a n d of b o u n d e d solutions. T h e n n , (x) - u ~ (x) ~ u ~ (x) - u~ (x)

~1 (x) - uL (x) u ~ (x) - uL ( x ) '

for a c o n s t a n t ~t. l~ow a t x = 0, choose u ~ (0) so n e a r to UL(O ) t h a t I)~1 is d e t e r m i n e d s m a l l e r t h a n a p r e s c r i b e d p o s i t i v e n u m b e r ~. N o w for each n u m b e r - N 2 t h e r e is a n abscissa ~ such t h a t u ~ ( x ) < - . u w h e r e v e r u ~ (x) is d e f i n e d for x > 2 . T h u s one call choose N 2 so large t h a t , for x > 2 , [u~ (x) - u ~ ( x ) ] / [ u ~ (x) - uL(x)] is a r b i t r a r i l y n e a r + 1.

B u t t h i s m e a n s t h a t t h e r a t i o I [ u ~ ( x ) - u ~ ( x ) ] / [ u l ( x ) - u L ( x ) ] t b e c o m e s s m a l l e r t h a n $.

T h u s lira inf. ] u u (x) - u 1 (x) I = 0.

x--> oo

W e c o m p l e t e t h e proof b y showing t h a t t h e r a t i o [u~r ( x ) - u~(x)]/'[u~r ( x ) - uL(x)] is m o n o t o n e l y decreasing. B u t t h e d e r i v a t i v e of t h i s r a t i o is e a s i l y f o u n d to be

[u~ (x) - u ~ (x)] [u~ (x) - u~ (x)]/[u~r (x) - u~ (x)] < 0.

T h e r e f o r e t h e r a t i o [u~ (x) - u 1 (x)]/[u 1 (x) - uL (x)] decreases m o n o t o n e l y to zero as x ~ ~ . T h u s ]im l u ~ ( x ) - u ~ ( x ) l = O . Q . E . D .

x-->~

W e can n o w p r o v e a n i m p o r t a n t r e s u l t c o n c e r n i n g a l m o s t p e r i o d i c s o l u t i o n s for t h e R i c c a t i e q u a t i o n (R) i n t e r i o r to t h e d o m a i n of d i s c o n j u g a c y . W e follow t h e m e t h o d i n t r o - d u c e d b y F a v a r d [5, Ch. 3]. S i m i l a r r e s u l t s are o b t a i n e d in [1] a n d [7].

T H E O R E M 9. L e t Ul(X ) a n d u ~ ( x ) be bounded solutions o/

u ' + u ~ + K (x) ~ O, ( R )

where K (x) is a real almost periodic /unction. I / u 1 (x) - u2(x ) > 5 > 0 on - er < x ~- cx~, then u 1 (x) and u 2 (x) are almost periodic a n d the modules o/ their /requencies are contained i n the module o/ K (x).

Proo/. Since u , ( x ) - u2(x ) > 5 > O, u l ( x ) is t h e u p p e r b o u n d e d s o l u t i o n a n d u2(x ) is t h e lower b o u n d e d s o l u t i o n of (R). F u r t h e r we m a k e t h e n o t a t i o n a l s i m p l i f i c a t i o n

_

Consider a sequence of real n u m b e r s {h~} a n d c o r r e s p o n d i n g t r a n s l a t e s of u l(x), t h a t is, u l ( x + h~). W e shall show t h a t , for s o m e s u b s e q u e n c e , u l ( x + h~) converges u n i f o r m l y on - cx~ < x < oo. This shows t h a t u 1 (x) is a l m o s t p e r i o d i c a n d a s i m i l a r a r g u m e n t w o u l d h o l d for u~(x).

Ii0 L A W R E N C E M A I % K U S A N D R I C H A R D A . M O O R E

Since t h e n u m b e r s u l ( 0 + h ~ ) a n d u2(0 + h~) a r e b o u n d e d sets, e x t r a c t a subse- q u e n e e (again called h.) for which

ul(O+h~)--~r162 u2(O +h,~)-+fl.

Also one can r e q u i r e

K (x + h~)+K* (x)

u n i f o r m l y on - oo < x < oo.

L e t u~' (x) a n d u~ (x) b e t h e s o l u t i o n s of

u ' + u 2 + K * (x) = 0 (R*)

w i t h i n i t i a l c o n d i t i o n s u* (0) = ~r u~'(0) = ft. Then, b y t h e c o n t i n u o u s d e p e n d e n c e of t h e s o l u t i o n s of a d i f f e r e n t i a l e q u a t i o n u p o n t h e coefficients, lim

ul(x + h,~)= u~(x)

a n d

it-~cla

lira u 2(x ~ ]z~) = u~ (x) w h e r e t h e c o n v e r g e n c e is u n i f o r m on each c o m p a c t i n t e r v a l . N o w inf ]u~(x + h~) - u 2(x + h,) I = 6 for each h,. I f

lug(.0)

- u ;

(x0) I <

6 a t s o m e p o i n t x0, t h e n for large n,

lul(xo+h~) u2(xo+h~)l<6,

which is false. T h u s inf lug(x) - u~(x) l > 5 > O. T h e r e f o r e u~(x) is t h e u p p e r b o u n d e d solution of (R*) a n d u~ (x) is t h e lower b o u n d e d s o l u t i o n of (R*).

W e n e e d t o show t h a t a s u b s e q u e n c e of u~ (x + h=) is C a u c h y in t h e m e t r i c space of real, b o u n d e d , c o n t i n u o u s f u n c t i o n s CB( -- c~, oo). S u p p o s e t h e c o n t r a r y . T h e n t h e r e e x i s t s

> 0 such t h a t : for each N 1 t h e r e a r e i n t e g e r s n 1 > N1, m I > ~/*1 a n d some x 1 a t w h i c h l ui (x 1 + h~,) - u l ( x I + hm,)[ > s. Choose a s e q u e n c e Nk-->oo a n d c o r r e s p o n d i n g i n t e g e r s

nk, mk > N k

a n d n u m b e r s x k a t w h i c h

lul (xl + h%) Ul (Xl § hmk) I > s.

E x t r a c t a s u b s e q u e n c e k/ (again called k) for which u 1 (x k +h=z)-->x a n d also u l (xk +h~k)-->~4=~. B u t consider t h e t r a n s l a t e s u 1 (x 4, x k 4. h%) a n d u 1 (x 4. x k 4. h~k ) a n d a g a i n e x t r a c t a s u b s e q u e n c e ki (again called k) so t h a t

K ( x 4-x k 4, h%)--> K* (x)

a n d

K ( x '

~- x k 4"hmk ) - ~ K (x). T h e n t h e u p p e r b o u n d e d s o l u t i o n s of t h e c o r r e s p o n d i n g e q u a t i o n s

~*

(/)*) a n d (J~*) a r e ~ ' (x) a n d d~ (x), r e s p e c t i v e l y , w i t h ~" (0) = h a n d .~' (0) = ~.

B u t

IK((x 4. xk) -+ h%) - K ( ( x 4. xk) + hmz) l<r]

for a n y p r e s c r i b e d ~1 > 0 a n d suf- f i c i e n t l y large k. T h e r e f o r e K* (x) = K* (x) a n d (/~*) is t h e s a m e as (R*). B u t t h e n ~* (x) = -~* (x) a n d ~ = ~ which is a c o n t r a d i c t i o n .

T h e r e f o r e u~ (x 4. h % ) ~ u~' (x) u n i f o r m l y on - co < x < co a n d u~ (x) is a l m o s t periodic:

F u r t h e r m o r e for each s e q u e n c e [h~} wi~h

K ( x + h ~ ) ~ K * ( x )

t h e r e is a s u b s e q u e n c e h~

such t h a t u 1 (x-4

hn,)-->u*~ (x)

w i t h u n i f o r m c o n v e r g e n c e on co < x < co. Therefore we a c t u a l l y m u s t h a v e

u~(x 4- h~)~U*l(X)

u n i f o r m l y on - c ~ < x < co. T h u s t h e m o d u l e of f r e q u e n c i e s of Ul(X ) is c o n t a i n e d in t h a t of

K(x).

Q . E . D .

U s i n g T h e o r e m 8, one can e a s i l y see t h a t t h e r e are no o t h e r a l m o s t p e r i o d i c s o l u t i o n s t h a n u~(x) a n d u 2(x).

O S C I L L A T I O N A~ID D I S C O ~ J U G A C Y FOI~ L I N E A R , D I F F E R E N T I A L E Q U A T I O N S 11 1

We now relate our results directly to t h e linear differential e q u a t i o n (L) a n d describe a distinguished solution basis for this equation.

T ~ E O ~ E M 10. Let

y" + ( - a + b p ( x ) ) y = O, (L)

with p (x) real almost periodic, belong to the interior o/the disconjugacy domain D. Then there is a solution basis o/ the /orm

x

yu(x) = e ~x exp f ~ u ( t ) d t

0

y L ( x ) - e - ~ e x p f c L ( t ) d t ,

o

where ~ > 0 and ~u (x), ~L (x) are almost periodic /unctions o/ mean zero. Also 2 ~ is mean o/ the width o/the band o~ bounded solutions o/ the associated Riccati equation and i" [~u (t) +

0

~L (t)] dt is almost periodic. Thus Yu (x) Yn (x) and its reciprocal are almost periodic.

Proo/. L e t uu(x) a n d uL(x) be the u p p e r ~nd lower almost periodic solutions of t h e Ricc~ti equation. T h e n define

X

y u ( x ) = e x p f u u ( t ) d t ~nd y L ( x ) = e x p f u L ( t ) d t .

0 0

These are clearly linearly i n d e p e n d e n t a n d thus form the required basis.

N o w uu(x) = ~ + ~ ( x ) , uL(x ) - :XL + eL(X) where (~u(X), ~L(x) are almost periodic with m e a n zero. T h e width of t h e b a n d B of b o u n d e d solutions of t h e l~iccati e q u a t i o n is

A ( x ) - u ~ ( x ) - u L ( x ) W

y ~ (x) yL (x)'

where the W r o n s k i a n W = y'~ (x)yL(x) yL (x) y~(x) is ~ non-zero constant. Therefore r

0 < c < y~(x)yL(x) < C for bounds c, C a n d for all x.

However, yu(x)yL(x) = exp {(:% + ~L)x + f [ r + r Since y , ( x ) y L ( x ) is

0

b o u n d e d as indicated, gu = - ~ L =Cr a n d t h e ~lmost periodic function A(x) has ~ m e a n of 2 ~ - 0 . Thus yu(x)yL(x) a n d its reciprocal are ~lmost periodic. B u t this means t h a t

i [ r + r is b o u n d e d a n d thus almost periodic. Q.E.D.

O

112 L A W R E : N C E M A R K U S A N D R I C H A R D A . M O O R E

C O R O L L A R Y 1. y~(X) (or yL(x)) is the unique (up to a constant factor) solution of (L) bounded on a negative (or positive) half-axis. Also i/ the product o/ two solutions o/ (L) is bounded, then the factors are (up to constant multiples) y~(x) a m d yL(x).

f 3"

I f c~(t)dt a n d f c L ( t ) d t a r e b o u n d e d , t h e n t h e y a r e e a c h a l m o s t p e r i o d i c . I n t h i s

0 0

e a s e t h e c a n o n i c a l s o l u t i o n basis h a s t h e f o r m

y~(x) = ~ F ~ ( x ) e ~x, yL(x) =uf~L(x)e ~x,

w h e r e tF~,(x) a n d ~ L ( X ) a r e a l m o s t p e r i o d i c f u n c t i o n s . M o r e o v e r o n e s h o w s e a s i l y t h a t

x 5r

b o t h t h e i n t e g r a l s f ~ ( t ) d t , f ~L(t)dt a r e b o u n d e d if o n e of t h e m is b o u n d e d .

0 0

2$

C O R O L L A R Y 2. There is a solution basis /or ( i ) of the form r e x p f dt/r 2, 0

r c x p f-dt//q~(t) 2, where r (x) is almost periodic and O < c < q ~ ( x ) < C .

~)

Proof. D e f i n e z(x) b y y ( x ) = t y~(x)yc(x)z = r w h e r e y(x) is a s o l u t i o n of (L.) T h e n w e c o m p u t e

(r ( x ) ~ ' ) ' - 4 ~ ( ~ ) ~ ~ = 0, W

9 t

w h e r e we c a n t a k e t h e W r o n s k i a n W = YuYL -YLY~ = 2. B u t t h i s e q u a t i o n c a n be s o l v e d e x p l i c i t l y t o y i e l d s o l u t i o n s e x p i -+ dt/cP (t)2" Q . E . D .

0

W e n e x t e x p l i c i t l y list c e r t a i n d a t a o n t h e m e a n s of t h e a l m o s t p e r i o d i c f u n c t i o n s d e s c r i b e d a b o v e .

C O R O L L A R Y 3 . m e a n [u,, (x) - u L (x)] = 2 m e a n u u (x) 2 = m e a n u L (x) 2 m e a n r (x) 2 = m e a n e L (x)~

m e a n r ( x ) - 2 = ~.

I[ mean p (x) = O, then ~2 + mean r (x) 2 = a and thus [mean d?~ (x)] 2 ~ a - ~ z , i = L o r u .

Proof. I n t h e t h e o r e m w e s h o w e d t h a t m e a n [u~ (x) - u L ( x ) ] = m e a n A (x) - 2 ~. N o w m e a n u~ (x) 2 = m e a n [~2 + ~ r (x) + r (x) 2] a n d m e a n u z (x) 2 - m e a n [:r + ~ r (x) + e L (x)2] 9 T h e n since m e a n Cu (x) - m e a n e L (x) = 0, m e a n [u u (x) 2 - uL (x) 2] = m e a n [r (x) e - e L (x)2] 9

t !

B u t u~(x) 2 - uL(x) 2 = u i . ( x ) - uu(x) w h i c h h a s a z e r o m e a n . N o w r - 2 - A ( x ) / W - A ( x ) / 2 a n d t h i s h a s a m e a n of ~.

O S C I L L A T I O N A N D D I S O O N J U G A C Y F O R L I N E A R D I ] ~ F E R E N T I A L E Q U A T I O N S 1 1 3

F i n a l l y f r o m t h e R i c e a t i e q u a t i o n

r § + r + b p ( x ) ) = 0 .

T a k i n g m e a n s one o b t a i n s ~2 + m e a n r (x) 2 - a = 0. Since [ m e a n r (x)] 2 < m e a n r (x) 2 one also has [ m e a n r 2 < a - ~2. Q . E . D .

F o r t h e e q u a t i o n (L) i n t e r i o r to D we s a y t h a t t h e n u m b e r s (~, - ~) a p p e a r i n g in t h e c a n o n i c a l s o l u t i o n s of T h e o r e m 10 a r e t h e c h a r a c t e r i s t i c e x p o n e n t s [10] of (L). T h e y s a t i s f y t h e u s u a l d e f i n i t i o n of c h a r a c t e r i s t i c e x p o n e n t s in t h a t

lim sup - log ] Yu (x) I = ~- 1

X-->~r X

a n d lira sup 1 log [yL(x)[ = - ~.

X-->oo X

U s i n g t h i s d e f i n i t i o n for t h e c h a r a c t e r i s t i c e x p o n e n t s we shall l a t e r s h o w t h a t t h e y a r e zero w h e n (L) belongs to t h e b o u n d a r y of D.

F o r t h e classical case of H i l l ' s e q u a t i o n , t h e s o l u t i o n s are e x p o n e n t i a l f u n c t i o n s m u l t i - plied b y p e r i o d i c functions. The following e x a m p l e shows t h a t t h e difficulties which are m e n t i o n e d in t h e a b o v e t h e o r e m a n d corol]aries do a c t u a l l y arise.

Consider t h e e q u a t i o n

y" + ( - a + b p ( x ) ) y - O , (L)

where a - ~2 _ m e a n Z (x) 2, b - - 1, a n d p (x) 2 :r (x) + Z (x) 2 + Z' (x) m e a n Z (x) ~" T h e n

(:)

t a k i n g ~ > 0 a n d Z (x) = n~ cos , p (x) is a n a l m o s t p e r i o d i c f u n c t i o n w i t h m e a n zero a n d (L) has a s o l u t i o n

y ( x ) = e x p ~ x + 1 sin x = e x p o~x+ z(t)dt .

n = l n 0

)

Since t h e c h a r a c t e r i s t i c e x p o n e n t of 9(x) is ~ > 0, \x~(lim xl ~oZ(t)dt = 0 , t h e e q u a t i o n (L) belongs t o t h e i n t e r i o r of D, of. T h e o r e m 15. T h e n i t is e a s y to see t h a t t h e r e is no s o l u t i o n basis of t h e f o r m e x p { ( - 1)i~x § where :~(x) a r e a l m o s t periodic, i = 1, 2. F o r one

X

would t h e n r e q u i r e t h a t g (x) = c e x p ( e x + :~1 (x)), for c * 0, a n d ~1 (x) = f Z (t) d t which

0

is n o t b o u n d e d . I t is u n k n o w n w h e t h e r such a d i f f i c u l t y can occur if p (x) is, say, a t r i g o - n o m e t r i c p o l y n o m i a l .

LE~aNA. Let

y" + ( - a + bp(x))y = O. (L)

1 1 4 L A W R E N C E M A R K U S A N D R I C H A R D A . M O O R E

with p (x) a real almost periodic/unction, be in the interior o/ D. Then the characteristic expo- nent :r and the quantities 6 = inf I uu (x) - uL (x) ] (/or the upper and lower almost periodic

- o r

solutions o/ the associated Riccati equation), and # = sup [ uu (x) - uL (x) [ are continuous,

- r 1 6 2

strictly increasing/unctions along each line b = const., as a increases a w a y / r o m the boundary ol D.

Proo]. L e t (ao, bo) lie i n t h e i n t e r i o r of D a n d let u~(x), uL(x ) be t h e u p p e r a n d lower a l m o s t periodic solutions of t h e associated R i c c a t i e q u a t i o n . T h e n A ( x ) = Uu(X ) - u L ( x ) has a m e a n of 2 ~ 0 where cr o is t h e c o r r e s p o n d i n g characteristic e x p o n e n t . B y T h e o r e m 8,

~, 6, a n d # are s t r i c t l y i n c r e a s i n g along t h e line b = const, a n d we n e x t show t h a t t h e y are c o n t i n u o u s f u n c t i o n s .

Consider t h e a l m o s t periodic f u n c t i o n s

wl(x) = tuu(x) + ( 1 - t)uL(x)

a n d w 2 ( x ) = ( 1 t ) u u ( x ) + t u L ( x ) for O < t < l .

T h e n a c a l c u l a t i o n shows t h a t b o t h of these a l m o s t periodic f u n c t i o n s are solutions of t h e e q u a t i o n

w' + w ~ + [ - a o + boP(X ) + t(1 - t)(u=(x) - uL(x)) 2] = 0.

Also w l ( x ) - w 2 ( x ) = ( 2 t - - 1 ) [ u u ( x ) - u L ( x ) ] . T h e n t h e w i d t h Aw(x ) of t h e b a n d of b o u n d e d solutions of

w' + w ~ + [ - a 0 + b o p ( x ) + c ] = 0 , where 0 < c < t(1 - t) inf {u~ - uL) 2, satisfies

- ~ <x<~x~

Aw(x) ~ (2t - 1 ) l u g ( x ) - u L ( x ) ] = (2t - 1)A(x).

F o r a p r e s c r i b e d ~ > 0 choose t so n e a r 1 t h a t A ( x ) - A w ( x ) < s w h e n e v e r c > 0 is sufficiently small. Therefore ~, 6, a n d # are c o n t i n u o u s from below along b = const. B u t w i t h i n a fixed n e i g h b o r h o o d of (a0, bo) , c c a n be chosen i n d e p e n d e n t l y of (a', b0) so t h a t t h e b a n d w i d t h of

w' + w 2 + [ - a' + boP(X)/= 0 a n d t h a t of w' + w e + [ - a ' + c ~ b o p ( x ) / = 0

differ b y less t h a n r Therefore 0r 6, a n d # are c o n t i n u o u s along b = b 0. Q . E . D . THEOREM l l . Let

y" + ( - a + b p ( x ) ) y = 0 , (L)

O S C I L L A T I O N A N D D I S C O N J U G A C Y F O R L I N E A R , D I F F E R E N T I A L E Q U A T I O N S 1 1 5

/or a real almost periodic p (x), belong to the interior o/ D. On the interior o/ D the real/unc- tions ~ (a, b) and ~ (a, b) are continuous. Let D~, and Eo, be the subsets o/ the interior o/ D wherein ~ > ~o and 5 > do, respectively. Then each D~ and Eo is a convex set which is rela- tively closed in the interior o/ D. Also D ~ (or E~,) is properly contained in the interior o[

Da~ (or E~) whenever 0~ 2 < a-1 (or de < c~1). On the relative boundary o/ D~ (o/Ee) the charac- teristic exponent (the inf lug(x) uL(x) l) is exactly ~ (or 5).

Proo/. L e t (al, bl) a n d (a 2, b2) lie i n D~ (or lie i n Ee). L e t u~)(x), ur be t h e u p p e r a l m o s t periodic solutions a n d u~)(x), u(~)(x) be t h e lower a l m o s t periodic solutions of t h e c o r r e s p o n d i n g R i c c a t i e q u a t i o n s . Consider w= (x) = tu~)(x) + (1 - t) u~)(x) a n d w~(x) = tu(~)(x) + (1 - t)u~)(x) for 0 < t < 1. A c o m p u t a t i o n shows t h a t w~(x) satisfies

w' + w 2 + { - [ t a I + (1 - t ) a 2 ] +[tb~ + (1 - t ) b 2 ] p ( x ) ) } + t(1 - t ) [ u ~ ) ( x ) - u ~ ) ( x ) ] ~ = 0.

T h u s t h e u p p e r a l m o s t periodic solution of

w' 4- w 2 =-{ --[ta x + (1 --t)a2] +[tb t + (1 t)b~]p(x)} = 0 (Rt) lies a b o v e w~ (x). S i m i l a r l y the lower a l m o s t periodic s o l u t i o n of t h e last. e q u a t i o n (Rt) lies below wL(x ). T h u s t h e b a n d w i d t h A t of (Rt) satisfies

A t > t A r l ) ( x ) ~- ( l - - t) A(2J(x).

T h u s Eo is convex. Since m e a n 1/k t = ~t, t h e characteristic e x p o n e n t c o r r e s p o n d i n g to (Rt), D~ is convex.

B y t h e l e m m a , t h e characteristic e x p o n e n t corresponding to a b o u n d a r y p o i n t of D~, i n t e r i o r to D, is e x a c t l y ~. Also on t h e relative b o u n d a r y of E~, inf l uu(x) - u L ( x ) I = 6.

~ r < x - o r

T h u s b o t h D~ a n d E~ are r e l a t i v e l y closed i n t h e i n t e r i o r of D. T h e enclosure relations m e n t i o n e d i n t h e t h e o r e m are t h e n e l e m e n t a r y .

F i n a l l y we verify t h a t e(a, b) a n d d (a, b) are c o n t i l m o u s on t h e interior of D. B u t this follows easily from t h e enclosure a n d c o n v e x i t y conditions. Q.E.D.

One can f u r t h e r describe D~ b y n o t i n g t h a t it m u s t c o n t a i n t h e sector b o u n d e d b y t h e r a y s

a = cr '~ -~ b inf p (x)

a n d a = cr 2 " b sup p(x).

o o < x < ~

Similarly Es m u s t c o n t a i n t h e sector b o u n d e d b y t h e rays a = 6 2 + b inf p(x)

a n d a = b 2 + b sup p (x).

S -- 563802. Acta mathematica. 96. I m p r i m 5 le 23 o c t o b r e I956.

1 1 6 LAVCRENCE M A R K U S A N D R I C H A R D A. M O O R E

Also each domain D~ and E~ is invariant under translations, and even limits of such, of the equation (L).

Later, cf. Theorem 14, we shall show t h a t (~-+0 near the boundary of D and thus t h a t each E~ lies interior to D. However, although it is very likely true, we have not been able to show t h a t :r near the boundary of D.

We conclude this section with a result for a forced or non-homogeneous differential equation.

THEOREM 12. Let

y " + ( - a + b p ( x ) ) y = O, (L) where p(x) and p' (x) are real almost periodic /unctions, lie interior to D. I / /(x) is almost periodic, then the equation

y " + ( a + b p ( x ) ) y - /(x) (F) has a unique bounded solution and this is almost periodic.

P r o @ Since the homogeneous equations (L*) have no (non-trivial) bounded solutions, (F) has at most one bounded solution. Moreover, b y F a v a r d ' s theory [5, Ch. 3], if there is a bounded solution y(x) for which y'(x) is also bounded, then y(x) is almost periodic.

A solution basis for the homogeneous equation is qS(x)exp i -+ dt//r (t)2 where r is

0

almost periodic and 0-~'-c < r C. Construct the Green's function

G(x,~) r 1 6 2 dt t

2

Consider the solution of (F) given b y

y(x) = ~ G(x, ~)/(~)d~.

o o

Tile integral exists s i n c e / ( x ) is bounded and G(x, ~) has au exponential decrease at both

-t- c<).

From the definition of q~(x), cf. Corollary 2 of Theorem 10, we see t h a t ~b'(x) and

~ " ( x ) are also almost periodic. Then one can differentiate tile expression for y(x) to see t h a t y(x) is a solution of (F).

Since y(x) and y" (x) are bounded, so is y' (x) bounded, as is required. Q.E.D.

O S C I L L A T I O N A N D D I S C O N J U G A C Y F O R L I N E A I ~ D I F F E R E b [ T I A L E Q U A T I O N S 117

4. Boundary of the disconjugacy domain

W e first give a c r i t e r i o n for d i s t i n g u i s h i n g b e t w e e n t h e i n t e r i o r a n d t h e b o u n d a r y of D b y t h e b e h a v i o r of t h e s o l u t i o n s of (L).

T t t E O R E ] g 13. Let

y" + ( - a + b p ( x ) ) y = O, (L)

where p (x) is real almost periodic, be disconjugate. T h e n (L) belongs to the interior o/ D i/ and only i/ there is a solution basis y ~ ( x ) y L ( x ) o/ (L) /or which 0 < y u ( x ) y L ( x ) < C.

Proo/. I f (L) lies i n t e r i o r t o D, t h e n b y T h e o r e m 10 t h e r e is a s o l u t i o n basis of t h e r e q u i r e d t y p e .

C o n v e r s e l y let yo(x), Yl (x) b e a basis whose p r o d u c t is b o u n d e d , as a b o v e . T h e n t h e f u n c t i o n s Uo(X ) = y o ( x ) / y o ( x ) a n d u l ( x ) = y~ ( x ) / y l ( x ) are s o l u t i o n s of t h e a s s o c i a t e d R i c - c a t i e q u a t i o n

u' -+- n ~ + ( - a + b p ( x ) ) - O. ( R)

L e t us w r i t e u l ( x ) > u 0(x) a n d u l ( x ) - u o ( x ) > 2 / C . Consider t h e f u n c t i o n w ( x ) =

~-[u I (x) - u 0 (x)]. A c o m p u t a t i o n y i e l d s

w' + w 2 + ( - a + bp (x)) + 88 [u 1 (x) - u o (x)] 2 = 0.

T h u s t h e e q u a t i o n w' + w ~ + ( - a + b p ( x ) ) § 1 / C 2 - 0

has a b o u n d e d solution. T h e r e f o r e y" + ( - a + b p (x) + 1 / C 2)y - 0 is d i s c o n j u g a t e a n d (L) lies i n t e r i o r to D. Q . E . D .

COROLLARY. I n the interior o / D no (non-trivial) solution o / t h e corresponding linear di//erential equation (L) has a square which is bounded. However, i/ on the boundary o/ D, the product o/ two positive solutions o/ (L) is bounded (even only positive with a bounded product on a hall-axis), then the two solutions are linearly dependent.

W e can n o w c o m p l e t e t h e discussion of (~(a, b) which was b e g u n in T h e o r e m 11.

TI-IEOREM 14. Let

y " + ( - a + b p ( x ) ) y = 0 , (L)

where p (x) is real almost periodic, belong to D. T h e n on the boundary o/ D lim inf I uu (x) - uL (x) I = 0

X - ~ • r

/or any two (possibly coincident) bounded solutions o/ the associated Riccati equation (R).

There~ore ~ (a, b) = 0 on the boundary o/ D and/urthermore (5 is continuous on all o/ D.

* -- 5 6 3 8 0 2 .

1 1 8 LAWRE:NCE M A R K U S A N D R I C H A R D A. M O O R E

Proo/. S u p p o s e t h e p o i n t (ao, bo) y i e l d s a p o i n t i n t e r i o r to D b u t (a 0 - ~ , b0) for a c e r t a i n e > 0 y i e l d s an o s c i l l a t o r y e q u a t i o n (L). If, for (a0, b0), t h e w i d t h of t h e b a n d for t h e a s s o c i a t e d R i c e a t i e q u a t i o n is A ( x ) ~ 2~"~, t h e n t h e p o i n t (a 0 - ~ , b0) lies i n t e r i o r t o D, as follows f r o m t h e c a l c u l a t i o n occuring in t h e proof of T h e o r e m 13. T h e r e f o r e 2~ (x~ < 2 ~"v for s o m e x a n d c~-->0 n e a r t h e b o u n d a r y of D.

N o w for (a, b) on the b o u n d a r y of D, t h e b a n d for t h e a s s o c i a t e d R i c c a t i e q u a t i o n n l u s t lie i n t e r i o r t o t h e b a n d for (a + e, b), v > 0. B u t for t h e w i d t h A~ (x) of t h i s b a n d one has lira inf A~(x) - 5(a + e, b). Since (~(a + c, b ) ~ 0 as ~ - + 0 , as we h a v e t h e d e s i r e d result.

Q . E . D .

W e n e x t t u r n to t h e p r o b l e m of c o m p u t i n g t h e c h a r a c t e r i s t i c e x p o n e n t ~ on t h e b o u n d a r y of D.

T H E O R E M 1 5 . Let

n' + u 2 ~ ( - a + - b p ( x ) ) - O , (R)

/or real almost periodic p (x), correspond to the boundary o/ D. I/<~ solution o/ (R) is almost periodic, it must have zero mean. Also there can be at most one almost periodic solution o / ( R ) . Proof. L e t u ( x ) be a n a l m o s t p e r i o d i c s o l u t i o n of (R) a n d s u p p o s e m e a n u (x) = ~ > 0.

C o n s i d e r t h e a u x i l i a r y e q u a t i o n

z' - 2 u ( x ) z - 1.

T h e n a solution is z(x) = f [exp f 2u(s)ds]dt, which is b o u n d e d on a r i g h t half-axis,

s a y x > 0. Define w(x) by z ( x ) = - w ( x ) e x p f u(t)dt. A n e a s y c o m p u t a t i o n shows t h a t

0

w(x) is a p o s i t i v e s o l u t i o n of

y" ~ ( - a + b p ( x ) ) y - 0 . (L)

B u t t h e n t h e p r o d u c t of t w o p o s i t i v e s o l u t i o n s of (L) is b o u n d e d on a r i g h t h a l f - a x i s a n d , b y t h e C o r o l l a r y t o T h e o r e m 13, t h e y m u s t be l i n e a r l y d e p e n d e n t solutions. B u t t h i s is i m p o s s i b l e since / u ( s ) d s - + § c~ as x ~ o c a n d so w(x)-~O as x - + c ~ . F r o m t h i s c o n t r a -

0

d i c t i o n we c o n c l u d e t h a t m e a n u ( x ) ~ O.

A s i m i l a r c o n t r a d i c t i o n arises f r o m t h e s u p p o s i t i o n t h a t m e a n u ( x ) < 0 . H e r e one uses t h e f u n c t i o n

z 1 (x) [exp ~ 2~ (s)ds]dt

cr t

O S C I L L A T I O N A N D D I S C O N J U G A C Y F O R L I N E A R D I F F E R E N T I A L E Q U A T I O N S 1 1 9

which satifies t h e same a u x i l l i a r y e q u a t i o n , a n d which is b o u n d e d on a left half-axis.

A g a i n define w l ( x ) b y zl(x ) =Wl(X) exp f u ( t ) d t a n d observe, t h a t w l ( x ) is a positive

0

solution of (L). T h e n t h e same r e a s o n i n g as i n t h e earlier case shows t h a t m e a n u(x) - 0 . If there were two a l m o s t periodic solutions of (R) t h e n these differences would be a positive a l m o s t periodic f u n c t i o n of zero m e a n a n d this is impossible. Q . E . D .

COROLLARY. Let y(x) be a positive solution o/ (L), /or the boundary o/ D,

such that u ( x ) - y ' ( x ) / y ( x ) is almost periodic. Then the characteristic exponent lira sup ( l / x ) log l y ( x ) ] - 0.

X-~cx~

T h e following e x a m p l e shows t h a t a n e q u a t i o n (L) on t h e b o u n d a r y of D need n o t h a v e a (non-trivial) b o u n d e d solution. Our e x a m p l e does h a v e a s o l u t i o n y(x) > 0 such t h a t y' ( x ) / y ( x ) is a l m o s t periodic. W h e t h e r or n o t there is always such a s o l u t i o n i n this s i t u a t i o n is u n k n o w n . W e shall h a v e f u r t h e r c o m m e n t s o n this d i l e m m a later.

Consider y" ~ [ - Z (x) 2 ^, Z' (x)] y ⁼ 0 where Z (x) ⁼ ~ ^-1 ^- cos ' x Here t a k e a =

n :: 1 n 2 , n . "

m e a n Z (x) 2, b - 1, p (x) - - X (x) ~ - Z' (x) § m e a n X (x) 2. T h e e q u a t i o n is d i s c o n j u g a t e since a positive solution is y(x) = exp i X (t)dt" Since lim sup (1~Ix) log y(x) = m e a n X(x) = 0, t h e

0 X-->vr

e q u a t i o n belongs to t h e b o u n d a r y of D. W e also n o t e t h a t lim sup y ( x ) - ~ oo, lira inf y (x) - 0.

If Y ( x ) were a b o u n d e d solution, say Y ( x ) > 0 for x > x o, t h e n 0 < Y ( x ) < cy(x) for x > x 0 a n d a c o n s t a n t c. B u t t h e n lim inf W ( x ) = 0 where W ( x ) is t h e W r o n s k i a n of Y ( x )

z->r

a n d y(x). However, W(x) is c o n s t a n t . T h u s there are no (non-trivial) b o u n d e d solutions.

T h e n e x t t h e o r e m shows t h a t one can always o b t a i n a b o u n d e d s o l u t i o n for (L) on t h e b o u n d a r y of D, m e r e l y b y t r a n s l a t i n g to a l i m i t e q u a t i o n (L*).

THEOREM 16. Let

y " + ( - a + b p ( x ) ) y = O, (L)

where p ( x ) and p' (x) are real almost periodic /unctions, belong to the boundary o/ D. Then there exists p* (x) C H { p (x)} such that

y" §

(

- a + bp* (x))y = 0 (L*)

has a positive bounded solution.

Proo/. F o r each g > 0 a solution basis yl (x), y2(x) o f y " -i- ( - a § bp(x) - e)y - 0 yields a general solution