On solutions of the differential equation = О

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V II (1974) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I : PRACE MATEMATYCZNE X Y II (1974)

Z. P

(Poznan)

On solutions of the differential equation у {п)—а (х )У = О

In [2] and [3] we proved theorems concerning the asymptotic prop­

erties of some integrals of the differential equations y " — a(x) у —

and y(n) — a{x)y

(foi* п ^ З ) , where a(x) is a complex function. In this paper we obtain a more complete result in the case of a real a (x).

We shall prove the following

T

. Suppose that n ^ 2 and that

(

) a{x) is a real function and a(x) Ф

for x > x 0, (

) there exists the derivative a (2n~l\x) for x ^ x 0.

Furthermore suppose that we have for x oo

(3) (a~ll2nf n) = o(xll2~nln~1~ex) and — o(l)

for m —

, ..., n —

and

(

) (a[i-»],»«)' = 0{x~llmh r l~ex)

for n > 4 and m = 3 , . . . , n — 1, with some e > 0.

Then the differential equation

(5) y(n) — a( œ)y =

has for х ф x Q the integrals y k{x), к =

, ..., n, such that

X

yk{x) ^ a[1~nV2n(x)expek f a lln(t)dt for x-> oo,

Xq

where ek = e2kniln.

I f re Eka lln(x) >

for X > x 0, then y ^ (x ) ~ e la vln{x)yk(x) for x oo,

v =

, ..., n —

, and lim |y ^ (a?)| = oo for v =

,

, n —

.

430 Z. P ol n i a k o w s k i

I f r eeka 1,n(x) <

, then limyk(x) =

.

x -> o o

If a(x) < 0 we assume A rg a1In(x) — izjn.

Let us notice that hypotheses (3) and (4) are satisfied by the function a(x) = xp for real p > —n and by a(x) = exP for arbitrary real p.

C

. From the above theorem we obtain the following oscillation theorem :

I f hypotheses (l)-(4 ) are satisfied, n ^ 2, if im ek

in the case of a(x) > 0 and if im ekemln Ф 0 in the case of a(x) < 0, then the differential equation (5) has for x ^ x 0 real integrals y*k{x) and y k*(x) such that

where

and

Ук№) = (cosBk(x) + ôk(x))\all~n]l2n(x)\expCk(x)

(X~> o o ),

Ук*(я) = (sm Bk{x) + r]k(x))\a[1- nV2n(x)\ex.-pCk(x)

X X

B k(x) — ime* j a 1,n(t)dt, Ck(x) —теек J a1/n(t)dt

x0 x0

lini<5Æ(a?) = Ит^(ж) = 0.

X -+ C O X -+ O Q

B y (3) we get Ит|.Вй(а?)| = Ит|(7А.(ж)| = oo. The functions yt(x ) and

X -> o o X -> o o

y*k {x) have infinitely many zeros with the lim it point £ = oo.

We introduce the following definition : We shall say that the function g(x) has the property Ш at the point £ = oo with the constant К and for x > x 0 if there exist К >

and a point x 0 such that we have

X

lim|<

(a?)| = oo and j \g'(x)\dt < K\g(x)\ for x ^ x 0 x0

or

OO

lim<

(a?) = 0 , g(x) Ф 0 and J \g'(t)\dt < K\g(x)\ for x ^ x 0.

x->oo x

L

1. Suppose that n^2> and that the functions f(x ), h(x), <pv(x) (v = 1 , . . . , n —

) and гр(х) are locally bounded and locally integrable for x > x 0. We set

X

gv{x) = exp^A Jh (t)d t/ rvj for v — 1, n — 1 and x ф x0, x0

where A and rv are some complex numbers and А ф 0, rv Ф 0.

Solutions of the differential equation

431

Moreover, suppose that

(

) if n > 3, then re(A/rw)

0 for v =

, ..., n —

,

(7) lim/(a?) = s (|s| <

),

X-+OQ

(

) h(x) >

for x > x0 and

O O

J h(x)dx — 00, X

(9) if n > 3, then lim

v(a?) =

X-+CQ for v =

, w —

, (

) lim

> tl_

(a?) =

ifte (X lrn_j)

O O

шгй J 7i(a?) |ç’n_

(a?)| <

if refA/r^j) =

, oo

(

) f \y>{x)\dx < oo.

Xq

We set for xx (with sufficiently large xl > x 0, defined in the proof) :

00 n — 1

Jo (y) = f y>{t)y(t)dt+ £ K v{y),

X r = l

where

Pi

K x {y) = J g'i (ti)<Pi (tx) y (tf) dtx ; if n > 3, йе?г

■г. (У) Яг if 2) 4>г (fz) У (^г)

>

if n > 4, йетг

01

, Рг ,

1

r ^(^i) /*

2(У

0 1

(Я) У

J 03

Pv

• • J g ' M v M y t t J d t v - d t 1

av

for v = 3 , . . . , n — l . We assume here aj = ж1? Д = if re(A/ri) > 0 and ctj = tj_x, =

if re(A/r^) < 0 (j = 1, w —1,

= ®)-

Tftew Йе integral equation

( 12 ) У(®) = / И + ^

(

И)

üas /or x ^ x x a solution y(x ) such that limy (a?) = s. I f f(x ) is continuous X-+00

for x > , йе?г ÿ (x) is also continuous for x ^ xx.

432 Z. P o ln i a k o w s k i

P ro o f. It is easy to see, applying l’Hospital’s rule, that the functions gv(x) (for v = 1, n —2 if 3 and for v = n — 1 if re(A/rn_x) Ф 0) have property Л at the point £ = oo with some constant К > ] A/rJ / |re(A/rv) \ and foi* x ^ xQ.

We choose a number de (0,1) and х х ф х 0 such that we have for

X

> x x

OO

l/ (® )l< 1+1*1, ^f Iv(t)|<««a/», Ж

for © = 1, — 2 if ^ 3, i(^)l ^ ôjnK n~1 if re(A/rn-1) Ф 0 and

oo

j Ht)\<Pn-i(t)\dt < ô\rn_x\ln\X\Kn~2 if re(A/rft_1) = 0.

a?

Setting t/^(y) = (y)) for © = 1 , 2 , . . . and x ^ x x we obtain

|e7

(l)I < ôln-\-(n — 1 )д / п ^ ô and

|<7В(1)| < d

for © =

,

, ... and x > x x.

OO

We assume y(x) = f(x ) + J v(f)i the series being uniformly convergent

«=0

for x > x x. We infer from this that if f(x ) is continuous for x ^ xx, then y{x) is also continuous for x > x x. The function y (x ) is bounded for x > x x since

O O

\y(x)\ <

+

*|) (l+ 2 ^à ’+l) = (

+ |s|)/(l- Ô).

V = 0

It is easy to see that y(x ) satisfies the integral equation (

) l^cause

OO

f[x )-{ -J0{y) = f{x) -\-J0{f) + Jo Jv if))

V=Q oo

= /(fl?)+J0(/) + J £ j v+i(f) =ÿ(®).

V = 0

Applying in (

) in the case of re(X{rv) Ф

the l ’Hospital’s rule in the formulation of Theorem C in [1], p. 20, we obtain that limJo(ÿ) = 0, by (9), (10) and (11), and lim^a?) = s. æ_*°°

£ C— >00

L

(Compare Lemma 5 in [3]). Suppose that n >

, that there exist A (

-

)(

), a%n~2)(x), a ^ +v~l)(x) (© = 0,1, ..., n —

), that XA(x) >

and an(x) Ф 0 for x ^ x 0.

I f n — 2 we set ôx<px(x) = r xA ' ф гхА (ip — a x/a2) + 1, where ôx = 1 if

те(Л/гх) >

and <5^ =

if re(X/rx) <

. (ip(x) will be defined below.)

Solutions of the differential equation

433

I f n ^ 3 we define the functions cpv{x) (v = 1, . . . , n — 1) as follows:

Ôl<Pl = Ф 10 + ^ 1 ^ ,

( JL о

àjVi = Ф/о ~ ^ - i,o -Г/^фу-

,о + riA W for j = 2, ..., n - 1 ,

x ^ x 0, where Sj = ( — !)*•/ and Ц is the number of indices v such that re(A/rv) < 0 and l ^ v ^ j (j =

, ..., n — 1). A and rv are defined as in Lemma 1.

The functions <Pj0 (j = 1, n — 1) are defined by the equalities (14) = 0 } v - 0 j - i , v - r j A 0 j - hv,

j = n — 1 , . . . , 2 and v = j —

, ...,

, x > xQ, where =

, and (15) %l- 1,v- 0n^ i , v = r 1 . . . r n_1bv1 v = 0 ,1 , . . . , n - 2, x ^ x 0, where bv = { — T)j~v~1(aj A n~l Janf i ~v~l\

j = v + l

The functions Ajv are defined for x ^ x 0 as follows :

% 0 = 1 , = r x . . . r 5Aj for j = 1, ..., n - 1 ,

(16) .

% , = % - hv + r3 A % _ hv + ^ А % _ х^ _ г for 1 < v < j < n - 1.

Suppose that there exists for x > x 0 a solution у (x) of the integral equation П

(12) w ithf(x) = l,h ( x ) = 1 IXA(x),ip{x) = r x ... rn_1^ ( — l ) v(avAn~1ja n){v)

V = 0

and with <pv (x) defined as above. Then y (x) satisfies for x ^ x 0 the differential equation

П

(17) У > „У м («0 = ° .

v — 0

P roo f. For n > 3 is like that of Lemma 5 in [3]. In the case n = 2 we prove similarly.

L

3 (Compare Lemma 6 in [3]). Suppose that n ^ 2, that there exist J .(

n_

)(aî), a (f n~2\x), ^ п+®_

)(ж) (v = 0, 1 , . . . , n — l ) , that ÂA(x) >

and an(x) Ф

for х ф x 0. Furthermore suppose that there are satisfied hypotheses (

) and (

)-(11) of Lemma 1 for h(x), <pv(x) and ip{x) defined as in Lemma

.

Then the differential equation (17) has for x~^ x Q an integral y(x ) such that Пт^(ж) =

.

Ж — *oo

P ro o f. By Lemma 1 the integral equation (

) under our hypotheses and for f(x )

has for х ^ х г a continuous solution y(x) such that Пт^(ж) =

. By Lemma 2 the function y(x) satisfies for x > x x the

X->oo

differential equation (17) and may be extended to the point x 0.

434 Z. P o l n i a k o w sk i

L

4 (Compare Lemma 7 in [3]). Suppose that n > 3, that the functions %v(x) (j =

, . . . , n —

, v =

, ..., n —

, v < j) are defined by

relations (16) and that

lim = 0 (s = 1, . . . , n ) , A^s)A s~2 = о (X) ( e = 2 , ...,w ),

Х-+ОЭ

(18)

{A'Y A

= o(X) for x-> oo,

where the function X (x ) is defined and different from

for large x.

Then

(19a) = i

+ (<,

) i

+ o(X), ( vLjvA r A~l + o(X) for 8 = 1 ,

(19b) A s- v- 4 $ = \ }V K ’ J ’

\o(X) for s =

, . .

n + v — j , x -+■ oo.

Ljv is the v-th fundamental symmetric function of the numbers r x, ..., r$.

P ro o f. We shall prove (19a) and (19b) successively for v =

, n — ¹

It is easy to see that they hold for v = j = ¹

Suppose that (19a) and (19b) are true for v = 1 and for some index j

< j < n — 2). We shall prove that they are true for v = 1 and for j +

. By (16) we obtain

A - % +1>1 = A - % 1 + rj+1A ~ % 1 + rj+1A

= {Lji + r j+0 A -1 + Lj2 A ' A - 1 + rj+1Lj l A ' A " 1 + o{X)

= Lj+ hlA~x -\-Li+ltiA 'A ~ l -\-o(X) for oo,

= А 8~2Щ1> + rj+1A 8~2(A % 1)W + rj+1A s~2A(8\

s =

, ..., n —j, where

A 8- 2{A%XY8) = = o(X)

m= 0 ' '

for x — > oo, by hypothesis. We have namely

j^(m) [L jx{A 'Y A -x + o{X) = o (X ) îoT8 = m = l ,

j LjxA ^ A s~2A ,Jro (X ) = o (X ) for s = m >

. This proves that (19a) and (19b) are true for v = 1 and for j =

, ...

..., n —

.

Suppose now that n > 4 and that they are true for the index v ^

(and for v < j < n —

). It is easy to prove that (19a) holds for v + 1 instead of v, and for j

+ 1. Belation (19b) is also true for these indices and for s = 1. For s > 2 we obtain

15+1, 15+1

=

, Ü+1

2

Solutions of the differential equation

435 1?+1

where Wt = -d(A^ ... A {ki> v+l\ V Jcim = s (Я- are constants). If him = 0

m = l

or

for some i and for m =

, v +

, then J s-t,-

TFi = (A ')8~2 {A 'f A -1

— o(X) for x -» oo. If for some i and m = M is &г-т > 2, then

v + l

^ s— v— 2 ~w^ — J~l A^kim^ Ак^т ~г ' A ^ ^ ' AkiM~2 — o(X) for x — > oo.

?n= 1

т Ф М

Suppose that (19a) and (19b) are true for the index -y-fl and for some index j (

<'y + l < j < n —

). We prove that they are true for the indices v + 1 and j + 1. In the case of (19a) we obtain

A - ’ ~ % +liV+1 = A - ’ - % iW + rw A - ’’- % ,„ +1 + r1+1A -'’- % v

= Zl ^ +lA - 1 + ^ A 2) r ly W A 'A - ' + rl+ l(v + l) L j> w A 'A - 1 + + ri +

ri

Ц

ч

1

А

^o(X)

= Lj + w l A - '+ (* A } L j+ h w A ' A~l + o(X ).

In the case of (19b) and s —

we get

12

t

= A - ’ -% _ w + r ^ l A -'’- l {A'%,w + A % [w ) + Jr ri+i A v 1(A A%jV)

= (V “b 1)-£/,«

.1-4- .d

+î^+iXÿ„A -d

+ îy+

î?iÿtJ-d A l -\-o{X)

= (,y + l)-iÿ +

,^+i-d A l j rO (X ).

Relation (19b) for s > 2 we prove as in the case v = 1. The proof of rela­

tions (19a) and (19b) for v ^ 2 and n — 3 is similar.

L

5. Suppose that the functions <&jv (0 < v < j < w — 1 , w > 3) are defined for x ^ x 0 by relations (14) and that

(20) As~v~10^_i>v = o{X ) as x -^ oo (® = 9 ,

, ..., v-\-l and v =

,

, ..., n —

),

(

) lim-d

J .(s)

(s = 1 , n).

X - X X >

Then

(22) As~v~l <f>$ = o{X) as x -^ oo

(s = 0 ,

, . . . , «? +

and 0 < v < j < n —

), w;7iere the function X (x) is defined as in Lemma

.

9 — Roczniki PTM — Prace M atem atyczne XVII.

436 Z. P ol n i a k o w s k i

The proof of Lemma 5 is like that of Lemma

in [3].

L

6 . Suppose that there exists the derivative a ^ ix ) for x ^ x 0 and w > 2, and that a(x) satisfies hypotheses (1), (3) and (4). We set for x > xQ (compare Lemma 14 in [3])

X X

F (x ) = all~n]l2nex-pek J a lln(t)dt, F ±(x) = a[

~n]/

nexp( - ek J a lln(t)dt),

Xq Xq

where ek = e2kmln with some к

< к < n).

Then we have for x-> oo, m = 1, n :

(23a) F (n){x) = F (x )a mIn{e% + e%-1m { n - m ) ( a - lln)'l2 + a - lln-o(X)}, (23b) F M (x ) = F 1(x )amln{ ( - l ) me% +

+( — l) m_1 e™

m (n — m) {a~l,n)r

-f a~1[n • о (X)}, where X = x~l In- 1-6x with some e >

.

P ro o f. It is easy to see that Lemma

is true for n =

. We assume that n ^ 3. We obtain

m x

F im) (*) = T J ( - % / ,

V — 0 Xq

where

Emv = a [v- 1]!n(all- n]l2n)(m- v\

X X

Cv = a[1~vVnexj> ek J a 1/n(t)dt jexp | — ek J al!n(t)d t^ v\

Xq x0

In the proof of Lemma 16 in [3] we proved that

Cv = ( - l f e l a ^ + i - i y ^ e l - ' i a - ^ y a ^ W o i X ) for a?-* oo.

As in the proof of Lemma 16 in [3] we show that

a[1+n~2m]!2nEmv = о(

) for v =

, m — 1 ,

^ a [3+n-2m]i2nEmv = о (X) for v =

, .. •, m -

, (a - lln) 'a [3+n- 2m]l2nEmv = o(X ) for v = 1, m - 1 . Since a[1+n~2m]l2nEm>m_1 = (n — l)(a ~ 1,n)' /2, we obtain

X

a [i+n-

m]/

»expCfc r a lln{t)dtF[m) = m ( n - l) ( a ~ lln)'Cm_1l2 +

Xq

+ Cm + o(X) = ( - l ) ” -

e ? -

m ( » - l ) ( e -

," )V ,"/2+-

+ ( - ! ) ”* a1'" + (

)“ (” )

(o-1'”)' a1,n + о (X ).

Solutions of the differential equation

437

From this there follows (23b). The proof of relation (23a) is similar.

F ro o f of th e th eo rem . For a given le

< It < n) we substitute y(x) = w (x)a[1~n]!2n(x)ex-pek f a 1/n(t)dt into (

) (ek = e21cniln) and we

x0

obtain the differential equation П

(24) ^ а ф ^ ( х ) = 0,

V — 0

where

X X

®o = (®tl-nl/2n(®)expefc J a lln{t)dijn) — a[1+n]l2n (x) ex-p ek J a 1/n(t)dt,

x 0 Xq

l \ x

av = r j |а

-в]/

пехрей j a lln(t)dt^n~v) foi* v =

, ..., n .

x 0

Suppose that

if a(x) >

, 0 =

if a(x) <

for x > &0. We assume first n = 2. (The case a(x) > 0 we proved in [

].) The functions

<Pi{x) and y>(x) defined as in Lemma 2 with A (x) = ek l a~ll2(x), X = Xk

— e(k+oi2)m^ r ^ _

12f and formed in the case of the differential equation (24) are of the from

ip(x) = ek 1a~ll4(a~lli)" /2, q>i(x) = a“3/4 (a~1/4)" jê .

Then lim <fi(x) =

and a ll2(x)<p1(x) = ekxp(x)j2 — o(x~l \nT1~sx) for

x->oo

x — > oo, by (3) (compare [2], p. 245), and hypotheses (10) for h(x) — \all2(x)\ ,

and (

) are satisfied.

Suppose that n > 3. If there exists the (unique) index v =

< v0 K n ~

) such that

(25) j,Qg(2k+e-{-v0-nl2)niln __

we set l/rv = ± — e2vmln £0I. v _ ...^ n — 2 and v A v0, 1 jrVo =

- <?(*-№» and

Irn_x =

- e2v°~iln.

If (25) holds for v0 = n —

or if does not exist a positive integer v0 satisfying (25) we set 1 jrv =

— e2vm,n for v =

, ..., n — 1. Assuming X = Xk = ^2fe+0)m/n we obtain re(X/rv) Ф 0 for v =

, ...,% —

and hypothesis (

) of Lemma 1 is satisfied.

We prove, as in the proof of the theorem in [3], p. 169-171, that the functions <pv(x) (v = l , . . . , n — l ) and \p(x) defined as in Lemma

with A (x) = sk 1a~lln(x), with X and rv defined as above, and formed in the case of the differential equation (24), satisfy the conditions limç^Æ) = 0

# x->oo

and tp(%) = o(X ) for x-> oo (X = æ-

ln-1_£æ with some s >

).

4 3 8

Z. P o l n i a k o w sk i

We shall prove that a lln{x)cpn_ 1(x) — o{X) for æ -*■ oo. If bv(x) are defined as in (15) for v = 0, — 2 and if F ( x ) ,F 1{x) are defined as in Lemma

? then

K = ek 2 ( - 1 Г - 1 " j ? [3 m~ )

j = v+ 1 ' ' m = 0 ' '

(compare the proof of the theorem in [3], p. 170). By Lemma

we get

F { n - v - m - l ) F (m) = a i - v - i y n e j - v - l ^ fi- l (w _ ^ m _ 1 ) + 1 ) ( a ~ l l n) ' / 2 - f

+ a~1/n • о(X)}• {( - 1 )ma l!n + { - 1 )п- геъ1т { п - т ) (a~lln) 'allnj2 + о(X)}

= <x

{( — l) m«1/w -f ( — l ) m~l £frl {v + i) [m — (n — v — i)/2] X x ( a - 1/w)'a1/n + o(X)}, since {(a-

/№ )'}

a1/n = 4{(а

= o(X) and (a ~lln)' = o( 1) for a?-* oo compare Lemma 13 in [3]). We obtain

4 = ( 4 1)et «,,'‘ + ( * + i ) { - ( „ * 2) + (» » --* -1 ) ( 4 i) / 2 } ( » - ,,”)V'"+

+ ° ( x > = ( , ” i)«*el," + ( * 7 1) (t, “ 2) ( “'‘ ,,")'®1,n+<>(X) for *-*■ ° ° >

since

1

if j = « +

,

0

if j >

; +

. Similarly we get for s = 1, . . . , v + 1

n j + s - v - 1

j(s) = gk ( i ) i —V—1 (^\ V h + s ~ v ~ 1 \ F ( . n + s - v - m - l ) F (m)

^ ' w=o ' m ’

= a [s- v~1]lnssk- v- 1 {( - 1 )meka l!n + ( - l ) m~l (v + 1 - s) X

X [m — (n + s — v — 1)/2] (a~lln) 'a lln + о (X)},

J — V — 1

2 ( - 1 )’

m=0

j - V - m

A s - v - l b (s) =

llnY alln + о(X) f o r $ = l ,

o(X) for s = 2, ...,г? + 1 if v ^ l .

Let ns observe that conditions (18) are satisfied for A (x) defined as above and for X — x~1ln~1~Bx (compare Lemma 13 in [3]). By Lemma 4 and (15) we obtain

A 8 v 1 Ф{ п-Х^ = о (X ) for v =

, — 2 and s = 0, ... ? v + 1 since L n_1>v = г г . .. (compare the proof of the theorem in [3], p. 170). By Lemma 5 we obtain A~l <Pj0 = o(X) and <P'j0 — o(X) for j = 1, . . . , n — 1, x -> 00, and by (13) we get

A~l (pv{x) = o(X) for v — 1, n — 1.

Solutions of the differential equation

439

Then hypotheses (8 )-(ll) for h(x) = \alln(x)\ and for <pv(x) and tp(x) defined as above, are satisfied.

Applying Lemma 3 for we obtain that the differential equation (24) has for x > x 0 an integral w(x) such that 1шШ(ж) = 1. We set

x->oo

X

y k{x) — м (х )а 11~пУ2п(х)ех-рек f a 1,n(t)dt.

x 0

We complete the proof as in the proof of the theorem in [3].

P ro o f of C o ro lla ry . Suppose that ayk(x) = y l(x ) + iyl*(x), where a = 1 if a(x) > 0, a = e(n_1)îïî/2n if a(x) < 0. We get for x-> oo

(yt + iyl*) (cos B k - i sinB*) | a[n~1]l2n\exp (-<?*)-> 1, (

a) (

/*cosPA. + ^ :,!sinBft)|a[w“1]/

n|exp ( _ CrA) _ 1 _> 0?

(26b) { - y t s in B k + yl*eo sB k)\a[n- 1]l2n\ex-p{-Ck) 0.

Multiplying (26a) by cosB k and (26b) by — sinB*. and adding we obtain 2/*|а[’г~1]/2,г|ехр( - C k) — ^{cosBk -+}

^for

Multiplying (26a) by sinBfe and (26b) by cosBfc and adding we obtain yl*

— Ck) — BinBk ->

for

Applying 1’Hospital’s rule we obtain from (3) a~lln = o(x) and lim|Bfc(a?)| = lim|C*(a?)| =

X—>00

Ж — >co

References

[1] Z. P o ln ia k o w sk i,

Ann. Polon. Math.

5 (1958), p. 1-24.

[2] —

= 0, Comment. Math. (Prace Mat.) 13 (1970), p. 241-248.

[3] —

= 0, ibidem 14 (1970), p. 151-172.

INSTITUTE OF MATHEMATICS OF THE POLISH ACADEMY OF SCIENCES INSTYTUT MATEMATYCZNY PO LSKIE J AKADEMII NAUK