On some linear differential equationsIn [1] and [2] we proved theorems concerning asymptotic properties (for of integrals of the differential equations т/п) =

R O C Z N IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO Séria I : P R A C E M A TEM A TY CZN E X I X (1977)

Z. P

(Poznan)

On some linear differential equations

In [1] and [2] we proved theorems concerning asymptotic properties (for x->oo) of integrals of the differential equations т/п) — a(x)y =

and y^n) — b(x)y' — a(x)y

(w >

). Applying the same method of the proof as in the above papers we may prove the analogous properties of integrals of some differential equation of the form

П — 1

(1) y(n)- У cv(x)y(v) = 0 {æ>cc0).

v=0

In this paper we shall prove the theorem of this kind (Theorem 2) applying another method of the proof based on the properties of integrals of a system of linear differential equations. We prove the following

Th e o r e m 1

. Suppose that

(

) bmv{x) are for x ^ x0 continuous functions complex valued, m ,v = 1, ...,n , n ^ 2, (3) blx(x) = 0, bvv(x) Ф 0 for v = 2 , and x ^ x 0, (4a) |&

(a?)| < A

|re

(^)l some constant K 2^ l , or

(4b) reb22(x) = 0 for

if n ^ 3, then there exist K v > 1 such that

(5) \bvv{x)\ < K v\rzbvv{x)\ for v — 3, n and x ^ xQ1 (

)

00

f lbvv(x)ldx =

Xq

4

oo for v = 2, n, 3 (7)

„1

„ = 2

$ , Ь^1Ьт = y V c g 1, where

ts~\3

Б^](й

and

tf = u Clÿ(œ)\0 as x->oo,

(

)

00

J \bvj(x)\B[$(x)dx < oo

*0

CO

and J \brv(x)\Cl$(x)dx <

xo

for v = 2, n ’j j = 1, n; v Ф r, v ф у , s = 0 , r = 1 in case (4a) and r = 1 ,2 in case (4b).

Then the system of differential equations ft

(9) um — ^ b mv(œ)uv, m = 1, n and x ^ x0

v—l

has an integral йг, ..., йп such that lim щ(х) = 1, lim um{x) — 0 for

x—>oo x->oo

m = 2, ...,n .

We write f(x)\Q if f(x) is a non-increasing function and tends to 0.

Th e o r e m 2.

Suppose that

and that

(10) cv(x) are for œ > x0 continuous functions real valued

for v = 0, ..., n —1, (11) c0{x) Ф 0 and there exists the continuous derivative c0(x) for х ф x0.

(12⁾

(13) (14)

(15)

|(c0 lln(x))'\\0 as #->oo and J |(c0 ll2n(x))'\2dx < oo, xo

14® n)ln(œ)cv{x)110 as x-+oo,

GO

f \ciï+1- n)ln(x)(côlln(x))'cv{x)\dx < oo,

*0

J

2n),n(x)cl{x)\dx <

x0

for

1, ..., n —1.

{We assume Argc)jn{x) — iz/n if c0(x) < 0.)

Then the differential equation (1) has for x > xQ n linearly independent integrals yk(x), h = 1, ...,n , such that

n —l

Ук(я)

n)/2n(æ )expf {ekcl0ln(t) + (lln ) ^ e vk+1c{f +1 n),n{t)cv(t)}dt,

Xq V=1

2/iw)H c f n(x)yk(x), m = 1 ,..., n - 1 , as x->oo, where ek — e2kmln.

I f ieekcl,n(x) > 0 for x ^ x 0, then lim l2/*m,(a?)| = oo.

X -> 0 0

I f ro>ekc)jn{x) < 0 for x0, then limi/j^a?) = 0 , m = 0,1, ...

x->oo

. . . , n —1.

E e m a rk 1. The conditions of Theorem 2 aie satisfied by the functions cv(x) = aA, where p 0 > —n and p v < { n — v —1I2)p0jn —1 /2 for v — 1, ...

. . . , n —l.

E e m a rk 2. It is well known that if W(yx, ..., yn) denotes the Wronski determinant of the integrals yx, ..., уn, then W(yx, ..., yn) =

= A exp f cn_x{t)dt. We have W{yx, ..., yn) = J ^ ± y {xn)... у%п\ where

Xq

vx, . . . , vn is some permutation of the numbers 0,1, ...,n —1. From Theorem 2 there follows that for every term of the above sum we have

X

y(xl) ... з#»> ~ Яр exp f cn_x(t)dt as x-+oo, where Яр are constants different from 0. *o

As in [1] and [2] we may prove the following

Co r o l l a r y

(Oscillation theorem). Suppose that hypotheses of The­

orem 2 are satisfied and im ekclJ n(x) # 0 for x ^ x 0. Then the differential equation (1) has two real integrals ykx{x) and yk2(%) such that we have for x > x0

УыЧх) = {cos (Eft(a?) + (2^ + 0) W7t/n) + 0кт{х)}\с<£т+1~п)12п{х)\ехрСк{х), У&(®) = {sin (Вк(х) + (2

+ 6) miz{n) + r]km{x)}\c$m+1- n),2n {x)\ expGk(x),

m = 0,1, ..., n —1, where B k(x) = imAft(a?), Ck{x) = reAk(x), n—1

*0 ®=1

в = 0 if c0(x) > 0, Q = 1 if C

(

) < 0, and lim ô ^ x ) = lim ^^(a?) = 0.

From (13) we obtain that Лк{х) ~ skclln(x) as

By the fhst hypothesis in (12) we get cy1/n(a?) = o(x) as

and lim |AA(a?)| =

It follows that the functions y ^ (x ) and y(ÿ { x ) have infinitely many zeros X^OO which tend to

The remaining part of the proof is similar to that of Corollary in [1].

We write B k(x) + (2& + в)тт:1п instead of B k{x).

We shall prove several lemmas.

Lem m a

1. Suppose that h(x) is a continuous function, different from 0,

3 3

IM®)|<-£|reft(a?)|, M®)/M®) = Л isB s(x), bz(x)lh{x) isCs(æ) f or

s = 0 s= 0

a?>a?0, where E s(a?)|0 and Gs(x)\0 os а?->оо.

OO 00

Moreover, suppose that f \h(x)\dx =

J

0 0 X 0 Xq

J \b2(x)\Bs{x)dXg< oo for s = 0 , ...,3 . We set g(x)

*0

\bx{x)\Cs(x)dx < oo,

X

= exp f h(t)dt.

Xq OO

Then there exists the integral

OO /?

Xq a

where a = x0, p — t if reh(x) > 0 «md

= t, p =

if reh(x) < 0 for

P roof. Since \g'{x)\l \g{x)\’ = \h(x)\lreh(x), it is easy to see that

(16⁾

X

as #->oo and f \g'(t)\dt ^ K\g(x)\

00

g(x)-^0 as x->oo and J \g'(t)\dt < K\g(x)\

X

if re h(x) > 0,

if гей (a?) < 0 for x ^ x0. The analogous properties of the function l/g(x) we prove setting —h(x) instead of h(x).

In the case reh(x) > 0 we obtain integrating by parts and applying l’Hospital’s rule

oo t oo t 3

/ IM _1I f lgh2jdt1d t ^ j Ig'g~2\ f \gb2\ ^ B sdtxdt

Xq Xq Xq Xq S ~ 0

3 3 OO t 3 00 00

y j

I f f 'i r W / 1 г '|а д 1й*1|“ + ^ / I

S = 0 *J = 0 t *0 S = 0 Xq t /

3 3 X 3 00

< 2 j \g\’B,GHdt + E £ f \b,\Bedt

s = 0 S j= 0 x-+oo a:0 s= 0 x0

3 oo

= J \b*\B s d t <

s = 0 x 0 *

In the case гей.(a?) < 0 we get

OO OO 3 00 oo

/ IM -1! / Ig b ^ ld ^ d t^ ^ J \bxGsg~l\ f \g'\dtxdt

Xq t S — 0 Xq t

3 OO

< К У f \bx\Csdt <

S = 0 Xq

Lem m a 2.

Suppose that bx(x) is a real function, continuous and dif-

oo 3

ferent from 0 for x ^ x 0, \ f b1(x)dxj = oo, b2(x)/b1(x) = isas{x), bz(x)

3 X^q S = 0

= isps(x), where as(a?)|0 as x~>oo, fts(x) are non-increasing and non-

8=0 X

negative, (3s(x) < M for x ^ x 0. We set g(x) = expi J bx{t)dt. Then there

exists the integral

(17)

g(t)b2(t)b3{t)dt = f{x) for x ^ x 0.

X

__ 3

Moreover, \f(x)\ < él/б I f £ as(x) for x ^ xQ.

00

s= 0 3

P roof. We have b2b3jb1 = £ isas £ isfis — E ^ where S8 = ^ a j k{8)

S =0 S = 0 S=0 t>=0

and <5в|0 as x-^oo. (Tc0(s), . k3(s) is some permutation of the numbers

3 V = 0i

0, Moreover, ôs(x) ^ M £ av(x) for x ^ x 0. We shall prove the existence of integrals

j 9(t)bi(t) ô3(t)dt for s = 0, 3.

We set

f s{x, £) = (1 lg{x)) J gb± ôsdt,

*

X t

C(æ) = j J b ^ d l j = z and G(t) — j

&1(i1)d#1 j = a for £ ^ x

x0 Xq

and x , t ^ x 0. Then

cm

/.(» , f) =/,(C ,-i(*), f) = i / а.(0_1(<г))е"<—, й<г

0(1)-0

= <5 / 0

where ô = sgnb^x), о = о^ + я and 0 _ 1(C,(aî)) = a?.

It is easy to see that if ç>(a?)jO as æ-»oo, then

00 oo

J J <p(x)cosx dx J < 9?(0) and 0 < f <p(x)sinxdxs^2<p(0).

о о

We infer from this that there exists the limit

00 00

l i m / e ( a ? , Ç) = â f ôs(C_1((T] +z))eiôaid(тг = (l/flr(a?)) f g{t)b1{t)da(t)dt = f 8(x)

f-wx> 0 x

and we have

Л И

= = es{z)ôs(x),

where |0e(«)| < VE for

> C(x0) — 0 and s = 0, 3.

then

Since by (17) we have f(x) = ] g{t)bx{t) £ i8ôs(t)dt = g{x) £ i8f s(x),

s

S** 0

l/(«)l < 1ЛИ1 < ^5 ôs(æ) < 4*/5 M JT av{œ)

s—O

for a? > a 0.

OO 00

Lemma 3.

I f f \<p(t)\dt < oo and f( x

= f <p(t)dt for x ^ f x 0,~ then

Xq X 3

there exist functions f 8(x), s — 0, .

such that f(x) = JT isf s(œ), f 8(œ)\0

oo S= 0

as i о о and f s{x) ^ / \<p{t)\dt.

P ro of. We set x

Л И = i f {lreçj(t)l+re<p(t)}dt,

X

OO

/ i H = i f {iim(p(t)l-him(p(t)}dt,

X

00

Л И

= i f {lre<p(t)l-re<p(t)}dt,

X

00

Л И = i f {lim<p(t)l—im<p(t)}dt.

X

Suppose now that the functions bmv(x), m , v = l , . . . , n , satisfy hypotheses of Theorem 1. We set, for x ^ x 0, gx{x) = 1, дт {я)

X

= e x p (- f bmm(t)dt) for m = 2, n,

9m(®) f gm(t)bmj(t)u(t)dt for Ш =£ j ,

0

for m —j ,

m ,j = 1, ..., n, where am = xQ, pm = x if reômm < 0 and am = x, 0m =

if re&mm > 0,

П

(19) Jjg(tt) = У Xv for * = 1, 2 ,... and w, j = 1, ..., w,

»=i (18) JffiM =

where Àv = 1 if reôw < 0 and /„ = —1 if rebvv^ 0.

Lem m a

4. Suppose that the functions bmv(x), m ,v =

n, satisfy hypotheses (2), (3), (4a) and (5)-(8) of Theorem 1.

Then there exists хг > x0 such that

(20) |J£ !l < 2 - ‘

for œ > œx, m = 1, ..., n and Те = 0,1, ..., where (1) and J lmi(u) are defined by relations (18) and (19).

P ro o f by in d u ctio n . By (7) and Lemma 1 there exists œx > a?0 such that we have for a? ^ xx and v = 2, ..., n ; j — 1 ,..., n j v Ф у.

00

@ V

\Ь^1Ът \ ^ Ц 2 { п - 1 ) К т and J Iblvg~x\ J \gvbvj\dtxdt < | ( w - l ) " 2

Ж a v

where av = œx, (iv = t if rebvv < 0 and av = t, = oo if re6w > 0.

Suppose that 7s ^ 2 and |J£\| < 2“ s for every s < Tc and a? > %x.

Then

■ A? = t)=2 where

W ° J ( 4 ‘r 1,)l = I J f f i ( 2 V g v r 2I))|

5 = 1

V

oo pv n

=

S»».-1 /

Ж a„ 5 = 1

j¥ = V ft II' woo /3,

22~k^ J \blvg~l \ j \gvbvj\dtxd t ^ 2 - k/ ( n - l)

5 = 1 ^X

i^v

and relation (20) follows for m — 1.

In the case 2 < m < n we have

J j a = 2 ) ^ H Î - U) = 2 ^ W ( « ) / я«ът^ ! г пт

5 = 1

5Vïft jVm

and for a? > a?!

WSl < а - У te‘ (®)l J « 2 ^-*,

fcm “m

since \gmiœ)\ f \ ^7ft

m\dt < ï m, by (4a), (5) and (6) (compare also (16)).

Similarly we prove (20) for Tc = 0 (and m — 2 , . . . , n). In the case am Tc = 1 we get for cox

И 1., 1 < 2 ' И ? , ( 5 Я ,ж У ’ / I

Ъц дт'\ J \ÿjbn

2

and for 2 < m < n

n n Pm

Æ i K £ \9 m (œ )I / \9mbmjlbmm\d t^ ( n - 2 ) / 2 ( n - l) < 2 - \

y=2 J=2 am

jVm

Lemma 5.

Suppose that the functions bmv(x), m, v = 1, n, satisfy hypotheses (2), (3), (4b) and (5)-(8) of Theorem 1. Then there exists x2 > x0 such that for x ^ x2 there hold relation (20) for m = 1, ..., n and & = 0,1, ..., and the relations

3 3

(21a) J™ = j [ V a fe(«) + es (1* ) j r ,£M(æ) = Им+Яи> 4 = 1,2,

s = 0 s = 0

3 3

( 2 1 b ) + = St*3+8tM, 4 = 0 ,1 ,...,

s = 0 s = 0

where aks{x)\ 0, ^ s(a?)jO as x->oo, aks ^ 2~k~2, f ks ^ 2~k~2,

and |0£| < 22“ fcVr5.

P ro o f by in d u ctio n . We choose x2^ cs0 such that we have for x > x%

CO

|B g ( æ ) |< 2 - 4, |С |? И |< 2 - 6, / |61, | ^ ? * < 2 - 9/ ( » - l ) ,

• X

со

J |621|B[ÿ<j«s;2-5/ ( » - i ) ( s = o , . . . , 3 ) CC

and in the case n > 3 (cf. the proof of Lemma 4):

/ l^ & r 1! / ISfv&„i |d«1d # < 2 " 4( n - l ) - 2 and Ъ^Ът < K n - 1 ) " 1 Ify 1,

Ж a v

v = 3, ..., n; j = 1, ..., n; © # j and r = 1,2, where av = x2, fiv =■ t in the case re&w < 0 and av = t, f v =

in the case reôw > 0.

Suppose that Tc > 2 and that relations (20), (21a) and (21b) hold for the indices Tc—1, ..., 0 instead of h. Then,

oo

J I M ? - 11) = / »«(И *_1,»+Я *_],1)<й.

X

By Lemma 2 we get for x > x2

oo oo 3 3

£C ^{X S =} 0 S = О I

By hypothesis we obtain for x >

2

oo oo 3 oo 3

/ I » A -

I < » = / l&12C . l i > W A < 2 3-V 5 /

x x s= 0 a: s = 0

< 2~k~2f(n—l)

and by Lemma 3

oo 3

f bljt% _ 1>4dt = JjV y B O »),

X S= 0

where yj^jO as x-+oo and y|f2] < 2~k~2j(n — 1), s = 0, It follows that

(22)

Л5, Й ? - 11) = ^ i V Ë + e ^ s g i .

s = 0 s= 0

In the ease n > 3 we get for v = 3, ..., n as in the proof of Lemma 4

n OO Pv

И 0„1( 4 ? “ 1,)1 < 2 г- й2 ' / Ifti.ff»1! /

3 = 1 X a v

ЗФп By Lemma 3 we infer that

(23) J K V r 11) = i > r l ë ,

s = 0

where уЩ о as x->oo and уЦ < 2~k~2l(n — 1), * = 0 , ...,3 .

By (19) for m = j = 1, (22) and (23) we obtain relation (21a). Since

3

0 < ak8 < 2~k~2, then |UtA1| < 2~fe_3/2. Moreover, we have |5ÏÜ2| = )вА B $ |

3 = 0

< 2~*~V5 and we get from this relation (20) for m = 1.

Similarly we have for m = 2 OO

X

By Lemma 2 we get for œ > x2

oo oo 3 3

I /

9bbn%k_ltldt\

x x s = 0 s=0

We obtain as above

/ = / 1»я в*_1| ^ в Ё 1* < 2 3- У б 2 ' /

X X S = 0 S = 0 X

< 2 “ * “ 2/( w - l)

and by Lemma 3

oo 3

ÆT^®) / 02&21%fc-l,2^ = 02_1 (®) ** 45 >

X s = o

where <5$ jO as &->oo and <5jJ < 2~k~2j(n —l), s = 0, 3.

И — P ra c e M atem aty czn e 19 z. 2

It follows that

■tfiVu-11) = г г » У « 0 + 0 » У е й ».

s= 0 s = 0

If n ^ 3, then we get as above for v = 3 ,..., n

n oo Pv

i=x £C av

ЗФ V and by Lemma 3 we obtain that

3

AV(J?rn) = г г » . у «0(®>,

s —0

where as a?->oo and <5j$ < 2~k~2j{n — 1), s = 0,

By (19) for m = 2 and j = 1 we get as above relation (21b) and relation (20) for m = 2.

In the case we prove relation (20) for m = 3 ,..., n as in the proof of Lemma 4. It remains to prove relations (20), (21b) for к — 0 and relations (20), (21a) and (21b) for к = 1.

Suppose first that к = 0. By Lemma 2 there exists the integral

OO __ 3

Г

Since [Jrj,°1,|

4Vo <7|f, there are satisfied relation (21b) (with

X 8 — 0

Pos = 0) and (20) for m = 2.

If n > 3, then we have for m = 3, ..., n

fjm

Æ l ! < \9m{®) I / \9mbmilbmm\dt < 1/2 ( w -1 ) < 2°.

am

In the case к = 1 we get

oo 3 oo

/ |6„|Ü0<« < 2" V 5/(« - 1 ) .

X S — О X

If n > 3, then

и. п OO (S,v

t>=3 v = 3 x a y

< 2~4/(n —1).

By (19) for m = j — 1 we infer from this that \J[^\ < 2~3 (w > 2)

and relation (20) is true for m = 1. By Lemma 3 is also satisfied (21a)

with O^æ) = 0.

Similarly we have ( 0 U® =

if n = 2,

n n oo Pv

\hv9vl\ J IdvKild^dt

V=3 V=3 X av

, < { n - 2 ) 2 ~ * / ( n - l ) 2 < 2 ~ 3 i f w ^ 3

and there are satisfied relation (20) for m = 2 and (21b) with в* (a?) = 0.

If n > 3, then we prove relation (20) for m = 3, ..., n as in the

proof of Lemma 4. ^

L

6. I f /?„(а?)|0 as æ->oo, v = 0,1, and the series £ /5v(a?)

= fl(cc) is convergent for a?>a?0, then /9(a;)|0 as æ->oc. v=0 P roof. It is easy to see that (i(oc) is a non-increasing function. For a given e > 0 we choose N ^ 1 such that OQ /?„(a?0) < e/2.

oo v = N

Then ^ /?„(<»)< e/2 for œ ^ æ 0. There exists aq ^ a;0 such that

N —l v = N N —l oo

2 j8e(®)< e/2 for a? > aq. We obtain /?(a?) = ] } p v(œ) + J J /5„(а?)< e/2 + e/2

t ’ = 0 r = 0 t > « = V

= e for sc > aq.

P ro o f of T heorem 1. Integrating the system of differential equations (9) we obtain the system of integral equations

(24) um(x) = c J g m(æ) + Xm^ J l2 [ u v{œ)), m = 1, ...,n ,

«=i

where J ^ v{uv) is defined by (18) and cm are constants. ^

By Lemmas 4 and 5 there exists aq > x0 such that the series Jj*|(l), m ~ 1, ...,n , are uniformly convergent for ж > aq. We set

= i - 2 ’J S Id ) , 1

üm(œ) = V j£ l ( l ) for w = 2 ,

The functions üm(a?), m = 1, satisfy for a;> a q the inequal­

ities |üm(® )|<2.

We shall prove that there exist the integrals in (24) for um(æ) = um(x) if r

We consider first the case (4a). By (19) we get fori; = 2, ..., n

OO OO 00

=

x x k—Q

oo pv n oo

= л „ / ь . л 1 / д.{ьл + 2 *

£C a- i = l fc=l

J¥=V

and by Lemmas 1 and 4 there exists the above integral. We set here av = x1, = t if rebvv < 0 and av = t, =

if re&OT > 0.

If теЪт т > 0, then we have for m = 2 , n; v = 1 and v Ф m

OO oo

f \gmbmvüv\d t^ 2 J \g'mbmJ b mm\dt < oo ( x ^ x j ,

X X

by (7) (compare also (16)).

In the case (4b), by Lemma 5 we get

oo 3 3

®.(e) = - У 4 ? а ) = -г г Ч ^ У ^ А М -в 'и У с й ’м ,

k—0 s = 0 s = 0

oo oo

where p8{x) = £

and 6*(x) = £ 6l(x) for x > хг. We have |0*(a?)|

_ *=o *=o

< 8^6 for х ^ х г and by Lemma 6 we obtain that ps(x)\0 as

s

— 0, ..., 3. By (7), Lemma 2 and (8) we infer from this that there exists

the integral f b12û2dt =

x oo

If n ^ 3, then we prove the existence of integrals J blvüvdt = (uv) for v — 3, ..., n, a? > xx, as in the case (4a). x

By Lemma 2 there exists the integral / g2b21dt. By Lemma 1 there

exist the integrals æ

oo oo n OO Pv oo

/ = у ч J / Я.ь„ y

x к^{= 0} j = l x av к = ¹

j^ v

for v — 1, ..., n and v Ф 2. We infer from this that there exist for those indices v the integrals g

{x)Jl°J(üv) = J g2b2vuvdt.

X

If n ^ 3 we prove as in the case (4a) the existence of the integrals

OO

/ gmbmvuvdt for those indices m ^ 3 for which is re&mm > 0 and for v

X

= 1, n; v Ф m.

It is easy to see that the functions um(x) satisfy the system of integral equations (24) for cx — 1 and cm = 0, m = 2 , By (19) we have namely for m — 1

n n 00 oo

1 - y <*£!(*.) = i-yj£j(A„y<n*.’ (i)) = 1 -У 4 Г '( 1 ) = »,

v=2 v=2 k= 0 k= 0

and similarly for m = 2, ..., n.

The expression \{uv) is of the form

J lmv(üv) =g{œ ) f <p(t)dt (\g(x)\ = 1 )

X

if m = 1 and in the case (4b) if m = 2 . Then lim J jj\(uv) = 0 for v = 1, ...

x->oo

In the remaining cases, by (7) we get applying l’Hospital’s rule

Pm Pm

l^rnv (^e)l

\dm (^01 J Igm^mv^vl^mmldt ^ ^ K m\gm (^)||^ lf/ml l^mvl^mml dt J —>-0

am am

as x->oc. We set here am = x1} (im — x if reЪт т < 0 and am = x, (im = oo if re&mm > 0.

By (24) we obtain limü^æ) = 1, lim um(x) = 0 for m = 2 ,

X->oo X-+CO

Lem m a

7. I f n is a positive integer, к

1, n — 1, ek

e2kniln

n

£ С - Л = n U m

for

..

then wm

^

r = l V=l

P roof. We have П

V=l

(i

ÿ c - i +i ÿ 4 - i wj

V=1

1

= № ) wj ^ 4 - ? 4-1 = № ) J ] щ £ 4 -T = щ

j = l V — l j = l 17 = 1

since ek = ek.

Lem m a 8.

I f ek is defined as in Lemma

then

2 ! v4 - i

v—l

пЦер - 1 )

{nV)

if p ф 0(modw), if p = O(modw).

The proof of the first equality follows from the identity П

= (1 — xn+1)/(l — x)2 — (w +l)æ n/(l — x) (x Ф 1)

V=1

in which we set x = ep.

Lemma 9.

I f

as x->oo, s

then there exist for suffi­

ciently large x functions fis(x), s =

such that

as x-^oo and

3 3

(25) l / l l + 2 ' * , « ,W ) = 1 +

s=0 s=0

for large x.

P ro o f. We choose x0 such that as(x) < 1/8 for ж > ж0, s = 0, ..., 3, and that there exist for ж > ж0 functions /?в„(ж), where fi8V{x)\0 as æ->oo

3 3

and ( — isa3(x))v = JT1 isp8V(x) for v = 1, 2, ..., s = 0, ..., 3. We have fisvioo) — a”3, where = v, and in the above sum appear

з^{= о}

4e_1 terms. Then we get Psv(x) < ^ “ '•(i)® = {\)2~v for s = 0, .. . , 3 ;

00

v = 1, 2 , . . . ; ж > ж0, and the series ^ &о(ж) are convergent for ж > ж0.

0 = 1

00

The functions Д,(ж) = J / /#8„(ж), $ = 0, . ..,3 , are non-increasing, non-

0 = 1

negative and satisfy relation (26) for a? > ж0. By Lemma 6 they tend to 0 as ж-> со.

P ro o f of T h eorem 2. We write the differential equation (1) in the form of the following system of differential equations:

= 0, (26)

V

< «

; 0+1 7

n —1

*3 + 1’

3=0

where zx — у and zv+1 = у(t,) for v = 1, ..., n —1.

Por a given k (1 < Jc < n) we substitute into (26) for ж > x0

X

zv{æ) = wv(a?)£^~1c(0“ 1" n+2*)/2” (a?)exp J jeftcJM(J) +

*0 ^ n-l

3 = 1

where

=

=

and we obtain for -y

1, ..., n —1

{(4-1- n+2e)/2n(a?))/

+

+ ( l / n ) J ? An~J'~1(x)cJ (c П—1 u)]J + w'v(x)ek~1c(0~1~n+2v)/2n(x )

3 = 1

+i(*)evA - n+2v)/2n(x)- Multiplying the above equation by ек ”с{* l~2v),2n(æ) we get

. n —1

wv{x)^n + 1 — Щ А ' (ж)/2 +1 ⁺ (1 ^/п) An~j (ж) Cj (ж) J ⁺

3 = 1

+ A(x)w'v(x) = wv+1(x).

From the last equation in (26) we obtain similarly

ft — 1

wJæ){{l-n)A'(œ)l2+l + (lln) £ АпЧ((в)ц(а})} + A{x)w’n{oo)

П —1

= i(«).

j=i

j=0

(27)

The obtained result we may write in the following form:

Aw'v = avwv + wv+1, v = l , . . . , n —l , 71—1

Awn + ^

3=i

n—X

where — n —l) A '/2 — (1/n) ^ An 3Cj.

For a given m (1 < m < w) we multiply the v-th equation in (27) i=i by e^l1! , v = 1, n — 1, and the last equation in (27) by . Adding the obtained % equations and setting » e^l1! = ?што we get for

V — 1

m = 1 ,..., n

n n—1 « —1

nAu'm = J ? eV m-lWv+l+£m'-\ £

u+l

t?= l

« = 0 V=1

= ÿ ( c - > , + c i 1)® „ + c t.i1ÿ ; = « ! + / » ,.

r = l r = l

Applying Lemmas 7 and 8 we obtain

n. n n n

Sx = У (С-Л+4Г-2.) y $ - l +1» i = У% У(4-11«,+СЛ)«Й

V=1 з = 1 3 = 1 t?=l

n n n n

= = y % y « r - / A ' -

y = l t?=-l y=*l V=1

n —1

- } l + (*. +

) A

+ (

/») y

- C - \}

V=1

= nA’ ÿ %/ ( e „ - j - l ) - { l + (!/«) У ^ " ' Ч - С Л }

3 = 1

ЗФт

n — 1 n

S , = < й - \ у ^ " - 'с ву ^ - ,+1

Ю=1 3 = 1

n n —1

=

- 1. y % У # - Ч * Г ' +

+ « » У ' ^ - Ч й - Г 1-

- J + l

«=1 W m.

n—1

3 = ¹ v = l з Ф т

We obtain the system of differential equations (9), where we have (writing in and $ 2 the index v instead of j and conversely)

71— 1

A ' l(em- v- l ) A + £ sJ~v+1An~j~l Cj if v + m ,

j = i

W —2

n J A + a / n ) у « г / - 1 - 1 )An- ^ 1cj if v = m, ' J = 1

and rjv = s^zl — 1.

We shall prove that the functions bmv(x) satisfy hypotheses of The­

orem 1. It is easy to see that Ь1г(х) = 0 for x ^ x0. By (13) we get for v = 2, n: bm(x) ~ r ] v/A(x) as x-+oo. Then bvv(x) Ф 0 for sufficiently large x and hypothesis (3) is satisfied.

Trom the first relation in (12) there follows A(x) = o(x) as x-^oo and hypothesis (6) is also satisfied.

We have

rjv/A(x) = (e41~v)™ln- l )e2knilnclln(x) and

av = SbTg^t]v/A (x)) — (27cф 0 + 1 — v)nln —^{t c}/2,

where в = 0 if c0(x) > 0 and 0 = 1 if c0(x) < 0 for x > x0.

Suppose now that

(29) 2& + Ô+1 — v Ф 0 (modw).

Then \rjv/A(x)\ == L v\re(rjv/A(x)) |, where

L v = l/|cosa„| = l/|sin(2& + 0 + l — v)ujn\.

Let us notice that from the identity

rebvv = TG(r]vIA)re(bvvAlr]v)-im (f]vIA)im(bvvAlr]v)

we obtain rebm ^TQ{r}vfA) as x-^oo. It follows that \bvv\ ^ K\rzbvv\ for sufficiently large x and those v for which (29) holds. We have here К

> т а х 1 и and hypotheses (4a) and (6) are satisfied.

V

If there exists the (unique) index v = v0 (2 < v0 < n) such that 2k + + 0 + 1 — v0 = 0(modw), then те(г)„[А(ов)) = 0. Moreover, we have

arg (x) —1) = (j +1) (2k + в + 1 — v0) n/n — 6n —

,

T e A n~ i ~1( x ) ( s ^ ~ J ï1 —

1) = 0 for j = 1, — 2 and

and we get mb4 4 = 0 for x ^ x 0.

We may assume v0 = 2. In the caae n > 3 this follows from the fact that, as we shall prove below, the functions brv(x) satisfy the first

(28) bm =

conditions in (7) and (8) for r = 1, ..., n and conditions (7), (8) remain satisfied if we write the index 2 instead of a fixed index ц (3 < /л < n) and conversely. Then hypothesis (4b) is satisfied.

By (28) we get for r = 1, ..., n; v = 2, ..., n-, and г Ф v

n —1

A(x)brv(x)/Vv = (orv + iTrv)\A'(x)\ + £ (^К] + ^ ]) Ц п“ *(®)с*(<»)|,

k= 1

where arv, rrv, and are real constants. Then (30) brv{x)/bvv{x)

n—1

= {orvAiTn )(pv{x)\A'(x)\+<pv{x) £ {a[^ A irf})- \A n- k{x)ck{x)\

&=i

where <pv{x) = ?]v/A(x)bvv{x). By (28) and (13) we get

3

A(x)bvv{x)/rjv = 1 + ] ? i sD l*](x),

S—0

where D ^ (x )\0 as a?-*oo, and by Lemma 9 we obtain that there exist for large x functions (x), s = 0, ..., 3, such that 9p[f](a?)|0 as x-+oo

3

and <pv(x) = 1 + is<p№(x). By (30), (12) and (13) first hypothesis in (7)

s = 0

is satisfied. Moreover, there exist positive constants and such that we have for large x

n —1 n —1

fc= l fl= 1

for r = 1, ..., n; v = 2, ..., w; j = 1, ..., n; г Ф v; v Ф j ; s = 0, ..., 3.

We infer from this that

n —1

K \B % < M „N ,j [\A V I\A \+2 \A’\У \A *-k- 1ck\ + fc= 1

n —1 n —1

+ У 2 u s* - ‘ - ' - 4 < g .

k = l ii—1

By the second relation in (12), relations (14), (15) and the inequality ab < i ( a 2 + 62) we obtain that the first hypothesis in (8) is satisfied. In a similar way we prove that the second hypotheses in (7) and (8) are satisfied.

Applying Theorem 1 we obtain that the system of differential equations (9), where bmv(x) are defined by (28), has for sufficiently large x an integral um(x), m = l , . . . , n , such that lim ux{x) = 1,

x-+xx>

lùïi йт (х) = 0 for m = 2, ...., n.

Setting гот (х) — £ e” ^ l uv{x) we infer that the functions wm(œ) satisfy for large x the system (27) and lim wm(x) = 1 for m = 1 «=i

x->oo

(cf. Lemma 7). Then the function

X n — 1

Ук(х) = щ (х)с^~п)12п{х)ехр f {ekclQln{t) + (lln) ^ АпЧ~1{Ь )с ^ Ш

X0 3 = 1

satisfies for large x the differential equation (

) and has with its deriva­

tives y ^ {x ), m = 1, n —1, the desired asymptotic properties. We have namely for m — 1, n

y P = * „ + . = ® „ +

# < r * +I” ,B» e x p / ( - ) *

~e%c™tnyk as <

->

.

The integral

/* (a?) may be extended to the point a?0.

We shall prove indirectly that if Jc± Ф Jc2, then the integrals yk (x) and ykz(x) are linearly independent. Suppose namely that ykl(x) = oyk2(x).

Then from the asymptotic relation there follows the existence of the integral

OO

/ lco/n(^)l(l + -\-i<P2(t))dt,

Xq

where <px(x) and

o2(œ) are real functions which by (13) tend to 0 as a?->oo,

and we get f \clJ n{t)\dt <

. We obtain the contradiction since from the first relation in (

) there follows с^11п(х) = o(x) as x~>oo and \clJ n(x)\

>

Jx for large x.

Finally we have y ^ +1H x)/y^ (x) ~ ekcl,n(x) as x->oo, m =

,1 , • ••

— 1 (in the case m = n — 1 we apply for the proof (1) and (13)).

Setting y ^ +lHx) ly ^ (x) — Y (x) we get

re Y = re ( Y /ек c)jn) re ( ek clJ n) — im ( Y Iek cJM) im ( ek c)jn) .

Since lim im( Yfekc}jn)

and im ( cJ/n)/re( clJ n) = const, we

X-+O0

obtain that

re Y (x) = |у1т)ИГ/1Йга)И1 ~ reskclln{x) as x-+oo.

X

Applying l’Hospital’s rule we get In\yikl)(x)\ ~ reek fclln(t)dt-+ +

or

x'0

—

as X-+

. It follows that lim tyffiix)] =

if reeÆcJ/n(a?) > 0 and

x-yoo

lim y ^ ix ) =

if ree c)jn{x) <

for x ^ x0, m — 0,1, ..., n — 1.

X -> O Q

References

[1] Z. P o ln iak o w sk i, On solutions of the differential equation y(n) —a(x)y = 0, Comm. Math. (Prace Mat.) 17 (1974), p. 429-439.

[2] — On the differential equation y№ —b(x)y' — a (x)y = 0, this volume, p. 311-321.

IN ST Y T U T M A TEM A TY CZN Y P O L S K I E J A K A D E M II N A U K

M A TH EM A TIC A L IN S T IT U T E O F T H E P O L IS H A C A D EM Y OF S C IE N C E S