On some system of linear differential equations

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PEACE MATEMATYCZNE X X (1978)

|Zb ig n ie w Po l n ia k o w s k i

O n some system of linear differential equations

In [1], Theorem 2, we considered asymptotic properties (as ar->oo) of integrals of the differential equation

П—1

y ( n ) _ s ? cv{ x)y W

= 0, v = 0

which may be written in the form of the system

Zv = Z

,+1 for

1,

1

П

Z'n )

7 = 1

where

=

In this paper we shall generalize (Theorem 1) this result for the case of the system

n

(1)

1, .. .,

and

From Theorem 1 we obtain an oscillation theorem (Theorem 2).

In Theorem 3 we consider asymptotic properties of integrals of an inhomo­

geneous system of linear differential equations.

We shall introduce the following definitions. Suppose th a t the func­

tions

= 1 are defined for

We set for

W —1

B ( x )

= / 7

8

8

1

8=1

П — 1

(2 )

J 7 «Î3+1 (®) 4 ? (®) for

. .. ,

s = 1

where

K s

=

e = l , 1,

Kn =

—(w-j-l)/2w +v/w

and

П П

a*(») = (lfn) £ <%~vavp(x)Pp {x)l(ïv{x), Je = l , . . . , n ,

' v = l p = l

where efc = e2ft7U/n (cf. ( 1 1 )).

Th e o r e m 1.

Let us suppose that

(3) avp(x) are real valued and continuous functions for х ф x 0 «mÆ

у

»3> = 1 » •••»

(4) avp(x) Ф 0 and there exist continuous avp(x) for х ф x0, ( 5 ) |Б~ 1 ,пл^р/л15Р| |0 as х->оо,

(6) / |В 1/л(«)| (*)/л^р(*))2 < оо,

х 0

where hypotheses (4)-(6) are satisfied for p = r + 1 (v = 1, 1) and v — Щ p = 1. Moreover, suppose that we have for v, p — 1, ..., n, except the indices occurring in hypotheses (4)-(6)

(7) IВ llnavpfiplpv I jO as x-+oo,

00

( 8 ) / |В - 1 '“(< )К < *)/У *Ш *)) 2 |<И<

*0

Then the system (1) has n integrals {ylk, ..., ynk], h = 1 , ..

we 7m w /o r x->oo

X

(9) y vk(x) ~ e vk~1pv(x )e x p J ak ( t) d t.

*0

The system of the functions yvk(x), v, Je — 1, n, is fundamental.

If теекВ 11п{х)> 0, then lim \yvk(x)\ = oo; if reekB lln(x) < 0, then

#—►00

lim yvk(x) = 0 for r = 1 , ..., n.

x-*oo

E e m a r k 1. By (11), hypotheses of Theorem 1 are satisfied by the functions ayp(x) = хг”р, where B > —n,

p - i

r*P < ]£ rS)S+i - ( p - r - l / 2 ) E / n - l /2 if p > v + 1 ,

8 — v

v -1

rvp< - £ r8,e+ i - ( P - r - 1 /2 ) В In - 1 / 2

8 = P

rvv< B j 2 n —1/2.

n— 1

We set here В = £ ^.a+i + ^ni-

S = 1

if p < v ,

E c m a r k 2. I t is well known th a t if У и У 12 ••• У m Уп1 Уп2 ••• Упп

X

then D = A exp f (an + ... + anJ dt = 2 ± у щ У2к2 •.. Упкп, where kn is some permutation of the numbers 1, x0 n. From Theorem 1 it follows

X

th a t for every term of the above sum we have ylk ... упкп ^ Лр ехр f (axx + ann) dt as x-^oo, where Xp are constants different from 0 . Namely xo we have fix(x) ... fin(x) = 1 and ax(x) + ... + an(x) = alx(x) + . . . + ann(x).

Th e o r e m

2. Suppose that hypotheses of Theorem 1 are satisfied. Moreover, suppose that im ekB lln(x)

0 for x ^ x 0. Then the system of differential equations ( 1 ) has two real integrals {y , ..., y*k} and {y*k , y*£} such that

У * М = {со 8 (Бл(а?) + У + 0Рк{а>)]\р,(а>)\ехрОл(а>),

У*1 (®) = {sin (Bk {x) + tvk) + r}vk (a?)} \p, (x) I exp Gk (x) for x ^ x 0,

X X

where B k(x) = im / ak(t)dt, Ck (x) = re J ak(t)dt, tyk = 2Jc(v— l ) n l n +

гг0 Xq

+Argf}r{x) and lim ôvk{x) = l i m ^ A .(a?) = 0 for v = 1 , ..., n. The funo-

X-+00 Z-~>OQ

tions ytu(x) and y*k {x) have infinitely many zeros which tend to oo.

Th e o r e m 3.

Suppose that the hypotheses of Theorem

are satisfied.

Moreover, suppose that the functions f r{x) are continuous for x ^ x 0, f v(x)

= o(Blln(x)j3v(x))for v — 1 n, x->oo andreekB lln(x) Ф 0 fo r k = 1 , ...

n, x > x Q. Then the system of the differential equations П

( 10 ) y'„ = ^ a vj(x)yj + f v(x), v = l i

has an integral {yy{x)} such that yp{x) — o[($v(x)) as x-+oo, v = 1 , . . . , n . P r o o f of T h e o re m 1. We obtain from (2)

n — 1

/ Ш = П K n - K = < * -» )/» ,

8 = 1

K

- K = (p - v ) l n + E , - E p , 8 = 1 , . . . , n - 1,

V >

s, where

0 if

and similarly for F7p. We infer from this 0 if v, p > s

*

1 £ II _{- 1} if v < s < p 1 if p < s < r I t follows th a t

( И )

fB lln if p = v + 1 {v = 1 ,. . . , n — 1) and p = 1, v = n ,

p - 1

a~l+l if p > v + 1 ,

S = v

v - 1

®vp ftp I ft

a^ - ^ f ] a s_s+l if p < v .

' s= p

For a fixed index 1c ( ! < / . ; < n) we substitute into (1)

X

y v{x) = w,

4"1 ft,(ж)exp J a*(<)d«, v = 1,

*0 and we obtain the system

П

еГ1 (ft + afcft) + evk- l pvwv = V avp8%-l Pp wp , n

p=i

(

12

«V = -(«fc + ^ /ft)w „ + ]?a>Vp 4 " ( f t / f t ) ^ .

p

=

We multiply the v-th equation in (12) by e^_\ (for a fixed m satisfying the inequality 1 < m < %), r = 1, ..., n. Adding the obtained n equa-

n *

tions and setting £ C - i wf = num we gel V = 1

n n n

H U m 7Ш,к И т &m— l ( fiv I f i v ) ~~Ь 1 ®*vp ( f t p I f t v ) ^ p •

r = X t> = 1 p = 1

n

By Lemma 7 in [1] we have wv = £ s*l{+1Uj and

• i=i

n n

in = - n a kum- ^ B vmiM ^ /ft) ^ e “r f +1% +

v = l 3 = 1

n n n

+ У СЛ У «.„«Г'(АЖ) У 4 - { +1щ

V ~ l p = * 1 J = 1

n w n n n

У Ч H 4 3 f i l P ' + У Ч У ^ C ‘, « r v , - t 4 f c A .

J=-l >’=1 J = 1 v= 1 ^5 = 1

Prom the last sum we evaluate, applying the first relation in ( 1 1 ), the following sum (for v = n, p — 1 and p = v + 1 (v = 1 , . . n — 1 ), with fixed j)

П —1

£m -1l eA: П a n l f i l l fin + s m - l s k £v 3+1 a v ,v + lfiv + ilfiv

V = 1

V - l n - j + l

&vn_1 ^11 ^{m - j _}

0 if m Фу,

n ^ 1ekB lln if m = j . Since

П j

S A l f i . = 0 >

we obtain the system (13)

where

П

um = bmj (x) Uj, m — 1 ,

/=i ?

(14) ^ (a?)

/ /» /*< «

| ( - 1 /») y < E ? / » ; / f t + ( i / » ) у * у Ч - Д ^ Й + Ч р / Ш if »»* i>

I y=i r=i i?=i

A, (») + (!/») У * У* ( с л - i )<

,

/

.

*’=1 £> = 1 if m = j .

71 71

We set here xm(a?) = (вт“ \ —l)e*.B 1 /n(a?)> in the sum ^V* we omit v = l p = l

the terms for v = n, p — 1 and p = v + 1 (v = 1 , n — 1 ), and we split the expression for ak{x) into two sums:

П П

ak{x) = (1

v p ~ l

П —1

+ (1/W)|ejfc

V = 1

P = l

We shall show th a t the functions bmj(x) defined by (14) satisfy hypo­

theses of Theorem 1 in [1]. I t is easy to see th a t bn {x) = 0 for x > x 0.

By the first equality in (11) and (7) we obtain th a t bmm(x) ~ Ят(х) as x->oo, m = 2, n. By (5) we get jB~l,n{x))' = o(l) and B~1,n(x) = о (a?)

C O o o

as x->oo. We infer from this th a t J \Blln(x)\dx = / !&„»,» (а?) |Д® = oo,

*0 *0

14 — Roczniki PTM Prace Mat. XX.2

for m = 2, n, and hypothesis ( 6 ) of Theorem. 1 in [1] is satisfied.

Moreover, by (4) we get bmm(x) Ф 0 for sufficiently large x , and hypo­

thesis (3) of Theorem 1 in [1] is also satisfied.

We set a = argAm(fl?) = {2кф 0+ 1 — т)п/п — n/2, where 6 = 0 if B(x) > 0 and 6 — 1 if B(x) < 0 for x ^ x 0. Suppose th a t we have for some m (2 < m < n)

(15) 2 fc + 6 -f-l — m ф 0-(mod n).

Then |Am(a?)| = L m\veXm(x)\, where L m = l/|co sa| = l/|sin(2fc + 6 -f + 1 - m )n [ n \. From the equality re bmm = re (bmm/Am ) r e Xm - im (Ът т / Am ) im Xm we get таЬтт(х) ~ reAm(x) as x-+oo, since limim(DM / 4 ) = 0 and

X -+ O 0

\hnXm(x) /reXm(x)\ < L m. Choosing K m such th a t K m > Lm we have | 6 mm(æ)j

^ К т\теЬтт(х)\ for sufficiently large x. If (15) holds for m = 2 , . . . , n then hypotheses (4a) and (5) in [ 1 ] are satisfied.

Suppose now th a t there exists an (unique) index m = m0 (2 < m0 < n) such th a t 2fc-f-6-fl —m 0 = 0 (m o d ^ ). Then геЯ^о(ж) = 0 and we have for v, p = 1, .. . , n and v Ф p

= (p — r)(27c+ 6 + 1 —w 0 )u /^ + sgn(r—p)*rc/2.

By ( 1 1 ) we obtain th a t r e ^ e ^ l j — l)e%~vavp{iplfiv} = 0, and by (14) we get rabmomo(x) = 0 for x ^ x 0. As in the proof of Theorem 2 in [1] we may assume th a t m0 ~ 2. In this case hypotheses (4b) and (5) of Theorem 1 in [ 1 ] are satisfied.

By (14) we obtain for r = 1, ..., n-f m = 2, . . . , n and г Ф m:

n n n

(16) Ьгп(х)Цт(х) = У + X X r„ „ B - ' I ' a . M P

v

= 1

3

= £ i sI)rms(x),

s = о

where arv and xrvp are complex constants and by (5) and (7) we get A w ( # ) i0 as Setting <pm(oo) = Xm(x)lbmm(x) we obtain

(17) brm (x)(pm(x)l?.m{x).

By (14) and (7) we get l/<pm(x) = bmm(x)/Am(x) = 1 + £ isE ms(x)}

8 = 0

where E ms(x) jO as x->oo. By Lemma 9 in [1] we infer th a t there exist, for large x, functions q>m3(x), s = 0, ..., 3, such th a t (pms(x) jO as x-^-oo

. 3

and <pn (x) = i + 2 ’«* < ;Pmsix). By (16) and (17) first hypothesis in (7) of

8 = 0

Theorem 1 in [1] is satisfied. Moreover, there exists a positive constant M

such th a t we have for large x (r, j = 1 , ..., w; m = 2 , . . . , w; r # m, m # j )

3 3

■^rm < ( ^ - ^ г т 8И ) ( 1 + £<Pms(3>))

8 = 0 8= 0

< j t f i B - № ( * ) i{ 2 'i/s;//9,i + f f I f \a*pPplPvi\

v = l »> = 1 p —1

(cf. (7) in [1]) and

n n n

M < M \ Z 1Л/А1 + I f I f I s A /A l} •

» = 1 P = 1 N

By the inequality ( £ av)2 < N ]? eft (which follows from the inequality

|>e=l J> = 1

of Schwartz) we get

n n n

|6 to .| bw < n*M*is-1'"!I21 tf.W .Y+If I * |«*A,/AI»}.

v=l p = 1

By ( 6 ) and ( 8 ) we obtain th a t first hypothesis in ( 8 ) of Theorem 1 in [ 1 ] is satisfied. In a similar way we show th a t second hypotheses in (7) and ( 8 ) in [1] are satisfied. Applying Theorem 1 in [1] we obtain th a t the system of differential equations (13) has for sufficiently large x an integral {йт (ж)}, m — 1 , . . . , n, such th a t lim щ(х) — 1 and lim um(x)

n X-+CQ да-к»

= 0 for m = 2, .. . , n. Setting wm(x) = £ efffff^l uv(x) for m = 1, ..., n,

»,=i

wo infer th a t the functions wm{x) satisfy, for large x , the system ( 1 2 ) and we have lim ïüm(x) = 1 for m = 1 , . . . , n . The functions yPk{x) =

x-+oo

X

wv(x)evk~1(ïv(x)expf ak {t)dt, v = l (the index Tc is fixed) satisfy,

x0

for large x, the system of differential equations ( 1 ) and have the desired asymptotic properties. We have

D(x)

У и • • У In d u • • d ln

Ут • • Упп • d nn

X n

exp

x0 v = l

X

where dwk(x) = (l//?v(a?))exp( — f ak(t)dt) yvk(x) (compare Eemark 2 ) and lim dvk{x) = e^-1, by (9). Since the determinant |s*-1| is different from 0, X-+0O

we obtain th a t B(x) Ф 0 for sufficiently large x and the system of the functions yvk{x) is fundamental.

By the first equality in (11) and (7), we obtain th a t ak (x) ~ ekB lln(x)

as Х-+ЭО. Suppose th a t reskB lln(x) Ф 0. Setting y = ekB lln(x) and

im y/rey — G ( = const) we get re a fc = re y { re (a fc/y) — Gim(akly)}. Since

lim im {akjy) — 0, we obtain th a t reafc(a?) ^ reekB lln(x) as x->oc. By (2)

«->00

and (5) we get

Te(ak

+ p'Jpv) =те(екВ 11п(х)){теак1те{£кВ 11п) +

n — 1

( 2 * V8^S,3 +1

= re(£t B ,"‘H ) ( l + o(X)) oo

as æ-+oo. We proved above th a t f \B1,n(t)\dt =

Since

X Xq

— const • \Bl!n(æ)\, then |геел J B 1,n(t)dt | f

as x-^oo. Applying l ’Hôspital’s

rule we obtain x°

x0

re

(ak{t) + pl(t)ipv(t))dt ~теек

B lln(t)dt as x

«o and by (9)

X

\yvk{œ)\ ~ exP re / {ak + p'vipv)dt

Xq

o o i f

xe{ekB lfn(x)) > 0, 0

ге(елВ 1 /и(ж)) < 0 . P r o o f of T h e o re m 2. We set yvk{x) = y*k{x) -f iy*k (x) for v = 1, ...

. . . , n and fixed 1c, and we obtain for x-+oo

(У*к + iy*l) {cos (Bk + tvk) - i sin(Bk + tvk)} |1 /& | exp( - C k)->1, (18) {y*k cos (Bk + tvk) + y*t sin (B k + *„ft)}

exp ( - C k) - 1 - > 0, (19) { - y * k sin (Bk + tvk) + ytu cos (Bk + t,k)} |1 IP,I exp ( - C k) ->0.

Multiplying relation (18) by cos(Bk + tvk) and (19) b j —sm(Bk + tvk) and adding, we obtain th a t

У*к И I 1 /& {oo) I exp ( —Ck {x)) - cos (Bk {x) + tvk) ->6 .

Multiplying relation (18) by sin ^-M * * ) and (19) by cos (Bk -\-tvk) and adding, we get

у*к

(®) I1 /А (®) I exp (

- sin

+

-*0.

As in the proof of Theorem 1 we show th a t im afc(a?) ^ imskB lln(x) as a?->oo. Then applying l’Hôspital’s rule we obtain th a t \Bk(x)\

' • X X

= |im f ak{t)dt\ ~ |imeÆ j B lln(t)dt\ ->oo as x — >oo.

Xq Xq

We shall apply l’Hôspital’s rule for complex valued functions in the following formulation. Suppose th at

lim \g{x)\ =

#-►00

X

and j \g'(t)\dt^K \g(x)\

Xq

( 20 a)

0 Г

00

( 20 b) Iim|$r(a?)| = 0 , g(x) ^ 0 and J \g'(t)\dt < K\g{x)\ for x ^ x 0,

X-+CQ T

f'(x ) and g'(x) are continuons and in the case ( 20 b) we have lim/(æ) = 0 .

Then ®^°°

( 2 1 ) lim \f{x)lg{x)\^K lim .\f'{x)/g'{x)\

X-WO X-+00

and

( 22 ) hypothesis lim f'(x)lg'(x) = s implies lim f(x)/g(x) = s.

X-+OQ X-+CO

Let us notice th a t the inequalities in (20a) or (20b) hold if \g'(x)\

< K \g(x)\' or \g'(x)\ ^ —K\g(x)\'j respectively.

P r o o f of T h e o re m 3. We shall prove the convergence of the inte- OO

grals / \Dmk(t)fm(t) /D(t)\dt, m,Jc = 1, n, in the ease r eekB lln(x) > 0, where ч

D(x) =

Ухх

Ут

Ут

exp I У avv(t)dt

4 V = l

and [ylk, ynk}, ~k = 1 , ..., n, form a fundamental system of integrals of the homogeneous differential equation (1). We have for a?->oo and m , 1c — 1 , ... , n:

(23) Dmk(x)lD(x) ~ Qmkg{x)IPm{x),

X

where g(x) = ex p ( — j ak(t)dt) and gmk are constants different from 0 .

XQ

This follows from the fact th a t, for example,

' Pz Pr exn

X П

r ^V

. . . e.

for

and

П П

= 1 , J£ 4 (® ) =

V = \ V = 1

Then there exist M > 0 and x x ^ x 0 such th a t \Dmk(x)/I)(x)\ <

< M\g'(x)\!\ak(x)f}m(x)\ for x ^ x x. Let us observe th a t we have in th e case reekB lln > 0

(24) \9'(х)\1(-\д(х)\')

= \ak(x)\lTQak(x) ~ \Blln(x)\/reekB l,n(x) = M x< oo

(compare the proof of Theorem 1). Then setting K x = М х+1 we obtain for sufficiently large x z > x x

CO oo

/ | 0 '(О 1 <Й< - K x f \g(t)\'dt = K x{\g{x2) \ - lim \g{x)\) = K x\g{x2)\<

x2 X2 x~*°°

We obtain from this for x > x 2

OO CO

f №mkfmlD\dt< M f \ g ' f j a kpm\dt < oo,

X X

since lim \fn {x)lak{x)pm{x)\ = lim |/т (а?)/-Я 1 /пИ & *(я )1 = 0 , by hypo-

X-+OQ Х-УОО

thesis.

The functions yv(x) = yp(x), r = 1, n, where

n n o :

y,№) = y , M I {DmkW mWIDifydt, fc—X m=1 ад.

satisfy the system of the differential equations (10). We set here ak = x 0 if твекВ 1,п(х) < 0, ak = oo if reekB lln(x) > 0 for x ^ x 0. By (24) in the case TGskB 1,n > 0 and analogously in the case ieekB lln < 0, we obtain

X

th a t the function g(x) = exp ( — J ak(t)dt) satisfies hypothesis ( 20 a) or (20b) with some К {К > M x).

We complete the proof of Theorem 3 applying l’Hôspital’s rule.

We obtain

lim (y,k{x)IP,{a>))

х-у о о

f №mkfmlD)dt = evk 1 lim f {DmkfJD )dtlg{x)

ak x~*°° °k

= ~

4 _1 lim /w (x)/ak (x) Pm (x) = 0 ,

x->oo

by (23) and hypothesis.

Reference

[1] Z. P o ln ia k o w s k i, On some linear differential equations, Comm. Math. (Praee Mat.) 19 (1977), p. 323-341.

MATHEMATICAL INSTITUTE OF THE POLISH ACADEMY OF SCIENCES