On the differential equation

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 the differential equation y {n)— Ъ ( х ) у ' — a ( x ) y = О

I n this paper we shall consider asymptotic properties (for x->oo) of integrals of the differential equation

(1) y (n) — b(æ)y' — a(œ)y = 0 ( œ ^ x 0)

for n > 2. The functions a (x) and Ъ(х) are real. Theorem 1 is a general­

ization of the theorem proved in [2].

T

1. Suppose that

(2) a(x) ф 0 for x ^ x 0,

(3) there exist the derivatives a^n~l\ x ) and Ъ(п\ х ) for x ф x 0.

Furthermore suppose that we have for

and with some s >

(4) (a~ll2nf v^ ~ o (xll2~vln~1~ex) for v = 1,

(5) (a11-®1/”®)' = o(a?-1/®ln_1-e£c) for n > 4 and v = 3, . . . , n — 1, (6) ba[2~n]ln = o( X) and V a^~nVn = o { X ) ,

where X = Æ-1 ln -1 -e £P.

Then the differential equation (1) has for x > x0 n linearly indepen­

dent integrals yk(x), Jc = 1, . . . , n , such that

X

yk(x) ~ a[1~n]l2n(x)expek ( a lln(t)dt for a?->oo, x0

where ek — e2kmln.

I f xeekalln(x) > 0 for x > x Q, then lim \yk{x)\ = oo. I f r eekalln(x) < 0,

then lim ^ (æ ) = 0 . a5_>0°

X -+ C Q

Le t us observe that the conditions of Theorem 1 are satisfied b y the functions a(x) = a? and b(x) = x9, where p > — n and q <

{ n ~ 2 ) p / n — l .

As in [2] we m ay form ulate the following

C

(Oscillation theorem). I f hypotheses of Theorem 1 are satisfied and if imekalln(x)

0 for x > x0, then the differential equation (1) has for x ^ x 0 two real integrals y*k{x) and y*jf{x) such that

y*k(x) = (cos5fc(a?) + <5A;(a?))|a[1- n[/2n(a?)|exp(7A;(a;) and

yl*(x) == {&mBk(x) + œk(x))\all- n]l2n(x)\expCk(æ) f o r x ^ x 0,

X X

where = im ek f alln(t)dt, Ck(x) = reek f a1/n(t) dt and lim ôk(x)

Xq Xq X->0O

— lim cok(x) = 0 . The functions yk{x) and yk*{x) have infinitely many

Xr^OQ

zeros with the limit point £ = oo.

T

2. Suppose that

(7) there exist the derivatives a(n ^(x) and

(x) for x ^ x { and that we have for

(8)

-!])<»> = o(x1/2~vln~1 ex) for v = 1, ..., n, (9) = 0(x~1/vln~1~‘x) for n > 4, v = S, ..., n —1,

(10) ajb = o(X).

I f Ъ{х)> 0 for x ^ x 0 and n = 2m + 1 , then there exist two integrals yk(x), Jc = m and h — 2m, of the differential equation (1) such that

X

(11) yk{a;) Ь~п12[п~1](х)ехрт)к J bll[n~1]{t)dt for x-^oo,

Xq

where r\k = ц follows r\m = —1, rj2m = 1.

I f b(x) > 0 for x ^ xQ and n — 2m, then there exists an integral y(x) of (1) such that (11) holds for ÿ = yk, Tc = 2m—1, r\2

_

= 1*

I f b{x) < 0 for x > x0 and n = 2m, then there exists an integral y(x) of (1) such that (11) holds for у — yk and Тс = m — 1.

Let ns notice that in the above cases the values 7]kbllln~^(x) are real. If r)kbll[n~1](x) > 0, then lim |yfc(a?)| = oo. If rjkbll[n~1](x) < 0, then lim yk(x) = 0. x_>0°

x—xx>

T

3. Suppose that

(12) there exist the derivatives a(n-1)(&) and Ы2п~^(х) for x ^ x 0, (13a) b(x) > 0 for х ^ х 0 and n Ф 4m + 3, m = 0 ,1 ,..., where n is as in (1) or

(13b) b(x) < 0 for x > X

and n

4m +1, m = 1 , 2 , ...,

(14)

о

for v = l , . . . , n , with some e > 0 ,

(15) ajb = o(X) for X-+ 00 .

Then the differential equation (1) has for x ^ x 0 an integral y(x) such that lim^(a?) = 1 .

x—>oo

The conditions of Theorems 2 and 3 are satisified by the functions a{x) = xP and b(x) = x9, where q > l — n and q > p - \- l.

P ro o f of T heorem 1. For a given к (1 < к < n) we set, for x ^ x0,

X

F(x) = a[1~n]l2n(x)expek f a1/n(t)dt and we substitute y(x) = w(x)F(x)

x0

into (1). We obtain the differential equation П

J £ a v(x)w(v) = 0 (x > x0),

U = 0

«0

= F in) - bF ^' - a F ,

= n P n~^ - bF, av = for V = 2 , ..., n.

Suppose that 6 = 0 if a(x) > 0 and в — 1 if a(x) < 0 for x ^ x0.

We set A

= e^1a~lln(x) and

_ в( 2 *+вн/»ш (if a(x) < 0 we suppose Arga 1 /"(<r) = я jn.)

We assume the values rv, v = 1 , . . . , n —l, as in the proof of the theorem in [2]. Then re(A/rJ Ф 0 for v = 1, n — 2 iî n ^ S , and hypo­

thesis ( 6 ) of Lemma 1 in [2] is satisfied.

We define the functions y(x), <pv(x), v —l , . . . , n —l, bv(x) for v = 0 , n — 2, 0jv and % v as in Lemma 2 in [ 2 ] for av(x) from the dif­

ferential equation (16) and for A (x), A and rv defined as above. We shall prove that %p(x) = o(X), a}ln(x)<pv{x) = o(X) and <pv{x) = o(l) for v

= 1 , . . . , n —1 and x-y oo.

We obtain ip(x) = у)*(х)+гг ... rn_xEk{ - b F 'F 1-\-(bFF1y}, where y>*(x) denotes the value of the function y (x) for b (x) = 0 , and

«

F x(x) = a[ 1 _nl/ 2 n(<r)exp(— ek j alln(t)dty

Xq

We have y>*(x) = o(X), what we show as in the proof of the theorem Ь [2] in the case n — 2, and as in the proof of the theorem in [1], p. 171, in the case n ^ S .

By hypothesis and by Lemma 14 in [1] we obtain

- b F 'F 1 + (bFF1)' = b 'F F x + b F F \ = b'a^~n]ln Abal2~n]ln-0( 1) = o(X) and we get гр (x) = o(X).

(

16

where

Suppose that n = 2 . If 9 ?*(a?) denotes the function q>x{x) for b(x) = 0, then we have as before

<Pi{œ) = ô1{r1A' + r1A (y )-a 1/a2) + 1}

= <P*\ (®) + dxrxA [rxek{b'a~112 + 6 • О(1)) + &},

where |<5X| = 1. We proved in the proof of the theorem in [ 2 ] that all2(x)

<p*(oc) — o(X) and <p*(a>) = o(l) as x->oo. Applying l’Hôspital’s rule we obtain from (4) that a~ll2(x) = o(œ). By hypothesis we get all2(x)(px(x)

= o{X) and q>x{x) = o(l).

Suppose now n ^ 3 . If b*(x), 0*v{x) and denotes the function bv{x), 0 jv{œ) and % v(x) respectively, for b(x) = 0 , then

b0(x) = b*0(œ )-£kb(x)all- n]ln(æ), / bv{a>) = b*(x) for v = 1, . . n — 2 and

3tn-i,*(®) = K - i ,v ( œ) for v = 0 , . . . , n - 2 . By hypothesis we get

A ~l b о = A ~ 1b*0+ o (X ),

К = b*r + o(X) as x->oo, since {ball~nVny = b'a[1- n]ln+ ( n - l) b a l2- nVn(a~llny = o(X).

Then

0 = А °-'Ф :% 0 +

(Х) for в = 0 ,1, Фп-i.v = for V = 1, n - 2 .

We proved in the proof of the theorem in [ 2 ] that A s~v~l = о (X) for v = 0, — 2 and s = 0, ..., v +1. This proves that also ^{A e~v~l} Ф(*}_ltV

= о (X) for the above indices.

By (4) we have for a?->oo

S

A s~lA& = ek-s« [1~s]/n J ? (а- и ы р (a-H

nÿs-v) = 0 (1)?

V = 0

s = 1 , ..., n, and by Lemma 6 in [ 2 ] we obtain that As~v~10fv) = o(X) for s = 0, ..., v-\-\ and 0 < v < j ^ n — 1. In particular we have ^{A ~ l i} Pj0

= o(X) and Ф'0 = o{X) for j = 1 , 1. By (13) in [ 2 ] we infer from this that A ~ l {x)(pv{x) = ekalln(x)(pv{x) = o(X) and ^<pv(x) = o( 1 ) for

= 1 , n —1, since ^a~l,n(x ) = o(x).

This shows that hypotheses of Lemma 3 in [ 2 ] are satisfied. Applying

this lemma we obtain that the differential equation (16) has an integral w(x) such that lim w{x) = 1. We set, for x > a?0, yk{x) = ïô{x)all~n]l2n(x) x

X

xexpe* J a1,n(t)dt and we complete the proof as in the proof of the theorem in [1].

L

. Suppose that n ^ 2 and that the function b(x) satisfies hy­

pothesis (8). I /

G(x) = b ($) expc f b1/[n 1](t)dt

4

with some constant c and a non-negative integer p , then G^{x) ~ cvbvl[n~^(x)G{x) as x->oo, v = 1, n.

P roof. In the case p = 0 we apply Lemma 12 in [1] for cbllln~1]{x) instead of b(x). (By Lemma 13 in [1] for b~ll[n~1](x) instead of a~lln(x) we have (&-1/t”~d)(h£[1-d/i>-i] — o(l) for j — 1, . . . , n — 1).

If p > 0, then

V X

G{v){x) = ^ (expo j bll[n~1](t)dtj i)Pj (x)J

3=0 X

where

b-pl²[n-

1]^) for j = 0, P j ^ ~ ' (b-pl2ln~ï](œ))U)

£ А{ [ J

b~m n-^{œ )fv

i s = l

for j = 1, V, where vis > 0, £ vis = j, p are constants.

S = 1

We complete the proof as in the proof of Lemma 14 in [1] for cÿlln-i] (д.) instead of ekalln(x) and for p instead of n — 1.

P ro o f of T heorem 2. We set

X

F 2(x) = b~nl2[n~1](x)expr)k f bllln~1](t)di for x ^ x 0

and we substitute y(x) = w (x)F2(x) into (1). We obtain the differential equation

П

(17) ^ a v{x)w{v) = 0 { x ^ x 0),

v=0 10 — prace M a te m a tyczne 19 z . 2

where

a, = & p - b F , - a F „ a, = n S^ ~ l)- b F „

= for v = 2 ,

Let be в = 0 if b{x) > 0 and в = 1 if b{x) < 0 for х > a?0. We assume Arg&1/[n-1](a?) = п/(п-~1) if b (x) < 0. We set

A = е<2*+вМ/(»-Ч A (a?) = and re_j = 1.

If 3, then 1 jrv — 1 — e2vniKn-i) for v z= X, n — 2.

Let ns notice that by onr hypotheses A is real. Moreover, we have 1 Jr9 = — 2i sin V 71 ■ - ет{1{п~г) for v = 1, — 2. Then ге(А/гв) ^ 0 for

n —1

v

— 1, n —1 and hypothesis (6) of Lemma 1 in [2] is satisfied.

We define the functions y>{x), <pv(x) for

bv(x) for

= 0, ..., n — 2, $>jv{x) and %v(x) as in Lemma 2 in [2] for av(x) = av(x) and for A, A{x) and rv defined as above. We shall prove that ip(x) = o(X) and <pv(x) = o(l) for v = 1, ..., n —1, as a?->oo.

Setting F 3(x) = ô[2-nl/2[TC_1J (x)exp[ — t]k f bll[n~^(t)dt) for x ^ x 0 we

Xq

obtain as in the proof of the theorem in [1], p. 171, П

Ч>(«) = г 1 . . . г п_ , \ У 1 { - 1 У ^ у 1 * Г ’>Ж,)М-ЬЖ'гЖ ,-а Ж ,Ж , + (ЪЖ,Ж,у}

17=0

= r ,_ ,{ ( -l)<">Æ’aÆl*> + [ » / 2 ( » - l ) ] i 7 i - i , i 6in » - 4 - a / » } . We have

n Ж3ж р =*

17=0

where

JS — j[2»-»-2l/2[n-l[^[2-n]/2[»-l]j(«-t7)^

X X

Cv = b[l~v]lln’~1^ex-pr]k J ô 1/[n-11(ÿ)d#{exp|— J

x0 x0

In particular,

c0

0

-r}kb^n~l\ o%

As in the proof of Lemma 16 in [1] we prove that

c, = ( - l ) ' ’r , l b 'l l » - '4 ( - i r - ' $ [ v l - 1l ( n - l W l b + 0 (X)

for v = 3 t ...,n . (We replace the function ekalln(æ) by r]kbl,ln~1](œ).

Then Cv — e\~vrj°k~1G*, where C* denotes the function Gv from Lemma 16 in [1].) By (41) in [1] for b~1/2ln~1] instead of b, for m — n — 2 and j = n — v, we obtain

JE , j [ 2 t > - n - 2 ] / 2 [ n - l ] V 1 Д j [ f c ( i ) - n + 2 ] / 2 [ n - l ]

k(i) 8 = 1

if v(8 > 0 for s = 1, ..., Jc(i) and vis = 0 for the remaining indices s. X{ are constants.

By (8) we get E v = o(l) for v = 0 , . . . , n — 1 since k ( i) ^ n — v.

Similarly we obtain

m

j g ^ £ l / [ n - l ] _ _ у 1' ^ ^ [k (i)-2 n + 2 v + 2 ]l2 [n -l] J ~ J ( f r - l l 4 n - l ] ÿ o is)

i 8 = 1

= o(X) for v = 0, n — 2,

and k(i)

E v b'lb = —2 (п —1)(Ь~т п ~1]У ^ ^ Ь 1 т ~2п+^ +1]121п~1] f j (b~1/2ln- li)(vis) i

— o(X) for v — 0, n — 1.

8 = 1

We infer from this that

JEVCV

o{X) for v = 0 , n — 2,

( —1 )n~1bll[n~1]E n_1+ o (X ) for v = n —1, ( + ( - l ) ” - 1 (“ ) [ l/ ( « —l)]b 'lb _{+ o(X)}

for V = П.

Since = [(2 —w)/2(%—1)]&'/&> we get П

= J ^ ( ” ) К С , = ( - 1 Г ч » » 1Л" - 11 + ( - 1 ) ж- 1[» /2 (» - 1 )]» '/Н -о (Х )

V = 0

a n d

V»(®) = —r1 . . . r n_1a /b + o {X ) = o(X), Suppose that n = 2 . We have

where l^j = 1.

ài<Pi = rxA ' + r1A (y — ai/cti)+l

=

0

1

We obtain for n > 3 П j=i

n j = 1

= J ? ( - î ) ^ 1 (?) £ f 7 1) & r ,- l)&ï' - i i=i

By Lemma we get

lim 6»W = J ; ( - 1 ) ' - (?) J j < ( V ) - 1 = » - 1 ■

j = l г=0

Similarly we obtain for v = 1, ..., n — 2 П

M®) = ^ (?) 4

j = v + l

J —V—1

j*= v+ l

By Lemma we have

i —0

j - v - l

IbnA-’ K = 4| £ ( - D ' f T * ) Ш— ‘ = ( + )

j = ® + l г= 0

and analogously (compare also the proof of the theorem in [1], p. 170)

П j+8—V —1

lim A8~vb^ = ^ ( -I)'"® -1 ( - l ) i ^ + s ~ v_1j = 0

J=«+l г=0

for S = 1, 17+1.

We have

<P»-M = •••»■ »-A for v = 0, . . .,n — 2.

By (8) we obtain that lim A^8) J .s-1 = 0 and by Lemma 7 in [1]

Х-+О Э

we get

lim .A v0 n- hv =

l —r1 . . . r n_1(n —l) for v = 0 ? -bn-i.e- ri * " r»-i(,, + 1) for ® = 1 ,

where L n_1>v is the i>-th fundamental symmetric function of the num­

bers rlf rn_x.

The numbers lfr v are roots of the equation W(x) = 0, where П —1

W(æ) = ( i«î- l ) { ( æ- l ) ”- 1- ( - i r - 1} = 0 ^ ( - 1 ) » - ' - 1(<|“ 1)®' + (-1 )*®

fb — 1 t?«0

= æ f j («-1/Л ,).

U —1

We infer from this that n — 1

8

v + 1

for V = 0,

for V

n —1,

where $n_u if the j-th fundamental symmetric function of the numbers ljr t , l/yn_! (compare also [1], p. 170).

It follows that lim A ~v0 n_l v = 0 for v = 0, n — 2. By Lemma 7 cc->oo

in [1], we obtain lim A8~v0 ^ ll>v = 0 for v = 0, w —2, and s = 1, ...

a;-»-oo

. . . , v + l . Applying Lemma 8 in [1], we get lim <Pj0 = lim АФ'^ = 0

>00 X-XX)

for j = 1, Since = о (a?) and Ay — o(l), we obtain that lim <pv(x) = 0 for v = 1, ..., л —1, by (19) in [1]. Applying Lemma 3

X -+ O Q

in [2], we obtain that the differential equation (17) has for a? > a?0 an integral w(x) such that lim w(x) = 1 . We set

X -+ C O

X

yk(x) = w(x)b~nl2ln~1](x)expr]k J bllln~1](t)dt

X (f

and we complete the proof as in the proof of the theorem in [1].

P ro o f of T heorem 3. Let us notice that from (14) there follow the relations

(18) (1 /&)<"> = o{X) and = 0(1)

for v = 0, ..., n — 2 and x->oo.

We have namely

(l/6)(«) = (b~ll[n~1]f i8).

г s = l

In the above products there appear some combinations of indices

n —l

vis, s = 1 , n —1 such that vi8^ 0 and £ vtа — n’ h are constants.

By (14) we obtain S=1

n —1

(1 /Ь)(п) = ^ Х , Ц о { х 1- Щ8\и-1- ех)

and from this there follows the first relation in (18). Similarly we have (1 ibÿn- v-v = £ $ [ ] (b~ll[n- 1]f is),

i 8=1

ft—1

where £ — n — v —1, and

8 = 1

n—l

frvl[n-l] ^ 1) _ bll~Vi8i,ln~^ (b~l![n~^ŸVis\

i s = l

We obtain from this the second relation in (18) since

£ [ l - vi s ] / [ r a - l ] / £ - l / [ n - l ] \ ( t’i8) _ ^ 0 Г Vis

Io(l) for vi3> 0.

We set rv = e-2miKn~1) for v — 1, ..., n — 1, A{x) — [ —Ь(ж)]~1/[”“ ч, A =

if &(#) > о for x >

and A — 1 if b(x) < 0.

We shall prove that under our hypotheses we have гe(A/r„) Ф 0 for v = 1, n — 1. In the case of b{x) > 0 we have re(A/rr) _ ree( 2 t>+i)m/(n—о — o if there exists a poitsive integer hx such that (2v -{-l)/(n — 1) = (2fc1+ l)/2 , then 2 (2 v + l) = (2&j + l)(w — 1). It follows n = 2 m j + l with some positive integer mx, 2 v + l = w 1(2/c1+ l ) and т г = 2m + 1 with some positive integer m. We infer from this that the relation re(A/rw) = 0 holds (for some v) if and only if n = 4m+ 3. In the case of b(x) < 0 we proceed similarly.

Then by (13a) and (13b) hypothesis (6) of Lemma 1 in [2] is satisfied.

We define the functions y{x) and <pv{x), v — 1, ..., n —l, as in Lemma 2 in [2] with A (x ),rv and A defined as above, with a0(x) = — a(x), ax(x)

— —b(x), an(x) = 1 and av(x) — 0 for the remaining indices v We shall prove that ip(x) — o(X) and cpv{x) = o(l) for v = 1, ..., n —l, as x^>oo.

We obtain

y> = afb + ( - l f - ^ l / b ) ^

and by (15) and (18) we infer that the function ip(x) has the required property.

Suppose that n = 2. Then

<5i <Pi = ^ A ' + V i A i y t — a J a J + l

= ( - W - ( l / 6 ) ( V + ft ) + l = ( - l/ f t ) '- y / & = o ( l ) , since 1 lb = o(x) (|<5Х| = 1 ) .

Suppose that n&z 3. If the functions bv(x), v = 0, . . .,n — 2, 0 jv(cf) and % v(x) are defined as in Lemma 2 in [2], then by (18) we have

6„ = X + ( — 1}<»*(1/Ь)<—« = 1 + 0(1),

A .-’ b, = ( - l ) “- ”e",i,(" - 1)i)',[" ' 11(l/ft)('‘^’ “ ‘i = o(l)

for v — 1, ..., n — 2, and

д в - г>£(s) _ ^ _ ^ n - v e t v - s ) n i l { n - l ) ^ [ v - s ] l [ n - l ) ^ i ^ ( n + 8 - v - i ) __

for v = 0, n — 2 and s = 1, ..., -y+l? since = o(so), by (14).

Let ns observe that if L n_l v is the w-th fundamental symmetric function of the numbers rx, гп_г, then L n_l v = 0 for v = 1, — 2 and L n_l>n_x ==rx ... rn-1 = 1. By Lemma 7 in [1] we get

= ^ - ' 5 4 * 1 . . . «-„..A— »<•> = 0(1)

for v = 0, — 2 and s = 0, « + 1 . By Lemma 8 in [1] we obtain 0jo = o(l) and A0'jо = o(l) for j = 1, ..., n — 1. In virtue of (13) in [2]

we obtain <pv(ct)) = o(l) for v = 1, ..., w—1.

We complete the proof applying Lemma 3 in [2].

References

[1] Z. P o ln iak o w sk i, On the differential equation y(n) — ay = 0, Comm. Math.

(Praoe Mat.) 14 (1970), p. 151-172.

[2] — On solutions of the differential equation y(n) — ay = 0, ibidem 17 (1974), p.

429-439.

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