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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNF. XXI (1979)

Z. POLNIAKOWSKI

On some ordinary differential equations

The purpose of this paper is to prove asymptotic properties (for x ->• oo) of integrals of some ordinary, non-linear differential equations. In Section 1 we consider a system of differential equations (Theorem 1) and the differ­

ential equation ÿ = F ( x , y ) (Theorem 2). Theorem 3 is an application of Theorem 2. In Section 2 we prove Theorem 4 concerning the differential equation y {n) = F(x, y, y ', y("-1)), n ^ 2, and Theorem 5 as an application of Theorem 4. The functions occurring in this paper are complex valued.

1. We shall formulate the following

Theorem 1. Suppose that n > 1, the functions av(x) are continuous, ay (x) Ф 0 for x ^ x0 and v = 1, . . . , n, and

(1) \av (x)| ^ К jre av (x)| for x ^ Xo, v = 1, n and some К ^ 1,

(2) 1 ^ = 0°

f o r v = u

Moreover, suppose that there exists r\ > 0 and complex values cv such that f , ( x, y x, ..., y„), v = 1 , . . . , n , are continuous functions of the variable x and satisfy the inequalities

(3) \ f v ( x , y f , . . . , y * ) - f ( x , y t * , . . . , y * * ) \ ^ A £ |y * -y * * | 7=1

for x ^ x 0, \yj-Cj\ ^ ц, Iyf — Cj\ ^ q and |yf*-Cj\ ^ rj (j = 1 , . . . , n) with some X satisfying the inequality 0 < X < 1/nK (the variables yj are complex);

(4) lim / v(x, c b c„) = cv, v = 1 , n.

00

Then the system (or the differential equation in the case n = 1) (x) y'v + yv = f v( x , y i , . . . , y n),

(5) V = 1 , n,

(2)

220 Z. P o l n i a ko ws k i

has for sufficiently large x an integral ( y i, s u c h that lim yv(x) = cv

Г 1 X - * C O

f o r V = 1, П .

Re ma r k 1. We obtain from Theorem 1 l’Hospital’s rule if (5) is replaced by (23) and if f ( x , y b y„) = f.(x). Then this theorem can be treated as a generalization of l’Hospital’s rule.

Re ma r k 2. System (23) can be generalized in the form

00

(*) yv = (pv {x)+ j N v( x , t ) f ( x , y ^ t ) , ..., y„{t))dt, v = l

*o

where lim <pv(x) = 0 and the functions N v(x, t) satisfy conditions of Toplitz:

x->ao 00

(i) lim J |iVv(x, t)\dt ^ К < oo,

X - * a o v X0

(ii) lim N v{x, t) = 0 almost uniformly for t ^ x0,

X - * a o

00

(iii) lim f N v( x, t ) dt — 1.

x->cc v.

x 0

As in the proof of Theorem 1 one can prove the existence, for large x, of a solution {ÿi, ...,ÿ„} of system (*) such that lim yv(x) = cv.

JC-+00 Theorem 2. Suppose that

(6) r(x) Ф 0 and there exists a continuous r'(x) for x ^ x 0, Moreover, suppose that there exists an ц > 0 such that

(7) F (x , y) is a continuous function of the variable x for x ^ x0 and

|y -r (x )| ^ *)|r(x)|,

Fy(x,r(x)) is continuous, different from 0 and

(8) IF y (x ,r)\ ^ K |re F y (x , r)\ for x ^ Xo and some K ^ 1,

00

(9) f |F ,(x , r(x))|dx = oo,

*0

(10) lim F(x, r)/ {rFy(x, r)} = 0,

X - + 0 0

(11) lim r'/{rFy{x, r)} = 0,

JC-^OO

(12) |ln Fy(x, y ) - l n F y (x, r)| ^ Ai/2

for x > x 0, \y — r(x)| ^ 17 |r (x)I and for some (0 < < 1 /К).

Then the differential equation

(13) y' = F( x , y )

has for sufficiently large x an integral y(x) such that y ( x ) ~ r(x) for x -+ 00.

(3)

On some ordinary differential equations 221

Theorem 3. Suppose that the functions bY(x), v = 1 are continuous for x ^ x0, a and v = 1, p, are re-mbers and a Ф 0. Moreover, suppose that

(14) there exists a continuous derivative b'(x) for x ^ x 0, (15) b(x) Ф 0 and |b(a-1)/a(x)| sc Х |г е т к-1Ь(а_1)/я(х)|

for x ^ x0 and some К ^ 1, where тк = e2knl/a with some integer k, (16) lim b'(*)b1/a_2(x) = 0,

JC-+QO

(17) lim b^ x) ^ '7* - 1 (x) = 0 for v = 1 ,

X ~ >00

p

Then the differential equation y' = y* — b( x) + Z Ьу( х) у ^ has, for suf-

V = 1

ficiently large x, an integral j;(x) such that ÿ(x) ~ тfcb1/a(x) for x- >oo.

We assume here Arg b1/oc(x) = (l/а) Arg b(x). Let us notice that hy­

potheses of Theorem 3 are satisfied by the functions b(x) = x" and bv(x)

= x°v, where we have о (1—а)/а < 1 and av < o ( \ — /L/а) for v = l , . . . , p . Lemma. Suppose that n ^ 1 and for every v (1 ^ v ^ n) the function gy(x) satisfies the condition

X

(18a) lim |0v(x)| = oo and j \g'v (t)| dt Ф К19v (*)l

X - + C O Xq

or

00

(18b) lim gv (x) = 0, gv(x) # 0 and J \g[ (t)\dt ^ K\ gv(x)\

x-oo *

for x ^ x 0, with some К ^ 1;

(19) cpv {x) are continuous for x ^ x0 and lim <pv(x) = 0 for v = 1,..., n.

Py

We set Ji0)(z) — \A/gv(x)\ J \gy (t)\ z (t) d t , where oty = x 0, /?v = x in the av

case (18a), av = x, = oo in the case (18b) and A satisfies the inequality 0 < A < 1/nK.

Then the system (or the integral equation in the case n = 1):

(20) zv = <pv(x) + J<0)( Z ZJ)> v = 1» •••>

J= i

has for x ^ x0 a solution { z 1, ..., z„), defined by (22), such that lim zv(x) = 0

(4)

222 Z. P ol n i akows k i

Proof. We set for x ^ x 0, v = l a n d m = 0 , l , . . . , J<m + 1)(z) П

= Y Л 0)№ т)(z )) and we shall prove by induction for m = 0,1, . . . , the j= i

inequality

(2 1) 0 ^ 4 И)(1) ^ {nXKT + \ x ^ x0 and v =

By (18a) or (18b) we have 0 ^ J<0)(1) ^ and inequality (21) holds for m = 0. Suppose that it is true for some m ^ 0. Then J|ra + 1)(l)

^ (пЖ Г + 1- « 40)(1) ^ (nAK)m+1.

There exists an M > 0 such that |q>v (x)| ^ M for x ^ x0 and v = 1,..., n . We set

00 n

(22) zv(x) = (pv(x)+ Y Z Jvm)W , v = l , . . . , n ,

m = 0 j =1

the series being uniformly convergent for x ^ x 0, by (21). Moreover, we have |zv(x)| ^ M + M n 2 AK/(1 — nAK) for x ^ x0 and v = 1,. . . , n. The func­

tions zv(x) satisfy system (20). Namely we have

n n n 00 n

(pv(x)+Ji 0)( Y M x)) = <M*)+ Z 4 0)W + Z 4 0)( Z Z Jjm)((pk))

j = 1 j = 1 7 = 1 m = 0 fc = 1

я со n n

= <M*)+ X Л°'(о>;)+ I I I Л0,№ ’Ы )

j = 1 m = 0 fc = l j = l

П GO И

= <М*)+ I 40,W+ I I 4m+1,M = ?,(*)•

j = 1 m = 0 k=1

Setting Lv = lim |zv (x)| and assuming x -► oo in (20) we obtain

X-+00

n n n n

Lv ^ AK Y Lj. Then ^ Lv ^ nAK Y А/ and (1 — nAK) Y A ^ 0. We ob-

j= 1 v = l j = 1 V — 1

tain form this Lv. = 0 for v = 1, . . . , n.

P r o o f o f T h e o r e m 1. We set gv (x) = exp J

X 0 M O and v = 1, .. ., n . Since

for x ^ x0

IflfvMflfvl = ~ r ~ ln Шax = re (g'v/gv), we obtain

\д’Ш \ =

Ov /re — = |av|/re ax

! av

By (1) the functions re av and \gv\' have a constant (the same) sign for x ^ x0 and we have \g[\ ^ X||gfv|'|. By (1) and (2) for every v (1 ^ v ^ n)

(5)

On some ordinary differential equations 223

the function gv(x) satisfies hypothesis (18a) in the case re a, > 0 or (18b) in the case re av < 0.

First suppose that cv = 0 for v = 1 , n. We choose an x l ^ x0 such that we have for x ^ х г

|/ v(x, 0 , 0)| ^ f f ( l - nAK) / 4K for v = 1, . . . , n and

|l/g v(x)| ^ rj(l — nAK)/4 if reav(x) > 0.

We obtain formally from (5) the system of integral equations (23) yv = yv/gv{x)+ôv Pv(fv ( x , y u . . . , y n)), v = 1 , и,

where Pv {z) = (l/gfv(x)) J g'v (t)z(t)dt, ôv = 1, av = x 1? = x in the case

a v

re a, > 0 and Sv = —1, av = x, f}v = oo in the case re av < 0; yv are constants. We assume yv = 1 if re Oy > 0 and yv = 0 if re av < 0.

We set, for x ^ Xx and v = l , . . . , n ,

yv0(x)

= Vv/^vW,

У*,т + 1 (*) = y v o ( x ) + ô y P v ( f v ( x , y l m , Ш = 0, 1, ...

n

By (4) and (3) we obtain \fv(x, y 10, ..., yn0)l ^ |/ v(x, 0 , . . . , 0)| + A £ bv o HO

v = 1

for x oo, v = 1,. . . , n. Then there exist integrals Pv( / v(x, y10( x) ,..., yn 0(x))).

Applying l’Hospital’s rule, we obtain

lim \Pv{fy{x, y10, . . . , y n0»| ^ lim \ f v{x, y l0, •••, yno)g'v(^)l/|l^v(^)Г|

X - + C G X - + O D

^ К lim / v( x , y10, . . . , y n0) = 0.

jc- +oo

We obtain from this that lim yv A (x) = 0. Similarly we prove by induction x-^oo

for m = 1 , 2 , . . . the existence of integrals Pv (fv(x, y lm, y nm)).

We shall show by induction for m = 0, 1,... that

(24) iFvm-^vol ^ Ц/2 and |yvm| ^ ц

for x ^ Xi and v = 1 , ... ,n. We have |yv0| ^ tj(l — nAK)/4 < rj/2 and (24) holds for m = 0. Suppose that it is true for some m (m ^ 0). Then by (3) we get

n n

l / v ( * , y i m, . . . , F nM)| ^ | / v ( x , 0 , . . . , 0 ) | + 2 Y М + Я Y \У}т-Уя\

J= i j=i

^ g (1 — nAK)/4K + nArj (1 — nAK)/4 + nArj/2

^ r j { l - nAK) / 2K + nAr]/2 = rj/2K,

bv.m+l-yvol = \PV(fv(x,yim,---,ynm))\ ^ {п/2К) \PV (1)| ^ Y]/2.

We obtain |yv,m+il < *U since |yv0| < r\/2.

(6)

224 Z. Po l n i a k o ws k i

Suppose that Jim)(z) are defined as in lemma, with Xi instead of x 0.

П

We set (pj(x) = £ \fv(x, 0, . . . , 0)\ /nX + \yjo\. We shall prove by induction

V = 1

for m = 0, 1, . . . and x ^ x b v = 1, . . . , n the inequalities (25) \yv,m + i - y vm\ ^ J i m)( i < P j ) .

i= i We obtain for m = 0

bvl-Jvol = |^v (/v (^C, ^10, У»о))| ^ 40) (|/v( x , y10,...,y„o)lA)

^ Ji0)(\fv (x, 0 , . . . , 0)|/я +

Z

tool) = 4 0)( i <Pj)-

}= 1 i= 1

Suppose that (25) holds for some m (m ^ 0). Then

\Уч,т + 2 .Vv.m+ll |Pv (,/v (x, У1,т + 1 > • • • 5 Уп,т + l) fv (x , У lm » • • •> ,Улт))|

< 4 0)( Z \yj,m+l-yjm) j=l

« 4 0,( ï Jjr»( Î %)) = J f +1’( t %)•

j= 1 s= 1 s= 1

By (4) we have lim (pj(x) = 0 for j — 1 , . . . , и . Since <Pj(x) are contin-

*“►00

uous for x ^ xl5 they are bounded for these x. By (25) and (21) we

00

infer that the series £ |yv,m + i — yVml are uniformly convergent for x ^ x 1?

m= 0

v = l , . . . , n , and there exist the limits functions yv = lim y vm = yv0 +

00 m - x x >

+

Z

(fv.m+i-fvm)- Since the functions yvm are continuous for x ^ x b then m= o

yv are also continuous for these x. Moreover, they satisfy the inequalities

\yv\ ^ rj for x ^ x t . By lemma we have

00 00 n

iTvl ^ \yvo l +

Z

Ijv.m+l-yvml < <Pv +

Z

4 M)(

Z

0 for X 00 .

m= 0 m = 0 j=l

We shall prove that the equalities

lim Pv(/v(x, y lm, ..., y„J) = Pv(/v(x, T i , ÿ„)) m~* oo

hold uniformly for x ^ x 1? v = 1 , .. ., n. For a given s > 0 we choose an index M such that |yvm~Fvl ^ £ for m ^ M, v = 1,..., n and x ^ x x. Then

I^V (f> (x, y Un, ..., Упт)) — Py (fv (х, У\, ..., >’„))) ^ 2.|PV( Z \УМ~~ Fjl)|

j= 1

^ иДе|РД1)| ^ nXKs < e.

(7)

On some ordinary differential equations 2 2 5

We obtain from this that ÿv satisfy system (23). It is easy to see that the functions / v(x, ÿi ( x ) , ÿ„(x)) are continuous for л: ^ Xi. Then by (23) the functions ÿv(x) are differentiable and satisfy system (5).

In the case cv Ф 0 we substitute into (5) yv(x) = ux,(x) + cv, f v{x, y b y„)

= f * ( x , Mb un) + cv and we obtain the system

(26) av(x)t<, + wv = / v*(x, uu ..., un), v = 1, —, n,

where the functions f * ( x , i q , ..., u„) (instead of f v(x, ..., y„)) satisfy hy­

potheses of Theorem 1 for cv = 0. By the proved part of this theorem we obtain that there exists for x ^ Xi an integral {йь ...,м„} of system (26) such that lim mv(x) = 0 for v = l , . . . , n . We set ÿv(x) = t7v(x)+cv.

JC-+ 00

P r o o f of T h e o r e m 2. We substitute y = r(x)u into (13) and we obtain the differential equation

(27) hu' + u = f ( x , u ) ,

where h(x) = — 1 /Fy(x, r) and f ( x , u ) = u + hF(x,ru)/r — (hr'/r)u.

We shall show that the functions h(x) = аДх) and f ( x , u ) = f i ( x , u ) satisfy hypotheses of Theorem 1 for n = c t = 1. By (8) and (9), hypotheses (1) and (2) are satisfied. By (12) and the inequality le2 —1| ^ 2|z|, true for

|z| ^ 1/2, we get Fy{x, y)/Fy{x, r) = eex^ where

|0| < 1/2, and \Fy(x, y)/Fy( x , r ) - l \ = |e0Ai - l | ^

We choose к satisfying the inequality k x < к < l / K and x t (Xi > x 0) such that \hr'/r\ ^ k — k t for x ^ x b what by (11) is possible. Then

\fu(x,u)\ = |1 + hFy (x, ru)-hr'/r\ ^ |Fy(x, y)/Fy(x, r ) - 1| + \hr'/r\

k^~\~k— Aj — к

for x ^ xi and \u —1| ^ rj, and hypothesis (3) is satisfied. Finally, by (10) and (1 1) we get

/ ( x , 1) = 1 — F(x, r)/{rFy(x, r)} — hr'/r -*■ 1 for x -► oo

and hypothesis (4) is satisfied. We complete the proof by applying Theorem 1.

P r o o f o f T h e o r e m 3. We shall show that hypotheses of Theorem 2

p

are satisfied by the functions F (x, y) = y*— b{ x) + £ bv(x)yPv and r(x)

p V = 1

= гkb l,x(x). We obtain Fj,(x, r) = aF' * + £ fiv bv rPv~ l ~ ar*- 1 for x -> oo,

V = 1

by (17). By (16) we get b(1_a)/a(x) = o(x) for x - » o o if a Ф 1, and

00

j |^(a-i)/a^l^x _ qq ^ e function Fy( x , r ) satisfies hypothesis (9).

-vo

15 — Prace Matematyczne 21.1

(8)

226 Z. P o ln ia k o w sk i

By the equality re С = re D re (C/D) —im D im (C/D) for C = Fy(x,r) and D = ar*-1 we obtain re C ~ re D for x -> o o , since by (15) we have jim D/re D\ ^ K . Then hypothesis (8) is satisfied for some K x { K x > K) and for sufficiently large x. By (17) we obtain lim F(x,r)/{rFy(x,r)}

p х-*ж

= lim £ bvrPv/ar“ = 0. By (16) we get lim r'/{rFy(x, r)} = lim r'/ar*

X - * GO V = 1

1 * X - * C O x - * o o

= xkcc~ 2 lim Ь'Ь1/я-2 = 0 and hypotheses (10) and (11) are satisfied.

X-> GO

We choose X2 satisfying the inequality 0 < X2 < 1/Xj. Suppose that y satisfies the condition

(28) \ y/r(x)~l\ ^ rj, x ^ x i,

where rj = min (1/2, A2/8 |a —1|) if a Ф 1 and t] = 1/2 if a = 1. Then there exist constants Afj and M 2 such that 0 < Mj ^ |y/r| ^ M 2. We obtain In {Fy (x, y)/ar*- 1 } = In { { y / r f ~ l + g (x , y)}, where by (17) we have

g(x, y) = X o"vbvb^v/ot_ 1 (y/r)/5v'~1 -► 0 for x -► oo

V — 1

and y satisfying (28); crv are constants.

By the inequality |1п(Л+В)| ^ |ln A\ + 2 \B/A\ (\B/A\ ^ 1/2, A # 0) for A = (y/r)a~ 1 and В = q(x ,y), we obtain

(29) |ln {Fy(x, y)/aF_1}| ^ jin (y/r)a _1| + 2M |g(x, y)|

for x ^ Xx and у satisfying (28), where M = max |y/r|1_a and Xi ^ x 0 is so x'i- X j

chosen that |{?(x, у)(у/г)1-я| ^ 1/2 for these x and y.

We choose x 2 ^ x x such that

И х , y)| ^ A2/16M and |ln {Fy(x, r)/ar*- 1 }| ^ A2/8

for x ^ x 2 and у satisfying (28). By (29) we obtain for these x and у

|ln Fy{x, y) —In Fy{x, r)| ^ jin {Fy(x,y)/ara_1}| + |ln {Fy{x, r)/ar“~ 1}|

^ |(a— 1) In (y/r)\ + 2M |e (x, y)| + |ln {Fy (x, r)/ar*~

^ A2/4 + A2/8 + A2/8 = Л2/2,

since by (28) we have y/r = 1 + 6rj (|0| ^ 1), and in the case а # 1 we get

|ln(y/r)| = |ln(l + 0rç)| ^ 2rj ^ Я2/4 |а -1 |.

Then hypothesis (12) is satisfied. We complete the proof by applying Theorem 2.

2. In this section we shall prove two theorems.

Theorem 4. S u p p o s e th a t n ^ 2 a n d

(30) r(x) Ф 0 a n d th e r e e x i s t s a c o n tin u o u s r'(x) f o r x ^ x0,

(9)

On some ordinary differential equations 2 2 7

there exists rjx > 0 such that F(x, y x, ..., yn), —— Fy (x, r, 0 , 0 ) and dx

Fyv(x, r, 0 , 0 ) , v = 2 are continuous functions of the variable x and Fn ( x , r , 0 , ..., 0) Ф 0 for x ^ x 0, b i - r | ^ rji\r\, \yv\ ^ r\x \rhy~v\, v = 2 ,...

. . ., n, where h(x) = {Fyi (x, r, 0 , . . . , 0)}-1/". The variables yv are complex.

Moreover, suppose that

(31) \h{x)\ ^ K|re Emh(x)\ for m = 1 , . . . , n , and x ^ x 0, where em = e2m7t,/";

(32) lim h' = lim Лг'/г = 0,

(33) lim F(x, r, 0 , 0 )/{rFyi (x, r, 0 , . . . , 0)} = 0.

Let us suppose that we have for x ^ x 0, |yi — r| ^ rjx |r|, \yv\ ^ rj x\rhl v|, v = 2 , . . . , n , and with some Xx (0 < Xx < 1/nK):

(34) |ln Fyi (x, y x, ..., y„) —In Fyi (x, r , 0 , 0)| ^ Xx/2, (35) \Fyv( x , y i , . . . , y n)hn' v + 1\ ^ X x, v = 2, ...,n.

Then the differential equation

(36) Ул) = F ( x , y , / " - 11)

has for sufficiently large x an integral ÿ(x) such that ÿ(x) ~ r(x) and ÿ*v)(x) = o(rh~v), v = 1 , n — 1, for x -*■ oo.

Let us notice that hypothesis (31) is satisfied if, for example, h(x)

= crh*(x), where a is a complex number, h* (x) is a real valued function and we have re(em<r) # 0 for m = 1 ,..., л.

Th e o r e m 5. Suppose that n ^ 2 and the functions bvs (x) are continuous for x ^ x 0, a and f vs are real numbers, а Ф 0, ^ 1 for v = 2 , . . . , n and s = l , . . . , p v. Moreover, suppose that

(37) there exist continuous derivatives b'(x) and b'Xs(x) for x ^ x 0 and s = 1,

(38) b(x) Ф 0 and \b(1~ct)lna\ ^ К |re ({?em4 1~a)/"b(1_a)/"a)| for x ^ x 0 , m — 1 ,..., n and some integer k, where em = e2wtl/n, zk = e2kKl,<x, q = 1 if a > 0 and q = e~Kl,n if a < 0,

(39) lim b, b{ 1 - * ) l n a - 1 = 0, (40)

(41)

lim b XsbPls/a 1 = 0 for s = 1 ,..., p x,

X - + O 0

lim Vu b * = o if a # 1, x-^oo

Hm ~ - { Ь и Ь Ри~ 1) = 0 i/ a = 1, s = 1 ,..., p x, JC-*-GO ax

(10)

228 Z. P o ln ia k o w sk i

(42) lim hvsb(v 1)(<x ^vs/^+zW01 1 = о for v = 2 , . . . , « , JC-► 00

and s = 1 ,..., pv.

Then the differential equaiion

(43) / ”’ = / - ( . ( * ) + £ £ Ы х ) ( / , - ,У ”

V = 1 s = 1

has /or sufficiently large x an integral ÿ(x) suc/i that ÿ(x) ~ xkb ll<x(x) and y v) (x) = о (bv(ot_ 1>/na + v = 1 ,..., n — 1, for x -> oo.

Let us observe that hypothesis (38) is satisfied if, for example, b{1~a)lna(x)

= oB(x), where a is a complex number, B(x) is a real valued function and we have re (Q8mxik1~a)ln а) Ф 0 for m = l , . . . , n .

P r o o f o f T h eo r e m 4. The differential equation (36) can be written in the form

y' = yv + 1, v = 1, n 1, y'n = F(x, у и . . . , у п), П

where yv = y v_1). Substituting yv = (1 /n) Y £ j - \ u j r h l ~v, v = 1 ,..., и, where j=i

ek

=

ê кш/п, we obtain the system

(44)

h

X

У - j w; + (hr7r + ( l- v ) /i') 2 £ ) -1U j = X £ Vj - i U j ,

j= i J=i J=i

v = l , . . . , n - l ,

h Y t f - l U j+ ( h r '/r + ( ln ) h ') Y = n h n F / r .

j =i j= i

For a . given m (1 ^ m ^ n) we multiply the v-th equation in (44) by 4 2 7 , v = l , . . . , n , and we add the obtained n equations. It is easy to show that

Y £m-Vl X fij-ÎMj = «Êm-1! Um

v = 1 j = 1

and

и - l

X 4-7 X = X

Uj

X £v m- X

U)

=

nUm~

X

"T

v = 1 j= 1 j - 1 v = 1 j = 1 j= 1

Then we obtain

4 - 1 fcHm + 4 (hr'/r) Ц , + Л' X C m j U j = Um ~ ( l/п) £ U, + h" F / r, j=i

(45) 4 - 1 Лм^ + Мт = ( l /и) X U j - h " F / r + Y d m j ( x ) U j ,

j= i ;= i

m = l , . . . , n , where <rm; are constants and lim dmj(x) = 0, by (32). System

(11)

On some ordinary differential equations 229

(45) can be written in the form

(46) av(x )u '+ u v = / v(x, wl5 ..., u„), v = l , . . . , n ,

ft ft

where а^(х) = - t % l \ h ( x ) and f v (x, ux, ..., u„) = ( l /и) X U j - h n F / r + X d v j U j .

j = i i = i

We shall prove that the functions ^ and / v satisfy hypotheses of Theorem 1.

By (31) hypothesis (1) is satisfied. By (32) we obtain h(x) = o(x) for n

x -*■ oo and hypothesis (2) is also satisfied. We have um = X 1 yv hv_1/r,

V = 1

m = l , . . . , w , and the domain defined by the inequalities ly^ < »/i H, l>’vl ^ ^/i Irh1 v|, v = 2 , . . . , « , is transformed into the domain |wm—1| ^ ra/i for m = 1 ,..., n. We have namely

\Um-H = I t ^ \ y j r - l \ + t l^v hV~ l/r\ ^ ПГ,Х.

v = 1 v = 2

We set wji = r\. Then f v (x, üu ..., u„) are continuous functions of the variable x for x ^ x 0 and |uv —1| ^ ц.

Next, we have

- ^ - / v(x, ux, ..., u„) = \ / n - h n FyJ n — X е4“Л hn ~ j + 1 F / n + drm.

dum ,= 2 J

By (34) we show, as in the proof of Theorem 2, that |1 — hnFyi\ ^ X x . We choose values X (Ai < X < 1 /пК) and x x ^ x 0 such that we have |dvm(x)|

^ X X x for' x ^ x x and v, m = 1 ,..., n. By (35) we get

—д fx C^» , • • • ? W„)

^ А1/и+(и —1) A^/i + A —Ai = A for x ^ Xj and |wv —1| ^ ц.

Then hypothesis (3) is satisfied for cv = 1. Finally, we have f v(x, 1 ,..., 1) П

= 1 — hnF(x, r, 0 , . . . , 0)/r+ X d v j -* 1, by (33) and (32).

j= 1

By Theorem 1 system (46) has an integral { tq ,..., U„} such that П

lim Uv (x) = 1, v = l , . . . , n . We set for x ^ x x: ÿv = (1 /и) X e}!* ûj rhi ~v,

X - 0 0 j=1

v = 1, and y(x) = >h(x).

P r o o f o f T h eo r e m 5. We shall show that hypotheses of Theorem 4 are satisfied by the functions

F ( x , y l , . . . , y n) = /1—Ь(х)+ X Z M x ) / vs and r(x) = ткЬ1/а(х).

V = 1 s = 1

(12)

230 Z. P o ln ia k o w sk i

P i

We have Fn (x, r, 0 , 0 ) = ar3-1 + Z P u b ^ r11^ 1 and Fyv{x, r, 0 , 0 )

S= 1

= Z ^vsbvs^ vs-1 for v = 2 ,..., n.

s = 1

By (40) we get Fyi( x ,r ,0 ,...,0 ) ~ ar*-1 and ft(x) = {Fyi( x , r , 0, ...,0 ) } -1/"

~ a -i/»d-«)/- for x -> oo. As in the proof of Theorem 3 we show that re /i(x) ~ re (a ~ 1/n r(1_3t)/n) for x -> oo. By (38) there exists (K1 > К ) such that the function h(x) satisfies hypothesis (31) for sufficiently large x and instead of K.

If а Ф 1, then

- n h ' h~ n ~ 1 = - ^ - F y i( x ,r ,0 , ...,0 ) = a ( a - l ) r V “ 2 + Z Р и Ф и ^ ls_1)'

dx s= l

= r' f ~ 2 |a ( a —1)+ Z ^ s№ s- l ) b i / ir a } + Z Pisb'u^ 1^ 1

s= 1 s = 1

= 0 ( r V - 2)+ I

s = 1

for x -> oo, by (40). Moreover, we have for x -*■ cc

hn + lr V - 2 _ а -(п + 1)/пг/ г(1-*)/И- 1 _ cb/ b( l-«)/«" 1 _>o, by (39), /in + 1b'ls r ^ - i ~ c j ^ ^ ^ - ^ - U O , by (41), and we obtain that lim /г'(х) = 0.

X~> 00

If а = 1, then lim h{x) = 1 and

—nhrh~n~i ~ Fj, (x, r, 0 , 0 ) = £ by (41).

dx s=i

By (39) we get

/ir'/r ~ a -i/" r' r(i-a)/"-i ^ c2 Ь'Ь(1"а)/па_1 -> 0 for x -» oo and hypothesis (32) is satisfied.

By (40) we obtain for x -+ oo

F ( x , r , 0 , ...,0 )/{r F yi(x, r ,0 , ...,0 )} ~ Z b u ^ ^ - y c t ^ O

S= 1

and hypothesis (33) is satisfied.

We choose Яг satisfying the inequality 0 < Ях < 1/nK. Suppose that the variables yv satisfy the inequalities

(47) |y i/r—1| ^ rj, |yv| ^ rj \rhl ~v\ for v = 2 ,..., n, x ^ x 0,

(13)

On some ordinary differential equations 231

where ц = min {1/2, Aly/8 |a—1|} if a Ф 1 and t] — 1/2 if a = 1. We obtain for x ^ x 0 and yv satisfying (47)

|ln Fyi (x, y l5 ..., yn) In Fyi (x, r, 0 , 0)|

= |ln {Fyi{ x , y1, . . . , y n)/ ar*_1} - l n {Fyi{x, r, 0 , 0)/ar3t_1}|

^ \ln { ( y J r f - ' + m Z

S = 1

+ |ln {Fyi (x, r, 0, ...,0)/arï_ 1 }|

and as in the proof of Theorem 3 we show that hypothesis (34) is satisfied for some x x ^ x0 instead of x 0 . Finally, we obtain for x ^ x 0 and yv, v = 2 satisfying (47):

\Fyv( x , y1, . . . , y n)hn~v + 1\ = \hn~v + 1 £ Pvsbvsfv^ 11

S = 1

^ |/l"~v+1 £ jSvs 1 bvs r^v." 1 fc(1 - v)^vS- D|.

s = 1

Since

J j H - v + l ^ V ^ vs- 1 ^(1- V) (/ Îvs- 1 ) _ b v s r ^ v s _ 1 /j" + ( l _ v) ^VS ^

~ c3 bvsb( v - 1 ) i a - 1 ) ^ l m + p ^ a - 1 -» 0, by (42), then hypothesis (35) is satisfied for sufficiently large x and yv satisfying (47).

We complete the proof applying Theorem 4.

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK

MATHEMATICAL INSTITUTE OF THE POLISH ACADEMY OF SCIENCES

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