**ROCZNIKI POLSKIEGO TO W A RZYSTW A M A T E M A T Y C ZN E G O **
**Séria I: PRACE M A T E M A T Y C ZN E XXVII (1988)**

Ja r o s l a w Mi k o l a j s k i (Poznan)

**On the linear systems of differential equations ** **with a bounded number of zeros of their solutions**

**Abstract. **The paper contains a certain sufficient condition under which all nontrivial
*solutions o f the linear system o f n ordinary differential equations have a bounded number of *
zeros o f the first component. F or this purpose we transform the considered system to a system
*o f n — 1 differential equations.*

1. Introduction. Nonoscillation o f systems o f ordinary differential equa

tions is investigated by many authors. In particular, Z. Ratajczak [5 ], [6] has given some criteria o f nonoscillation of linear systems. In other papers, there are considered some sufficient conditions in order that components of all nontrivial solutions o f systems o f differential equations have a bounded number of zeros. Z. Butlewski [1], [2 ] has given sufficient conditions under which components o f all nontrivial solutions of linear homogeneous systems of two and three differential equations have at most one and two zeros, respectively. In [4], these results have been extended to the linear homo

geneous system of four differential equations. A generalization of this prob

lem for the linear homogeneous system of two differential equations and an extension to the linear nonhomogeneous system of two differential equations are contained in [3].

The present paper contains a generalization o f these results and an
extension to any linear system of *n* differential equations *(n* ^ 3), i.e.,

П

(1.1) *x'i = Z OijMxj + fii*)» * *i = 1,2,*
*j=* i

where

(1.2)

*aijtf* e C1 [r0, oo) for *i , j* = 1,2,
*ain(t) Ф*0 for *t* e [r 0, oo)

W e give a sufficient condition under which the nontrivial first components of all solutions o f this system have a bounded number o f zeros. Applying results from [3], Corollary 6 is o f great importance. Analogous conditions for the remaining components can be given.

A function / : *[T,* oo) -+ *R* is called *nonoscillatory* if
3 V / ( I ) #0.

T j e f T , oo) t e t T j . a o )

**2. ** Lemma. Let *conditions* (1.2) *hold. Assume that there exists a solution *
*(£i,* £2, ..., £„) *of the linear homogeneous system*

(2.1) *y'i* = X ay(f).ty, i = 1, 2, ..., *n,*

*j=***1**

*corresponding to the system of differential equations* (1.1) *such that* ^ (f) *Ф*0

y b r t e [ t 0 , **0 0**). T / je n t/ie *substitution*

**(2.2)**

***1 = £1 (0 .f Zi**
r*0

*Xi = Zi(t)* *f Zxds **+ **Zi, * *i* *= 2, 3, ..., n, *

*o

*transforms the system* (1.1) *to the system of equations*

(2.3)

*where*

(2.4)

*n-*1

*A* ** = Z MOzj- + 0i(O, ** *** = 1, **

**2 , . . . ,**

**л-1,**

* j=*1

1

£ _ 1 bn

*b*

**L k = 2** ^1 л

^ln
*a ln*

V1 *^lj &кп\ * *a lj a ln* ,

**Z flulaw-—— ** **-**

^{+ a XJ}*_k=2* V a l *n * *J * *a l **n*

*a*,:

b,i = — £1- 6, aln

b y *aij*

*g *^{1}* =*

**<*ln**

## r

**V ^ I k ^ k n***1*

**O l n \ r**

^{V }

^{f I}**Z-- + ** **— /**

^{1}

**+ Z f** **l** **u/k+/i**

= 2 aln а1я k = 2

01= *~ — f l + f*

*a l **n*

*for i, j —* 2, 3, ..., *n —* 1. *Moreover,*

(2.5) ^{1}

*n-*1

*■ n - *

**I «ij(0Zj-/,(t)].**

“ i n W *j=***2**

P r o o f. Let (£j, £2, £„) be the solution of the system (2.1) appearing in the assumption of the lemma. In the system of differential equations (1.1) we change variables according to formulae (2.2). W e get

**(2.6)**

### £i(*)*i = Z flij(0f/+/i(t),

**7**= **2**

*n*

*z [ = - & ( t ) z*1+ Z *<kj{t)Zj + f A*0, i = 2, 3, ..., n.

7 = 2

Hence equation (2.5) foliowis. Differentiating the first equation of (2.6) and
eliminating the variable *z„,* we obtain the system

**(2.7)**

£1*(t)*

*n— 1*

+ 1

7 = 2

«1»(0 7 «1ЛО

Jk= 2

Zi +

*L aik(t) akJ(t*) ---*— —— ---— —— + aM(r)

k = 2 *Oln(t)* *dln(t)* *Zi —*

*у* Q l*k(t)<*kn(t)* + Д**1**^{Л О}

### /Л 0 + Z a«(0/*W + /i(t)S

*k=2*

*Z =* ^{Qin (f)}

Я1ЛО M O - 6 ( 0

И-1

^ 1 + z

7=2

### Я

^{1}

### Л

^{0}

M O

### /Л0+7Л0,

*i **=* 2, 3, ..., *n —* 1.

This system is o f the form (2.3) with coefficients defined by formulae (2.4).

**3. ** **T****heorem****. ***L e t* **(iq, ***u 2,* . . . , *u*n) *be a s o lu tio n o f th e system o f d iffe re n tia l *
*equ a tion s* (1.1) *w ith co e ffic ie n ts sa tisfyin g co n d itio n s* (1.2). *A ssum e th a t th ere *
*exists a s o lu tio n* (£ l5 £2, . . . , £„) *o f th e system o f d iffe re n tia l eq u a tion s* (2.1) *such *
*th at* M O ^ O *f o r t e [ t 0,* 0 0). *I f f o r ev ery s o lu tio n* (ul5 i?2, u „ - i ) *° f th e*
*system o f d iffe re n tia l eq u a tio n s* (2.7) *th e f u n c tio n v t e ith e r is id e n tic a lly eq u a l to *
*z e ro o r has at m ost m z e ros in th e in te rv a l* (*t*0, 0 0), *th en the fu n c tio n* iq *e ith e r is *
*id e n tica lly eq u a l to z e ro o r has at m ost* m+ 1 *z e ro s in th e in te rv a l [ t 0,* 0 0).

P r o o f. Take the solution ( iq , *u 2,* . . . , *u*n) o f the system o f differential
equations (1.1) and the solution (<6, £2, ..., £„) o f the system (2.1) appearing
in the assumptions o f the theorem. Let *c* = ( tq /£ i) 0 o ) and

*% = щ - с £ ь* i = l ,2, ...,n .

Hence *(rjx, rj2,* . *rj*„) is a solution o f the system (1.1). Since

>/. ( 0 = f l ( 0 I ('/ !/ < ?■ )'( s ) d s ,

*0

it follows by Lemma 2 that the function *{ rj j*^x)' is the first component o f a
solution o f the system (2.7).

W e have

= *$ ( r h / £ i)'(s )d s ].*

'o Write

*U ( t ) = c + j ( r ] 1/^l )'(s )d s , * *t e [ t 0,* oo).

*o

Differentiating the function *U ,* we obtain *U '* = (ih/£iX- Thus *I T* is the first
component o f a solution o f the system (2.7). By the assumption, the function
*U '* either is identically equal to zero or has at most *m* zeros in the interval
(r0, oo). Hence the function *U* either is identically equal to zero or has at
most *m +*1 zeros in the interval [t 0, oo). Obviously, the function *u x* has the
same property.

**4.** Co r o l l a r y. *L e t ( u x, u 2,* . . . , *un) be a s o lu tio n o f th e system o f d iffere n *
*tia l e q u a tion s* (1.1) *w ith c o e ffic ie n ts sa tisfyin g c o n d itio n s* (1.2). *A ssum e th a t *
*th e re e x ists a s o lu tio n* (£ 1? £2, •••, £„) *o f the system o f d iffe re n tia l eq u a tion s*
(2.1) *such th a t th e f u n c t io n * *£x * *is n o n o s cilla to ry . I f f o r every s o lu tio n *
*( v x , v 2,* . . . , u„_i) *o f th e system o f d iffe re n tia l e q u a tion s* (2.7) *th e fu n c tio n v x is *
*e ith e r id e n tic a lly eq u a l to z e ro o r n o n o s cilla to ry , th en th e fu n c tio n u x has th e *
*same p ro p e rty .*

This corollary follows immediately from the proof of Theorem 3.

**5. P****roposition****. ** *L e t co n d itio n s* **(1.2) ** *hold. A ssum e th a t th ere e x is ts a*
*s o lu tio n* (£ l5 £2, £„) *o f th e system o f d iffe re n tia l eq u a tion s* (2.1) *such th a t*

£ i( 0 # 0 *f o r t e [ t 0,* 0 0).

(a) *I f th e system o f d iffe re n tia l e q u a tion s* (2.7) *has no so lu tion s w ith the *
*f ir s t c o m p o n e n t id e n tic a lly eq u a l to z e ro , then th e system* (1.1) *has th e same *

*p ro p e rty .*

(b) *I f f x* = 0 *and th e system o f d iffe re n tia l eq u a tio n s* (2.7) *has no n o n triv ia l *
*s o lu tio n s w ith th e f ir s t c o m p o n e n t id e n tic a lly eq u a l to z e ro , th en th e system*

(1.1) *has th e sam e p ro p e rty .*

P r o o f. Assertion (a) follows from the proof o f Theorem 3.

Ad (b). Suppose that *{u x, u 2, . . . , un)* is a solution o f the system of

equations (1.1) such that the function is identically equal to zero. W e have

t *t*

(0 = £i (*t*) f о*ds, * *Щ* (*t*) = & *(t) f ods + щ* (0, *i = 2,* 3, ..., *n,*

*0 'o

in the interval [ f 0, oo). By Lemma 2, (0, n2, *u3,* ..., « „ - i ) is a solution of the
system (2.7) and

1 *n~1*

(5.1) *u„ =*

### Z

^{°}

^{u uj}

^{-}*a ln j=2*

By assumption, the functions *u2, u3,* ..., u„_ *t* are identically equal to zero
and, by formula (5.1), *un* has the same property.

6. Co r o l l a r y. *Let* (u 1? *u2,* ..., *u„) be a solution of the system of differen*

*tial equations* (1.1) *with coefficients satisfying the conditions*
*a i j , f e C n* 1 **[t0, oo) ** *for i , j **= 1, *2, . . . , *n, *

*aln( t ) ^ 0 * *for t e [ t 0,* **oo).**

*Assume that there exist functions Çki* e C " -fe_ 1 [ t 0, * oo) for к* = 0, 1, ...,

*n —*2,

*i —*1, 2, ...,

*n — k such that*

(i) £ u(07* 0*for t E*[ t 0, oo),

### (ü) & = Z

^{aiajZkj,}*}=* i
*where*

**(****6****.****1****)**

*aoij = Oij * *( h j =* 1, 2, ..., n),

fl( k + l ) l l —

a,( k + l ) U

£kl

£kl
*n **— **k*

*S*

*l akp(n-k)*

*z*s \ , A k l(n -k )

a k l p \ ’ k l Skp 1 +

, p = 2 \ a k l ( n - k ) / a k l ( n - k )

^ k l - f t l (fc = 0, 1, ..., n —3),

**V ** Л *ak\jakp{n-k)* \ *akljakl(n-k)* ,

*la aklp\akpj* I + a k l j

L p = 2 \ а к 1 (л -к ) / a k l ( » - k )

fl(k + l ) i l

(fc = 0, 1 ,..., *n -*3, *j = 2,* 3 ,. .. , n - f c - 1 ),

*а к Ц п - к ) p * K

~ S k i — Ski

“ k l(n - k )

*(к =* 0, 1, ..., *n —* 3, i = 2, 3, ..., *n — к —* 1),
a(k+l)ü ^{^kij}

*a kljaki(n-k)*
*akl(n-k)*

*(к = 0, 1, ..., n —3, i, j = 2, 3,* *., n — k — 1)*

6 — Prace M atem atyczne 27.2

*and*

*aki(n-k)(0* # 0 *for t e [ t 0,* oo) (к = 0, 1, ..., w -2).

*Define the functions*

*f o i = f* (/ = 1, 2, ..., и),

*{к =* 0, 1, *n -*3),
(/с = 0, 1, *n* — 3, *i =* 2, 3, *n — k —* 1),

**(6.2) ** **F =**

*ü(n-2)1*2

(a) // f/ie *function F has at most m zeros in the interval* (*t0*, oo), *then ux *
*has at most m + n zeros in the interval* [ f 0, oo).

(b) *I f F =* 0, *then either ux = 0 or ux has at most* /7—1 *zeros in the *
*interval* [ f0, oo).

P r o o f. W e apply Lemma 2 to the system of differential equations (1.1) n- 2 times and bring it to the linear system of two differential equations

Such a system was considered in paper [3]. Let *( U x, U*2) be a solution of this
system. By Theorem 2 in [3 ]:

(a) if the function *F* defined by (6.2) has at most m zeros in the interval
(r0, oo), then *U x* has at most m-f 2 zeros in the interval [r0, oo),

(b) if *F* = 0, then either *JJX* = 0 or *U x* has at most one zero in the
interval [ f 0, oo).

Applying now Theorem 3 and Proposition 5(a) o f the present paper *n — 2 *
times, we obtain the assertion.

7. R em a rk . The considerations o f the present paper are true under some weaker assumptions concerning the coefficients of the system of differential equations (1.1). Namely, in Sections 2-5 it suffices to assume that

and the remaining coefficients are from C [r0, oo). In Section 6 it suffices to assume that

**2**

**Щ =**

** I 0,„-2WW + **

^{I}** = 1** **.** ** 2.**

“ i j . / i e C1 [t 0, oo) for 7 = 2, 3,...., *n*

and the remaining coefficients are from C [r 0, oo). Moreover, in C orol

lary 6 it suffices to assume about the functions *Çki (к* = 0, 1, *n —*2,
*i =* 1, 2...*n - k ) :*

£ki e C " -k“2[ t 0, oo) for *к* = 0, 1, n —3
and the remaining functions are from C1 [ f 0, oo).

8. Ex a m p l e s, (a) Take the system o f differential equations
*x\ ~ (5 — e‘) x l —4x2 + x3,*

(8.1) *x 2 ~* (1 — *t)* * i + *tx2 +* + 1,

*3 *= (2 — к)е1 x l + t x 3 + et.*

The linear homogeneous system corresponding to this system has the solu

tion

*Ш* * = е\ * *{ 2(t) = e‘, * *Ш* * = е21.*

Applying the substitution (2.2) to the system (8.1), we obtain a system o f the form (2.7). The first equation o f the system is

**(8.2) ** **z\ = ( и З - ф г ( н 4 ) е 'Ч 1 .**

Solving this equation, we get

*z1 —* [1 + c e x p( ^ 2 + 4r — e*)] *e~\ * *c e R .*
Write

We have

Z c(r) = 1 + c e x p (^ r + 4t — *e*).*

*Z'c(t) = c ( t + 4 — é)* exp (^ r -1- *4t — e*),*

so the function Z ' for *с Ф* 0 has two zeros in the interval [ — 4, oo). Hence Z c
for any *c e R* has at most three zeros in this interval and all solutions of
equation (8.2) have the same property. By Theorem 3 and Proposition 5 (a),
the first component of any solution of the system (8.1) has at most four zeros
in the interval [ — 4, oo).

(b) Take the system o f differential equations

* i = — *é x t + x 3,*
* X2 = t2 Xi *+

* 3 = *é xx* +

1 — *t2 — é — té *
*t{*1 — e*)
*e2t* *x 2* +

*■x2 +* *t + r*

*■3»

*t* (e* — 1) *ï —é* ^{* 3}
(8.3)

for r e [ 1, oo). This system has the solution

*Ш* * =* U «2 (') = ». *Ш* * =*

Form the system of equations with coefficients (6.1) for *k =* 0. W e have

By Corollary 6, the first component o f any nontrivial solution of the system (8.3) has at most two zeros in the interval [1, oo).

[1 ] *Z. B u tl e w s k i, O calkach rzeczywistych rôwnan rôznipzkowych liniowych zwyczajnych, *
Wiadom. Mat. 44 (1938), 17-81.

[2 ] *—, Sur les intégrales non oscillatoires d'un système de trois équations différentielles linéaires, *
Ann. Polon. Math. 18 (1966), 95-106.

[3 ] *J. M i k o l a j s k i , On a certain property o f the linear system o f two differential equations, *
Funct. et Approx. 16, in print.

[4 ] *—, Z. R a t a jc z a k , On the nonoscillator y solutions o f a system o f fou r linear homogeneous *
*differential equations, Fasc. Math. 13 (1981), 51-59.*

[5 ] *Z. R a t a jc z a k , 0 rozwiqzaniach monotonicznych ukladu n rôwnan rôzniczkowych liniowych *
*jednorodnych, Zesz. Nauk. Polit. Pozn. Matem. 3 (1967), 59-65.*

Г61 *—, On monotone solutions o f à system o f n linear nonhomoqeneous differential equations,*
Fasc. Math. 10 (1978), 47-52.

**(IN STYTU T M ATEM AT YK I, P O LIT E C H N IK A P O Z N A N S K A )**

**INSTITUTE OF MATHEMATICS, T E C H N IC A L UNIVERSITY, P O Z N A N**

*t ( e ' ~*

### 1)

This system has the solution
*e*^{,}*21*

£ ll(0 = l, Él2(0 = t.

**References**