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

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) Ф0 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) Ф0

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,

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-—— -

_k=2 V a l n J a l n

a,:

b,i = — £1- 6, aln

b y aij

g ¹ =

<*ln

r

Z-- + — /

+ 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 ) z1+ Z <kj{t)Zj + f A0, 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 lk(t)<*kn(t) + Д1^{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

Л

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. Theorem. L e t (iq, u 2, . . . , un) 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 (t0, 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, . . . , un) 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. Proposition. 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 оds, 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

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* 0for t E[ t 0, oo),

(ü) & = Z

}= 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

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)12

(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, U2) 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 +

= 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