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Mem. Dierential Equations Math. Phys. 6(1995), 127{129

G.Giorgadze

ON OSCILLATORY PROPERTIES OF THE

n

-TH ORDER SYSTEM OF DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

(Reported on November 27, 1995) Consider the system

x 0

i(t) =fi t;x1(i1(t));:::;xn(in(t)); (i= 1;:::;n); (1) wheren2, the vector function (fi)ni=1:R+Rn!Rnsatises the local Caratheodory conditions,ij:R+!Rare nondecreasing and

lim

t!+1

ij(t) = +1 (i;j= 1;:::;n); i;i+12C0(R+;R) (i= 1;:::;n 1): Dene:R+!R+by

(t) = infs:s2R+;st;ij()t; for 2[s;+1[ (i;j= 1;:::;n) :

Denition 1.

A continuous vector functionX= (Xi)ni=1: [t0;+1[!Rn witht02

R

+is said to be a proper solution of the system (1) if it is locally absolutely continuous on [(t0);+1[, almost everywhere on this interval the equality (1) is fullled, and

supkx(s)k:s2[t;1[ >0; for t2[t0;+1[

Denition 2.

A proper solution of the system (1) is said to be oscillatory if every component of this solution has a sequence of zeroes tending to +1. Otherwise the solution is said to be nonoscillatory.

Denition 3.

We say that the system (1) has the property Aprovided its every proper solution is oscillatory ifnis even, and either is oscillatory or satises

jx

i(t)j#0; for t"+1; (i= 1;:::;n); (2) ifnis odd.

Denition 4.

We say that the system (1) has the property B provided its every proper solution either is oscillatory or satises either (2) or

jx

i(t)j"+1; for t"+1; (i= 1;:::;n) (3) ifnis even, and either is oscillatory or satises (3) ifnis odd.

1991 Mathematics Subject Classication. 34K15.

Key words and phrases. Oscillatory solution, nonoscillatory solution, property A, propertyB.

(2)

128

We will assume that there existi2f0;1gsuch that

( 1)ifi(t;x1;:::;xn)signxi+1pi(t)jxi+1j;

( 1)nfn(t;x1;:::;xn)signx1g(t;jx1j); for t2R+; x1;:::;xn2R; (4) where the functiong 2 Kloc(R+R+;R+) is nondecreasing in the second argument,

pi2L

loc(R+;R+) and

+1

Z

0 p

i(t)dt= +1 (i= 1;:::;n 1): (5) Besides, introduce the notation

=

n

X

i=1

i :

i(t) =i 1;i(t) (i= 2;:::;n); 1(t) =n+1(t) =n1(t):

ji(t) =

j

j 1(:::(i+1(t)):::); if 1i<jn+ 1;

t; if i=j (i= 1;:::;n);

ji(t) = infs:s2R+;k i(s)t(k=i;:::;j) (1ijn);

I

0= 1; Ij(s;t;pi+j 1;:::;pi) =

=

s

Z

t p

i+j 1(i+j 1;i())(i+j0 1;i()Ij 1(;t;pi+j 2;:::;pi)d;

J

0= 1; Jj(t;s;pi;:::;pi+j 1) =

t

Z

s p

i()Jj 1(i+1();i+1(s);pi+1;:::;pi+j 1d); (i= 1;:::;n 1; j= 1;:::;n i):

Note that the functionsji :R!R+are increasing,

k i(t)ji(t) (1ijkn);

ji(t)t (1ijn); for t2R;

and the expressionsIj(s;t;pi+j 1;:::;pi) andJj(t;s;pi;:::;pi+j 1) have the meaning it;si+j 1;i(0) (i= 1;:::;n 1;j= 1;:::;n i).

Theorem -1.

Suppose that the conditions(4) and (5) are fullled, is odd and for everyl2f1;:::;n 1gsuch thatl+nis odd, the equation

v

0(t) =In l l1(t);tl;pn 1;:::;plg n1 (t);zl(n+1;1 (t))n10(t); (6) withzl(t) =Jl(t;l1(0);p1;:::;pl)

J

0(l1(t);0;pl) ,tl=n 1;l(0), has no positive proper solution. In the case wherenis odd, let, moreover,

+1

Z

n1 (0)

I n 1

;

n 1;1(0);pn 1;:::;p1g n1 ();cn10()d= +1; for c>0: (7) Then the system(1) has the propertyA.

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129

Theorem 0.

Suppose that the conditions(4) and (5) are fullled, is even and for everyl2f1;:::;n 2gsuch thatl+nis even, the equation(6) has no positive proper solution. Let, moreover,

+1

Z

(0) g t;cJ

n 1(1(t);n1 (0);p1;:::;pn 1)dt= +1: (8) for anyc>0, and, in the case wherenis even, the condition(7) be fullled. Then the system(1) has the propertyB.

Consider now the case where the inequalities

( 1)ifi(t;x1;:::;xn)signxi+1pi(t)jxi+1j (i= 1;:::;n;xn+1=x1) (9) are fullled, wherepi2Lloc(R+;R+) (i= 1;:::;n) and (5) holds.

Theorem 1.

Suppose that(12) is fullled, is odd and for everyi2f1;:::;n 1g such thati+nis odd, the inequalites

lim

t!+1

sup In i(t;ti;pn 1;:::;pi)

I n i 1

i+1(t);i+1(ti);pn 1;:::;pi+1

+1

Z

t I

n i 1

i+1(s);i+1(ti);pn 1;:::;pi+1

J

i

n+1(s);i1(0);p1;:::;pi

J 0

i1(n+1 (s));0;pi pn(ni(s))ni0(s)ds>1 (10) and

i1(n+1;i (t))t; textfor t(0) (11) hold, whereti=n 1;i(0). In the case wherenis odd, let, moreover,

+1

Z

n1 (0)

I n 1

;

n 1;1(0);pn 1;:::;p1pn n1 ()n10()d= +1 (12) Then the system(1) has the propertyA.

Theorem 2.

Suppose that(12) is fullled and for everyi2f1;:::;n 1gsuch that

i+nis even, the inequalities(13) and (14) are fullled. Let. moreover,

+1

Z

(0) J

n 1

1(t);n 1;1(0);p1;:::;pn 1pn(t)dt= +1;

and, in the case wherenis odd,(15) hold. Then the system (1) has the propertyB. Author's address:

Georgian Technical University 72, Kostava St., Tbilisi 380093 Georgia

参照

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