ON OSCILLATORY PROPERTIES OF THE

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

Denition 1.

A continuous vector function^X= (^Xi)ⁿⁱ⁼¹: [^t0;+¹[^!Rn with^t02

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

sup^kx(^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, property^B.

128

We will assume that there exist^i2f0;1^gsuch that

( 1)^ifi(^t;x1;:::;xn)sign^xi+1pi(^t)^jxi+1j;

( 1)^nfn(^t;x1;:::;xn)sign^x1g(^t;jx1j)^; for ^t2R+; ^x1;:::;xn2R; (4) where the function^{g 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 1^i<jn+ 1^;

t; if ⁱ=^j (ⁱ= 1^;:::;n)^;

ji(^t) = inf^s:^{s2R+;k i}(^s)^t(^k=^i;:::;j) (1^ijn)^;

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 functions^ji :^R!R+are increasing,

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

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

and the expressions^Ij(^s;t;^{pi+j 1;:::;pi}) and^Jj(^t;s;^{pi;:::;pi+j 1}) have the meaning i^{t;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 every^l2f1^;:::;n 1^gsuch that^l+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) with^zl(^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 property^A.

129

Theorem 0.

Suppose that the conditions(4) and (5) are fullled, is even and for every^l2f1^;:::;n 2^gsuch that^l+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 any^c>0, and, in the case whereⁿis even, the condition(7) be fullled. Then the system(1) has the property^B.

Consider now the case where the inequalities

( 1)^ifi(^t;x1;:::;xn)sign^xi+1pi(^t)^jxi+1j (ⁱ= 1^;:::;n;^xn+1=x1) (9) are fullled, where^pi2Lloc(^R+;R+) (ⁱ= 1^;:::;n) and (5) holds.

Theorem 1.

Suppose that(12) is fullled, is odd and for every^i2f1^;:::;n 1^g 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, where^ti=^{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 property^A.

Theorem 2.

Suppose that(12) is fullled and for every^i2f1^;:::;n 1^gsuch 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 property^B. Author's address:

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