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GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 3, 1998, 201-212

OSCILLATORY PROPERTIES OF SOLUTIONS OF
IMPULSIVE DIFFERENTIAL EQUATIONS WITH
SEVERAL RETARDED ARGUMENTS
D. D. BAINOV, M. B. DIMITROVA, AND V. A. PETROV

Abstract. The impulsive differential equation
x′ (t) +

m
X

pi (t)x(t − τi ) = 0,

t 6= ξk ,

i=1

∆x(ξk ) = bk x(ξk )
with several retarded arguments is considered, where pi (t) ≥ 0, 1 +

bk > 0 for i = 1, . . . , m, t ≥ 0, k ∈ N. Sufficient conditions for the
oscillation of all solutions of this equation are found.

§ 1. Introduction
In the past two decades the number of investigations of the oscillatory
and nonoscillatory behavior of solutions of functional differential equations
has been growing constantly. The greater part of works on this subject published up to 1977 are given in [1]. In the monographs [2] and [3], published
in 1987 and 1991 respectively, the oscillatory and asymptotic properties of
solutions of various classes of functional differential equations are systematically studied.
The first work in which the oscillatory properties of impulsive differential
equations with retarded argument of the form
x′ (t) + p(t)x(t − τ ) = 0,
∆x(tk ) = bk x(tk )

t 6 = tk ,

(∗)

are investigated is the paper of Gopalsamy and Zhang [4]. In it the authors give sufficient conditions for the oscillation of all solutions under the
1991 Mathematics Subject Classification. 34A37.

Key words and phrases. Oscillation of solutions, impulsive differential equations with
several retarded arguments.

201
c 1998 Plenum Publishing Corporation
1072-947X/98/0500-0201$15.00/0 

202

D. D. BAINOV, M. B. DIMITROVA, AND V. A. PETROV

assumption that p(t) is a positive function, and sufficient conditions are
found for the existence of a nonoscillatory solution if p(t) ≡ p = const > 0.
In [5] more general conditions for the oscillation of solutions of equation
(∗) are found, when this equation has a retarded argument (τ > 0). In [6]
similar results are obtained when equation (∗) has an advanced argument
(τ < 0).
In the present work sufficient conditions for the oscillation of solutions of
impulsive differential equations with several retarded arguments are found.
§ 2. Preliminary notes

Consider the impulsive differential equation with several retarded arguments
x′ (t) +

m
X

pi (t)x(t − τi ) = 0,

t=
6 ξk ,

i=1

(1)

∆x(ξk ) = bk x(ξk ),
where ∆x(ξk ) = x(ξk+ ) − x(ξk ), together with the impulsive differential
inequalities
x′ (t) +

m
X

pi (t)x(t − τi ) ≤ 0,

t 6= ξk ,

i=1

(2)

∆x(ξk ) = bk x(ξk )
and
x′ (t) +

m
X

pi (t)x(t − τi ) ≥ 0,

6 ξk ,
t=

i=1

(3)

∆x(ξk ) = bk x(ξk )
provided that the following conditions are met:
A1. 0 < τ1 < · · · < τm .
A2. The sequence {ξk } is such that
0 < ξ1 < ξ2 < · · · ,

lim ξk = ∞.

k→∞

A3. pi ∈ C(R+ , R+ ), i = 1, 2, . . . , m.
A4. bk > −1 for k ∈ N.
Remark 1. If 1 + bk ≤ 0 for an infinite number of k ∈ N, then every

nonzero solution x(t) of equation (1) is oscillatory, since x(ξk+ ) = (1 +
bk )x(ξk ). Therefore the case bk < −1 is not interesting for consideration
and condition A4 is quite natural.

IMPULSIVE DIFFERENTIAL EQUATIONS

203

∞
Consider the sequences {ξn }∞
n=1 (the jump points) and {ξn + τi }n=1 ,
i = 1, 2, . . . , m, and let
∞
∞
{ξn }∞
n=1 ∩ {ξn + τ1 }n=1 ∩ · · · ∩ {ξn + τm }n=1 = ∅.

Let {tk }∞
1 be the sequence with the following properties:
(i) t1 < t2 < · · · .

∞
∞
(ii) {tk }1∞ = {ξn }∞
1 ∪ {ξn + τ1 }1 ∪ · · · ∪ {ξn + τm }1 .
Clearly lim tk = ∞.
k→∞

Definition 1. The function x(t) is said to be a solution of equation (1)
if:
1. x(t) is continuous in [−τ, ∞) \ {ξn }∞
n=1 , and is continuous from the
left for t = ξn , n = 1, 2, . . . .
2. x(t) is differentiable in (0, ∞) \ {tk }∞
1 .
∞

3. x(t) satisfies equation (1) for t ∈ (0, t1 ) and t ∈ ∪ (ti , ti+1 ).
i=1

4. x(ξk+ ) = (1 + bk )x(ξk ).

Analogously solutions of the inequalities (2) and (3) are defined. Details
about the general theory of differential equations with impulses can be found
in [7].
Definition 2. The solution x(t) of the inequality (2) is said to be eventually positive if there exists t0 > 0 such that x(t) > 0 for t ≥ t0 .
Definition 3. The solution x(t) of the inequality (3) is said to be eventually negative if there exists t0 > 0 such that x(t) < 0 for t ≥ t0 .
Definition 4. The solution x(t) of equation (1) is said to be oscillatory if the set of its zeros is unbounded above, otherwise it is said to be
nonoscillatory.
Let the function u(t) satisfy conditions 1, 2, and 4 from Definition 1 and
let u(t) be nonincreasing in (tk , tk+1 ), k = 0, 1, 2, . . . (t0 = 0). Define the
function ψ(t) by
Y
ψ(t) =
(1 + bk ).
(4)
0≤ξk