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) + ∑_{i=1}^{m} p_{i}(t)x(t − τ_{i}) = 0, t ≠ ξ_{k},

Δx(ξ_{k}) = b_{k}x(ξ_{k})

with several retarded arguments is considered, where p_{i}(t) ≥ 0, 1 + b_{k} > 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

\begin{matrix}x′(t) + p(t)x(t − τ) = 0, t ≠ t_{k}, \\ Δx(t_{k}) = b_{k}x(t_{k}) \end{matrix}  (∗)

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

201 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.

1991 Mathematics Subject Classification. 34A37.

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

1072-947X/98/0500-0201$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

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

\begin{matrix}x′(t) + ∑_{i=1}^{m} p_{i}(t)x(t − τ_{i}) = 0, t ≠ ξ_{k}, \\ Δx(ξ_{k}) = b_{k}x(ξ_{k}), \end{matrix}  (1)

where Δx(ξ_{k}) = x(ξ_{k}^{+}) − x(ξ_{k}), together with the impulsive differential inequalities

\begin{matrix}x′(t) + ∑_{i=1}^{m} p_{i}(t)x(t − τ_{i}) ≤ 0, t ≠ ξ_{k}, \\ Δx(ξ_{k}) = b_{k}x(ξ_{k}) \end{matrix}  (2)

and

\begin{matrix}x′(t) + ∑_{i=1}^{m} p_{i}(t)x(t − τ_{i}) ≥ 0, t ≠ ξ_{k}, \\ Δx(ξ_{k}) = b_{k}x(ξ_{k}) \end{matrix}  (3)

provided that the following conditions are met:

A1. 0 < τ_{1} < ⋯ < τ_{m}.

A2. The sequence {ξ_{k}} is such that

0 < ξ_{1} < ξ_{2} < ⋯,  lim_{k→∞} ξ_{k} = ∞.

A3. p_{i} ∈ C(ℝ_{+}, ℝ_{+}), i = 1, 2, …, m.

A4. b_{k} > −1 for k ∈ ℕ.

Remark 1. If 1 + b_{k} ≤ 0 for an infinite number of k ∈ ℕ, then every nonzero solution x(t) of equation (1) is oscillatory, since x(ξ_{k}^{+}) = (1 + b_{k})x(ξ_{k}). Therefore the case b_{k} < −1 is not interesting for consideration and condition A4 is quite natural.

Consider the sequences {ξ_{n}}_{n=1}^{∞} (the jump points) and {ξ_{n} + τ_{i}}_{n=1}^{∞}, i = 1, 2, …, m, and let

{ξ_{n}}_{n=1}^{∞} ∩ {ξ_{n} + τ_{1}}_{n=1}^{∞} ∩ ⋯ ∩ {ξ_{n} + τ_{m}}_{n=1}^{∞} = ∅.

Let {t_{k}}_{1}^{∞} be the sequence with the following properties:

(i) t_{1} < t_{2} < · · · .

(ii) {t_{k}}_{1}^{∞} = {ξ_{n}}_{1}^{∞} ∪ {ξ_{n} + τ_{1}}_{1}^{∞} ∪ ⋯ ∪ {ξ_{n} + τ_{m}}_{1}^{∞}. Clearly lim_{k→∞} t_{k} = ∞.

Definition 1. The function x(t) is said to be a solution of equation (1) if:

1. x(t) is continuous in [−τ, ∞) \ {ξ_{n}}_{n=1}^{∞}, and is continuous from the

left for t = ξ_{n}, n = 1, 2, ….

2. x(t) is differentiable in (0, ∞) \ {t_{k}}_{1}^{∞}.

3. x(t) satisfies equation (1) for t ∈ (0, t_{1}) and t ∈ ∪^{∞}_{i=1}(t_{i}, t_{i+1}).

4. x(ξ_{k}^{+}) = (1 + b_{k})x(ξ_{k}).

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 t_{0} > 0 such that x(t) > 0 for t ≥ t_{0}.

Definition 3. The solution x(t) of the inequality (3) is said to be eventually negative if there exists t_{0} > 0 such that x(t) < 0 for t ≥ t_{0}.

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 (t_{k}, t_{k+1}), k = 0, 1, 2, … (t_{0} = 0). Define the function ψ(t) by

ψ(t) = ∏_{0≤ξ_{k}<t}(1 + b_{k}).  (4)

(If t ≤ ξ_{1}, ψ(t) = 1.) Let

v(t) = \frac{u(t)}{ψ(t)}, t ≥ 0.  (5)

We shall establish some properties of v(t) which we shall use in the proof of the main results. To this end we need the following result.

Lemma 1 ([8], p. 330). Let f have the properties:

(i) f ∈ C([a, b], ℝ).

(ii) There exists a finite derivative f′(x) in (a, b) with the possible ex-

ception of countably many points. (iii) f′ ∈ L_{1}(a, b).

Then f is absolutely continuous and

∫_{a}^{x} f′(t) dt = f(x) − f(a), x ∈ [a, b].

Lemma 2. The function v(t) defined by (5) has the following properties:

(i) v ∈ C([0, ∞), ℝ).

(ii) v(t) is nonincreasing in [0, ∞).

(iii) v(t) is differentiable for t ∈ ∪^{∞}_{i=0}(t_{i}, t_{i+1}) and if 0 ≤ a < b then

∫_{a}^{b} v′(t) dt = v(b) − v(a).

Proof. (i) We have to prove only that v(t) is continuous at the points ξ_{1}, ξ_{2}, . . .. Since u(t) and ψ(t) are continuous from the left at these points it suffices to prove that lim_{t→ξ_{k}+} v(t) = v(ξ_{k}) but this follows easily from the

following relations:

lim_{t→ξ_{k}+} v(t) = lim_{t→ξ_{k}+} \frac{u(t)}{ψ(t)} = \frac{lim_{t→ξ_{k}+} u(t)}{lim_{t→ξ_{k}+} ψ(t)} = \frac{u(ξ_{k}^{+})}{∏_{i=1}^{k}(1 + b_{i})} =

= \frac{(1 + b_{k})u(ξ_{k})}{∏_{i=1}^{k}(1 + b_{i})} = \frac{u(ξ_{k})}{∏_{i=1}^{k−1}(1 + b_{i})} = \frac{u(ξ_{k})}{ψ(ξ_{k})} = v(ξ_{k}).

(ii) Since u(t) is nonincreasing in (t_{i}, t_{i+1}), i = 0, 1, …, and ψ(t) is constant in each of these intervals, we conclude that v(t) is nonincreasing in (t_{i}, t_{i+1}), i = 0, 1, . . .. The fact that v(t) is continuous implies that v(t) is nonincreasing in [0, ∞).

(iii) The fact that the derivative v′(t) exists at each point t ∈ ∪^{∞}_{i=0}(t_{i}, t_{i+1})

is obvious. The already proved properties (i) and (ii) imply that v′(t) is summable in any finite interval. Then (iii) follows immediately from Lemma 1.

§ 3. Main Results

Theorem 1. Suppose that:

1. Conditions A1—A4 hold.

2. There exists an unbounded increasing sequence {γ_{n}} such that for each n ∈ ℕ,

∫_{γ_{n}−τ_{1}}^{γ_{n}} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}ds ≥ 1.  (6)

Then:

1. Inequality (2) has no eventually positive solution.

2. Inequality (3) has no eventually negative solution.

3. Each solution of equation (1) is oscillatory.

Proof. First we shall prove that inequality (2) has no eventually positive solutions. Suppose that this is not true and let x(t) be an eventually positive solution of (2). Without loss of generality we may assume that x(t) > 0 for t ≥ −τ_{m}. We define the function y(t) by y(t) = x(t)ψ(t)^{−1}, where ψ(t) was defined by (4). Then the inequality (2) takes the form

y′(t) + ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}y(t − τ_{i}) ≤ 0, t ≠ t_{k}, t ≥ 0. (7)

Since x(t) > 0, (2) implies that x(t) is nonincreasing in each interval (t_{i}, t_{i+1}), i = 0, 1, . . .. Thus x(t) satisfies the same conditions as the function u(t). Then (5) and Lemma 2 imply that y(t) is a continuous nonincreasing function. Let N be a large enough integer such that γ_{n} ≥ τ_{1} for n ≥ N. We integrate (7) from γ_{n} −τ_{1} to γ_{n} and by making use of assertion (iii) of Lemma 2 obtain

y(γ_{n}) − y(γ_{n} − τ_{1}) + ∫_{γ_{n}−τ_{1}}^{γ_{n}} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}y(s − τ_{i})ds ≤ 0.

Since y(t) is nonincreasing,

y(s − τ_{i}) ≥ y(γ_{n} − τ_{i}) ≥ y(γ_{n} − τ_{1}), i = 1, 2, …, m,

for s ∈ [γ_{n} − τ_{1}, γ_{n}]. Then

y(γ_{n})+y(γ_{n} − τ_{1}) { ∫_{γ_{n}−τ_{1}}^{γ_{n}} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds − 1}≤0. (8)

The last inequality and the fact that y(t) is positive imply that for n ≥ N the following inequality holds:

∫_{γ_{n}−τ_{1}}^{γ_{n}} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds < 1,

which contradicts (6).

In order to prove that (3) has no eventually negative solution it suffices to note that if x(t) is a solution of (3), then −x(t) is a solution of (2). From assertions 1 and 2 it follows that equation (1) has neither eventually positive nor eventually negative solutions. Thus each solution of (1) is oscillatory.

Theorem 2. Suppose that:

1. Conditions A1—A4 hold.

2. There exists an unbounded increasing sequence {γ_{n}} such that for each n ∈ ℕ and for some i = i(n) ∈ {1, …, m} the inequalities

∫_{γ_{n}−τ_{i}}^{γ_{n}} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}ds ≥ 1  (9)

hold.

Then:

1. Inequality (2) has no eventually positive solution.

2. Inequality (3) has no eventually negative solution.

3. Each solution of equation (1) is oscillatory.

The proof of Theorem 2 is analogous to the proof of Theorem 1.

Theorem 3. Let conditions A1—A4 hold and a sequence of disjoint intervals {(α_{n}, β_{n})}_{n=1}^{∞} (α_{1} > 0) exist such that β_{n}−α_{n} ≥ 2τ_{m}, lim_{n→∞}(β_{n}− α_{n}) = ∞ and

lim inf_{t→∞} ∫_{t−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}ds > \frac{1}{e}  (10)

for t ∈ ∪^{∞}_{n=1}(α_{n} + τ_{m}, β_{n}). Then:

1. Inequality (2) has no eventually positive solution.

2. Inequality (3) has no eventually negative solution.

3. Each solution of equation (1) is oscillatory.

Proof. Suppose that x(t) is an eventually positive solution of (2). Without loss of generality we may assume that x(t) > 0, t ≥ −τ_{m}. We define the function y(t) as in Theorem 1. Then inequality (2) takes form (7), and the function y(t) is continuous, positive, and nonincreasing for t ≥ 0. Choose K such that

lim inf_{t→∞} ∫_{t−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds > K > \frac{1}{e},

t ∈ ∪^{∞}_{n=1}(α_{n} + τ_{m}, β_{n}).

Then there exists an integer N_{1} such that

∫_{t−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≥ K  (11)

for t ∈ ∪^{∞}_{n=N_{1}}(α_{n}+τ_{m}, β_{n}). We integrate (7) from β_{n}−τ_{1} to t and by making

use of Lemma 2_{(iii)} we obtain

y(t) − y(β_{n} − τ_{1}) + ∫_{β_{n}−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}y(s − τ_{i})ds ≤ 0,

t ∈ [β_{n} − τ_{1}, β_{n}], n ≥ N_{1}. Since β_{n} − τ_{m} − τ_{1} > 0 and y(t) is nonincreasing for t ≥ 0, we have

y(s − τ_{i}) ≥ y(s − τ_{1}) ≥ y(t − τ_{1}), i = 1, 2, …, m,

for s ∈ [β_{n} − τ_{1}, t] and t ∈ [β_{n} − τ_{1}, β_{n}]. Then

y(t) − y(β_{n} − τ_{1}) + y(t − τ_{1}) ∫_{β_{n}−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≤ 0.

Having in mind that y(t) is a positive function, from the last inequality we obtain for n ≥ N_{1}

y(t − τ_{1})A_{n}(t) ≤ y(β_{n} − τ_{1}) , t ∈ [β_{n} − τ_{1}, β_{n}],  (12)

where

A_{n}(t) = ∫_{β_{n}−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds.

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

Analogously, from the inequality

y(β_{n}) − y(t) + ∫_{t}^{β_{n}}∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1}y(s − τ_{i}) ds ≤ 0

we obtain

y(β_{n} − τ_{1})B_{n}(t) ≤ y(t), t ∈ [β_{n} − τ_{1}, β_{n}],  (13)

where

B_{n}(t) = ∫_{t}^{β_{n}}∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds.

Since A_{n}(t) and B_{n}(t) are continuous for t ∈ [β_{n} − τ_{1}, β_{n}] and

A_{n}(t) + B_{n}(t) = ∫_{β_{n}−τ_{1}}^{β_{n}} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≥ K,

there exists γ_{n} ∈ (β_{n} − τ_{1}, β_{n}) such that A_{n}(γ_{n}) ≥ \frac{K}{2} and B_{n}(γ_{n}) ≥ \frac{K}{2} for n ≥ N_{1}. Then from (12) and (13) we obtain

\frac{y(γ_{n} − τ_{1})}{y(γ_{n})} ≤ \frac{4}{K^{2}} , n ≥ N_{1} , γ_{n} ∈ (β_{n} − τ_{1}, β_{n}).  (14)

On the other hand, it follows from (7) that

y′(t) + y(t) ∑_{i=1}^{m} p_{i}(t) ∏_{t−τ_{i}≤ξ_{k}<t}(1 + b_{k})^{−1} ≤ 0

for t ≥ τ_{m} and in particular for t ∈ ∪^{∞}_{n=N_{1}}(α_{n} + τ_{m}, β_{n}), or

\frac{y′(t)}{y(t)} + ∑_{i=1}^{m} p_{i}(t) ∏_{t−τ_{i}≤ξ_{k}<t}(1 + b_{k})^{−1} ≤ 0.

From Lemma 1 applied to the function ln y(t) it follows that

ln \frac{y(t)}{y(t − τ_{1})} + ∫_{t−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≤ 0

for t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + τ_{1}, β_{n}]. From (11) we obtain

ln \frac{y(t − τ_{1})}{y(t)} ≥ K , t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + τ_{1}, β_{n}]

or equivalently

\frac{y(t − τ_{1})}{y(t)} ≥ e^{K} ≥ eK , t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + τ_{1}, β_{n}].

Since y(t − τ_{1}) ≥ eKy(t) , t ∈ ∪^{∞}_{n=N_{1}}(α_{n} + τ_{m} + τ_{1}, β_{n}), it follows from (7)

that

y′(t) + eKy(t) ∑_{i=1}^{m} p_{i}(t) ∏_{t−τ_{i}≤ξ_{k}<t}(1 + b_{k})^{−1} ≤ 0

and, as above,

ln \frac{y(t)}{y(t − τ_{1})} + eK ∫_{t−τ_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≤ 0

for t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + 2τ_{1}, β_{n}] or

ln \frac{y(t − τ_{1})}{y(t)} ≥ eK^{2} , t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + 2τ_{1}, β_{n}].

Thus

\frac{y(t − τ_{1})}{y(t)} ≥ e^{eK^{2}} ≥ e^{2}K^{2} , t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + 2τ_{1}, β_{n}].

Repeating the above procedure, we arrive at

\frac{y(t − τ_{1})}{y(t)} ≥ (eK)^{r} , t ∈ ∪^{∞}_{n=N_{1}}[α_{n} + τ_{m} + rτ_{1}, β_{n}].

If r is sufficiently large, then (eK)^{r} > \frac{4}{K^{2}} . Since lim_{n→∞}(β_{n} − α_{n}) = ∞ there exists an integer N_{2} ≥ N_{1} such that [β_{n} − τ_{1}, β] ⊆ [α_{n} + τ_{m} + rτ_{1}, β_{n}] for n ≥ N_{2}. It follows from (14) that

\frac{4}{K^{2}} ≥ \frac{y(γ_{n} − τ_{1})}{y(γ_{n})} ≥ (eK)^{r} > \frac{4}{K^{2}} , n ≥ N_{2}.

The contradiction obtained shows that (2) has no eventually positive solutions.

The proof of assertions 2 and 3 is carried out as in Theorem 1.

Remark 2. The assertion of Theorem 3 is still valid if p(t) > 0 only on ∪^{∞}_{n=1}(α_{n}, β_{n}). The proof with insignificant changes is the same.

Remark 3. Theorem 3 generalizes the result of Gopalsamy and Zhang in several directions. The trivial generalization (the procedure is very familiar for delay equations without impulses) is that we have several delays while it is one in [4], and we impose the integral condition (10) on p(t) only on a sequence of disjoint intervals and not on the whole axis. The nontrivial extension is that there are no restrictions on the inter-jump distance (in [4] it is greater than the delay) and also that b_{k} ∈ (−1, ∞) while in [4], b_{k} ∈ (0, ∞). In the case of one delay and integral condition on the whole axis (10) takes the form

lim inf_{t→∞} ∫_{t−τ}^{t} p(s) ∏_{s−τ≤ξ_{k}<s}(1 + b_{k})^{−1} ds > \frac{1}{e}.  (15)

If the sequence {b_{k}} is bounded from above and the number of jumps in [t − τ, t] is also bounded, then

l = lim inf_{t→∞} ∏_{t−τ≤ξ_{k}<t}(1 + b_{k})^{−1} > 0.

It is easy to see that

lim inf_{t→∞} ∫_{t−τ}^{t} p(s) ∏_{s−τ≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≥ l lim inf_{t→∞} ∫_{t−τ}^{t} p(s) ds.

Since

lim inf_{t→∞} ∏_{t−τ≤ξ_{k}<t}(1 + b_{k})^{−1} = \frac{1}{lim sup_{t→∞} ∏_{t−τ≤ξ_{k}<t}(1 + b_{k})},

the integral condition (15) can be replaced by

lim inf_{t→∞} ∫_{t−τ}^{t} p(s) ds > \frac{lim sup_{t→∞} ∏_{t−τ≤ξ_{k}<t}(1 + b_{k})}{e}.  (16)

For the sake of comparison the corresponding condition in [4] is

lim inf_{t→∞} ∫_{t−τ}^{t} p(s) ds > \frac{1 + sup_{k} b_{k}}{e}  (17)

with b_{k} > 0 and ξ_{k+1} − ξ_{k} > τ. If ξ_{k+1} − ξ_{k} > τ, then (16) takes the form

lim inf_{t→∞} ∫_{t−τ}^{t} p(s) ds > \frac{1 + lim sup_{k} b_{k}}{e} ,

which is again slightly better than (17).

Analogously to Theorem 3 we can prove the following theorem:

Theorem 4. Let conditions A1—A4 hold and there exist a sequence of disjoint intervals {(α_{n}, β_{n})}_{n=1}^{∞} (α_{1} > 0) such that β_{n} − α_{n} ≥ 2τ_{m}, lim_{n→∞}(β_{n} − α_{n}) = ∞, and there exist K > 0 and N_{1} such that the inequalities

∫_{t−τ_{i}}^{t} p_{i}(s) ∏_{s−τ_{i}≤ξ_{k}<s}(1 + b_{k})^{−1} ds ≥ K > \frac{1}{e}  (18)

hold for any n ≥ N_{1} and t ∈ (α_{n} + τ_{m}, β_{n}) and for some i = i(n) ∈ {1, 2, …, m}. Then:

1. Inequality (2) has no eventually positive solution.

2. Inequality (3) has no eventually negative solution.

3. Each solution of equation (1) is oscillatory.

Acknowledgement

The present investigation was supported by the Bulgarian Ministry of Education, Science and Technologies under Grant MM–511.

The authors are grateful to Dr. P. S. Simeonov for careful reading of the manuscript and for useful suggestions.

The authors are also grateful to the referee for valuable suggestions.

References

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2. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation theory of differential equations with deviating arguments. Pure Appl. Math. 110, Marcel Dekker, New York, 1987.

3. I. Györi and G. Ladas, Oscillation theory of delay differential equations with applications. Clarendon Press, Oxford, 1991.

4. K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses. J. Math. Anal. Appl. 139(1989), 110–122.

5. D. D. Bainov, M. B. Dimitrova, and P. S. Simeonov, Oscillatory properties of the solutions of impulsive differential equations with retarded argument and nonconstant coefficients. (To appear).

6. D. D. Bainov, M. B. Dimitrova, and P. S. Simeonov, Sufficient conditions for oscillation of the solutions of a class of impulsive differential equations with advanced argument. (To appear).

7. D. D. Bainov and P. S. Simeonov, Systems with impulse effect. Stability, theory and applications. Ellis Horwood Series in Mathematics and Its Applications, Ellis Horwood, Chichester, 1989.

8. E. Hewitt and K. Stromberg, Real and abstract analysis. Springer-Verlag, Berlin etc. 1965.

(Received 14.05.1996; revised 15.05.1997)

Authors’ addresses:

Drumi D. Bainov Medical University of Sofia P.O. Box 45, 1504 Sofia Bulgaria

Margarita B. Dimitrova Technical University, Sliven Bulgaria

Vasil A. Petrov Technical University, Plovdiv Bulgaria