GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 2, 1999, 99-106

SUFFICIENT CONDITIONS FOR THE OSCILLATION OF BOUNDED SOLUTIONS OF A CLASS OF IMPULSIVE

DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH A CONSTANT DELAY

D. D. BAINOV AND M. B. DIMITROVA

Abstract. Sufficient conditions are found for oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay. Some asymptotic properties are studied for the bounded solutions.

1. Introduction

The last twenty years have seen a significant increase in the number of papers devoted to the oscillation theory of differential equations with a deviating argument. The main part of these investigations is given in the monographs [1], [2], [3].

On the other hand, the last decade has been marked by a growing interest in impulsive differential equations due to their various applications in science and technology. In the monographs [4] and [5] numerous aspects of their qualitative theory are studied. However, the oscillation theory of impulsive differential equations has not yet been worked out.

In the present paper we obtain sufficient conditions for the oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay and fixed moments of the impulse effect.

2. Preliminary Notes

We consider the impulsive differential equations of second order

\begin{matrix}(r(t)y′(t))′ − ∑_{i=1}^{n} p_{i}(t)y(t − h_{i}) = 0, t ≠ τ_{k}, k ∈ ℕ, \\ Δy′(τ_{k}) = y′(τ_{k} + 0) − y′(τ_{k} − 0) = β_{k}y(τ_{k}), \\ Δy(τ_{k}) = y(τ_{k} + 0) − y(τ_{k} − 0) = 0, \end{matrix}  (1)

99 under the initial conditions

1991 Mathematics Subject Classification. 34A37.

Key words and phrases. Oscillation of bounded solutions, impulsive differential equations.

1072-947X/99/0300-0099$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

\begin{matrix}y(t) = φ(t), t ∈ [−h, 0], h = max\{h_{i} : i ∈ ℕ_{n}\}, \\ y′(0) = φ′(0) = y′_{0}. \end{matrix}  (2)

Here ℕ_{n} = {1, 2, …, n}; {τ_{k}}_{k=1}^{∞} is a monotone increasing unbounded sequence of positive numbers; {β_{k}}_{k=1}^{∞} is a sequence of positive numbers; h_{i}, i ∈ ℕ_{n}, are positive constants, \overline{ℝ}_{+} = [0, +∞); ℝ_{+} = (0, +∞); y′(τ_{k} − 0) = y′(τ_{k}).

We denote by P C(\overline{ℝ}_{+}, ℝ) the set of all functions u: \overline{ℝ}_{+} → ℝ which are continuous for t ∈ \overline{ℝ}_{+}, t ≠ τ_{k} (k ∈ ℕ), continuous from the left for t ∈ ℝ_{+} and having a discontinuity of first kind at the points τ_{k} ∈ ℝ_{+} (k ∈ ℕ).

Let us introduce the following conditions:

H1. φ ∈ C^{2}([−h, 0], ℝ).

H2. p_{i} ∈ P C(\overline{ℝ}_{+}, ℝ_{+}), i ∈ ℕ_{n}.

H3. r ∈ P C(\overline{ℝ}_{+}, ℝ_{+}), r(τ_{k} + 0) > 0, k ∈ ℕ.

Definition 1. We shall call a solution of equation (1) with the initial conditions (2) any function y: [−h, +∞) → ℝ for which the following conditions are fulfilled:

1. If −h ≤ t ≤ 0, y(t) = φ(t).

2. If 0 < t ≤ τ_{1}, the solution y(t) coincides with the solution of problem (1), (2) without impulse effect.

3. If τ_{k} < t ≤ τ_{k+1}, k ∈ ℕ, the solution of problem (1), (2) coincides with the solution of the integro-differential equation

r(t)y′(t) = r(τ_{k} + 0)y′(τ_{k} + 0) + ∫_{τ_{k}}^{t} ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds

with the initial conditions (2).

Definition 2. The solution y(t) of problem (1), (2) is said to be oscillatory if for each a > 0 we have

{t : y(t) > 0, t > a} ≠ ∅ and {t : y(t) < 0, t > a} ≠ ∅.

Otherwise, the solution y(t) is called nonoscillatory.

3. Main Results

Theorem 1. Let the following conditions hold:

1. Conditions H1–H3 are fulfilled.

2. lim_{t→+∞} R(t) = +∞, where R(t) = ∫_{0}^{t} \frac{ds}{r(s)}.

3. ∫^{∞}R(s) ∑_{i=1}^{n} p_{i}(s)ds = +∞.

Then all bounded solutions of equation (1) either tend to zero as t → +∞, or oscillate.

Proof. Let y(t) be a positive bounded solution of equation (1) for t ≥ t_{1} > 0. It is clear that y(t − h_{i}) > 0 for t ≥ t_{2} = t_{1} + h. This fact, (1) and condition H2 imply that the function r(t)y′(t) increases in the set M = [t_{2}, τ_{s}) ∪

[ ⋃_{i=s}^{∞}(τ_{i}, τ_{i+1})], where τ_{s−1} < t_{2} < τ_{s}. On the other hand, r(τ_{k})Δy′(τ_{k}) =

β_{k}r(τ_{k})y(τ_{k}) > 0 for τ_{k} > t_{2} and therefore r(t)y′(t) is an increasing function for t ≥ t_{2}.

The following cases are possible:

Case 1. Let r(t)y′(t) > 0 for t ≥ t_{2}. Since r(t)y′(t) is an increasing function for t ≥ t_{2}, there exist a constant c > 0 and a point t_{3} ≥ t_{2} such that

y′(t) ≥ \frac{c}{r(t)},  t ≥ t_{3}.  (3)

We integrate (3) from t_{3} to t (t ≥ t_{3}) and obtain

y(t) ≥ y(t_{3}) + ∫_{t_{3}}^{t} \frac{c}{r(s)} ds.  (4)

Now (4) and condition 2 of the theorem imply that lim_{t→+∞} y(t) = +∞, which

contradicts the assumption that y(t) is a bounded solution.

Case 2. Let r(t)y′(t) < 0 for t ≥ t_{2}. Therefore y′(t) < 0, t ≥ t_{2}. On the other hand, y(t) > 0 for t ≥ t_{2}. Then it follows that there exists a finite limit lim_{t→+∞} y(t) ≥ 0. The assumption that r(t)y′(t) < 0 and the fact that

r(t)y′(t) is an increasing function for t ≥ t_{2} lead to the existence of a finite limit lim_{t→+∞} r(t)y′(t) ≤ 0.

Suppose that lim_{t→+∞} r(t)y′(t) = c_{1} < 0, i.e., there exists a point t̄ ≥ t_{2}

such that for t ≥ t̄ we have

y′(t) ≤ \frac{c_{1}}{2r(t)}.  (5)

Integrating (5) from t̄ to t (t > t̄), we obtain

y(t) ≤ y(t̄) + ∫_{t̄}^{t} \frac{c_{1}}{2r(s)} ds.  (6)

Thus (6) and condition 2 of the theorem yield lim_{t→+∞} y(t) = −∞, which contradicts the assumption that y(t) is a positive solution of equation (1).

Therefore

lim_{t→+∞} r(t)y′(t) = 0.  (7)

We integrate (1) from t_{2} to t and arrive at

r(t)y′(t) = r(t_{2})y′(t_{2}) + ∑_{t_{2}≤τ_{i}<t}β_{i}r(τ_{i})y(τ_{i}) + ∫_{t_{2}}^{t} ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds. (8)

Passing to the limit in (8) as t → +∞ and bearing in mind (7), we obtain

r(t_{2})y′(t_{2}) = − ∑_{t_{2}≤τ_{i}<∞} β_{i}r(τ_{i})y(τ_{i}) − ∫_{t_{2}}^{∞}∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds. (9)

We divide (8) by r(t) > 0, integrate the equality obtained from t_{2} to t and obtain

y(t) = y(t_{2}) + r(t_{2})y′(t_{2})[R(t) − R(t_{2})] +

+ ∫_{t_{2}}^{t} [R(t) − R(s)] ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds +

+ ∫_{t_{2}}^{t} \frac{1}{r(s)} ∑_{t_{2}≤τ_{i}<s} β_{i}r(τ_{i})y(τ_{i}) ds.  (10)

It follows from (9) and (10) that

y(t) = y(t_{2}) − [R(t) − R(t_{2})]{ ∑_{i=1}^{n} ∫_{t_{2}}^{∞}p_{i}(s)y(s − h_{i}) ds +

+ ∑_{t_{2}≤τ_{i}<∞} β_{i}r(τ_{i})y(τ_{i})\} + ∫_{t_{2}}^{t} \frac{1}{r(s)} ∑_{t_{2}≤τ_{i}<s} β_{i}r(τ_{i})y(τ_{i}) ds +

+ ∫_{t_{2}}^{t} [R(t) − R(s)] ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds.  (11)

SUFFICIENT CONDITIONS FOR THE OSCILLATION 103

Hence

y(t) ≤ y(t_{2}) − [R(t) − R(t_{2})]{ ∑_{t_{2}≤τ_{i}<∞} β_{i}r(τ_{i})y(τ_{i}) +

+ ∫_{t_{2}}^{∞}∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds}[R(t) − R(t_{2})] ∑_{t_{2}≤τ_{i}<∞} β_{i}r(τ_{i})y(τ_{i}) +

+ ∫_{t_{2}}^{t} [R(t) − R(s)] ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds.

The latter inequality implies the relation

y(t) ≤ y(t_{2}) + ∫_{t_{2}}^{t} [R(t_{2}) − R(t)] ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds −

− [R(t) − R(t_{2})] ∫_{t}^{∞}∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds +

+ ∫_{t_{2}}^{t} [R(t) − R(s)] ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds =

= y(t_{2}) + ∫_{t_{2}}^{t} [R(t_{2}) − R(s)] ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds −

− [R(t) − R(t_{2})] ∫_{t}^{∞}∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds,

i.e.,

y(t) ≤ y(t_{2}) + R(t_{2}) ∫_{t_{2}}^{t} ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds −
− ∫_{t_{2}}^{t} R(s) ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds.  (12)

It follows from (8) and (12) that

y(t) ≤ y(t_{2}) + R(t_{2})r(t)y′(t) − R(t_{2})r(t_{2})y′(t_{2}) −

− R(t_{2}) ∑_{t_{2}≤τ_{i}<t} β_{i}r(τ_{i})y(τ_{i}) − ∫_{t_{2}}^{t} R(s) ∑_{i=1}^{n} p_{i}(s)y(s − h_{i}) ds.

Therefore

y(t) ≤ y(t_{2}) − R(t_{2})r(t_{2})y′(t_{2}) − ∫_{t_{2}}^{t} R(s)y(s − h̄) ∑_{i=1}^{n} p_{i}(s) ds, (13)

where h̄ = min{h_{i} : i ∈ ℕ_{n}}.

Now, from y(t) > 0 for t ≥ t_{2} and from the fact that y(t) is a decreasing function in [t_{2}, +∞) we have inf_{s∈[t_{2},t]} y(s − h̄) = y(t − h̄). Thus (13) yields the

inequality

y(t) ≤ y(t_{2}) − R(t_{2})r(t_{2})y′(t_{2}) − y(t − h̄) ∫_{t_{2}}^{t} R(s) ∑_{i=1}^{n} p_{i}(s) ds.

If we suppose that lim_{t→+∞} y(t) = c > 0, then the latter inequality gives

lim_{t→+∞} y(t) = −∞ as t → +∞, which contradicts the fact that y(t) is a

bounded positive solution of equation (1). Therefore lim_{t→+∞} y(t) = 0.

Theorem 2. Let the following conditions hold:

1. Conditions H1–H3 are fulfilled.

2. ∫_{0}^{∞} \frac{dt}{r(t)} = +∞.

3. lim sup_{t→+∞} \frac{1}{r(t)} ∫_{t−h̄}^{t} (s − t + h̄) ∑_{i=1}^{n} p_{i}(s) ds > 1,

where h̄ = min{h_{i} : i ∈ ℕ_{n}}.

Then all bounded nontrivial solutions of equation (1) are oscillatory.

Proof. Let y(t) be a bounded nonoscillatory solution of equation (1). Without loss of generality we may assume y(t) > 0 for t ≥ t_{0} ≥ 0. Then y(t − h_{i}) > 0 for t ≥ t_{0} + h = t_{1}. Analogously to the proof of Theorem 1 we arrive at r(t)y′(t) ≤ 0, t ≥ t_{1}.

Integrate (1) from s to t (t > s ≥ t_{1}) and obtain

r(t)y′(t) = r(s)y′(s) + ∑_{s≤τ_{i}<t} β_{i}r(τ_{i})y(τ_{i}) + ∫_{s}^{t} ∑_{i=1}^{n} p_{i}(σ)y(σ − h_{i})dσ. (14)

SUFFICIENT CONDITIONS FOR THE OSCILLATION 105

Now we integrate (14) from t − h̄ to t, t ≥ t_{1} + h̄, and derive

r(t)y′(t)h̄ = ∫_{t−h̄}^{t} r(s) dy(s) + ∫_{t−h̄}^{t} ∑_{s≤τ_{i}<t} β_{i}r(τ_{i})y(τ_{i}) ds +
+ ∫_{t−h̄}^{t} [σ − t + h̄] ∑_{i=1}^{n} p_{i}(σ)y(σ − h_{i}) dσ.  (15)

From (15) we obtain

0 ≥ r(t)y(t) − r(t − h̄)y(t − h̄) − ∫_{t−h̄}^{t} y(s) dr(s) +

+ ∫_{t−h̄}^{t} ∑_{s≤τ_{i}<t} β_{i}r(τ_{i})y(τ_{i}) ds + ∫_{t−h̄}^{t} [σ − t + h̄] ∑_{i=1}^{n} p_{i}(σ)y(σ − h_{i}) dσ ≥

≥ r(t)y(t) − r(t − h̄)y(t − h̄) − y(t − h̄)[r(t) − r(t − h̄)] +

+ ∫_{t−h̄}^{t} [σ − t + h̄] ∑_{i=1}^{n} p_{i}(σ)y(σ − h_{i}) dσ =

=r(t)y(t) − r(t)y(t − h̄) + ∫_{t−h̄}^{t} [σ − t + h̄] ∑_{i=1}^{n} p_{i}(σ)y(σ − h_{i}) dσ. (16)

Since y(t) is a nonincreasing function in [t_{1}, +∞), we have y(σ − h_{i}) ≥ y(σ − h̄), σ ∈ [t − h̄, t] and inf_{t−h̄≤σ≤t} y(σ − h̄) = y(t − h̄).

Then (16) implies

0 ≥ r(t)y(t) − r(t)y(t − h̄) + y(t − h̄) ∫_{t−h̄}^{t} [σ − t + h̄] ∑_{i=1}^{n} p_{i}(σ) dσ. (17)

Divide (17) by r(t)y(t − h̄) > 0 and obtain

\frac{y(t)}{y(t − h̄)} + [\frac{1}{r(t)} ∫_{t−h̄}^{t} (σ − t + h̄) ∑_{i=1}^{n} p_{i}(σ) dσ − 1] ≤ 0.

The latter inequality contradicts condition 3 of Theorem 2.

Acknowledgement

The authors thank the referee for his useful remarks and comments. The present investigation was supported by the Bulgarian Ministry of

Education, Science and Technologies under Grant MM–511.

References

1. L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation theory for functional differential equations. Pure and Applied Mathematics 190, Marcel Dekker, New York, 1995.

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

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

4. D. D. Bainov and P. S. Simeonov, Systems with impulse effect: stability, theory and applications. Ellis Horwood, Chichester, 1989.

5. D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations: asymptotic properties of the solutions and applications. World Scientific Publishers, Singapore, 1995.

(Received 30.05.1996; revised 28.02.1997)

Authors’ addreses:

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

M. B. Dimitrova Technical University 8800 Sliven Bulgaria