Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 16, 1-11, http://www.math.u-szeged.hu/ejqtde/

Oscillation of the Solutions of Neutral Impulsive Differential-Difference Equations of First Order

M.B.Dimitrova

Technical University, 8800 Sliven, Bulgaria

D.Mishev

Department of Mathematics, Prairie View A & M University, Prairie View, TX 77446, USA

AMS (MOS) Subject Classification. 34K37.

Key words: Oscillation of the solutions, neutral impulsive differential-difference equations

Abstract

Sufficient conditions for oscillation of all solutions of a class of neutral impulsive differential-difference equations of first order with deviating argument and fixed moments of impulse effect are found .

1. Introduction

The impulsive differential equations describe processes which are characterized as continuous, as jump-wise change of the phase variables describing the process. They are adequate mathematical models of processes and phenomena studied in theoretical physics, chemical technology, population dynamics, technique and economics. That is why, the impulsive differential equations are an object of intensive investigation.

In the recent two decades the number of investigations of the oscillatory and nonoscillatory behavior of the solutions of functional differential equations is constantly growing. Greater part of the works on this subject published by 1977 are given in [19]. In the monographs [18] and [17] published in 1987 and 1991 respectively, the oscillatory and asymptotic properties of the solutions of various classes of functional differential equations are systematically studied.

The investigation of impulsive differential equations of neutral type is still not well studied. Let us note that in contrast to [9] the present paper deals with the oscillatory properties of more general homogeneous impulsive differential equation. In the works [1]— [6],[8],[16],[17] more general necessary and sufficient conditions for oscillation and nonoscillation of the solutions of impulsive differential equations of first and second order are found.

While qualitative theory for retarded and advanced differential equations has been well developed over the last twenty years, (see,for example, [17],[18] and [19] ), only in resent years has much effort been devoted to the study of neutral differential equations (see, for example, [7] and [10]-[16] ).

In the present paper, we establish sufficient conditions for oscillation of all solutions of a class of neutral impulsive differential-difference equations of first order with deviating argument and fixed moments of impulse effect.

2. Preliminary notes

Consider the impulsive differential-difference inequalities of neutral type with a constant delay:

\begin{matrix}[y(t) + p(t)y(t − τ)]′ + q(t)y(t − σ) < 0,  t ≠ τ_{k}, k ∈ N, \\ Δ[y(τ_{k}) + p_{k}y(τ_{k} − τ)] + q_{k}y(τ_{k} − σ) < 0,  k ∈ N, \end{matrix}  (1)

and

\begin{matrix}[y(t) + p(t)y(t − τ)]′ + q(t)y(t − σ) > 0,  t ≠ τ_{k}, k ∈ N, \\ Δ[y(τ_{k}) + p_{k}y(τ_{k} − τ)] + q_{k}y(τ_{k} − σ) > 0,  k ∈ N, \end{matrix}  (2)

and corresponding to it equation

\begin{matrix}[y(t) + p(t)y(t − τ)]′ + q(t)y(t − σ) = 0,  t ≠ τ_{k}, k ∈ N, \\ Δ[y(τ_{k}) + p_{k}y(τ_{k} − τ)] + q_{k}y(τ_{k} − σ) = 0,  k ∈ N, \end{matrix}  (3)

where τ, σ ∈ R_{+}, R_{+} = (0, +∞); τ > σ; τ_{1}, τ_{2}, … τ_{k}, … are the moments of impulse effect; p_{k} and q_{k} are constants (k ∈ N).

Here Δ[y(τ_{k}) + p_{k}y(τ_{k} − τ)] = y(τ_{k} + 0) + p_{k}y(τ_{k} − τ + 0) − y(τ_{k} − 0) − p_{k}y(τ_{k} − τ − 0). We suppose that y(τ_{k} − 0) = y(τ_{k}) and y(τ_{k} − τ − 0) = y(τ_{k} − τ) for k ∈ N.

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

Introduce the following conditions:

H1. 0 < τ_{1} < τ_{2} < … and lim_{k→+∞} τ_{k} = +∞.

H2. p ∈ P C(R_{+}, R) and p_{k} = p(τ_{k} − 0) = p(τ_{k}) for k ∈ N.

H3. q ∈ C(R_{+}, R_{+}) and q_{k} ≥ 0 for k ∈ N.

H4. p(t) ≤ −1 for t ∈ R_{+}.

H5.

∫_{0}^{+∞}q(s)ds + ∑_{k=1}^{+∞} q_{k} = +∞.

Definition 1 A function y : [−τ, +∞) → R is said to be a solution of (3) with initial function φ ∈ C([−τ, 0], R) if y(t) = ϕ(t) for t ∈ [−τ, 0], y ∈ P C(R_{+}, R), z(t) = y(t) + p(t)y(t−τ) is continuously differentiable for t ∈ R_{+}, and y(t) satisfies (3) for all sufficiently large t ≥ 0.

Definition 2 The nonzero solution y(t) of the equation (3) is said to be nonoscillating if there exists a point t_{0} ≥ 0 such that y(t) has a constant sign for t ≥ t_{0}. Otherwise the solution y(t) is said to oscillate.

Definition 3 The solution y(t) of the equation (3) is said to be regular, if it is defined on some interval [T_{y}, +∞) ⊂ [t_{0}, +∞) and

sup {|y(t)|: t ≥ T } > 0 for each T ≥ T_{y}.

Definition 4 The regular solution y(t) of the equation (3) is said to be eventually positive(eventually negative), if there exists t_{1} > 0 such that y(t) > 0 (y(t) < 0) for t ≥ t_{1}.

3. Main results

Theorem 1 Let the following conditions hold:

1. Conditions H1 — H5 are met.

2.

lim inf_{t→+∞} [ ∑_{t≤τ_{k}<t+τ−σ}(1 − \frac{q_{k}}{p_{k_{1}}} ) ∫_{t}^{t+τ−σ} \frac{−q(s)}{p(s + τ − σ)}ds] > \frac{1}{e}

where e = exp, p_{k_{1}} = p(τ_{k} + τ − σ).

Then:

1. The inequality (1) has no eventually positive solution.

2. The inequality (2) has no eventually negative solution.

3.All solutions of the equation (3) are oscillatory.

Proof. First of all,we shall prove that the inequality (1) has no eventually positive solution. Let us suppose the opposite, i.e., there exists a solution y(t) of inequality (1) and a number t_{0} > 0 such that y(t) is defined for t ≥ t_{0} and y(t) > 0 for t ≥ t_{0}.

Set

\begin{matrix}z(t) = y(t) + p(t)y(t − τ),  t≥ 0 \\ Δz(τ_{k}) = Δ[y(τ_{k}) + p_{k}y(τ_{k} − τ)],  k ∈ N. \end{matrix}  (4)

From (1) and conditions H2 and H3 it follows that

z′(t) < −q(t)y(t − σ) < 0

and

Δz(τ_{k}) < −q_{k}y(τ_{k} − σ) ≤ 0.

The last inequalities implies that z is a decreasing function for t ≥ t_{0}. Let us suppose that z(t) ≥ 0 for t ≥ t_{1} ≥ t_{0}. From (4) and condition H4 we obtain

y(t) ≥ −p(t)y(t − τ) ≥ y(t − τ)
Δy(τ_{k}) ≥ −p_{k}Δy(τ_{k} − τ) ≥ Δy(τ_{k} − τ),

i.e. y is a bounded function from below by m > 0.

Integrating (1) from t_{1} to t (t ≥ t_{1}), we obtain

z(t) − z(t_{1}) − ∑_{t_{1}≤τ_{k}<t} Δz(τ_{k}) + ∫_{t_{1}}^{t} q(s)y(s − σ)ds < 0

or

z(t) − z(t_{1}) + ∑_{t_{1}≤τ_{k}<t} q_{k}y(τ_{k} − σ) + ∫_{t_{1}}^{t} q(s)y(s − σ)ds < 0,

i.e.,

z(t) ≤ z(t_{1}) − m[ ∫_{t_{1}}^{t} q(s)ds + ∑_{t_{1}≤τ_{k}<t} q_{k}].

It follows from the above inequality after passing to limit as t → +∞ that lim_{t→+∞} z(t) = −∞, which contradicts the assumption that z(t) ≥ 0 for t ≥ t_{1}. Therefore, z(t) < 0, for t ≥ t_{1}.

From (4) we have that z(t) > p(t)y(t−τ), t ≥ t_{1}, i.e. z(t+τ −σ) > p(t+τ −σ)y(t−σ).

Multiplying both sides of the last inequality by \frac{q(t)}{p(t + τ − σ)} < 0 we obtain

\frac{q(t)}{p(t + τ − σ)}z(t + τ − σ)) < q(t)y(t − σ) < −z′(t)

Therefore

z′(t) + \frac{q(t)}{p(t + τ − σ)}z(t + τ − σ)) < 0,  t ≥ t_{1}, t ≠ τ_{k}  (5)

But z(τ_{k} + τ − σ) > p_{k_{1}}y(τ_{k} − σ), p_{k_{1}} = p(τ_{k} + τ − σ) or

\frac{q_{k}}{p_{k_{1}}} z(τ_{k} + τ − σ) < q_{k}y(τ_{k} − σ) < −Δz(τ_{k}),

i.e.,

Δz(τ_{k}) + \frac{q_{k}}{p_{k_{1}}} z(τ_{k} + τ − σ) < 0,  k ∈ N.  (6)

Denote τ − σ = l, s(t) = \frac{−q(t)}{p(t + l)} > 0 for t ≥ t_{1} and s_{k} = \frac{−q_{k}}{p_{k_{1}}} > 0, k ∈ N. Then from

(5) and (6) follows that

\begin{matrix}z′(t) − s(t)z(t + l) < 0,  t ≥ t_{1}, t ≠ τ_{k} \\ Δz(τ_{k}) − s_{k}z(τ_{k} + l) < 0,  τ_{k} ≥ t_{1},  k ∈ N. \end{matrix}  (7)

We shall prove that the impulsive differential-difference inequality (7) has no eventually negative solution. Let us suppose the opposite, i.e., there exists a solution z(t) of inequality (7) and a number t_{2} > 0 such that z(t) is defined for t ≥ t_{2} and z(t) < 0 for t ≥ t_{2}.

We divide (7) by z(t) < 0, (t ≥ t_{2}) and obtain

\frac{z′(t)}{z(t)} − s(t)\frac{z(t + l)}{z(t)} > 0,  t ≥ t_{2}, t ≠ τ_{k}  (8)

Denote

w(t) = \frac{z(t + l)}{z(t)}, t ≥ t_{2}

From the fact that z(t) is a decreasing function for t ≥ t_{2} it follows the inequality w(t) > 1 for t ≥ t_{2}.

Integrating (8) from t to t + l, (t ≥ t_{2}) we obtain

ln\frac{z(t + l)}{z(t)} + ∑_{t≤τ_{k}<t+l} ln\frac{z(τ_{k})}{z(τ_{k} + 0)} > ∫_{t}^{t+l}s(u)\frac{z(u + l)}{z(u)}du.  (9)

From

Δz(τ_{k}) = z(τ_{k} + 0) − z(τ_{k}) < s_{k}z(τ_{k} + l) < s_{k}z(τ_{k})

implies

z(τ_{k} + 0) < (1 + s_{k})z(τ_{k}).

Then

ln\frac{z(τ_{k})}{z(τ_{k} + 0)} < ln\frac{1}{1 + s_{k}} ,  k ∈ N.  (10)

From (9) and (10) we obtain

∫_{t}^{t+l} \frac{z(u + l)}{z(u)}s(u)du < ln[\frac{z(t + l)}{z(t)} ∏_{t_{1}≤τ_{k}<t+l} \frac{1}{1 + s_{k}}]

or

\frac{z(t + 2l)}{z(t)} ∫_{t}^{t+l}s(u)du < ln[\frac{z(t + l)}{z(t)} ∏_{t≤τ_{k}<t+l} \frac{1}{1 + s_{k}}]  (11)

Using the inequality e^{x} > ex for x > 0 and (11) we find that

\frac{z(t + l)}{z(t)∏_{t_{1}≤τ_{k}<t+l}(1 + s_{k})} > e^{\frac{z(t+2l)}{z(t)} ∫_{t}^{t+l}s(u)du} > e\frac{z(t + 2l)}{z(t)} ∫_{t}^{t+l}s(u)du

or

\frac{1}{e} > [ ∏_{t≤τ_{k}<t+l}(1 + s_{k})]\frac{z(t + 2l)}{z(t + l)} ∫_{t}^{t+l}s(u)du

i.e.

\frac{1}{e} > ∏_{t≤τ_{k}<t+l}(1 + s_{k}) ∫_{t}^{t+l}s(u)du

The last inequality contradicts condition 2 of Theorem 1. Thus z(t) < 0 will not hold for all t ≥ t_{2}, and therefore (1) has no eventually positive solution.

In order to prove that (2) has no eventually negative solution, it is enough to note that if y(t) is a solution of (2), then −y(t) is a solution of (1).

It follows from assertions 1 and 2 of Theorem 1 that the equation (3) has neither eventually positive nor eventually negative solution. Therefore, each regular solution of (3) is oscillatory.

The proof of the theorem is complete.

Corollary 1 Let the following conditions hold:

1. Conditions H1 — H5 are met.

2. There exists a constant r > 0 such that

\frac{q(t)}{p(t + τ − σ)} ≤ −r

for t ∈ R_{+}.

3.

lim inf_{t→∞}[ ∏_{t≤τ_{k}<t+τ−σ}(1 − \frac{q_{k}}{p_{k_{1}}} )] > \frac{1}{er(τ − σ)}

where p_{k_{1}} = p(τ_{k} + τ − σ).

Then:

1. The inequality (1) has no eventually positive solution.

2. The inequality (2) has no eventually negative solution. 3.All solutions of the equation (3) are oscillatory.

Proof. Let y(t) be an eventually positive solution of the inequality (1) for t ≥ t_{0}, (t_{0} > 0). Then, proceeding as in proof of Theorem 1, we conclude (7). From (7) and from condition 2 of Corollary 1 we obtain

\begin{matrix}z′(t) − rz(t + l) < 0 t ≥ t_{1}, t ≠ τ_{k} \\ Δz(τ_{k}) − r_{k}z(τ_{k} + l) < 0,  τ_{k} ≥ t_{1} k ∈ N, \end{matrix}  (12)

where

r_{k} = −\frac{q_{k}}{p_{k_{1}}} , k ∈ N.

We shall prove that the impulsive differential-difference inequality (12) has no eventually negative solution. Let us suppose the opposite, i.e., there exists a solution z(t) of inequality (12) and a number t_{2} > t_{1} such that z(t) < 0 for t ≥ t_{2}. Then it follows from (12) that z(t) is decreasing function for t ≥ t_{2}.

We divide (12) by z(t) < 0, integrate from t to t + l, (t ≥ t_{2}) and obtain

ln\frac{z(t + l)}{z(t)} − ∑_{t≤τ_{k}<t+l} ln\frac{z(τ_{k} + 0)}{z(τ_{k})} > r ∫_{t}^{t+l} \frac{z(u + l)}{z(u)}du  (13)

Analogously to the proof of Theorem 1 we obtain that

r ∫_{t}^{t+l} \frac{z(u + l)}{z(u)}du < ln[\frac{z(t + l)}{z(t)} ∏_{t≤τ_{k}<t+l}(1 + r_{k})^{−1}]

or

rl\frac{z(t + 2l)}{z(t)} < ln[\frac{z(t + l)}{z(t)} ∏_{t≤τ_{k}<t+l}(1 + r_{k})^{−1}]  (14)

Using the monotonicity of the function z(t) for t ≥ t_{2} we find that

\frac{z(t + 2l)}{z(t)} > 1 and \frac{z(t + l)}{z(t)} > 1. From two last inequalities, (14) and from the inequality

e^{x} > ex for x > 0 we find that

\frac{z(t + l)}{z(t)} ∏_{t≤τ_{k}<t+l}(1 + r_{k})^{−1} > e^{rl \frac{z(t+2l)}{z(t)}} > erl\frac{z(t + 2l)}{z(t)};

∏_{t≤τ_{k}<t+l}(1 + r_{k})^{−1} > erl\frac{z(t)z(t + 2l)}{z(t + l)z(t)},

or

∏_{t≤τ_{k}<t+l}(1 + r_{k})^{−1} > erl\frac{z(t + 2l)}{z(t + l)}.

Using the monotonicity of the function z(t) for t ≥ t_{2} we find that \frac{z(t + 2l)}{z(t + l)} > 1.

Therefore

∏_{t≤τ_{k}<t+l}(1 + r_{k})^{−1} > erl

or

∏_{t≤τ_{k}<t+l}(1 + r_{k}) < \frac{1}{rle}

The last inequality contradicts condition 3 of Corollary 1. In order to prove that (2) has no eventually negative solution, it is enough to note that

if y(t) is a solution of (2), then −y(t) is a solution of (1).

It follows from assertions 1 and 2 of Corollary 1 that the equation (3) has neither eventually positive nor eventually negative solution. Therefore, each regular solution of (3) is oscillatory.

The proof of the corollary is complete.

Theorem 2 Let the following conditions hold:

1. Conditions H1 — H5 are met. 2.

lim sup_{k→+∞} [ ∫_{τ_{k}−τ+σ}^{τ_{k}} \frac{−q(s)}{p(s + τ − σ)}ds + ∑_{τ_{k}−τ+σ≤τ_{i}<τ_{k}}(−\frac{q_{i}}{p_{i}})] > 1.

Then:

1. The inequality (1) has no eventually positive solution.

2. The inequality (2) has no eventually negative solution.

3. All solutions of the equation (3) are oscillatory.

Proof. Let y(t) be an eventually positive solution of the inequality (1) for t ≥ t_{0}, (t_{0} > 0). Then, proceeding as in proof of Theorem 1, we conclude (7).

We shall prove that the impulsive differential-difference inequality (7) has no eventually negative solution.

Let us suppose the opposite, i.e., there exists a solution z(t) of inequality (7) and a number t_{2} > t_{1} such that z(t) < 0 for t ≥ t_{2}, t_{2} ≥ t_{1}.

Integrating (7) from τ_{k} − l to τ_{k} (τ_{k} ≥ t_{2} + l, l = τ − σ), we obtain

∫_{τ_{k}−l}^{τ_{k}} s(u)z(u + l)du − z(τ_{k}) + z(τ_{k} − l) + ∑_{τ_{k}−l≤τ_{i}<τ_{k}} Δz(τ_{i}) > 0

or

z(τ_{k}) ∫_{τ_{k}−l}^{τ_{k}} s(u)du − z(τ_{k}) + z(τ_{k} − l) + ∑_{τ_{k}−l≤τ_{i}<τ_{k}} s_{i}z(τ_{i} + l) > 0

i.e.,

z(τ_{k})[ ∫_{τ_{k}−l}^{τ_{k}} s(u)du + ∑_{τ_{k}−l≤τ_{i}<τ_{k}} s_{i} − 1] > −z(τ_{k} − l) > 0

From the last inequality and from z(t) < 0 we obtain

∫_{τ_{k}−l}^{τ_{k}} s(u)du + ∑_{τ_{k}−l≤τ_{i}<τ_{k}} s_{i} < 1

The last inequality contradicts condition 2 of Theorem 2.

In order to prove that (2) has no eventually negative solution, it is enough to note that if y(t) is a solution of (2), then −y(t) is a solutionof (1).

It follows from assertions 1 and 2 of Theorem 2 that the equation (3) has neither eventually positive nor eventually negative solution. Therefore, each regular solution of (3) is oscillatory.

The proof of the theorem is complete.

Acknowledgement

The present investigation was supported by professor M.K.Grammatikopoulos. The authors would like to thank the referee for the useful comments and corrections.

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(Received September 8, 2004)

Email addresses:

dimichev@pvamu.edu
MBDimitrova@tu-sliven.com