Abstract

Our aim in this study, is to develop a linearized oscillation theory for nonlinear neutral delay impulsive differential equations. Precisely, we prove that a certain nonlinear neutral delay impulsive differential equation has the same oscillatory character as its associated linear impulsive equation.

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INTRODUCTION

Recently, linearized oscillation theory for first order differential equations with and without impulses have been discussed by Ladde et al. (1987), Bainov and Hristova (1987), Bainov and Simeonov (1998), Zhang et al. (2004), Xia et al. (2007) and Agarwal et al. (2000). However, there appear to be little or no results in linearized oscillations for first order nonlinear neutral delay impulsive differential equations. In this study, we develop some results on linearized oscillation theory which parallels the so-called linearized stability theory of differential and difference equations. Roughly speaking, we prove that certain nonlinear neutral delay impulsive differential equations have the same oscillatory character as the associated linear neutral delay impulsive differential equations.

Before the formulation of the problem considered in this study, we present some basic definitions and concepts that will be useful in our discussions throughout.

Let, S: = {t_k}_k∈E denote the set of time points of impulses, where E represents a subscript set which can be the set of natural numbers N or the set of integers Z and satisfy the properties:

C1.1: If {t_k}_k∈E is defined with E = N, then 0 <t₁<t₂< ... and

C1.2: If {t_k}_k∈E is defined with E = Z, then t₀≤0<t₁,t_k<t_{k + 1} for k ∈ Z, k≠0 and

Let, f: R x R → R and f_k: R → R, k ∈ Z be continuous functions and let x: R → R, then

(1.1)

Where,

Definition 1.1: The function x = φ(t) is a solution of (1.1) in the interval J: = (α, β) if

•	φ(t) is differentiable in J, t≠tk, k ∈ N and satisfies the condition n'(t) = f(t, φ(t)) for all t ∈ J, t≠tk and k ∈ N

•	φ(t) satisfies the relation n(tk + 0) - φ(tk - 0) = fk (φ(tk - 0)), tk ∈ J and k ∈ Z

Definition 1.2: A solution x is said to be finally positive, if there exists T≥0 such that x(t) is defined for t≥T and x(t) > 0 for all t≥T:

•	Finally, negative, if there exists T≥0 such that x(t) is defined for t≥T and x(t) < 0 for all t≥T

•	Non-oscillatory, if it is either finally positive or finally negative

•	Oscillatory, if it is neither finally positive nor finally negative (Lakshmikantham et al., 1989)

Usually, the solution x(t) for t ∈ J, t∉S of a given impulsive differential equation or its first derivative x'(t) is a piece-wise continuous function with points of discontinuity t_k, t_k ∈ J∩S. Therefore, in order to simplify the statements of the assertions, we introduce the set of functions PC and PC^r, which are defined as follows:

Let, r ∈ N, D ⊂ R and the sequence S be fixed.

Definition 1.3: PC (D, R) is the set of those functions which are continuous for all t ∈ D, t∉S, ∀k ∈ N and have discontinuity of the first kind for t ∈ S and k ∈ N.

Definition 1.4: PC^r (D, R) is the set of those functions which are r-times continuously differentiable for all t ∈ D, t∉S, ∀k ∈ N and have discontinuity of the first kind for t∈S and k∈N (Bainov and Simeonov, 1998; Lakshmikantham et al., 1989).

To specify the points of discontinuity of functions belonging to PC or PC^r, we shall sometimes use the symbols PC^r (D, R; S) and PC^r (D, R; S), r∈N.

Now let us consider the nonlinear neutral delay impulsive differential equation.

(1.2)

where, τ>0 and σ, q_k≥0 and associate with it, the following hypotheses:

(1.3)

where, I₀ = [t₀, ∞) and

(1.4)

ug (u)>0 for u≠0, g (u)≤u for u≥0 and g (u) ≥ u for u≤0,

(1.5)

uh (u)>0 for u≠0 and

(1.6)

Always when at least one of the conditions (1.5) or (1.6) holds, we will refer to Eq (1.7):

(1.7)

as linearized in respect of Eq. (1.1).

The following lemma and theorem extracted from Bainov and Simeonov (1998), are needed in establishing the oscillatory conditions of the problem in question. They may also have further applications in analysis.

Lemma 1.1: Let, p∈ [0, 1), τ ∈ (0, ∞), t₀ ∈ R, x ∈ C([t₀ - τ, ∞), R₊) and assume that for every ε>0 there exists a t_ε≥t₀ such that

(1.8)

Then,

Consider the linear impulsive differential equation with delay

(1.9)

together with the corresponding inequalities

(1.10)

And

(1.11)

We assume that the following condition is fulfilled:

C1.3: p ∈ PC(R₊, R) and τ ≥ 0.

Our aim, here is to establish the following results.

Theorem 1.1: Let condition C1.3 be fulfilled and let there exist a sequence of disjoint intervals J_n = [ζ_n, η_n] with η_n-ζ_n = 2τ, such that:

•	For each n ∈ N, t ∈ Jn and tk ∈ Jn

 

(1.12)

 

 

•	There exists v1 ∈ N such that for n≥v1

 

 

(1.13)

 

Then,

•	The inequality (1.10) has no finally positive solution

•	The inequality (1.11) has no finally negative solution

•	Each regular solution of Eq. (1.9) is oscillatory

Corollary 1.1: Consider the Eq. (1.14):

(1.14)

Let, p_j, q_i ∈(0, ∞) and q_i0 ∈ [0, ∞), τ_j, σ_i≥0 for 1≤j≤M and 1≤i≤ N and assume that

(1.15)

Then the following statements are equivalent:

•	The Eq. (1.14) has a finally positive solution

•	The characteristic system

 

(1.16)

 

where, λ and μ<1 are constants, has a solution (λ, μ) ∈ R x (-∞, 0].

•	The inequality

 

(1.17)

 

has a non-increasing finally positive solution

•	There exists an ε0 ∈ (0, 1) such that for every ε ∈ [0, ε0], the inequality

 

 

has a non-increasing finally positive solution.

RESULTS

Our aim in this study, is to establish conditions for the oscillation of all solutions of Eq. (1.2) in terms of the oscillation of all the solutions of Eq. (1.7) and vice versa.

We recall (Bainov and Simeonov, 1998) that every solution of Eq. (1.7) oscillates if and only if the characteristic equation:

(2.1)

where,

and

has no real roots. As p₀ ∈ [0, 1), Eq. (2.1) has no roots in [0, ∞) and so every solution of Eq. (1.7) oscillates if and only if Eq. (2.1) has no negative roots.

Notice that Eq. (2.1) is built from the characteristic system

with the solution (λ, μ) satisfying the relation

Theorem 2.1: Let conditions (1.3-1.6) be satisfied and that Eq. (2.1) has no real roots. Then every regular solution of Eq. (1.2) oscillates.

Proof: Let us assume on the contrary, that Eq. (1.2) has a non-oscillatory solution x(t). We further assume that x(t) is finally positive. The case where x(t) is finally negative is similar and is omitted. Set

Then finally,

(2.2)

and so z (t) is a decreasing function. We claim that x (t) is bounded. Otherwise, there exists a sequence of points

{t_n}^∞_n=1 for which

and

Then from conditions (1.4) and (1.5)

which contradicts the fact that z (t) is decreasing. Thus, x (t) is bounded and so

(2.3)

But then, the condition L < 0 cannot occur since

(2.4)

follows from L<∞.

On the other hand,

The

and

This contradicts condition (2.4), hence, L≥0.

If L>0, then

Since, x is bounded, 0 < x(s) ≤ M, hence,

by Eq. (1.6) and what is more, h (u) > 0, u >0, Therefore, h (u)≥mu and 0≤u≤M, which implies

(2.5)

This contradicts the statement

hence, L = 0. Thus,

(2.6)

From Eq. (2.6) follows

if t > , hence,

where, ε ∈ (0; 1 + p₀) is given. Then by Lemma 1.1,

(2.7)

Set

From the hypotheses and Eq. (2.7), it follows that

and

(2.8)

where, Q_k = q_k. We integrate both sides of Eq. (2.8) from t to +∞, with sufficiently large t ∈ [T_l, ∞) and from Eq. (2.6) obtain

(2.9)

Again, set

(2.10)

Then finally, w (t) > 0 and

Hence,

(2.11)

Substituting Eq. (2.10) and (2.11) into Eq. (2.9), we obtain, for t sufficiently large,

(2.12)

Clearly,

Let us first assume that p₀>0. Then, for any ε ∈ (0, 1). Eq. (2.12) yields

(2.13)

By virtue of Corollary 1.1, Eq. (2.1) has a real root. This is a contradiction and the proof is complete when p₀>0.

Next, we assume that p₀ = 0. Then, the inequality (2.13) reduces to

(2.14)

with its characteristic system as

Clearly, the solution (λ, μ) of the system satisfies the relation

This enables us to build the characteristic equation

of the corresponding inequality (2.14). As it stands, the equation has no negative roots. Hence, by theorem 1.1 (i), the inequality (2.14) cannot have a finally positive solution.

This contradicts the fact that w(t) > 0 and the proof of theorem 2.1 is complete.

The following result is a partial converse to theorem 2.1.

Theorem 2.2: Consider the neutral impulsive differential equation

(2.15)

where,

(2.16)

Assume that there exist positive numbers q₀ and δ such that

(2.17)

and

(2.18)

Suppose, that Eq. (2.1) has a real root. Then, Eq. (2.15) has a finally positive solution on [t₀ - r, ∞], where

Proof: Let us assume that condition (2.18) holds with 0≤h(u)≤u for 0≤u≤δ. The case where, 0≥h(u)≥u for -δ≤u≤0 is similar and is omitted. Choose d, d₀ ∈ (0, δ) and consider the solution x (t) of Eq. (2.15) with the initial conditions,

and

Consequently, x (t) is left continuous on [t₀ - r, ∞) and satisfies the equation

(2.19)

almost everywhere on [t₀, ∞).

Now, we prove that x (t)>0 for t≥t₀ - r. First, we claim that as long as x (t)>0, it remains strictly less then δ. Otherwise, there exists a T_l such that

Set

Then, for t₀≤t≤T_l

and so

Hence,

which is a contradiction.

Now assume conversely, that there exists T > t₀ such that

(2.20)

From Eq. (2.19) and conditions (2.17) (2.18) and (2.20), we have

almost everywhere, on [t₀, ∞). By our assumption, the characteristic equation

is assumed to have a real root, say λ₀. As p₀ ∈ (0, 1) and q₀q_0k>0, it is seen that λ₀<0. Therefore,

is a positive, left continuous and non-increasing solution of Eq. (1.7). From Corollary 1.1, it follows that x(t)>0 for all t≥t₀ and the proof of Theorem 2.2 is complete.

A combination of Theorem 2.1 and 2.2 yields the following linearized oscillation result for neutral impulsive differential equations.

Corollary 2.1: Assume that conditions (1.6, 2.16-2.18) are satisfied and suppose

Then every solution of Eq. (2.15) oscillates if and only if every solution of Eq. (1.2) oscillates if and only if Eq. (2.1) has no negative real roots.

CONCLUSION

Certain nonlinear neutral impulsive differential equations with deviating arguments have the same oscillatory character as the associated linear neutral impulsive differential equations with deviating arguments.

Precisely, we have been able to establish the necessary conditions for the oscillation of all solutions of the nonlinear neutral delay impulsive differential equations in terms of the oscillatory conditions of the solutions of the corresponding linear neutral impulsive differential equations with deviating arguments and vice versa.

How to cite this article:

I.O. Isaac and Zsolt Lipcsey . Linearized Oscillations in Nonlinear Neutral Delay Impulsive Differential Equations.
DOI: https://doi.org/10.36478/jmmstat.2009.1.7
URL: https://www.makhillpublications.co/view-article/1994-5388/jmmstat.2009.1.7