OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH PERTURBATION

In this paper, we present some new sufficient conditions for the oscillation of all solutions of a second order perturbed differential equation with impulses. These results extend some results for the differential equation without impulses. Examples are provided to illustrate our results. (r(t)( (u(t))|u ' (t)| α−1 u ' (t)) ' +Q(t,u(t)) = P(t,u(t),u ' (t)), t 6 �k; �(r(t) (u(t))|u ' (t)| α−1 u ' (t)) |t=θk +Qk(�k,u) = 0, t ∈ R + , k ∈ N


Introduction
In this paper, we are concerned with the oscillatory behavior of solutions of impulsive differential equations of the form where t ≥ t 0 , ∆z(t) | t=θ = z(θ + )−z(θ − ) and z(θ ± ) := lim Let P LC(J, R) denote the set of all real valued functions u(t) defined on J ⊂ [t 0 , ∞) such that u(t) is continuous for all t ∈ J except possibly at t = θ k , where u(θ ± k ) exist and u(θ k ) = u(θ − k ).Throughout this paper, according to the results, we assume the following conditions hold: (H1) r : [t 0 , ∞) → (0, ∞) is a continuously differentiable function, ψ : R → R + is a continuously differentiable function, and α is a positive real number.
(H3) P ∈ P LC([t 0 , ∞) × R × R, R), and there exists a continuous function , and there exists a continuous function By a solution of equation ( 1) on an interval J ⊂ R, we mean a continuously differentiable function u(t) which is defined on J such that and which satisfies the equation (1).We consider only those solutions u(t) of equation ( 1) which satisfy sup{|u(t)| : t ≥ T u } > 0 for all T u ≥ t 0 .That is equation (1) has a solution which is nontrivial for large t.Such a solution of equation ( 1) is called oscillatory if it has arbitrarily large zeros; Otherwise, it is said to be nonoscillatory.
In recent years the oscillation theory of impulsive differential equations emerging as an important area of research, since such equations have applications in control theory, physics, biology and population dynamics.There are few papers have devoted to the oscillation criteria of second order differential equations with impulses see for example [4,5,8] and the references cited therein.In [1,6,7,10,11] the authors obtained some oscillation results for special cases of equation (1) without impulse effect.
Motivated by this observation, in this paper, we present sufficient conditions for the oscillation of all solutions of equation (1).The results obtained in this paper extend that of in [7] and [8].

Main Results
In this section we present some sufficient condition for the oscillation of all solutions of equation ( 1).We begin with the following theorem.
Theorem 1. Assume that q(t) − p(t) ≥ 0 for all t ≥ t 0 and conditions (H 1 ), (H 2 ), (H 3 ) hold.Suppose that there exists a constant L > 0 such that If there exists a non-increasing differentiable function ρ and where Proof.Suppose that there exist a non-oscillatory solution u(t) of equation (1).We may assume that u(t) is eventually positive.(The case u(t) eventually negative is similar, so that we can omit it.)Define and Differentiating (5) and using (H 2 ), (H 3 ) and equation ( 1), we have From ( 2) and ( 7), we obtain, By using the inequality we get from ( 8) that and ( 6), then integrating the inequality (10) from t 1 to t, we obtain That is, Taking lim sup t→∞ in the last inequality, we obtain ω(t) → −∞ in view of (4).
Integrating from t 3 to t, by using (11), we obtain ds Now taking limit as t → ∞ and using (3), we get a contradiction.The proof is complete.
From Theorem 1, we have the following corollary.
and lim sup then every solution of equation (1) is oscillatory.
Proof.Let u(t) be a non-oscillatory solution on [t 0 , ∞) of equation ( 1).Without loss of generality, we may assume that u(t) is eventually positive.We consider the following three cases for the behavior of u ′ (t).Case 1: u ′ (t) > 0 for t ≥ t 1 for some t 1 ≥ t 0 .Proceeding as in the proof of Theorem 1 we get ( 12) Dividing the last inequality by ρ α (t)r(t) and then integrating from t 1 to t, we have Letting t → ∞, from (17) and (18), we obtain a contradiction.Case 2: u ′ (t) changes sign, then there exists a sequence From (12), we have This implies that u ′ (t) < 0 for all t ≥ α n which is a contradiction.Case 3: u ′ (t) < 0 for t ≥ t 1 .Because of condition (3) we obtain a contradiction from the proof of Theorem 1.This completes the proof.Theorem 4. Assume that conditions (H 1 ), (H 2 ), (H 4 ) and (2) hold with ψ(u) ≡ 1 .If ρ(t) is a positive non-decreasing differentiable function such that and lim sup where Proof.Let u(t) be a non-oscillatory solution of equation (1).Without loss of generality we may assume that u(t) > 0 for all t ≥ t 1 ≥ t 0 .Differentiating (5) and using (H 2 ), (H 4 ) and ( 2), and then proceeding as in the proof of Theorem 1, we obtain Integrating the inequality (21) from t 1 to t, and using (11), we have Taking lim sup t→∞ in the last inequality, we obtain ω(t) → −∞ in view of (20).
Hence there exists t 2 ≥ t 1 such that u ′ (t) < 0 for t ≥ t 2 .Then Proceeding similarly as in Theorem 1 and using condition (19), we obtain a contradiction.
The proof is complete.
The following result is an immediate consequence of Theorem 4.
We conclude this section with the following Theorem.
Proof.Proof is similar to that of Theorem 3, and hence the details are omitted.

Examples
Example 6.1.Consider the following second order impulsive differential equation Here Now, it is easy to see that all conditions of Theorem 1 are satisfied and hence every solution of equation ( 28) is oscillatory.
Example 6.2.Consider the following second order impulsive differential equation Here r(t) = 1 t , θ k = 2 k , t 0 = 1, ψ(u) = u −2 , α = 1.If we choose f (x) = x 3 and ρ(t) = t, then Now, it is easy to see that all conditions of Theorem 3 are satisfied and hence every solution of equation ( 28) is oscillatory.
Remark 6.1.The results obtained in this paper reduces to that of [7] when the impulse are dropped.Remark 6.2.It would be interesting to extend the results obtained in this paper to delay and neutral type impulsive differential equations.

,
then every solution of equation (1) is oscillatory.