Interval Oscillation Criteria for Nonlinear Impulsive Differential Equations with Variable Delay

In this paper, the interval qualitative properties of a class of second order nonlinear differential equations are studied. For the hypothesis of delay being variable τ(t), an " interval delay function " is introduced to estimate the ratio of functions x(t − τ(t)) and x(t) on each considered interval, then Riccati transformation and H functions are applied to obtain interval oscillation criteria. The known results gained by Huang and Feng [Comput. under the assumption of constant delay τ are developed. Moreover, examples are also given to illustrate the effectiveness and non-emptiness of our results.

However, almost all of interval oscillation results for the impulsive equations in the existing literature were established only for the case of "without delay", in other words, for the case of "with delay" the study on the interval oscillation is very scarce.To the best of our knowledge, Huang and Feng [4] gave the first research in this subject recently.They considered the second order impulsive differential equations with constant delay of the form and established some interval oscillation criteria which developed some known results for the equations without delay in [1,5,13].
Later, by idea of [4], Guo et al. [3] studied the delay case of mixed nonlinear impulsive differential equations (1.3) as follows (r(t)Φ α (x ′ (t))) ′ + p 0 (t)Φ α (x(t)) .5)where Φ γ (s) = |s| γ−1 s and They corrected some errors in proof of [4] (cf.Remark 2.4 in [3]) and obtained some results which developed some known results of [2,6,10].In 2014, Zhou et al. [14] investigated interval qualitative properties of a class of nonlinear impulsive differential equations under three factors -impulse, damping and delay of the form where φ γ (s) = |s| γ−1 s and γ is positive.As the comment above, the delay considered in [3,4,14] is constant.It is natural to ask if it is possible to research the interval oscillation of the impulsive equations with variable delay.In fact, when there is no impulse, the interval oscillation of differential equations with variable delay of the form in the linear (γ = 1) and the superlinear (γ > 1) cases has been studied by Sun [12].Some techniques to estimate the unknown function x(τ(t))/x(t) on each considered interval were used in [12], which inspires us to consider more complex problem.
In this paper, motivated mainly by [4,12], we consider the following second order nonlinear impulsive differential equations with variable delay where {θ k } denotes the impulsive moments sequence with 0 By introducing an "interval delay function" and discussing its zero points on intervals of impulse moments, we estimate the function of ratio of x(t − τ(t)) and x(t) on each considered interval, then making use of Riccati transformation and H functions (introduced first by Philos [11]), we establish some interval oscillation criteria, which generalize or improve the results of [4].Moreover, we also give two examples to illustrate the effectiveness and non-emptiness of the results.

Main results
We first introduce some definitions and assumptions.
Let I ⊂ R be an interval, a functional space PLC(I, R) is defined as follows: PLC(I, R) := {y : I → R | y is continuous on I \ {t i } and at each t i , y(t + i ) and y(t − i ) exist, and the left continuity of y is assumed, i.e. y(t − i ) = y(t i ), i ∈ N}.Throughout the paper, we always assume that the following conditions hold: x ≥ η for all x ∈ R \ {0}; and there exists a nonnegative constant τ such that 0 ≤ τ(t) ≤ τ for all t ≥ t 0 and For the discussion of impulse moments of x(t) and x(t − τ(t)) on two intervals [c j , d j ] (j = 1, 2), we need to consider the following possible cases for k(c j ) < k(d j ) and the possible cases for k(c j ) = k(d j ) In order to save space, throughout the paper, we study (1.8) under the case of combination of (S 1 ) with ( S1 ) only.The discussions for other cases are similar and omitted.
We define a function (called "interval delay function"): (A 4−1 ) There is one zero point See Figure 2; or (A 4−3 ) There is not any zero point such that D k (t) = 0.This case must lead to D k (t) > 0 for all t ∈ (θ k , θ k+1 ].See Figure 3. In Remark 2.2, the case (A 4−1 ) is more complex to consider than other two cases for the estimation of x(t) .We study (1.8) under the assumption (A 4−1 ) only throughout the paper.The discussions for cases (A 4−2 ) and (A 4−3 ) are similar and omitted.Lemma 2.3.Assume that for any T ≥ t 0 , there exist c 1 , If x(t) is a positive solution of (1.8), then there exist the following estimations of x(t−τ(t)) x(t) : Proof.From (1.8), (2.1) and (A 1 ), we obtain, Next, we give the proof of cases (a) and (b) only.For other cases, the proof is similar and will be omitted.Case (a).
Integrating both sides of above inequality from t − τ(t) to t, we obtain Using the impulsive condition of (1.8) and the monotone properties of x ′ (t), we get In addition, Similarly to the analysis of (2.2) and (2.3), we have From (2.5) and (2.7), we get In view of (A 2 ), we have On the other hand, using similar analysis of (2.2) and ( 2.3), we get (2.9) From (2.8) and (2.10), we obtain Lemma 2.4.Assume that for any T ≥ t 0 , there exist c 2 , The proof of Lemma 2.4 is similar to that of Lemma 2.3 and will be omitted.
Lemma 2.5.Assume that for any T ≥ t 0 there exist c 1 , (2.1) holds.Let x(t) be a positive solution of (1.8) and u(t) be defined by , then, there are the following estimations of u(t): From the proof of Lemma 2.3, we know that x ′ (t) is nonincreasing.In view of x(θ i−1 ) > 0, we obtain Using similar analysis on (c 1 , θ k(c 1 )+1 ], we can get (2.15) Lemma 2.6.Assume that for any T ≥ t 0 , there exist c 2 , (2.12) holds.Let x(t) be a negative solution of (1.8) and u(t) be defined by ) , then, the estimations (g), (h) in Lemma 2.5 are correct with the replacement of [c 1 , The proof of Lemma 2.6 is similar to that of Lemma 2.5 and will be omitted.We introduce a space Ω(c, d) as follows In order to save a space, we define , for j = 1, 2.
Lemma 2.7.Assume that for any T ≥ t 0 , there exist c 1 , (2.1) hold.Let x(t) be a positive solution of (1.8) and u(t) be defined by (2.13).

.18)
Proof.Differentiating u(t) and in view of (1.8) and condition (A 1 ), we obtain, for t ̸ = θ k , , multiplying both sides of (2.19) by w 2 1 (t) and then integrating it from c 1 to d 1 , we obtain Using the integration by parts formula on the left side of above inequality and noting the condition w 1 (c 1 ) = w 1 (d 1 ) = 0, we obtain where Meanwhile, for t = θ k , k = 1, 2, . . ., we have Using the integration by parts on the left-hand side and noting the condition w 1 (c 1 ) = w 1 (d 1 ) = 0, we obtain where V(w 1 (t), u(t)) is defined by (2.21).Thus This is (2.18).Therefore we complete the proof.
The proof of Lemma 2.8 is similar to that of Lemma 2.7 and will be omitted.
For convenience in the expression below, we use the following notations. (2.25) where t i are zero points of D i (t Theorem 2.9.Assume that for any T ≥ t 0 , there exist c j , ) and (2.12) hold.If there exist w j (t) ∈ Ω(c j , d j ) such that, for k(c j ) < k(d j ), j = 1, 2, ) which contradicts condition (2.26) for j = 1.
When the cases k(c 1 ) = k(d 1 ) holds, from Lemma 2.3 and Lemma 2.5 we easily obtain Therefore we complete the proof.
In formula (2.25), t i and t k(d j ) are zero points of D i (t) and D k(d j ) (t) respectively.In general, it is not easy to solve these zero points from D i (t) = 0 and D k(d j ) (t) = 0.This makes the calculation of (2.25) more difficult.To overcome this difficulty it is need to avoid these points being upper limit (or lower limit) of integrals.So, we give the following theorem.
Theorem 2.11.Assume that for any T ≥ t 0 , there exist c j , (2.30) Proof.The proof is similar to that of Theorem 2.9, only the estimations of on (t i , θ i+1 ], (θ i , t i ), (t k(d j ) , d j ) and (θ k(d j ) , t k(d j ) ) need to be modified.
(2.34) Therefore we complete the proof.
In the following we will establish Kamenev-type interval oscillation criteria for (1.8) by the ideas of Philos [11].
Let D = {(t, s) : for any T ≥ t 0 .Noticing whether there are impulsive moments of x(t) in [c j , δ j ] and [δ j , d j ] or not, we should consider the following four cases Moreover, in the discussion of the impulse moments of x(t − τ(t)), it is necessary to consider the following three cases: In the following theorems, we only consider the case of combination of (S 5 ) with ( S5 ).For the other combinations, similar conclusions can be given and their details will be omitted here.
For convenience in the expression below, we define, for j = 1, 2, that Theorem 2.12.Assume that for any T ≥ t 0 , there exist c j , (2.1) and (2.12) hold.If there exists a pair of (H 1 , H 2 ) ∈ H such that, for j = 1, 2, then (1.8) is oscillatory.
Proof.Assume, to the contrary, that x(t) is a nonoscillatory solution of (1.8).If x(t) is positive solution, we choose the interval [c 1 − τ, d 1 ] to consider.Defining function u(t) as in (2.13) and using the same proof as in Lemma 2.5, we can get (2.19).Multiplying both sides of (2.19) by H 1 (t, c 1 ) and integrating it from c 1 to δ 1 , we have ] and using the integration by parts on the left-hand side of above inequality, we obtain ) (2.37) Substituting (2.37) into (2.36),we have (2.38) Since (2.39) On the other hand, using same method as in Lemma 2.3, we can get estimations of the function x(t) in several sub-intervals of [c 1 , δ 1 ] and then we obtain θ k(δ 1 ) (2.40) From (2.39) and (2.40), we have Multiplying both sides of (2.19) by H 2 (d 1 , t) and using similar process to the above, we obtain (2.43)Meanwhile, by using same way as in Lemma 2.5, we get the estimations of u(θ i ) .

3 2 u 3 4 u 3 ( 3 − u) 3 (
(u − 1) 3 (u − 3 2 ) ) D k (t) has at most one zero point on (θ k , θ k+1 ] for any k = 1, 2, . . .The situation for zero points of D k (t) on (θ k , θ k+1 ] may be very complicated.That is to say, the number of zero points of D k (t) on (θ k , θ k+1 ] may be arbitrary.Assumption (A 4 ) is just a simple situation for D k (t).The research of other complex situations will be left to the reader.In (A 4 ), the assumption that D k (t) has at most one zero point on (θ k , θ k+1 ] may be divided into three cases as follows. Multiplying both sides of (2.19) by w 2 1 (t) and integrating it from c 1 to d 1 , we obtain .22) Therefore, from (2.20)-(2.22),we can get (2.17).If k(c 1 ) = k(d 1 ), there is no impulsive moment in [c 1 , d 1 ].
then a pair of functions H 1 , H 2 is said to belong to a function set H, defined by (H 1 , H 2 ) ∈ H, if there exist h 1 , h 2 ∈ L loc (D, R) satisfying the following conditions: (δ 1 , c 1 ) and H 2 (d 1 , δ 1 ) respectively and adding them, we get < 0, we can choose interval [c 2 , d 2 ] Particularly, if a n,i = b n,i , for n ∈ N, i = 1, 2, condition (3.4) becomes .46) then (1.8) is oscillatory.