# Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay

Authors: Zhonghai Guo; Xiaoliang Zhou; Wu-Sheng Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the following second order mixed nonlinear impulsive differential equations with delay(r(t)Φα(x′(t)))′+p0(t)Φα(x(t))+∑i=1npi(t)Φβi(x(t-σ))=e(t),t≥t0,t≠τk,x(τk+)=akx(τk),x′(τk+)=bkx′(τk),k=1,2,…, where Φ*(u)=|u|*-1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence, and τk+1-τk>σ. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.


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## Body

## 1. Introduction

We consider the following second order impulsive differential equations with delay(1.1)(r(t)Φα(x′(t)))′+p0(t)Φα(x(t))+∑i=1npi(t)Φβi(x(t-σ))=e(t),t≥t0,t≠τk,x(τk+)=akx(τk),x′(τk+)=bkx′(τk),k=1,2,…,
where Φ*(u)=|u|*-1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence, and τk+1-τk>σ, for all k∈ℕ.LetJ⊂ℝ be an interval, and we define(1.2)PLC⁡(J,R):={y:J⟶R∣yiscontinuouseverywhereexcepteachτkatwhichy(τk+)andy(τk-)existandy(τk-)=y(τk),k∈N}.
For given t0 and ϕ∈PLC([t0-σ,t0],ℝ), we say x∈PLC([t0-σ,∞),ℝ) is a solution of (1.1) with initial value ϕ if x(t) satisfies (1.1) for t≥t0 and x(t)=ϕ(t) for t∈[t0-σ,t0].A solution of (1.1) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory.Impulsive differential equation is an adequate mathematical apparatus for the simulation of processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth. Because it has more richer theory than its corresponding without impulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).In the last decades, there is constant interest in obtaining new sufficient conditions for oscillation or nonoscillation of the solutions of various impulsive differential equations, see, for example, [1–9] and the references cited therein.In recent years, interval oscillation of impulsive differential equations was also arousing the interest of many researchers. In 2007, Özbekler and Zafer [10] investigated the following equations:(1.3)(m(t)φα(y′))′+q(t)φβ(y)=f(t),t≠θi,Δ(m(t)φα(y′))+qiφβ(y)=fi,t=θi,(i∈N),(1.4)(m(t)y′)′+s(t)y′+q(t)φβ(y)=f(t),t≠θi,Δ(m(t)y′)+qiφβ(y)=fi,t=θi,(β≥1),
where φ*(u)=|u|*-1u, β≥α, {qi} and {fi} are sequences of real numbers. In 2009, they further gave a research [11] for equations of the form(1.5)(r(t)φα(x′))′+p(t)φα(x′)+q(t)φβ(x)=e(t),t≠θi,Δ(r(t)φα(x′))+qiφβ(x)=ei,t=θi,
and obtained some interval oscillation results which improved and extended the earlier ones for the equations without impulses.For the mixed type Emden-Fowler equations(1.6)(r(t)x′(t))′+p(t)x(t)+∑i=1npi(t)|x(t)|αi-1x(t)=e(t),t≠τk,x(τk+)=akx(τk),x′(τk+)=bkx′(τk),k∈N,
Liu and Xu [12] established some interval oscillation results. Recently, Özbekler and Zafer [13] investigated the more general cases(1.7)(r(t)Φα(x′(t)))′+q(t)Φα(x(t))+∑k=1nqk(t)Φβk(x(t))=e(t),t≠θi,x(θi+)=aix(θi),x′(θi+)=bix′(θi),
where β1>⋯>βm>α>βm+1>⋯>βn>0.However, for the impulsive equations, almost all of interval oscillation results 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 [14] gave the first research in this direction recently. They considered second order delay differential equations with impulses(1.8)x′′(t)+p(t)f(x(t-τ))=e(t),t≥t0,t≠tk,k=1,2,…,x(tk+)=akx(tk),x′(tk+)=bkx′(tk),k=1,2,…
and established some interval oscillation criteria which developed some known results for the equations without delay or impulses [15–17].Motivated mainly by [13, 14], in this paper, we study the interval oscillation of the delay impulsive (1.1). By using some inequalities, Riccati transformation and ℋ functions (introduced first by Philos [18]), we establish some interval oscillation criteria which generalize and improve some known results. Moreover, examples are considered to illustrate the main results.


## 2. Main Results

Throughout the paper, we always assume that the following conditions hold:(A1)
the exponents satisfy thatβ1>⋯>βm>α>βm+1>⋯>βn>0;(A2)
r(t)∈C([t0,∞),(0,∞)) is nondecreasing, e(t),pi(t)∈PLC([t0,∞),ℝ),i=0,1,…,n;(A3)
{ak} and {bk} are real constant sequences such that bk≥ak>0,k∈ℕ.It is clear that all solutions of (1.1) are oscillatory if there exists a subsequence {ki} of {k} such that aki≤0 for all i∈ℕ. So, we assume ak>0 for all k∈ℕ in condition (A3).In this section, intervals[c1,d1] and [c2,d2] are considered to establish oscillation criteria. For convenience, we introduce the following notations (see [12]). Let(2.1)k(s)=max⁡{i:t0<τi<s},rj=max⁡{r(t):t∈[cj,dj]},Ω(cj,dj)={wj∈C1[cj,dj]:wj(t)≢0,wj(cj)=wj(dj)=0},j=1,2.For two constantsc,d∉{τk} with c<d and k(c)<k(d) and a function φ∈C([c,d],ℝ), we define an operator Q:C([c,d],ℝ)→ℝ by(2.2)Qcd[φ]=φ(τk(c)+1)bk(c)+1α-ak(c)+1αak(c)+1α(τk(c)+1-c)α+∑i=k(c)+2k(d)φ(τi)biα-aiαaiα(τi-τi-1)α,
where ∑st=0 if s>t.In the discussion of the impulse moments ofx(t) and x(t-σ), we need to consider the following cases for k(cj)<k(dj)(S1)
τk(cj)+σ<cj and τk(dj)+σ>dj,(S2)
τk(cj)+σ<cj and τk(dj)+σ<dj,(S3)
τk(cj)+σ>cj and τk(dj)+σ>dj,(S4)
τk(cj)+σ>cj and τk(dj)+σ<dj,and the cases fork(cj)=k(dj)(S̅1)
τk(cj)+σ<cj,(S̅2)
cj<τk(cj)+σ<dj,(S̅3)
τk(cj)+σ>dj.Combining (S*) with (S̅*), we can get 12 cases. In order to save space, throughout the paper, we study (1.1) under the case of combination of (S1) with (S̅1) only. The discussions for other cases are similar and omitted.The following preparatory lemmas will be useful to prove our theorems. The first is derived from [19] and second is from [20].Lemma 2.1.
For any given n-tuple{β1,β2,…,βn} satisfying  (A1), then there exists an n-tuple  {η1,η2,…,ηn} such that
(2.3)∑i=1nβiηi=α,∑i=1nηi=λ,0<ηi<1,
where λ∈(0,1].Lemma 2.2.
SupposeX and  Y are nonnegative, then
(2.4)λXYλ-1-Xλ≤(λ-1)Yλ,λ>1,
where equality holds if and if X=Y.Letα>0, B≥0, A>0 and y≥0. Put(2.5)λ=1+1α,X=Aα/(α+1)y,Y=(αα+1)αBαA-α2/(α+1).
It follows from Lemma 2.2 that(2.6)By-Ay(α+1)/α≤αα(α+1)α+1Bα+1Aα.Theorem 2.3.
Assume that for anyT≥t0, there exist  cj,dj∉{τk},j=1,2, such that  T<c1<d1≤c2<d2 and for  j=1,2(2.7)pi(t)≥0,t∈[cj-σ,dj]∖{τk},i=0,1,2,…,n;(-1)je(t)≥0,t∈[cj-σ,dj]∖{τk}.
If there exist wj(t)∈Ω(cj,dj) and ρ(t)∈C1([cj,dj],(0,∞)) such that, for k(cj)<k(dj), j=1,2,
(2.8)∫cjτk(cj)+1Wj(t)(t-τk(cj)-σ)α(t-τk(cj))αdt+∑i=k(cj)+1k(dj)-1[∫τiτi+σWj(t)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1Wj(t)(t-τi-σ)α(t-τi)αdt]+∫τk(dj)djWj(t)(t-τk(dj))αbk(dj)α(t+σ-τk(dj))αdt+∫cjdjρ(t)(p0(t)|wj(t)|α+1-r(t)(|wj′(t)|+|ρ′(t)||wj(t)|(α+1)ρ(t))α+1)dt≥ρjrjQcjdj[|wj|α+1],
where ρj is maximum value of ρ(t) on [cj,dj] and, for k(cj)=k(dj), j=1,2,
(2.9)∫cidi(Wj(t)(t-ci)α(t-ci+σ)α+ρ(t)(p0(t)|wj(t)|α+1-r(t)(|wj′(t)|+|ρ′(t)||wj(t)|(α+1)ρ(t))α+1)dt≥0,
where Wj(t)=η0-η0|e(t)|η0∏i=1nηi-ηi(pi(t))ηi|wj(t)|α+1 with η0=1-∑i=1nηi and η1,η2,…,ηn are positive constants satisfying conditions of Lemma 2.1, then (1.1) is oscillatory.Proof.
Assume, to the contrary, thatx(t) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(t)>0 and x(t-σ)>0 for t≥t0. In this case the interval of t selected for the following discussion is [c1,d1]. Define
(2.10)u(t)=ρ(t)r(t)Φα(x′(t))Φα(x(t)),t∈[c1,d1].
It follows, for t≠τk, that
(2.11)u′(t)=-ρ(t)p0(t)-ρ(t)[∑i=1npi(t)Φβi-α(x(t-σ))+|e(t)|Φα(x(t-σ))]Φα(x(t-σ))Φα(x(t))+ρ′(t)ρ(t)u(t)-α(ρ(t)r(t))1/α|u(t)|(α+1)/α.
Now, let
(2.12)v0=η0-1|e(t)|Φα(x(t-σ)),vi=ηi-1pi(t)Φβi-α(x(t-σ)),i=1,2,…,n,
where η1,η2,…,ηn are positive constants satisfying conditions of Lemma 2.1 and η0=1-∑i=1nηi. Employing in (2.11) the arithmetic-geometric mean inequality (see [20])
(2.13)∑i=0nηivi≥∏i=0nviηi
and in view of (2.3), we have that
(2.14)u′(t)≤-ρ(t)p0(t)-ρ(t)ψ(t)Φα(x(t-σ))Φα(x(t))-α(ρ(t)r(t))1/α|u(t)|(α+1)/α+ρ′(t)ρ(t)u(t),
where
(2.15)ψ(t)=η0-η0|e(t)|η0∏i=1nηi-ηi(pi(t))ηi.
First, we consider the casek(c1)<k(d1).
In this case, we assume impulsive moments in[c1,d1] are τk(c1)+1,τk(c1)+2,…,τk(d1). Choosing w1(t)∈Ω(c1,d1), multiplying both sides of (2.14) by |w1(t)|α+1 and then integrating it from c1 to d1, we obtain
(2.16)(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1∫τiτi+1+∫τk(d1)d1)u′(t)|w1(t)|α+1dt≤(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1∫τiτi+1+∫τk(d1)d1)(ρ′(t)ρ(t)u(t)-α(ρ(t)r(t))1/α|u(t)|(1+α)/α)|w1(t)|α+1dt-(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1[∫τiτi+σ+∫τi+στi+1]+∫τk(d1)d1)xα(t-σ)xα(t)W1(t)dt-∫c1d1ρ(t)p0(t)|w1(t)|α+1dt,
where W1(t)=ρ(t)ψ(t)|w1(t)|α+1. Using the integration by parts formula in the left-hand side of above inequality and noting the condition w1(c1)=w1(d1)=0, we obtain
(2.17)∑i=k(c1)+1k(d1)|w1(τi)|α+1[u(τi)-u(τi+)]≤(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1∫τiτi+1+∫τk(d1)d1)×[(α+1)Φα(w1(t))w1′(t)u(t)+ρ′(t)ρ(t)u(t)|w1(t)|α+1-α(ρ(t)r(t))1/α|u(t)|(1+α)/α|w1(t)|α+1]dt-(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1[∫τiτi+σ+∫τi+στi+1]+∫τk(d1)d1)xα(t-σ)xα(t)W1(t)dt-∫c1d1ρ(t)p0(t)|w1(t)|α+1dt,≤(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1∫τiτi+1+∫τk(d1)d1)×[(α+1)|w1(t)|α|u(t)|(|w1′(t)|+|ρ′(t)||w1(t)|(α+1)ρ(t))-α(ρ(t)r(t))1/α|u(t)|(1+α)/α|w1(t)|α+1]dt-(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1[∫τiτi+σ+∫τi+στi+1]+∫τk(d1)d1)xα(t-σ)xα(t)W1(t)dt-∫c1d1ρ(t)p0(t)|w1(t)|α+1dt.
Letting y=|w1(t)|α|u(t)|, B=(α+1)(|w1′(t)|+|ρ′(t)||w1(t)|/(α+1)|ρ(t)|), A=α/(ρ(t)r(t))1/α and using (2.6), we have for the integrand function in above inequality that
(2.18)(α+1)|w1(t)|α|w1′(t)||u(t)|-αr1/α(t)|u(t)|(1+α)/α|w1(t)|α+1≤ρ(t)r(t)(|w1′(t)|+|ρ′(t)||w1(t)|(α+1)ρ(t))α+1.
In view of the impulse condition in (1.1) and the definition of u we have, for t=τk, k=1,2,…, that
(2.19)u(τk+)=bkαakαu(τk).
From (2.19), we have
(2.20)∑i=k(c1)+1k(d1)|w1(τi)|α+1[u(τi)-u(τi+)]=∑i=k(c1)+1k(d1)(1-biαaiα)|w1(τi)|α+1u(τi).
Therefore, we get
(2.21)∑i=k(c1)+1k(d1)(1-biαaiα)|w1(τi)|α+1u(τi)≤∫c1d1ρ(t)r(t)(|w1′(t)|+|ρ′(t)||w1(t)|(α+1)ρ(t))α+1dt-(∫c1τk(c1)+1+∑i=k(c1)+1k(d1)-1[∫τiτi+σ+∫τi+στi+1]+∫τk(d1)d1)xα(t-σ)xα(t)W1(t)dt-∫c1d1ρ(t)p0(t)|w1(t)|α+1dt.
On the other hand, for t∈[c1,d1]∖{τi},
(2.22)(r(t)Φα(x′(t)))′=e(t)-p0(t)Φα(x(t))-∑i=1npi(t)Φβi(x(t-σ))≤0.
Hence r(t)Φα(x′(t)) is nonincreasing on [c1,d1]∖{τk}.
Because there are different integration intervals in (2.21), we will estimate x(t-σ)/x(t) in each interval of t as follows.
Case 1.
t∈(τi,τi+1]⊂[c1,d1], for i=k(c1)+1,…,k(d1)-1.
Subcase 1.
Ifτi+σ<t≤τi+1, then (t-σ,t)⊂(τi,τi+1]. Thus there is no impulsive moment in (t-σ,t). For any s∈(t-σ,t), we have
(2.23)x(s)-x(τi+)=x′(ξ1)(s-τi),ξ1∈(τi,s).
Since x(τi+)>0, r(s) is nondecreasing, function Φα(·) is an increasing function and r(t)Φα(x′(t)) is nonincreasing on (τi,τi+1), we have
(2.24)Φα(x(s))≥r(ξ1)r(s)Φα(x(s))>r(ξ1)r(s)Φα(x′(ξ1)(s-τi))=r(ξ1)Φα(x′(ξ1))r(s)(s-τi)α≥r(s)Φα(x′(s))r(s)(s-τi)α=Φα(x′(s)(s-τi)),ξ1∈(τi,s).
Therefore,
(2.25)x′(s)x(s)<1s-τi.
Integrating both sides of the above inequality from t-σ to t, we obtain
(2.26)x(t-σ)x(t)>t-τi-σt-τi,t∈(τi+σ,τi+1].
Subcase 2.
Ifτi<t<τi+σ, then τi-σ<t-σ<τi<t<τi+σ. There is an impulsive moment τi in (t-σ,t). For any t∈(τi,τi+σ), we have
(2.27)x(t)-x(τi+)=x′(ξ2)(t-τi),ξ2∈(τi,t).
Using the impulsive condition of (1.1) and the monotone properties of r(t), Φα(·) and r(t)Φα(x′(t)), we get
(2.28)Φα(x(t)-aix(τi))=r(ξ2)Φα(x′(ξ2))r(ξ2)(t-τi)α≤r(τi)Φα(x′(τi+))r(ξ2)(t-τi)α=r(τi)Φα(bix′(τi)(t-τi))r(ξ2).
Since x(τi)>0, we have
(2.29)Φα(x(t)x(τi)-ai)≤r(τi)r(ξ2)Φα(bix′(τi)x(τi)(t-τi)).
In addition,
(2.30)x(τi)>x(τi)-x(τi-σ)=x′(ξ3)σ,ξ3∈(τi-σ,τi).
Using the same analysis as (2.24) and (2.25), we have
(2.31)x′(τi)x(τi)<1σ.
From (2.29) and (2.31) and note that the monotone properties of Φα(·) and r(t), we get
(2.32)Φα(x(t)x(τi)-ai)<r(τi)r(ξ2)Φα(biσ(t-τi))≤Φα(biσ(t-τi)).
Then,
(2.33)x(t)x(τi)<ai+biσ(t-τi).
In view of (A3), we have
(2.34)x(τi)x(t)>σσai+bi(t-τi)≥σbi(t+σ-τi)>0.
On the other hand, similar to the above analysis, we get
(2.35)x′(s)x(s)<1s-τi+σ,s∈(τi-σ,τi).
Integrating (2.35) from t-σ to τi, where t∈(τi,τi+σ), we have
(2.36)x(t-σ)x(τi)>t-τiσ≥0.
From (2.34) and (2.36), we obtain
(2.37)x(t-σ)x(t)>t-τibi(t+σ-τi),t∈(τi,τi+σ).
Case 2 (t∈[c1,τk(c1)+1]).
Sinceτk(c1)+σ<c1, then t-σ∈[c1-σ,τk(c1)+1-σ]⊂(τk(c1),τk(c1)+1-σ]. So, there is no impulsive moment in (t-σ,t). Similar to (2.26) of Subcase 1, we have
(2.38)x(t-σ)x(t)>t-τk(c1)-σt-τk(c1),t∈[c1,τk(c1)+1].
Case 3 (t∈(τk(d1),d1]).
Sinceτk(d1)+σ>d1, then t-σ∈(τk(d1)-σ,d1-σ]⊂(τk(d1)-σ,τk(d1)). Hence, there is an impulsive moment τk(d1) in (t-σ,t). Making a similar analysis of Subcase 2, we obtain
(2.39)x(t-σ)x(t)>t-τk(d1)bk(d1)(t+σ-τk(d1))≥0,t∈(τk(d1),d1].
From (2.21), (2.26), (2.37), (2.38), and (2.39) we get
(2.40)∑i=k(c1)+1k(d1)(1-biαaiα)|w1(τi)|α+1u(τi)<∫c1d1ρ(t)r(t)(|w1′(t)|+|ρ′(t)||w1(t)|(α+1)ρ(t))α+1dt-∫c1τk(c1)+1W1(t)(t-τk(c1)-σ)α(t-τk(c1))αdt-∑i=k(c1)+1k(d1)-1[∫τiτi+σW1(t)(t-τi)αbiα(t+σ-τi)αdt-∫τi+στi+1W1(t)(t-τi-σ)α(t-τi)αdt]-∫τk(d1)d1W1(t)(t-τk(d1))αbk(d1)α(t+σ-τk(d1))αdt-∫c1d1ρ(t)p0(t)|w1(t)|α+1dt.
On the other hand, fort∈(τi-1,τi]⊂[c1,d1], i=k(c1)+2,…,k(d1), we have
(2.41)x(t)-x(τi-1)=x′(ξ)(t-τi-1),ξ∈(τi-1,t).
In view of x(τi-1)>0 and the monotone properties of Φα(·), r(t)Φα(x′(t)) and r(t), we obtain
(2.42)Φα(x(t))>Φα(x′(ξ))Φα(t-τi-1)≥r(t)r(ξ)Φα(x′(t))Φα(t-τi-1).
This is
(2.43)ρ(t)r(t)Φα(x′(t))Φα(x(t))<ρ1r(ξ)(t-τi-1)α.
Letting t→τi-, we have
(2.44)u(τi)=ρ(t)r(τi)Φα(x′(τi))Φα(x(τi))<ρ1r1(τi-τi-1)α,i=k(c1)+2,…,k(d1).
Using similar analysis on (c1,τk(c1)+1], we get
(2.45)u(τk(c1)+1)<ρ1r1(τk(c1)+1-c1)α.
Then from (2.44), (2.45), and (A3), we have
(2.46)∑i=k(c1)+1k(d1)(biαaiα-1)|w1(τi)|α+1u(τi)<ρ1r1[|w1(τk(c1)+1)|α+1θ(c1)+∑i=k(c1)+2k(d1)|w1(τi)|α+1ζ(τi)]=ρ1r1Qc1d1[|w1|α+1],
where θ(c1)=(bk(c1)+1α-ak(c1)+1α)/ak(c1)+1α(τk(c1)+1-c1)α and ζ(τi)=(biα-aiα)/aiα(τi-τi-1)α.
From (2.40) and (2.46), we obtain
(2.47)∫c1τk(c1)+1W1(t)(t-τk(c1)-σ)α(t-τk(c1))αdt+∑i=k(c1)+1k(d1)-1[∫τiτi+σW1(t)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1W1(t)(t-τi-σ)α(t-τi)αdt]+∫τk(d1)d1W1(t)(t-τk(d1))αbk(d1)α(t+σ-τk(d1))αdt+∫c1d1ρ(t)p0(t)|w1(t)|α+1dt-∫c1d1ρ(t)r(t)(|w1′(t)|+|ρ′(t)||w1(t)|(α+1)ρ(t))α+1dt<ρ1r1Qc1d1[|w1|α+1].
This contradicts (2.8).
Next we consider the casek(c1)=k(d1). From the condition (S̅1) we know that there is no impulsive moment in [c1,d1]. Multiplying both sides of (2.47) by |w1(t)|α+1 and integrating it from c1 to d1, we obtain
(2.48)∫c1d1u′(t)|w1(t)|α+1dt≤-∫c1d1α(ρ(t)r(t))1/α|u(t)|(α+1)/α|w1(t)|α+1dt-∫c1d1xα(t-σ)xα(t)W1(t)dt-∫c1d1ρ(t)p0(t)|w1(t)|α+1dt.
Similar to the proof of (2.21), we have
(2.49)∫c1d1[xα(t-σ)xα(t)W1(t)+ρ(t)p0(t)|wj(t)|α+1-ρ(t)r(t)(|wj′(t)|+|ρ′(t)||wj(t)|(α+1)ρ(t))α+1]dt≤0.
Using same way as Subcase 1, we get
(2.50)x(t-σ)x(t)>t-c1t-c1+σ,t∈[c1,d1].
From (2.49) and (2.50) we obtain
(2.51)∫c1d1[W1(t)(t-c1)α(t-c1+σ)α+ρ(t)p0(t)|wj(t)|α+1-ρ(t)r(t)(|wj′(t)|+|ρ′(t)||wj(t)|(α+1)ρ(t))α+1]dt<0.
This contradicts condition (2.9).
Whenx(t)<0, we can choose interval [c2,d2] to study (1.1). The proof is similar and will be omitted. Therefore, we complete the proof.Remark 2.4.
In article [14], the authors obtained the following inequalities:
(2.52)x(t)x(tj)>ajtj+τ-tτ≥0,t∈(tj,tj+τ),
See [14, equation (2.9)],
(2.53)x(t)-τx(tj)>t-tjτ≥0,t∈(tj,tj+τ).
See [14, equation (2.10)].
Dividing [14, equation (2.10)] by [14, equation (2.9)], they obtained
(2.54)x(t)-τx(t)>t-tjaj(tj+τ-t)≥0,t∈(tj,tj+τ).
See [14, equation (2.11)]
This is an error. Moreover, similar errors appeared many times in the later arguments, for example, in inequalities (2.15), (2.19), and (2.20) in [14]. Moreover, the above substitution can lead to some divergent integrals, for example, the integrals in (2.22), (2.24) in [14]. Therefore, the conditions of their Theorems 2.1–2.5 must be defective. In the proof of our Theorem 2.3, this error is remedied.Remark 2.5.
Whenσ=0, that is, the delay disappears, (1.1) reduces to (1.7) studied by Özbekler and Zafer [13]. In this case, our result with ρ(t)=1 is Theorem 2.1 of [13].Remark 2.6.
Whenσ=0, that is, the delay disappears in (1.1) and α=1, our result reduces to Theorem 2.1 of [12].Remark 2.7.
Whenak=bk=1 for all k=1,2,… and σ=0, that is, both impulses and delay disappear in (1.1), our result with α=1 and ρ(t)=1 reduces to Theorem 1 of [21].In the following we will establish a Kong-type interval oscillation criteria for (1.1) by the ideas of Philos [18] and Kong [22].LetD={(t,s):t0≤s≤t}, H1,H2∈C1(D,ℝ), then a pair function H1,H2 is said to belong to a function set ℋ, defined by (H1,H2)∈ℋ, if there exist h1,h2∈Lloc(D,ℝ) satisfying the following conditions:(C1)
H1(t,t)=H2(t,t)=0,H1(t,s)>0,H2(t,s)>0fort>s;(C2)
(∂/∂t)H1(t,s)=h1(t,s)H1(t,s),(∂/∂s)H2(t,s)=h2(t,s)H2(t,s).We assume that there existcj,dj,δj∉{τk,k=1,2,….}(j=1,2) such that T<c1<δ1<d1≤c2<δ2<d2 for any T≥t0. Noticing whether or not there are impulsive moments of x(t) in [cj,δj] and [δj,dj], we should consider the following four cases, namely, (S5) k(cj)<k(δj)<k(dj); (S6) k(cj)=k(δj)<k(dj); (S7) k(cj)<k(δj)=k(dj) and (S8) k(cj)=k(δj)=k(dj). Moreover, in the discussion of the impulse moments of x(t-σ), it is necessary to consider the following two cases: (S̅5) τk(δj)+σ>δj and (S̅6) τk(δj)+σ≤δj. In the following theorem, we only consider the case of combination of (S5) with (S̅5). For the other cases, similar conclusions can be given and the proofs will be omitted here.For convenience in the expression below, we define, forj=1,2,(2.55)Π1,j=:1H1(δj,cj)×{∫cjτk(cj)+1H̃1(t,cj)(t-τk(cj)-σ)α(t-τk(cj))αdt+∑i=k(cj)+1k(δj)-1[∫τiτi+σH̃1(t,cj)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1H̃1(t,cj)(t-τi-σ)α(t-τi)αdt]+∫τk(δj)δjH̃1(t,cj)(t-τk(δj))αbk(δj)α(t+σ-τk(δj))αdt+∫cjδjρ(t)p0(t)H1(t,cj)dt-1(α+1)α+1∫cjδjρ(t)r(t)H1(t,cj)|h1(t,cj)+ρ′(t)ρ(t)|α+1dt∫cjτk(cj)+1H̃1(t,cj)(t-τk(cj)-σ)α(t-τk(cj))αdt},Π2,j=:1H2(dj,δj){∫δjτk(δj)+σH̃2(dj,t)(t-τk(δj))αbk(δj)α(t+σ-τk(δj))αdt+∫τk(δj)+στk(δj)+1H̃2(dj,t)(t-τk(δj)-σ)α(t-τk(δj))αdt+∑i=k(δj)+1k(dj)-1[∫τiτi+σH̃2(dj,t)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1H̃2(dj,t)(t-τi-σ)α(t-τi)αdt]+∫τk(dj)djH̃2(dj,t)(t-τk(dj))αbk(dj)α(t+σ-τk(dj))αdt-1(α+1)α+1∫δjdjρ(t)r(t)H2(dj,t)|h2(dj,t)+ρ′(t)ρ(t)|α+1dt+∫δjdjρ(t)p0(t)H2(dj,t)dt∫δjτk(δj)+σH̃2(dj,t)(t-τk(δj))αbk(δj)α(t+σ-τk(δj))αdt},
where H̃1(t,cj)=H1(t,cj)ψ(t), H̃2(dj,t)=H2(dj,t)ψ(t) and ψ(t)=η0-η0|e(t)|η0∏i=1nηi-ηi(pi(t))ηi with η0=1-∑i=1nηi and η1,η2,…,ηn are positive constants satisfying conditions of Lemma 2.1.Theorem 2.8.
Assume  (2.7) holds. If there exists a pair of  (H1,H2)∈ℋ such that
(2.56)Π1,j+Π2,j>ρjrjH1(δj,cj)Qcjδj[H1(⋅,cj)]+ρjrjH2(dj,δj)Qδjdj[H2(dj,⋅)],j=1,2,
then (1.1) is oscillatory.Proof.
Assume, to the contrary, thatx(t) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(t)>0 and x(t-σ)>0 for t≥t0. In this case the interval of t selected for the following discussion is [c1,d1]. Similar to the proof of Theorem 2.3, we can get (2.14) and (2.19). Multiplying both sides of (2.14) by H1(t,c1) and integrating it from c1 to δ1, we have
(2.57)∫c1δ1H1(t,c1)u′(t)dt≤∫c1δ1H1(t,c1)(ρ′(t)ρ(t)u(t)-α(ρ(t)r(t))1/α|u(t)|(1+α)/α)dt-∫c1δ1H̃1(t,c1)xα(t-σ)xα(t)dt-∫c1δ1H1(t,c1)ρ(t)p0(t)dt,
where H̃1(t,c1)=H1(t,c1)ρ(t)ψ(t), ψ(t)=η0-η0|e(t)|η0∏i=1nηi-ηi(pi(t))ηi with η0=1-∑i=1nηi and η1,η2,…,ηn are positive constants satisfying conditions of Lemma 2.1.
Noticing impulsive momentsτk(c1)+1,τk(c1)+2,…,τk(δ1) are in [c1,δ1] and using the integration by parts formula on the left-hand side of above inequality, we obtain
(2.58)∫c1δ1H1(t,c1)u′(t)dt=(∫c1τk(c1)+1+∫τk(c1)+1τk(c1)+2+⋯+∫τk(δ1)δ1)H1(t,c1)du(t)=∑i=k(c1)+1k(δ1)(1-biαaiα)H1(τi,c1)u(τi)+H1(δ1,c1)u(δ1)-(∫c1τk(c1)+1+∫τk(c1)+1τk(c1)+2+⋯+∫τk(δ1)δ1)H1(t,c1)h1(t,c1)u(t)dt.
Substituting (2.58) into (2.57), we obtain
(2.59)∫c1δ1H̃1(t,c1)xα(t-σ)xα(t)dt≤∑i=k(c1)+1k(δ1)(biαaiα-1)H1(τi,c1)u(τi)-H1(δ1,c1)u(δ1)+(∫c1τk(c1)+1+∫τk(c1)+1τk(c1)+2+⋯+∫τk(δ1)δ1)H1(t,c1)×[|h1(t,c1)+ρ′(t)ρ(t)||u(t)|-α(ρ(t)r(t))1/α|u(t)|(α+1)/α]dt-∫c1δ1ρ(t)p0(t)H1(t,c1)dt.
Letting A=α/(ρ(t)r(t))1/α, B=|h1(t,c1)+ρ′(t)/ρ(t)|, y=|u(t)| and using (2.6) to the right-hand side of above inequality, we have
(2.60)∫c1δ1H̃1(t,c1)xα(t-σ)xα(t)dt≤∑i=k(c1)+1k(δ1)(biαaiα-1)H1(τi,c1)u(τi)-H1(δ1,c1)u(δ1)+1(α+1)α+1∫c1δ1ρ(t)r(t)H1(t,c1)|h1(t,c1)+ρ′(t)ρ(t)|α+1dt-∫c1δ1ρ(t)p0(t)H1(t,c1)dt.
Similar to the proof of Theorem 2.3, we need to divide the integration interval [c1,δ1] into several subintervals for estimating the function x(t-σ)/x(t). Using the methods of (2.26), (2.37), (2.38), and (2.39) we estimate the left-hand side of above inequality as follows:
(2.61)∫c1δ1H̃1(t,c1)xα(t-σ)xα(t)dt>∫c1τk(c1)+1H̃1(t,c1)(t-τk(c1)-σ)α(t-τk(c1))αdt+∑i=k(c1)+1k(δ1)-1[∫τiτi+σH̃1(t,c1)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1H̃1(t,c1)(t-τi-σ)α(t-τi)αdt]+∫τk(δ1)δ1H̃1(t,c1)(t-τk(δ1))αbk(δ1)α(t+σ-τk(δ1))αdt.
From (2.60) and (2.61), we have
(2.62)∫c1τk(c1)+1H̃1(t,c1)(t-τk(c1)-σ)α(t-τk(c1))αdt+∑i=k(c1)+1k(δ1)-1[∫τiτi+σH̃1(t,c1)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1H̃1(t,c1)(t-τi-σ)α(t-τi)αdt]+∫τk(δ1)δ1H̃1(t,c1)(t-τk(δ1))αbk(δ1)α(t+σ-τk(δ1))αdt-1(α+1)α+1∫c1δ1ρ(t)r(t)H1(t,c1)|h1(t,c1)+ρ′(t)ρ(t)|α+1dt+∫c1δ1ρ(t)p0(t)H1(t,c1)dt<∑i=k(c1)+1k(δ1)(biαaiα-1)H1(τi,c1)u(τi)-H1(δ1,c1)u(δ1).
On the other hand, multiplying both sides of (2.14) by H2(d1,t) and using similar analysis to the above, we can obtain
(2.63)∫δ1τk(δ1)+σH̃2(d1,t)(t-τk(δ1))αbk(δ1)α(t+σ-τk(δ1))α+∫τk(δ1)+στk(δ1)+1H̃2(d1,t)(t-τk(δ1)-σ)α(t-τk(δ1))αdt+∑i=k(δ1)+1k(d1)-1[∫τiτi+σH̃2(d1,t)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1H̃2(d1,t)(t-τi-σ)α(t-τi)αdt]+∫τk(d1)d1H̃2(d1,t)(t-τk(d1))αbk(d1)α(t+σ-τk(d1))αdt+∫δ1d1ρ(t)p0(t)H2(d1,t)dt-1(α+1)α+1∫δ1d1ρ(t)r(t)H2(d1,t)|h2(d1,t)+ρ′(t)ρ(t)|α+1dt<∑i=k(δ1)+1k(d1)(biαaiα-1)H2(d1,τi)u(τi)+H2(d1,δ1)u(δ1).
Dividing (2.62) and (2.63) by H1(δ1,c1), and H2(d1,δ1) respectively, and adding them, we get
(2.64)Π1,1+Π2,1<1H1(δ1,c1)∑i=k(c1)+1k(δ1)(biαaiα-1)H1(τi,c1)u(τi)+1H2(d1,δ1)∑i=k(δ1)+1k(d1)(biαaiα-1)H2(d1,τi)u(τi).
Using the same methods as (2.46), we have
(2.65)∑i=k(c1)+1k(δ1)(biαaiα-1)H1(τi,c1)u(τi)≤ρ1r1Qc1δ1[H1(⋅,c1)],∑i=k(δ1)+1k(d1)(ribiαaiα-1)H2(d1,τi)u(τi)≤ρ1r1Qδ1d1[H2(d1,⋅)].
From (2.64), (2.65), we can obtain a contradiction to the condition (2.56).
Whenx(t)<0, we choose interval [c2,d2] to study (1.1). The proof is similar and will be omitted. Therefore, we complete the proof.Remark 2.9.
Whenσ=0, that is, the delay disappears and α=1 in (1.1), our result Theorem 2.8 reduces to Theorem 2.2 of [12].


## 3. Examples

In this section, we give two examples to illustrate the effectiveness and nonemptiness of our results.Example 3.1.
Consider the following delay differential equation with impulse:(3.1)x′′(t)+μ1|x(t-π12)|3/2x(t-π12)+μ2|x(t-π12)|-1/2x(t-π12)=-sin⁡(2t),t≠τk,x(τk+)=akx(τk),x′(τk+)=bkx′(τk),t=τk,
where τk:τn,1=2nπ+5π/18,τn,2=2nπ+11π/18,n∈ℕ and μ1,μ2 are positive constants.For anyT>0, we can choose large n0 such that T<c1=2nπ+π/6,d1=2nπ+π/3,c2=2nπ+π/2,d2=2n+2π/3,n=n0,n0+1,…. There are impulsive moments τn,1 in [c1,d1] and τn,2 in [c2,d2]. From τn,2-τn,1=π/3>π/12 and τn+1,1-τn,2=5π/3>π/12 for all n>n0, we know that condition τk+1-τk>σ is satisfied. Moreover, we also see the conditions (S1) and (2.7) are satisfied.We can chooseη0=η1=η2=1/3 such that Lemma 2.1 holds. Let w1(t)=w2(t)=sin(6t) and ρ(t)=1. It is easy to verify that W1(t)=3(μ1μ2)1/3|sin(2t)|1/3sin2(6t). By a simple calculation, the left side of (2.8) is the following:(3.2)∫c1τk(c1)+1W1(t)(t-τk(c1)-σ)α(t-τk(c1))αdt+∑i=k(c1)+1k(d1)-1[∫τiτi+σW1(t)(t-τi)αbiα(t+σ-τi)αdt+∫τi+στi+1W1(t)(t-τi-σ)α(t-τi)αdt]+∫τk(d1)d1W1(t)(t-τk(d1))αbk(d1)α(t+σ-τk(d1))αdt+∫c1d1ρ(t)p0(t)|w1(t)|α+1dt-∫c1d1ρ(t)r(t)(|w1′(t)|+|ρ′(t)||w1(t)|(α+1)ρ(t))α+1dt=∫2nπ+π/62nπ+5π/18W1(t)t-2(n-1)π-11π/18-π/12t-2(n-1)π-11π/18dt+∫2nπ+5π/182nπ+π/3W1(t)t-2nπ-5π/18bn,1(t-2nπ-5π/18+π/12)dt-∫2nπ+π/62nπ+π/336cos⁡26tdt=∫π/65π/18W1(t)t+47π/36t+25π/18dt+∫5π/18π/3W1(t)t-5π/18bn,1(t-7π/36)dt-3π≈3μ1μ23(0.199+0.715bn,1)-3π.
On the other hand, we have(3.3)Qc1d1[w12]=274πbn,1-an,1an,1.
Thus if(3.4)3μ1μ23(0.199+0.715bn,1)≥3π+274πbn,1-an,1an,1,
the condition (2.8) is satisfied in [c1,d1]. Similarly, we can show that for t∈[c2,d2] the condition (2.8) is satisfied if(3.5)3μ1μ23(0.057+0.003bn,2)≥3π+274πbn,2-an,2an,2.
Hence, by Theorem 2.3, (3.1) is oscillatory, if (3.4) and (3.5) hold. Particularly, let ak=bk, for all k∈ℕ, condition (3.4) and (3.5) become(3.6)μ1μ23(0.199+0.715bn,1)≥π,μ1μ23(0.057+0.003bn,2)≥π.Example 3.2.
Consider the following equation:(3.7)x′′(t)+μ1p1(t)|x(t-23)|3/2x(t-23)+μ2p2(t)|x(t-23)|-1/2x(t-23)=e(t),t≠τk,x(τk+)=akx(τk),x′(τk+)=bkx′(τk),k=1,2,…,
where μ1,μ2 are positive constants; τk:τn,1=9n+3/2, τn,2=9n+5/2, τn,3=9n+15/2, τn,4=9n+17/2(n=0,1,2,…) and τk+1-τk>σ=2/3. In addition, let
(3.8)p1(t)=p2(t)={(t-9n)3,t∈[9n,9n+3],33,t∈[9n+3,9n+6],(9n+9-t)3,t∈[9n+6,9n+9],e(t)=(t-9n-3)3,t∈[9n,9n+9].For anyt0>0, we choose n large enough such that t0<9n, and let [c1,d1]=[9n+1,9n+3], [c2,d2]=[9n+7,9n+9], δ1=9n+2 and δ2=9n+8. It is easy to see that condition (2.7) in Theorem 2.8 is satisfied. Letting H1(t,s)=H2(t,s)=(t-s)3, we get h1(t,s)=-h2(t,s)=3/(t-s). By simple calculation, we have(3.9)Π1,1=3μ1μ23{∫9n+19n+3/2(t-9n-1)3(9n+3-t)(t-9n)2t-9n-1/6t-9n+1/2dt+∫9n+3/29n+2(t-9n-1)3(9n+3-t)(t-9n)2t-9-3/2bn,1(t-9n-5/6)dt}-98=3μ1μ23(∫13/2u2(u-1)3(3-u)(u-1/6)u+1/2du+∫3/22u2(u-1)3(3-u)(u-3/2)bn,1(u-5/6)du)-98≈3μ1μ23(0.32+0.290bn,1)-98,Π2,1=3μ1μ23{∫9n+29n+13/6(9n+3-t)4(t-9n)2t-9n-3/2bn,1(t-9n+5/6)dt+∫9n+13/69n+5/2(9n+3-t)4(t-9n)2t-9n-13/6t-9n-3/2dt+∫9n+5/29n+3(9n+3-t)4(t-9n)2t-9n-5/2bn,2(t-9n-11/6)dt}-98=3μ1μ23{∫213/6u2(3-u)4(u-3/2)bn,1(u+5/6)dt+∫13/65/2u2(3-u)4(u-13/6)u-3/2dt+∫5/23u2(3-u)4(u-5/2)bn,2(u-11/6)dt}-98≈3μ1μ23(0.102bn,1+0.056+0.005bn,2)-98.
Then the left-hand side of the inequality (2.56) is(3.10)Π1,1+Π2,1≈3μ1μ23(0.376+0.392bn,1+0.005bn,2)-94.Becauser1=r2=1, τk(c1)+1=τk(δ1)=τn,1=9n+3/2∈(c1,δ1) and τk(δ1)+1=τk(d1)=τn,2=9n+5/2∈(δ1,d1), it is easy to get that the right-hand side of the inequality (2.56) for j=1 is(3.11)r1H1(δ1,c1)Qc1δ1[H1(⋅,c1)]+r1H2(d1,δ1)Qδ1d1[H2(d1,⋅)]=bn,1-an,14an,1+bn,2-an,24an,2.
Thus (2.56) is satisfied with j=1 if(3.12)3μ1μ23(0.376+0.392bn,1+0.005bn,2)>94+bn,1-an,14an,1+bn,2-an,24an,2.Whenj=2, with the same argument as above we get that the left-hand side of inequality (2.56) is(3.13)Π1,2+Π2,2=3μ1μ23{∫715/2(u-9)2(u-3)(u-7)3(u-19/6)u-5/2du+∫15/28(u-9)2(u-3)(u-7)3(u-15/2)bn,3(u-41/6)du+∫849/6(u-3)(9-u)5(u-15/2)bn,3(u-41/6)du+∫49/617/2(u-3)(9-u)5(u-19/6)u-5/2du+∫17/29(u-3)(9-u)5(u-17/2)bn,4(u-47/6)du}-94≈3μ1μ23(0.400+0.724bn,3+0.001bn,4)-94,
and the right-hand side of the inequality (2.56) is(3.14)r2H2(δ2,c2)Qc2δ2[H1(⋅,c2)]+r2H2(d2,δ2)Qδ2d2[H2(d2,⋅)]=bn,3-an,34an,3+bn,4-an,44an,4.
Therefore, (2.51) is satisfied with j=2 if(3.15)3μ1μ23(0.400+0.724bn,3+0.001bn,4)>94+bn,3-an,34an,3+bn,4-an,44an,4.
Hence, by Theorem 2.8, (3.7) is oscillatory if(3.16)3μ1μ23(0.376+0.392bn,1+0.005bn,2)>94+bn,1-an,14an,1+bn,2-an,24an,2,3μ1μ23(0.400+0.724bn,3+0.001bn,4)>94+bn,3-an,34an,3+bn,4-an,44an,4.
Particularly, when ak=bk, for all k∈ℕ, condition (3.16) becomes(3.17)3μ1μ23(0.376+0.392bn,1+0.005bn,2)>94,3μ1μ23(0.400+0.724bn,3+0.001bn,4)>94.



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*Source: 351709-2012-06-10.xml*