# Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

Authors: Zhonghai Guo; Xiaoliang Zhou; Wu-Sheng Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We study the following second-order super-half-linear impulsive differential equations with delay[r(t)φγ(x′(t))]′+p(t)φγ(x(t-σ))+q(t)f(x(t-σ))=e(t), t≠τk, x(t+)=akx(t),x′(t+)=bkx′(t),
t=τk, where t≥t0∈ℝ, φ*(u)=|u|*-1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence with τ1<τ2<⋯<τk<⋯, limk→∞τk=∞, and τk+1-τk>σ. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.


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## Body

## 1. Introduction

We consider the following second-order super-half-linear impulsive differential equations with delay(1.1)[r(t)φγ(x′(t))]′+p(t)φγ(x(t-σ))+q(t)f(x(t-σ))=e(t),t≠τk,x(t+)=akx(t),x′(t+)=bkx′(t),t=τk,k=1,2,…,
where t≥t0∈ℝ, φ*(u)=|u|*-1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence with τ1<τ2<⋯<τk<⋯, limk→∞τk=∞, and τk+1-τk>σ.LetJ⊂ℝ be an interval, 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 nonimpulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).In the last decades, there has been an increasing interest in obtaining sufficient conditions for oscillation and/or nonoscillation of different classes impulsive differential equations with delay (constant or variable), 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, see [10–15].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 [16] gave the first research in this direction recently. They considered the second-order delay differential equations with impulses
(1.3)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 equation without delay or impulses [13, 17, 18].Motivated mainly by [16], in this paper, we study the interval oscillation of the delay impulsive equation (1.1). By using classical inequalities, Riccati transformation, and two classes of functions (introduced first by Philos [19]), we establish some interval oscillation criteria which generalize and improve some known results of [13, 16–18]. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.


## 2. Main Results

Throughout the paper, we always assume that the following conditions hold:(A1)
r(t)∈C([t0,∞),(0,∞)) and is nondecreasing, p(t),q(t),e(t)∈PLC([t0,∞),ℝ);(A2)
γ is a quotient of odd positive integers ak, bk are real constants satisfying bk≥ak>0, k=1,2,…;(A3)
f∈C(ℝ,ℝ),  xf(x)>0 and there exist some positive constants η and α such that f(x)/φα(x)>η for all x≠0 with α≥γ.We introduce the following notations at intervals[c1,d1] and [c2,d2](2.1)k(s):=max⁡{i:t0<τi<s},Mj:=max⁡{r(t):t∈[cj,dj]},j=1,2,Ωj(cj,dj):={w∈C1[cj,dj]∣w(t)≢0,w(cj)=w(dj)=0},j=1,2.For two constantsc,d∉{τk} with c<d, 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)+1γ-ak(c)+1γak(c)+1γ(τk(c)+1-c)γ+∑i=k(c)+2k(d)‍φ(τi)biγ-aiγaiγ(τi-τi-1)γ,
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 for k(cj)=k(dj);(S-1)
τk(cj)+σ<cj;   (S-2) cj<τk(cj)+σ<dj;   (S-3) τk(cj)+σ>dj.Combining (S*) with (S-*), 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 (S-1) 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 [20] and the second from [21].Lemma 2.1.
Letλ and δ be positive real numbers with λ>δ. Then
(2.3)AXλ-δ+BX-δ≥λδ-δ/λ(λ-δ)(δ/λ)-1Aδ/λB1-δ/λ,
for all A,B≥0 and X>0.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.Letγ>0, B≥0, A>0 and y≥0. Put
(2.5)λ=1+1γ,X=Aγ/(γ+1)y,Y=(γγ+1)γBγA-γ2/(γ+1).
It follows from Lemma 2.2 that
(2.6)By-Ay(γ+1)/γ≤γγ(γ+1)γ+1Bγ+1Aγ.Lemma 2.3.
Assume that for anyT≥t0, there exists cj,dj∉{τk}, j=1,2, such that T<c1<d1≤c2<d2 and
(2.7)p(t),q(t)≥0,t∈[c1-σ,d1]∪[c2-σ,d2]∖{τk},e(t)≤0,t∈[c1-σ,d1]∖{τk},e(t)≥0,t∈[c2-σ,d2]∖{τk}.
If x(t) is a nonoscillatory solution of (1.1), then there exist the following estimations of x(t-σ)/x(t):
(2.8)(a)fort∈(τi+σ,τi+1],x(t-σ)x(t)>t-τi-σt-τi,(b)fort∈(τi,τi+σ),x(t-σ)x(t)>t-τibi(t+σ-τi),(c)fort∈[cj,τk(cj)+1],x(t-σ)x(t)>t-τk(cj)-σt-τk(cj),(d)fort∈(τk(dj),dj],x(t-σ)x(t)>t-τk(dj)bk(dj)(t+σ-τk(dj)),
where i=k(cj),…,k(dj)-1, j=1,2.Proof.
Without loss of generality, we assume thatx(t)>0 and x(t-σ)>0 for t≥t0. In this case the selected interval of t is [c1,d1]. From (1.1) and (2.7), we obtain
(2.9)[r(t)φγ(x′(t))]′=e(t)-p(t)φγ(x(t-σ))-q(t)f(x(t-σ))≤0.
Hence r(t)φγ(x′(t)) is nonincreasing on the interval [c1,d1]∖{τk}.
Case (a) (if τi+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
(2.10)x(s)-x(τi+)=x′(ξ1)(s-τi),ξ1∈(τi,s).
Since x(τi+)>0, the function φγ(·) is an increasing function and r(s)φγ(x′(s)) is non-increasing on (τi,τi+1), we have
(2.11)φγ(x(s))>φγ(x′(ξ1)(s-τi))=r(ξ1)φγ(x′(ξ1))r(ξ1)(s-τi)γ≥r(s)φγ(x′(s))r(ξ1)(s-τi)γ,ξ1∈(τi,s).
From (2.11) and the conditions r(s)>0 and r(s) is nondecreasing, we have
(2.12)φγ(x(s))>φγ(x′(s))(s-τi)γ=φγ(x′(s)(s-τi)),ξ1∈(τi,s).
Thus
(2.13)x′(s)x(s)<1s-τi.
Integrating both sides of above inequality from t-σ to t, we obtain
(2.14)x(t-σ)x(t)>t-τi-σt-τi,t∈(τi+σ,τi+1].
Case (b) (if τi<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
(2.15)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), φγ(·) and r(t)φγ(x′(t)), we get
(2.16)φγ(x(t)-aix(τi))=r(ξ2)φγ(x′(ξ2))r(ξ2)(t-τi)γ≤r(τi+)φγ(x′(τi+))r(ξ2)(t-τi)γ=r(τi)φγ(bix′(τi)(t-τi))r(ξ2).
Since x(τi)>0, we have
(2.17)φγ(x(t)x(τi)-ai)≤r(τi)r(ξ2)φγ(bix′(τi)x(τi)(t-τi)).
In addition,
(2.18)x(τi)>x(τi)-x(τi-σ)=x′(ξ3)σ,ξ3∈(τi-σ,τi).
Similar to the analysis of (2.11)–(2.14), we have
(2.19)x′(τi)x(τi)<1σ.
From (2.17) and (2.19) and note that the monotone properties of φγ(·) and r(t), we get
(2.20)x(t)x(τi)<ai+biσ(t-τi).
In view of (A3), we have
(2.21)x(τi)x(t)>σσai+bi(t-τi)≥σbi(t+σ-τi)>0.
On the other hand, using similar analysis of (2.11)–(2.19), we get
(2.22)x′(s)x(s)<1s-τi+σ,s∈(τi-σ,τi).
Integrating (2.22) from t-σ to τi, where t∈(τi,τi+σ), we have
(2.23)x(t-σ)x(τi)>t-τiσ≥0.
From (2.21) and (2.23), we obtain
(2.24)x(t-σ)x(t)>t-τibi(t+σ-τi),t∈(τi,τi+σ).
Case (c) (t∈[c1,τk(c1)+1]). Since τk(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 (2.14) of Case (a), we have
(2.25)x(t-σ)x(t)>t-τk(c1)-σt-τk(c1),t∈[c1,τk(c1)+1].
Case (d) (t∈(τk(d1),d1]). Since τk(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 Case (b), we obtain
(2.26)x(t-σ)x(t)>t-τk(d1)bk(d1)(t+σ-τk(d1))≥0,t∈(τk(d1),d1].
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.Theorem 2.4.
Assume that for anyT≥t0, there exists cj,dj∉{τk}, j=1,2, such that T<c1<d1≤c2<d2 and (2.7) holds. If there exists wj(t)∈Ωj(cj,dj) (j=1,2) such that, for k(cj)<k(dj),
(2.27)∫cjτk(cj)+1‍Wj(t)(t-τk(cj)-σ)γ(t-τk(cj))γdt+∑i=k(cj)+1k(dj)-1[∫τiτi+σ‍Wj(t)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍Wj(t)(t-τi-σ)γ(t-τi)γdt]+∫τk(dj)dj‍Wj(t)(t-τk(dj))γbk(dj)γ(t+σ-τk(dj))γdt-∫cjdj‍(r(t)|wj′(t)|γ+1)dt≥MjQcjdj[wjγ+1],
and for k(cj)=k(dj),
(2.28)∫cjdj‍(Wj(t)(t-ci)γ(t-ci+σ)γ-r(t)|wj′(t)|γ+1)dt≥0,
where Wj(t)=(p(t)+αγ-γ/α(α-γ)γ/α-1ηγ/αqγ/α(t)|e(t)|1-γ/α)wjγ+1(t), 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].
We define(2.29)u(t)=r(t)φγ(x′(t))xγ(t),fort∈[c1,d1].
Differentiatingu(t) and in view of (1.1) we obtain, for  t≠tk,
(2.30)u′(t)=-p(t)xγ(t-σ)xγ(t)-q(t)f(x(t-σ))xγ(t)-|e(t)|xγ(t)-γr1/γ(t)|u(t)|(1+γ)/γ≤-{p(t)+ηq(t)xα-γ(t-σ)+|e(t)|x-γ(t-σ)}xγ(t-σ)xγ(t)-γr1/γ(t)|u(t)|(1+γ)/γ.
Putting A=ηq(t), B=|e(t)|, X=x(t-σ), λ=α, and δ=γ, by Lemma 2.1, we see that
(2.31)u′(t)≤-(p(t)+αγ-γ/α(α-γ)(γ/α)-1(ηq(t))γ/α|e(t)|1-(γ/α))xγ(t-σ)xγ(t)-γr1/γ(t)|u(t)|(1+γ)/γ=-xγ(t-σ)xγ(t)R(t)-γr1/γ(t)|u(t)|(1+γ)/γ,
where R(t)=p(t)+αγ-γ/α(α-γ)γ/α-1(ηq(t))γ/α(t)|e(t)|1-γ/α.
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 a w1(t)∈Ω1(c1,d1), multiplying both sides of (2.31) by w1γ+1(t) and then integrating it from c1 to d1, we obtain
(2.32)∫c1τk(c1)+1‍u′(t)w1γ+1(t)dt+∫τk(c1)+1τk(c1)+2‍u′(t)w1γ+1(t)dt+⋯+∫τk(d1)d1‍u′(t)w1γ+1(t)dt≤-∫c1τk(c1)+1‍γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)dt-∫τk(c1)+1τk(c1)+2‍γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)dt-⋯-∫τk(d1)d1‍γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)dt-∫c1τk(c1)+1‍xγ(t-σ)xγ(t)W1(t)dt-∫τk(c1)+1τk(c1)+1+σ‍xγ(t-σ)xγ(t)W1(t)dt-∫τk(c1)+1+στk(c1)+2‍xγ(t-σ)xγ(t)W1(t)dt-⋯-∫τk(d1)-1+στk(d1)‍xγ(t-σ)xγ(t)W1(t)dt-∫τk(d1)d1‍xγ(t-σ)xγ(t)W1(t)dt,
where W1(t)=R(t)w1γ+1(t). Using the integration by parts on the left side of above inequality and noting the condition w1(c1)=w1(d1)=0, we obtain
(2.33)∑i=k(c1)+1k(d1)‍w1γ+1(τi)[u(τi)-u(τi+)]≤∫c1τk(c1)+1‍[(γ+1)|w1γ(t)w1′(t)||u(t)|-γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)]dt+∑i=k(c1)+1k(d1)-1‍∫τiτi+1‍[(γ+1)|w1γ(t)w1′(t)||u(t)|-γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)]dt+∫τk(d1)d1‍[(γ+1)|w1γ(t)w1′(t)||u(t)|-γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)]dt-∫c1τk(c1)+1‍xγ(t-σ)xγ(t)W1(t)dt-∑i=k(c1)+1k(d1)-1‍[∫τiτi+σ‍xγ(t-σ)xγ(t)W1(t)dt+∫τi+στi+1‍xγ(t-σ)xγ(t)W1(t)dt]-∫τk(d1)d1‍xγ(t-σ)xγ(t)W1(t)dt.
Letting y=|w1γ(t)||u(t)|, B=(γ+1)|w1′(t)|, A=γ/r1/γ(t) and using (2.6), for the integrand function in above inequality we have that
(2.34)(γ+1)|w1γ(t)w1′(t)||u(t)|-γr1/γ(t)|u(t)|(1+γ)/γw1γ+1(t)≤r(t)|w1′(t)|γ+1.
Meanwhile, for t=τk, k=1,2,…, we have
(2.35)u(τk+)=(bkak)γu(τk).
Hence
(2.36)∑i=k(c1)+1k(d1)‍w1γ+1(τi)[u(τi)-u(τi+)]=∑i=k(c1)+1k(d1)‍aiγ-biγaiγw1γ+1(τi)u(τi).
Therefore, we get
(2.37)∑i=k(c1)+1k(d1)‍aiγ-biγaiγw1γ+1(τi)u(τi)≤∫c1d1‍r(t)|w1′(t)|γ+1dt-∫c1τk(c1)+1‍xγ(t-σ)xγ(t)W1(t)dt-∑i=k(c1)+1k(d1)-1‍[∫τiτi+σ‍xγ(t-σ)xγ(t)W1(t)dt+∫τi+στi+1‍xγ(t-σ)xγ(t)W1(t)dt]-∫τk(d1)d1‍xγ(t-σ)xγ(t)W1(t)dt.
On the other hand, for t∈(τi-1,τi]⊂[c1,d1], i=k(c1)+2,…,k(d1), we have
(2.38)x(t)-x(τi-1)=x′(ξ)(t-τi-1),ξ∈(τi-1,t).
In view of x(τi-1)>0, and note that the monotone properties of φγ(·), r(t)φγ(x′(t)), and r(t), we obtain
(2.39)φγ(x(t))>φγ(x′(ξ))φγ(t-τi-1)≥r(t)r(ξ)φγ(x′(t))φγ(t-τi-1).
This is
(2.40)r(t)φγ(x′(t))φγ(x(t))<r(ξ)(t-τi-1)γ.
Let t→τi-, it follows
(2.41)u(τi)=r(τi)φγ(x′(τi))φγ(x(τi))<M1(τi-τi-1)γ,i=k(c1)+2,…,k(d1).
Similar analysis on (c1,τk(c1)+1], we can get
(2.42)u(τk(c1)+1)<M1(τk(c1)+1-c1)γ.
Then from (2.41), (2.42), and (A2), we have
(2.43)∑i=k(c1)+1k(d1)‍biγ-aiγaiγw1γ+1(τi)u(τi)<M1[w1γ+1(τk(c1)+1)θ(c1)+∑i=k(c1)+2k(d1)‍w1γ+1(τi)ζ(τi)]=M1Qc1d1[w1γ+1],
where θ(c1)=(bk(c1)+1γ-ak(c1)+1γ)/(ak(c1)+1γ(τk(c1)+1-c1)γ) and ζ(τi)=(biγ-aiγ)/(aiγ(τi-τi-1)γ). From (2.37) and (2.43) and applying Lemma 2.3, we obtain
(2.44)∫c1τk(c1)+1‍W1(t)(t-τk(c1)-σ)γ(t-τk(c1))γdt+∑i=k(c1)+1k(d1)-1‍[∫τiτi+σ‍W1(t)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍W1(t)(t-τi-σ)γ(t-τi)γdt]+∫τk(d1)d1‍W1(t)(t-τk(d1))γbk(d1)γ(t+σ-τk(d1))γdt-∫c1d1‍r(t)|w1′(t)|γ+1dt<M1Qc1d1[w1γ+1].
This contradicts (2.27).
Next we consider the casek(c1)=k(d1). By the condition (S-1) we know there is no impulsive moment in [c1,d1]. Multiplying both sides of (2.31) by w1γ+1(t) and integrating it from c1 to d1, we obtain
(2.45)∫c1d1‍u′(t)w1γ+1(t)dt≤-∫c1d1‍γr1/γ(t)|u(t)|(γ+1)/γw1γ+1(t)dt-∫c1d1‍xγ(t-σ)xγ(t)W1(t)dt.
Using the integration by parts on the left-hand side and noting the condition w1(c1)=w1(d1)=0, we obtain
(2.46)∫c1d1‍[(γ+1)w1γ(t)w1′(t)u(t)-γr1/γ(t)w1γ+1(t)|u(t)|(γ+1)/γ]dt-∫c1d1‍xγ(t-σ)xγ(t)W1(t)dt≥0.
It follows that
(2.47)∫c1d1‍[(γ+1)|w1γ(t)w1′(t)||u(t)|-γr1/γ(t)w1γ+1(t)|u(t)|(γ+1)/γ]dt-∫c1d1‍xγ(t-σ)xγ(t)W1(t)dt≥0.
Letting A=(γ/r1/γ(t))w1γ+1(t), B=(γ+1)|w1γ(t)w1′(t)|, and y=|u(t)| and applying the inequality (2.6), we get
(2.48)∫c1d1‍[r(t)|w1′(t)|γ+1-xγ(t-σ)xγ(t)W1(t)]dt≥0.
Using same way as Case (a) we get
(2.49)x(t-σ)x(t)>t-c1t-c1+σ,t∈[c1,d1].
From (2.48) and (2.49) we obtain
(2.50)∫c1d1‍(W1(t)(t-c1)γ(t-c1+σ)γ-r(t)|w1′(t)|γ+1)dt<0.
This contradicts our assumption (2.28).
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 2.5.
Whenγ=1, p(t)=0, f(x)=|x|α-1x and the delay term σ=0, (1.1) reduces to that studied by Liu and Xu [13]. Therefore our Theorem 2.4 generalizes Theorem 2.1 of [13].Remark 2.6.
Whenγ=α=1, r(t)=1 and p(t)=0, (1.1) reduces to the (1.3) studied by Huang and Feng [16]. Therefore our Theorem 2.4 extends Theorem 2.1 of [16].Remark 2.7.
Whenak=bk=1, for all k=1,2,..., the impulses in (1.1) disappear, Theorem 2.4 reduces to the main results of [17, 18].In the following we will establish a Kemenev type interval oscillation criteria for (1.1) by the ideas of Philos [19] 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 exists h1,h2∈Lloc(D,ℝ) satisfying the following conditions:(A4)
H1(t,t)=H2(t,t)=0,  H1(t,s)>0, H2(t,s)>0 for t>s;(A5)
(∂/∂t)H1(t,s)=h1(t,s)H1(t,s),    (∂/∂s)H2(t,s)=h2(t,s)H2(t,s).We assume there existcj,dj,δj∉{τk,k=1,2,…} (j=1,2) which satisfy 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,  (S1)
k(cj)<k(δj)<k(dj);   (S2) k(cj)=k(δj)<k(dj);(S3)
k(cj)<k(δj)=k(dj);   (S4) k(cj)=k(δj)=k(dj).Moreover, in the discussion of the impulse moments ofx(t-σ), it is necessary to consider the following two cases:  (S-1)
τk(δj)+σ>δj;    (S-2) τk(δj)+σ≤δj.In the following theorem, we only consider the case of combination of (S1) with (S-1). For the other cases, similar conclusions can be given and their proofs will be omitted here.For convenience in the expressing blow, we define(2.51)Π1,j=:1H1(δj,cj){∫cjτk(cj)+1‍H~1(t,cj)(t-τk(cj)-σ)γ(t-τk(cj))γdt+∑i=k(cj)+1k(δj)-1‍[∫τiτi+σ‍H~1(t,cj)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~1(t,cj)(t-τi-σ)γ(t-τi)γdt]+∫τk(δj)δj‍H~1(t,cj)(t-τk(δj))γbk(δj)γ(t+σ-τk(δj))γdt-1(γ+1)γ+1∫cjδj‍r(t)Hj(t,cj)|hj(t,cj)|γ+1dt{(t-τk(cj)-σ)γ(t-τk(cj))γ}},Π2,j=:1H2(dj,δj){∫δjτk(δj)+σ‍H~2(dj,t)(t-τk(δj))γbk(δj)γ(t+σ-τk(δj))γdt+∫τk(δj)+στk(δj)+1‍H~2(dj,t)(t-τk(δj)-σ)γ(t-τk(δj))γdt+∑i=k(δj)+1k(dj)-1‍[∫τiτi+σ‍H~2(dj,t)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~2(dj,t)(t-τi-σ)γ(t-τi)γdt]+∫τk(dj)dj‍H~2(dj,t)(t-τk(dj))γbk(dj)γ(t+σ-τk(dj))γdt-1(γ+1)γ+1∫δjdj‍r(t)H2(dj,t)|h2(dj,t)|γ+1dt},
where H~1(t,cj)=H1(t,cj)R(t), H~2(dj,t)=H2(dj,t)R(t)  (j=1,2) and R(t)=p(t)+αγ-γ/α(α-γ)γ/α-1ηγ/αqγ/α(t)|e(t)|1-γ/α.Theorem 2.8.
Assume that for anyT≥t0, there exists cj,dj∉{τk}, j=1,2, such that T<c1<d1≤c2<d2 and if there exists a pair of (H1,H2)∈ℋ such that
(2.52)Π1,j+Π2,j>MjH1(δj,cj)Qcjδj[H1(⋅,cj)]+MjH2(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]. Using the same proof as in Theorem 2.4, we can get (2.31). Multiplying both sides of (2.31) by H1(t,c1) and integrating it from c1 to δ1, we have
(2.53)∫c1δ1H1(t,c1)u′(t)dt≤-∫c1δ1H1(t,c1)γr1/γ(t)|u(t)|(1+γ)/γdt-∫c1δ1H~1(t,c1)xγ(t-σ)xγ(t)dt,
where H~1(t,c1)=H1(t,c1)(p(t)+αγ-γ/α(α-γ)γ/α-1ηγ/αqγ/α(t)|e(t)|1-γ/α). Noticing impulsive moments τk(c1)+1,τk(c1)+2,…,τk(δ1) are in [c1,δ1] and using the integration by parts on the left-hand side of above inequality, we obtain
(2.54)∫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)‍aiγ-biγaiγH1(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 (2.54) into (2.53), we have
(2.55)∫c1δ1H~1(t,c1)xγ(t-σ)xγ(t)dt≤∑i=k(c1)+1k(δ1)‍biγ-aiγaiγH1(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)|-γr1/γ(t)|u(t)|(γ+1)/γ]dt.
Letting A=γ/r1/γ(t), B=|h1(t,c1)|, y=|u(t)| and using (2.6) to the right side of above inequality, we have
(2.56)∫c1δ1‍H~1(t,c1)xγ(t-σ)xγ(t)dt≤∑i=k(c1)+1k(δ1)‍biγ-aiγaiγH1(τi,c1)u(τi)-H1(δ1,c1)u(δ1)+1(γ+1)γ+1∫c1δ1‍r(t)H1(t,c1)|h1(t,c1)|γ+1dt.
Because there are different integration intervals in (2.56), we need to divide the integration interval [c1,δ1] into several subintervals for estimating the function x(t-σ)/x(t). Using Lemma 2.2, (2.6), we get estimation for the left-hand side of above inequality as follows,
(2.57)∫c1δ1‍H~1(t,c1)xγ(t-σ)xγ(t)dt>∫c1τk(c1)+1‍H~1(t,c1)(t-τk(c1)-σ)γ(t-τk(c1))γdt+∑i=k(c1)+1k(δ1)-1‍[∫τiτi+σ‍H~1(t,c1)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~1(t,c1)(t-τi-σ)γ(t-τi)γdt]+∫τk(δ1)δ1‍H~1(t,c1)(t-τk(δ1))γbk(δ1)γ(t+σ-τk(δ1))γdt.
From (2.56) and (2.57), we have
(2.58)∫c1τk(c1)+1‍H~1(t,c1)(t-τk(c1)-σ)γ(t-τk(c1))γdt+∑i=k(c1)+1k(δ1)-1‍[∫τiτi+σ‍H~1(t,c1)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~1(t,c1)(t-τi-σ)γ(t-τi)γdt]+∫τk(δ1)δ1‍H~1(t,c1)(t-τk(δ1))γbk(δ1)γ(t+σ-τk(δ1))γdt-1(γ+1)γ+1∫c1δ1‍r(t)H1(t,c1)|h1(t,c1)|γ+1dt<∑i=k(c1)+1k(δ1)‍biγ-aiγaiγH1(τi,c1)u(τi)-H1(δ1,c1)u(δ1).
Multiplying both sides of (2.31) by H2(d1,t) and using similar analysis to the above, we can obtain
(2.59)∫δ1τk(δ1)+σ‍H~2(d1,t)(t-τk(δ1))γbk(δ1)γ(t+σ-τk(δ1))γ+∫τk(δ1)+στk(δ1)+1‍H~2(d1,t)(t-τk(δ1)-σ)γ(t-τk(δ1))γdt+∑i=k(δ1)+1k(d1)-1‍[∫τiτi+σ‍H~2(d1,t)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~2(d1,t)(t-τi-σ)γ(t-τi)γdt]+∫τk(d1)d1‍H~2(d1,t)(t-τk(d1))γbk(d1)γ(t+σ-τk(d1))γdt-1(γ+1)γ+1∫δ1d1‍r(t)H2(d1,t)|h2(d1,t)|γ+1dt<∑i=k(δ1)+1k(d1)‍biγ-aiγaiγH2(d1,τi)u(τi)+H2(d1,δ1)u(δ1).
Dividing (2.58) and (2.59) by H1(δ1,c1) and H2(d1,δ1), respectively, and adding them, we get
(2.60)Π1,1+Π2,1<1H1(δ1,c1)∑i=k(c1)+1k(δ1)‍biγ-aiγaiγH1(τi,c1)u(τi)+1H2(d1,δ1)∑i=k(δ1)+1k(d1)‍biγ-aiγaiγH2(d1,τi)u(τi).
On the other hand, similar to (2.43), we have
(2.61)∑i=k(c1)+1k(δ1)‍biγ-aiγaiγH1(τi,c1)u(τi)≤M1Qc1δ1[H1(⋅,c1)],∑i=k(δ1)+1k(d1)‍biγ-aiγaiγH2(d1,τi)u(τi)≤M1Qδ1d1[H2(d1,⋅)].
From (2.60) and (2.61), we can obtain a contradiction to the condition (2.52).
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 2.9.
LetH1(t,s)=H2(t,s)=H(t,s), h1(t,s)=2h1*(t,s)/H(t,s) and h2(t,s)=-2h2*(t,s)/H(t,s), the conditions (A4), (A5) can be changed into  (A6)
H(t,t)=0,  H(t,s)>0,  for  t>s;(A7)
(∂/∂t)H(t,s)=2h1*(t,s)H(t,s),    (∂/∂s)H(t,s)=-2h2*(t,s)H(t,s).We know that (A6) and (A7) are the main assumptions used in [13, 16] to obtain Kemenev type oscillation criteria. Therefore, Theorem 2.8 is a generalization of Theorem 2.3 in [13] and Theorem 2.5 in [16].


## 3. Examples

In this section, we give two examples to illustrate the effectiveness and nonemptiness of our results.Example 3.1.
Consider the following equation(3.1)[φγ(x′(t))]′+ν1sin⁡tφγ(x(t-π12))+ν22cos⁡tφα(x(t-π12))=-cos⁡2t,t≠τk,i,x(t+)=akx(t),x′(t+)=bkx′(t),t=τk,i,
where τk,i=2kπ+(i-1)(π/2)+(-1)(i-1)(π/8),  i=1,2,  k=1,2,..., t≥0, γ=3, α=6, ν1 and ν2 are positive constants.
For anyT>0, we can choose large n0∈ℕ such that
(3.2)T<c1=2nπ+π12,d1=2nπ+π4,c2=2nπ+π3,d2=2nπ+π2,n=n0,n0+1,….
There are impulsive moments τn,1=2nπ+π/8 in [c1,d1] and τn,2=2nπ+3π/8 in [c2,d2]. From τn,2-τn,1=π/4>π/12 and τn+1,1-τn,2=13π/8>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.
Letw(t)=sin12t. It is easy to get that W1(t)=(ν1sint+2ν2costcos2t)sin412t. In view of ∑i=k(c1)+1k(d1)-1‍=0 as k(c1)+1>k(d1)-1, by a simple calculation, the left side of (2.27) is the following
(3.3)∫c1τk(c1)+1‍W1(t)(t-τk(c1)-σ)γ(t-τk(c1))γdt+∑i=k(c1)+1k(d1)-1‍[∫τiτi+σ‍W1(t)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍W1(t)(t-τi-σ)γ(t-τi)γdt]+∫τk(d1)d1‍W1(t)(t-τk(d1))γbk(d1)γ(t+σ-τk(d1))γdt-∫c1d1‍r(t)|w1′(t)|γ+1dt=∫2nπ+π/122nπ+π/8‍W1(t)(t-2(n-1)π-3π/8-π/12t-2(n-1)π-3π/8)3dt+∫2nπ+π/82nπ+π/4‍W1(t)(t-2nπ-π/8bn,1(t+π/12-2nπ-π/8))3dt-∫2nπ+π/122nπ+π/4‍124cos⁡412tdt=∫π/12π/8‍W1(t)(t+37π/24t+13π/8)3dt+∫π/8π/4‍W1(t)(t-π/8bn,1(t-π/24))3dt-124∫π/12π/4‍cos⁡412tdt≈0.671ν1+1.326ν2+bn,1-3(0.008ν1+0.002ν2)-64π.
On the other hand, we have
(3.4)Qc1d1[w4]=(24π)3bn,13-an,13an,13.
Thus condition (2.27) is satisfied for t∈[c1,d1] if
(3.5)0.671ν1+1.326ν2+bn,1-3(0.008ν1+0.002ν2)-64π≥(24π)3bn,13-an,13an,13.
Similarly, for t∈[c2,d2] we can get the following condition
(3.6)0.012ν1+0.003ν2+bn,2-3(0.012ν1+0.001ν2)-64π≥(24π)3bn,23-an,23an,23
which ensures (2.27). Hence, by Theorem 2.4, if (3.5) and (3.6) hold, (3.1) is oscillatory. It is easy to see that (3.5) and (3.6) may be satisfied when ν1 or ν2 is large enough. Particularly, let an,i=bn,i, for i=1,2 and n=1,2,…, condition (3.5) and (3.6) become a simple form
(3.7)0.671ν1+1.326ν2+bn,1-3(0.008ν1+0.002ν2)≥64π,0.012ν1+0.003ν2+bn,2-3(0.012ν1+0.001ν2)≥64π.Example 3.2.
Consider the following equation(3.8)x′′(t)+μ1p(t)x(t-23)+μ2q(t)x2(t-23)=e(t),t≠τk,x(t+)=akx(t),x′(t+)=bkx′(t),t=τk,
where τk: τn,1=8n+1/2, τn,2=8n+3/2, τn,3=8n+9/2, τn,4=8n+(11/2) (n=0,1,2,…), γ=1, α=2, σ=2/3, bk≥ak>0, μ1,μ2 (>0) is a constant. Clearly, τk+1-τk>σ. In addition, let
(3.9)p(t)=q(t)={(t-8n)3,t∈[8n,8n+2],(8n+4-t)3,t∈[8n+2,8n+4],(t-8n-4)3,t∈[8n+4,8n+6],(8n+8-t)3,t∈[8n+6,8n+8],e(t)={(t-8n-2)3,t∈[8n,8n+4],(8n+6-t)3,t∈[8n+4,8n+8].
For any t0>0, we choose n large enough such that t0<8n and let [c1,d1]=[8n,8n+2], [c2,d2]=[8n+4,8n+6], δ1=8n+1 and δ2=8n+5. Then p(t), q(t), and e(t) on [c1,d1] and [c2,d2] satisfy (2.7). Let H1(t,s)=H2(t,s)=(t-s)3, then h1(t,s)=-h2(t,s)=3/(t-s). By simple calculation, we get
(3.10)Π1,1=:1H1(δ1,c1){∫c1τk(c1)+1‍H~1(t,c1)(t-τk(c1)-σ)γ(t-τk(c1))γdt+∑i=k(c1)+1k(δ1)-1‍[∫τiτi+σ‍H~1(t,c1)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~1(t,c1)(t-τi-σ)γ(t-τi)γdt]-γγ(γ+1)γ+1∫c1δ1‍r(t)H1(t,c1)|h1(t,c1)|γ+1dt}+∫τk(δ1)δ1‍H~1(t,c1)(t-τk(δ1))γbk(δ1)γ(t+σ-τk(δ1))γdt=∫8n8n+1/2‍(t-8n)3[μ1(t-8n)3+2μ21/2(t-8n)3/2(8n+2-t)1/2]t-8n+11/6t-8n+5/2dt+∫8n+1/28n+1‍(t-8n)3[μ1(t-8n)3+2μ21/2(t-8n)3/2(8n+2-t)1/2](t-8n-1/2)bn,1(t-8n+1/6)dt-98=μ1∫01/2‍u6(u+11/6)u+5/2du+2μ21/2∫01/2‍u9/2(2-u)1/2(u+11/6)u+5/2du+μ1bn,1∫1/21‍u6(u-1/2)u+1/6du+2μ21/2bn,1∫1/21‍u9/2(2-u)1/2(u-1/2)u+1/6du-98≈(0.0008+0.0005bn,1)μ1+(0.0081+0.1268bn,1)μ21/2-98,Π2,1=:1H2(d1,δ1){∫δ1τk(δ1)+σ‍H~2(d1,t)(t-τk(δ1))γbk(δ1)γ(t+σ-τk(δ1))γ+∫τk(δ1)+στk(δ1)+1‍H~2(d1,t)(t-τk(δ1)-σ)γ(t-τk(δ1))γdt+∑i=k(δ1)+1k(d1)-1‍[∫τiτi+σ‍H~2(d1,t)(t-τi)γbiγ(t+σ-τi)γdt+∫τi+στi+1‍H~2(d1,t)(t-τi-σ)γ(t-τi)γdt]-γγ(γ+1)γ+1∫δ1d1‍r(t)H2(d1,t)|h2(d1,t)|γ+1dt}+∫τk(d1)d1‍H~2(d1,t)(t-τk(d1))γbk(d1)γ(t+σ-τk(d1))γdt=∫8n+18n+7/6‍(8n+2-t)3[μ1(t-8n)3+2μ21/2(t-8n)3/2(8n+2-t)1/2]t-8n-1/2bn,1(t-8n+1/6)dt+∫8n+7/68n+3/2‍(8n+2-t)3[μ1(t-8n)3+2μ21/2(t-8n)3/2(8n+2-t)1/2]t-8n-7/6t-8n-1/2dt+∫8n+3/28n+2‍(8n+2-t)3[μ1(t-8n)3+2μ21/2(t-8n)3/2(8n+2-t)1/2]t-8n-3/2bn,2(t-8n-5/6)dt-98=μ1bn,1∫17/6‍u3(2-u)3(u-1/2)u+1/6du+2μ21/2bn,1∫17/6‍u3/2(2-u)7/2(u-1/2)u+1/6du+μ1∫7/63/2‍u3(2-u)3(u-7/6)u-1/2du+2μ21/2∫7/63/2‍u3/2(2-u)7/2(u-7/6)u-1/2du+μ1bn,2∫3/22‍u3(2-u)3(u-3/2)u-5/6du+2μ21/2bn,2∫3/22‍u3/2(2-u)7/2(u-3/2)u-5/6du-98≈(0.0389+0.0754bn,1+0.0087bn,2)μ1+(0.0394+0.1281bn,1+0.0046bn,2)μ21/2-98.
Then the left side of the inequality (2.52) is
(3.11)Π1,1+Π2,1≈(0.0497+0.0759bn,1+0.0087bn,2)μ1+(0.0475+0.2541bn,1+0.0046bn,2)μ21/2-94.
BecauseM1=M2=1, τk(c1)+1=τk(δ1)=τn,1=8n+1/2∈(c1,δ1) and τk(δ1)+1=τk(d1)=τn,2=8n+3/2∈(δ1,d1), it is easy to get that the right side of the inequality (2.52) for j=1 is
(3.12)M1H1(δ1,c1)Qc1δ1[H1(⋅,c1)]+M1H2(d1,δ1)Qδ1d1[H2(d1,⋅)]=bn,1-an,14an,12+29(bn,2-an,2)8an,2.
Thus (2.52) is satisfied for j=1 if
(3.13)(0.0497+0.0759bn,1+0.0087bn,2)μ1+(0.0475+0.2541bn,1+0.0046bn,2)μ21/2>94+bn,1-an,14an,12+29(bn,2-an,2)8an,2.
When j=2, with the similar argument above we get that the left side of inequality (2.52) is
(3.14)Π1,2+Π2,2=μ1∫01/2‍u6(u+11/6)u+5/2du+2μ21/2∫01/2‍u9/2(2-u)1/2(u+11/6)u+5/2du+μ1bn,3∫1/21‍u6(u-1/2)u+1/6du+2μ21/2bn,3∫1/21‍u9/2(2-u)1/2(u-1/2)u+1/6du+μ1bn,4∫17/6‍u3(2-u)3(u-1/2)u+1/6du+2μ21/2bn,4∫17/6‍u3/2(2-u)7/2(u-1/2)u+1/6du+μ1∫7/63/2‍u3(2-u)3(u-7/6)u-1/2du+2μ21/2∫7/63/2‍u3/2(2-u)7/2(u-7/6)u-1/2du+μ1bn,4∫3/22‍u3(2-u)3(u-3/2)u-5/6du+2μ21/2bn,4∫3/22‍u3/2(2-u)7/2(u-3/2)u-5/6du-94≈(0.0497+0.0759bn,3+0.0087bn,4)μ1+(0.0475+0.2541bn,3+0.0046bn,4)μ21/2-94
and the right side of the inequality (2.52) is
(3.15)M2H2(δ2,c2)Qc2δ2[H1(⋅,c2)]+M2H2(d2,δ2)Qδ2d2[H2(d2,⋅)]=bn,3-an,34an,32+29(bn,4-an,4)8an,4.
Therefore (2.52) is satisfied for j=2 if
(3.16)(0.0497+0.0759bn,3+0.0087bn,4)μ1+(0.0475+0.2541bn,4+0.0046bn,4)μ21/2>94+bn,3-an,34an,32+29(bn,4-an,4)8an,4.
Hence, by Theorem 2.8, (3.8) is oscillatory if (3.13) and (3.16) hold. Particularly, let an,i=bn,i, for i=1,2,3,4, condition (3.13) and (3.16) become a simple form
(3.17)(0.0497+0.0759bn,1+0.0087bn,2)μ1+(0.0475+0.2541bn,1+0.0046bn,2)μ21/2>94,(0.0497+0.0759bn,3+0.0087bn,4)μ1+(0.0475+0.2541bn,3+0.0046bn,4)μ21/2>94.



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*Source: 285051-2012-08-13.xml*