We establish several oscillation criteria for a class of third-order nonlinear dynamic equations with a damping term and a nonpositive neutral coefficient by using the Riccati transformation. Two illustrative examples are presented to show the significance of the results obtained.

National Natural Science Foundation of China116714066150317161403061China Postdoctoral Science Foundation2015M582091Program of Cultivation for Outstanding Young Scholars Sponsored by Guangdong ProvinceZX03240302Natural Science Foundation of Shandong ProvinceZR2016JL021ZR2016AB04Program of Cultivation for Young Scholars Sponsored by Shunde Polytechnic2015-KJZX080Linyi UniversityLYDX2015BS001Foundation for Young Teachers of Qilu Normal University2016L06051. Introduction

In this paper, we are concerned with the oscillation of a class of third-order damped dynamic equations of neutral type(1)rtφγzΔΔtΔ+dtφγzΔΔt+ft,xht=0on a time scale T satisfying supT=∞, where t∈[t0,∞)T, φγ(u)=|u|γ-1u, and z(t)=x(t)-p(t)x(g(t)). Throughout, we suppose that the following conditions are satisfied:

γ≥1 is a constant.

r∈Crd([t0,∞)T,(0,∞)), d∈Crd([t0,∞)T,R), -d/r is positively regressive (i.e., r(t)-(σ(t)-t)d(t)>0), and(2)∫t0∞e-d/rt,t0rt1/γΔt=∞.

p∈Crd([t0,∞)T,[0,∞)) and limt→∞p(t)=p0, where 0≤p0<1.

g,h∈Crd([t0,∞)T,T), g(t)≤t, h(t)≤t or h(t)≥t, limt→∞gt=limt→∞ht=∞, and there exists a sequence {ck}k≥0 such that limk→∞ck=∞ and g(ck+1)=ck.

f∈C([t0,∞)T×R,R) and there exists a function q∈Crd([t0,∞)T,(0,∞)) such that uf(t,u)>0 and f(t,u)/(|u|γ-1u)≥q(t) for u≠0.

The theory of time scales, which was firstly introduced by Hilger in [1, 2], has been enriched by researchers; see, for instance, [3, 4], monographs [5, 6], and the references cited therein. During the past decade, a great deal of interest in oscillation of solutions to different classes of dynamic equations on time scales has been shown; we refer the reader to [7–23].

Yu and Wang [23] studied a third-order dynamic equation(3)1a2t1a1txΔtα1Δα2Δ+qtfxt=0,α1α2=1.Agarwal et al. [8, 10], Candan [12], Erbe et al. [13], Hassan [15], and Li et al. [18] considered a third-order retarded dynamic equation(4)atrtxΔtΔγΔ+ft,xτt=0.Saker et al. [21] studied a second-order damped dynamic equation(5)atxΔtΔ+ptxΔσt+qtf∘xσ=0,whereas Qiu and Wang [20] considered a second-order damped dynamic equation(6)rtφγxΔtΔ+ptφγxΔt+ft,xgt=0,where γ>0, φγ(u)=|u|γ-1u, and(7)gt≥σt,0<γ<1,t,γ≥1.Han et al. [14] and Qiu [19] investigated the third-order dynamic equations with nonpositive neutral coefficients(8)rtzΔΔtΔ+qtxγht=0,rΔ≥0,(9)r1tr2tzΔtγ2Δγ1Δ+ft,xht=0,respectively, where z(t)=x(t)-p(t)x(g(t)).

In this paper, using the Riccati transformation, we obtain some sufficient conditions which ensure that every solution x of (1) either is oscillatory or converges to a finite number asymptotically. We do not impose restrictive assumption d≥0 in our results. To illustrate the significance of new results, two examples are provided in the last section. In what follows, all functional inequalities are assumed to hold for all sufficiently large t. Without loss of generality, we can deal only with eventually positive solutions of (1).

Definition 1.

A solution x of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is termed nonoscillatory.

Definition 2.

Equation (1) is said to be almost oscillatory if all its solutions either are oscillatory or converge to zero asymptotically.

2. Auxiliary ResultsLemma 3 (see [<xref ref-type="bibr" rid="B19">19</xref>, Lemma 2.1]).

Suppose that x is an eventually positive solution of (1) and there exists a constant a≥0 such that limt→∞z(t)=a. Then(10)limt→∞xt=a1-p0.

Lemma 4.

If x is an eventually positive solution of (1), then there exists a sufficiently large T∈[t0,∞)T such that, for t∈[T,∞)T,(11)rtzΔΔtγe-d/rt,TΔ<0,zΔΔt>0,zΔt>0or zΔt<0.

Proof.

Let x be an eventually positive solution of (1). From (C3) and (C4), there exist a t1∈[t0,∞)T and a constant p1 such that p0<p1<1, x(t)>0, x(g(t))>0, x(h(t))>0, and p(t)≤p1 for t∈[t1,∞)T. By virtue of (1) and (C5), we conclude that(12)rtφγzΔΔte-d/rt,t1Δ=-ft,xhte-d/rσt,t1<0.We claim that there exists a t2∈[t1,∞)T such that zΔΔ(t)>0 for t∈[t2,∞)T. Otherwise, assume that zΔΔ(t)<0 for t∈[t2,∞)T. Then, by (12) and (C2), limt→∞zΔ(t)=-∞. It follows from zΔΔ(t)<0 and zΔ(t)<0 that limt→∞z(t)=-∞. Hence, there exists a t3∈[t2,∞)T such that z(t)<0 for t∈[t3,∞)T; that is,(13)xt<ptxgt≤p1xgt,t∈t3,∞T.By (C4), there exists a sufficiently large integer k0 such that ck∈[t3,∞)T for k≥k0. For k≥k0+1, we have(14)xck<p1xgck=p1xck-1<p12xgck-1=p12xck-2<⋯<p1k-k0xgck0+1=p1k-k0xck0,which yields limk→∞x(ck)=0 and so limk→∞z(ck)=0, which contradicts the fact that limt→∞z(t)=-∞. Therefore, zΔΔ(t)>0 and hence zΔ(t)>0 or zΔ(t)<0. The proof is complete.

Lemma 5 (see [<xref ref-type="bibr" rid="B19">19</xref>, Lemma 2.3]).

If x is an eventually positive solution of (1), then z is eventually positive or limt→∞x(t)=0.

Lemma 6.

Let x be an eventually positive solution of (1) and suppose that z and zΔ are eventually positive. Assume also that(15)∫t0∞qthγte-d/rσt,t0Δt=∞.Then there exists a sufficiently large T∈[t0,∞)T such that, for t∈[T,∞)T,(16)zhtzt≥minhtt,1.

Proof.

Let x be an eventually positive solution of (1) and assume that there exists a t1∈[t0,∞)T such that z(t)>0 and zΔ(t)>0 for t∈[t1,∞)T. Define(17)yt=zt-tzΔt,t∈t1,∞T.Then, by Lemma 4, for t∈[t1,∞)T,(18)yΔt=zΔt-zΔt+σtzΔΔt=-σtzΔΔt<0.We can prove that y is eventually positive. If not, then there exists a t2∈[t1,∞)T such that y(t)<0 for t∈[t2,∞)T. Hence, we conclude that(19)zttΔ=tzΔt-zttσt=-yttσt>0,t∈t2,∞T,which implies that z(t)/t is strictly increasing on [t2,∞)T. Since limt→∞h(t)=∞, there exists a t3∈[t2,∞)T such that, for t∈[t3,∞)T, h(t)≥h(t3)≥t2, and so(20)zhtht≥zht3ht3.Using (C5), we have f(t,x(h(t)))≥q(t)xγ(h(t))≥q(t)zγ(h(t)). Therefore, by virtue of (1) and Lemma 4, for t∈[t3,∞)T,(21)rtzΔΔtγe-d/rt,t3Δ=-ft,xhte-d/rσt,t3≤-qtzγhte-d/rσt,t3.Integrating (21) from t3 to t, we get(22)rtzΔΔtγe-d/rt,t3-rt3zΔΔt3γ≤-∫t3tqszγhse-d/rσs,t3Δs≤-zγht3hγt3∫t3tqshγse-d/rσs,t3Δs.It follows from (15) that(23)rt3zΔΔt3γ≥zγht3hγt3∫t3tqshγse-d/rσs,t3Δs⟶∞,t⟶∞,which is a contradiction. Hence, y is eventually positive. Then, there exists a sufficiently large T∈[t1,∞)T such that, for t∈[T,∞)T,(24)zttΔ=-yttσt<0,so z(t)/t is strictly decreasing on [T,∞)T. If h(t)≥t, then z(h(t))/z(t)≥1. If h(t)≤t, then z(h(t))/z(t)≥h(t)/t. Therefore, we arrive at (16). This completes the proof.

Lemma 7.

Assume that all assumptions of Lemma 6 are satisfied. For t∈[t1,∞)T, define(25)ut=AtrtzΔΔtγe-d/rt,t1zγt,where A∈Crd1([t1,∞)T,(0,∞)). Then, u satisfies(26)uΔt+Atαtqte-d/rσt,t1-Φt≤0,where(27)Φt=AΔtuAσt-γAtδtuAσ1+γ/γt,αt=minhttγ,1,δt=∫t1te-d/rs,t1rs1/γΔs.

Proof.

Suppose that all assumptions of Lemma 6 hold. Differentiating (25) and using (1), we have(28)uΔt=AtzγtrzΔΔγe-d/r·,t1Δt+AzγΔtrzΔΔγe-d/r·,t1σt=-Atzγtft,xhte-d/rσt,t1+AΔtzγt-AtzγΔtzγtzγσtrzΔΔγe-d/r·,t1σt.If h(t)≥t, then(29)ft,xht≥qtxγht≥qtzγht≥qtzγt.Assume now that h(t)≤t. It follows from Lemma 6 that(30)ft,xht≥qtzγht≥httγqtzγt.Hence, we have(31)uΔt≤-Atαtqte-d/rσt,t1+AΔtuAσt-AtzγΔtzγtuAσt.Using Pötzsche chain rule (see [5, Theorem 1.90] for details), we obtain(32)zγΔt≥γzγ-1tzΔt,which yields(33)zγΔtzγt≥γzγ-1tzΔtzγt=γzΔtzt.From Lemma 4, we conclude that(34)zΔt=zΔt1+∫t1tzΔΔsΔs=zΔt1+∫t1trszΔΔsγe-d/rs,t11/γe-d/rs,t1rs1/γΔs≥rtzΔΔtγe-d/rt,t11/γ∫t1te-d/rs,t1rs1/γΔs=δtrtzΔΔtγe-d/rt,t11/γ,which implies that(35)zΔtzt≥δtrtzΔΔtγe-d/rt,t11/γ1zt≥δtrzΔΔγe-d/r·,t11/γσt1zσt=δtuAσ1/γt.It follows now from (31) that(36)uΔt≤-Atαtqte-d/rσt,t1+AΔtuAσt-γAtδtuAσ1+γ/γt.The proof is complete.

Lemma 8.

Assume that x is an eventually positive solution of (1) and zΔ is eventually negative. If(37)∫t0∞qte-d/rσt,t0Δt=∞,then limt→∞x(t)=0.

Proof.

Since zΔ is eventually negative, z is either eventually positive or eventually negative. If z is eventually negative, then there exist a constant c<0 and a T∈[t0,∞)T such that z(t)<c for t∈[T,∞)T, which causes a contradiction as in the proof of Lemma 4. Thus, z is eventually positive.

Taking into account the fact that x>0, by Lemma 4, there exists a t1∈[t0,∞)T such that zΔΔ(t)>0 for t∈[t1,∞)T. We prove that limt→∞z(t)=b≥0. Otherwise, there exists a t2∈[t1,∞)T such that z(t)<0 for t∈[t2,∞)T, and a similar contradiction can be obtained. Suppose that b>0. It follows from the proof of Lemma 4 and z(h(t))>b that(38)rtzΔΔtγe-d/rt,t1Δ≤-qtzγhte-d/rσt,t1<-bγqte-d/rσt,t1.Integrating (38) from t1 to t, t∈[σ(t1),∞)T, we have(39)rtzΔΔtγe-d/rt,t1≤rt1zΔΔt1γ-bγ∫t1tqse-d/rσs,t1Δs⟶-∞,t⟶∞,which contradicts the fact that zΔΔ(t)>0. Hence, b=0 and so limt→∞x(t)=0 when using Lemma 3. This completes the proof.

3. Main Results

Let D={(t,s)∈T2:t≥s≥t0}. Define(40)H=Ht,s∈CrdD,0,∞:Ht,t=0,Ht,s>0,H2Δt,s≤0,t>s≥t0,where H2Δ is the Δ-partial derivative of H with respect to s.

Theorem 9.

Assume that (15) holds and there exist two functions A∈Crd1([t0,∞)T,(0,∞)) and H∈H such that, for all sufficiently large t1∈[t0,∞)T and for some t2∈[t1,∞)T,(41)limsupt→∞1Ht,t2∫t2tHt,sAsαsqse-d/rσs,t1-11+γ1+γH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγΔs=∞,where α and δ are as in Lemma 7. Then every solution x of (1) is oscillatory or limt→∞x(t) exists (finite).

Proof.

Suppose that (1) has a nonoscillatory solution x. Without loss of generality, let x be eventually positive. From Lemma 5, it follows that z is eventually positive or limt→∞x(t)=0. Assume that z is eventually positive. By Lemma 4, there exists a t1∈[t0,∞)T such that either zΔ(t)>0 or zΔ(t)<0 for t∈[t1,∞)T. Let zΔ(t)>0 for t∈[t1,∞)T. Define u by (25). Then, by Lemma 7, (26) holds. It follows from (26) that, for some t2∈[t1,∞)T,(42)∫t2tHt,sAsαsqse-d/rσs,t1Δs≤-∫t2tHt,suΔsΔs+∫t2tHt,sΦsΔs=Ht,t2ut2+∫t2tH2Δt,suσs+Ht,sΦsΔs=Ht,t2ut2+∫t2tH2Δt,sAσsuAσs+Ht,sΦsΔs,where(43)H2Δt,sAσsuAσs+Ht,sΦs=H2Δt,sAσs+Ht,sAΔsuAσs-γHt,sAsδsuAσ1+γ/γs.Let λ=(1+γ)/γ,(44)aλ=γHt,sAsδsuAσ1+γ/γs,bλ-1=γ1+γH2Δt,sAσs+Ht,sAΔsγHt,sAsδsγ/1+γ.Using the inequality (a variation of the well-known Young inequality)(45)λabλ-1-aλ≤λ-1bλ,a,b≥0,we deduce that(46)H2Δt,sAσsuAσs+Ht,sΦs≤1γγ1+γH2Δt,sAσs+Ht,sAΔsγHt,sAsδsγ/1+γ1+γ=11+γ1+γH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγ.Therefore, we obtain(47)∫t2tHt,sAsαsqse-d/rσs,t1Δs≤Ht,t2ut2+11+γ1+γ∫t2tH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγΔs,which implies that(48)1Ht,t2∫t2tHt,sAsαsqse-d/rσs,t1-11+γ1+γH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγΔs≤ut2.This contradicts (41). Thus, zΔ(t)<0 for t∈[t1,∞)T, and so limt→∞z(t) exists. By Lemma 3, limt→∞x(t) exists. The proof is complete.

From Lemma 8 and Theorem 9, we have the following corollary.

Corollary 10.

Assume that (37) is satisfied and there exist two functions A∈Crd1([t0,∞)T,(0,∞)) and H∈H such that, for all sufficiently large t1∈[t0,∞)T and for some t2∈[t1,∞)T,(49)limsupt→∞1Ht,t2∫t2tHt,sAsαsqse-d/rσs,t1-11+γ1+γH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγΔs=∞,where α and δ are as in Lemma 7. Then (1) is almost oscillatory.

Theorem 11.

Assume that (15) holds and there exists a function A∈Crd1([t0,∞)T,(0,∞)) such that, for all sufficiently large t1∈[t0,∞)T and for some t2∈[t1,∞)T,(50)limsupt→∞∫t2tAsαsqse-d/rσs,t1-11+γ1+γAΔs1+γAsδsγΔs=∞,where α and δ are as in Lemma 7. Then conclusion of Theorem 9 remains intact.

Proof.

Proceeding as in the proof of Theorem 9, assume that zΔ(t)>0 for t∈[t1,∞)T. Let u be defined by (25). By virtue of Lemma 7, we arrive at (36). Let λ=(1+γ)/γ,(51)aλ=γAtδtuAσ1+γ/γt,bλ-1=γ1+γAΔtγAtδtγ/1+γ.Using (45), we conclude that(52)uΔt≤-Atαtqte-d/rσt,t1+11+γ1+γAΔt1+γAtδtγ,which yields(53)Atαtqte-d/rσt,t1-11+γ1+γAΔt1+γAtδtγ≤-uΔt.Let t2∈[t1,∞)T. It follows from (53) that(54)∫t2tAsαsqse-d/rσs,t1-11+γ1+γAΔs1+γAsδsγΔs≤ut2,which contradicts (50). Therefore, zΔ(t)<0 for t∈[t1,∞)T. Along the same lines as in Theorem 9, we complete the proof.

If (37) holds, then we have the following corollary on the basis of Lemma 8 and Theorem 11.

Corollary 12.

Assume that (37) is satisfied and there exists a function A∈Crd1([t0,∞)T,(0,∞)) such that, for all sufficiently large t1∈[t0,∞)T and for some t2∈[t1,∞)T,(55)limsupt→∞∫t2tAsαsqse-d/rσs,t1-11+γ1+γAΔs1+γAsδsγΔs=∞,where α and δ are as in Lemma 7. Then conclusion of Corollary 10 remains intact.

Remark 13.

If d≥0, then it is not difficult to see that(56)0<e-d/rσt,t1≤1,and so (41) and (50) can be simplified to(57)limsupt→∞1Ht,t2∫t2tHt,sAsαsqs-11+γ1+γH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγΔs=∞,(58)limsupt→∞∫t2tAsαsqs-11+γ1+γAΔs1+γAsδsγΔs=∞,respectively.

Remark 14.

If h(t)≥t, then α(t)=1 and we do not impose restrictive condition (15) in our theorems and corollaries.

Remark 15.

Our results complement and improve those obtained by Han et al. [14] since we do not impose specific restrictions on rΔ.

4. Examples

The following examples are presented to show applications of the main results.

Example 1.

Consider the third-order equation(59)1tφγzΔΔtΔ+λt2φγzΔΔt+1t2xγt2=0,where t∈1,∞R, z(t)=x(t)-(t-1)x(t-1)/(2t), γ≥1, and 0≤λ<2. It is clear that r(t)=1/t, p(t)=(t-1)/(2t), d(t)=λ/t2, g(t)=t-1, and h(t)=t/2<t. Then, p0=1/2 and α(t)=1/2γ. Let q(t)=1/t2. Since -d(t)/r(t)=-λ/t, we have(60)e-d/rt,t0=1tλ,e-d/rt,t1=t1tλ,∫t0∞e-d/rt,t0rt1/γΔt=∫1∞1tλ-1/γdt=∞,∫t0∞qthγte-d/rσt,t0Δt=12γ∫1∞tγ+λ-2dt=∞,δt=∫t1te-d/rs,t1rs1/γΔs=t1λ/γ∫t1tdssλ-1/γ=Otγ-λ+1/γ.Hence, assumptions (C1)–(C5) and (15) hold. Let H(t,s)=(t-s)2 and A(t)=t2. If(61)3γ-4-λ+λ-1γ<0,we obtain(62)limsupt→∞1Ht,t2∫t2tHt,sAsαsqse-d/rσs,t1-11+γ1+γH2Δt,sAσs+Ht,sAΔs1+γHt,sAsδsγΔs=limsupt→∞1t-t22∫t2tt-s2sλ2γt1λ-Os3γ-2+λ-1/γ1+γ1+γds=∞.That is, (41) is satisfied. By virtue of Theorem 9, we deduce that every solution of (59) either is oscillatory or converges to a finite number asymptotically. Furthermore, if (61) holds and 1≤λ<2, then(63)∫t0∞qte-d/rσt,t0Δt=∫1∞tλ-2dt=∞,which implies that (59) is almost oscillatory by using Corollary 10.

Example 2.

Consider the third-order equation(64)tφγzΔΔtΔ-φγzΔΔt+tβ+tλxγt+1=0,where t∈[2,∞)Z, z(t)=x(t)-x(t-2)/t, β,γ,λ∈R, and γ≥1. It is easy to see that r(t)=t, p(t)=-1/t, d(t)=-1, g(t)=t-2, and h(t)=t+1>t. Then, p0=0 and α(t)=1. Since -d(t)/r(t)=1/t, we get(65)e-d/rt,t0=t2,e-d/rt,t1=tt1,∫t0∞e-d/rt,t0rt1/γΔt=121/γ∫2∞Δt=∞,δt=∫t1te-d/rs,t1rs1/γΔs=1t11/γ∫t1tΔs=t-t1t11/γ.Obviously, conditions (C1)–(C5) are satisfied. Let q(t)=tβ or tλ and A(t)=1. If β≥0 or λ≥0, then(66)∫t0∞qte-d/rσt,t0Δt≥∫2∞2t+1Δt=∞,limsupt→∞∫t2tAsαsqse-d/rσs,t1-11+γ1+γAΔs1+γAsδsγΔs≥t1limsupt→∞∫t2t1s+1Δs=∞.That is, both (37) and (55) hold. By Corollary 12, we conclude that (64) is almost oscillatory for β≥0 or λ≥0.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (Grant nos. 11671406, 61503171, and 61403061), China Postdoctoral Science Foundation (Grant no. 2015M582091), Program of Cultivation for Outstanding Young Scholars Sponsored by Guangdong Province (Grant no. ZX03240302), Natural Science Foundation of Shandong Province (Grant nos. ZR2016JL021 and ZR2016AB04), Program of Cultivation for Young Scholars Sponsored by Shunde Polytechnic (Grant no. 2015-KJZX080), Doctoral Scientific Research Foundation of Linyi University (Grant no. LYDX2015BS001), Applied Mathematics Enhancement Program of Linyi University, and Foundation for Young Teachers of Qilu Normal University (Grant no. 2016L0605).

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