# Oscillatory and Asymptotic Behavior of Nonlinear Functional Dynamic Equations of Third Order

Authors: Taher S. Hassan; Adel A. Attiya; Mohammad Alshammari; Amir Abdel Menaem; Ayékotan Tchalla; Ismoil Odinaev
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The purpose of this work is to derive sufficient conditions for the oscillation of all solutions of the third-order functional dynamic equationp2ξϕγ2p1ξϕγ1yΔξΔΔ+pξϕβygξ=0, on a time scale T. In addition, we present some Hille-type conditions for generalized third-order dynamic equations that improve and extend significant contributions reported in the literature without imposing time-scale restrictions. An example is given to demonstrate the essential results.


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## Body

## 1. Introduction

Oscillatory criteria of solutions to dynamic equations on time scales are gaining interest due to their applications in engineering and natural sciences. Eventually, this kind of study aids in comprehending the geometric behavior of solutions. We are interested in the oscillatory and asymptotic behavior of the third-order functional dynamic equation in the form of(1)p2ξϕγ2p1ξϕγ1yΔξΔΔ+pξϕβygξ=0,on an arbitrary time scale T with supT=∞, where ξ∈ξ0,∞T≔ξ0,∞∩T, ξ0≥0, ξ0∈T, ϕθu≔uθsgnu, θ>0, γ1,γ2,β>0, p∈CrdT,ℝ+, and g∈CrdT,T such that limξ⟶∞gξ=∞ and gΔξ≥0 on T and pi∈CrdT,ℝ+, i=1,2, such that
(2)∫ξ0∞Δτpi1/γiτ=∞,i=1,2.By a solution of equation (1), we mean a nontrivial real–valued function y∈Crd1Ty,∞T for some Ty≥ξ0 for a positive constant ξ0∈T such that p1ξϕγ1yΔξ,p2ξϕγ2p1ξϕγ1yΔξΔ∈Crd1Ty,∞T, and yξ satisfies equation (1) on Ty,∞T, where Crd is the space of right-dense continuous functions. Solutions that disappear near infinity will not be considered. A solution y of (1) is oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonnoscillatory. We presume that the reader has a basic understanding of time scales and notation; see [1–4] for providing a great introduction to time scale calculus.Hille [5] investigated the oscillatory behavior of second-order linear differential equation
(3)y′′ξ+pξyξ=0,and shown that if
(4)liminfξ⟶∞ξ∫ξ∞pτdτ>14,then every solution of (3) is oscillatory. The results in [6–13] generalized the Hille-type criterion for different forms of second-order dynamic equations. Regarding third-order dynamic equations, [14–18, 32] established several Hille-type oscillation criteria for different forms of third-order dynamic equations under some restrictive times, which ensure that the solutions are either oscillatory or nonoscillatory and converge to a finite limit under various restrictive conditions. Recently, Hassan et al. [19] improved the Hille-type criteria in [14–18, 32] for equation (1) with β=γ≔γ1γ2, see ([19], in Discussions and Conclusions) for a good comparison between these results. Some of these results in [19] are as follows.Theorem 1 (see [19]).
Assume thatβ=γ≔γ1γ2, every solution of equation (1) is either oscillatory or convergent if
(1)(5)liminfξ⟶∞H1γ2ξ,ξ0∫ξ∞H2ψτ,ξ0H1στ,ξ0γ2pτΔτ>γ2γ2lγ21−γ2γ2+1γ2+1,0<γ2≤1.
(2)(6)liminfξ⟶∞H1γ2ξ,ξ0∫ξ∞H2ψτ,ξ0H1τ,ξ0γ2pτΔτ>γ2γ2lγ2γ2−1γ2+1γ2+1,γ2≥1,where
(7)ψξ≔minξ,gξandl≔liminfξ⟶∞H1ξ,ξ0H1σξ,ξ0,Hiv,u≔ϕγi−1∫uvϕγi−1−1Hi−1τ,upi−1τΔτ,i=1,2,3,with
(8)H0v,u≔1p21/γ2v,p0=γ0=1.The reader is directed to papers [20–33] as well as the sources listed therein.Contrary to [14–16, 19], we are concerned in this paper in deducing sufficient oscillation criteria that guarantee that all solutions of nonlinear third-order dynamic equation (1) are oscillatory when β=γ and gξ≤ξ without imposing restrictive on the time scales. This solves an open problem posed in [1] (Remark 3.3). Furthermore, we will propose certain Hille-type conditions for generalized third-order dynamic equation (1) for the cases β≥γ,β≤γ,gξ≤ξ, and gξ≤ξ that improve and extend relevant significant contributions reported in [14–16, 18, 19] without extra imposing time-scale constraints. All functional inequalities reported in this paper are considered to hold eventually, that is, for all sufficiently large ξ.


## 2. Main Results

Throughout this paper, we let(9)α≔lllβ,β≥γ,γ,β≤γ,(10)yiξ≔piξϕγiyi−1ξΔ,i=1,2,withy0ξ=y.Before stating the main results, we will present some preliminary lemmas to aid in the proving of the main results.Lemma 2.
Assume that equation (1) has a solution y such that
(11)−1iyiξ>0,i=0,1,2,onξ0,∞T.Then, forv∈u,∞T⊆ξ0,∞T,(12)ϕγyuy2v≥H2γ2u,v.Proof.
Suppose, without losing generality, thatygξ>0 on ξ0,∞T. From (1) and (10), we have for ξ0,∞T,(13)y2ξΔ=−pξϕβygξ<0.
From (13), we get for u≤τ≤v and u,τ,v∈ξ0,∞T,(14)y1v−y1τ=∫τvy2ω1/γ2H0ω,τΔω≥y2v1/γ2∫τvH0ω,τΔω=y2v1/γ2H1v,τ.
Therefore,(15)−y1τ≥y2v1/γ2H1v,τ.So,
(16)−yΔτ≥y2v1/γH1v,τp1τ1/γ1.
Integrating the previous inequality with respect toτ from u to v yields that
(17)yu−yv≥y2v1/γ∫uvH1v,τp1τ1/γ1Δτ=y2v1/γH21/γ1u,v.
Thus,(18)yγu≥y2vH2γ2u,v.
Thus, (12) holds for u,v∈ξ0,∞T. The proof is now complete.Lemma 3.
Assume that equation (1) has a solution y such that
(19)yiξ>0,i=0,1,2,onξ0,∞T,
then forv∈u,∞T⊆ξ0,∞T,(20)y1vH1v,uΔv<0,(21)ϕγ2y1vy2v>H1γ2v,u,(22)ϕγyvy2v>H2γ2v,u,and
(23)ϕγ1yvy1v>H2v,uH1v,u.Proof.
Suppose, without losing generality, thatygξ>0 on ξ0,∞T. By dint of (13), y2ξΔ<0 on ξ0,∞T. Therefore,
(24)y1v>y1v−y1u=∫uvϕγ2−1y2τH0τ,uΔτ≥ϕγ2−1y2v∫uvH0τ,uΔτ=ϕγ2−1y2vH1v,u.
Thus, (21) holds for u,v∈ξ0,∞T. From (24), we have
(25)yΔv>ϕγ−1y2vH1v,up1v1/γ1.
Replacingv by s in (25) and integrating with respect to s from u to v gives
(26)yv≥∫uvϕγ−1y2τH1τ,up1τ1/γ1Δτ≥ϕγ−1y2v∫uvH1τ,up1τ1/γ1Δτ=ϕγ−1y2vH21/γ1v,u.
Thus, (22) holds for u,v∈ξ0,∞T. By virtue of (24), there is a v∈u,∞T such that
(27)y1vH1v,uΔv<0forv∈u,∞T⊆ξ0,∞T.
Hence, forv∈u,∞T,
(28)yv>∫uvϕγ1−1y1τH1τ,uH1τ,up1τ1/γ1Δτ≥ϕγ1−1y1vH1v,u∫uvH1τ,up1τ1/γ1Δτ=ϕγ1−1y1vH1v,uH21/γ1v,u.
Thus, (23) holds for v∈u,∞T. The proof is now complete.


### 2.1. Asymptotic Behavior

In this subsection, we debate the asymptotic behavior of the solutions of equation (1) for both cases β≥γ and β≤γ.Theorem 4.
Assume thatl>0 and for sufficiently large T∈ξ0,∞T,
(29)liminfξ⟶∞H1γ2ξ,T∫ξ∞H2βψτ,TH1ατ,T1/γ1pτΔτ>γ2γ2lγ2γ2−1γ2+1γ2+1.
If equation (1) has a nonoscillatory solution yξ, then yiξ and i=0,1,2 converge.Proof.
Suppose, without losing generality, thatyξ and ygξ are eventually positive, by virtue of (1), we deduce that yiξ and i=1,2 are eventually of one sign and by (2), and we can easily see that y2ξ is eventually positive, see [34], part I of the proof of Theorem 4]. Therefore, we consider the following two cases:
Iy1ξ is eventually positive. In this case, there is a ξ1∈ξ0,∞T such that
(30)yiξ>0,i=0,1,2,andy2ξΔ<0onξ1,∞T.
Consider(31)zξ≔y2ξy1ξγ2.
Hence,(32)zΔξ=y2ξΔy1ξγ2+1y1ξγ2Δy2σξ=y2ξΔy1ξγ2−y1ξγ2Δy1ξγ2y2σξy1σξγ2.(1)
If0<γ2≤1, by means of the Pötzsche chain rule and the definitions of H0ξ,ξ1 and zξ, we obtain for ξ∈ξ2,∞T(33)zΔξ≤y2ξΔy1ξγ2−γ2y1σξγ2−1y1ξΔy1ξγ2zσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1y1ξy1σξ1−γ2y2ξ1/γ2y1ξzσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1y1ξy1σξ1−γ2z1/γ2ξzσξ.
Using the fact thaty1ξ/H1ξ,ξ1Δ<0 for ξ∈ξ2,∞T⊆ξ1,∞T, we get
(34)y1ξy1σξ1−γ2≥H1ξ,ξ1H1σξ,ξ11−γ2.
Therefore, (33) becomes
(35)zΔξ≤y2ξΔy1ξγ2−γ2H0ξ,ξ1H1ξ,ξ1H1σξ,ξ11−γ2z1/γ2ξzσξ.
Now, for anyε>0, there exists a ξ3∈ξ2,∞T such that
(36)H1ξ,ξ1H1σξ,ξ1≥l−εandH1γ2ξ,ξ1zξ≥H−εforξ∈ξ3,∞T,where
(37)H≔liminfξ⟶∞H1γ2ξ,ξ1zξ,0≤H≤1.
Substituting (36) into (35), we get for ξ∈ξ3,∞T,
(38)zΔξ≤y2ξΔy1ξγ2−H−ε1+1/γ2γ2H0ξ,ξ1H1γ2ξ,ξ1H1σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−ε1−γ2γ2H0ξ,ξ1H1ξ,ξ1H1γ2σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−ε1−γ2−1H1γ2ξ,ξ1Δ.(2)
Ifγ2≥1, by means of the Potzsche chain rule and the definitions of H0ξ,ξ1 and zξ, we obtain for ξ∈ξ2,∞T(39)zΔξ≤y2ξΔy1ξγ2−γ2y1ξγ2−1y1ξΔy1ξγ2zσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1y2ξ1/γ2y1ξzσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1z1/γ2ξzσξ.
Substituting (36) into (39), we get for ξ∈ξ3,∞T,
(40)zΔξ≤y2ξΔy1ξγ2−H−ε1+1/γ2γ2H0ξ,ξ1H1ξ,ξ1H1γ2σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−εγ2−1γ2H0ξ,ξ1H1γ2ξ,ξ1H1σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
Combining (38) with (40), we conclude that for γ2>0 and ξ∈ξ3,∞T,
(41)zΔξ≤y2ξΔy1ξγ2−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
From (1), we see that
(42)zΔξ≤−yγ1gξy1ξβ/γ1y1ξβ/γ1−γ2pξ−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
Ifβ≥γ, by the fact that y2ξ>0 for ξ∈ξ3,∞T, then
(43)y1ξβ/γ1−γ2≥y1ξ0β/γ1−γ2≕k1>0forξ≥ξ3,whereas if β≤γ, by the fact that y1ξ/H1ξ,ξ1Δ<0 for ξ∈ξ3,∞T, then
(44)y1ξH1ξ,ξ1β/γ1−γ2≥y1ξ1H1ξ2,ξ1β/γ1−γ2≕k2>0forξ≥ξ3.
Now, consider the case whengξ≤ξ. From (23) and using the fact that y1ξ/H1ξ,ξ1Δ<0, we deduce that
(45)yγ1gξ≥y1gξH1gξ,ξ1H2gξ,ξ1≥y1ξH1ξ,ξ1H2gξ,ξ1.
Next consider the case whengξ≥ξ. From (23) and using the fact that y1ξ>0, we conclude that
(46)yγ1gξ≥yγ1ξ≥y1ξH1ξ,ξ1H2ξ,ξ1.
By combining (45) with (46), we have
(47)yγ1gξy1ξ≥H2ψξ,ξ1H1ξ,ξ1forξ∈ξ3,∞T.
From (43), (44), and (47), we get for ξ∈ξ3,∞T,(48)yγ1gξy1ξβ/γ1y1ξβ/γ1−γ2≥kH2βψξ,ξ1H1αξ,ξ11/γ1,where k≔mink1,k2. Substituting (47) into (42), we obtain for ξ∈ξ3,∞T,
(49)zΔξ≤−kH2βψξ,ξ1H1αξ,ξ11/γ1pξ−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
Integrating (49) from ξ to v, we have
(50)zv−zξ≤k∫ξvH2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ−H−ε1+1/γ2l−εγ2−11H1γ2ξ,ξ1−1H1γ2v,ξ1.
Taking into consideration thatz>0 and passing to the limit as v⟶∞, we conclude that
(51)k∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤zξ−H−ε1+1/γ2l−εγ2−11H1γ2ξ,ξ1.
Multiplying both sides of (51) by H1γ2ξ,ξ1, we get for ξ∈ξ3,∞T,
(52)kH1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤H1γ2ξ,ξ1zξ−H−ε1+1/γ2l−εγ2−1.
Taking theliminf of both sides of the last inequality (52) as ξ⟶∞, we obtain
(53)kliminfξ⟶∞H1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤H−H−ε1+1/γ2l−εγ2−1.
By dint of the fact thatk,ε>0 is arbitrary, we deduce that
(54)liminfξ⟶∞H1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤H−H1+1/γ2lγ2−1.
Setting(55)X≔Hlγ2γ2−1/1+γ2andY≔γ21+γ2γ2l−γ22γ2−1/γ2+1,and λ≔γ2+1/γ2>1. Then, using inequality (see [35])
(56)λXYλ−1−Xλ≤λ−1Yλ,we achieve
(57)H−H1+1/γ2lγ2−1≤γ2γ2lγ2γ2−1γ2+1γ2+1.
Thus, (54) becomes
(58)liminfξ⟶∞H1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤γ2γ2lγ2γ2−1γ2+1γ2+1,as a result of which there is a contradiction with (29).
IIy1ξ is eventually positive. In this case, there is a ξ1∈ξ0,∞T such that
(59)−1iyiξ>0,i=0,1,2,andy2ξΔ<0onξ1,∞T.
By virtue of (59), it is easy to see that yiξ and i=0,1,2 converge. This completes the proof.Remark 5.
[1] The deduction of Theorem 4 keeps intact if assumption (29) is replaced by
(60)∫ξ0∞H2βψτ,TH1ατ,T1/γ1pτΔτ=∞.
2) If either(61)∫ξ0∞pτΔτ=∞;∫ξ0∞1p2τ∫τ∞pωΔω1/γ2Δτ=∞or
(62)∫ξ0∞1p1τ∫τ∞1p2ω∫ω∞psΔs1/γ2Δω1/γ1Δτ=∞,then nonoscillatory solutions of the investigated equation (1) are convergent to zero, see [35], [Theorem 4].


### 2.2. Oscillation Criteria

In this subsection, we establish oscillation criteria of the solutions of equation (1) when β=γ and gξ≤ξ on ξ0,∞T. This solves an open problem posed in [1].Theorem 6.
Assume that (29) and either
(63)limsupξ⟶∞∫gξξH2γ2gτ,gξpτΔτ>1,or
(64)limsupξ⟶∞∫gξξ1p1τ∫τξ1p2ω∫ωξpsΔs1/γ2Δω1/γ1Δτ>1,hold. Then, all solutions to equation (1) are oscillatory.Proof.
Suppose, without losing generality, thatyξ and ygξ are eventually positive, by virtue of (1), we deduce that yiξ and i=1,2 are eventually of one sign and by (2), we can easily see that y2ξ is eventually positive. Therefore, we consider the following two cases:
Iy1ξ is eventually positive. The same proof as in part I of the proof of Theorem 4 hence is omitted.
IIy1ξ is eventually positive. In this case, there is a ξ1∈ξ0,∞T such that (59) for ξ∈ξ1,∞T.
Let (63) hold. Integrating (1) form gξ to ξ gives
(65)y2gξ≥y2gξ−y2ξ=∫gξξpτϕγygτΔτ.
By using Lemma2 and setting u=gτ and v=gξ, we obtain
(66)ϕγygτ≥H2γ2gτ,gξy2gξ.
Substituting (66) into (65) yields
(67)y2gξ≥y2gξ∫gξξH2γ2gτ,gξpτΔτ.
We get a contradiction with (63) by taking the limsup on both sides of the last inequality as ξ⟶∞.
Let (64) hold. Integrating (1) form τ to ξ, we get
(68)y2ω≥y2ω−y2ξ=∫ωξpsyγgsΔs≥yγgξ∫ωξpsΔs.
Subsequently,(69)y1ωΔ≥yγ1gξ1p2ω∫ωξpsΔs1/γ2.
Integrating the last inequality formτ to ξ, we conclude that
(70)−y1τ≥y1ξ−y1τ≥yγ1gξ∫τξ1p2ω∫ωξpsΔs1/γ2Δω.
As a result,(71)−yΔτ≥ygξ1p1τ∫τξ1p2ω∫ωξpsΔs1/γ2Δω1/γ1.
Integrating again the last inequality formgξ to ξ, we achieve
(72)ygξ≥ygξ∫gξξ1p1τ∫τξ1p2ω∫ωξpsΔs1/γ2Δω1/γ1Δτ.
We get a contradiction with (64) by taking the limsup on both sides of the last inequality as ξ⟶∞. The proof is complete.Example 7.
Consider the third-order delay dynamic equation(73)14ξ2ϕ214103ξ3ϕ3yΔξΔΔ+1ξ17ϕ6yξ2=0,ξ∈1,∞.It is obvious that condition (2) is fulfilled. Now,
(74)limsupξ⟶∞∫gξξH2γ2gτ,gξpτΔτ=limsupξ⟶∞∫ξ/2ξ411τ17τ24−ξ248dτ=66454835+64ln2,liminfξ⟶∞H1γ2ξ,ξ0∫ξ∞H2βψτ,TH1ατ,T1/γ1pτΔτ=liminfξ⟶∞ξ2−12∫ξ∞245τ2/4−14τ2−121τ17dτ=16.Therefore, the conditions (63) and (29) are satisfied. Then, all solutions to equation (73) are oscillatory, according to Theorem 6.


## 2.1. Asymptotic Behavior

In this subsection, we debate the asymptotic behavior of the solutions of equation (1) for both cases β≥γ and β≤γ.Theorem 4.
Assume thatl>0 and for sufficiently large T∈ξ0,∞T,
(29)liminfξ⟶∞H1γ2ξ,T∫ξ∞H2βψτ,TH1ατ,T1/γ1pτΔτ>γ2γ2lγ2γ2−1γ2+1γ2+1.
If equation (1) has a nonoscillatory solution yξ, then yiξ and i=0,1,2 converge.Proof.
Suppose, without losing generality, thatyξ and ygξ are eventually positive, by virtue of (1), we deduce that yiξ and i=1,2 are eventually of one sign and by (2), and we can easily see that y2ξ is eventually positive, see [34], part I of the proof of Theorem 4]. Therefore, we consider the following two cases:
Iy1ξ is eventually positive. In this case, there is a ξ1∈ξ0,∞T such that
(30)yiξ>0,i=0,1,2,andy2ξΔ<0onξ1,∞T.
Consider(31)zξ≔y2ξy1ξγ2.
Hence,(32)zΔξ=y2ξΔy1ξγ2+1y1ξγ2Δy2σξ=y2ξΔy1ξγ2−y1ξγ2Δy1ξγ2y2σξy1σξγ2.(1)
If0<γ2≤1, by means of the Pötzsche chain rule and the definitions of H0ξ,ξ1 and zξ, we obtain for ξ∈ξ2,∞T(33)zΔξ≤y2ξΔy1ξγ2−γ2y1σξγ2−1y1ξΔy1ξγ2zσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1y1ξy1σξ1−γ2y2ξ1/γ2y1ξzσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1y1ξy1σξ1−γ2z1/γ2ξzσξ.
Using the fact thaty1ξ/H1ξ,ξ1Δ<0 for ξ∈ξ2,∞T⊆ξ1,∞T, we get
(34)y1ξy1σξ1−γ2≥H1ξ,ξ1H1σξ,ξ11−γ2.
Therefore, (33) becomes
(35)zΔξ≤y2ξΔy1ξγ2−γ2H0ξ,ξ1H1ξ,ξ1H1σξ,ξ11−γ2z1/γ2ξzσξ.
Now, for anyε>0, there exists a ξ3∈ξ2,∞T such that
(36)H1ξ,ξ1H1σξ,ξ1≥l−εandH1γ2ξ,ξ1zξ≥H−εforξ∈ξ3,∞T,where
(37)H≔liminfξ⟶∞H1γ2ξ,ξ1zξ,0≤H≤1.
Substituting (36) into (35), we get for ξ∈ξ3,∞T,
(38)zΔξ≤y2ξΔy1ξγ2−H−ε1+1/γ2γ2H0ξ,ξ1H1γ2ξ,ξ1H1σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−ε1−γ2γ2H0ξ,ξ1H1ξ,ξ1H1γ2σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−ε1−γ2−1H1γ2ξ,ξ1Δ.(2)
Ifγ2≥1, by means of the Potzsche chain rule and the definitions of H0ξ,ξ1 and zξ, we obtain for ξ∈ξ2,∞T(39)zΔξ≤y2ξΔy1ξγ2−γ2y1ξγ2−1y1ξΔy1ξγ2zσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1y2ξ1/γ2y1ξzσξ=y2ξΔy1ξγ2−γ2H0ξ,ξ1z1/γ2ξzσξ.
Substituting (36) into (39), we get for ξ∈ξ3,∞T,
(40)zΔξ≤y2ξΔy1ξγ2−H−ε1+1/γ2γ2H0ξ,ξ1H1ξ,ξ1H1γ2σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−εγ2−1γ2H0ξ,ξ1H1γ2ξ,ξ1H1σξ,ξ1≤y2ξΔy1ξγ2−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
Combining (38) with (40), we conclude that for γ2>0 and ξ∈ξ3,∞T,
(41)zΔξ≤y2ξΔy1ξγ2−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
From (1), we see that
(42)zΔξ≤−yγ1gξy1ξβ/γ1y1ξβ/γ1−γ2pξ−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
Ifβ≥γ, by the fact that y2ξ>0 for ξ∈ξ3,∞T, then
(43)y1ξβ/γ1−γ2≥y1ξ0β/γ1−γ2≕k1>0forξ≥ξ3,whereas if β≤γ, by the fact that y1ξ/H1ξ,ξ1Δ<0 for ξ∈ξ3,∞T, then
(44)y1ξH1ξ,ξ1β/γ1−γ2≥y1ξ1H1ξ2,ξ1β/γ1−γ2≕k2>0forξ≥ξ3.
Now, consider the case whengξ≤ξ. From (23) and using the fact that y1ξ/H1ξ,ξ1Δ<0, we deduce that
(45)yγ1gξ≥y1gξH1gξ,ξ1H2gξ,ξ1≥y1ξH1ξ,ξ1H2gξ,ξ1.
Next consider the case whengξ≥ξ. From (23) and using the fact that y1ξ>0, we conclude that
(46)yγ1gξ≥yγ1ξ≥y1ξH1ξ,ξ1H2ξ,ξ1.
By combining (45) with (46), we have
(47)yγ1gξy1ξ≥H2ψξ,ξ1H1ξ,ξ1forξ∈ξ3,∞T.
From (43), (44), and (47), we get for ξ∈ξ3,∞T,(48)yγ1gξy1ξβ/γ1y1ξβ/γ1−γ2≥kH2βψξ,ξ1H1αξ,ξ11/γ1,where k≔mink1,k2. Substituting (47) into (42), we obtain for ξ∈ξ3,∞T,
(49)zΔξ≤−kH2βψξ,ξ1H1αξ,ξ11/γ1pξ−H−ε1+1/γ2l−εγ2−1−1H1γ2ξ,ξ1Δ.
Integrating (49) from ξ to v, we have
(50)zv−zξ≤k∫ξvH2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ−H−ε1+1/γ2l−εγ2−11H1γ2ξ,ξ1−1H1γ2v,ξ1.
Taking into consideration thatz>0 and passing to the limit as v⟶∞, we conclude that
(51)k∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤zξ−H−ε1+1/γ2l−εγ2−11H1γ2ξ,ξ1.
Multiplying both sides of (51) by H1γ2ξ,ξ1, we get for ξ∈ξ3,∞T,
(52)kH1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤H1γ2ξ,ξ1zξ−H−ε1+1/γ2l−εγ2−1.
Taking theliminf of both sides of the last inequality (52) as ξ⟶∞, we obtain
(53)kliminfξ⟶∞H1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤H−H−ε1+1/γ2l−εγ2−1.
By dint of the fact thatk,ε>0 is arbitrary, we deduce that
(54)liminfξ⟶∞H1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤H−H1+1/γ2lγ2−1.
Setting(55)X≔Hlγ2γ2−1/1+γ2andY≔γ21+γ2γ2l−γ22γ2−1/γ2+1,and λ≔γ2+1/γ2>1. Then, using inequality (see [35])
(56)λXYλ−1−Xλ≤λ−1Yλ,we achieve
(57)H−H1+1/γ2lγ2−1≤γ2γ2lγ2γ2−1γ2+1γ2+1.
Thus, (54) becomes
(58)liminfξ⟶∞H1γ2ξ,ξ1∫ξ∞H2βψτ,ξ1H1ατ,ξ11/γ1pτΔτ≤γ2γ2lγ2γ2−1γ2+1γ2+1,as a result of which there is a contradiction with (29).
IIy1ξ is eventually positive. In this case, there is a ξ1∈ξ0,∞T such that
(59)−1iyiξ>0,i=0,1,2,andy2ξΔ<0onξ1,∞T.
By virtue of (59), it is easy to see that yiξ and i=0,1,2 converge. This completes the proof.Remark 5.
[1] The deduction of Theorem 4 keeps intact if assumption (29) is replaced by
(60)∫ξ0∞H2βψτ,TH1ατ,T1/γ1pτΔτ=∞.
2) If either(61)∫ξ0∞pτΔτ=∞;∫ξ0∞1p2τ∫τ∞pωΔω1/γ2Δτ=∞or
(62)∫ξ0∞1p1τ∫τ∞1p2ω∫ω∞psΔs1/γ2Δω1/γ1Δτ=∞,then nonoscillatory solutions of the investigated equation (1) are convergent to zero, see [35], [Theorem 4].


## 2.2. Oscillation Criteria

In this subsection, we establish oscillation criteria of the solutions of equation (1) when β=γ and gξ≤ξ on ξ0,∞T. This solves an open problem posed in [1].Theorem 6.
Assume that (29) and either
(63)limsupξ⟶∞∫gξξH2γ2gτ,gξpτΔτ>1,or
(64)limsupξ⟶∞∫gξξ1p1τ∫τξ1p2ω∫ωξpsΔs1/γ2Δω1/γ1Δτ>1,hold. Then, all solutions to equation (1) are oscillatory.Proof.
Suppose, without losing generality, thatyξ and ygξ are eventually positive, by virtue of (1), we deduce that yiξ and i=1,2 are eventually of one sign and by (2), we can easily see that y2ξ is eventually positive. Therefore, we consider the following two cases:
Iy1ξ is eventually positive. The same proof as in part I of the proof of Theorem 4 hence is omitted.
IIy1ξ is eventually positive. In this case, there is a ξ1∈ξ0,∞T such that (59) for ξ∈ξ1,∞T.
Let (63) hold. Integrating (1) form gξ to ξ gives
(65)y2gξ≥y2gξ−y2ξ=∫gξξpτϕγygτΔτ.
By using Lemma2 and setting u=gτ and v=gξ, we obtain
(66)ϕγygτ≥H2γ2gτ,gξy2gξ.
Substituting (66) into (65) yields
(67)y2gξ≥y2gξ∫gξξH2γ2gτ,gξpτΔτ.
We get a contradiction with (63) by taking the limsup on both sides of the last inequality as ξ⟶∞.
Let (64) hold. Integrating (1) form τ to ξ, we get
(68)y2ω≥y2ω−y2ξ=∫ωξpsyγgsΔs≥yγgξ∫ωξpsΔs.
Subsequently,(69)y1ωΔ≥yγ1gξ1p2ω∫ωξpsΔs1/γ2.
Integrating the last inequality formτ to ξ, we conclude that
(70)−y1τ≥y1ξ−y1τ≥yγ1gξ∫τξ1p2ω∫ωξpsΔs1/γ2Δω.
As a result,(71)−yΔτ≥ygξ1p1τ∫τξ1p2ω∫ωξpsΔs1/γ2Δω1/γ1.
Integrating again the last inequality formgξ to ξ, we achieve
(72)ygξ≥ygξ∫gξξ1p1τ∫τξ1p2ω∫ωξpsΔs1/γ2Δω1/γ1Δτ.
We get a contradiction with (64) by taking the limsup on both sides of the last inequality as ξ⟶∞. The proof is complete.Example 7.
Consider the third-order delay dynamic equation(73)14ξ2ϕ214103ξ3ϕ3yΔξΔΔ+1ξ17ϕ6yξ2=0,ξ∈1,∞.It is obvious that condition (2) is fulfilled. Now,
(74)limsupξ⟶∞∫gξξH2γ2gτ,gξpτΔτ=limsupξ⟶∞∫ξ/2ξ411τ17τ24−ξ248dτ=66454835+64ln2,liminfξ⟶∞H1γ2ξ,ξ0∫ξ∞H2βψτ,TH1ατ,T1/γ1pτΔτ=liminfξ⟶∞ξ2−12∫ξ∞245τ2/4−14τ2−121τ17dτ=16.Therefore, the conditions (63) and (29) are satisfied. Then, all solutions to equation (73) are oscillatory, according to Theorem 6.


## 3. Conclusions

(1)
The important point to note here is that the proposed results in Theorem6 are new for third-order dynamic equation (1) and also, these results, in a special case, have answered an open problem stated by [1] (Remark 3.3), which is concerned with studying the sufficient conditions that guarantee that all solutions of third-order delay dynamic equations oscillate(2)
In comparison to the results in the cited papers [14–16, 18], Hille-type criteria for dynamic equation (1) in the cases β≥γ,β≤γ,gξ≤ξ, and gt≥ξ have been developed, and the results in this study are a significant improvement; for more details, see ([19], in Discussions and Conclusions). Moreover, our results improve and expand upon those described in [19], see the following details(i)
Ifβ=γ and 0<γ2≤1, then condition (29) becomes(75)liminfξ⟶∞H1γ2ξ,T∫ξ∞H2ψτ,TH1τ,Tγ2pτΔτ>γ2γ2lγ21−γ2γ2+1γ2+1.By dint of(76)H1γ2ξ,T∫ξ∞H2ψτ,TH1τ,Tγ2pτΔτ≥H1γ2ξ,T∫ξ∞H2ψτ,TH1στ,Tγ2pτΔτ,Hille-type criterion (75) improves (5);
(ii)
Condition (29) reduces to (6) in the case where β=γ and γ2≥1(3)
The asymptotic behavior of solutions is viable to dynamic equation (1) for both gξ≤ξ and gξ≥ξ, whereas the oscillation conditions are viable to dynamic equation (1) for gξ≤ξ. As a result, oscillations can be ensured by a delay in equations(4)
The results presented here are for equation (1) on an unbounded above arbitrary time scale; therefore, they are applicable to different of time scales(5)
It would be interesting to define Hille-type criteria for the third-order dynamic equation (1) under noncanonical assumptions(77)∫ξ0∞Δτpi1/γiτ<∞,i=1,2.



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