# Some Oscillation Criteria for a Class of Third-Order Delay Differential Equations

Authors: Miao Wang; Lifei Dong
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study oscillatory properties of solutions to a class of third-order differential equationatrtx′t′α′+ptrtx′t′α+qtfxσt=0, where fx/xβ≥k>0 and α and β are quotients of odd positive integers. By using the generalized Riccati technique, we obtain some oscillation and asymptotic criteria when α≥β and α<β. Finally, some examples are given to show the effectiveness of the criteria obtained.


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## Body

## 1. Introduction

Oscillation phenomena take part in different models from real-world applications (see, e.g., the papers [1, 2]), where oscillation and delay situations take part in models from mathematical biology when their formulation includes cross-diffusion terms. The oscillation theory of differential equations is an active area of research. Many scholars have studied the oscillation of solutions to various classes of differential equations and made some progress, especially for the first-order and second-order equations (see, e.g., the monographs [3, 4], the papers [5, 6], and the references cited therein). Third-order differential equations originate in many different fields of applied mathematics and physics, for example, the deflection of a buckling beam with a fixed or varying cross section, three-layer beam, electromagnetic waves, and the rising tide caused by gravitational blowing. The theory of third-order differential equations can be applied to many fields, such as to biology, population growth, engineering, generic repression, control theory, and climate model. Recently, a great deal of interest in asymptotic properties and oscillatory behavior of solutions to different classes of third-order delay differential equations has been manifested (see, e.g., the papers [7–19] and the references cited therein).In particular, by using the Riccati transformation technique, the oscillation properties of the equation(1)ru′γ−1u′′t+qyσβ−1yσt=0,have been considered by Liu et al. [20] under the conditions γ≥β, r′t>0, and σ′t>0. Based on the properties of comformable fractional differential and integral, Feng and Meng have considered the following fractional-order dynamic equation on time scales [21]:(2)atrtxαtαγα+ptrtxαtαγ+qtfxt=0,and obtained some oscillation criteria where fx/xγ≥L>0 for x≠0, and γ≥1 is a quotient of two odd positive integers. By using the integral averaging technique, the equation(3)r2τr1τy′τα′′+ϕτ,y′δτ+qτfyστ=0,has been studied by Moaaz et al. [22] under the conditions that fx/xβ≥k2, α≥β, and σt is nondecreasing. However, most of the oscillation criteria have been obtained under the condition α≥β or α<β. Motivated by the above works, we further study the oscillation criteria of third-order equation when α≥β and α<β .In this paper, we mainly study the oscillation solution of the following third-order delay differential equation:(4)atrtx′t′α′+ptrtx′t′α+qtfxσt=0,where t≥t0 and α is a quotient of odd positive integers. Throughout this paper, we assume that(A1)
at∈C1t0,∞),(0,∞, rt∈C2(t0,∞),(0,∞), pt∈C(t0,∞),(0,∞), qt∈C(t0,∞),(0,∞).(A2)
σt∈Ct0,∞,ℝ, σt≤t, σ′t≥0, and limt⟶∞σt=∞.(A3)
f∈Cℝ,ℝ and fx/xβ≥k>0, and β is a quotient of odd positive integers.A functionxt is called a solution of (4) if xt satisfies (4) on Tx,∞ and if xt, rtx′t and rtx′t′α∈C1Tx,∞,ℝ where Tx≥t0. We consider solutions of (4) which satisfy supxt:T≤t<∞,Tx≥tx. A solution of (4) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (4) is said to be oscillatory if all its solutions are oscillatory.


## 2. Preliminaries

In this section, we give the following lemmas which are used as tools in establishing new oscillation criteria for (4).We introduce the following notation:(5)vt=exp∫t0tpsasds,ηt,t0=∫t0t1rsds,vt,t0=∫t0t1vsas1/αds,Rt,t0=∫t0tvs,t0rsds,Qt=λ1,ifβ>α;1,ifβ=α;λ2Rβ−α/αt,t1,ifβ<α,where λ1 and λ2 are positive real constants.Lemma 1.
Assume thatxt is the positive solution of (4) and(6)∫t0∞1vsas1/αds=∞,(7)∫t0∞1rsds=∞.
Then, one of the following two cases holds.(I)
xt>0, x′t>0, rtx′t′>0.(II)
xt>0, x′t<0, rtx′t′>0 for t≥t1≥t0, where t1 is sufficiently large.Proof.
Letxt be the positive solution of (4). Without loss of generality, we may assume that there exists a sufficiently large t1 such that xt>0 and xσt>0.
From (4), we obtain(8)vtatrtx′t′α′=vtatrtx′t′α′+vtptrtx′t′α=−vtqtfxσt<0.
Hence,vtatrtx′t′α is strictly decreasing on t1,∞. Because at>0, vt>0, we can deduce that rtx′t′ is eventually of one sign.
We can claim thatrtx′t′>0 for t≥t2≥t1. In fact, if rtx′t′≤0, according to the monotonicity of vtatrtx′t′α, we have(9)vtatrtx′t′α≤vt2at2rt2x′t2′α=−M,for M>0. Integrating (9) from t2 to t, we obtain(10)rtx′t≤rt2x′t2−∫t2tM1/αvsas−1/αds.
Lettingt⟶∞, from (6), we have rtx′t⟶−∞. Thus, there exists a sufficiently large t3 such that rtx′t<0 for t≥t3≥t2. Furthermore, we have rtx′t≤rt3x′t3. Dividing by rt and integrating on t3,t, we have(11)xt≤xt3+rt3x′t3∫t3t1rsds.
From (7), we have limt⟶∞xt=−∞. This contradicts xt>0, which implies rtx′t′>0. We complete the proof.Lemma 2.
Assume thatxt satisfies case (II). Suppose(12)∫t0∞1rv∫u∞1ruau∫u∞vsqsds1/αdudv=∞,and then limt⟶∞xt=0.Proof.
Letxt be an eventually positive solution of equation (4). Because xt has property (II), we obtain limt⟶∞xt=l≥0.
We claiml=0. Otherwise, if we assume l>0, then it follows that xt≥l for t≥t1≥t0.
From (A3), we have(13)vtatrtx′t′α′=−vtqtfxσt≤−kvtqtxβσt≤0.
Integrating the above inequality ont,∞, we have(14)−vtatrtx′t′α≤−limt⟶∞vtatrtx′t′α−k∫t∞vsqsxβσsds≤−k∫t∞vsqsxβσsds≤−klβ∫t∞vsqsds,which means(15)−rtx′t′≤−k1/αlβ/α1vtat∫t∞vsqsds1/α.
Integrating inequality (15) on t,∞, we obtain(16)rtx′t≤limt⟶∞rtx′t−k1/αlβ/α∫t∞1vuau∫u∞vsqsds1/αdu≤−k1/αlβ/α∫t∞1vuau∫u∞vsqsds1/αdu.
Further integrate the above inequality ont1,t to get(17)xt≤xt1−k1/αlβ/α∫t1t1rv∫v∞1vuau∫u∞vsqsds1/αdudv.
Lettingt⟶∞, by (12), we can deduce that limt⟶∞xt=−∞, which leads to a contradiction. So, we have l=0, and then limt⟶∞xt=0. We complete the proof.Lemma 3.
Assume (6) and (7) hold. If xt is a positive solution of equation (4) with case (I) for t≥t1≥t0, where t1 is sufficiently large, then for t∈t1,∞, the following inequality holds:(18)x′t≥vt,t1rtvtatrtx′t′α1/α,xβ−αt≥m,ifβ>α,1,ifβ=α,MRβ−αt,t1,ifβ<α,where m, M are positive real constants.Proof.
Letxt be a positive solution of (4). We assume that there exists a t1≥t0 such that xt>0 and xσt>0 for t≥t1. By Lemma 1, we know(19)rtx′t≥rtx′t−rt1x′t1=∫t1trsx′s′ds≥∫t1tvsasrsx′s′α1/αvsas1/αds.
From (13), we can see that vtatrtx′t′α is decreasing on t1,∞. Hence,(20)rtx′t≥vtatrtx′t′α1/α∫t1t1vsas1/αds=vtatrtx′t′α1/αvt,t1.
Thus,(21)x′t≥vt,t1rtvtatrtx′t′α1/α.
Sincext is positive and increasing, we have xt≥xt1=m1 for t≥t1≥t0. Moreover, due to the monotonicity of vtatrtx′t′α, we have(22)vtatrtx′t′α≤vt1at1rt1x′t1′α=M1,where M≥0 for t≥t2≥t1≥t0. Hence,(23)rtx′t′≤M11/α1vtat1/α.
Integrating the above inequality fromt1 to t, we have(24)rtx′t≤rt1x′t1+M11/α∫t1t1vsas1/αds=rt1x′t1+M11/αvt,t1≤rt1x′t1vt,t1+M11/αvt,t1≤rt1x′t1vt2,t1+M11/αvt,t1=M1∗vt,t1,where M1∗=rt1x′t1/vt2,t1+M11/α. Thus,(25)x′t≤1rtM1∗vt,t1.
Furthermore,(26)xt≤xt1+M1∗∫t1t1rsvs,t1ds≤xt1Rt,t1+M1∗Rt,t1≤xt1Rt2,t1+M1∗Rt,t1=M2Rt,t1,where M2=xt1/Rt2,t1+M1∗. Then, we have(27)xβ−αt≥m,ifβ>α,1,ifβ=α,MRβ−αt,t1,ifβ<α,for t≥t2≥t1, where m=m1β−α, M=M2β−α. The proof is complete.Lemma 4.
(see [23, Theorem 41]). Assuming that A and B are nonnegative real numbers, then(28)Aγ+γ−1Bγ−γABγ−1≥0γ>1,


## 3. Oscillation Results

In this section, we establish some new oscillation criteria for (4). For the following theorem, we introduce a class of function ℛ. Let(29)D=t,s:t0≤s≤t,D0=t,s:t0≤s<t.The functionH∈CD,ℝ is said to belong to the class ℛ, if(i)
Ht,t=0 for t≥t0, Ht,s>0 for t,s∈D0.(ii)
H has a continuous and nonpositive partial derivative ∂Ht,s/∂s on D0 with respect to s.Theorem 1.
Assume that (6), (7), and (12) hold. If there exists a function ρt∈C1t0.∞,R+ such that for all sufficiently large T, the following condition holds:(30)limsupt⟶∞∫Ttkρsvsqs−ρsα+1α+1αrσsβσ′svσs,t1Qσsαρ+′sρsα+1ds=∞,where ρ+′t=max0,ρ′t, then every solution of equation (4) is oscillatory or tends to zero.Proof.
Letxt be a nonoscillatory solution of (4). Without loss of generality, we may assume that there exists a sufficiently large t1 such that xt>0 and xσt>0 for t>t1. According to Lemma 1, xt is either the case of (I) or the case of (II).
Now we assumext satisfies the (I); define(31)wt=ρtvtatrtx′t′αxβσtt≥t1.
Then,wt>0, and(32)w′t=ρ′tρtwt+ρtvtatrtx′t′α′xβσt−ρtβvtatrtx′t′αx′σtσ′txβ+1σt.
From (13), we know(33)w′t≤ρ′tρtwt−kρtvtqt−ρtβvtatrtx′t′αx′σtσ′txβ+1σt≤ρ′tρtwt−kρtvtqt−βx′σtσ′txσtwt.
By Lemma3 and (A2), it is obvious that(34)x′σt≥vσt,t1rσtvσtaσtrσtx′σt′α1/α≥vtatrtx′t′α1/αvσt,t1rσt.
Thus,(35)w′t≤ρ′tρtwt−kρtvtqt−βσ′txσtvtatrtx′t′α1/αvσt,t1rσtwt.
So, we can obtain(36)w′t≤ρ′tρtwt−kρtvtqt−βσ′tvσt,t1rσtρ1/αtxβ−α/ασtw1+1/αt.
By Lemma3, we have(37)w′t≤ρ+′tρtwt−kρtvtqt−βσ′tvσt,t1rσtρ1/αtQσtw1+1/αt.
By Lemma4, let(38)γ=1+1α,A=βσ′tvσt,t1Qσtrσtρ1/αt1γwt,B=rσtρ1/αtβσ′tvσt,t1Qσtα/γρ+′tρtα1γα.
Then,(39)ρ+′tρtwt−βσ′tvσt,t1rσtρ1/αtQσtw1+1/αt≤ρtα+1α+1αrσtβσ′tvσt,t1Qσtαρ+′tρtα+1.
Combining (30) and (37), we obtain(40)w′t≤−kρtvtqt+ρtα+1α+1αrσtβσ′tvσt,t1Qσtαρ+′tρtα+1.
Integrating (39) from t1 to t,(41)∫t1tkρsvsqs−ρsα+1α+1αrσsβσ′svσs,t1Qσsαρ+′sρsα+1ds≤wt1−wt<wt1.
Lettingt⟶∞, we have(42)limsupt⟶∞∫t1tkρsvsqs−ρsα+1α+1αrσsβσ′svσs,t1Qσsαρ+′sρsα+1ds≤wt1<∞,which contradicts condition (30).
Ifyt satisfies (II), then from Lemma 2, it is obvious that limt⟶∞xt=0. The proof is complete.Theorem 2.
Assume that (6), (7), and (12) hold. If there exist functions ρt∈C1t0.∞,R+ and H∈ℛ such that(43)limsupt⟶∞1Ht,t1∫t1tkHt,sρsvsqs−αrσsβHt,sσ′svσs,tQσsαh+t,sα+1α+1ρsds=∞,for t≥t1≥t0, where ht,s=Ht,sρ+′s/ρs+Hs′t,s, h+t,s=max0,ht,s, then every solution of equation (4) is oscillatory or tends to zero.Proof.
Letxt be a nonoscillatory solution of (4). Without loss of generality, we may assume that there exists a sufficiently large t1 such that xt>0 and xσt>0 for t>t1. According to Lemma 1, xt is either the case of (I) or the case of (II). Now we assume xt satisfies (I). Let wt be defined as in Theorem 1; by (37), we have(44)kρtvtqt≤−w′t+ρ+′tρtwt−βσ′tvσt,t1rσtρ1/αtQσtw1+1/αt.
Substitutingt with s in the above inequality, multiplying each side by Ht,s, and integrating from t1 to t, we obtain(45)k∫t1tHt,sρsvsqsds≤−∫t1tHt,sw′sds+∫t1tHt,sρ+′sρswsds−∫t1tHt,sβσ′svσs,t1rσsρ1/αsQσsw1+1/αsds≤Ht,t1wt1+∫t1tHt,sρ+′sρs+Hs′t,swsds−∫t1tHt,sβσ′svσs,t1rσsρ1/αsQσsw1+1/αsds≤Ht,t1wt1+∫t1th+t,sws−Ht,sβσ′svσs,t1rσsρ1/αsQσsw1+1/αsds.
By Lemma4, let(46)γ=1+1α,A=Ht,sβσ′svσs,t1rσsρ1/αsQσs1/γws,B=rσsρ1/αsHt,sβσ′svσs,t1Qσsα/γh+t,sγα.
Then,(47)h+t,sws−Ht,sβσ′svσs,t1rσsρ1/αsQσsw1+1/αs≤αrσsβHt,sσ′svσs,t1Qσsαh+t,sα+1α+1ρs.
Combining (45) and (47), we have(48)k∫t1tHt,sρsvsqsds≤Ht,t1wt1+∫t1tαrσsβHt,sσ′svσs,t1Qσsαh+t,sα+1α+1ρsds.
BecauseHs′t,s≤0 for t≥t1≥t0, we can get Ht,t0≥Ht,t1>0. Hence,(49)∫t1tkHt,sρsvsqs−αrσsβHt,sσ′svσs,t1Qσsαh+t,sα+1α+1ρsds≤Ht,t1wt1≤Ht,t0wt1.
DividingHt,t0, we have(50)1Ht,t0∫t1tkHt,sρsvsqs−αrσsβHt,sσ′svσs,t1Qσsαh+t,sα+1α+1ρsds≤wt1.
Lettingt⟶∞, we have(51)limsupt⟶∞1Ht,t0∫t1tkHt,sρsvsqs−αrσsβHt,sσ′svσs,t1Qσsαh+t,sα+1α+1ρsds≤wt1<∞,which contradicts (43).
Ifyt satisfies (II), then from Lemma 2, it is obvious that limt⟶∞xt=0. The proof is complete.


## 4. Examples

Example 1.
Consider the equation(52)tα−1/2x″tα′+1tx″tα+1t2xβtβext/β+1=0,where β,α>1, at=tα−1/2, rt=1, pt=1/t, qt=1/t2, fx=xβex+1, σt=t/β. Then, we have(53)1≤vt=e∫t0tps/asds=e∫t0t1/sα+1/2ds=e2/1−2α1/tα−1/2−1/t0α−1/2≤e2.
Now we check conditions (6), (7), and (12).(54)∫t0∞1vsas1/αds≥e−2/α∫t0∞1s1−1/2αds=∞,∫t0∞1rsds=∫t0∞ds=∞,∫t0∞1rv∫u∞1ruau∫u∞vsqsds1/αdudv≥∫t0∞∫v∞1uα−1/2∫u∞1s2ds1/αdudv=∫t0∞∫v∞1u1+1/2αdudv=2α∫t0∞1v1/2αdv=∞.
So, conditions (6), (7), and (12) hold. Because v∞,t0=∞, there exists a sufficiently large T≥t1 such that vt,t1≥vt/β,t1≥1 for t∈T,∞. Furthermore, fx/xβ≥k=1, and letting ρt=t, we have the following three situations.
Forβ>α,(55)limsupt⟶∞∫t1tkρsvsqs−ρsα+1α+1αrσsβσ′svσs,t1Qσsαρ+′sρsα+1ds≥limsupt⟶∞∫t1T1s−1α+1α+1αλ1vs/β,t1α·1sαds+limsupt⟶∞∫Tt1s−1α+1α+1αλ1α·1sαds=∞.
Forβ=α, from the above, it is obvious that (30) holds.
Forβ<α,(56)vt,t1≤∫t1t1a1/αsds≤∫t1t1s1−1/2αds≤2αt1/2α,Rt,t1≤∫t1t2α·s1/2αds≤2α·t1/2α+1,limsupt⟶∞∫t1tkρsvsqs−ρsα+1α+1αrσsβσ′svσs,t1Qσsαρ+′sρsα+1ds≥limsupt⟶∞∫t1∞1s−1α+1α+1αλ2vs/β,t1αRα−βsβ,t1·1sαds≥limsupt⟶∞∫t1T1s−2α2λ2αα+1α+12αβ1vs/β,t1α·1sβ/2α+β−1/2ds+limsupt⟶∞∫Tt1s−2α2λ2αα+1α+12αβ·1sβ/2α+β−1/2ds=∞.
From above, we know(57)limsupt⟶∞∫t1tkρsvsqs−ρsα+1α+1αrσsβσ′svσs,t1Qσsαρ+′sρsα+1ds=∞.
Therefore, by Theorem1, every solution of (52) is oscillatory or tends to zero.Example 2.
Consider the equation(58)tα−1/21tx′t′α′+1t1tx′t′α+1t2xβtβex2t/β=0,where at=tα−1/2, rt=1/t, pt=1/t, qt=1/t2, fx=xβex2, σt=t/β, which satisfy conditions (A1)–(A3). Similar to the Example 1, we can easily verify that (6), (7), and (12) hold. Furthermore, we choose ρt=t, Ht,s=t−s. By Theorem 2, we have ht,s=t/s−2 and(59)limsupt⟶∞1Ht,t1∫t1tkHt,sρsvsqs−αrσsβHt,sσ′svσs,tQσsαh+t,sα+1α+1ρsds=∞.
Then, by Theorem2, we know that any solution of (58) is oscillatory or tends to zero.


## 5. Conclusions

In this work, by using the generalized Riccati technique, we establish some new oscillation criteria of (4) and give some examples to verify the oscillation criteria. These new oscillation criteria complement some known results for third-order delay differential equations. Furthermore, in future work we will study the forced oscillatory behavior of this equation with forcing term.



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