# On the Oscillation Criteria for Fourth-Orderp-Laplacian Differential Equations with Middle Term

Authors: Dandan Yang; Chuanzhi Bai
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the oscillatory properties of the solutions of a class of fourth-orderp-Laplacian differential equations with middle term. The new oscillation criteria obtained by using the theory of comparison with first- and second-order differential equations and a refinement of the Riccati transformations. The results in this paper improve and generalize the corresponding results in the literatures. Three examples are provided to illustrate our results.


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## Body

## 1. Introduction

In this paper, we are concerned with the oscillation behavior of solutions of the fourth-orderp-Laplacian differential equations with middle term
(1)rtz′′′tp1−2z′′′t′+ptz′′′tp1−2z′′′t+qtxσtp2−2xσt=0,where t≥t0, pi>1i=1,2, are real numbers, zt≔xt+atxτt, r∈C1t0,∞, p,q,a∈Ct0,∞, rt>0, qt>0, r′t≥0, 0≤at≤a0<∞, τ∈C1t0,∞, σ∈Ct0,∞, τ′t>0, τt≤t, and limt⟶∞τt=limt⟶∞σt=∞. Furthermore, we investigate (1) under the condition.
(2)H1limt⟶∞∫t0t1rsexp−∫t0spξrξdξ1/p1−1ds=∞.A functionz∈C′′′tx,∞, tx≥t0, is called a solution of equation (1), if rtz′′′tp1−2z′′′t∈C1[tx,∞, and x satisfies (1) on tx,∞.Since the fourth-order delay differential equations described many real-life applications, such as models related to physical, chemical, and biological phenomena, in the last decades, a lot of research has been done on the oscillatory behavior of fourth-order delay differential equations, see [1–14] and the references cited therein. On the other hand, the p-Laplacian differential equations there are some important applications in continuum mechanics and elasticity theory [15–23], the oscillatory behavior of the solutions of fourth-order differential equations with p-Laplacian like operator have been investigated in recent years by using different methods and various techniques, for example, [24–26].In this paper, motivated by [26–28], we will give some new sufficient conditions for that oscillatory behavior of (1). In Section 2, we will provide some lemmas that will help us to prove our main results. In Section 3, based on the comparison with first- and second-order differential equations and a refinement of the Riccati transformations, we establish some new oscillation criteria of (1).


## 2. Preliminaries

First, we give the following lemmas that can discuss our main results.Lemma 1 ([1], Lemma 2.21).
Letxt be a positive and n-times differentiable function on an T,∞ with its nth derivative xnt nonpositive on T,∞ and not identically zero on any interval of the form T′,∞, T′≥T. Then, there exists an integer l, 0≤l≤n−1, with n+l odd such that for some large T∗≥T,
(3)−1l+jxj>0onT∗,∞j=l,l+1,⋯,n−1,xi>0onT∗,∞i=1,2,⋯,l−1whenl>1.Lemma 2 ([29]).
Let the functionx satisfies xit>0, i=0,1,⋯,n, and xn+1t<0 eventually. Then,
(4)xttn/n!≥x′ttn−1/n−1!.Lemma 3 ([30], Lemma 2.2.3).
Leth∈CnT,∞ and ht>0. Assume that hnt is of a fixed sign, on t0,∞, hnt not identically zero and that there exists a t1≥t0 such that, for all t≥t1, hn−1thnt≤0. If we have limt⟶∞ht≠0, then there exists tλ≥t0 such that
(5)ht≥λn−1!tn−1hn−1t,for every λ∈0,1 and t≥tλ.Lemma 4 ([31], Lemma 2.3).
Assume thatα is a quotient of odd positive integers; V>0 and U∈ℝ are constants. Then
(6)Uy−Vyα+1/α≤ααα+1α+1Uα+1V−α.The following lemma will be used in the proof of our main results in the next section.Lemma 5.
Letxt be an eventually positive solution of (1); if (H1) holds, then z′>0 and z′′′>0.Proof.
Sincext is an eventually positive solution of (1), then xt>0, xτt>0, and xσt>0 for t≥t1≥t0. Thus, zt>0, t≥t1. From (1), we obtain
(7)rtz′′′tp1−2z′′′t′+ptz′′′tp1−2z′′′t=−qtzσtp2−1zσt<0.
Multiplying byexp∫t0tps/rsds on both sides of the above equation, we get
(8)ddtexp∫t0tpsrsdsrtz′′′tp1−2z′′′t=−qtexp∫t0tpsrsdszσtp2−1zσt<0.
Thus(9)exp∫t0tpsrsdsrtz′′′tp1−2z′′′t,is decreasing, and hence, z′′′t is eventually of one sign. Hence, we assert that z′′′t>0 for any t≥t1. Otherwise, if z′′′t<0 for any t≥t1, we get by (8) that
(10)exp∫t0tpsrsdsrtz′′′tp1−2z′′′t≤exp∫t0t1psrsdsrt1z′′′t1p1−2z′′′t1=−Mp1−1exp∫t0t1psrsds,where M=rt11/p1−1z′′′t1>0. From (10), one has
(11)−z′′′tp1−1≥Mp1−1rtexp−∫t1tpsrsds,that is,
(12)z′′′t≤−M1rtexp−∫t1tpsrsds1/p1−1.
Consequently,(13)z′′t≤z′′t1−M∫t1t1rsexp−∫t1spξrξdξ1/p1−1ds⟶−∞,ast⟶∞.
Sozt is an eventually negative function which contradicts zt>0. Thus, we have z′′′t>0. From (1), we get
(14)rtz′′′tp1−1′=r′tz′′′tp1−1+p1−1rtz′′′tp1−2z4t≤0,from which it follows that
(15)z4t≤0,t≥t1≥t0.
By (15) and Lemma 1 (set n=4 and l=3), one has that z′t>0, t≥t1≥t0. The proof is completed.Lemma 6 ([32]).
Letα∈0,1 be a quotient of two positive integers. Assume that R is a positive continuous function on t0,∞. If
(16)liminft⟶∞∫σttRsds>1e,then the first-order delay differential equation
(17)y′t+Rtyασt=0,is oscillatory.


## 3. Main Results

In the following theorem, we then by using a comparison strategy involving first-order differential equations to provide an oscillation criterion for equation (1).For convenience, let(18)At=1aτ−1t1−τ−1τ−1t3τ−1t3aτ−1τ−1t,t≥t0.Theorem 7.
Assume that (H1) and
(19)H2τ−1τ−1t3τ−1t3aτ−1τ−1t<1,t≥t0,hold. If the differential equation
(20)ω′t+μ6p2−1qtt3p2−1exp∫t0tps/rsdsAp2−1σtrp2−1/p1−1τ−1σtexpp2−1/p1−1∫t0τ−1σtps/rsdsωp2−1/p1−1τ−1σt=0,is oscillatory for some μ∈0,1, then the differential equation (1) is oscillatory.Proof.
Assume that (1) has a nonoscillatory solution in t0,∞. Without loss of generality, we may let x be an eventually positive solution of (1). Then, there exists a t1≥t0 such that xt>0, xτt>0, and xσt>0 for t≥t1. Let
(21)ωt≔rtexp∫t0tpsrsdsz′′′tp1−1>0fromLemma5,which having in mind (1) gives
(22)ω′t+qtexp∫t0tpsrsdsxσtp2−1=0.
From the definition ofzt, one has
(23)xt=1aτ−1tzτ−1t−xτ−1t.
By repeating the same process, we have(24)xt=zτ−1taτ−1t−1aτ−1tzτ−1τ−1taτ−1τ−1t−xτ−1τ−1taτ−1τ−1t≥zτ−1taτ−1t−1aτ−1tzτ−1τ−1taτ−1τ−1t.
Setn=3 in Lemma 2, we obtain zt≥1/3tz′t, which implies that zt/t3 is nonincreasing. Moreover, by the fact τt≤t that gives
(25)τ−1t3zτ−1τ−1t≤τ−1τ−1t3zτ−1t.
Combining (24) and (25), which yields
(26)xt≥1aτ−1t1−τ−1τ−1t3τ−1t3aτ−1τ−1tzτ−1t=Atzτ−1t.
Between equations (1) and (26), we obtain
(27)ω′t+qtexp∫t0tpsrsdsAp2−1σtzp2−1τ−1σt≤0.
Sincez is positive and increasing (by Lemma 5), we have limt⟶∞zt=0. So, from Lemma 3, one has
(28)zt≥μ6t3z′′′t,
for someμ∈0,1. It follows between (27) and (28) that, for all μ∈0,1, ω is a positive solution of the first-order delay differential inequality
(29)ω′t+μ6p2−1qtt3p2−1exp∫t0tps/rsdsAp2−1σtrp2−1/p1−1τ−1σtexpp2−1/p1−1∫t0τ−1σtps/rsdsωp2−1/p1−1τ−1σt≤0.
It is well known (see [33] and Theorem 7) that the corresponding equation (20) also has a positive solution, which is a contradiction. The theorem is proved.Corollary 8.
Assume that (H1) and (H2) hold, and p1≥p2. If
(30)liminft⟶∞∫τ−1σttμ6p2−1qss3p2−1exp∫t0spξ/rξdξAp2−1σsrp2−1/p1−1τ−1σsexpp2−1/p1−1∫t0τ−1σspξ/rξdξds>1e,
for someμ∈0,1, then the differential equation (1) is oscillatory.Proof.
From Lemma6, we know that (30) implies the oscillatory of (20).Theorem 9.
Assume that (H1) and (H3) a∈Ct0,∞, 0≤at<a0<1 hold. If the differential equation
(31)ω′t+μ6p2−1qtσt3p2−1exp∫t0tps/rsds1−aσtp2−1rp2−1/p1−1σtexpp2−1/p1−1∫t0σtps/rsdsωp2−1/p1−1σt=0,is oscillatory, then every solution of equation (1) is oscillatory.Proof.
Assume the contrary thatx is an eventually positive solution of equation (1). Thus, we may suppose that xt, xτt, and xσt are positive for all t≥t1 that are sufficiently large. From zt=xt+atxτt and z′t>0 (by Lemma 5), we obtain by xt≤zt and τt≤t that
(32)xt=zt−atxτt≥zt−atzτt≥1−atzt.
Combining (21) and (32), one has
(33)ω′t+qtexp∫t0tpsrsds1−aσtp2−1zσtp2−1≤0.
By using (28) and (33), we get that
(34)ω′t+μ6p2−1qtσt3p2−1exp∫t0tpsrsds1−aσtp2−1z′′′σtp2−1≤0.
From (21), (34) implies that
(35)ω′t+μ6p2−1qtσt3p2−1exp∫t0tps/rsds1−aσtp2−1rp2−1/p1−1σtexpp2−1/p1−1∫t0σtps/rsdsωp2−1/p1−1σt≤0.
It is well known (see [33] and Theorem 7) that the corresponding equation (31) also has a positive solution, which is a contradiction. The theorem is proved.Corollary 10.
Assume that (H1) and (H3) hold, and p1≥p2. If
(36)liminft⟶∞∫σttμ6p2−1qsσs3p2−1exp∫t0spξ/rξdξ1−aσsp2−1rp2−1/p1−1σsexpp2−1/p1−1∫t0σspξ/rξdξds>1e,
for someμ∈0,1, then the differential equation (1) is oscillatory.Proof.
From Lemma6, we know that (36) implies the oscillatory of (31).Lemma 11.
Assume that (H1) holds, x is an eventually positive solution of (1), and
(37)∫t2∞hsqsexp∫t0spξrξdξσs3p1−1s3p1−1−2p1−1p1p1rsh′sp1ε1s2hsp1−1exp−∫t0spξ/rξdξds=∞,for some constants ε1∈0,1. Then, we have z′′<0.Proof.
Our proof by reduction to the absurd. Assume thatz′′t>0. From Lemma 2, we obtain
(38)z′tzt≤3t.
Integrating the above equality fromσt to t, one find that
(39)zσtzt≥σt3t3.
Letht=z′t in Lemma 3, then
(40)z′t≥ε12t2z′′′t,for all ε1∈0,1 and every sufficiently large t. Now, we define a function ϕ by
(41)ϕt≔htexp∫t0tps/rsdsrtz′′′tp1−1zp1−1t>0.
By differentiating (41) and using the inequalities (39) and (40), we get
(42)ϕ′t≤h′thtϕt−htqtexp∫t0tpsrsdsσt3p1−1t3p1−1`zp2−p1σt−p1−1ε12t2exp−1/p1−1∫t0tps/rsdsh1/p1−1tr1/p1−1tϕ1+1/p1−1.
Sincez′t>0, there exist a t2≥t1 and a constant M>0 such that zt>M, for all t≥t2. Without loss of generality, we may let M≥1. By using Lemma 4 with
(43)U=h′h,V=p1−1ε12t2exp−1/p1−1∫t0tps/rsdsh1/p1−1tr1/p1−1t,we obtain
(44)ϕ′t≤−Mp2−p1htqtexp∫t0tpsrsdsσt3p1−1t3p1−1+2p1−1p1p1rth′tp1ε1t2htp1−1exp−∫t0tps/rsds.
This implies that(45)∫t2thsqsexp∫t0spξrξdξσs3p1−1s3p1−1−2p1−1p1p1rsh′sp1ε1s2hsp1−1exp−∫t0spξ/rξdξds≤ϕt2,≤∫t2tMp2−p1hsqsexp∫t0spξrξdξσs3p1−1s3p1−1−2p1−1p1p1rsh′sp1ε1s2hsp1−1exp−∫t0spξ/rξdξds≤ϕt2,which contradicts (37). The proof is completed.For convenience, let(46)θt=λp2−1/p1−1∫t∞1rξ∫ξ∞qsexp∫tspξrξdξσssp2−1ds1/p1−1dξ.Theorem 12.
Assume that (H1) and (37) hold for some ε1∈0,1 and p2≥p1. If
(47)y′′t+θtyt=0,is oscillatory, then (1) is also oscillatory.Proof.
We use the reduction to the absurd arguments. Assume that (1) has a nonoscillatory solution in t0,∞. Without loss of generality, we only need to be concerned with positive solutions of equation (1). Then, there exists a t1≥t0 such that xt>0, xτt>0, and xσt>0 for t≥t1. From Lemmas 4 and 11, one has that
(48)z′t>0,z′′t<0,z′′′t>0,for t≥t2, where t2 is sufficiently large. Integrating (8) from t to ρ, we obtain
(49)exp∫t0ρpsrsdsrρz′′′ρp1−1=exp∫t0tpsrsdsrtz′′′tp1−1−∫tρqsexp∫t0spξrξdξzp2−1σsds.
By using Lemma3 in [26] together with (48), we have
(50)zσtzt≥λσtt,for all λ∈0,1. This coupled with (49); we can arrive at
(51)exp∫t0ρpsrsdsrρz′′′ρp1−1−exp∫t0tpsrsdsrtz′′′tp1−1+λp2−1∫tρqsexp∫t0spξrξdξσssp2−1zp2−1sds≤0.
Sincez′>0, then (51) reduced to the following inequality
(52)exp∫t0ρpsrsdsrρz′′′ρp1−1−exp∫t0tpsrsdsrtz′′′tp1−1+λp2−1zp2−1t∫tρqsexp∫t0spξrξdξσssp2−1ds≤0.
Takingρ⟶∞ in (52), one has
(53)−exp∫t0tpsrsdsrtz′′′tp1−1+λp2−1zp2−1t∫t∞qsexp∫t0spξrξdξσssp2−1ds≤0,from which we readily obtain
(54)z′′′t≥λp2−1/p1−1r1/p1−1texp−1p1−1∫t0tpsrsdszp2−1/p1−1t×∫t∞qsexp∫t0spξrξdξσssp2−1ds1/p1−1.
Integrating the above inequality fromt to ∞, we obtain
(55)−z′′t≥λp2−1/p1−1zp2−1/p1−1t×∫t∞1rξ∫ξ∞qsexp∫tspξrξdξσssp2−1ds1/p1−1dξ,which implies that
(56)z′′t≤−θtzp2−1/p1−1t.
Now, if we defineγt=z′t/zt, then γt>0 for t≥t1, and
(57)γ′t=z′′tzt−z′tzt2.
By using (56) and the definition of γt, we see that
(58)γ′t≤−θtzp2−1/p1−1tzt−γ2t.
Sincez′t>0, there exists a constant M>0 such that zt≥Mp2−1, for all t≥t2, where t2 is sufficiently large. Without loss of generality, we may let M≥1. Then, by (56), one has
(59)γ′t+γ2t+θt≤γ′t+γ2t+Mp2−p1θt≤0.
It is well known (see [34]) that the differential equation (47) is oscillatory if and only if there exists a t3>maxt1,t2 such that (59) holds, which is a contradiction. The theorem is proved.For convenience, let(60)βt=∫t∞1rξ∫ξ∞qsexp∫tspξrξdξds1/p1−1dξ.Theorem 13.
Assume that (H1) and (37) hold for some ε1∈0,1, p2≥p1, and σ′t≥1. If
(61)1σ′ty′t′+βtyt=0,is oscillatory, then (1) is also oscillatory.Proof.
As in the proof of Theorem12, one has (49). Hence, it follows between σ′t≥0 and z′t≥0 that
(62)exp∫t0ρpsrsdsrρz′′′ρp1−1−exp∫t0tpsrsdsrtz′′′tp1−1+zp2−1σt∫tρqsexp∫tspξrξdξds≤0.
By (62), similar to (56), we obtain
(63)z′′t≤−βtzp2−1/p1−1σt.
We now defineκ by
(64)κt=z′tzσt,then κt>0 for t≥t1, and
(65)κ′t=z′′tzσt−z′tz2σtz′σtσ′t≤z′′tzσt−z′tzσt2σ′t.
By using (63) and the definition of κt, similar to (59), we see that
(66)κ′t+βt+σ′tκ2t≤0.
It is well known (see [34]) that the differential equation in (61) is nonoscillatory if and only if there exists a t3>maxt1,t2 such that (66) holds, which is a contradiction. So, the theorem is proved.In the following, we employ the integral averaging technique to establish a Philos-type oscillation criterion for (1).Let(67)D=t,s∈R2:t≥s≥t0,D0=t,s∈R2:t>s≥t0.Corollary 14.
Assume that (H1) and (37) hold for some ε1∈0,1 and p1=p2. Let there exists a continuous function H:D⟶R such that Ht,s=0 for t≥t0,Ht,s>0, t,s∈D0, and H has a nonpositive continuous partial derivative with respect to the second variable in D. Suppose that there exists function h∈CD,R such that
(68)−∂∂sHt,s=ht,sHt,sforallt,s∈D.
If(69)limsupt⟶∞1Ht,t0∫t0tHt,sθs−14h2t,sds=∞,or
(70)liminft⟶∞t∫t∞θsds>14,then (1) is oscillatory.Example 15.
Consider the equation(71)z4t+1tz′′′t+q0t4xσ0t=0,where t≥1, zt=xt+a0xτ0t, q0>0, τ0∈a0−1/3,1, and σ0∈0,τ0. Let p1=p2=2, rt=1, at=a0, τt=τ0t, σt=σ0t, pt=1/t, and qt=q0/t4. It is easy to check that (H1) and (H2) are satisfied. Moreover, we have
(72)At=1a01−1a0τ03.
Ifq0>0 satisfies inequality (30), that is
(73)liminft⟶∞∫σ0/τ0ttμ6q0s4s3Aσ0sexp∫σ0/τ0ss1ξdξds=q0μ61a01−1a0τ03τ0σ0lnτ0σ0>1e,then equation (71) is oscillatory by Corollary 8.Example 16.
Consider the equation(74)t∣z′′′t∣z′′′t′+z′′′tz′′′t+q0t4xt2xt2=0,where t≥t0=1, zt=xt+1/3xt/4, and q0>0. Let p1=3, p2=2, rt=t, τt=t/4, σt=t/2, at=1/3, pt=1, and qt=q0/t4. It is easy to check that (H1) and (H3) are satisfied. Moreover, we find
(75)liminft⟶∞∫σttμ6p2−1qsσs3p2−1exp∫t0spξ/rξdξ1−aσsp2−1rp2−1/p1−1σsexpp2−1/p1−1∫t0σspξ/rξdξds=liminft⟶∞∫t/2tμ6q0/s4s/232/3ss/21/2exp1/2∫1s/21/ξdξds=μq036ln2.Thus, by Corollary10, all solutions of equation (74) are oscillatory if q0>36/μeln2.Example 17.
Consider the equation(76)t2z′′′t2z′′′t′+tz′′′t2z′′′t+q0t7xct2xct=0,where t≥t0=1, zt=xt+a0xτ0t, q0>0, and c∈0,1. Let p1=p2=4, rt=t2, σt=ct, pt=t, and qt=q0/t8. It is easy to verify that (H1) holds. Set hs=s6, for any q0>0; then, it is easy to verify that
(77)∫t2∞hsqsexp∫t0spξrξdξσs3p1−1s3p1−1−2p1−1p1p1rsh′sp1ε1s2hsp1−1exp−∫t0spξ/rξdξds=∫t2∞q0c9t01s−812ε13t01s2ds=∞,that is, (37) holds. Moreover, one has
(78)θt=λp2−1/p1−1∫t∞1rξ∫ξ∞qsexp∫tspξrξdξσssp2−1ds1/p1−1dξ=λ∫t∞c3ξ2∫ξ∞q0s7stds1/3dξ=35λcq0631t2.Thus, by Corollary14, we obtain that equation (76) is oscillatory if q0>max65/12λc3,812ε13c9.


## 4. Conclusions

With the help of the comparison strategies involving first- and second-order differential equations and a refinement of the Riccati transformations, some new criteria for oscillation of fourth-orderp-Laplacian differential equations with middle term are established. The results obtained here complement and extend some known results in [26].



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