# Oscillatory Behavior of Even-Order Half-Linear Neutral Differential Equations

Authors: S. Sangeetha; S. K. Thamilvanan; Ethiraju Thandapani
Journal: International Journal of Differential Equations
 (2022)

---

## Abstract

This paper discusses some sufficient conditions for oscillatory behavior of even-order half-linear neutral differential equation. An example is given to illustrate the main result.


---

## Body

## 1. Introduction

In this paper, we investigate the oscillatory behavior of even-order half-linear neutral differential equation(1)pγyn−1γα′+qγyn−1γα+rγxασγ=0,where n≥2 is an even integer and yγ=xγ+hγxζγ with α is the ratio of two odd positive integers.Throughout this paper, we assume that the following conditions are satisfied:K1 p, q, r ϵγ0,∞,ℝ,pγ≥0,qγ≥0,rγ≥0 and(2)Eγ=∫γ0γ1psexp−∫γ0sqιpιdι1/αds=∞,asγ⟶∞.K2h:γ0,∞,ℝ,0≤hγ<1 for all γ≥γ0, K3ζ,σ:γ0,∞,ℝ with ζγ≤γ,σγ≤γ and limγ⟶∞ζγ=limγ⟶∞σγ=∞.By a solutionx of (1), we mean a function xϵℂγx,∞,ℝ for some γx≥γ0 which has a property yϵℂ2γx,∞,ℝ and satisfies (1) on γx,∞.We consider the solution x of (1) on some half-line γx,∞ and satisfy the condition as follows:(3)supxγ:Γ≤γ≤∞>0for anyΓ≥γx.Such solutions can be called as oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is non-oscillatory.The oscillation problem for neutral differential (1) has been studied by many authors with various techniques. Most of the results are obtained for the case 0≤hι<∞, see [1–6], hι≥1, see [7, 8], and α≥1, see [2, 5, 6, 8–12]. The purpose of this paper is to extend the main results for α≤1 using the Philos function method. Furthermore, the oscillation conditions are obtained in order to make it easy to apply. Also, it is our hope that the present paper will be a good contribution to study the oscillatory behavior of solutions of the even-order neutral differential equations. An example is provided to illustrate the main result.


## 2. Main Results

To prove our main results, we use the following notations:(4)ψγ=1−hγ.We introduce a function class P by following Philos [13]. Let G0=γ,ιϵR2:γ>ι≥γ0 and G=γ,ιϵR2:γ≥ι≥γ0. We say that the function HϵℂG,ℝ belongs to the class P, denoted by HϵP if(i)
Hγ,γ=0 for γ≥γ0, and Hγ,ι>0 on γ,ιϵG0(ii)
H has a continuous and nonpositive partial derivative onG0 with respect to the second variableTheorem 1.
Let the conditionsK1−K3 hold. Let h, H : G⟶ℝ be continuous functions such that H belongs to the class P and(5)−∂H∂ιγ,ι=hγ,ιHγ,ιfor allγ,ιϵG0.If there exists a positive functionρϵℂ1γ0,∞,ℝ such that for some β≥1,(6)limγ⟶∞sup1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι=∞,for sufficiently large γ≥Γ, where(7)βγ=ρ′γργ−qγpγ,Fγ=μγn−2ρ1/αγp1/αγn−2,Ψγ=ργrγψασγσγγαn−1,Φγ,ι=−hγ,ιHγ,ι+βιHγ,ι.for some μϵ0,1, then every solution of (1) is oscillatory.Proof.
Letxγ be a positive solution of (1). Then, yγ is positive for all γ≥γ0. From equation (1), we get that(8)pγyn−1γα′+qγyn−1γα<0.
Letuγ=pγyn−1γα; then, from (8), we get that(9)u′γ+qγpγuγ<0,exp∫γ0γqιpιdιuγ′<0,which says that exp∫γ0γqι/pιdιuγ is decreasing and does not change its sign which leads to two cases:(I)yn−1γ<0(II)yn−1γ>0
Now, we consider that Case (I) holds. Ifγ≥Γ, then we obtain(10)exp∫γ0γqιpιdιpγyn−1γα≤exp∫γ0γqιpιdιpΓyn−1Γα≡−Mαexp∫γ0γqιpιdι,where M=p1/αΓyn−1Γ>0. Now, we have(11)yn−1γ≤−Mαpγexp−∫Γγ0qιpιdι1/α.
Integrating the last inequality with respect toγ, we get(12)∫Γγyn−1tdt≤−M∫Γγ1pιexp−∫Γιqupudu1/αdι.
Asγ⟶∞, we have limγ⟶∞yn−2γ=−∞, and consequently, we see that limγ⟶∞yγ=−∞ which is a contradiction.
Now, let us take Case (II).
Sinceyn−1>0, from Lemma 1 of [5], we have y′>0. From the definition of yγ, we see that(13)xγ=yγ−hγxζγ≥1−hγyγ, (14)xσγ≥ψσγyσγ.
Then, (1) becomes(15)pγyn−1γα′+qγyn−1γα+rγψασγyασγ≤0.
We consider the Riccati transformation(16)wγ=ργpγyn−1γαyαγ.
We see thatwγ>0, and from (16), we get(17)w′γ≤ρ′γργ−qγpγwγ−rγργψασγyασγyαγ−αy′γpγργyn−1γαyα+1γ.
From (18), we get(18)w′γ≤ρ′γργ−qγpγwγ−Ψγ−αwγy′γyγ.
The above inequality can be written as(19)w′γ≤βγwγ−Ψγ−αFγw1+1/αγ.
Multiplying (19) by Hγ,ι and integrating from Γtoγ, we have for some β≥1 and for all γ≤Γ≤γ0(20)∫ΓγHγ,ιΨιdι≤−∫ΓγHγ,ιw′ιdι+∫ΓγHγ,ιβιwιdι−αβ∫ΓγHγ,ιw1+1/αιFιdι−αβ−1β∫ΓγHγ,ιw1+1/αιFιdι.
Integrating by parts yields(21)∫ΓγHγ,ιw′ιdι=−Hγ,ΓwΓ−∫Γγ∂Hγ,ι∂ιwιdι.
Substituting (5) and (21) into (20), we get(22)∫ΓγHγ,ιΨιdι≤Hγ,ΓwΓ+∫ΓγΦγ,ιwιdι−αβ∫ΓγHγ,ιw1+1/αιFιdι−αβ−1β∫ΓγHγ,ιw1+1/αιFιdι.
Now, by applying the inequality,Au−Bu1+1/α≤αα/α+1α+1Aα+1/Bα (22), we get(23)∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≤Hγ,ΓwΓ−αβ−1β∫ΓγHγ,ιw1+1/αιFιdι.
So, for everyγ≥Γ, we obtain(24)∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≤Hγ,ΓwΓ,which is a contradiction.
Next, oscillation result follows immediately from Theorem1.Corollary 1.
Let the assumptions of Theorem1 be holded except that the condition (4) is replaced by(25)limγ⟶∞sup1Hγ,ι∫ΓγHγ,ιΨιdι=∞,limγ⟶∞sup1Hγ,ι∫ΓγΦα+1γ,ιHαγ,ιFαιdι<∞.Then, every solution of (1) is oscillatory.Theorem 2.
Let the conditionsK1−K3 and (2) be satisfied. Let H and h be same as in Theorem 1. We suppose that(26)0≤infι≥γ0liminfγ⟶∞Hγ,ιHγ,γ0≤∞.If there exist functionsϕϵCγ0,∞,ℝ and ρϵℂ1γ0,∞,ℝ such that for some β≥1(27)limsupγ⟶∞1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≥ΦΓ, (28)∫Γ∞FιΦ+1+1/αιdι=∞,for all sufficiently large γ2ϵγ1,∞⊆γ0,∞ and all Γ≥γ2, where ψγ,ι and Fγ are as in Theorem 1 and Φ+γ=maxϕγ,0, then every solution of (1) is oscillatory.Proof.
Letxγ be a non-oscillatory solution of (1), say xγ>0,xσγ>0 and xζγ>0 for γ≥γ1 for some γ1≥γ0. Proceeding as in the proof of Theorem 1, we again have two cases:(I)yn−1γ<0(II)yn−1γ>0
Forγ≥γ2, if Case (I) holds proceeding as in the proof of Theorem 1, we obtain a contradiction to the positivity of y.
Next, we assume case (II) holds, proceeding as in the proof of Theorem1; we again arrive at (23), which can be written as γ>Γ>γ0.(29)1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≤wΓ−1Hγ,Γ∫Γγαβ−1βHγ,ιw1+1/αιFιdι.
From (29), we have(30)limγ⟶∞sup1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≤wΓ−limγ⟶∞inf1Hγ,Γ∫Γγαβ−1βHγ,ιw1+1/αιFιdι.
In view of (27), it follows from (30) that(31)wΓ≥ΦΓ+limγ⟶∞inf1Hγ,Γ∫Γγαβ−1βHγ,ιw1+1/αιFιdι,for all γ≥γ3 for any β>1. Then, for all Γ≥γ3,(32)WΓ≥ϕΓ, (33)limγ⟶∞inf1Hγ,Γ∫γ4γHγ,ιw1+1/αιFιdι≤βwγ3−ϕγ3αβ−1<∞.
Now, we claim that(34)∫γ3∞w1+1/αιFιdι<∞.
If not, then(35)∫γ3∞w1+1/αιFιdι=∞.
By (26), there exists a constant ϵ>0 such that(36)infι≥γ0lim infγ⟶∞Hγ,ιHγ,γ0>ϵ.
On the other hand, by virtue of (35) for any positive number δ, there exists γ4>γ3 such that(37)∫γ4γw1+1/αιFιdι≥δϵfor allγ≥γ5.
Using integration by parts and taking (37) into account, we conclude that for all γ≥γ5,(38)1Hγ,γ4∫γ4γHγ,ιω1+1/αιFιdι=1Hγ,γ4∫γ4γHγ,ιd∫γ4ιω1+1/αsFsds=1Hγ,γ4∫γ4γ∫γ4ιω1+1/αsFsds−∂Hγ,ι∂ιdι≥δε1Hγ,γ4∫γ5γ−∂Hγ,ι∂ιdι≥δεHγ,γ5Hγ,γ4≥δεHγ,γ5Hγ,γ0.
It follows from (36) that(39)limt⟶∞infHγ,ιHγ,γ0≥ϵ>0,and hence, there exists γ6≥γ5 such that(40)Hγ,γ5Hγ,γ0≥ϵforγ≥γ6.
From the latter inequality and (32), we see that(41)1Hγ,γ4∫γ4γHγ,ιw1+1/αιFιdι≥δ,forγ≥γ6.
Sinceδ is an arbitrary positive constant, we obtain(42)limγ⟶∞inf1Hγ,γ4∫γ4γHγ,ιw1+1/αιFιdι=∞,which contradicts (33). Thus, (34) should hold and so by (32), we have(43)∫γ4∞Fιϕ+1+1/αιdι≤∫γ4∞w1+1/αιFιdι<∞,which contradicts (28). This completes the proof of the theorem.Theorem 3.
Let all the conditions of Theorem2 be satisfied except the condition (21) be replaced with(44)lim infγ⟶∞1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≥ΦΓ,then every solution of (1) is oscillatory.Proof.
The proof easily follows from the fact that(45)ΦΓ≤lim supγ⟶∞1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι≤lim infγ⟶∞1Hγ,Γ∫ΓγHγ,ιΨι−βαα+1α+1Φα+1γ,ιHαγ,ιFαιdι.
Hence the proof is complete.


## 3. Example

In this section, we provide an example to illustrate the main result.Example 1.
We consider the fourth-order neutral delay differential equation(46)y‴γ1/3′+12γy‴γ1/3+γ2x1/3γ−1=0,γ≥2,where yγ=xγ+1/2xγ−2.Hence, we haven=4,α=1/3,ζγ=γ−2,ζ−1γ=γ+1,ζ−1ζ−1γ=γ+2,σγ=γ−1,pγ=1,qγ=1/2γ,rγ=γ2.Simple calculation shows that by takingργ=γ, that Bγ=1/2γ,Fγ=μ/2γ,Φγ=1/8γ3γ−1/γ.LettingHγ,ι=γ−ι2, we see that HϵP, hγ,ι=2 and(47)Φγ,ι=γ22ι−3ι+5ι2.We chooseΓ=2, and the condition (46) becomes(48)limγ⟶∞sup1γ−22∫2γγ−22ι38ι−1ι−2344/3γ2/2ι−3γ+5ι/24/3γ−ι2/3μι=∞.Hence, all conditions of Theorem1 are satisfied, and therefore, every solution of (46) is oscillatory.It is noted that the conditions of Corollary1 are not satisfied, and hence, Theorem 1 improves Corollary 1.


## 4. Conclusion

In this paper, we have obtained criteria for the oscillation of all solutions of (1) using the Philos type method. An example is provided to illustrate the main result. It will be of interest to study the equation under the condition [14].(49)∫γ0∞1rγexp−∫γ0γpιrιdt1/αdr<∞.It is also interesting to extend the results of this paper to stochastic type neutral differential equations and differential equations with impulses, see [15, 16].



---

*Source: 3352789-2022-05-25.xml*