# Second-Order Neutral Differential Equations: Improved Criteria for Testing the Oscillation

Authors: Osama Moaaz; Ali Muhib; Saud Owyed; Emad E. Mahmoud; Aml Abdelnaser
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main purpose of this study is to establish new improved conditions for testing the oscillation of solutions of second-order neutral differential equationrlu′lγ′+qlxβσl=0, where l≥l0 and ul≔xl+pxϱl. By optimizing the commonly used relationship x>1−pu, we obtain new criteria that give sharper results for oscillation than the previous related results. Moreover, we obtain criteria of an iterative nature. Our new results are illustrated by an example.


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## Body

## 1. Introduction

In this study, we consider the neutral delay differential equation (NDDE) with second-order of the form(1)rlu′lγ′+qlxβσl=0,where l≥l0 and ul≔xl+pxϱl. Throughout the results, we always suppose γ,β∈ℚodd+≔a/b:a,b∈ℤ+are odd, p is a nonnegative constant, r∈Cl0,∞,0,∞, q∈Cl0,∞,0,∞, ϱ,σ∈Cl0,∞,ℝ, ql is not congruently zero for l≥l∗ large enough, ϱl≤l, σl≤l, liml⟶∞ϱl=liml⟶∞σl=∞, and(2)∫l0∞r−1/γξdξ=∞.By a solution of equation (1), we mean a x∈C1Tx,∞ for Tx≥l0, which has the feature ru′γ∈C1Tx,∞ and satisfies (1) on Tx,∞. We only take into account those solutions x that achieve the advantage supxl:l≥T>0, for all T≥Tx. If the solution of (1) is neither ultimately positive nor ultimately negative, then it is called an oscillatory solution; otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions oscillate.In real-world life problems, the NDDEs have interesting applications. The NDDEs appear in the modeling of the networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar, in the theory of automatic control, and others, see [1]. It is easy—in recent times—to observe the great development in the theory of oscillation for differential equations of different orders.In the following, we review some of the works that contributed to the development of the oscillation theory of second-order NDDEs were the motivation for this work.At studying the oscillatory behavior of NDDEs with canonical Case (2), the relationship between the solution and the corresponding function(3)xl>1−pulhas been commonly used in the literature. For canonical case (2), by using the Riccati technique, Xu and Meng [2] presented some oscillation criteria for (1) when γ=β. In the case, where 0<pt≤p0<∞, Baculikova and Dzurina [3] established the oscillation criteria for (1).Theorem 1 (see Corollary 2 in [3]).
Let0<β≤1, β≤γ,(4)σl≤ϱl≤l,ϱ′l≥ϱ0>0,ϱ∘σ=σ∘ϱ.
If(5)liminfl⟶∞∫ϱ−1σllG^s∫l1σlr−1/γζdζβds>1+p0βϱ0β/γ1e,then (1) is oscillatory, where G^l≔minql,qϱl.Using relationship (3), Grace et al. [4] studied the oscillatory behavior of solutions of (1) when γ=β and p<1. Moreover, they improved previous results in the literature.Theorem 2 (see Theorems3 and 6 in [4]).
If(6)liminfl⟶∞∫σllGsυ∗σsγds>1eor(7)limsupl⟶∞∫l1∞ϕsexp−∫σssdζr1/γζυ∗ζ−rsϕ+′sγ+1γ+1γ+1ϕγsds=∞,then (1) is oscillatory, where υl≔∫l1lr−1/γsds, ϕ∈Cl0,∞,0,∞, ϕ+′l≔maxϕ′l,0, and(8)Gl≔ql1−pσlγ,υl1∗l≔υl1l+1γ∫l1lυl1ζυγζdζ.Recently, Moaaz et al. [5, 6] generalized and complemented the results in [4]. They established the following criteria for oscillation of (1) with p<1.Theorem 3 (see Theorem2 in [6]).
Letβ=γ. If(9)liminfl⟶∞γψl∫l∞r−1/γζψγ+1/γζdζ>γγ+1γ+1/γ,then (1) is oscillatory, where(10)υ^l≔exp−γ∫σll1υ∗sr1/γsds,ψl≔∫l∞υ^ζGζdζ,and Gl and υ∗l are defined as in (8).The similar results as those above have been extended for even-order NDDEs in [7–11]. For the works that dealt with the noncanonical case, that is,(11)∫l0∞r−1/γξdξ<∞,see, for example, [12–14].The objective of this paper is to establish new oscillation criteria for the NDDE (1) by improving (3). The new relationship enables us to(i)
Create more effective criteria for studying neutral equations in both casesp<1 and p>1(ii)
Essentially take into account the influence of the delay argumentϱl that has been careless in all related results(iii)
Exclude some restrictions that are usually imposed on the coefficients of the neutral equations in the case wherep>1Moreover, we use an iterative technique to establish new oscillation criteria for the NDDE (1) when β=γ and p<1. One purpose of this paper is to further improve Theorems 2 and 1. The results reported in this paper generalize, complement, and improve those in [3–6]. To show the importance of our results, we provide an example.


## 2. Main Results I: New Relationship betweenx and u

For simplicity, we just write the functions without the independent variable, such asfl≔f and fgl=fg. Moreover, assuming(12)ϱ0≔l,ϱm≔ϱ∘ϱm−1,ϱ−m−1≔ϱ−1∘ϱ−m,form=1,2,…,ηl0l≔∫l0lr−1/γsds,B≔c1β−γ,ifγ≤β,c2ηl2β−γl,ifγ>β,where c1 and c2 are positive constants, the set of all eventually positive solutions of (1) is denoted by X+.Lemma 1 (see Lemma3 in [3]).
Letx∈X+. Then,(13)u>0,u′>0,ru′γ′≤0,for l≥l1, where l1 is sufficiently large.The following lemma is a direct observation from the Proof of 2.1 in [5].Lemma 2.
Ifx∈X+, then uβ−γl≥Bl, eventually.Lemma 3.
Letx∈X+ and p>0, and there exists an even positive integer n such that(14)p˜≔∑m=1n/21p2m−11−1pηl2ϱ−2mηl2ϱ−2m−1>0.
Then,(15)xl≥p˜lul.Proof.
Suppose thatx∈X+. Thus, xl, xϱl, and xσl are positive for all l≥l1, where l1 is sufficiently large. From Lemma 1, we see that (13) holds. Since r1/γu′′≤0, we have that(16)ul>∫l1l1r1/γξr1/γξu′ξdξ>r1/γlu′lηl1l,for all l≥l1. Using the definition of ul, we obtain(17)x=1puϱ−1−xϱ−1=1puϱ−1−1puϱ−2+1p2xϱ−2.
Repeating this procedure, we obtain(18)x=∑m=1n−1m+1pmuϱ−m+1pnxϱ−n>∑m=1n/21p2m−1uϱ−2m−1−1puϱ−2m,for l≥l2≥l1, where l2 is sufficiently large, and any even positive integer n. Taking (16) and ϱ−2m≥ϱ−2m−1 into account, we obtain(19)uϱ−2m<uϱ−2m−1ηl2ϱ−2mηl2ϱ−2m−1,for m=1,2,…,n/2. Combining (18) and (19), we obtain(20)x>∑m=1n/21p2m−11−1p0ηl2ϱ−2mηl2ϱ−2m−1uϱ−2m−1>p˜u.
This completes the proof.Lemma 4.
Letx∈X+ and p0<1. Then,(21)xl≥p^lul,for any odd positive integer n, where(22)p^≔1−p∑m=0n−1/2p2mηl1ϱ2m+1ηl1.Proof.
Proceeding as in the proof of Lemma3, we arrive at (16). From the definition of ul, we have that(23)x=u−pxϱ=u−puϱ+p2xϱ2.
Repeating this procedure, we obtain(24)x=∑m=0n−1mpmuϱm+pn+1xϱn+1≥∑m=0n−1/2p2muϱ2m−p2m+1uϱ2m+1,for l≥l2≥l1, where l2 is sufficiently large, and any odd n∈ℤ+. Since ϱ2m+1l≤ϱ2ml, we see that(25)uϱn≤⋯≤uϱ2m+1≤uϱ2m≤⋯≤u,for m=0,2,…,n−1/2, which with (24) gives(26)x≥∑m=0n−1/2p2m1−puϱ2m+1.
From (16), we find(27)uϱ2m+1>uηl1ϱ2m+1ηl1,which with (26) gives(28)x≥1−pu∑m=0n−1/2p2mηl1ϱ2m+1ηl1.
This completes the proof.Theorem 4.
Assume thatp0<1. If there exists a function θ∈C1l0,∞,0,∞ such that(29)limsupl⟶∞∫l1lθξqξp^βσξδξBσξ−1γ+1γ+1rξθ+′ξγ+1θγξdξ=∞,then (1) is oscillatory, where(30)η^l0l≔ηl0l+1γ∫l0lηl0ξηl0γσξqξp^βσξBσξdξ,δl≔exp−γ∫σll1r1/γξη^l1ξdξ.Proof.
Assume the contrary thatx is a nonoscillatory solution of (1). Without loss of generality, we suppose that x∈X+. Thus, xl, xϱl, and xσl are positive for all l≥l1, where l1 is sufficiently large. Using Lemma 4, we have that (21) holds. Using (1) and (21), we obtain(31)ru′γ′≤−qp^βσuβσ.
Using the chain rule and simple computation, we find(32)γr1/γu′γ−1ddlu−ηl1r1/γu′=−γr1/γu′γ−1ηl1r1/γu′′=−ηl1ru′γ′,which with (31) gives(33)ddlu−ηl1r1/γu′≥1γr1/γu′1−γηl1qp^βσuβσ≥1γr1/γu′1−γηl1qp^βσBσuγσ.
Integrating (33) from l1 to l, we obtain(34)u≥ηl1r1/γu′+1γ∫l1lr1/γξu′ξ1−γηl1ξqξp^βσξBσξuγσξdξ.
Sincer1/γlu′lγ′≤0, we have(35)uσ≥ηl1σr1/γσu′σ≥ηl1σr1/γu′.
Thus, (24) becomes(36)u≥ηl1l+1γ∫l1lηl0ξηl0γσξqξp^βσξBσξdξr1/γu′,that is,(37)u≥η^l1r1/γu′.
Integratingu′/u≤1/rl11/γη^l1 from σl to l, we find(38)lnuluσl≤∫σll1r1/γξη^l1ξdξ,that is,(39)uσl≥exp−∫σll1r1/γξη^l1ξdξul.
Next, we define the function(40)Θ≔θru′γuγ.
Clearly,Θl>0 for all l≥l1 and(41)Θ′=θ′θΘ+θru′γ′uγ−γθru′γuγ+1u′.
It follows from (31) and (39) that(42)Θ′=θ′θΘ+−θδqp^βσBσ−γθru′γuγ+1u′,from definition Θ, we have(43)Θ′≤θ′θΘ−θδqp^βσBσ−γr1/γθ1/γΘ1+1/γ.
Using the inequality (see Lemma 1.2 in [5]),(44)Aϕ−Bϕγ+1/γ≤γγγ+1γ+1Aγ+1Bγ,B>0,with A=θ′/θ,B=γ/r1/γθ1/γ, and ϕ=Θ, we obtain(45)Θ′≤−θδqp^βσBσ+1γ+1γ+1rθ+′γ+1θγ.
Integrating this inequality froml1 to l, we obtain(46)∫l1lθξqξp^βσξδξBσξ−1γ+1γ+1rξθ+′ξγ+1θγξdξ≤Θl1,which contradicts (29). This completes the proof.Theorem 5.
Assume that (14) holds for some even positive integer n. If there exists a function ϑ∈C1l0,∞,0,∞ such that(47)limsupl⟶∞∫l1lϑξqξp˜βσξδ˜ξBσξ−1γ+1γ+1rξϑ+′ξγ+1ϑγξdξ=∞,then (1) is oscillatory, where(48)η˜l0l≔ηl0l+1γ∫l0lηl0ξηl0γσξqξp˜βσξBσξdξ,δ˜l≔exp−γ∫σll1r1/γξη˜l1ξdξ.Proof.
To prove this theorem, it suffices to use (15) instead of (21) in the proof of Theorem 4.


## 3. Main Results II: Iterative Technique

Lemma 5.
Assume thatx∈X+, γ=β, and p0<1. Then,(49)ul≥φklr1/γlu′l,for k=0,1,…, where φ0l≔η^l1l and(50)φk+1l≔∫l1l1rsexp∫slqξp^γσξφkγσξdξ1/γds.Proof.
Suppose thatx∈X+. Thus, xl, xϱl, and xσl are positive for all l≥l1, where l1 is sufficiently large. From Lemma 1, we see that (13) holds. Now, we will prove (49) using induction.
Fork=1, proceeding as in proof of Theorem 4, we obtain that (31) and (37) hold. From (37), we obtain(51)u≥η^l1lr1/γu′=φ0lr1/γu′.
Next, we assume that (49) holds at k=n, that is, u≥φnr1/γu′. Thus, since r1/γu′γ′≤0, we find(52)uσ≥φnσr1/γσu′σ≥φnσr1/γu′.which with (31) gives(53)ru′γ′+qp^γσφnγσru′γ≤0.
If we setH≔ru′γ, (53) becomes(54)H′l+qp^γσφnγσHl≤0.
Applying the Grönwall inequality in (54), we obtain(55)Hs≥Hlexp∫slqξp^γσξφnγσξdξ,for l≥s≥l1, so(56)u′s≥r1/γlu′l1rsexp∫slqξp^γσξφnγσξdξ1/γ.
Integrating this inequality froml1 to l, we obtain(57)ul≥r1/γlu′l∫l1l1rsexp∫slqξp^γσξφnγσξdξ1/γds=φn+1lr1/γlu′l.
This completes the proof.Theorem 6.
Assume thatγ=β and p0<1. If(58)liminfl⟶∞∫σllqξp^γσξφkγσξdξ>1e,for some integers k≥0, then (1) is oscillatory, where p^ and φk are defined as in (22) and (50), respectively.Proof.
Assume the contrary thatx is a nonoscillatory solution of (1). Without loss of generality, we suppose that x∈X+. Thus, xl, xϱl, and xσl are positive for all l≥l1, where l1 is sufficiently large. From Lemma 5, we have that (49) holds. Proceeding as in the proof of Theorem 4, we arrive at (31). Combining (53) and (49), we obtain(59)ru′γ′l+qlp^γσlφkγσlrσlu′σlγ≤0.
If we setw≔ru′γ, we have that w is a positive solution of the delay differential inequality:(60)w′l+qlp^γσlφkγσlwσl≤0.
Using Theorem1 in [15], the associated DDE(61)w′l+qlp^γσlφkγσlwσl=0has also a positive solution. However, condition (58) ensures oscillation of (61), which is a contradiction. This completes the proof.Theorem 7.
Assume thatγ=β and p0<1. If there exists a function ρ∈C1l0,∞,0,∞ such that(62)limsupl⟶∞∫l1lρξqξp^γσξδ^kξ−1γ+1γ+1rξρ+′ξγ+1ργξdξ=∞,for some integers k≥0, then (1) is oscillatory, where(63)δ^kl≔exp−γ∫σll1r1/γξφkξdξ,and p^ and φk are defined as in (22) and (50), respectively.Proof.
Assume the contrary thatx is a nonoscillatory solution of (1). Without loss of generality, we suppose that x∈X+. Thus, xl, xϱl, and xσl are positive for all l≥l1, where l1 is sufficiently large. Now, we define the function ψ≔ρru′/uγ. Thus, ψl>0 and(64)ψ′=ρ′ρψ+ρru′γ′uγ−γρru′uγ+1γ+1.
From Lemma5, we have that (49) holds. By replacing (37) with (49) in the proof of Theorem 4, this part of proof is similar to that of Theorem 4, so we omit it.Now, we give an example to illustrate our main results.Example 1.
Consider the NDDE:(65)xl+pxμl′γ′+q0lγ+1xγλl=0,where q0>0 and μ,λ∈0,1. It is easy to verify that ηl0l=l, ϱml=μml, η^l1l=1+1/γp^0γq0λγl, and δl=λγ^, where(66)p^l=1−p∑m=0n−1/2p2mμ2m+1≔p^0,γ^≔γ1+1/γp^0γq0λγ,p˜l=∑m=1n/21p2m−11−1μp≔p˜0,γ^≔γ1+1/γp˜0γq0λγ.
Using Theorem4, we see that (65) is oscillatory if p<1 and(67)p^0γλγ^q0>γγ+1γ+1.
Using Theorem5, we see that (65) is oscillatory if(68)p˜0γλγ˜q0>γγ+1γ+1.Remark 1.
The best-known criteria for oscillation of NDDE (65) are(69)q01−pγλγ/1+1/γ1−pγq0λγ>γγ+1γ+1see4,Example3,(70)q0λγlnμλ>μ+pγeμsee3,Corollary2,for p<1 and p>1, respectively.
Giving values for the parametersp, μ, and λ, we can determine the lower bound of the parameter q0 to ensure that every solution of (65) is oscillatory. Table 1 shows the lower boundaries of the parameter q0 in different special cases of (65) when δ=1 by using conditions (69) and (67).
Let another particular case of (66), namely,(71)xl+2xμl″+q0l2x910l=0.
Conditions (70) and (68) reduce to q0>6.2587 and q0>1.5881, respectively.
So, our results improve the related results in [3, 4].Table 1
The lower boundaries of the parameterq0 in oscillation criteria.
(68)(70)p,λ,μ↓n=5n=9n=49—2/3,0.1,0.7555.33425.25295.24745.306100.5,0.5,0.8300.88440.88010.87990.882270.9,0.5,0.9002.34911.91891.68574.41130Remark 2.
Using the boundedness conditionp1≤pl≤p2, it will be easy to infer results similar to ours if p is a function in l.


## 4. Conclusion

This article is concerned with oscillatory behavior of a class of the neutral delay differential equation (NDDE) with second-order. By optimizing the commonly used relationshipx>1−pu, we obtained new criteria that give sharper results for oscillation than the previous related results. Moreover, we obtained criteria of an iterative nature. Furthermore, in the future work, we can try to get some oscillation criteria of (1) under case of zl=xl+clxνϱl, 0<ν<1.



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