# Third-Order Neutral Delay Differential Equations: New Iterative Criteria for Oscillation

Authors: Osama Moaaz; Emad E. Mahmoud; Wedad R. Alharbi
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This study is aimed at developing new criteria of the iterative nature to test the oscillation of neutral delay differential equations of third order. First, we obtain a new criterion for the nonexistence of the so-called Kneser solutions, using an iterative technique. Further, we use several methods to obtain different criteria, so that a larger area of the models can be covered. The examples provided strongly support the importance of the new results.


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## Body

## 1. Introduction

This study is concerned with developing new iterative criteria to test the oscillation of solutions of neutral delay differential equations NDDE of third order:(1)ddlrld2dl2zlα+qlxασl=0,where l≥l0, zl≔xl+plxτl is a corresponding function of x, α is a quotient of odd positive integers, r∈C1Τl0,0,∞, ∫l0∞r−1/αsds⟶∞ as l⟶∞, p∈CΤl0,0,p0, p0<1 is a positive real number, q∈CΤl0,0,∞, τ,σ∈C1Τl0,ℝ,τl<l,σl<l, liml→∞τl=liml→∞σl=∞, and Τl0≔l0,∞.By a solution of (1), we mean a nontrivial real function x∈CΤlx,ℝ for all lx≥l0, which has the properties z, r1z′, and r2r1z′′∈C1Τlx,ℝ, and satisfies (1) on Τlx. We only consider those solutions of (1) which exist on some half-line Τlx and satisfy the condition supxl:l≥l∗>0 for any l∗∈Τlx.If the solutionx is either ultimately positive or ultimately negative, then x is called nonoscillatory; otherwise, it is called an oscillatory solution. The equation itself is termed oscillatory if all its solutions oscillate. Solutions x whose corresponding function z satisfies zlz′l<0 are called Kneser solutions. We denote the class of all Kneser solutions of (1) with the symbol XK. Otherwise, XN denote to the class of all positive solutions of (1) whose z satisfies zlz′l>0.Delay differential equations as a subclass of functional differential equations take into account the dependence on the system’s past history, which results in predicting the future in a more reliable and efficient way. Neutral delay differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays, or in the theory of automatic control and in neuromechanical systems in which inertia plays a major role, see [1–6].For interesting methods, techniques, and results which are concerned with the study of oscillation of third-order NDDEs, we refer the reader to the works [7–16]. Furthermore, the studies [17–20] are concerned with the study of odd-order DDEs.At the beginning of any study of the oscillatory properties of solutions of differential equations, it is easy to notice the importance of classifying signs of derivatives of nonoscillatory solutions. For the positive solutions, based on the canonical condition∫l0∞r−1/αsds⟶∞ as l⟶∞, it follows from ([21], Lemma 1) that r1z′′>0 and there are two possible cases for z′l: either z′l>0 or z′l<0. By creating conditions that ensure XN and XK are empty sets, we can directly set the criteria for oscillation.There are numerous results interested in finding conditions that ensure classXN is empty, which include Hille and Nehari types and Philos type. Baculikova and Dzurina [7] established a condition of Hille and Nehari type and proved that if
(2)liminfl→∞∫l∞qsσ2αssαds>2ααα+1α+11−p0α,where p0<1 and r′l>0, then XN=∅. By comparison principles, Baculikova and Dzurina [8] proved that if the first-order DDE
(3)y′l+qlσl−l021−pσl2r1/αlαyσl=0,is oscillatory, then XN=∅. We can easily notice that the delay argument τl has been neglected in (2) and (3). Otherwise, by using the Riccati transformation, Thandapani and Li [16] guaranteed that class XN is empty if
(4)limsupl→∞∫l0l12α−1ρsQs−1+p0α/τ0α+1α+1ρ′sα+1ρsβ1σs,l0σ′sα=∞,where σ′l>0, τ′l≥τ0>0,Ql≔minql,qτl, and ρ∈Cl0,∞,0,∞. All previous results focused on the class XN only and proved that every solution that belongs to XK tends to zero.On the other hand, by establishing conditions for the nonexistence of Kneser solutions (XK=∅), Dzurina et al. [12] attained the oscillation of all solutions of (1) in the linear case α=1. They proved that if (4) and
(5)liminfl→∞∫τ−1ρllQh∫σhρh∫sρh1rududsdh>τ0+p0τ0e,hold, then equation (1) is oscillatory, where ρ∈Cl0,∞,0,∞ satisfying σ<ρ<τ.One purpose of this study is to further complement and improve the well-known results reported in the literature. In Section2, by using an iterative technique, we get analogous iterative estimates for Kneser solutions of (1). These iterative estimates enable us to establish new criteria that ensure the nonexistence of Kneser solutions. Further, criteria of an iterative nature help check the oscillation, even when the other criteria fail to apply. In Section 3, we use the Riccati transformation method and comparison principles to obtain different criteria which guarantee that XN=∅. Examples illustrating the new results are also given.For the sake of ease and assistance in presenting the main results, we provide the following notations and lemmas:(6)θv,u≔∫uvr−1/αhdh,foru<v,δ≔ll1for0<α≤1;21−αforα>1.Lemma 1 (see [7], Lemma 1).
All eventually positive solutionsx of (1) have the following properties:
Pz and z″ are positive, z′is of fixed sign, and rlz″lα is nonincreasing, for l large enough.Lemma 2 (see [16]).
Letϕ,φ∈0,∞. Then, δϕ+φα≤ϕα+φα for all α≥0.


## 2. Main Results 1: Iterative Technique

Lemma 3.
Assume thatx belongs to XK and there is a function ρ∈CΤl0,0,∞ with the property
(7)σl<ρl<τl.Ifτ−1u≤v, then
(8)zu≥Θnv,ur1/αvz″v,for n=0,1,⋯, where Θ0v,u≔∫uvθv,hdh,Ql≔minqτl,ql, and
(9)Θn+1v,u≔∫uv∫u2v1ru1expδτ0τ0+p0α∫τ−1u1vQhΘnαρh,σhdh1/αdu1du2.Proof.
Supposex belongs to XK. Thus, there is a l1≥l0 satisfying xl, and xσl and xτl are positive for l≥l1. As a direct result of Lemma 1, x achieves property P. Using induction, we will prove the iterative relationship (8).
Atn=0, since rlz″lα′≤0, we obtain
(10)∫uvr−1/αhrhz″hα1/αdh≥r1/αvz″vθv,u,foru<v,which in turn leads to
(11)−z′u≥z′v−z′u≥r1/αvz″vθv,u,foru<v.
As a result of integrating (11) over u,v, we get
(12)zu≥zu−zv≥r1/αvz″v∫uvθv,hdh≥Θ0v,ur1/αvz″v.
Next, we will prove (8) at n+1 based on the assumption that it is correct at n, that is,
(13)zu≥Θnv,ur1/αvz″v.
First, we have from (1) that
(14)qτlp0αxατσl=qτlp0αxαστl=−p0ατ′lrτlz″τlα′≤−p0ατ0rτlz″τlα′.
By exploiting Lemma2 with ϕ=xl and φ=p0xτl, we obtain
(15)δzαl≤δxl+p0xτlα≤xαl+p0αxατl,which, with (1) and (14), gives
(16)δQlzασl≤qlxασl+qτlp0αxατσl≤−rlz″lα′−p0ατ0rτlz″τlα′.
Bringing (13) with u=σ and v=ρ, and combining it with (16), we get
(17)rz″α+p0ατ0rτz″τα′l≤−δQlΘnαρl,σlrρlz″ρlα.
Secondly, we defineH as
(18)H≔rz″α+p0ατ0rτz″τα.
Sincerlz″lα≤0, we see that
(19)1+p0ατ0rz″αl≤Hl≤1+p0ατ0rτz″ταl.
Therefore,(20)τ0τ0+p0αHτ−1ρl≤rρlz″ρlα.
From (17) and (20), it follows that
(21)H′l+δτ0τ0+p0αQlΘnαρl,σlHτ−1ρl≤0.
It is easy to note from (17) that H′l≤0 and hence Hτ−1ρl≥Hl. Thus, (21) becomes
(22)H′l+δτ0τ0+p0αQlΘnαρl,σlHl≤0.
Using the Grönwall inequality, (22) becomes
(23)Hτ−1u≥Hvexpδτ0τ0+p0α∫τ−1uvQhΘnαρh,σhdh,forτ1u<v.
From (19), we have
(24)z″u≥r1/αvz″v1ruexpδτ0τ0+p0α∫τ−1uvQhΘnαρh,σhdh1/α.
Integrating (24) twice over u,v, we get
(25)−z′u≥r1/αvz″v∫uv1ru1expδτ0τ0+p0α∫τ−1u1vQhΘnαρh,σhdh1/αdu1,zu≥Θn+1v,ur1/αvz″v.
This completes the proof.Theorem 4.
Assume that there exists a functionρ∈CΤl0,0,∞ with the properties σl<ρl<τl and σl≤τρl. If
(26)liminfl→∞∫τ−1ρllQhΘnαρh,σhdh>τ0+p0αδτ0e,then XK=∅.Proof.
Supposex belongs to XK. As a direct result of Lemma 2, we get that (8) holds. By following the same approach as in proof of Lemma 2, we get the relationships from (14) to (21). Now, assume H is defined as in (18). From Lemma 1, we have that z″l>0, and hence, Hl>0, for l∈Τl1. Then, the delay inequality (21) has a positive solution. From Theorem 1 in [22], the associated equation of (21) is
(27)H′l+δτ0τ0+p0αQlΘnαρl,σlHτ−1ρl=0,has also a positive solution. However, it is well known from ([23], Theorem 2) that (34) implies oscillation of (27), a contradiction. This completes the proof.Example 1.
Assume the following NDDE of third order:(28)xl+p0xal″+q0l3xbl=0,where l≥1,a,b∈0,1, and p0 and q0 are positive. First, we need to calculate θ, Q, and Θn. Some careful calculations shows that Ql=q0/l3, θu1,u2=u1−u2,
(29)Θ0u1,u2=12u1−u22,Θku1,u2=11−μkaμku1μk1−μk2−μku12−μk−u11−μku2−12−μku22−μk,for k=1,2,⋯, where c≔a+b/2, λ0≔c−b2/2,
(30)μk≔aq0λka+p,λk+1=aμkμk−1cμk1μk−2b2−μk−μk−1μk−2c2−μk+bc1−μk.Next, if we setρl=a+b/2l, then applying Theorem 4 requires that
(31)b<mina,a22−a=a22−a.Then, it is easy to verify that(32)Θ0cl,bl=12c−b2l2=λ0l2,Θncl,bl=aμn−1μn−1−1cμn−11μn−1−2b2−μn−1−μn−1−1μn−1−2c2−μn−1+bc1−μn−1=λnl2.Now, we set(33)Hn≔liminfl→∞∫τ−1ρllQhΘnαρh,σhdh=q0λnlnac.In the following particular case,(34)xl+110x910l″+81110l3x12l=0,we note that H0=0.40864, H1=0.40982, and
(35)a+p0ae=0.9+0.10.9e=0.40875.Therefore, condition (26) is not satisfied when n=0, but satisfies when n=1. Thus, (34) has no Kneser solutions.Remark 5.
Very recently, Dzurina et al. ([12], Example 1) proved that (28) is oscillatory if
(36)q0>max8a+p0aea−b2ln2a/a+b,a+p0ab2.In the particular case (34), condition (36) reduces to 81.1>max81.323,4.4444=81.323 (not fulfilled). However, by using our results, the oscillation condition is 81.1>80.889 (fulfilled). Thus, equation (34) is oscillatory.


## 3. Main Results 2: An Improved Approach

Lemma 6.
Assume thatx belongs to XN on l1,∞, and r′l>0. Then, zl≥1/2l−l1z′l for all l≥l1.Proof.
Assume thatx belongs to XN on l1,∞. Then, it follows from Lemma 1 that there is a l1≥l0, such that zl, z′l, and z″l are positive and z″ is nonpositive, for all l≥l1. It is easy to conclude that
(37)∫l1l∫l1lz″sds=zl−zl1−z′l1l−l1,which, with the fact that z″≤0, gives
(38)12l−l12z″l≤zl−zl1−z′l1l−l1,(39)zl≥zl1+l−l1z′l1+12l−l2z″l.
Now, we define(40)Fl≔l−l1zl−12l−l12z′l.
From (39), we get
(41)F′l=zl−12l−l12z″l≥zl1+l−l1z′l1>0.
Thus, we have thatF is an increasing function with Fl1=0, and so, Fl>0 for all l≥l1. Therefore, from the definition of F, we get the required directly.Lemma 7.
Assume thatx belongs to XN on l1,∞, r′l>0, and pl≡p (constant). Then,
(42)xl≥1−pzl∑κ=0n−1/2p2κτ2κ+1l−l1l−l12,for any odd positive integer n and for all l≥l2, where l2 large enough and
(43)τ0≔t,τκ≔ττκ−1t,forκ=1,2,⋯.Proof.
Assume thatx belongs to XN on l1,∞. From the definition of z, we obtain
(44)xl=zl−pxτl=zl−pzτl+p2xτ2l.
Resuming this procedure, we get, for anyn odd positive integers,
(45)xl=∑κ=0n−1κpκzτκl+pn+1xτn+1l≥∑κ=0n−1/2p2κzτ2κl−p2κ+1zτ2κ+1l,for l≥l1≥l0, where l1 is sufficiently large. Since z′l>0 and τ2k+1l≤τ2kl for all k=0,1,⋯, (45) becomes
(46)xl≥1−p∑κ=0n−1/2p2κzτ2κ+1l.
From Lemma6, we get z′l/zl≤2/l−l1 for all l≥l1. By integrating this inequality from τ2κ+1l→l, we arrive at
(47)zτ2κ+1l≥τ2κ+1l−l1l−l12zl,for all l≥l2≥l1, which with (46) gives
(48)xl≥1−pzl∑κ=0n−1/2p2κτ2κ+1l−l1l−l12.
This completes the proof.By replacing the commonly used relationshipx>1−pz with the new relationship (42), we directly get the following results.Theorem 8.
Assume thatr′l>0 and pl≡p (constant). If
(49)liminfl→∞lαrl∫l∞qsσ2ssαds>2ααα+1α+1p∗α,then XN=∅, where
(50)p∗≔1−p∑κ=0n−1/2p2κτ2κ+1l−l1l−l12.Proof.
The proof of this theorem is similar to that of ([7], Corollary 1), and hence, we omit it.Theorem 9.
Assume thatr′l>0,pl≡p (constant) and p∗ is defined as (50). If
(51)limsupl→∞∫l0lp∗αρsqsσs−l2s−l22α−1α+1α+1ρ+′sα+1ραsθαs,l2ds=∞,then XN=∅.Proof.
Assume the contrary thatx belongs to XN on l1,∞. From Lemma 7, we get that (42) holds. Combining (1) and (42), we find
(52)rlz″lα′≤−p∗αqlzασl,for l≥l2≥l1. Now, we define
(53)wl≔ρlrlz″lαzαl.
Then,wl>0 for all l≥l1, and
(54)w′l=ρ′lρlwl+ρlrlz″lα′zαl−αρlrlz″lαzα+1lz′l.
From Lemma6, we get z′l/zl≤2/l−l2 for all l≥l2. By integrating this inequality from σl→l, we arrive at
(55)zσl≥σl−l2l−l22zl,for l≥l3≥l2. Since rz″α′≤0, we have
(56)z′l≥∫l2l1r1/αsr1/αsz″sds≥r1/αlz″lθl,l2.
From (52)–(56), we obtain
(57)w′l≤ρ+′lρlwl−p∗αρlqlσl−l2l−l22α−αθl,l2ρ1/αlw1+1/αl.
Next, we define(58)Φw=ρ+′ρw−αθρ1/αw1+1/α.
Then,Φ′w=ρ+′/ρ−α+1ρ−1/αθw1/α, and so Φ attains its maximum value on ℝ at w∗=ρρ+′/α+1ρθα and
(59)Φs≤maxw∈ℝΦs=1α+1α+1ρ+′α+1ραθα.
Combining (57) and (59), we find
(60)w′l≤−p∗αρlqlσl−l2l−l22α+1α+1α+1ρ+′lα+1ραlθαl,l2.
By integrating this inequality froml3→l, we arrive at
(61)∫l3lp∗αρsqsσs−l2s−l22α−1α+1α+1ρ+′sα+1ραsθαs,l2ds≤wl1,which contradicts (51). This contradiction completes the proof.Theorem 10.
Assume thatr′l>0, pl≡p (constant), and p∗ is defined as (50). If
(62)Y′l+qlp∗∫l2σlθs,l2dsαYσl=0,is oscillatory, then XN=∅.Proof.
Assume the contrary thatx belongs to XN on l1,∞. Proceeding as in the proof of Theorem 9, we obtain (52) and (56) hold for l≥l2≥l1. Integrating (56) from l2→l and taking into account the fact that rlz″lα′≤0, we get
(63)zl≥r1/αlz″l∫l2lθs,l2ds,which with (52) gives
(64)rlz″lα′+qlp∗∫l2σlθs,l2dsαrσlz″σlα≤0.
Thus, if we setY≔rz″α>0, then we note that Y is a positive solution of the differential inequality
(65)Y′l+qlp∗∫l2σlθs,l2dsαYσl≤0.
Thus, it follows from ([22], Theorem 1) that the corresponding equation (62) also has a positive solution, which is a contradiction. This contradiction completes the proof.Example 2.
Consider the third-order NDDE,(66)lxl+34x910l″3′+q0l6x312l=0,where l>0 and q0>0. Then, it is easy to verify that
(67)τ2κ+1l=9102κ+1l,p∗≔14∑κ=0n−1/2342κ9104κ+2.To apply Theorem8, condition (49) reduces to
(68)liminfl→∞lαrl∫l∞qsσ2ssαds=liminfl→∞l2∫l∞q0s614s3ds=q012143>6328p∗3.On the other hand, using Theorem10, the class XN of (66) is empty if the first-order DDE
(69)Y′l+2q0l940p∗3Y12l=0,is oscillatory. Based on Theorem 2.1 in [24], the first-order DDE (69) is oscillatory if
(70)liminfl→∞∫l/2l2q0940p∗31sds=2q0940p∗3ln2>1e.Remark 11.
In order to ensure that the classXN of (66) is empty, Table 1 compares between our criteria and the previous related one:Table 1
Criteria that ensure thatXN=∅ for (66).
Hille and Nehari type criterionC1Corollary 1 in [7]q0>6912.0C2Condition (68)q0>3266.8Using comparison theorems with 1sl-order DDEC3Theorem 2.4 in [8]q0>1491.0C4Condition (70)q0>704.69It is easy to notice that Theorem10 supports the most efficient condition. Thus, our results improve the results in [7, 8].


## 4. Conclusion

Most studies investigating the asymptotic behavior of solutions of DDEs provided only sufficient criteria. This was a major motivation for the multiplicity of methods and techniques used in the study, so that a larger area of models could be covered. The aim of this study is to develop new criteria that have an iterative nature, so that we can apply them when the other criteria fail. First, we established a criterion to ensure that there are no Kneser solutions of (1) based on the comparison with a first-order DDE. Further, we obtained various criteria ensuring that XN is empty. By combining the results in Sections 2 and 3, we can obtain new conditions for the oscillation of all solutions of (1). It would be interesting to extend our results to the NDDE
(71)r2lr1lz′lα1′α2′+qlfxσl=0.It is also interesting to eliminate some restrictions that are usually imposed on the coefficients of the NDDEs studied.



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