# On the Oscillation of Even-Order Nonlinear Differential Equations with Mixed Neutral Terms

Authors: Mohammed K. A. Kaabar; Said R. Grace; Jehad Alzabut; Abdullah Özbekler; Zailan Siri
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The oscillation of even-order nonlinear differential equations (NLDiffEqs) with mixed nonlinear neutral terms (MNLNTs) is investigated in this work. New oscillation criteria are obtained which improve, extend, and simplify the existing ones in other previous works. Some examples are also given to illustrate the validity and potentiality of our results.


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## Body

## 1. Introduction

Recently, numerous research studies have been carried out concerning the oscillatory behavior of the differential equations with a linear neutral term. Some previous notable studies include the investigation of even-order quasilinear neutral functional differential equations’ oscillation (DEqsOs) [1] (see also [2–4]), 3rd-order neutral delay dynamic equations on time scales [5], 2nd-order nonlinear neutral delay differential equation solutions’ asymptotic behavior [6] (see also [7]), and 2nd-order superlinear Emden-Fowler neutral DEqsOs [8]. On one hand, higher-order neutral delay DEqsOs was studied in [9]. On the other hand, even-order of DEqsOs and nonlinear neutral DEqsOs with variable coefficients were investigated in [10, 11], respectively. A neutral functional delay differential equation was investigated in the sense of fractional calculus [12] (for more information about the applications of fractional calculus, refer to [13]).However, differential equations’ oscillation with nonlinear neutral terms has been rarely studied in literature. For the case of differential equations with a sublinear neutral term [14–16], Grace et al. [17] proposed differential equations with both sublinear and super-linear neutral terms, where a second-order half-linear differential equation of the following form was investigated:
(1)rtyn−1tα′+qtxγτ1t=0,where n>0 is an even integer, and
(2)yt=xt+p1txβτ2t−p2txδτ2t.From Equations (1) and (2), the following are assumed:
(i)
α, β, γ, and δ are the ratios of two positive odd integers with α≥1(ii)
p1,p2,q:t0,∞⟶ℝ+ are continuous functions(iii)
τk:t0,∞⟶ℝ are continuous functions; τkt≤t and τkt⟶∞ as t⟶∞ for k=1,2(iv)
ht=τ2−1τ1t≤t, and ht⟶∞ as t⟶∞Let us suppose that(3)A∗t,t0≔∫t0tt−sn−2As,t0ds⟶∞ast⟶∞,for which
(4)At,t0≔∫t0tr−1/αsds⟶∞ast⟶∞.A continuous functionx satisfying Equation (1) on t∗,∞, t∗≥t0, is said to be a solution of Equation (1) on t∗,∞ where yt is defined in (2). We only consider those solutions x of (1) which satisfy
(5)supxt:t≥t∗>0forallt∗≥t∗.A solutionx of (1) is said to be oscillatory if there exists a sequence ξn such that xξn=0 and
(6)limn⟶∞ξn=∞.Otherwise, it is callednonoscillatory. Equation (1) is said to be an oscillatory (or nonoscillatory) equation if all its solutions are oscillatory (or nonoscillatory).According to the best of our knowledge, the higher-order differential equations with nonlinear neutral terms have not been studied yet in any other research work. Inspired by the above studies, the oscillation of the proposed differential equations in (1) is investigated in this paper. New oscillation results for Equation (1) are obtained by comparing with the first-order delay differential equations whose oscillatory characters are well-known via an integral criterion. All results in this work are totally new, and more general oscillation results can be obtained by extending our obtained results to more general differential equations with both sublinear and super-linear neutral terms. As a result, a special research interest is hopefully stimulated from our work for possible general investigation of various neutral differential equations’ classes, particularly the ones with sublinear and/or super-linear neutral terms.This article consists of the following sections: our main results are investigated in Section2. Two illustrative examples are given in Section 3. Then, a short conclusion of our work is provided in Section 4.


## 2. Main Results

Some oscillation criteria for Equation (1) are studied when β<1 and δ>1.To obtain our results, the following lemma is needed:Lemma 1 ([5]).
LetX and Y be two nonnegative real numbers. Then, the following inequality is obtained:
(7)Xλ+λ−1Yλ−λXYλ−1≥0forλ>1,≤0for0<λ<1,where equality holds if and only if X=Y.
In what follows, we let(8)g1t≔1−βββ/1−βpβ/β−1tp11/1−βt,g2t≔δ−1δδ/1−δpδ/δ−1tp21/1−δt,Qt≔qtp2ht−γ/δ,for t≥t1 for some t1≥t0, where p:t0,∞⟶0,∞ is a continuous function.Theorem 2.
Letβ<1 and δ>1, conditions (i)-(iv), and (3) hold, and let p∈Ct0,∞,0,∞ such that
(9)p2t≠0isboundedandlimt⟶∞g1t+g2t=0,and the equation
(10)z′t+CqtAγτ1tzγ/ατ1t=0,is oscillatory for all constant C>0. Let us assume that there exist constants μi, i=1,2,3, and φ∈0,1 such that
(11)1≤μ1≤μ2≤μ3,(12)μ3ht≤t,and the equations
(13)Z′t+Qtμ2−μ1n−2n−2!hn−2tAμ3ht,μ2htγ/δZγ/αδμ3ht=0,X′t+Qtφμ2−μ1n−2!hn−1tγ/δXγ/αδμ2ht=0,are oscillatory, and
(14)∫t0∞QsA∗hs,t0γ/δds=∞.Then, every solution xt of Equation (1) is oscillatory, or
(15)limt⟶∞xt=∞.Proof.
Without loss of generality, the solutionxt of Equation (1) is assumed to be positive and xτ1t>0 for t≥t1 for some t1≥t0 (i.e., a nonoscillatory solution). From Equation (1), we have the following: xτ2t>0and
(16)rtyn−1tα′=−qtxγτ1t≤0.
Hence,rtyn−1tα is nonincreasing with a constant sign. Namely, yn−1t>0 or yn−1t<0 for t≥t2 for some t2≥t1, so the following four cases are examined separately:
(a)
yt>0 and yn−1t<0(b)
yt>0 and yn−1t>0(c)
yt<0 and yn−1t>0(d)
yt<0 and yn−1t<0
Let us first consider the case (a). Sinceyn−1t<0 for t≥t2, we obtain the following:
(17)rtyn−1tα≤−c,for some positive constant c, i.e.,
(18)yn−1t≤−crt1/α,for t≥t2. Integrating the last inequality n−1-times and by condition (3), we conclude that
(19)limt⟶∞yn−1t=−∞,which is a contradiction.
Next, let us consider the case (b). It is obvious that(20)yt=xt+ptxτ2t−p2txδτ2t+p1txβτ2t−ptxτ2t.
From Equation ((2)) of yt, i.e., we obtain the following:
(21)xt=yt−ptxτ2t−p2txδτ2t−p1txβτ2t−ptxτ2t.
If we apply the first inequality in (7) with λ=δ>1, X=p21/δtxτ2t, and
(22)Y=1δptp2−1/δt1/δ−1,then we have
(23)ptxτ2t−p2txδτ2t≤δ−1δδ/1−δpδ/δ−1tp21/1−δt≕g2t.
In a similar manner, by applying the second inequality in (7) with λ=β<1, X=p11/βtxτ2t, and
(24)Y=1βptp1−1/βt1/β−1,we obtain the following:
(25)p1txβτ2t−ptxτ2t≤1−βββ/1−βpβ/β−1tp11/1−βt≕g1t.
By using (21) and (23), (25) turns out that
(26)xt≥yt−g1t−g2t=1−g1t+g2tytyt.
Sinceyt in nondecreasing, we have the following: yt≥c0 for some c0>0. Hence, (26) turns that
(27)xt≥1−g1t+g2tc0yt.
Now, we see(28)xt≥c1yt,from (9) and (27) for some c1∈0,1. (28) implies that Equation (1) turns to be
(29)rtyn−1tα′+c1γqtyγτ1t≤0.
There exists a constantθ0∈0,1 such that
(30)yτ1t≥θ0n−1!τ1n−1tyn−1τ1t,for t≥t1 (see [16, 18, 19]). By setting wt=rtyn−1tα, we obtain the following:
(31)yτ1t≥θ0n−1!τ1n−1tr−1/ατ1tw1/ατ1t.
By using (31), (29) turns that
(32)w′t≤−Kτ1n−1tr−1/ατ1tγqtwγ/ατ1t,where
(33)K=c1θ0n−1!γ.
From Corollary 1 in [20], it can easily be concluded that there exists a positive solution wt of Equation (10) with limt⟶∞wt=0, which contradicts the fact that Equation (10) is oscillatory.
Now, let us consider the cases whenyt<0 for t≥t2. Suppose that
(34)vt=−yt=−xt−p1txβτ2t+p2txδτ2t≤p2txδτ2t,which implies
(35)xτ2t≥vtp2t1/δ,or
(36)xt≥vτ2−1tp2τ2−1t1/δ.
On the other hand, we obtain the following:(37)rtvn−1tα′=qtxγτ1t≥qtvτ2−1τ1tp2τ2−1τ1tγ/δ=Qtvγ/δht.
Now, let us consider the case (c). Clearly, we see thatvn−1t≤0 and either v′t<0 or v′t>0 for t≥t1. First, we assume that v′t<0 for t≥t1. It is easy to see that
(38)vμ3ht≥μ2−μ1n−2n−2!hn−2tvn−2μ2ht,
(refer to [18]). Now, we may express
(39)vn−2u1−vn−2u2=−∫u1u2r−1/αsrszn−1sα1/αds≥Au2,u1−r−1/αu2vn−1u2,for t1≤u1≤u2. By taking u1=μ2ht and u2=μ3ht for t≥t1 in inequality (39), we see that
(40)vn−2μ2ht≥Aμ3ht,μ2ht−r−1/αμ3htvn−1μ3ht.
By using (40), (38) turns out to be
(41)vμ3ht≥μ2−μ1n−2n−2!hn−2tAμ3ht,μ2ht×−r−1/αμ3htvn−1μ3ht.
By settingVt≔−rtvn−1tα for t≥t1, (41) turns that
(42)vht≥vμ3ht≥μ2−μ1n−2n−2!hn−2tAμ3ht,μ2ht×−V1/αμ3ht.
From (42) and (31), we obtain the following:
(43)-V′t≥Qtvγ/δht≥Qtμ2-μ1n-2n-2!hn-2tAμ3ht,μ2htγ/δ×Vγ/αδμ3ht,which implies
(44)V′t+Qtμ2−μ1n−2n−2!hn−2tAμ3ht,μ2htγ/δVγ/αδμ3ht≤0.
The proof can be easily completed by following the same steps as we did for the case (a) and hence is omitted.
Next, we assume thatv′t>0 for t≥t1. Clearly, we have the following:
(45)vn−2μ3ht≥−μ2−μ1htvn−1μ2ht.
There exists a constantθ1∈0,1 such that
(46)vht≥θ1n−2!hn−2tvn−2ht≥θ1n−2!hn−2tvn−2μ1ht,for t≥t1. Now, we see that
(47)vht≥θ1n−2!hn−2tvn−2μ1ht≥θ1n−2!hn−2tμ2−μ1ht−vn−1μ2ht.
The rest of the proof is similar to that of the above case and hence is omitted.
Finally, let us consider the case (d). Clearly, we havertv′tα>0 and so
(48)rtvn−1tα≥c2,or that
(49)vn−1t≥c2rt1/a,for some c2>0. Thus, we obtain the following:
(50)vt≥c21/αA∗t,t2for t≥t3≥t2. By using (50), (37) turns out
(51)rtvn−1tα′≥Qtvγ/δht≥Qtc21/αA∗ht,t2γ/δ.
The rest of the proof is trivial and hence is omitted. This completes the proof.☐Corollary 3.
Letβ<1 and δ>1, conditions (i)-(iv), and (3) hold, and let p∈Ct0,∞,0,∞ such that (9) holds. Assume that there exist real numbers μi, i=1,2,3 such that (11) is satisfied. If we have condition (14), then
(52)limt⟶∞∫τ1ttqsAγτ1sds=∞whenγ≤α,liminft⟶∞∫μ3httQshn−2sAμ3hs,μ2hsγ/δds>1en−2!μ2−μ1n−2γ/δwhenγ=αδ,limt⟶∞∫μ3httQshn−2sAμ3hs,μ2hsγ/δds=∞whenγ<αδ,limt⟶∞∫μ2htthn−1sμ2−μ1γ/δQsds=∞whenγ≤α.Then, Equation (1) is oscillatory.


## 3. Illustrative Examples

Two illustrative examples are presented in this section as follows:Example 1.
Consider the following second-order equation:(53)e−txt+1tx1/3t/2−x3t/2′′+34−536t+12t2+2t3e−4t/3xt/2=0.
Clearly,rt=e−t, p1t=pt=t−1, and p2t=1, and hence, there exists a t∗≥3 such that
(54)34−536t+12t2+2t3e−4t/3>0,for t≥t∗. The verification of all the conditions of Theorem 2 gives that every solution x of Equation (53) is oscillatory; otherwise, limt⟶∞xt=∞. It is worth mentioning that x1t=et is such a solution of Equation (53).Example 2.
Consider the following even-order equation:(55)e−txt+1tx1/3t/2−x3t/2n−1′+1te−t/2xt/2=0.
By noting thatrt=e−t, p1t=pt=t−1, p2t=1, and qt=e−t/2/t and letting μ1=1/8, μ2=1/4, and μ3=3/8, it can be easily seen that all the conditions of Corollary 3 hold, and hence, Equation (55) is oscillatory.


## 4. Conclusion

New results concerning the oscillation of NLDiffEq with MNLNTs have been successfully established in this paper. We have used novel technique which is based on a basic inequality and some comparison results to prove the main theorem. Demonstrating the validity and applicability of our results, two examples have been presented in this regard. It is worth mentioning that the oscillation of Equations (53) and (55) cannot be commented by previous works.



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