Next Article in Journal
Stability Analysis of an Age-Structured SEIRS Model with Time Delay
Next Article in Special Issue
Weaker Conditions for the q-Steffensen Inequality and Some Related Generalizations
Previous Article in Journal
An Optimization Model for the Temporary Locations of Mobile Charging Stations
Previous Article in Special Issue
On a New Half-Discrete Hilbert-Type Inequality Involving the Variable Upper Limit Integral and Partial Sums

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# New Comparison Theorems for the Nth Order Neutral Differential Equations with Delay Inequalities

by
Osama Moaaz
1,†,
Shigeru Furuichi
2,*,† and
Ali Muhib
1,3,†
1
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2
Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan
3
Department of Mathematics, Faculty of Education (Al-Nadirah), Ibb University, PO Box 70270, Ibb, Yemen
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(3), 454; https://doi.org/10.3390/math8030454
Submission received: 20 February 2020 / Revised: 15 March 2020 / Accepted: 18 March 2020 / Published: 22 March 2020

## Abstract

:
In this work, we present a new technique for the oscillatory properties of solutions of higher-order differential equations. We set new sufficient criteria for oscillation via comparison with higher-order differential inequalities. Moreover, we use the comparison with first-order differential equations. Finally, we provide an example to illustrate the importance of the results.

## 1. Introduction

This work is concerned with studying the oscillatory behavior of the higher-order neutral differential equation
$( ( z n − 1 ς ) α ) ′ + f ( ς , x ( σ ( ς ) ) ) = 0 , ς ≥ ς 0 > 0 ,$
where $n ≥ 2$ is an even natural number and $z ( ς ) = x ( ς ) + p ( ς ) x ( τ ( ς ) )$. Through the paper, we assume that $0 < α < 1$ is a ratio of odd positive integers, $p ∈ C 1 ( [ ς 0 , ∞ ) , R + )$, with $lim ς → ∞ p ( ς ) = 0 , σ , τ ∈ C 1 ( [ ς 0 , ∞ ) , R )$ such that $σ ( ς ) ≤ ς , lim ς → ∞ σ ( ς ) = ∞$. $τ ( ς ) ≤ ς , lim ς → ∞ τ ( ς ) = ∞$ and there exists a nonnegative function h such that $f ( ς , x ) ≥ h ( ς ) x β$, $β$ is a ratio of odd positive integers.
If there exists a $ς x ≥ ς 0$ such that the real valued function $x$ is continuous, $( z ( n − 1 ) ) α ∈ C 1 ( [ ς x , ∞ ) , R )$ and satisfies (1), for all $ς ∈ [ ς x , ∞ )$, then x is said to be a solution of (1). We restrict our discussion to those solutions x of (1) which satisfy $sup x ς : ς 1 ≤ ς 0 > 0$ for every $ς 1 ∈ ς x , ∞$ and tacitly assume that (1) possesses such solutions.
Definition 1.
A solution x of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. The equation itself is termed oscillatory if all its solutions oscillate.
Differential equations are of great importance in various applied sciences, for example, kinetic chemistry [1,2], eigenproblems [3,4] and wave functions [5]. A neutral differential equation is an equation in which the delayed argument occurs in the highest derivative of the state variable. These equations has become an significant field of research due to the fact that such equations appear in many applications, for example population dynamics, mixing liquids, etc., see Hale [6].
The study of the problem of oscillation of differential equations with higher order has attracted the attention of many researchers in recent times. As a result of this interest, many different techniques and methods have been introduced to study the oscillation problem. Below we show some works closely related to this paper.
Moaaz et al. [7] and El-Nabulsi et al. [8] studied the equations
$r ς x ‴ ς α ′ + ∫ a b q ς , ξ f x g ς , ξ d ξ = 0 ,$
under the conditions
$∫ ς 0 ∞ 1 r 1 / α s d s < ∞$
and
$∫ ς 0 ∞ 1 r 1 / α s d s = ∞ ,$
respectively. Baculikova et al. [9] and Moaaz et al. [10,11] use the Riccati-technique to study the oscillation of quasi-linear neutral equation
$x ς + p ς x τ ς n − 1 γ ′ + q ς x γ σ ς = 0 , ς ≥ ς 0 ,$
Moaaz et al. [12] established the oscillation criteria for solutions of the even-order equation
$r ς z n − 1 ς ′ + ∫ a b q ς , s f x g ς , s d s = 0 ,$
where $z ( ς ) = x α ( ς ) + p ( ς ) x ( σ ( ς ) )$. Xing et al. [13] investigated the oscillation behavior of neutral equation
$r ς x ς + p ς x τ ς n − 1 γ ′ + q ς x γ σ ς = 0 , n ≥ 2 ,$
by using comparison technique with first-order delay equations. Baculikova and Dzurina [14] investigated the asymptotic properties and oscillation of the nth order neutral equations
$r ς x ς + p ς x τ ς n − 1 ′ + q ς x σ ς = 0 , n ≥ 3 .$
Elabbasy et al. [15] studied the asymptotic properties of solutions of even order neutral delay differential equations of the form
$r ς z n − 1 ς α 1 − 1 z n − 1 ς ′ + ∑ i = 1 m q i ς x σ i ς α i − 1 x σ i ς = 0 .$
We have greatly fewer results for oscillation of the neutral equation than for oscillation of the delay equation. Therefore, the importance of this study lies in linking between delay and neutral equations by comparison. This comparison enables us to benefit from the previous results—in the literature—of oscillation of delay equations. In this paper, we establish some new oscillation criteria via comparison with higher-order differential inequalities and with first-order differential equations. An example illustrating the results is also given.
Lemma 1
([16]). Let $f ∈ C n ς 0 , ∞ , 0 , ∞$. If the derivative $f n ς$ is eventually of one sign for all large ς, then there exist a $ς x$ such that $ς x ≥ ς 0$ and an integer $l , 0 ≤ l ≤ n ,$ with $n + l$ even for $f n ς ≥ 0$, or $n + l$ odd for $f n ς ≤ 0$ such that
$l > 0 i m p l i e s f k ς > 0 f o r ς ≥ ς x , k = 0 , 1 , … , l − 1 ,$
and
$l ≤ n − 1 i m p l i e s − 1 l + k f k ς > 0 f o r ς ≥ ς x , k = l , l + 1 , … , n − 1 .$
Lemma 2
([17]). If $a$ and b are nonnegative and $0 < δ < 1$, then
$a δ − δ a b δ − 1 − 1 − δ b δ ≤ 0 ,$
where equality holds if and only if $a = b$.
Lemma 3
([18]). Let $x ∈ C n ς 0 , ∞ , 0 , ∞ , x n − 1 ς x n ς ≤ 0$ for $ς ≥ ς x$ and assume that $lim ς → ∞ x ( ς ) ≠ 0$, then for every $θ ∈ 0 , 1$, there exists a $ς θ ∈ ς x , ∞$ such that
$x ς ≥ θ n − 1 ! ς n − 1 x n − 1 ς f o r a l l ς ∈ ς θ , ∞ .$
Remark 1.
We will prove the results when $x ( ς )$ eventually positive while at $x ( ς )$ is eventually negative the proof is similar, so we omit the details of that case in this paper.

## 2. Main Results

Theorem 1.
Assume that $β > 1$. For every constant $N > 0$ if the even-order differential inequality
$( ( z n − 1 ς ) α ) ′ + N h ς z σ ς ≤ 0$
has no positive solution, then Equation (1) is oscillatory.
Proof.
Assume to the contrary that x is a nonoscillatory solution of (1), say $x ς > 0$, $x τ ς > 0$ and $x σ ς > 0$ for $ς ≥ ς 1$ for some $ς 1 ≥ ς 0$. From (1) we have
$( ( z n − 1 ς ) α ) ′ ≤ − h ς x β σ ς < 0 ,$
from Lemma 1, there exists a $ς 2 ≥ ς 1$ such that
$z ′ ς > 0 and z n − 1 ς > 0 ,$
for $ς ≥ ς 2$. From definition $z ς$, we have
$x ς = z ς − p ς x α τ ς − p ς x τ ς + p ς x α τ ς − 2 p ς x τ ς .$
By using Lemma 2 with $a = p 1 / α ς x τ ς , b = α − 1 p α − 1 / α ς 1 / α − 1$ and $δ = α$, we get
$p ς x α τ ς − p ς x τ ς ≤ 1 − α α α / 1 − α p ς .$
From (5) and (6), we have
$x ς ≥ z ς + p ς x α τ ς − 2 p ς x τ ς − 1 − α α α / 1 − α p ς ,$
that is
$x ς ≥ z ς − 2 p ς x τ ς − 1 − α α α / 1 − α p ς .$
Since $x ς ≤ z ς$, we have
$x ς ≥ z ς − 2 p ς z τ ς − 1 − α α α / 1 − α p ς .$
Since $z ς > 0$ and $z ′ ς > 0$ on $ς 2 , ∞$, there exist a $ς 3 ≥ ς 2$ and a constant $μ > 0$ such that
$z ς ≥ μ ,$
for $ς ≥ ς 3$. From (9) and (10), we obtain
$x ς ≥ z ς 1 − 2 p ς − 1 − α α α / 1 − α p ς ≥ z ς 1 − 2 p ς − 1 − α α α / 1 − α p ς z ς ≥ z ς 1 − 2 p ς − 1 − α α α / 1 − α p ς μ ≥ z ς 1 − p ς 2 + 1 − α α α / 1 − α 1 μ ,$
for $ς ≥ ς 3$. From (11) and the fact that $lim ς → ∞ p ς = 0$, for any $λ ∈ 0 , 1$ there exists $ς λ ≥ ς 3$ such that
$x ς ≥ λ z ς for ς ≥ ς λ .$
Fix $λ ∈ 0 , 1$ and choose $ς λ$ by (12). Since $lim ς → ∞ σ ς = ∞$, we can choose $ς 5 ≥ ς λ$ such that $σ ς ≥ ς λ$ for all $ς ≥ ς 5$. Thus, from (12) we have
$x σ ς ≥ λ z σ ς ,$
for $ς ≥ ς 5$. From (13) and (1), we have
$( ( z n − 1 ς ) α ) ′ + λ β h ς z β σ ς ≤ 0 .$
It can be reformulated as follow
$( ( z n − 1 ς ) α ) ′ + λ β h ς z β − 1 σ ς z σ ς ≤ 0 , for ς ≥ ς 5 .$
By using (10), $z ′ ς > 0$ and $σ ς ≥ ς λ$, (15) become to
$( ( z n − 1 ς ) α ) ′ + λ β μ β − 1 h ς z σ ς ≤ 0 ,$
that is
$( ( z n − 1 ς ) α ) ′ + N h ς z σ ς ≤ 0 , for ς ≥ ς 5 ,$
where N = $λ β μ β − 1$. We conclude that (2) has a positive solution, which is a contradiction. The proof is complete. □
Theorem 2.
Assume that $0 < β < 1$. For every constant $J > 0$ if the even-order differential inequality
$( ( z n − 1 ς ) α ) ′ + J σ n − 1 ς β − 1 h ς z σ ς ≤ 0$
has no positive solution, then Equation (1) is oscillatory.
Proof.
Assume to the contrary that x is a nonoscillatory solution of (1), say $x ς > 0$, $x τ ς > 0$ and $x σ ς > 0$ for $ς ≥ ς 1$ for some $ς 1 ≥ ς 0$. We follow the same method as in proof Theorem 1, we get (14) which can be reformulated as follow
$( ( z n − 1 ς ) α ) ′ + λ β h ς z 1 − β σ ς z σ ς ≤ 0 , for ς ≥ ς 5 .$
Since $z n − 1 ς > 0$ and $z n − 1 ς$ is decreasing on $ς 5 , ∞ ⊂ ς 2 , ∞$, there exist a $ς 6$ such that $ς 6 ≥ ς 5$ and a constant $η > 0$ such that
$z n − 1 ς ≤ η ,$
for $ς ≥ ς 6$. Integrating (19) from $ς 6$ to $ς$ $n − 1$ times, we get
$z ς ≤ U ς n − 1 , ς ≥ ς 6 ,$
for some constant $U > 0$, and so,
$z σ ς ≤ U σ n − 1 ς , ς ≥ ς 7 ≥ ς 6 ,$
where we assume $σ ς ≥ ς 6$ for $ς ≥ ς 7$. By using (21), and (18), we have
$( ( z n − 1 ς ) α ) ′ + λ β U β − 1 σ n − 1 ς β − 1 h ς z σ ς ≤ 0$
or
$( ( z n − 1 ς ) α ) ′ + J ( σ n − 1 ς ) β − 1 h ς z σ ς ≤ 0 , for ς ≥ ς 7 ,$
where $J = λ β U β − 1$. We conclude that (17) has a positive solution, which is a contradiction. The proof is complete. □
Theorem 3.
Assume that $β = 1$. For any $λ ∈ 0 , 1$ if the even-order differential inequality
$( ( z n − 1 ς ) α ) ′ + λ h ς z σ ς ≤ 0$
has no positive solution, then Equation (1) is oscillatory.
The Proof of the previous theorem is produced directly from (14) with $β = 1$ and Theorem 1; so we omit the details.
Theorem 4.
Assume that $β > 1$. For every constant $N > 0$, if the first-order differential equation
$y ′ ς + θ 0 n − 1 ! N σ n − 1 ς h ς y 1 / α σ ς = 0$
is oscillatory for some $θ 0 ∈ 0 , 1$, then Equation (1) is oscillatory.
Proof.
Assume to the contrary that x is a nonoscillatory solution of (1), say $x ς > 0$, $x τ ς > 0$ and $x σ ς > 0$ for $ς ≥ ς 1$ for some $ς 1 ≥ ς 0$. We follow the same method as in proof Theorem 1, we get (16) for $ς ≥ ς 5$. Since $lim ς → ∞ z ς ≠ 0$, where $z ς > 0$ and $z ′ ς > 0$ on $ς 5 , ∞ ⊂ ς 2 , ∞$. By using Lemma 3 we have
$z ς ≥ θ n − 1 ! ς n − 1 z n − 1 ς ,$
for $ς ≥ ς 6$. From (16) and (25), we have
$( ( z n − 1 ς ) α ) ′ + θ n − 1 ! N σ n − 1 ς h ς z n − 1 σ ς ≤ 0 , ς ≥ ς 7 ≥ ς 6 .$
Set $y ς = ( z n − 1 ( ς ) ) α$, we have
$y ′ ς + θ n − 1 ! N σ n − 1 ς h ς y 1 / α σ ( ς ) ≤ 0 ,$
for $ς ≥ ς 7$. Thus $y ς$ is a positive solution of the above inequality. Integrating inequality (27) from $ς$ to $∞$ where $ς ≥ ς 7$, we have
$y ς ≥ ∫ ς ∞ θ n − 1 ! N σ n − 1 s h s y 1 / α σ s d s .$
From [19] (Theorem 1) we see that there exists a positive solution $y ς$ of Equation (24) with $lim ς → ∞ y ς = 0$, which contradicts (24), is oscillatory. The proof is complete. □
Theorem 5.
Assume that $0 < β < 1$. For every constant $J > 0$ if the first-order differential equation
$y ′ ς + θ 0 n − 1 ! J σ n − 1 ς β h ς y 1 / α σ ς = 0$
is oscillatory for some $θ 0 ∈ 0 , 1$, then Equation (1) is oscillatory.
The Proof of the previous theorem is produced directly from (22) and (25) and Theorem 4, so we omit the details.
Theorem 6.
Assume that $β = 1$. For any $λ ∈ 0 , 1$ if the first-order differential equation
$y ′ ς + θ 0 n − 1 ! λ σ n − 1 ς h ς y 1 / α σ ς = 0$
is oscillatory for some $θ 0 ∈ 0 , 1$, then Equation (1) is oscillatory.
The Proof of the previous theorem is produced directly from (14) with $β = 1$, (25) and Theorem 4, so we omit the details.
Corollary 1.
Assume that $β ≥ 1$. If
$lim ς → ∞ ∫ σ ς ς σ n − 1 s h s d s = ∞ ,$
then Equation (1) is oscillatory.
Corollary 2.
Assume that $0 < β < 1$. If
$lim ς → ∞ ∫ σ ς ς σ n − 1 s β h s d s = ∞ ,$
then Equation (1) is oscillatory.
Example 1.
Consider the differential equation
$x ς + 1 ς x ς 4 n − 1 α ′ + 3 ς 2 x 2 ς 3 = 0 , ς ≥ ς 0 > 0 a n d n > 2 .$
From (30) we have $0 < α < 1 , β = 3 , p ς = 1 / ς , τ ς = ς / 4$, $σ ς = ς / 3$ and $h ς = 3 / ς 2$. By using Corollary 1 we find
$∫ σ ς ς σ n − 1 s h s d s = ∫ ς / 3 ς s 3 n − 1 3 s 2 d s = 1 3 n − 2 n − 2 3 n − 2 − 1 3 n − 2 ς n − 2 ,$
therefore
$lim ς → ∞ ∫ σ ς ς σ n − 1 s h s d s = ∞ ,$
then (30) is oscillatory.

## 3. Conclusions

Several techniques have been used in studying the oscillation of solutions of higher-order differential equations. Unusually, our method here is based on presenting new comparison theorems that compare the higher-order neutral equation with higher or first-order delay inequality or equation. Through our results, Theorems 1–6 guaranteed the oscillation of the solutions of Equation (1) by ensuring the oscillation of one of the inequalities (2), (17) and (23) or Equations (24), (28) and (29).

## Author Contributions

Formal analysis, S.F.; Investigation, O.M.; Supervision, O.M.; Writing–original draft, A.M.; Writing–review & editing, O.M., S.F. and A.M. These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

## Funding

The authors received no direct funding for this work.

## Acknowledgments

The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 16K05257.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Lotka, A.J. Contribution to the Theory of Periodic Reactions. J. Phys. Chem. 1909, 14, 271–274. [Google Scholar] [CrossRef] [Green Version]
2. Jantschi, L.; Bolboaca, S.D. First order derivatives of thermodynamic functions under assumption of no chemical changes revisited. J. Comput. Sci. 2014, 5, 597–602. [Google Scholar] [CrossRef]
3. Miyajima, S. Verified solutions of delay eigenvalue problems. Appl. Math. Comput. 2017, 303, 211–225. [Google Scholar] [CrossRef]
4. Jantschi, L. The Eigenproblem Translated for Alignment of Molecules. Symmetry 2019, 11, 1027. [Google Scholar] [CrossRef] [Green Version]
5. Arellano, H.F.; Blanchon, G. Exact scattering waves off nonlocal potentials under Coulomb interaction within Schrodinger’s integro-differential equation. Phys. Lett. B 2019, 789, 256–261. [Google Scholar] [CrossRef]
6. Hale, J.K. Theory of Functional Differential Equations; Springer: New York, NY, USA, 1977. [Google Scholar]
7. Moaaz, O.; Elabbasy, E.M.; Bazighifan, O. On the asymptotic behavior of fourth-order functional differential equations. Adv. Differ. Equ. 2017, 2017, 261. [Google Scholar] [CrossRef] [Green Version]
8. El-Nabulsi, R.A.; Moaaz, O.; Bazighifan, O. New Results for Oscillatory Behavior of Fourth-Order Differential Equations. Symmetry 2020, 12, 136. [Google Scholar] [CrossRef] [Green Version]
9. Baculikova, B.; Dzurina, J.; Li, T. Oscillation results for even-order quasilinear neutral functional differential equations. Electron. J. Differ. Equ. 2011, 143, 1–9. [Google Scholar]
10. Moaaz, O.; Awrejcewicz, J.; Bazighifan, O. A New Approach in the Study of Oscillation Criteria of Even-Order Neutral Differential Equations. Mathematics 2020, 8, 197. [Google Scholar] [CrossRef] [Green Version]
11. Moaaz, O.; Dassios, I.; Bazighifan, O. Oscillation Criteria of Higher-order Neutral Differential Equations with Several Deviating Arguments. Mathematics 2020, 8, 412. [Google Scholar] [CrossRef] [Green Version]
12. Moaaz, O.; Elabbasy, E.M.; Muhib, A. Oscillation criteria for even-order neutral differential equations with distributed deviating arguments. Adv. Differ. Equ. 2019, 2019, 297. [Google Scholar] [CrossRef] [Green Version]
13. Xing, G.; Li, T.; Zhang, C. Oscillation of higher-order quasi-linear neutral differential equations. Adv. Differ. Equ. 2011, 2011, 45. [Google Scholar] [CrossRef] [Green Version]
14. Baculikova, B.; Dzurina, J. Oscillation theorems for higher order neutral differential equations. Appl. Math. Comput. 2012, 219, 3769–3778. [Google Scholar] [CrossRef]
15. Elabbasy, E.M.; El-Nabulsi, R.A.; Moaaz, O.; Bazighifan, O. Oscillatory Properties of Solutions of Even-Order Differential Equations. Symmetry 2020, 12, 212. [Google Scholar] [CrossRef] [Green Version]
16. Philos, C.G. A new criterion for the oscillatory and asymptotic behavior of delay differential equations. Bull. Acad. Polonaise Sci. 1981, 39, 61–64. [Google Scholar]
17. Hardy, G.H.; Littlewood, I.E.; Polya, G. Inequalities; Reprint of the 1952 edition; Cambridge University Press: Cambridge, UK, 1988. [Google Scholar]
18. Agarwal, R.P.; Grace, S.R.; O’Regan, D. Oscillation Theory for Difference and Functional Differential Equations; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000. [Google Scholar]
19. Philos, C.G. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. (Basel) 1981, 36, 168–178. [Google Scholar] [CrossRef]

## Share and Cite

MDPI and ACS Style

Moaaz, O.; Furuichi, S.; Muhib, A. New Comparison Theorems for the Nth Order Neutral Differential Equations with Delay Inequalities. Mathematics 2020, 8, 454. https://doi.org/10.3390/math8030454

AMA Style

Moaaz O, Furuichi S, Muhib A. New Comparison Theorems for the Nth Order Neutral Differential Equations with Delay Inequalities. Mathematics. 2020; 8(3):454. https://doi.org/10.3390/math8030454

Chicago/Turabian Style

Moaaz, Osama, Shigeru Furuichi, and Ali Muhib. 2020. "New Comparison Theorems for the Nth Order Neutral Differential Equations with Delay Inequalities" Mathematics 8, no. 3: 454. https://doi.org/10.3390/math8030454

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.