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

# Oscillatory Properties of Solutions of Even-Order Differential Equations

by
Elmetwally M. Elabbasy
1,†,
2,*,†,
Osama Moaaz
1,† and
Omar Bazighifan
3,†
1
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2
Athens Institute for Education and Research, Mathematics and Physics Divisions, 10671 Athens, Greece
3
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2020, 12(2), 212; https://doi.org/10.3390/sym12020212
Submission received: 24 December 2019 / Revised: 13 January 2020 / Accepted: 19 January 2020 / Published: 2 February 2020

## Abstract

:
This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using Riccati transformation and the integral averaging technique, we obtain a new oscillation criteria. This new theorem complements and improves some known results from the literature. An example is provided to illustrate the main results.

## 1. Introduction

Neutral differential equations are used in numerous applications in technology and natural science. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [1], and therefore their qualitative properties are important.
Very recently, some scholars have been attracted by the problems of the oscillations of differential equations and made relative advances therein, as in [2,3,4,5,6,7,8,9,10,11].
Delay differential equations are often studied in one of two cases
$∫ t 0 ∞ r - 1 / α 1 s d s = ∞$
or
$∫ t 0 ∞ r - 1 / α 1 s d s < ∞ ,$
which are said to be in the canonical or noncanonical form, respectively, see [12]. For the canonical form, many authors in [13,14,15,16] studied the asymptotic behavior of the solutions of equation
$r t z n - 1 t α ′ + q t x β τ t = 0 .$
In the noncanonical form, Li and Rogovchenko [17] studied the asymptotic properties of solutions of higher-order neutral differential (2) under the assumptions that allow applications to even- and odd-order equations with delayed and advanced arguments.
This paper is motivated by several recent studies [3,7,9,10] of such higher order equations. Using the integral averaging technique and the Riccati transformation, we study the asymptotic properties of solutions of even order neutral delay differential equations of the form
$r t z n - 1 t α 1 - 1 z n - 1 t ′ + ∑ i = 1 m q i t y σ i t α i - 1 y σ i t = 0 ,$
where n is an even natural number and m is a natural number. In this paper, we assume that $α i$ are positive integers, $α i + 1 > α i ,$ $r ∈ C 1 t 0 , ∞ , R ,$ $r t > 0 , r ′ t ≥ 0 , a ∈ C t 0 , ∞ , 0 , 1 ,$ $lim t → ∞ a t = ∞ , q i , σ i ∈ C t 0 , ∞ , R ,$ $q i > 0 , σ i t ≤ t$, and $lim t → ∞ σ i t = ∞$ for all $i = 1 , . . . , m$. During the following results, for clarity of presentation, we study only the case where $m = 3$. Moreover, we denote, for convenience, that
$z t : = y t + a t y τ t , σ t : = min σ i t , i = 1 , 2 , 3 B t : = ∫ t 0 t 1 r 1 / α 1 t d t , F + t : = max F t , 0 , A i t : = q i t 1 - a σ i t α i , for all i = 1 , 2 , 3 . m 1 : = α 3 + α 2 - 2 α 1 α 2 - α 1 , m 2 : = α 3 + α 2 - 2 α 1 α 3 - α 1$
and
$A t = A 1 t + m 1 A 2 t 1 / m 1 m 2 A 3 t 1 / m 2 .$
We say that a function, y, is a solution of (3), we mean a non-trivial real function $z ( t ) ∈ C n - 1 [ t y , ∞ )$, $t y ≥ t 0$, satisfying (3) on $[ t y , ∞ )$ and which has the property $r t z n - 1 t α 1 ∈ C [ t y , ∞ )$. We consider only those solutions y of (3) which satisfy $sup { y ( t ) : t ≥ T } > 0 ,$ for any $T ≥ t y$. A solution of (3) is called oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory. Studying the functional differential equations, the continuity of all functions, and $r ( t )$ > 0 are sufficient conditions for the existence of one or more solutions of the equation.
To establish our main results, we make use of the following lemmas:
Lemma 1
([18]). Let C and D nonnegative real numbers. Then
$C μ + μ - 1 D μ - μ C D μ - 1 ≥ 0 , μ > 1 ,$
where the equality holds, if and only if, $C = D .$
Lemma 2
([19]). Let $h ∈ C n t 0 , ∞ , 0 , ∞ .$ If $h n t$ is eventually of one sign for all large $t ,$ then there exist a $t y > t 1$ for some $t 1 > t 0$ and an integer $m , 0 ≤ m ≤ n$ with $n + m$ even for $h n t ≥ 0$ or $n + m$ odd for $h n t ≤ 0$ such that $m > 0$ implies that $h k t > 0$ for $t > t y , k = 0 , 1 , . . . , m - 1$ and $m ≤ n - 1$ implies that $- 1 m + k h k t > 0$ for $t > t y , k = m , m + 1 , . . . , n - 1 .$
Lemma 3
([20]). Let $h t ∈ C n t 0 , ∞ , 0 , ∞ .$ If $h n - 1 t h n t ≤ 0$ for all $t ≥ t y ,$ then for every $θ ∈ 0 , 1 ,$ there exists a constant $M > 0$ such that
$h ′ θ t ≥ M t n - 2 h n - 1 t ,$
for all sufficiently large$t .$
This paper is concerned with the oscillatory behavior of a class of even-order neutral differential equations with multi-delays. Firstly, by using the Riccati transformations, we obtain a new oscillation criteria for this equation. Secondly, using the integral averaging technique, we establish a Philos type oscillation criterion. This new theorem complements and improves some known results in the literature. Finally, an example is provided to illustrate the main results.

## 2. Oscillation Criteria

In this section, we establish new oscillation results for Equation (3) using the Riccatti transformation.
Theorem 1.
Assume that (1) holds. If there exists a positive function$ρ ∈ C 1 t 0 , ∞ , 0 , ∞$such that
$∫ t 0 ∞ ρ s A s - 1 α 1 + 1 α 1 + 1 r α 1 + 1 t ρ + ′ t α 1 + 1 θ M σ ′ ( s ) σ n - 2 s ρ s r s α 1 d s = ∞ ,$
then all solutions of (3) are oscillatory.
Proof.
Assume that (3) has a nonoscillatory solution y. Without loss of generality, we may assume that there exists a $t 1 ∈ t 0 , ∞$ such that $y t > 0 ,$ $y τ t > 0$ and $y σ i t > 0$ for all $i = 1 , 2 , 3$ and $t ∈ t 1 , ∞$. It follows from Lemma 2 that
$z t > 0 , z ′ t > 0 , z n - 1 t > 0 and z n t < 0 ,$
for $t ≥ t 1$. Since $τ ( t ) ≤ t$ and $z ′ t > 0$, we find
$y t = z ( t ) - a t y τ t ≥ z ( t ) - a t z τ t ≥ z ( t ) - a t z t ≥ 1 - a t z t .$
Hence,
$y σ i t ≥ 1 - a σ i t z σ i t , i = 1 , 2 , 3 ,$
which, with (3), gives
$r t z n - 1 t α 1 ′ = - q 1 t y α 1 σ 1 t - q 2 t y α 2 σ 2 t - q 3 t y α 3 σ 3 t ≤ - A 1 t z α 1 σ 1 t - A 2 t z α 2 σ 2 t - A 3 t z α 3 σ 3 t .$
Using Lemma 3, we obtain
$z ′ θ σ t ≥ M σ n - 2 t z n - 1 σ t ≥ M σ n - 2 t z n - 1 t .$
Now, we define a generalized Riccati substitution $ω$ by
$ω t : = ρ t r t z n - 1 t α 1 z α 1 θ σ ( t ) .$
Then, $ω t > 0$. By differentiating (7), we obtain
$ω ′ t = ρ ′ t r t z n - 1 t α 1 z α 1 θ σ ( t ) + ρ t r t z n - 1 t α 1 ′ z α 1 θ σ ( t ) - α 1 θ ρ t r t z n - 1 t α 1 z ′ θ σ ( t ) σ ′ ( t ) z α 1 + 1 θ σ ( t ) .$
Since $z ′ t > 0$ and $σ t ≤ σ i t ,$ we have
$z σ i t ≥ z σ t , i = 1 , 2 , 3 .$
Hence, from (5), (6), and (8), we see that
$ω ′ t ≤ ρ ′ t ρ t ω t - ρ t A 1 t + A 2 t z α 2 - α 1 σ 2 t + A 3 t z α 3 - α 1 σ 3 t - α 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t z n - 1 t α 1 + 1 z α 1 + 1 θ σ ( t ) .$
This implies that
$ω ′ t ≤ ρ ′ t ρ t ω t - ρ t A 1 t + A 2 t z α 2 - α 1 σ 2 t + A 3 t z α 3 - α 1 σ 3 t - α 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t 1 / α 1 ω α 1 + 1 / α 1 t .$
Using Youngs inequality
$u v ≤ 1 c 1 u c 1 + 1 c 2 v c 2 , c 1 > 1 , c 2 > 1 , 1 c 1 + 1 c 2 = 1 ,$
with $u = m 1 A 2 t z α 2 - α 1 σ t 1 m 1 , v = m 2 A 3 t z α 3 - α 1 σ t 1 m 2$ and $c i = m i ,$ we obtain
$A 2 t z α 2 - α 1 σ t + A 3 t z α 3 - α 1 σ t ≥ m 1 A 2 t 1 m 1 m 2 A 3 t 1 m 2 .$
Combining (9) and (10), we have
$ω ′ t ≤ ρ ′ t ρ t ω t - ρ t A t - α 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t 1 / α 1 ω α 1 + 1 / α 1 t .$
If we set $μ = α 1 + 1 / α 1 ,$ then we find
$ω ′ t ≤ ρ ′ t ρ t ω t - ρ t A t - α 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t μ - 1 ω μ t .$
Now, using Lemma 1 with
$C = α 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t μ - 1 1 / μ ω t$
and
$D = r t ρ + ′ t μ α 1 θ M σ ′ ( t ) σ n - 2 t - 1 / μ 1 / μ - 1 ,$
we obtain
$ρ ′ t ρ t ω t - α 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t μ - 1 ω μ t ≤ 1 α 1 + 1 α 1 + 1 r α 1 + 1 t ρ + ′ t α 1 + 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t α 1 .$
Hence, from (11) and (12), we have
$ω ′ t ≤ 1 α 1 + 1 α 1 + 1 r α 1 + 1 t ρ + ′ t α 1 + 1 θ M σ ′ ( t ) σ n - 2 t ρ t r t α 1 - ρ t A t .$
Integrating from $t 1$ to t we find
$∫ t 1 t ρ s A s - 1 α 1 + 1 α 1 + 1 r α 1 + 1 t ρ + ′ t α 1 + 1 θ M σ ′ ( s ) σ n - 2 s ρ s r s α 1 d s ≤ ω t 1 - ω t < ω t 1 .$
This completes the proof. □
In Theorem 1, we can obtain different conditions for oscillation of all solutions of Equation (3) with different choices of $ρ t$. If we set $ρ ( t ) : = B α 1 σ ( t )$, then we obtain the following corollary.
Corollary 1.
Assume that (1) holds. If
$lim sup t → ∞ ∫ t 1 t B α 1 σ ( s ) A s - 1 α 1 + 1 α 1 + 1 α 1 r s σ ′ ( s ) r - 1 / α 1 σ ( s ) α 1 + 1 B σ ( s ) θ M r ( s ) σ ′ s σ n - 2 s α 1 d s = ∞ ,$
then all solutions of (3) are oscillatory.

## 3. Kamenev-Type Criteria

In the section theorem, we establish new oscillation results for Equation (3) using the integral averaging technique to establish the Philos-type.
Definition 1.
Let
$D = { t , s ∈ R 2 : t ≥ s ≥ t 0 } and D 0 = { t , s ∈ R 2 : t > s ≥ t 0 } .$
A kernel function$H ∈ C D , R$is said to belong to the function class ℑ, written by$H ∈ ℑ$, if
(i1)
$H t , t = 0$and$H t , s > 0 , t , s ∈ D 0$for$t ≥ t 0 ,$
(i2)
$H$has a nonpositive continuous partial derivative$∂ H / ∂ s$on$D 0$with respect to the second variable, and there exist functions$h ∈ C D 0 , R$and$δ ∈ C 1 t 0 , ∞ , 0 , ∞$such that
$- ∂ ∂ s H t , s δ s = H t , s A s ρ ′ t ρ t + h t , s .$
Theorem 2.
Assume that (1) holds. If there exist functions$ρ , δ ∈ C 1 t 0 , ∞ , 0 , ∞$such that (13) and
$lim sup t → ∞ 1 H t , t 1 ∫ t 1 t H t , s δ s ρ s A s - Θ s d s = ∞ ,$
hold, where
$Θ s : = h t , s α 1 + 1 α 1 + 1 r s ρ s θ M H t , s δ s σ ′ ( s ) σ n - 2 s α 1 ,$
then every solution of (3) is oscillatory.
Proof.
Proceeding as in the proof of Lemma 1, we obtain (11). Multiplying (11) by $H t , s δ s$ and integrating from $t 1$ to t, we find
$∫ t 1 t H t , s δ s ρ s A s d s ≤ ∫ t 1 t H t , s δ s ρ ′ t ρ t ω s d s - ∫ t 1 t H t , s δ s ω ′ s d s - ∫ t 1 t H t , s δ s α 1 θ M σ ′ ( s ) σ n - 2 s ρ s r s μ - 1 ω μ s d s . ≤ - ∫ t 1 t - ∂ ∂ s H t , s δ s - H t , s δ s ρ ′ t ρ t ω s d s - H t , s δ s ω s t 1 t - ∫ t 1 t H t , s δ s α 1 θ M σ ′ ( s ) σ n - 2 s ρ s r s μ - 1 ω μ s d s .$
This implies that
$∫ t 1 t H t , s δ s ρ s A s d s ≤ - ∫ t 1 t h t , s ω s d s + H t , t 1 δ t 1 ω t 1 - ∫ t 1 t H t , s δ s α 1 θ M σ ′ ( s ) σ n - 2 s ρ s r s μ - 1 ω μ s d s .$
Using Lemma 1 with
$C = α 1 θ M σ ′ ( s ) σ n - 2 s H t , s δ s 1 / μ ρ s r s 1 / α 1 + 1 ω s ; D = h t , s α 1 ρ s r s μ α 1 α 1 θ M σ ′ ( s ) σ n - 2 s H t , s δ s α 1 1 / μ ,$
we have
$∫ t 1 t H t , s δ s ρ s A s - Θ s d s ≤ H t , t 1 δ t 1 ω t 1 ,$
which contradicts (14). Theorem 2 is proved. □
Corollary 2.
If the condition (14) in Theorem 2 is replaced by the following conditions:
$lim sup t → ∞ 1 H t , t 1 ∫ t 1 t H t , s δ s ρ s A s d s = ∞$
and
$lim sup t → ∞ 1 H t , t 1 ∫ t 1 t Θ s d s < ∞ ,$
then every solution of (3) is oscillatory.
Example 1.
Consider the differential equation
$t y t + 1 2 y t 3 ′ ′ + y t 2 + y 2 t + y 3 t 4 = 0 ,$
where$t ≥ 1$. Note that$r t = t ,$$n = 2 ,$$α 1 = 1 ,$$α 2 = 2 ,$$α 3 = 3 ,$$a = 1 / 2 ,$$τ t = t / 3 ,$$σ 1 t = t / 2 ,$$σ 2 t = t ,$$σ 3 t = t / 4$, and$q i t = 1$. Hence, we have$m 1 = m 2 = 3 ,$$A k t = 2 - k$
$A s = 1 4 18 3 + 1 2$
and
$∫ t 0 ∞ 1 r 1 / α 1 t d t = ∫ t 0 ∞ 1 t d t = ∞ .$
If we set$ρ t = 1 ,$then condition (4) is satisfied. Therefore, from Theorem 1, every solution of Equation (18) is oscillatory.

## 4. Conclusions

In this work, by using the generalized Riccati transformation technique and the integral averaging technique, we establish a new oscillation criteria for (3) under (1). Further, in future work, we can attempt to find some oscillation criteria of Equation (3), if $z t = y t - a t y τ t$.

## Author Contributions

The authors have contributed equally and significantly in this paper. 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 authors thank the reviewers for for their useful comments, which led to the improvement of the content of the paper.

## Conflicts of Interest

There are no competing interests between the authors.

## References

1. Hale, J.K. Theory of Functional Differential Equations; Springer: New York, NY, USA, 1977. [Google Scholar]
2. Bazighifan, O.; Cesarano, C. Some New Oscillation Criteria for Second-Order Neutral Differential Equations with Delayed Arguments. Mathematics 2019, 7, 619. [Google Scholar] [CrossRef] [Green Version]
3. Bazighifan, O.; Elabbasy, E.M.; Moaaz, O. Oscillation of higher-order differential equations with distributed delay. J. Inequal. Appl. 2019, 55, 55. [Google Scholar] [CrossRef] [Green Version]
4. Cesarano, C.; Pinelas, S.; Al-Showaikh, F.; Bazighifan, O. Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations. Symmetry 2019, 11, 628. [Google Scholar] [CrossRef] [Green Version]
5. Cesarano, C.; Bazighifan, O. Qualitative behavior of solutions of second order differential equations. Symmetry 2019, 11, 777. [Google Scholar] [CrossRef] [Green Version]
6. El-Nabulsi, R.A.; Bazighifan, O.; Moaaz, O. New Results for Oscillatory Behavior of Fourth-Order Differential Equations. Symmetry 2020, 12, 136. [Google Scholar] [CrossRef] [Green Version]
7. Elabbasy, E.M.; Cesarano, C.; Bazighifan, O.; Moaaz, O. Asymptotic and oscillatory behavior of solutions of a class of higher order differential equation. Symmetry 2019, 11, 1434. [Google Scholar] [CrossRef] [Green Version]
8. Moaaz, O. New criteria for oscillation of nonlinear neutral differential equations. Adv. Differ. Equ. 2019, 2019, 484. [Google Scholar] [CrossRef] [Green Version]
9. 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]
10. Moaaz, O.; Elabbasy, E.M.; Muhib, A. Some new oscillation results for fourth-order neutral differential equations. Adv. Differ. Equ. 2019, 2019, 297. [Google Scholar] [CrossRef] [Green Version]
11. Moaaz, O.; Elabbasy, E.M.; Shaaban, E. Oscillation criteria for a class of third order damped differential equations. Arab J. Math. Sci. 2018, 24, 16–30. [Google Scholar] [CrossRef]
12. Trench, W.F. Canonical forms and principal systems for general disconjugate equations. Trans. Amer. Math. Soc. 1973, 189, 319–327. [Google Scholar] [CrossRef]
13. Baculíková, B.; Džurina, J. Oscillation theorems for higher order neutral differential equations. Appl. Math. Comput. 2012, 219, 3769–3778. [Google Scholar] [CrossRef]
14. Baculikova, B.; Dzurina, J.; Li, T. Oscillation results for evenorder quasilinear neutral functional differential equations. Electron. J. Differ. Equ. 2011, 143, 1–9. [Google Scholar]
15. Chatzarakis, G.E.; Elabbasy, E.M.; Bazighifan, O. An oscillation criterion in 4th-order neutral differential equations with a continuously distributed delay. Adv. Differ. Equ. 2019, 336, 1–9. [Google Scholar]
16. Xing, G.; Li, T.; Zhang, C. Oscillation of higher-order quasilinear neutral differential equations. Adv. Differ. Equ. 2011, 2011, 45. [Google Scholar] [CrossRef] [Green Version]
17. Li, T.; Rogovchenko, Y.V. Asymptotic behavior of higher-order quasilinear neutral differential equations. In Abstract and Applied Analysis; Hindawi: London, UK, 2014; Volume 2014, p. 395368. [Google Scholar]
18. Liu, S.; Zhang, Q.; Yu, Y. Oscillation of even-order half-linear functional differential equations with damping. Comput. Math. Appl. 2011, 61, 2191–2196. [Google Scholar] [CrossRef] [Green Version]
19. Agarwal, R.P.; Grace, S.R.; O’Regan, D. Oscillation Theory for Difference and Functional Differential Equations; Kluwer Acad. Publ.: Dordrecht, The Netherlands, 2000. [Google Scholar]
20. Philos, C. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay. Arch. Math. 1981, 36, 168–178. [Google Scholar] [CrossRef]

## Share and Cite

MDPI and ACS Style

Elabbasy, E.M.; El-Nabulsi, R.A.; Moaaz, O.; Bazighifan, O. Oscillatory Properties of Solutions of Even-Order Differential Equations. Symmetry 2020, 12, 212. https://doi.org/10.3390/sym12020212

AMA Style

Elabbasy EM, El-Nabulsi RA, Moaaz O, Bazighifan O. Oscillatory Properties of Solutions of Even-Order Differential Equations. Symmetry. 2020; 12(2):212. https://doi.org/10.3390/sym12020212

Chicago/Turabian Style

Elabbasy, Elmetwally M., Rami Ahmad El-Nabulsi, Osama Moaaz, and Omar Bazighifan. 2020. "Oscillatory Properties of Solutions of Even-Order Differential Equations" Symmetry 12, no. 2: 212. https://doi.org/10.3390/sym12020212

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