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Article

# Oscillation Results of Emden–Fowler-Type Differential Equations

by
Omar Bazighifan
1,*,†,
Taher A. Nofal
2,† and
Mehmet Yavuz
3,*,†
1
2
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
3
Department of Mathematics and Computer Sciences, Necmettin Erbakan University, 42090 Konya, Turkey
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2021, 13(3), 410; https://doi.org/10.3390/sym13030410
Submission received: 31 January 2021 / Revised: 25 February 2021 / Accepted: 1 March 2021 / Published: 3 March 2021

## Abstract

:
In this article, we obtain oscillation conditions for second-order differential equation with neutral term. Our results extend, improve, and simplify some known results for neutral delay differential equations. Several effective and illustrative implementations are provided.

## 1. Introduction

The neutral differential equations are of great importance in life, due to their contribution to many applications in science and technology. We find this type of equations in applications of engineering in-flight vibration, medicine represented in the heartbeat, physics, see [1,2]. For this, many researchers have been interested in studying the oscillations of Emden-Fowler neutral differential equations, see [3].
This manuscript focuses on the oscillatory properties of solutions to second-order differential equation with neutral term:
where $y ≥ y 0$ and $ξ 2 y = ξ 1 y + α 1 y ξ 1 α 2 y .$ Our novel outcomes are obtained by considering the following suppositions:
$α 2 , β ı ∈ C y 0 , ∞ , R , β ı ′ y > 0 , α 2 y ≤ y , lim y → ∞ α 2 y = ∞ , lim y → ∞ β ı y = ∞ , f y , ξ 1 ∈ C y 0 , ∞ × R , R , ζ ∈ C y 0 , ∞ , R , α 1 y ∈ 0 , 1 , ξ 1 f y , ξ 1 > 0 for all ξ 1 ≠ 0 , f y , ξ 1 ≥ ∑ ı = 1 j ζ ı y ξ 1 r 2 , j ≥ 1 , ı = 1 , 2 , . . , j , ς and α 1 are positive functions ; r 1 , r 2 are quotient of odd positive integers .$
The following relations are satisfied:
and
In the context of oscillation theory, it has been the object of many researchers who have investigated this notion for Emden–Fowler neutral differential and difference equations; the reader can refer to Reference [4,5,6,7,8,9,10,11,12,13,14,15,16].
The authors in Reference [5,10,14,15] considered equation of the form
The authors used Riccati technique that differs from the one we used in this article. Moreover, they also studied the equation under the condition $ϖ y 0 = ∞$ which is different from our condition .
Liu et al. [17] proved that the Equation (5) is oscillates or $lim y → ∞ ξ 1 y = 0$ if
$∫ y 0 ∞ η 1 s ζ s 1 − α 1 β ı s r 2 − 2 r 2 η 1 ′ s 2 ς r 2 / r 1 β ı s 8 r 2 M r 1 − r 2 r 1 β ı r 2 − 1 s β ı ′ s η 1 s d s = ∞ ,$
and
$∫ y 0 ∞ 1 ς y φ y ∫ y 0 y φ s ζ s d s 1 / r 1 d y = ∞ ,$
where $η 1 , φ ∈ C y 0 , ∞ , 0 , ∞$ and $η 1 > 0$ and
$α 1 ′ y ≥ 0 , α 2 ′ y ≥ 0 and lim y → ∞ α 1 y = C .$
On the other hand, in Reference [18], the authors obtained oscillation criteria for equation
where
and used the comparing technique to ensure that the equation is oscillatory if
$∫ y 0 ∞ η 1 s ζ s 1 − α 1 β ı s r 2 − η 1 ′ s ε 1 + 1 ς ε 1 s ε 1 + 1 ε 1 + 1 m η 1 s β ı ′ s ε 1 d s = ∞ ,$
and
$∫ y 0 ∞ ϖ ε 2 s ζ s 1 − α 1 s r 2 − K ς 1 − ε 2 + 1 r 1 s ϖ s d s = ∞ ,$
where $η 1 ∈ C y 0 , ∞ , 0 , ∞$ and $ε 1 = min r 1 , r 2$, $ε 2 = max r 1 , r 2$,
$ε 1 y = β y if r 2 ≥ r 1 y if r 2 < r 1 and m = 1 r 1 = r 2 0 < m ≤ 1 r 1 ≠ r 2 .$
Saker [13] explained that the Equation (5) is oscillatory if
$∫ y ∞ ε 2 s ζ s 1 − α 1 β ı s ∫ y β ı y ς − 1 / r 1 x d x ∫ y y ς − 1 / r 1 x d x r 2 d s = ∞ ,$
and
$∫ y ∞ 1 ς s ∫ y s ϖ r 2 x ζ x 1 − α 1 x r 2 d x 1 / r 1 d s = ∞ ,$
where
$ε 2 y = 1 if r 2 = r 1 c 2 ∫ y y ς − 1 / r 1 s d s r 2 − r 1 if r 2 < r 1 c 1 if r 2 > r 1 ,$
and $c 1 ,$$c 2$ are any positive constants.
In Reference [8,9,19,20,21,22], the authors used the comparing technique with first-order to ensure that Equation (5) is oscillatory and also explained that the equation
is oscillatory if and only if
$∫ y 0 y X ˜ s d s = ∞ ,$
and
$lim y → ∞ ∫ β ı y y X ˜ s d s = ∞ ,$
where
and
From the above, we can observe the following:
-
The conditions that [17] obtained guarantee that all solutions of (5) are either oscillatory or tends to zero;
-
To apply the results in Reference [13,17,18], we must ensure that $η 1 y$ and $α 2 y$ are nondecreasing.
The purpose of this paper is to extend, improve, and simplify results in [13,17,18] and establish new oscillation criteria for (1). Our results improved results in Reference [17] so that the condition guarantee that all solution of (1) is oscillatory, and without imposing restrictions on the derivatives of $η 1 y$ and $α 2 y$. In addition, we simplify the results in Reference [13,18].

## 2. Oscillation Criteria

Theorem 1.
If
$∫ y 0 ∞ 1 ς x ∫ y 0 x ϖ r 2 β ı s X s d s 1 / r 1 d x = ∞ ,$
where
$X y = ∑ ı = 1 j ζ ı y 1 − α 1 β ı y ϖ α 2 β ı y ϖ β ı y r 2 ,$
then Equation (1) is oscillatory.
Proof.
Let $ξ 1$ be a nonoscillatory solution of Equation (1); then, and for all $y ≥ y 1$. Since , we see that and
$ς y ξ 2 ′ y r 1 ′ = − f y , ξ 1 β ı y ≤ − ∑ ı = 1 j ζ ı y ξ 1 r 2 β ı y < 0 .$
Then, $ς y ξ 2 ′ y r 1$ is strictly decreasing on $y 0 , ∞$ and of one sign.
Let $ξ 2 ′ y < 0$ for $y ≥ y 1$. Hence,
$ξ 2 y ≥ − ∫ y ∞ 1 ς 1 / r 1 s ς 1 / r 1 s ξ 2 ′ s d s ≥ − ς 1 / r 1 y ξ 2 ′ y ϖ y .$
Since
$ς y ξ 2 ′ y r 1 ′ < 0 ,$
we find
$ς y ξ 2 ′ y r 1 ≤ ς y 1 ξ 2 ′ y 1 r 1 : = − K < 0 ,$
for all $y ≥ y 1$; thus,
$ξ 2 y ≥ K 1 / r 1 ϖ y for all y ≥ y 1 .$
From (19), we see
$d d y ξ 2 y ϖ y ≥ 0 .$
We can also note that
and
Then,
From (18), we have
$ς y ξ 2 ′ y r 1 ′ ≤ − ∑ ı = 1 j ζ ı y ξ 2 r 2 β ı y 1 − α 1 β ı y ϖ α 2 β ı y ϖ β ı y r 2 .$
Combining (19) with (20) yields
$ς y ξ 2 ′ y r 1 ′ ≤ − K r 2 / r 1 ϖ r 2 β ı y X y .$
By integrating (21) from $y 1$ to y, we find
$ς y ξ 2 ′ y r 1 ≤ − K r 2 / r 1 ∫ y 1 y ϖ r 2 β ı s X s d s .$
Integrating (22) from $y 1$ to y, we obtain
$ξ 2 y ≤ ξ 2 y 1 − K r 2 / r 1 2 ∫ y 1 y 1 ς x ∫ y 1 x ϖ r 2 β ı s X s d s 1 / r 1 d x .$
From (17) and (23), we find that
Now, let $ξ 2 ′ y > 0$. Then, $ξ 2 y > ξ 2 α 2 y > ξ 1 α 2 y$; hence,
From (18), we find
$ς y ξ 2 ′ y r 1 ′ ≤ − ∑ ı = 1 j ζ ı y 1 − α 1 β ı y r 2 ξ 2 r 2 β ı y .$
Since $ϖ ′ y < 0$, we see that
$ϖ α 2 β ı y ≥ ϖ β ı y for y ≥ y 2 ≥ y 1 .$
Then, and from (24), we get
$ς y ξ 2 ′ y r 1 ′ ≤ − ∑ ı = 1 j ζ ı y 1 − α 1 β ı y r 2 ξ 2 r 2 β ı y .$
Integrating (25) from $y 1$ to y, we obtain
$ς y ξ 2 ′ y r 1 ≤ ς y 2 ξ 2 ′ y 2 r 1 − ∫ y 2 y ∑ ı = 1 j ζ ı s 1 − α 1 β ı s r 2 ξ 2 r 2 β ı s d s ≤ ς y 2 ξ 2 ′ y 2 r 1 − ξ 2 r 2 β ı y 2 ∫ y 2 y ∑ ı = 1 j ζ ı s 1 − α 1 β ı s r 2 d s ≤ ς y 2 ξ 2 ′ y 2 r 1 − ξ 2 r 2 β ı y 2 ∫ y 2 y ∑ ı = 1 j ζ ı s 1 − α 1 β ı s ϖ α 2 β ı y ϖ β ı y r 2 d s ≤ ς y 2 ξ 2 ′ y 2 r 1 − ξ 2 r 2 β ı y 2 ∫ y 2 y X s d s .$
Since $ϖ ′ y < 0$, we obtain
$∫ y 2 y ϖ r 2 β ı s X s d s ≤ ϖ r 2 β ı y 2 ∫ y 2 y X s d s .$
From (17), we see that $∫ y 2 y ϖ r 2 β ı s X s d s$ must be unbounded. Hence, from (27), we obtain
$∫ y 2 y X s d s → ∞ as y → ∞ .$
From (26), we find
while this is a contradiction to $ξ 2 ′ y > 0$. Thus, the theorem is proved. □
Theorem 2.
If
is oscillatory and
$∫ y 0 ∞ X s d s = ∞ .$
Then, Equation (1) is oscillatory.
Proof.
Let $ξ 1$ be a nonoscillatory solution of Equation (1); then, $ξ 1 y > 0$ and $ξ 2 ′ y$ is of one sign eventually.
Let $ξ 2 ′ y < 0$ for all $y ≥ y 1$. Integrating (20) from $y 1$ to y, we see that
$ς y ξ 2 ′ y r 1 ≤ ς y 1 ξ 2 ′ y 1 r 1 − ∫ y 1 y ξ 2 r 2 β ı s X s d s .$
Since $β ı ′ y > 0$, we find
$ς y ξ 2 ′ y r 1 ≤ − ξ 2 r 2 β ı y ∫ y 1 y X s d s ,$
so
$ξ 2 ′ y ≤ − ξ 2 r 2 / r 1 β ı y 1 ς y ∫ y 1 y X s d s 1 / r 1 .$
Hence, we have that for equation
From Reference Lemma 1 in [20], Equation (29) has a positive solution. This indicates that is no positive solution of (1), while this is a contradiction.
Let $ξ 2 ′ y > 0$ for all $y ≥ y 1$. Thus, we find (30) leads to (28), and then the remainder of the proof is the same as the proof of Theorem 1. Thus, the theorem is proved. □
Next, we establish new oscillation conditions for Equation (1) according to the results obtained by Reference [8,20].
Corollary 1.
Let $0 < r 2 < r 1$ and
$∫ y 0 y 1 ς s ∫ y 1 y X z d z 1 / r 1 d s = ∞ ;$
then, Equation (1) is oscillatory.
Corollary 2.
Let $r 2 > r 1$ and
$∫ y 0 ∞ ∑ ı = 1 j ζ ı s 1 − α 1 β ı s ϖ α 2 β ı s ϖ β ı s r 2 d s = ∞ .$
If
$lim sup y → ∞ r 2 θ ′ β ı y β ı ′ y r 1 θ ′ y < 1 ,$
and
$lim inf y → ∞ 1 θ ′ y e − θ y X ˜ y > 0 ,$
where $θ y ∈ C 1 y 0 , ∞ , R ,$$θ ′ y > 0$ and $lim y → ∞ θ y = ∞ .$ Then, Equation (1) is oscillatory.

## 3. Applications

Through this section, we will provide some examples and applications that support the validity of the results and conditions that we got in our article.
Example 1.
Consider the equation
$y 2 ξ 1 y + μ y ξ 1 y − ε 1 ′ ′ + a y − ε 2 ξ 1 y − ε 2 = 0 , y ≥ y 0 ,$
where $μ , ε 1 , a$, and $ε 2$ are positive real numbers, and $y 0 = max μ + ε 1 , ε 2 + ε 1 .$
We have that $r 1 = r 2 = 1 ,$ $ς y = y 2 ,$ $α 1 y = μ / y ,$ $α 2 y = y − ε 1 ,$ $ζ y = a y − ε 2$, and $β ı y = y − ε 2$. This gives that $ϖ y = y − 1$ and
$X y = a y − ε 2 1 − μ y − ε 2 − ε 1 .$
Since $y > μ + ε 1$, we have that $μ / y < y − ε 1 / y$, and hence $α 1 y < ϖ y / ϖ α 2 y$. Note that,
$∫ y 0 ∞ 1 x 2 ∫ y 0 x 1 − μ y − ε 2 − ε 1 d s d x = ∞ .$
By Theorem 1, Equation (33) is oscillatory.
Example 2.
Let the equation
$e − y ξ 1 y + e − y ξ 1 α 2 y ′ 11 / 3 ′ + 1 y + 1 ξ 1 y − 2 7 / 3 + 1 2 + y ξ 1 y − 3 7 / 3 = 0 .$
Let, $r 1 = 11 / 3 , r 2 = 7 / 3 , ς y = e − y , α 1 y = e − y ,$$β ı y = y − j + 1$ with index $j = 1 , 2$ and
$ϖ y 0 = ∫ y 0 ∞ ς − 1 / r 1 v d v = 11 3 e 3 y / 11 − 1 .$
Thus, the condition of Theorem 1 hold, and therefore, each solution of (34) is oscillatory.
Example 3.
Let the equation
$e y ξ 1 y + e − y ξ 1 ε 1 y ′ ′ + e a y ξ 1 r 2 ε 2 y = 0 , y > 0 ,$
where $ε 1 , ε 2 ∈ 0 , 1$ and $r 2 > 1 .$ Let, $r 1 = 1 , ς y = e y , α 1 y = e − y ,$$ζ y = e a y , α 2 y = ε 1 y$ and $β ı y = ε 2 y$. This gives that $ϖ y = e − y$ and
$X y = e a y 1 − e − ε 1 ε 2 y r 2 .$
Moreover, we find
$∫ y 0 ∞ 1 ς x ∫ y 0 x ϖ r 2 β ı s X s d s 1 / r 1 d x = ∫ y 0 ∞ e − x ∫ y 0 x e a y e − ε 2 y 1 − e − ε 1 ε 2 y r 2 d s d x = ∞ ,$
if $a > r 2 ε 2 + 1$. Thus, by Theorem 1, every solution of (35) oscillates if $a > r 2 ε 2 + 1$.
Remark 1.
We see that in Example 3; thus, the results in Reference [13,17,18] cannot be applied in Equation (35).

## 4. Conclusions

In this work, we have presented many of the oscillatory properties of the second-order neutral differential equation (1). Our results improve, unify, and extend some known results for differential equations with neutral term. In future work, we will discuss oscillation conditions of these equations by using integral averaging technique and Riccati technique.

## Author Contributions

Conceptualization, O.B., T.A.N. and M.Y. These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

Not applicable.

## Acknowledgments

The authors thank the reviewers for their useful comments, which led to the improvement of the content of the paper. (Taher A. Nofal) Taif University Researchers Supporting Project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

## Conflicts of Interest

The authors declare no conflict of interest.

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Bazighifan, O.; Nofal, T.A.; Yavuz, M. Oscillation Results of Emden–Fowler-Type Differential Equations. Symmetry 2021, 13, 410. https://doi.org/10.3390/sym13030410

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Bazighifan O, Nofal TA, Yavuz M. Oscillation Results of Emden–Fowler-Type Differential Equations. Symmetry. 2021; 13(3):410. https://doi.org/10.3390/sym13030410

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Bazighifan, Omar, Taher A. Nofal, and Mehmet Yavuz. 2021. "Oscillation Results of Emden–Fowler-Type Differential Equations" Symmetry 13, no. 3: 410. https://doi.org/10.3390/sym13030410

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