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Article

# Oscillation Results for Higher Order Differential Equations

by 1,*,†, 2,† and
1
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
2
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Axioms 2020, 9(1), 14; https://doi.org/10.3390/axioms9010014
Received: 22 December 2019 / Revised: 23 January 2020 / Accepted: 25 January 2020 / Published: 3 February 2020

## Abstract

:
The objective of our research was to study asymptotic properties of the class of higher order differential equations with a p-Laplacian-like operator. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided.

## 1. Introduction

In this work, we are concerned with oscillations of higher-order differential equations with a p-Laplacian-like operator of the form
$r t y n − 1 t p − 2 y n − 1 t ′ + q t y τ t p − 2 y τ t = 0 .$
We assume that $p > 1$ is a constant,$r ∈ C 1 [ t 0 , ∞ ) , R ,$$r t > 0 , q , τ ∈ C [ t 0 , ∞ ) , R ,$$q > 0 , τ t ≤ t ,$$lim t → ∞ τ t = ∞$ and the condition
$η t 0 = ∞ ,$
where
$η t : = ∫ t ∞ d s r 1 / p − 1 s .$
By a solution of (1) we mean a function y$∈ C n − 1 [ T y , ∞ ) , T y ≥ t 0 ,$which has the property $r t y n − 1 t p − 2 y n − 1 t ∈ C 1 [ T y , ∞ ) ,$and satisfies (1) on $[ T y , ∞ )$. We consider only those solutions y of (1) which satisfy $sup { y t : t ≥ T } > 0 ,$for all $T > T y .$A solution of (1) is called oscillatory if it has arbitrarily large number of zeros on $[ T y , ∞ ) ,$ and otherwise it is called to be nonoscillatory; (1) is said to be oscillatory if all its solutions are oscillatory.
In recent decades, there has been a lot of research concerning the oscillation of solutions of various classes of differential equations; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].
It is interesting to study Equation (1) since the p-Laplace differential equations have applications in continuum mechanics [14,25]. In the following, we briefly review some important oscillation criteria obtained for higher-order equations, which can be seen as a motivation for this paper.
Elabbasy et al. [26] proved that the equation
$r t y n − 1 t p − 2 y n − 1 t ′ + q t f y τ t = 0 ,$
is oscillatory, under the conditions
$∫ t 0 ∞ 1 r p − 1 t d t = ∞ ;$
$∫ 𝓁 0 ∞ ψ s − 1 p p ϕ p s n − 1 ! p − 1 ρ s a s p − 1 μ s n − 1 p − 1 − p − 1 ρ s a 1 / p − 1 s η p ( s ) d s = + ∞ ,$
for some constant $μ ∈ 0 , 1$and
$∫ 𝓁 0 ∞ k q s τ s p − 1 s p − 1 d s = ∞ .$
Agarwal et al. [2] studied the oscillation of the higher-order nonlinear delay differential equation
$y n − 1 t α − 1 y n − 1 t ′ + q t y τ t α − 1 y τ t = 0 .$
where $α$ is a positive real number. In [27], Zhang et al. studied the asymptotic properties of the solutions of equation
$r t y n − 1 t α ′ + q t y β τ t = 0 t ≥ t 0 .$
where $α$ and $β$ are ratios of odd positive integers, $β ≤ α$ and
$∫ t 0 ∞ r − 1 / α s d s < ∞ .$
In this work, by using the Riccati transformations, the integral averaging technique and comparison principles, we establish a new oscillation criterion for a class of higher-order neutral delay differential Equations (1). This theorem complements and improves results reported in [26]. An illustrative example is provided.
In the sequel, all occurring functional inequalities are assumed to hold eventually; that is, they are satisfied for all $t$large enough.

## 2. Main Results

In this section, we establish some oscillation criteria for Equation (1). For convenience, we denote that $F + t : = max 0 , F t ,$
$B t : = 1 n − 4 ! ∫ t ∞ θ − t n − 4 ∫ θ ∞ q s τ s s p − 1 d s r θ 1 / p − 1 d θ$
and
$D s : = r s δ s h t , s p p p H t , s A s μ s n − 2 n − 2 ! p − 1 .$
We begin with the following lemmas.
Lemma 1
(Agarwal [1]). Let $y ( t ) ∈ C m t 0 , ∞$ be of constant sign and $y m t ≠ 0$ on $t 0 , ∞$ which satisfies $y t y m t ≤ 0 .$Then,
$I$ There exists a $t 1 ≥ t 0$ such that the functions $y i t , i = 1 , 2 , … , m − 1$ are of constant sign on $t 0 , ∞ ;$
$II$ There exists a number $k ∈ 1 , 3 , 5 , … , m − 1$ when m is even, $k ∈ { 0 , 2 , 4 , … , m − 1 }$ when m is odd, such that, for$t ≥ t 1$,
$y t y i t > 0 ,$
for all $i = 0 , 1 , … , k$ and
$− 1 m + i + 1 y t y i t > 0 ,$
for all $i = k + 1 , … , m .$
Lemma 2
(Kiguradze [15]). If the function y satisfies $y ( j ) > 0$for all $j = 0 , 1 , … , m ,$ and $y m + 1 < 0 ,$ then
$m ! t m y t − m − 1 ! t m − 1 y ′ t ≥ 0 .$
Lemma 3
(Bazighifan [7]). Let $h ∈ C m t 0 , ∞ , 0 , ∞ .$ Suppose that $h m t$ is of a fixed sign, on $t 0 , ∞$, $h m t$ not identically zero, and that there exists a $t 1 ≥ t 0$ such that, for all $t ≥ t 1 ,$
$h m − 1 t h m t ≤ 0 .$
If we have $lim t → ∞ h t ≠ 0 ,$ then there exists $t λ ≥ t 0$ such that
$h t ≥ λ m − 1 ! t m − 1 h m − 1 t ,$
for every $λ ∈ 0 , 1$ and $t ≥ t λ$.
Lemma 4.
Let $n ≥ 4$be even, and assume that y is an eventually positive solution of Equation (1). If (2) holds, then there exists two possible cases for$t ≥ t 1 ,$where$t 1 ≥ t 0$is sufficiently large:
Proof.
Let y be an eventually positive solution of Equation (1). By virtue of (1), we get
$r t y n − 1 t p − 2 y n − 1 t ′ < 0 .$
From ([11] Lemma 4), we have that $y n − 1 t > 0$ eventually. Then, we can write (4) in the from
$r t y n − 1 t p − 1 ′ < 0 ,$
which gives
$r ′ t y n − 1 t p − 1 + r t p − 1 y n − 1 t p − 2 y n t < 0 .$
Thus, $y n t < 0$ eventually. Thus, by Lemma 1, we have two possible cases $C 1$and$C 2$. This completes the proof. □
Lemma 5.
Let y be an eventually positive solution of Equation (1) and assume that Case$C 1$ holds. If
$ω t : = δ t r t y n − 1 t p − 1 y p − 1 t ,$
where $δ ∈ C 1 t 0 , ∞ , 0 , ∞ ,$then
$ω ′ t ≤ δ + ′ t δ t ω t − δ t q t τ n − 1 t t n − 1 p − 1 − p − 1 μ t n − 2 n − 2 ! δ t r t 1 / p − 1 ω p / p − 1 t .$
Proof.
Let y be an eventually positive solution of Equation (1) and assume that Case $C 1$ holds. From the definition of $ω$, we see that $ω t > 0$for$t ≥ t 1 ,$and
$ω ′ t ≤ δ ′ t r t y n − 1 t p − 1 y p − 1 t + δ t r t y n − 1 t p − 1 ′ y p − 1 t − δ t p − 1 y ′ t r t y n − 1 t p − 1 y p t .$
Using Lemma 3 with $m = n − 1 , h t = y ′ t$, we get
$y ′ t ≥ μ n − 2 ! t n − 2 y n − 1 t ,$
for every constant $μ ∈ ( 0 , 1 )$.From (5) and (7), we obtain
$ω ′ t ≤ δ ′ t r t y n − 1 t p − 1 y p − 1 t + δ t r t y n − 1 t p − 1 ′ y p − 1 t − δ t p − 1 μ t n − 2 n − 2 ! r t y n − 1 t p y p t .$
By Lemma 2, we have
$y t y ′ t ≥ t n − 1 .$
Integrating this inequality from $τ t$ to t, we obtain
$y τ t y t ≥ τ n − 1 t t n − 1 .$
Combining (1) and (8), we get
$ω ′ t ≤ δ ′ t r t y n − 1 t p − 1 y p − 1 t − δ t q t y p − 1 τ t y p − 1 t − δ t p − 1 μ t n − 2 n − 2 ! r t y n − 1 t p y p t .$
From (9) and (10), we obtain
$ω ′ t ≤ δ + ′ t δ t ω t − δ t q t τ n − 1 t t n − 1 p − 1 − p − 1 μ t n − 2 n − 2 ! δ t r t 1 / p − 1 ω p / p − 1 t .$
It follows from (11) that
$δ t q t τ n − 1 t t n − 1 p − 1 ≤ δ + ′ t δ t ω t − ω ′ t − p − 1 μ t n − 2 n − 2 ! δ t r t 1 / p − 1 ω p / p − 1 t .$
This completes the proof. □
Lemma 6.
Let y be an eventually positive solution of Equation (1) and assume that Case$C 2$ holds. If
$ψ t : = σ t y ′ t y t ,$
where $σ ∈ C 1 t 0 , ∞ , 0 , ∞ ,$then
$σ t B t ≤ − ψ ′ t + σ ′ t σ t ψ t − 1 σ t ψ 2 t .$
Proof.
Let y be an eventually positive solution of Equation (1) and assume that Case $C 2$ holds. Using Lemma 2, we obtain
$y t ≥ t y ′ t .$
Thus we find that $y / t$is nonincreasing, and hence
$y τ t ≥ y t τ t t .$
Since $y > 0$, (1) becomes
$r t y n − 1 t p − 1 ′ + q t y p − 1 τ t = 0 .$
Integrating that equation from t to , we see that
$lim t → ∞ r t y n − 1 t p − 1 − r t y n − 1 t p − 1 + ∫ t ∞ q s y p − 2 τ s = 0 .$
Since the function $r y n − 1 p − 1$ is positive $r > 0 and y n − 1 > 0$ and nonincreasing $r y n − 1 p − 1 ′ < 0$, there exists a $t 2 ≥ t 0$ such that $r y n − 1 p − 1$ is bounded above for all $t ≥ t 2$, and so $lim t → ∞ r t y n − 1 t p − 1 = c ≥ 0 .$ Then, from (15), we obtain
$− r t y n − 1 t p − 1 + ∫ t ∞ q s y p − 2 τ s ≤ − c ≤ 0 .$
From (14), we obtain
$− r t y n − 1 t p − 1 + ∫ t ∞ q s y s p − 1 τ s p − 1 s p − 1 d s ≤ 0 .$
It follows from $y ′ t > 0$that
$− y n − 1 t + y t r 1 / p − 1 t ∫ t ∞ q s τ s s p − 1 d s 1 / p − 1 ≤ 0 .$
Integrating the above inequality from t to $∞$for a total of $n − 3$ times, we get
$y ″ t + ∫ t ∞ θ − t n − 4 ∫ θ ∞ q s τ s s p − 1 d s r θ 1 / p − 1 d θ n − 4 ! y t ≤ 0 .$
From the definition of $ψ t$, we see that $ψ t > 0$for$t ≥ t 1 ,$and
$ψ ′ t = σ ′ t y ′ t y t + σ t y ″ t y t − y ′ t 2 y 2 t .$
It follows from (16) and (17) that
$σ t B t ≤ − ψ ′ t + σ ′ t σ t ψ t − 1 σ t ψ 2 t .$
This completes the proof. □
Definition 1.
Let
$D = { t , s ∈ R 2 : t ≥ s ≥ t 0 } and D 0 = { t , s ∈ R 2 : t > s ≥ t 0 } .$
We say that a function $H ∈ C D , R$ belongs to the class ℜ if
$i 1 H t , t = 0$for $t ≥ t 0 , H t , s > 0 , t , s ∈ D 0 .$
$i 2 H$has a nonpositive continuous partial derivative $∂ H / ∂ s$ on $D 0$ with respect to the second variable.
Theorem 1.
Let $n ≥ 4$be even. Assume that there exist functions $H , H * ∈ ℜ ,$$δ , A , σ , A * ∈ C 1 t 0 , ∞ , 0 , ∞$ and $h , h * ∈ C D 0 , R$ such that
$− ∂ ∂ s H t , s A s = H t , s A s δ ′ t δ t + h t , s .$
and
$− ∂ ∂ s H * t , s A * s = H * t , s A * s σ ′ t σ t + h * t , s .$
If
$lim sup t → ∞ 1 H t , t 0 ∫ t 0 t H t , s A s δ s q s τ n − 1 s s n − 1 p − 1 − D s d s = ∞ ,$
for some constant $μ ∈ 0 , 1$and
$lim sup t → ∞ 1 H * t , t 0 ∫ t 0 t H * t , s A * s σ s B s − σ s h * t , s 2 4 H * t , s A * s d s = ∞ ,$
then every solution of (1) is oscillatory.
Proof.
Let y be a nonoscillatory solution of Equation (1) on the interval $t 0 , ∞$. Without loss of generality, we can assume that $y$is an eventually positive. By Lemma 4, there exist two possible cases for$t ≥ t 1 ,$where$t 1 ≥ t 0$is sufficiently large.
Assume that $C 1$ holds. From Lemma 5, we get that (6) holds. Multiplying (6) by $H t , s A s$ and integrating the resulting inequality from $t 1$to t, we have
$∫ t 1 t H t , s A s δ s q s τ n − 1 s s n − 1 p − 1 d s ≤ − ∫ t 1 t H t , s A s ω ′ s d s + ∫ t 1 t H t , s A s δ ′ s δ s ω s d s − ∫ t 1 t H t , s A s p − 1 μ s n − 2 n − 2 ! δ s r s 1 / p − 1 ω p / p − 1 s d s$
Thus
$∫ t 1 t H t , s A s δ s q s τ n − 1 s s n − 1 p − 1 d s ≤ H t , t 1 A t 1 ω t 1 − ∫ t 1 t − ∂ ∂ s H t , s A s − H t , s A s δ ′ t δ t ω s d s − ∫ t 1 t H t , s A s p − 1 μ s n − 2 n − 2 ! δ s r s 1 / p − 1 ω p / p − 1 s d s$
This implies
$∫ t 1 t H ( t , s ) A s δ s q s τ n − 1 s s n − 1 p − 1 d s ≤ H t , t 1 A t 1 ω t 1 + ∫ t 1 t h t , s ω s d s − ∫ t 1 t H t , s A s p − 1 μ s n − 2 n − 2 ! δ s r s 1 / p − 1 ω p / p − 1 s d s .$
Using the inequality
$β U V β − 1 − U β ≤ β − 1 V β , β > 1 , U ≥ 0 and V ≥ 0 ,$
with $β = p / p − 1 ,$
$U = p − 1 H t , s A s μ s n − 2 n − 2 ! p − 1 / p ω s δ s r s 1 / p$
and
$V = p − 1 p p − 1 h t , s p − 1 δ s r s p − 1 H t , s A s μ s n − 2 n − 2 ! p − 1 p − 1 / p ,$
we get
$h t , s ω s − H t , s A s p − 1 μ s n − 2 n − 2 ! δ s r s 1 / p − 1 ω p / p − 1 ≤ δ s r s H t , s A s μ s n − 2 n − 2 ! p − 1 h t , s p p ,$
which with (23) gives
$∫ t 1 t H t , s A s δ s q s τ n − 1 s s n − 1 p − 1 − D s d s ≤ H t , t 1 A t 1 ω t 1 ≤ H t , t 0 A t 1 ω t 1 .$
Then
$1 H t , t 0 ∫ t 0 t H t , s A s δ s q s τ n − 1 s s n − 1 p − 1 − D s d s ≤ A t 1 ω t 1 + ∫ t 0 t 1 A s δ s q s τ n − 1 s s n − 1 p − 1 d s < ∞ ,$
for some $μ ∈ 0 , 1$, which contradicts (20).
Assume that Case $C 2$ holds. From Lemma 6, we get that (13) holds. Multiplying (13) by $H * t , s A * s$, and integrating the resulting inequality from $t 1$to t, we have
$∫ t 1 t H * t , s A * s σ s B s d s ≤ − ∫ t 1 t H * t , s A * s ψ ′ s d s + ∫ t 1 t H * t , s A * s σ ′ s σ s ψ s d s − ∫ t 1 t H * t , s A * s σ s ψ 2 s d s = H * t , t 1 A * t 1 ψ t 1 − ∫ t 1 t H * t , s A * s σ s ψ 2 s d s − ∫ t 1 t − ∂ ∂ s H * t , s A * s − H * t , s A * s σ ′ t σ t ψ s d s .$
Then
$∫ t 1 t H * t , s A * s σ s B s d s ≤ H * t , t 1 A * t 1 ψ t 1 + ∫ t 1 t h * t , s ψ s d s − ∫ t 1 t H * t , s A * s σ s ψ 2 s d s .$
Hence we have
$∫ t 1 t ( H * t , s A * s σ s B s − σ s h * t , s 2 4 H * t , s A * ) d s ≤ H * t , t 1 A * t 1 ψ t 1 ≤ H * t , t 0 A * t 1 ψ t 1 .$
This implies
$1 H * t , t 0 ∫ t 0 t ( H * ( t , s ) A * ( s ) σ s B s − σ s h * t , s 2 4 H * t , s A * ) d s ≤ A * t 1 ψ t 1 + ∫ t 0 t A * s σ s B s d s < ∞$
which contradicts (21). Therefore, every solution of (1) is oscillatory. □
In the next theorem, we establish new oscillation results for Equation (1) by using the comparison technique with the first-order differential inequality:
Theorem 2.
Let $n ≥ 2$be even and $r ′ t > 0$. Assume that for some constant $λ ∈ 0 , 1$, the differential equation
$φ ′ t + q t r τ t λ τ n − 1 t n − 1 ! p − 1 φ τ t = 0$
is oscillatory. Then every solution of (1) is oscillatory.
Proof.
Let (1) have a nonoscillatory solution y. Without loss of generality, we can assume that $y t > 0$ for$t ≥ t 1 ,$where$t 1 ≥ t 0$is sufficiently large. Since $r ′ t > 0$, we have
$y ′ t > 0 , y n − 1 t > 0 and y n t < 0 .$
From Lemma 3, we get
$y t ≥ λ t n − 1 n − 1 ! r 1 / p − 1 t r 1 / p − 1 t y n − 1 t ,$
for every $λ ∈ 0 , 1$. Thus, if we set
$φ t = r t y n − 1 t p − 1 > 0 ,$
then we see that $φ$ is a positive solution of the inequality
$φ ′ t + q t r τ t λ τ n − 1 t n − 1 ! p − 1 φ τ t ≤ 0 .$
From [22] (Theorem 1), we conclude that the corresponding Equation (24) also has a positive solution, which is a contradiction.
Theorem 2 is proved. □
Corollary 1.
Assume that (2) holds and let $n ≥ 2$be even. If
$lim t → ∞ inf ∫ τ t t q s r τ s τ n − 1 s p − 1 d s > n − 1 ! p − 1 e ,$
then every solution of (1) is oscillatory.
Next, we give the following example to illustrate our main results.
Example 1.
Consider the equation
$y 4 t + γ t 4 y 9 10 t = 0 , t ≥ 1 ,$
where $γ > 0$ is a constant. We note that $n = 4 , r t = 1 , p = 2 , τ t = 9 t / 10$ and$q t = γ / t 4$. If we set $H t , s = H * t , s = t − s 2 ,$$A s = A * s = 1 ,$$δ s = t 3 ,$$σ s = t , h t , s = t − s 5 − 3 t s − 1$ and $h * t , s = t − s 3 − t s − 1$ then we get
$η s = ∫ t 0 ∞ 1 r 1 / p − 1 s d s = ∞$
and
$B t = 1 n − 4 ! ∫ t ∞ θ − t n − 4 ∫ θ ∞ q s τ s s p − 1 d s r θ 1 / p − 1 d θ = 3 γ / 20 t 2 .$
Hence conditions (20) and (21) become
$lim sup t → ∞ 1 H t , t 0 ∫ t 0 t H t , s A s δ s q s τ n − 1 s s n − 1 p − 1 − D s d s = lim sup t → ∞ 1 t − 1 2 ∫ 1 t 729 γ 1000 t 2 s − 1 + 729 γ 1000 s − 729 γ 500 t − s 2 μ 25 + 9 t 2 s − 2 − 30 t s − 1 d s = ∞ if γ > 500 / 81$
and
$lim sup t → ∞ 1 H * t , t 0 ∫ t 0 t H * t , s A * s σ s B s − σ s h * t , s 2 4 H * t , s A * s d s = lim sup t → ∞ 1 t − 1 2 ∫ 1 t 3 γ 20 t 2 s − 1 + 3 γ 20 s − 3 γ 10 t − s 4 9 − 630 t s − 1 + t 2 s − 2 d s = ∞ if γ > 5 / 3 .$
Thus, by Theorem 1, every solution of Equation (29) is oscillatory if $γ > 500 / 81$.

## 3. Conclusions

In this work, we have discussed the oscillation of the higher-order differential equation with a p-Laplacian-like operator and we proved that Equation (1) is oscillatory by using the following methods:
• The Riccati transformation technique.
• Comparison principles.
• The Integral averaging technique.
Additionally, in future work we could try to get some oscillation criteria of Equation (1) under the condition $∫ t 0 ∞ 1 r 1 / p − 1 t d t < ∞$. Thus, we would discuss the following two cases:
$C 1 y t > 0 , y n − 1 t > 0 , y n t < 0 , C 2 y t > 0 , y n − 2 t > 0 , y n − 1 t < 0 .$

## Author Contributions

The authors claim to 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

The authors declare no conflict of interest.

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MDPI and ACS Style

Park, C.; Moaaz, O.; Bazighifan, O. Oscillation Results for Higher Order Differential Equations. Axioms 2020, 9, 14. https://doi.org/10.3390/axioms9010014

AMA Style

Park C, Moaaz O, Bazighifan O. Oscillation Results for Higher Order Differential Equations. Axioms. 2020; 9(1):14. https://doi.org/10.3390/axioms9010014

Chicago/Turabian Style

Park, Choonkil, Osama Moaaz, and Omar Bazighifan. 2020. "Oscillation Results for Higher Order Differential Equations" Axioms 9, no. 1: 14. https://doi.org/10.3390/axioms9010014

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