Oscillation Results for Higher Order Differential Equations

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Article

Oscillation Results for Higher Order Differential Equations

Choonkil Park^1,∗,† , Osama Moaaz^2,† and Omar Bazighifan^2,†

1 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt;

[email protected] (O.M.); [email protected] (O.B.)

* Correspondence: [email protected]

† These authors contributed equally to this work.

Received: 22 December 2019; 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 ap-Laplacian-like operator. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided.

Keywords:ocillation; higher-order; differential equations;p-Laplacian equations

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−2y⁽ⁿ⁻¹⁾(t) 0

+q(t)|y(τ(t))|^p−2y(τ(t)) =0. (1) We assume that p > 1 is a constant,r ∈ C¹([t0,∞),R),r(t) > 0, q,τ ∈ C([t0,∞), R),q >

0, τ(t)≤t, limt→∞τ(t) =∞and the condition

η(t0) =_∞, (2)

where

η(t):= Z _∞

t

ds r^1/(p−1)(s)^.

By a solution of (1) we mean a functiony ∈ Cⁿ⁻¹[Ty,∞), Ty ≥ t0, which has the property r(t)y⁽ⁿ⁻¹⁾(t)^p−2y⁽ⁿ⁻¹⁾(t) ∈ C¹[Ty,∞), and satisfies (1) on [Ty,∞). We consider only those solutionsyof (1) which satisfy sup{|y(t)| : t ≥ T} > 0, for allT > Ty. A solution of (1) is called oscillatory if it has arbitrarily large number of zeros on [Ty,∞), 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 [1–24].

It is interesting to study Equation (1) since thep-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−2y⁽ⁿ⁻¹⁾(t) 0

+q(t)f(y(τ(t))) =0,

Axioms2020,9, 14; doi:10.3390/axioms9010014 www.mdpi.com/journal/axioms

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is oscillatory, under the conditions

Z _∞ t₀

1

r^p⁻¹(t)^dt=_∞;

additionally, Z _∞

`₀ ψ(s)− ¹

p^pφ^p(s)((n−1)!)^p−1ρ(s)a(s)

((p−1)µsⁿ⁻¹)^p−1 − (p−1)ρ(s) a^1/(p−1)(s)η^p(s)

!

ds= +_∞,

for some constantµ∈(0, 1)and

Z _∞

`₀ kq(s)^τ(s)^p−1

s^p−1 ds=_∞.

Agarwal et al. [2] studied the oscillation of the higher-order nonlinear delay differential equation

y⁽ⁿ⁻¹⁾(t)^α−1y⁽ⁿ⁻¹⁾(t) 0

+q(t)|y(τ(t))|^α⁻¹y(τ(t)) =0.

whereα is a positive real number. In [27], Zhang et al. studied the asymptotic properties of the solutions of equation

hr(t)y⁽ⁿ⁻¹⁾(t)^αⁱ

0

+q(t)y^β(τ(t)) =0‚ t≥t₀. whereαandβare ratios of odd positive integers,β≤αand

Z _∞

t₀ r^−1/α(s)ds<_∞. (3)

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 alltlarge enough.

2. Main Results

In this section, we establish some oscillation criteria for Equation (1). For convenience, we denote thatF+(t):=max{0,F(t)},

B(t):= ¹ (n−4)!

Z _∞

t (θ−t)ⁿ⁻⁴





 R_∞

θ q(s)^τ(s)_s ^p−1ds r(θ)







1/(p−1)

dθ

and

D(s):= ^r(s)δ(s)|h(t,s)|^p p^ph

H(t,s)A(s)µ_(n−2)!^sⁿ⁻² ip−1. We begin with the following lemmas.

Lemma 1(Agarwal [1]). Let y(t) ∈ C^m[t0,∞) be of constant sign and y^(m)(t) 6= 0on[t0,∞)which satisfies y(t)y^(m)(t)≤0.Then,

(I)There exists a t₁≥t₀such that the functions y⁽ⁱ⁾(t), i=_{1, 2, ...,}m−1are of constant sign on[t₀,∞);

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(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≥t₁,

y(t)y⁽ⁱ⁾(t)>_0, for all i=0, 1, ...,k and

(−1)^m+i+1y(t)y⁽ⁱ⁾(t)>0, for all i=k+1, ...,m.

Lemma 2(Kiguradze [15]). If the function y satisfies y^(j)>0for all j=0, 1, ...,m,and y^(m+1)<0,then m!

t^my(t)−(m−1)!

t^m−1 y⁰(t)≥0.

Lemma 3(Bazighifan [7]). Let h∈C^m([t₀,∞),(0,∞)).Suppose that h^(m)(t)is of a fixed sign, on[t₀,∞), h^(m)(t)not identically zero, and that there exists a t1≥t0such that, for all t≥t1,

h^(m−1)(t)h^(m)(t)≤0.

If we havelim

t→∞h(t)6=0,then there exists t_λ≥t0such that h(t)≥ ^λ

(m−1)!t^m−1

h^(m−1)(t), for everyλ∈(0, 1)and t≥t_λ.

Lemma 4. Let n≥4be even, and assume that y is an eventually positive solution of Equation (1). If (2) holds, then there exists two possible cases for t≥t₁,where t₁≥t₀is sufficiently large:

(C₁) y⁰(t)>0,y⁰⁰(t)>0, y⁽ⁿ⁻¹⁾(t)>0, y⁽ⁿ⁾(t)<0, (C₂) y^(j)(t)>_0,y^(j+1)(t)<₀for all odd integer

j∈ {1, 2, ...,n−3},y⁽ⁿ⁻¹⁾(t)>0, y⁽ⁿ⁾(t)<0.

Proof. Letybe an eventually positive solution of Equation (1). By virtue of (1), we get

r(t)y⁽ⁿ⁻¹⁾(t)^p−2y⁽ⁿ⁻¹⁾(t) 0

<0. (4)

From ([11] Lemma 4), we have thaty⁽ⁿ⁻¹⁾(t)>0 eventually. Then, we can write (4) in the from

r(t)y⁽ⁿ⁻¹⁾(t)^p−1 0

<_0, which gives

r⁰(t)y⁽ⁿ⁻¹⁾(t)^p−1+r(t) (p−1)y⁽ⁿ⁻¹⁾(t)^p−2y⁽ⁿ⁾(t)<0.

Thus,y⁽ⁿ⁾(t)<0 eventually. Thus, by Lemma1, we have two possible cases(C1)and(C2). This completes the proof.

Lemma 5. Let y be an eventually positive solution of Equation (1) and assume that Case(C₁)holds. If

ω(t):=δ(t)





 r(t)

y⁽ⁿ⁻¹⁾(t)^p−1 y^p−1(t)





, (5)

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whereδ∈C¹([t0,∞),(0,∞)),then

ω⁰(t)≤ ^δ

+0 (t)

δ(t) ^ω(t)−δ(t)q(t)

τⁿ⁻¹(t) tⁿ⁻¹

^p−1

− (p−1)µtⁿ⁻²

(n−2)!(δ(t)r(t))^1/(p−1)^ω

p/(p−1)

(t). (6) Proof. Lety be an eventually positive solution of Equation (1) and assume that Case(C1) holds.

From the definition ofω, we see thatω(t)>0 fort≥t1, and

ω⁰(t) ≤ δ⁰(t)^r

(t)y⁽ⁿ⁻¹⁾(t)^p−1 y^p−1(t) +δ(t)

r(t)y⁽ⁿ⁻¹⁾(t)^p−1 0

y^p−1(t)

−δ(t)

(p−1)y⁰(t)r(t)y⁽ⁿ⁻¹⁾(t)^p−1

y^p(t) ^.

Using Lemma3withm=n−1, h(t) =y⁰(t), we get y⁰(t)≥ ^µ

(n−2)!tⁿ⁻²y⁽ⁿ⁻¹⁾(t), (7)

for every constantµ∈(0, 1). From (5) and (7), we obtain

ω⁰(t) ≤ δ⁰(t)^r(t)|(^y⁽ⁿ⁻¹⁾^(t))|^p⁻¹

y^p⁻¹(t) +δ(t)

r(t)|(^y⁽ⁿ⁻¹⁾^(t))|^p⁻¹⁰

y^p⁻¹(t)

−δ(t)^(p−1)µt_(n−2)!ⁿ⁻²^r(t)|(y⁽ⁿ⁻¹⁾(t))|^p

y^p(t) .

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By Lemma2, we have

y(t) y⁰(t) ≥ ^t

n−1. Integrating this inequality fromτ(t)tot, we obtain

y(τ(t)) y(t) ≥ ^τ

n−1(t)

tⁿ⁻¹ . (9)

Combining (1) and (8), we get

ω⁰(t) ≤ δ⁰(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)µt_(n−2)!ⁿ⁻²^r(t)|(^y⁽ⁿ⁻¹⁾^(t))|^p

y^p(t) .

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From (9) and (10), we obtain

ω⁰(t)≤ ^δ

0+(t)

δ(t) ^ω(t)−δ(t)q(t)

^p−1

− (p−1)µtⁿ⁻²

(n−2)!(δ(t)r(t))^1/(p−1)^ω

p/(p−1)(t). (11)

It follows from (11) that

δ(t)q(t)

^p−1

≤ ^δ

0+(t)

δ(t) ^ω(t)−ω⁰(t)− (p−1)µtⁿ⁻²

(n−2)!(δ(t)r(t))^1/(p−1)^ω

p/(p−1)(t).

This completes the proof.

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Lemma 6. Let y be an eventually positive solution of Equation (1) and assume that Case(C2)holds. If

ψ(t):=σ(t)^y

0(t)

y(t)^, ⁽¹²⁾

whereσ∈C¹([t0,∞),(0,∞)),then

σ(t)B(t)≤ −ψ⁰(t) +^σ

0(t)

σ(t)^ψ(t)− ¹ σ(t)^ψ

2(t). (13)

Proof. Letybe an eventually positive solution of Equation (1) and assume that Case(C2)holds. Using Lemma2, we obtain

y(t)≥ty⁰(t). Thus we find thaty/tis nonincreasing, and hence

y(τ(t))≥y(t)^τ(t)

t . (14)

Sincey>0, (1) becomes

r(t)y⁽ⁿ⁻¹⁾(t)^p−1 0

+q(t)y^p−1(τ(t)) =0.

Integrating that equation fromt to∞, we see that

t→lim∞

r(t)y⁽ⁿ⁻¹⁾(t)^p−1

−r(t)y⁽ⁿ⁻¹⁾(t)^p−1+ Z _∞

t q(s)y^p−2(τ(s)) =0. (15) Since the function r

y⁽ⁿ⁻¹⁾p−1

is positive h

r>0 andy⁽ⁿ⁻¹⁾>0i

and nonincreasing

r

y⁽ⁿ⁻¹⁾p−10

<0

!

, there exists a t2 ≥ t0 such that r

y⁽ⁿ⁻¹⁾p−1

is bounded above for all t≥t2, and so limt→∞

r(t)y⁽ⁿ⁻¹⁾(t)^p−1

=c≥0. Then, from (15), we obtain

−r(t)y⁽ⁿ⁻¹⁾(t)^p−1+ Z _∞

t q(s)y^p−2(τ(s))≤ −c≤0.

From (14), we obtain

−r(t)y⁽ⁿ⁻¹⁾(t)^p−1+ Z _∞

t q(s)y(s)^p−1^τ(s)^p−1 s^p−1 ds≤0.

It follows fromy⁰(t)>0 that

−y⁽ⁿ⁻¹⁾(t) + ^y(t) r^1/(p−1)(t)

Z _∞ t q(s)

τ(s) s

p−1

ds

!1/(p−1)

≤0.

Integrating the above inequality fromt to∞for a total of(n−3)times, we get

y⁰⁰(t) + R_∞

t (θ−t)ⁿ⁻⁴

R_∞

θ q(s)

τ(s) s

p−1

ds r(θ)

!^1/(p−1) dθ

(n−4)! y(t)≤0. (16)

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From the definition ofψ(t), we see thatψ(t)>0 fort≥t1, and

ψ⁰(t) =σ⁰(t)^y

0(t)

y(t) +σ(t)^y

00(t)y(t)−(y⁰(t))²

y²(t) ^. ⁽¹⁷⁾

It follows from (16) and (17) that

σ(t)B(t)≤ −ψ⁰(t) +^σ

0(t)

σ(t)^ψ(t)− ¹ σ(t)^ψ

2(t). This completes the proof.

Definition 1. Let

D={(t,s)∈R²^:^t≥s≥t0}and D0={(t,s)∈R²^:^t>s≥t0}. We say that a function H∈C(D,R)belongs to the class<if

(i1) H(t,t) =0for t≥t0,H(t,s)>0, (t,s)∈D0.

(i2)H has a nonpositive continuous partial derivative∂H/∂s on D0with respect to the second variable.

Theorem 1. Let n ≥ 4 be even. Assume that there exist functions H,H∗ ∈ <, δ,A,σ,A∗ ∈ C¹([t₀,∞),(0,∞))and h,h∗∈C(D₀,R)such that

− ^∂

∂s(H(t,s)A(s)) =H(t,s)A(s)^δ

0(t)

δ(t) +h(t,s). (18) and

− ^∂

∂s(H∗(t,s)A∗(s)) =H∗(t,s)A∗(s)^σ

0(t)

σ(t) +h∗(t,s). (19) If

lim sup

t→∞

1 H(t,t0)

Z _t t₀

"

H(t,s)A(s)δ(s)q(s)

τⁿ⁻¹(s) sⁿ⁻¹

^p−1

−D(s)

#

ds=_∞, ₍₂₀₎

for some constantµ∈(0, 1)and lim sup

t→∞

1 H∗(t,t₀)

Z t

t₀ H∗(t,s)A∗(s)σ(s)B(s)− ^σ(s)|h∗(t,s)|² 4H∗(t,s)A∗(s)

!

ds=_∞, (21)

then every solution of (1) is oscillatory.

Proof. Let y be a nonoscillatory solution of Equation (1) on the interval [t0,∞). Without loss of generality, we can assume thatyis an eventually positive. By Lemma4, there exist two possible cases fort≥t1, wheret1≥t0is sufficiently large.

Assume that(C1)holds. From Lemma5, we get that (6) holds. Multiplying (6) byH(t,s)A(s) and integrating the resulting inequality fromt₁tot, we have

Z _t

t₁ H(t,s)A(s)δ(s)q(s)

^p−1 ds

≤ − Z t

t₁ H(t,s)A(s)ω⁰(s)ds+ Z t

t₁ H(t,s)A(s)^δ

0(s)

δ(s)^ω(s)ds

− Z _t

t₁ H(t,s)A(s) (p−1)µsⁿ⁻²

(n−2)!(δ(s)r(s))^1/(p−1)^ω

p/(p−1)

(s)ds

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Thus Z _t

^p−1 ds

≤H(t,t1)A(t1)ω(t1)− Z t

t₁

− ^∂

∂s(H(t,s)A(s))−H(t,s)A(s)^δ

0(t) δ(t)

ω(s)ds

− Z _t

(n−2)!(δ(s)r(s))^1/(p−1)^ω

p/(p−1)

(s)ds

This implies Z _t

^p−1 ds

≤ H(t,t₁)A(t₁)ω(t₁) + Z _t

t₁|h(t,s)|ω(s)d(s) (22)

− Z t

(n−2)!(δ(s)r(s))^1/(p−1)^ω

p/(p−1)(s)ds.

Using the inequality

βUV^β⁻¹−U^β ≤(β−1)V^β, β>1, U≥0 andV≥0, (23) withβ= p/(p−1),

U=

(p−1)H(t,s)A(s) ^µs

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)_(n−2)!^µsⁿ⁻² ^p−1







(p−1)/p

,

we get

|h(t,s)|ω(s)−H(t,s)A(s) (p−1)µsⁿ⁻²

(n−2)!(δ(s)r(s))^1/(p−1)^ω

p/(p−1)

≤ ^δ(s)r(s)

H(t,s)A(s)_(n−2)!^µsⁿ⁻² ^p−1

|h(t,s)|

p p

,

which with (23) gives Z t

^p−1

−D(s)

!

ds ≤ H(t,t1)A(t1)ω(t1)

≤ H(t,t0)A(t₁)ω(t₁).

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Then

1 H(t,t0)

Z _t

t₀ H(t,s)A(s)δ(s)q(s)

^p−1

−D(s)

! ds

≤A(t₁)ω(t₁) + Z _t₁

t₀ A(s)δ(s)q(s)

^p−1 ds

<_∞,

for someµ∈(0, 1), which contradicts (20).

Assume that Case (C₂)holds. From Lemma 6, we get that (13) holds. Multiplying (13) by H∗(t,s)A∗(s), and integrating the resulting inequality fromt1tot, we have

Z _t

t₁ H∗(t,s)A∗(s)σ(s)B(s)ds ≤ − Z _t

t₁H∗(t,s)A∗(s)ψ⁰(s)ds+ Z _t

t₁ H∗(t,s)A∗(s)^σ

0(s) σ(s)^ψ(s)ds

− Z _t

t₁

H∗(t,s)A∗(s) σ(s) ^ψ

2(s)ds

= H∗(t,t₁)A∗(t₁)ψ(t₁)− Z t

t₁

2(s)ds

− Z _t

t₁

− ^∂

∂s(H∗(t,s)A∗(s))−H∗(t,s)A∗(s)^σ

0(t) σ(t)

ψ(s)ds.

Then Z _t

t₁ H∗(t,s)A∗(s)σ(s)B(s)ds ≤ H∗(t,t₁)A∗(t₁)ψ(t₁) + Z _t

t₁|h∗(t,s)|ψ(s)d(s)

− Z t

t₁

2(s)ds.

Hence we have Z _t

t₁ H∗(t,s)A∗(s)σ(s)B(s)−^σ(s)|h∗(t,s)|² 4H∗(t,s)A∗

!

ds ≤ H∗(t,t1)A∗(t1)ψ(t1)

≤ H∗(t,t0)A∗(t1)ψ(t1). This implies

1 H∗(t,t0)

Z _t

t₀ H∗(t,s)A∗(s)σ(s)B(s)−^σ(s)|h∗(t,s)|² 4H∗(t,s)A∗

! ds

≤ A∗(t1)ψ(t1) + Z _t

t₀ A∗(s)σ(s)B(s)ds<_∞ 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 r⁰(t) > 0. Assume that for some constant λ ∈ (0, 1), the differential equation

ϕ⁰(t) + ^q(t) r(τ(t))

λτⁿ⁻¹(t) (n−1)!

!p−1

ϕ(τ(t)) =0 (24) is oscillatory. Then every solution of (1) is oscillatory.

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Proof. Let (1) have a nonoscillatory solutiony. Without loss of generality, we can assume thaty(t)>0 fort≥t₁, wheret₁≥t₀is sufficiently large. Sincer⁰(t)>0, we have

y⁰(t)>0, y⁽ⁿ⁻¹⁾(t)>0 andy⁽ⁿ⁾(t)<0. (25) From Lemma3, we get

y(t)≥ ^λt

n−1

(n−1)!r^1/p−1(t)^r

1/p−1

(t)y⁽ⁿ⁻¹⁾(t), (26)

for everyλ∈(0, 1). Thus, if we set

ϕ(t) =r(t)^hy⁽ⁿ⁻¹⁾(t)ⁱ^p−1>0, then we see thatϕis a positive solution of the inequality

ϕ⁰(t) + ^q(t) r(τ(t))

λτⁿ⁻¹(t) (n−1)!

!p−1

ϕ(τ(t))≤0. (27) From [22] (Theorem 1), we conclude that the corresponding Equation (24) also has a positive solution, which is a contradiction.

Theorem2is proved.

Corollary 1. Assume that (2) holds and let n≥2be even. If

t→lim∞inf Z _t

τ(t)

q(s) r(τ(s))

τⁿ

−1

(s)^p−1ds> ((n−1)!)^p−1

e , (28)

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 t⁴y

9 10t

=0, t≥1, (29)

whereγ>0is a constant. We note that n=4, r(t) =1, p=2, τ(t) =9t/10and q(t) =γ/t⁴. If we set H(t,s) =H∗(t,s) = (t−s)²,A(s) = A∗(s) =1,δ(s) =t³,σ(s) =t, h(t,s) = (t−s) 5−3ts⁻¹ and h∗(t,s) = (t−s) 3−ts⁻¹

then we get

η(s) = Z _∞

t₀

1

r^1/(p−1)(s)^ds=∞ and

B(t) = ¹ (n−4)!

Z _∞ t

(θ−t)ⁿ⁻⁴





 R_∞

θ q(s)^τ(s)_s ^p−1ds r(θ)







1/(p−1)

dθ

= 3γ/

20t² .

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Hence conditions (20) and (21) become

lim sup

t→∞

1 H(t,t₀)

Z t

t₀ H(t,s)A(s)δ(s)q(s)

^p−1

−D(s)

! ds

=lim sup

t→∞

1 (t−1)²

Z _t 1

729γ

1000t²s⁻¹+^729γ

1000s−^729γ 500 t− ^s

2µ

25+9t²s⁻²−30ts⁻¹ ds

=_∞ (ifγ>500/81) and

lim sup

t→∞

1 H∗(t,t₀)

Z t

t₀ H∗(t,s)A∗(s)σ(s)B(s)− ^σ(s)|h∗(t,s)|² 4H∗(t,s)A∗(s)

! ds

=lim sup

t→∞

1 (t−1)²

Z _t 1

3γ

20t²s⁻¹+^3γ 20s−^3γ

10t− ^s 4

9−630ts⁻¹+t²s⁻² ds

=_∞ (ifγ>5/3).

Thus, by Theorem1, every solution of Equation (29) is oscillatory ifγ>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:

1. The Riccati transformation technique.

2. Comparison principles.

3. The Integral averaging technique.

Additionally, in future work we could try to get some oscillation criteria of Equation (1) under the conditionR_∞

t₀ 1

r^1/⁽^p⁻¹⁾(t)dt<∞. Thus, we would discuss the following two cases:

(C₁) y(t)>0, y⁽ⁿ⁻¹⁾(t)>0, y⁽ⁿ⁾(t)<0, (C₂) 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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