EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2499-2508
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Oscillatory properties Test for Even-Order Differential
Equations of Neutral type

Alanoud Almutairi
1 Department of Mathematics, Faculty of Science, University of Hafr Al Batin, P.O. Box
1803, Hafar Al Batin 31991, Saudi Arabia

Abstract. This paper presents a study on the oscillatory behavior of solutions to even-order neu-
tral differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using
techniques from first-order delay differential equations, Riccati technique and integral averages
technique. The results of this work contribute to a deeper understanding of even-order differential
equations and their connections to various branches of mathematics and practical sciences. The
findings emphasize the importance of continued research in this area.

2020 Mathematics Subject Classifications: 34K10, 34K11

Key Words and Phrases: Oscillation conditions, neutral, even-order, differential equation

1. Introduction

Differential equations are characterized by many important advantages that contribute
to many practical applications in this life. It is involved in the aviation industry, especially
in controlling vibrational motion, in medicine and in civil engineering in building bridges;
see [2, 3, 5–7] .

The p -Laplace equations have some significant applications in elasticity theory and
continuum mechanics. The oscillation theory of equations has undergone many research
contributions by many researchers, especially the study of approximate and oscillatory
behavior; see [1, 4, 8, 10, 10, 11, 16, 17].
The aim of this work is to study the oscillatory properties of the solutions of even-order
delay differential equations(

a (ι)w(β−1) (ι)
)′

+
r∑

i=1

bi (ι)φ (ξ (zi (ι))) = 0, (1)

where β ≥ 2 and
w (ι) = |ξ (ι)|p−2 ξ (ι) + ς (ι) ξ (γ (ι)) (2)

DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4945

Email address: amalmutairi@uhb.edu.sa (A. Almutairi)

https://www.ejpam.com 2499 © 2023 EJPAM All rights reserved.



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2500

also ∫ ∞

ι0

1

a (s)
ds = ∞. (3)

We also suppose the following conditions:
(H1)a, ς ∈ C ([ι0,∞) , [0,∞)) , bi ∈ C ([ι0,∞) ,R+) , a (ι) > 0, a′ (ι) ≥ 0, 0 ≤ ς (ι) < 1,

(H2)γ ∈ C ([ι0,∞) , (0,∞)) , γ (ι) ≤ ι, limι→∞ γ (ι) = ∞; zi ∈ C ([ι0,∞) ,R) , i = 1, 2, ..., r

(H3)φ ∈ C (R,R) , φ (ξ) ≥ |ξ|p−2 ξ for ξ ̸= 0, zi (ι) ≤ ι, z′i (ι) > 0, limι→∞ zi (ι) = ∞.

(H4)β and p are positive integers, β is even, p > 1.

k

Definition 1. A solution of (1) is said oscillatory if it has arbitrarily large zeros on
[ιξ,∞), and otherwise is called to be nonoscillatory.

Definition 2. Eq. (1) is called to be oscillatory if all its solutions are oscillatory.

Definition 3. [14] Let

L0 = {(ι, s) : ι > s > ι0} and L = {(ι, s) : ι ≥ s ≥ ι0} .

A function W ∈ C (L,R) is said to belong to the function class ς, written by W ∈ ς, if

(i) W (ι, s) > 0 on L0 and W (ι, s) = 0 for ι ≥ ι0 with (ι, s) /∈ L0;

W (ι, s) has a continuous and nonpositive partial derivative ∂W/∂s on L0 and ςi ∈
C (L0,R) such that

∂W (ι, s)

∂s
= −g(ι, s)

√
W (ι, s).

In recent times, some conditions and properties have been found for delay and neutral
differential equations of different orders; see [9, 11–15, 18].
Agarwal et al. [2] used the Riccati method to obtain conditions for the oscillation of
equation [∣∣∣ξ(β−1) (ι)

∣∣∣α−1
ξ(β−1) (ι)

]′
+ b (ι) |ξ (γ (ι))|α−1 ξ (γ (ι)) = 0.

Elabbasy et al. [8] used the comparison method to obtain comparison for the oscillation
of equation [

a (ι)
∣∣∣(ξ(β−1) (ι)

)∣∣∣p−2
ξ(β−1) (ι)

]′
+ b (ι)φ (ξ (γ (ι))) = 0, p > 1,

under ∫ ∞

ι0

1

a1/(p−1) (s)
ds = ∞



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2501

Bazighifan et al. in [13] considered the equation(
a (ι)φ

(
ξ(β−1) (ι)

))′
+ b (ι)φ (ξ (γ (ι))) = 0, (4)

where φ (s) = |s|p−2 s and obtained properties for oscillation of (4).
In [18]. Zhang et al. studied the oscillation of the solutions of equation(

a (ι)
(
ξ(β−1) (ι)

)α)′
+ b (ι) ξγ (γ (ι)) = 0,

under
∫∞
ι0
a−1/α (s) ds <∞.

In our current research, we applied three techniques with some auxiliary lemmas to
obtain several conditions and properties of the approximate and oscillatory behavior of
the studied equation. These techniques are the comparison method, Riccati method and
integral averages method.

2. Oscillation results

Now, we present some the lemmas:

Lemma 1. [7] If w ∈ Cβ ([ι0,∞) , (0,∞)) and w(β−1) (ι)w(β) (ι) ≤ 0 for ι ≥ ι0 , then for
every ν ∈ (0, 1) there exists a constant j > 0 such that

|w (νι)| ≥ jιβ−1
∣∣∣w(β−1) (ι)

∣∣∣ ,
for all large t.

Lemma 2. [4] Let w ∈ Cβ ([ι0,∞) , (0,∞)) and w(β−1) (ι)w(β) (ι) ≤ 0. If limι→∞w (ι) ̸=
0, then for every µ ∈ (0, 1) there exists a ιµ ≥ ι0 such that

|w (ι)| ≥ µ

(β − 1)!
ιβ−1

∣∣∣w(β−1) (ι)
∣∣∣ ,

for all ι ≥ ιµ.

Lemma 3. [3] Let w(ι) be an β times differentiable function on [ι0,∞) of constant sign
and w(β) (ι) ̸= 0 on [ι0,∞) which satisfies w (ι)w(β) (ι) ≤ 0. Then,

(I) there exists a ι1 ≥ ι0 such that the functions w(i) (ι) , i = 1, 2, ..., β−1 are of constant
sign on [ι0,∞) ;

(II) there exists a number l ∈ {1, 3, 5, ..., β − 1} when β is even, l ∈ {0, 2, 4, ..., β − 1}
when β is odd, such that, for ι ≥ ι1,

w (ι)w(i) (ι) > 0,

for all i = 0, 1, ..., l and
(−1)β+i+1w (ι)w(i) (ι) > 0,

for all i = l + 1, ..., β.



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2502

Lemma 4. Let ξ (ι) is an eventually positive solution of equation (1). Then

w (ι) > 0, w′ (ι) > 0, w(β−1) (ι) ≥ 0 and w(β) (ι) ≤ 0, (5)

for ι ≥ ι2.

Proof. Suppose ξ (ι) is an eventually positive solution of (1). Then w (ι) > 0 and

(
aw(β−1)

)′
(ι) = −

r∑
i=1

bi (ι)φ (ξ (zi (ι))) ≤ 0. (6)

Which means that a (ι)w(β−1) (ι) is decreasing and w(β−1) (ι) is eventually of one sign.
We claim that w(β−1) (ι) ≥ 0. Otherwise, if there exists a ι2 ≥ ι1 such that w(β−1) (ι) <
0 for ι ≥ ι2, and (

aw(β−1)
)
(ι) ≤

(
aw(β−1)

)
(ι2) = −L,

where L > 0. Integrating the above inequality from ι2 to ι we find

w(β−2) (ι) ≤ w(β−2) (ι2)− L

∫ ι

ι2

1

a (s)
ds.

Letting ι→ ∞, we have limι→∞w(β−2) (ι) = −∞, which contradicts the fact that w (ι) >
0. Hence, we obtain w(β−1) (ι) ≥ 0 for ι ≥ ι1.
From Eq. (1), we get

(
aw(β)

)
(ι) = −

(
a′w(β−1)

)
(ι)−

r∑
i=1

bi (ι)φ (ξ (zi (ι))) ≤ 0,

this implies that w(β) (ι) ≤ 0, ι ≥ ι1. From Lemma 3, we find (5) hold. The proof is
complete.

Theorem 1. Let
ξ′ (ι) + K̂ (ι) ξ (z (ι)) = 0, (7)

is oscillatory, where

K̂ (ι) : =
µzβ−1 (ι)

(β − 1)!a (z (ι))
K (ι) ,

K (ι) : =
r∑

i=1

bi (ι) (1− ς (z (ι))) ,

then Eq. (1) is oscillatory.



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2503

Proof. Let (1) has a nonoscillatory solution. From Lemma 4, we find (5) holds. From

w (ι) = |ξ (ι)|p−2 ξ (ι) + ς (ι) ξ (γ (ι)) ,

we see that

ξp−1 (ι) = w (ι)− ς (ι) ξ (γ (ι)) ≥ w (ι)− ς (ι)w (γ (ι)) ≥ w (ι)− ς (ι)w (ι)

≥ (1− ς (ι))w (ι)

and so
ξp−1 (zi (ι)) ≥ w (zi (ι)) (1− ς (zi (ι))) . (8)

From (8), we get
φ (ξ (zi (ι))) ≥ w (zi (ι)) (1− ς (zi (ι))) . (9)

From (1) and (9), we see(
aw(β−1)

)′
(ι) ≤ −

r∑
i=1

bi (ι)w (zi (ι)) (1− ς (zi (ι)))

≤ −w (z (ι))

r∑
i=1

bi (ι) (1− ς (zi (ι)))

= −K (ι)w (z (ι)) (10)

In view of Lemma 2, we obtain

w (ι) ≥ µ

(β − 1)!
ιβ−1w(β−1) (ι) ,

for all ι ≥ ι2 ≥ max {ι1, ιµ}. Thus, by using (10), we find(
a (ι)w(β−1) (ι)

)′
+
µzβ−1 (ι)K (ι)

(β − 1)!a (z (ι))

(
a (z (ι))w(β−1) (z (ι))

)
≤ 0.

Therefore, we get ξ (ι) = a (ι)w(β−1) (ι) is a positive solution of the differential inequality

ξ′ (ι) + K̂ (ι) ξ (z (ι)) ≤ 0.

From [14, Corollary 1], we see that (7) also has a positive solution, a contradiction. This
completes the proof.

Using Theorem 2.1.1 in [18], we get the following corollary.

Corollary 1. If

lim inf
ι→∞

∫ ι

z(ι)

zβ−1 (s)

a (z (s))
K (s) ds >

(β − 1)!

µe
, for µ ∈ (0, 1) ,

then (1) is oscillatory.



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2504

Theorem 2. If∫ ∞

ι0

(
ℏ (u)K (u)− 1

4ν

(
ℏ′ (u)
ℏ (u)

)2

b (u)

)
du = ∞, for ν ∈ (0, 1) , j > 0, (11)

then (1) is oscillatory, where ℏ ∈ C1 ([ι0,∞) ,R+) and

b (ι) :=
a (ι) ℏ (ι)

jzβ−2 (ι) z′ (ι)
.

Proof. Let (1) has a nonoscillatory solution. As in the proof of Theorem 1, we arrive
at (10). From Lemma 1 with ξ = w′, there exists a j > 0 and z (ι) ≤ ι such that

w′ (νz (ι)) ≥ jzβ−2 (ι)w(β−1) (z (ι))

≥ jzβ−2 (ι)w(β−1) (ι) . (12)

Define

ψ (ι) := ℏ (ι)
a (ι)w(β−1) (ι)

w (νz (ι))
> 0,

we have

ψ′ (ι) =
ℏ′ (ι)
ℏ (ι)

ψ (ι) + ℏ (ι)
(
a (ι)w(β−1) (ι)

)′
w (νz (ι))

− νℏ (ι)
a (ι)w(β−1) (ι)w′ (νz (ι)) z′ (ι)

(w (νz (ι)))2
.

From (10), we obtain

ψ′ (ι) ≤ ℏ′ (ι)
ℏ (ι)

ψ (ι)− ℏ (ι)K (ι)− ν
w′ (z (ι)) z′ (ι)

w (νz (ι))
ψ (ι) .

By using (12), we have

ψ′ (ι) ≤ ℏ′ (ι)
ℏ (ι)

ψ (ι)− ℏ (ι)K (ι)− ν
jzβ−2 (ι)w(β−1) (ι) z′ (ι)

w (νz (ι))
ψ (ι)

≤ ℏ′ (ι)
ℏ (ι)

ψ (ι)− ℏ (ι)K (ι)− ν
jzβ−2 (ι) z′ (ι)

a (ι) ℏ (ι)
ℏ (ι) a (ι)w(β−1) (ι)

w (νz (ι))
ψ (ι)

≤ ℏ′ (ι)
ℏ (ι)

ψ (ι)− ℏ (ι)K (ι)− ν

b (ι)
ψ2 (ι) . (13)

Using the inequality

ξw − uw
a+1
a ≤ aa

(a+ 1)a+1

ξa+1

ua
,

with ξ = ℏ′/ℏ, u = νjzβ−2 (ι) z′ (ι) / (a (ι) ℏ (ι)) and w = ψ (ι), we find

ψ′ (ι) ≤ −ℏ (ι)K (ι) +
1

4ν

(
ℏ′ (ι)
ℏ (ι)

)2 a (ι) ℏ (ι)
jzβ−2 (ι) z′ (ι)

. (14)



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2505

Integrating (14) from ι1 to ι we find∫ ι

ι1

(
ℏ (u)K (u)− 1

4ν

(
ℏ′ (u)
ℏ (u)

)2

b (u)

)
du ≤ ψ (ι1)− ψ (ι)

≤ ψ (ι1) ,

which contradicts (11). This completes the proof.

Theorem 3. If ℏ ∈ C1 ([ι0,∞) ,R+) such that

lim sup
ι→∞

1

W (ι, ι0)

∫ ι

ι0

W (ι, u)

(
ℏ (u)K (u)− 1

4ν
b (u) ς2 (ι, u)

)
du = ∞, (15)

where

ς (ι, s) =
ℏ′ (s)
ℏ (s)

− g (ι, s)√
W (ι, s)

,

then Eq.(1) is oscillatory.

Proof. Multiplying (13) by W (ι, s) and integrating both sides from ι2 to ι, we obtain∫ ι

ι2

W (ι, u) ℏ (u)K (u) du ≤ −
∫ ι

ι2

W (ι, u)ψ′ (u) du−
∫ ι

ι2

W (ι, u)
ν

b (u)
ψ2 (u) du

+

∫ ι

ι2

W (ι, u)
ℏ′ (u)
ℏ (u)

ψ (u) du

≤ W (ι, ι2)ψ (ι2)−
∫ ι

ι2

W (ι, u)
ν

b (u)
ψ2 (u) du

+

∫ ι

ι2

W (ι, u)ψ (u) ς (ι, u) du

which implies that∫ ι

ι2

W (ι, u) ℏ (u)K (u) du ≤ W (ι, ι2)ψ (ι2)

−
∫ ι

ι2

W (ι, u)
ν

b (u)

(
ψ2 (u)− b (u)

ν
ς (ι, u)ψ (u)

)
du

It follows that

1

W (ι, ι2)

∫ ι

ι2

W (ι, u)

(
ℏ (u)K (u)− 1

4ν
b (u) ς2 (ι, u)

)
du

≤ ψ (ι2)−
1

W (ι, ι2)

∫ ι

ι2

W (ι, u)
ν

b (u)

(
ψ (u)− 1

2ν
b (u) ς (ι, u)

)2

du,



A. Almutairi / Eur. J. Pure Appl. Math, 16 (4) (2023), 2499-2508 2506

which implies

lim sup
ι→∞

1

W (ι, ι2)

∫ ι

ι2

W (ι, u)

(
ℏ (u)K (u)− 1

4ν
b (u) ς2 (ι, u)

)
du ≤ ψ (ι2) .

From (15), we have a contradiction. This completes the proof.

Corollary 2. Let

0 < inf
s≥ι

(
lim inf
ι→∞

W (ι, s)

W (ι, ι0)

)
≤ ∞

and

lim sup
ι→∞

1

W (ι, ι0)

∫ ι

ι0

W (ι, u) b (u) ς2 (ι, u) du <∞.

If

lim sup
ι→∞

∫ ι

ι0

ς2 (s)

b (s)
ds = ∞

for ς ∈ C ([ι0,∞) ,R) and ς (ι) = max {ς (ι) , 0} , also,

lim sup
ι→∞

1

W (ι, ι0)

∫ ι

ι0

W (ι, u)

(
ℏ (u)K (u)− 1

4ν
b (u) ς2 (ι, u)

)
du ≥ sup

ι≥ι0

ς (ι) ,

then (1) is oscillatory.

Example 1. Consider the second-order equation:[
ι

(
ξ (ι) +

1

2
ξ
( ι
3

))′]′
+
b0
ι

(
ξ2 + ξ

) ( ι
2

)
= 0, ι ≥ 1, (16)

where b0 > 0 is a constant. Let β = p = 2, a (ι) = ι, ς (ι) = 1/2, γ (ι) = ι/3, b (ι) =
b0/ι, z (ι) = ι/2, φ (ξ) = ξ2 + ξ.
Now, we see that

K (ι) = b (ι) (1− ς (z (ι))) =
b0
2ι

and

b (ι) =
a (ι) ℏ (ι)

jzβ−2 (ι) z′ (ι)
=

2ι2

j
.

If we set ℏ = ι then any for constants j > 0, 0 < ν < 1∫ ∞

ι0

(
ℏ (u)K (u)− 1

4ν

(
ℏ′ (u)
ℏ (u)

)2

b (u)

)
du,

=

∫ ∞

ι0

(
b0
2

− 1

2νj

)
du

= ∞ if b0 > 1.

From Theorem 2, every solution of equation (16) is oscillatory if b0 > 1.



REFERENCES 2507

Example 2. Consider the fourth-order equation:[
ιw′′′ (ι)

]′
+
b

ι
ξ
( ι
3

)
= 0, ι ≥ 1, (17)

where w (ι) = ξ (ι) + 1
3ξ
(
ι
2

)
and b > 0 is a constant. Let β = 4, p = 2, a (ι) = ι, ς (ι) =

1/3, γ (ι) = ι/2, b (ι) = b0/ι, z (ι) = ι/3, φ (ξ) = ξ.
Thus, we see that ∫ ∞

a−1 (ι) dι = ∞.

By Theorem 3, every solution of equation (17) is oscillatory.

3. Conclusion

In conclusion, this study aimed at investigating the oscillatory properties of solutions
to even-order differential equations with a p-Laplacian. The findings of this paper con-
tribute to the understanding of the asymptotic and oscillatory behavior of such equations
and provide new oscillation criteria through the use of comparison methods with first-
order differential equations, Riccati technique and integral averages technique. This work
highlights the relevance of the theory of fourth-order differential equations to various fields
of mathematics and practical sciences, emphasizing the importance of continued research
in this area.

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