EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5714
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Criteria for Guaranteeing Oscillation of
Second-Order Differential Equations with Several

Delays

Faten Aldosari

Department of Mathematics, College of Science, Shaqra University, P.O. Box 15572,
Shaqra 11961, Saudi Arabia

Abstract. The primary objective of this work is to establish new criteria to guarantee the os-
cillation of solutions for second-order differential equations with p-Laplace type operator. New
prerequisites are presented in order to analyze the oscillatory features of the analyzed equations.
To support these findings, we employ a range of analysis tools, establishing new conditions to
address specific the problems that have hindered previous researches. More specifically, we obtain
results that both build upon and extend those discovered in earlier studies by applying the Ric-
cati transformation and the principles of comparison. Several examples are given to illustrate the
significance of our results.

2020 Mathematics Subject Classifications: 34C10, 34K11

Key Words and Phrases: Differential equations, Oscillation theorems, Second-order, Delay
terms

1. Introduction

In this article, we examine the p-Laplace type operator oscillation problem for second-
order differential equations

(
b (t) |ϖ′(t)|p−2ϖ′(t)

)′
+

n∑
i=1

qi
∣∣κp−2 (πi (t))

∣∣κ (πi (t)) = 0, t ≥ t0, (1)

where p > 1, ϖ (t) := κ (t) + y (t)κ (ζ (t)), b ∈ C ([t,∞), (0,∞)) , y ∈ C ([t,∞), [0,∞)) ,
qi ∈ C ([t,∞), [0,∞)) , ζ, πi ∈ C ([t,∞),R) , ζ (t) ≤ t, πi (t) ≤ t, limt→∞ ζ (t) = limt→∞ πi (t) =
∞, qi (t) does not vanish identically, i = 1, 2, ..., n, y (t) < 1 and∫ ∞

t0

b−1/(p−1) (s) ds = ∞. (2)

DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5714

Email address: faldosari@su.edu.sa (F. Aldosari)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 2 of 16

Definition 1. By a solution of (1), we mean a function κ ∈ C1 ([t,∞),R) , tκ ≥ t0, which

has the property b (t) (ϖ′ (t))(p−1) ∈ C1 ([t0,∞),R) , p > 1, and satisfies (1) on [tκ,∞). We
consider only those solutions κ of (1) which satisfy sup{|κ (t)| : t ≥ tκ} > 0, for all t > tκ.

Definition 2. κ is referred to as oscillatory if it is neither finally positive nor eventually
negative. Otherwise it is referred to as non-oscillatory. If every solution to the equation
oscillates, then the equation is said to be oscillatory.

In differential equations (DEs), which are the most successful models for studying
natural events, each dependant variable represents a quantity in the modeled phenomenon.
DEs have helped us understand a variety of complex events in our daily lives and are crucial
to many technical applications. These days, they are essential tools in applied sciences
and technology, used to study media, conversations, phone signals, and online purchasing
data. In a more traditional sense, astronomers used them to describe the motion of stars
and the orbits of planets. They also serve a variety of purposes in biology and medicine,
see [11].
Neutral differential equations (NDEs), a specific subset of functional differential equations,
have derivatives that depend on both the function’s derivatives from earlier periods and its
current values. This unique characteristic distinguishes NDEs from traditional differential
equations and establishes a distinct analytical framework. The relationship between NDEs
and FDEs is essential because they often arise in systems where past values and rates of
change influence future states. NDEs’ significance is particularly evident in fields like
control theory and signal processing since they represent systems with memory effects.
For instance, in mechanical systems with inertia, acceleration may be affected by both
velocity and current position. This association emphasizes the importance of NDEs in
accurately modeling and simulating dynamic systems. Moreover, the research of NDEs
complements that of DDEs because understanding one usually provides significant insights
into the other [2, 7, 9, 14]. These equations find use in a wide range of fields, including
problems requiring masses connected to a flexible, shaky rod [10, 13, 16].
The ordinary differential equation (ODE) is a crucial tool for understanding and modeling
a wide range of technical and natural systems. The complexity and diversity of real-world
events often necessitate the use of sophisticated arguments to obtain more comprehensive
and correct solutions, despite the widespread use of ODEs (see [15, 17]). The behavior of
many nonlinear systems is not well described by conventional linear differential equations,
which emphasizes the importance of including complex arguments into ODEs. Advanced
nonlinear dynamics may make these systems more realistically represented, improving
insights and predictions. Furthermore, perturbation methods can be used to analyze
systems that are subject to small perturbations, providing a means of understanding how
complex systems react in different contexts. Furthermore, stability analysis is essential for
determining the long-term behavior of ODE solutions, which is important in fields such
as control theory and epidemiology (see [12]).

In recent years, there has been a substantial advancement in the study of oscillation
conditions for higher-order equations, particularly second-order differential equations with



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 3 of 16

delays [1, 3]. This explains why the qualitative aspects of these equations are so fascinating.
Oscillation phenomena are present in many real-world models; for instance, mathematical
biology models that use cross-diffusion terms to construct oscillation and/or delay actions
are discussed in the publications [5, 6]. This methodology includes a detailed development
of the oscillation theory of this type of equation.

Alqahtani et al. [22] established asymptotic behavior for equations with several delays

(
b (t)

(
κ′ (t)

)α)′
+

n∑
i=1

qi (t)κα (πi (t)) = 0, α > 0.

In [8], the authors was able to provide some oscillation conditions for

(b(t)|κ′(t)|α−1κ′(t))′ + q(t)|κ[π(t)]|α−1κ[π(t)] = 0. (3)

Later contributions include studies by Sahiner and Wang [18, 20], Zhao and Meng [23] and
Xu and Weng [21] that focus on oscillation criteria and asymptotic behavior. Baculikova
and Dzurina’s recent study [4] is significant because it provides crucial new details on
oscillation conditions for second-order delay differential equations of type(

b (t)
(
(κ (t) + y (t)κ (ζ (t)))′

)α)′
+ q (t)κα (π (t)) = 0. (4)

Lastly, recent publications by Al-Jaser et al. [3] give additional helpful criteria for assessing
the asymptotic and oscillatory behavior of solutions. The first-order differential equations
and the second-order (4) differential equations are compared using established comparison
theorems.

In this research, we use comparison principles and Riccati transformations to obtain
the different conditions for oscillation of (1). Examples are provided to illustrate the main
findings. The format of this document is as follows. In the first section (Introduction), we
present the studied equation and the general conditions needed to reach the main results
of the paper. We also provide an overview of pertinent topics and the goal of this study.
The oscillation results discussed in the ”Oscillation Results” part will be derived using
a few relationships and findings that we present in Section 2. In Section 3, we provide
several examples to illustrate the significance of the obtained results. We summarize the
main conclusions of the paper in Section 4 and draw attention to an open question that
may be of interest to researchers in the considered field.

2. Oscillation Results

We start by listing a number of auxiliary lemmas and conditions that we will employ
in order to accomplish the primary findings.
For ease of use, we set the following notation:

Bt0 (t) : =

∫ t

t0

b−1/(p−1) (t) dt, p > 1,



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 4 of 16

B̃t0(t) : = Bt0 (t) +
1

(p− 1)

∫ t

t0

Bt1 (t)B
p−1
t0

(πi(t))

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) dt,

and

B̂ (t) := exp

(
− (p− 1)

∫ t

πi(t)

dt

B̃t0(t)b
1/(p−1) (t)

)
.

Lemma 1. [4] If κ be an eventually positive solution of (1), then

ϖ (t) > 0, ϖ′ (t) > 0,
(
b (t)

(
ϖ′ (t)

)(p−1)
)′

≤ 0, (5)

for t ≥ t1.

Lemma 2. [19] Let G,W > 0 be constants and

max
κ∈b

f = f (κ∗) = αα (α+ 1)−(α+1) G
α+1

Wα
, α ≥ 1, (6)

where κ∗ = (αG/ ((α+ 1)W ))α and f (κ) = Gκ −Wκ(α+1)/α.

Lemma 3. Let κ be an eventually positive solution of (1). Then

(
b (t)

(
ϖ′ (t)

)(p−1)
)′

≤ −
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1)ϖ(p−1) (πi (t)) , (7)

and
ϖ (t) ≥ B̃t1 (t) b

1/(p−1) (t)ϖ′ (t) , (8)

also, (
b (t)

(
ϖ′ (t)

)(p−1)
)′

≤ −
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t)ϖ(p−1) (t) . (9)

Proof. Let κ be an eventually positive solution of (1). (5) holds according to Lemma
1. Therefore, using the definition ofϖ (t), we get

κ (t) = ϖ (t)− y (t)κ (ζ (t))

≥ ϖ (t)− y (t)ϖ (ζ (t))

≥ ϖ (t) (1− y (t)) .

This suggests that (1)

(
b (t)

(
ϖ′ (t)

)(p−1)
)′

≤ −
n∑

i=1

qi (t)ϖ
(p−1) (πi (t)) (1− y (πi (t)))

(p−1) .



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 5 of 16

Since ϖ′ (t) > 0 and ∂
∂sπi (t) > 0, we obtain ϖ (πi (t)) > ϖ (πi (t)) and so

(
b (t)

(
ϖ′ (t)

)(p−1)
)′

≤ −
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1)ϖ(p−1) (πi (t)) .

Using basic computation and the chain rule, it is evident that

Bt1 (t)
(
b (t)

(
ϖ′ (t)

)(p−1)
)′

= (p− 1)
(
b1/(p−1) (t)ϖ′ (t)

)(p−1)−1
Bt1 (t)

(
b1/(p−1) (t)ϖ′ (t)

)′
= − (p− 1)

(
b1/(p−1) (t)ϖ′ (t)

)(p−1)−1 d

dt

(
ϖ (t)−Bt1 (t) b

1/(p−1) (t)ϖ′ (t)
)
.(10)

Combining (7) and (10), we obtain

d

dt

(
ϖ (t)−Bt1 (t) b

1/(p−1) (t)ϖ′ (t)
)

≥
(

1

(p− 1)
Bt1 (t)

(
b1/(p−1) (t)ϖ′ (t)

)2−p
)

(
n∑

i=1

qi (t) (1− y (πi (t)))
p−1ϖp−1 (πi (t))

)
.

Integrating this inequality from t1 to t, we have

ϖ (t) ≥ Bt1 (t) b
1/(p−1) (t)ϖ′ (t)

+
1

(p− 1)

∫ t

t1

Bt1 (t)

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1)

(
b1/(p−1) (t)ϖ′ (t)

)2−p
ϖ(p−1) (πi(t)) dt.(11)

From the monotonicity of b1/(p−1) (t)ϖ′ (t), we have

ϖ (t) = ϖ (t1) +

∫ t

t1

1

b1/(p−1) (t)

(
b1/(p−1) (t)ϖ′ (t)

)
dt ≥ Bt1 (t) b

1/(p−1) (t)ϖ′ (t) .

So, by
(
b1/(p−1) (t)ϖ′ (t)

)′ ≤ 0, (11) becomes

ϖ (t) ≥ Bt1 (t) b
1/(p−1) (t)ϖ′ (t)

+
1

(p− 1)

∫ t

t1

(Bt1 (t)
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1)

(
b1/(p−1) (t)ϖ′ (t)

)1−(p−1)
B

(p−1)
t1

(πi(t))[
b (πi(t))

(
ϖ′ (πi(t))

)(p−1)
]
)dt

≥ Bt1 (t) b
1/(p−1) (t)ϖ′ (t)

+
1

(p− 1)

∫ t

t1

(
b1/(p−1) (t)ϖ′ (t)

)2−p
Bt1 (t)B

(p−1)
t1

πi(t)
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1)

[
b1/(p−1) (t)ϖ′ (t)

](p−1)
dt

≥ b1/(p−1) (t)ϖ′ (t)

[
Bt1 (t) +

1

(p− 1)

∫ t

t1

Bt1 (t)B
p−1
t1

(πi(t))
n∑

i=1

qi (t) (1− y (πi (t)))
p−1 dt

]
.



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 6 of 16

≥ B̃t1(t)b
1/(p−1) (t)ϖ′ (t) ,

or
ϖ′ (t)

ϖ(t)
≤ 1

B̃t1(t)b
1/(p−1) (t)

.

Integrating from πi (t) to t, we find that

ϖ (πi (t))

ϖ (t)
≥ exp

(
−
∫ t

πi(t)

dt

B̃t1(t)b
1/(p−1) (t)

)
,

which with (7), gives(
b (t) (ϖ′ (t))(p−1)

)′
ϖ(p−1) (t)

≤ −
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1)

(
ϖ (πi (t))

ϖ (t)

)(p−1)

≤ −
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) .

The proof is complete.

Lemma 4. Let (1) have a positive solution. If

ξ (t) = x (t) b (t)

(
ϖ′ (t)

ϖ (t)

)p−1

> 0, (12)

then

ξ′ (t) ≤
x′+ (t)

x (t)
ξ(t)− x(t)

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 B̂ (t)− (p− 1)

(x (t) b (t))1/(p−1)
ξp/(p−1) (t) .

(13)

Proof. Let κ be a positive solution of equation (1). From Lemma 3, we have (9) holds.
Thus, when we differentiate ξ (t) we get

ξ′ (t) =
x′ (t)

x (t)
ξ (t) + x (t)

(b (t)ϖ′ (t))′

ϖ(p−1) (t)
− (p− 1)x (t) b (t)

(
ϖ′ (t)

ϖ(t)

)p

.

From (9) and (12), we see that

ξ′ (t) ≤
x′+ (t)

x (t)
ξ(t)−1x(t)

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t)− (p− 1)

(x (t) b (t))1/(p−1)
ξp/(p−1) (t) .

The proof is complete.



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 7 of 16

Theorem 1. If the equation

ω′ (t) + B̃
(p−1)
t1

(πi (t))

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) ω (πi (t)) = 0, (14)

is oscillatory, then (1) is oscillatory.

Proof. Let κ (t) > 0, that is κ (ζ (t)) > 0 and κ (πi (t)) > 0. From Lemma 3, we have

(7) and (8) hold. Using (7) and (8), we find ω (t) = b (t) (ϖ′ (t))(p−1) is a positive solution
of

ω′ (t) + B̃
(p−1)
t1

(πi (t))
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1) ω (πi (t)) ≤ 0.

By [12, Theorem 1], then also, the solution of the associated equation (14) is a positive,
and this a contradiction. The proof is complete.

Corollary 1. Let

lim sup
t→∞

∫ t

πi(t)
B̃

(p−1)
t1

(πi (t))

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) dt > 1,

∂

∂t
πi (t) ≥ 0, (15)

or

lim inf
t→∞

∫ t

πi(t)
B̃

(p−1)
t1

(πi (t))

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 dt >

1

e
, (16)

then all solutions of (1) is oscillatory.

Proof. As may be shown from [10, Theorem 2.1.1], (15) or (16) guarantee oscillation
of (14).

Lemma 5. Suppose πi is strictly growing in relation to t and

lim inf
t→∞

∫ t

πi(t)
B̃

(p−1)
t1

(πi (t))

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 dt ≥ δ, (17)

for some δ > 0, and (1) has an eventually positive solution κ. Then,

H (πi (t))

H (t)
≥ zn (δ) , n ≥ 0, (18)

where H (t) := b (t) (ϖ′ (t))(p−1), and

z0 (t) := 1 and zn (t) := exp (ρzn−1 (t)) . (19)

Proof. Let κ (t) > 0, κ (ζ (t)) > 0 and κ (πi (t)) > 0 for t ≥ t1. We conclude that ω is
a positive solution of (14) by following the same procedure as in the proof of Theorem 1.
We can demonstrate that (18) holds in a manner akin to that used in the proof of Lemma
1 in [19].



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 8 of 16

Lemma 6. Let (1) have a positive solution. If

σ(t) := ς(t)b(t)

(
ϖ′(t)

ϖ (πi (t))

)p−1

> 0, (20)

then

σ′ (t) ≤ −ς (t)
n∑

i=1

qi (t) (1− y (πi (t)))
p−1+

ς ′+ (t)

ς(t)
σ (t)−(p− 1) z

1/(p−1)
n (δ)π′i (t)

(ς (t) b (πi (t)))
1/(p−1)

σp/(p−1) (t) .

(21)

Proof. Let κ be a positive solution of equation (1). From Lemma 3, we obtain (7)
holds. By Lemma 5, we find

ϖ′ (πi (t))

ϖ′ (t)
≥
(
zn (δ) b (t)

b (πi (t))

)1/(p−1)

. (22)

Now, we differentiate σ (t) , we get

σ′(t) =
ς ′(t)

ς(t)
σ(t)+ς(t)

(b(t)(ϖ′(t))p−1)′

ϖ(p−1) (πi (t))
−(p− 1) ς(t)b(t)

(
ϖ′(t)

ϖ (πi (t))

)(p−1)(ϖ′ (πi (t))

ϖ (πi (t))

)
π′i (t) .

From (7), (20) and (22), we obtain

σ′ (t) ≤ −ς (t)
n∑

i=1

qi (t) (1− y (πi (t)))
p−1+

ς ′+ (t)

ς(t)
σ (t)−(p− 1) z

1/(p−1)
n (δ)π′i (t)

(ς (t) b (πi (t)))
1/(p−1)

σp/(p−1) (t) .

The proof is complete.

Theorem 2. Suppose πi is strictly growing in relation to t and (17) holds. If ς ∈
C1(I, (0,∞)) such that

lim
t→∞

sup

∫ t

t1

(
ς (t)

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 −

(
ς ′+ (t)

)p
b (πi (t))

ppzn (δ) ς(p−1) (t) (π′i(t))
p−1

)
= ∞, (23)

for some δ < 0 and n ≥ 0, where ς ′+(t) = max {0, ς ′(t)} and zn(δ) is defined as (19), then
all solutions of (1) is oscillatory.

Proof. Suppose κ (t) > 0, κ (ζ (t)) > 0 and κ (πi (t)) > 0. From Lemma 6, we have

(21) holds. Using Lemma 2 with W = (p− 1) z
1/(p−1)
n (δ) / (ς (t) b (πi (t)))

−1/(p−1) and
G = ς ′+ (t) /ς (t), (21) yield

σ′ (t) ≤ −ς (t)
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1) +

ς ′+ (t)p b (πi (t))

ppzn (δ) ςp−1 (t) (π′i (t))
p−1 .



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 9 of 16

Integrating this inequality from t1 to t, we find∫ t

t1

(
ς (t)

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) −

(
ς ′+ (t)

)p
b (πi (t))

ppzn (δ) ςp−1 (t) (π′i (t))
p−1

)
dt ≤ σ (t) .

A contradiction with condition (23) is then discovered. The proof is finished.

Theorem 3. If

lim
t→∞

sup

∫ t

t1

(
x (t)

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 B̂ (t)−

b (t)
(
x′+ (t)

)p
ppxp−1 (t)

)
dt = ∞, (24)

where x ∈ C1 (I, (0,∞)) and x′+ (t) = max {0, ψ′ (t)} , then (1) is oscillatory.

Proof. Let κ (t) > 0, that is κ (ζ (t)) and κ (πi (t)) are positive on [t0,∞). From Lemma
3, we have (7)-(9) hold. Next, we arrive at (13) using Lemma 2 with G = x′+ (t) /x (t) and

W = (p− 1) (x (t) b (t))−1/(p−1) (Lemma 4), the inequality(13) becomes

ξ′ (t) ≤ −x(t)
n∑

i=1

qi (t) (1− y (πi (t)))
p−1 B̂ (t) +

b (t)
(
x′+ (t)

)p
ppxp−1 (t)

.

Integrating this inequality from t1 to t, we have∫ t

t1

(
x (t)

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 B̂ (t)−

b (t)
(
x′+ (t)p

)
ppxp−1 (t)

)
dt ≤ ξ (t) .

This contradicts the condition (24). The proof is finished.

Now, we obtain some oscillation results for equation (1) using other methods.

Theorem 4. Let ∫ ∞

t0

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) dt = ∞, (25)

then, equation (1) is oscillatory.

Proof. Suppose κ (t) > 0, κ (ζ (t)) > 0 and κ (πi (t)) > 0, we can infer from Lemma 4
that (13) holds. if we set x (t) := 1, then (13) becomes

ξ′ (t) +

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) + (p− 1) / (b (t))1/(p−1) ξ

p
(p−1) (t) ≤ 0, (26)

or

ξ′ (t) +
n∑

i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) ≤ 0. (27)



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 10 of 16

Integrating (27) from t3 to t and using (25), we arrive at

ξ (t) ≤ ξ (t3)−
∫ t

t3

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) ds→ ∞ as t→ ∞.

This contradicts the conclusion that the evidence is complete because ξ (t) > 0.

Definition 3. Assume that the series of functions {ϑn (t)}∞n=0 is defined as

ϑn (t) =

∫ ∞

t
(p− 1) / (b (s))1/(p−1) ϑ

p
(p−1)

n−1 (s) ds+ ϑ0 (t) , t ≥ t0, n = 1, 2, 3, ..., (28)

and

ϑ0 (t) =

∫ ∞

t

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) dt, t ≥ t0,

where ϑn (t) ≤ ϑn+1 (t) , t ≥ t0.

Lemma 7. Let κ be a solution of equation (1) that becomes positive for suffciently large t.
Then ξ (t) ≥ ϑn (t) where limn→∞ ϑn (t) = ϑ (t) for t ≥ T ≥ t0 when ϑ (t) on [T,∞) and

ϑ (t) =

∫ ∞

t
(p− 1) / (b (s))1/(p−1) ϑ

p
(p−1)

(s) ds+ ϑ0 (t) , t ≥ T. (29)

Proof. Let κ be a solution of equation (1) that becomes positive for suffciently large
t. We get to (26) by using the same steps as in the proof of Theorem 4. The result of
integrating (26) from t to t′ is

ξ
(
t′
)
−ξ (t)+

∫ t′

t

n∑
i=1

qi (s) (1− y (πi (s)))
(p−1) B̂ (s) ds+

∫ t′

t
ξ

p
(p−1) (s) (p− 1) / (b (s))1/(p−1) ds ≤ 0.

This implies

ξ
(
t′
)
− ξ (t) +

∫ t′

t
ξ

p
(p−1) (s) (p− 1) / (b (s))1/(p−1) ds ≤ 0.

Then, we conclude that∫ ∞

t
ξ

p
(p−1) (s) (p− 1) / (b (s))1/(p−1) ds <∞ for t ≥ T, (30)

Otherwise, when t′ → ∞, ξ(t′) ≤ ξ(t) −
∫ t′

t ξ
p

(p−1) (s) (p− 1) / (b (s))1/(p−1) ds → −∞,
which contradicts ξ(t) > 0. Given that ξ(t) > 0 and ξ′(t) > 0, (26) indicates that

ξ (t) ≥
∫ ∞

t

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) dt+

∫ ∞

t
ξ

p
(p−1) (s) (p− 1) / (b (s))1/(p−1) ds



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 11 of 16

= ϑ0 (t) +

∫ ∞

t
ξ

p
(p−1) (s) (p− 1) / (b (s))1/(p−1) ds, (31)

or

ξ (t) ≥
∫ ∞

t

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) dt := ϑ0 (t) .

Consequently, ξ(t) ≥ ϑn(t), where n = 1, 2, 3, .... We obtain that ϑn → ϑ as n → ∞
since {ϑn(t)}∞n=0 is growing and bounded above. The monotone convergence theorem of
Lebesgue shows that when n→ ∞, (28) becomes (29).

Theorem 5. If

lim inf
t→∞

1

ϑ0 (t)

∫ ∞

t
ϑ

p
(p−1)

0 (s) (p− 1) / (b (s))1/(p−1) ds >
p− 1

p
p

p−1

, (32)

then all solutions (1) are oscillatory.

Proof. Assume that κ (t) > 0, meaning that both κ (ζ (t)) and κ (πi (t)) are positive.
Following the same steps as in the Lemma 7 proof, we get (31). Using (31), we discover

ξ(t)

ϑ0(t)
≥ 1 +

1

ϑ0(t)

∫ ∞

t
ϑ

p−
p−1

0 (s) (p− 1) / (b (s))1/(p−1)

(
ξ(s)

ϑ0(s)

) p
p−1

ds. (33)

If we consider µ = inft≥T (ξ(t)/ϑ0(t)), then µ ≥ 1 of course. We can observe using(32)
and (33) that

µ ≥ p

(
µ

p

) p
p−1

,

or
p− 1

p

(
µ

p

) p
p−1

+
1

p
≤ µ

p
.

It defies the predicted value of µ and p, hence, the proof is finished.

Theorem 6. Let

lim sup
t→∞

ϑn(t)

(∫ t

t0

b
− 1

(p−1) (s)ds

)(p−1)

> 1, (34)

then every solutions of (1) are oscillatory.

Proof. Assume that κ (t) > 0, meaning that both κ (ζ (t)) and κ (πi (t)) are positive.
From (12), we obtain

1

ξ (t)
=

1

b (t)

(
ϖ (t)

ϖ′ (t)

)(p−1)

=
1

b (t)

(
ϖ (T ) +

∫ t
T b

−1/(p−1) (s) b1/(p−1) (s)ϖ′ (s) ds

ϖ′ (t)

)(p−1)



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 12 of 16

≥ 1

b (t)

(
b1/(p−1) (t)ϖ′ (t)

∫ t
T b

−1/(p−1) (s) ds

ϖ′ (t)

)(p−1)

=

(∫ t

T
b−1/(p−1) (s) ds

)(p−1)

, (35)

for t ≥ T . So, from (35) we find

ξ (t)

(∫ t

t0

b−1/(p−1) (s) ds

)(p−1)

≤

(∫ t
t0
b−1/(p−1) (s) ds∫ t

T b
−1/(p−1) (s) ds

)(p−1)

,

and so

lim sup
t→∞

ξ (t)

(∫ t

t0

b
− 1

(p−1) (s)ds

)(p−1)

≤ 1,

which contradicts (34). Hence, the proof is finished.

Corollary 2. If∫ ∞

t0

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) exp

(∫ t

t0

ϑ
1

(p−1)

n (s) (p− 1) / (b (s))1/(p−1) ds

)
dt = ∞,

(36)
or∫ ∞

t0

(p− 1) / (b (t))1/(p−1) ϑ
1

(p−1)

n (t)ϑ0(t) exp

(∫ t

t0

(p− 1) / (b (s))1/(p−1) ϑ
1

(p−1)

n (s)ds

)
dt = ∞,

(37)
then all solutions (1) are oscillatory.

Proof. Suppose that κ (t) > 0, meaning that both κ (ζ (t)) and κ (πi (t)) are positive
on [t0,∞). (29) holds according to Lemma 7. (29) gives us

ϑ′ (t) = − (p− 1) / (b (t))1/(p−1) ϑ
p−
p−1

(t)−
n∑

i=1

qi (t) (1− y (πi (t)))
p−1 B̂ (t)

≤ − (p− 1) / (b (t))1/(p−1) ϑ
1

p−1

n (t)ϑ (t)−
n∑

i=1

qi (t) (1− y (πi (t)))
p−1 B̂ (t) .(38)

Hence,∫ t

T

n∑
i=1

qi (s) (1− y (πi (s)))
p−1 B̂ (s) exp

(∫ s

T
ϑ

1
(p−1)

n (t) (p− 1) / (b (t))1/(p−1) dt

)
ds ≤ ϑ (T ) <∞,

which contradicts (36).

Next, let M (t) =
∫∞
t (p− 1) / (b (s))1/(p−1) ϑ

p
p−1

(s) ds. Then, we obtain

M ′ (t) = − (p− 1) / (b (t))1/(p−1) ϑ
p

p−1
(t)



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 13 of 16

≤ − (p− 1) / (b (t))1/(p−1) ϑ
1

p−1

n (t)ϑ (t)

= − (p− 1) / (b (t))1/(p−1) ϑ
1

p−1

n (t) (M (t) + ϑ0 (t)) .

Consequently, we discover∫ ∞

T
(p− 1) / (b (t))1/(p−1) ϑ

1
p−1

n (t)ϑ0(t) exp

(∫ t

T
(p− 1) / (b (s))1/(p−1) ϑ

1
p−1

n (s)ds

)
dt <∞.

This runs counter to (37). The proof is finished.

3. Examples and Discussion

Example 1. Let the equation((
(κ (t) + y0κ (ζ0t))

′)(p−1)
)′

+
q0t

p−1 − ptp

t2p−1
κ(p−1) (εt) +

p

tp−1
κ(p−1) (εt) = 0, (39)

Let b (t) = 1, y (t) = y0, ζ (t) = ζ0t,
∑n

i=1 qi (t) = q0tp−1−ptp

t2p−1 + p
tp−1 and πi (t) = εt,where

ε, ζ0 ∈ (0, 1) .
It is easy to verify that

n∑
i=1

qi (t) (1− y (πi (t)))
p−1 =

q0
tp

(1− ε) [1− y0]
p−1 ,

Bt0 (t) =t and B̃t0(t) =Mt,

where
M := 1 + ε(p−1) q0

(p− 1)
(1− ε) [1− y0]

(p−1) .

By Corollary 1, we find (39) is oscillatory if(
M (p−1)ε(p−1)q0 (1− ε) [1− y0]

(p−1)
)
ln

1

ε
>

1

e
,

or

(p− 1) (M − 1)M (p−1) ln
1

ε
>

1

e
. (40)

Also, we find that

B̂t1 (t) = ε1/M ,

n∑
i=1

qi (t) (1− y (πi (t)))
(p−1) B̂ (t) =

q0 (1− y0)
(p−1) (1− ε)

tp
ε(p−1)/M ,

and∫ ∞

t

n∑
i=1

qi (s) (1− y (πi (s)))
p−1 B̂ (s) ds = q0 (1− y0)

p−1 (1− ε) ε(p−1)/M (p− 1)−1 t−p.



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 14 of 16

From Theorem 5, (39) is oscillatory if(
q0 (1− y0)

p−1 (1− ε)

p− 1
ε(p−1)/M

)1/(p−1)

> (p− 1) p−p/(p−1).

Example 2. Consider the differential equation(
κ (t) +

1

2
κ (ζ0t)

)′′
+
q0t− 2

t3
κ (1/3t) +

2

t3
κ (1/3t) = 0, (41)

where ζ0 ∈ (0, 1). Let b (t) = 1, y (t) = 1
2 , ζ (t) = ζ0t, qi (t) =

q0t−2
t3

+ 2
t3

and πi (t) = 1/3t.
From Theorem 5, we find the equation (41) is oscillatory if

q0
6

(
1 +

1

6
q0

)
ln 3 >

1

e
, (42)

thus, he equation (41) is oscillatory if q0 > 1.588.

Example 3. Let equation(
κ (t) +

1

2
κ
(
t

3

))′′
+
q0
t2
κ
(
t

2

)
= 0, q0 > 0. (43)

Let b (t) = 1, y (t) = 1
2 , ζ (t) =

t
3 , qi (t) =

q0
t2

and πi (t) =
t
2 .

It is easy to verify that
Bt0 (t) = t,

and

B̃t0 = t+
q0
4

∫ t

t0

dκ = t
(
1 +

qi0
4

)
.

Using Corollary 1, if

q0 +
q20
4
>

4

ln 2e
,

then (43) is oscillatory.

4. Conclusion

The oscillatory properties of neutral second-order differential equations with p-Laplace
type operator is thoroughly examined in this work. The analytical process is greatly sim-
plified by this modification. Under certain limits, we have defined some conditions that
effectively exclude the existence of positive solutions. Building on these discoveries, we
created new standards that ensure all solutions to the examined equations oscillate. This
contribution offers a strong basis for upcoming investigations and is essential for develop-
ing the theoretical framework of neutral differential equations. Furthermore, we included
illustrated instances that show the theoretical significance and practical implementation



F. Aldosari / Eur. J. Pure Appl. Math, 18 (1) (2025), 5714 15 of 16

of our criteria. These illustrations demonstrate how well our method works to solve chal-
lenging neutral differential equation situations. The study’s findings broaden the field’s
current theoretical frameworks and create new research opportunities. We suggest that
future research investigate the use of our techniques for higher-order equations, especially
odd-order ones.b (t)

(κ (t) +
n∑

i=1

yi (t)κ (ζi (t))

)(n−1)
(p−1)


′

+
n∑

i=1

qi (t)κ(p−1) (πi (t)) = 0.

Acknowledgements

The author would like to thank the deanship of scientific research at Shagra University
for supporting this work.

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