Fourth-order differential equations with neutral delay: Investigation of monotonic and oscillatory features

1.   Introduction

Differential equations (DEs) are a powerful tool that can be used to model and understand a wide variety of systems. It plays a crucial role in solving real-world problems in many fields; see ^[1,2]. During the 20th century, the rapid progress of science resulted in applications across biology, population studies, chemistry, medicine, social sciences, genetic engineering, economics, and more. Many of the phenomena that appear in these fields are modeled using delay differential equations (DDEs). This led to many disciplines being elevated, and significant discoveries were made with this type of mathematical modeling.

The DEs that have the delayed argument in the highest derivative of the state variable are known as neutral differential equations (NDEs). NDEs have an extremely diverse historical background. In reality, they have a wide range of uses in natural science, as in the process of chemical reactions. Time-delayed transitions may be seen in chemical reaction kinetics, especially in complex processes. These kinetics are described by NDDEs; see ^[3]. The presence of the delay term in NDDs, which expresses the need for historical information, expands the solution space, and complicates numerical methods. This led to studying the qualitative behavior of these equations because finding closed solutions is often impossible due to their complexity. In recent years, there has been significant research focused on the asymptotic behavior of solutions to DEs; see ^[4,5,6]. By examining the asymptotic properties, researchers can forecast the future behavior of systems modeled by DEs from simple physical processes to complicated biological and economic systems. This part of the study supports the practical use of theoretical models in a variety of scientific and engineering domains in addition to aiding in their refinement; see ^[7]. In recent years, one of the most significant branches of qualitative theory has been oscillation theory; it was introduced in a pioneering paper of Fite; see ^[8,9,10]. This theory answers a lot of questions regarding the oscillatory behavior and asymptotic properties of DE solutions.

Finding adequate criteria to guarantee that all DE solutions oscillate while eliminating positive solutions is one of the main objectives of oscillation theory; see ^[11,12,13]. One of the main characteristics of oscillation theory is the variety of mathematical and analytical approaches it uses; see ^[14,15,16]. Over the past decade, there has been significant progress in the study of the oscillatory properties of DEs; see ^[17,18,19]. This interest stems from the fact that comprehending mathematical models and the phenomena they describe is made easier by examining the oscillatory and asymptotic behavior of these models; also, see ^[20,21,22]. Moreover, oscillation theory is abundant with fascinating theoretical problems that require the tools of mathematical analysis. In recent decades, oscillation theory has attracted the attention of many researchers, resulting in numerous books and hundreds of studies on several kinds of functional DEs; see ^[23,24,25]. Due to the critical roles of NDDE in various fields, such as civil engineering and application-oriented research that can support research with the potential to develop the ship-building, airplane, and rocket industries, the study of the oscillatory properties of these equations has advanced significantly. This makes them extremely important practically in addition to their abundance of interesting analytical problems. For more recent results regarding the oscillatory properties of NDDE solutions; see ^[26,27].

1.1.   Fourth-order NDE

In this paper, we examine the oscillatory behavior of solutions to the neutral equation

where $\Omega \left(u\right) = x\left(u\right)$ $+$ $p\left(u\right) x\left(\zeta \left(u\right) \right)$. Here, we accomplish our important results by considering the next conditions:

(H$_{1}$) $p, \zeta \in C^{4}\left(\left[ u_{0}, \infty \right) \right)$, $0 < p\left(u\right) < p_{0} < \infty,$ $\zeta (u)\leq u,$ $\lim_{u\rightarrow \infty }\zeta (u) = \infty,$ and $\zeta (u)$ invertible;

(H$_{2}$) $\theta,$ $q\in C\left(\left[ u_{0}, \infty \right) \right)$, $q\left(u\right) > 0,$ $\theta (u)\leq u$, $\lim_{u\rightarrow \infty }\theta (u) = \infty$;

(H$_{3}$) $r\left(u\right) \in C^{1}\left(\left[ u_{0}, \infty \right) \right)$, $r\left(u\right) > 0$ and satisfy

By a solution of (1.1), we mean a function $x\in C^{3}\left(\left[ u_{x}, \infty \right) \right)$ for $u_{x}\geq u_{0}$, which has the properties $r\left(\Omega ^{\prime \prime \prime }\right) \in C^{1}\left(\left[ u_{x}, \infty \right) \right)$, and satisfies (1.1) on $\left[ u_{x}, \infty \right)$. We only take into account the solutions $x$ of (1.1) that satisfy $Sup\left \{ \left \vert x\left(u\right) \right \vert :u\geq u_{\ast }\right \} > 0$ for all $u_{\ast }$ $\geq$ $u_{x}$.

Definition 1.1. ^[22] A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on $[u_{0}, \infty)$; otherwise, it is called non-oscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

1.2.   Literature review

The majority of studies have focused on establishing a condition that assures excluding increasing positive solutions using a variety of techniques. The oscillations of higher-order NDE have been investigated by many researchers, and many techniques have been presented for establishing oscillatory criteria for these equations. A lot of research has been conducted regarding the canonical condition; see ^[28,29,30]. We will now outline some of the results from previous papers that have contributed to an important part in advancing research on fourth-order NDEs, particularly Moaaz et al. in ^[31] established criteria for oscillation of solutions of NDDE

by applying two Riccati substitutions in each case of the derivatives of the corresponding function $\Omega$. This criteria guarantees that all solutions oscillate under the canonical condition, where $\beta \geq \alpha$ and $0\leq p(u) < p_{0} < \infty.$

In ^[32], Bazighifan et al. obtained the Philos type the oscillation criteria to ensure oscillation of solutions of the equation

and by employing the well-known Riccati transformation, they established an asymptotic criterion that enhances and supplements previous results, where $0\leq p(u) < p_{0} < 1.$

To comprehend the asymptotic and oscillatory behavior of solutions to NDEs, it is essential to understand the relationship between the solution $x$ and its associated function $\Omega$. Through this relationship, many researchers discovered several criteria that simplified and enhanced their earlier results. Here, we will present some of the relationships identified from previous research. For $p\left(u\right) = p_{0}$, the conventional relationship:

is generally employed for second-order equations under the condition (1.2), while in ^[33,34] they applied the following relationship

in the non-canonical case. Moaaz et al. ^[35] in the canonical condition, they obtained some oscillation criteria for the next form of the equation

by optimizing the relationship (1.4), where $\alpha$, $\beta \in \mathbb{Q} _{odd}^{+}$. They provided for $p_{0} > 1$ the following relationship

for $m$ is even, while for $p_{0} < 1$ they provided the relationship

for $m$ is odd. In ^[36], Hassan et al. enhanced the relationship (1.5) by the next one

for $m$ an odd integer, when examined, the oscillatory properties of the equation

where $\alpha \in \mathbb{Q} _{odd}^{+}$. In ^[37], Moaaz et al. improved the relationship (1.4) to the following

and they obtained a criterion to ensure that there are no Kneser solutions of the equation of third-order NDDE

by comparing (1.7) with a first-order DDE (comparison technique).

Moaaz and Alnafisah ^[38] examined the oscillatory behavior of solutions to DE

and derived inequalities and relationships, by enhancing the relationship (1.4) considering the two cases $p_{0} > 1$ and $p_{0} < 1$ without restrictions on the delay functions. Then, using an improved approach, they obtained new monotonic properties for the positive solutions.

Recently, Bohner et al. ^[39], by considering two cases $\zeta \leq u$ and $\zeta \geq u$, studied the NDDE

and improved the relationship (1.5) by getting the next one

where

where $k\in \mathbb{N}$.

In addition, recently, for higher-order some research improved the relationship of (1.4). Among these research, Alnafisah et al. ^[40] presented the following relationship:

by investigating the asymptotic and oscillatory behaviors of solutions to the NDEs

where $n\geq 4$ and $\alpha$ is the ratio of two positive odd integers.

The key to our contribution to this work, We categorize the positive solutions to the studied equation based on the signs of its derivatives. After that, we obtain new monotonic properties in certain cases of positive solutions. Based on these properties, we discover the relationship between the solution and its corresponding function $\Omega$ of our Eq (1.1) in the two cases $p_{0} > 1$ and $p_{0} < 1$. Additionally, we use these new relationships to exclude positive solutions by obtaining some oscillation criteria. The results are illustrated by an example. These results obtained extend and improve upon previous findings in the literature, providing a more comprehensive framework for analyzing these equations.

The paper is organized as follows: In Section 2, we present the fundamental notation and definitions that will be used in our proofs. In Section 3, we present a series of lemmas that enhance the monotonicity properties of nonoscillatory solutions. In Section 4, we establish oscillation criteria for (1.1) as our main result. Lastly, we illustrate our results with an example. In conclusion, briefly discuss what we have done in this research and the results we have obtained.

2.   Preliminaries

In this section, we will display the following constants and functions that are used in this paper. The class of all positive non-oscillatory solutions to (1.1) is denoted by the symbol $S^{+}$.

Notation 2.1. For any integer $k\geq 0$. In order to present the results, we will need the following notation:

for $i = 1, 2, ...$.

Lemma 2.1. ^[41] Let $\digamma \in C^{\kappa }\left(\left[ u_{0}, \infty \right), \mathbb{R} ^{+}\right)$ and $\digamma ^{\left(\kappa \right) }$ be of constant sign, eventually. Then there are a $u_{x}\geq u_{0}$ and a $j\in \mathbb{Z}$, $0$ $\leq j\leq \kappa$, with $\kappa +j$ even for $\digamma ^{\left(\kappa \right) }\left(u\right) \geq 0$, or $\kappa +j$ odd for $\digamma ^{\left(\kappa \right) }\left(u\right) \leq 0$ such that

And $j\leq \kappa -1$ implies that $\left(-1\right) ^{j+l}\digamma ^{\left(l\right) }\left(u\right) > 0$ for$\ u\geq u_{x}$, $l = j, j+1, .......\kappa -1$.

Lemma 2.2. ^[42] Assume $\digamma \ $is stated in Lemma 2.1. If $\digamma ^{\left(\kappa -1\right) }\left(u\right) \digamma ^{\left(\kappa \right) }\left(u\right) \leq 0$, eventually, and $\lim_{u\rightarrow \infty }\digamma \left(u\right) \neq 0$, then, there exists $u_{k}\in \left[ u_{1}, \infty \right)$ for every $\epsilon \in (0, 1)$, such that

Lemma 2.3. ^[43] If the Eq (1.1) has a solution $x$ that is eventually positive, then

for $k\geq 0.$

Lemma 2.4. ^[41] If $h\in C^{\kappa }\left(\left[ u_{0}, \infty \right), \mathbb{R} ^{+}\right)$, $h^{\left(i\right) }\left(u\right) > 0$ for $i = 0, 1, 2, ..., \kappa,$ and $h^{\left(\kappa +1\right) }\left(u\right) \leq 0,$ then eventually,

3.   Results and discussion

We will introduce the next lemma that describes the behavior of positive solutions.

Lemma 3.1. Assume that $x\in S^{+}$. Then, eventually, we have two cases for $\Omega$ eventually:

Case (1)

Case (2)

Proof. Suppose that $x$ is a positive solution of (1.1); we obtain $\Omega ^{\left(4\right) }\left(u\right) \leq 0$ from (1.1). From the Lemma 2.1 Cases (1) and (2), and their derivatives, are obtained. □

Notation 3.1. We will refer to the symbol $\aleph _{1}$ as the class of all eventually positive solutions of Eq (1.1) whose corresponding function satisfies Case $\left(1\right)$ and $\aleph _{2}$ as the class of all eventually positive solutions of Eq (1.1) whose corresponding function satisfies Case $\left(2\right)$. Moreover, we will use the following notation throughout the proof of our lemmas.

Notation 3.2. For any positive integer $k$, we define the functions

and

and

For every $k\in$ $\mathbb{N} _{0},$ we assume that

It is clear that $\beta _{k}^{\ast }$ is positive. Our reasoning will usually rely on the obvious truth that a $u_{1}\geq u_{0}$ is large enough such that for fixed but arbitrary $\beta _{k}\in \left(0, \beta _{k}^{\ast }\right)$, we have

on $\left[ u_{1}, \infty \right)$.

3.1.   Asymptotic and monotonic properties

This section contains several lemmas regarding the asymptotic properties of solutions that are part of the classes $\aleph _{1}$ and $\aleph _{2}$.

Lemma 3.2. Suppose that $x\in S^{+}$. If $\Omega ^{\prime \prime }\left(u\right) > 0$ eventually, then,

(I) $\Omega \left(u\right) \geq \frac{1}{3}u\Omega ^{\prime }\left(u\right).$

However, if $\Omega ^{\prime \prime }\left(u\right) < 0$, eventually, then

(II)$\ \Omega \left(u\right) \geq \epsilon u\Omega ^{\prime }\left(u\right),$ for $\epsilon \in \left(0, 1\right).$

Proof. Suppose that $x$ is a positive solution of (1.1) and for $u\geq u_{1},$ $\Omega ^{\prime \prime }\left(u\right) > 0.$ By applying Lemma 2.4 with $F = \Omega$ and $\kappa \geq 3,$ we obtain

which gives (Ⅰ). Next, for $u\geq u_{1},$ $\Omega ^{\prime \prime }\left(u\right) < 0.$ Then, $u_{2} > u_{1}$ exists, such that

which gives (Ⅱ), for all $\epsilon \in \left(0, 1\right)$ and $u\geq u_{2}.$ □

3.1.1.   Properties in the case where $p(u) \leq p_{0} < 1$

Lemma 3.3. Let $\beta _{0}^{\ast } > 0$ and $x\in \aleph _{1}$. Then, for $u$ large enough,

(A$_{1}$) $\lim_{u\rightarrow \infty }r\left(u\right) \Omega ^{\prime \prime \prime }\left(u\right) = \lim_{u\rightarrow \infty }\Omega ^{\left(k\right) }\left(u\right) /\pi _{2-k}\left(u\right) = 0$, $k = 0, 1, 2$;

(A$_{2}$) $\Omega ^{\prime \prime }\left(u\right) /\pi _{0}\left(u\right)$ is decreasing;

(A$_{3}$) $\Omega ^{\prime }\left(u\right) /\pi _{1}\left(u\right)$ is decreasing;

(A$_{4}$) $\Omega \left(u\right) /\pi _{2}\left(u\right)$ is decreasing.

Proof. Assume that $x\in \aleph _{1}.$ From the definition of $\Omega$

we have

Since $\Omega \left(u\right) > x\left(u\right)$, $\Omega ^{\prime }\left(u\right) > 0$, and $\zeta \left(u\right) \leq u$, we have

which with (1.1) we have

(A$_{1}$): Since we have $\Omega ^{\prime \prime \prime }\left(u\right)$ as a non-increasing and positive function, then

Assume $\ell > 0$; then $r\left(u\right) \Omega ^{\prime \prime \prime }\left(u\right) \geq \ell > 0$, by integrating three times

From (3.1) with $R(u; 0) = \left(1-p\left(u\right) \right)$ and (3.3) we obtain

From (3.4) into (3.5) we obtain

By integrating the above inequality from $u_{3}$ to $u$, we obtain

which is

we find that there is a contradiction; therefore, $\ell = 0$. When $x\in \aleph _{1}$, we have $\Omega \left(u\right) \longrightarrow \infty$, $\Omega ^{\prime }\left(u\right) \longrightarrow \infty$ as $u\longrightarrow \infty$ and $\Omega ^{\prime \prime }\left(u\right) /\pi _{0}\left(u\right) \longrightarrow 0$ as $u\longrightarrow \infty$ such that $\Omega ^{\prime \prime }\left(u\right) > 0$ for $k = 2$ is increasing, then by l'Hôpital's rule we find that (A$_{1}$) satisfied.

(A$_{2}$): As $r\left(u\right) \Omega ^{\prime \prime \prime }\left(u\right)$ is nonincreasing in $\aleph _{1}$, we are able to say that

that is

Since $\Omega ^{\prime \prime }\left(u\right) > 0$, and $r\left(u\right) \Omega ^{\prime \prime \prime }\left(u\right)$ converges to zero by (A$_{1}$), there exists $u_{4} > u_{3}$ such that

so we obtain

Therefore,

then $\Omega ^{\prime \prime }\left(u\right) /\pi _{0}$ is decreasing; that proves (A$_{2}$).

(A$_{3}$): From (A$_{1}$) and (A$_{2}$), we have $\Omega ^{\prime \prime }\left(u\right) /\pi _{0}$ decreasing and tending to zero, then we find

then we obtain

for $u_{5} > u_{4}$. Hence

from that we arrive at (A$_{3}$).

(A$_{4}$): Likewise, since $\Omega ^{\prime }\left(u\right) /\pi _{1}\left(u\right)$ is decreasing and tends to zero, we obtain

then we arrive at

for $u_{6} > u_{5}$, so

that proves (A$_{4}$). □

Lemma 3.4. Let $\beta _{0}^{\ast } > 0$ and $x\in \aleph _{1}$. Then, the corresponding function $\Omega$ eventually satisfies

Proof. By using the facts that $\zeta ^{\left[ 2i+1\right] }\left(u\right) \leq \zeta ^{ \left[ 2i\right] }\left(u\right) < u$ and $\Omega ^{\prime }\left(u\right) > 0$, we find that

By utilizing $\left(\Omega \left(u\right) /\pi _{2}\left(u\right) \right) ^{\prime } < 0,$ we arrive at

this leads to

From this inequality in (2.1), we obtain

From this and (1.1) we obtain

This is completes the proof. □

Lemma 3.5. Let $\beta _{k}^{\ast } > 0$ for some $k\in N$, and $x\in \aleph _{1}$. Then, for $u$ large enough, (A$_{1}$)–(A$_{4}$) (in Lemma 3.3) hold.

Proof. Replacing inequality (3.5) in Lemma 3.2 (Ⅱ) and proceeding in the same manner, we obtain properties in (A$_{1}$)–(A$_{4}$). □

Lemma 3.6. Assume that $\beta _{k}^{\ast } > 0$ and $x\in \aleph _{2}.$ If $\Omega ^{\prime \prime }\left(u\right) < 0,$ eventually, then

Proof. Suppose that $x\in \aleph _{2}$. For $u\geq u_{1},$ $\Omega ^{\prime \prime }\left(u\right) < 0.$ From the facts $\Omega ^{\prime }\left(u\right) > 0$ and $(II),$ we obtain

and

Then, Eq (2.1) becomes

which together with (1.1) gives (3.9). □

Lemma 3.7. Let $\beta _{k}^{\ast } > 0$ for some $k\in$ $\mathbb{N}$ and $x\in \aleph _{1}$. Then,

Proof. Follows from Lemmas 3.3, 3.6, and 3.4 with (1.1) gives (3.10). □

3.1.2.   Properties in the case where $p(u) \geq p_{0} > 1$

Lemma 3.8. Let $x\in \aleph _{1}\cup \aleph _{2}$. Then there exists $k$ such that

Proof. From the definition $\Omega (u)$, we find that

hence, we obtain

After $k$ steps we arrive at there exists a $k$ such that

This concludes the proof. □

Lemma 3.9. Let $x\in \aleph _{1}\cup \aleph _{2}$. Then there exists $k$ such that (1.1) implies,

When

Proof. Suppose that $x\in \aleph _{1}.$ From the fact that $\Omega ^{\prime }\left(u\right) > 0$ and from $(I)$ in Lemma 3.2, we obtain

and

Then, from Lemma (3.8), Eq (3.11) becomes

in view of the fact that $\Omega \left(\zeta ^{\left[ -2i\right] }\left(u\right) \right) \geq \Omega \left(u\right),$ the above inequality becomes

which is together with (1.1), we obtain

Suppose that $x\in \aleph _{2}$. For $u\geq u_{1},$ $\Omega ^{\prime \prime }\left(u\right) < 0.$ From the facts $\Omega ^{\prime }\left(u\right) > 0$ and $(II)$ in Lemma 3.2, we obtain

and

Then, from Lemma (3.8), Eq (3.11) becomes

which is together with (1.1), we obtain

Followed by (3.14) and (3.15) gives (3.13). □

4.   Oscillation theorems

In the following theorem, we will obtain oscillation criteria for Eq (1.1) in the case where $p_{0} < 1$.

For clarity, we will define that:

Theorem 4.1. Assume that there is $\epsilon \in (0, 1)$ and $p_{0} < 1$ such that, if $x\in \aleph _{1},$ then

and, if $x\in \aleph _{2}$, then

where

Then, Eq (1.1) is oscillatory.

Proof. Assume the contrary, that (1.1) has a non-oscillatory solution $x$. Then, there exists a $u_{1}\geq u_{0}$ such that $x(u) > 0$, $x(\theta (u)) > 0$, and $x(\zeta (u)) > 0$ for $u\geq u_{1}$. There are two possible classes from Lemma 3.1: $\aleph _{1}$ and $\aleph _{2}$. Assume $\aleph _{1}$ holds. From (3.8) in Lemma 3.4 we have

Introduce Riccati substitutions

We observe that $\omega \left(u\right) > 0$ for $u\geq u_{1}$; by differentiating (4.4), we obtain

from (4.3) we obtain

From $(I)$ in Lemma 3.2, we have

then we obtain

by applying Lemma 2.2 for every $\epsilon \in \left(0, 1\right)$, we obtain

Therefore, from (4.5)–(4.7), we obtain

which is

By integrating the above inequality, from $u$ to$\infty,$ we obtain

since $\omega > 0$ and $\omega ^{\prime } < 0,$ we have

which is

Let $\lambda = \inf_{u\geq u_{\ast }}\omega \left(u\right) /\widetilde{M} _{1}\left(u\right),$ then from (4.8) we notice

which contradicts that$\lambda \geq 1$ in (4.8).

Assume $\aleph _{2}$ holds. From (3.9) in Lemma 3.6 we have

By introducing Riccati substitutions

Integrating (4.9) from $u$ to $\infty$, we obtain

From $(II)$ in Lemma 3.2, we have

hence

By using (4.12) in (4.11), we obtain

by integrating again this inequality from $u$ to $\infty$, we obtain

by differentiating $\varpi \left(u\right)$ in (4.10), we obtain

for $u\geq u_{2}.$

Then, from (4.13) and (4.10), we obtain

by integrating the above inequality from $u$ to $\infty$

from this we obtain

The rest of the proof is done as in the case $\aleph _{1}$. As a result, the theorem is established. □

The next theorem provides two oscillation conditions for Eq (1.1), which require that $p_{0} > 1$. These conditions are established by using a comparison method with a first-order equation, under the next constraints: $g\left(u\right) \leq \theta \left(u\right)$.

Theorem 4.2. Let $p_{0} > 1$ hold, if there exists $\epsilon \in \left(0, 1\right)$ and $k$ such that $\rho _{2}$ and $\hat{\rho}_{2}$ are defined, such that the two first-order DDEs

and when $g\left(u\right) \leq \theta \left(u\right)$

where

are oscillatory, then (1.1) is oscillatory.

Proof. Assume the contrary, that (1.1) has a non-oscillatory solution $x$. Then, there exists a $u_{1}\geq u_{0}$ such that $x(u) > 0$, $x(\theta (u)) > 0$, and $x(\zeta (u)) > 0$ for $u\geq u_{1}$. There are two possible classes from Lemma 3.1: $\aleph _{1}$ and $\aleph _{2}$. Assume $\aleph _{1}$ holds. Then from Eq (3.14) in Lemma 3.9, we have

From (4.7) and (4.16) we notice

If we set $\psi \left(u\right) = r\left(u\right) \Omega ^{\prime \prime \prime }\left(u\right)$, then $\psi \left(u\right)$ is a positive solution of the first-order delay differential inequality

By [44, Theorem 1], the DDE (4.14) also has a positive solution; this leads to a contradiction.

Assume $\aleph _{2}$ holds. Then from Eq (3.15) in Lemma 3.9, we have

Since $g\left(u\right) \leq \theta \left(u\right),$ we find

by integrating (4.18) from $u$ to $\infty$ we obtain

which is

Integrating (4.20) again from $u$ to $\infty$, we obtain

From $(II)$ in Lemma 3.2, we have

Let $\nu \left(u\right) = \Omega ^{\prime }\left(u\right)$ and by using (4.22) in (4.21), we find that $\nu \left(u\right)$ is a positive solution of the inequality

But according to [44, Theorem 1], the condition (4.15) also has a positive solution $\nu \left(u\right)$; this leads to a contradiction. □

Corollary 4.1. If $p_{0} > 1$ such that

and

Then (1.1) is oscillatory.

5.   Applications and discussion

Example 5.1. Let the fourth-order NDE

where $p_{0},$ $q_{0}$ are positive and $\delta$, $\beta \in \left(0, 1\right)$. We note that $\zeta ^{-1}\left(\theta \left(u\right) \right) = \frac{\beta u}{\delta }$, $r\left(u\right) = 1$. As a result, it is clear that $\zeta ^{\left[ -2i\right] }\left(u\right) = \delta ^{-2i}u$, $\zeta ^{ \left[ -2i+1\right] }\left(u\right) = \delta ^{-2i+1}u$. Thus, for $p_{0} > 1,$ we define

for $k > 0.$ Then, the condition (4.23) in Corollary 4.1, becomes

Also, we have

Then, the condition (4.24) simplifies to

which is

Using Corollary 4.1, Eq (5.1) is oscillatory if

Example 5.2. Now, consider the fourth-order NDE

where $p_{0}$, $q_{0}$ are positive, $\lambda$, $\mu \in \left(0, 1\right)$, $r\left(u\right) = 1,$ $\theta \left(u\right) = \mu u$, $q\left(u\right) = \frac{q_{0}}{u^{4}},$ $p\left(u\right) = p_{0}$ and $\zeta \left(u\right) = \lambda u.$

As a result, it is clear that $\pi _{0}\left(u\right) = u$, $\pi _{1}\left(u\right) = \frac{u^{2}}{2}$, $\pi _{2}\left(u\right) = \frac{u^{3}}{6}$ and $\pi _{2}\left(\zeta ^{\left[ 2i\right] }\left(u\right) \right) = \frac{ \left(\lambda ^{2i}u\right) ^{3}}{6},$ $\pi _{0}\left(\zeta ^{\left[ 2i \right] }\left(u\right) \right) = \lambda ^{2i}u$, then for $p_{0} < 1,$ we define

and

By applying condition (4.1) in Theorem 4.1 when $x\in \aleph _{1},$ we see

this implies

which get all solutions of (5.5) are oscillatory if

By applying condition (4.2) in Theorem 4.1 when $x\in \aleph _{2},$ we see

which obtain all solutions of (5.5) are oscillatory if

Remark 5.1. Consider a special case of Eq (5.5) in the form

when talking $\epsilon = p_{0} = 0.5$. By using our conditions (5.6) and (5.7), then Eq (5.8) is oscillatory if

By applying [32, Corollary 1], we see that

Then, by choosing $\theta \left(t\right) = t^{4}$ and $\phi \left(t\right) = t^{2}$, we find that (5.8) is oscillatory if

While [4, Theorem 2.1] ensures the oscillation of Eq (5.8) if

Figures 1 illustrates the efficiency of the conditions (5.9) in studying the oscillation of the solutions of (5.8) for values of $\mu \in \left(0, 1\right).$ Thus, our results present a better criterion for oscillation.

6.   Conclusions

Finding conditions that exclude each of the cases of the derivatives of the positive solution is often the foundation of the idea of establishing oscillation criteria for differential equations. This study examines the oscillatory behavior of fourth-order NDEs in the canonical case. The relationship between the solution and the corresponding function is vital to the oscillation theory of NDEs. Therefore, we improve these relationships by applying the modified monotonic properties of positive solutions. The conditions that we obtained using these relationships subsequently proved that there are no positive solutions in categories $\aleph _{1}$ and $\aleph _{2}$. Then, using the newly deduced relationships and properties, we employed a number of approaches by using different techniques, including recatti and comparison techniques, to develop a set of oscillation criteria. Additionally, we provided examples that illustrated and clarified the importance of our results; they were compared with some previous results in the literature. In the future, we can try to develop new conditions that ensure that every solution of (1.1) is oscillatory in the noncanonical case.

Author contributions

H. Salah, C. Cesarano, and E. M. Elabbasy: Conceptualization, methodology, writing-original draft; M. Anis, S. S. Askar, and S. A. M. Alshamrani: Formal analysis, investigation, writing-review and editing. All authors have read and approved the final version of the manuscript for publication

Acknowledgments

The authors present their appreciation to King Saud University for funding this research through Researchers Supporting Project number (RSPD2024R533), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that there is no conflict of interest.