Novel iterative criteria for oscillatory behavior in nonlinear neutral differential equations

1.   Introduction

In this paper, we investigate the oscillatory properties of nonlinear second-order neutral differential equations (NDEs) of the form:

where $\omega \left(\mathrm{s}\right) = \mathrm{y}\left(\mathrm{s}\right) + \mathrm{u}\left(\mathrm{s}\right) \mathrm{y}\left(\tau \left(\mathrm{s} \right) \right).$ The following hypotheses are assumed throughout this study:

$\left({{\bf{Hyp.}}}1\right)$ $0 < \alpha \leq 1,$ $\alpha \geq \beta$ are ratios of odd positive integers;

$\left({{\bf{Hyp.}}}2\right)$ $\mathrm{h}\in C\left(\left[\mathrm{s} _{0}, \infty \right) \times \left(a, b\right), \mathbb{R} \right)$ and $\mathrm{h}(\mathrm{s}, \ell)\geq 0;$

$\left({{\bf{Hyp.}}}3\right)$ $\mathrm{u}\in C\left(\left[\mathrm{s} _{0}, \infty \right), \left(0, \infty \right) \right),$ $0\leq \mathrm{u} \left(\mathrm{s}\right) < 1,$ $\tau \in C^{1}([\mathrm{s}_{0}, \infty), \mathbb{R}),$ $\sigma \in C^{1}(\left[\mathrm{s}_{0}, \infty \right) \times \left(a, b\right), \mathbb{R}),$ $\tau (\mathrm{s})\leq \mathrm{s},$ $\sigma (\mathrm{s}, \ell)\leq \mathrm{s},$ $\sigma$ has nonnegative partial derivatives with respect to $\mathrm{s}$ and nondecreasing with respect to $\ell,$ $\lim_{\mathrm{s} \rightarrow \infty }\tau (\mathrm{s}) = \infty,$ and $\lim_{\mathrm{s} \rightarrow \infty }\sigma (\mathrm{s}, \ell) = \infty$ for $\ell \in \left[a, b\right]$;

$\left({{\bf{Hyp.}}}4\right)$ $\kappa \in C\left(\left[\mathrm{s} _{0}, \infty \right), \mathbb{R} ^{+}\right)$ satisfies the noncanonical case. That is

where

$\left({{\bf{Hyp.}}}5\right)$ $\mathrm{u}(\mathrm{s}) < \xi (\mathrm{s})/\xi (\tau (\mathrm{s})).$

Below we provide some basic definitions ^[1]:

$\left(\mathrm{i}\right)$ A function $\mathrm{y}\left(\mathrm{s}\right) \in C([\mathrm{s}_{\mathrm{y}}, \infty), \mathbb{R})$, $\mathrm{s}_{\mathrm{y}}\geqslant \mathrm{s}_{0}$, is said to be a solution of (1.1) which has the property $\kappa \left(\mathrm{s} \right) (\omega ^{\prime }\left(\mathrm{s}\right))^{\alpha }\in C^{1}[\mathrm{s}_{\mathrm{y}}, \infty)$, and it satisfies (1.1) for all $\mathrm{s}\in \lbrack \mathrm{s}_{\mathrm{y}}, \infty)$. We consider only those solutions $\mathrm{y}(\mathrm{s})$ of (1.1) that are defined on a half-line $[\mathrm{s}_{\mathrm{y}}, \infty)$ and satisfy the condition

$\left(\mathrm{ii}\right)$ A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory.

$\left(\mathrm{iii}\right)$ The Eq (1.1) is said to be oscillatory if all its solutions are oscillatory.

The study of differential equations (DEs) is a cornerstone of mathematical analysis, particularly in understanding dynamic systems that arise in various scientific and engineering applications. Among these, (NDEs) play a critical role in modeling phenomena where the derivative of the unknown function depends not only on the function itself but also on its delayed or advanced argument. In recent decades, there has been a growing interest in the qualitative analysis of such equations, particularly regarding their oscillatory behavior. This interest stems from the fact that oscillatory solutions often represent stable or periodic phenomena in real-world systems; see ^[2,3,4].

Oscillation theorems are pivotal in the analysis of DEs, as they provide critical insight into the nature of solutions, particularly in identifying whether these solutions exhibit oscillatory behavior over time. These theorems are essential tools for mathematicians and scientists alike, helping to predict and understand the dynamics of various physical, biological, and engineering systems. Historically, oscillation criteria have been developed and refined to handle a wide array of DEs, from simple linear forms to more intricate nonlinear systems. In recent years, there has been significant progress in extending these classical theorems to accommodate the growing complexity of DEs, including those with non-standard, non-canonical forms. These advancements reflect the continuous evolution of mathematical methods and the increasing sophistication of the systems being studied, making oscillation theorems more relevant and applicable than ever before in addressing contemporary challenges across multiple disciplines; see ^[5,6,7,8,9].

Second-order non-linear neutral differential equations (NDEs) with deviating arguments constitute a specialized class of DEs that have garnered significant attention due to their broad applications in physics, engineering, and biological systems. These equations are distinguished by terms involving delays or advanced arguments, adding layers of complexity to their analysis and necessitating advanced mathematical techniques for understanding their behavior. While previous studies have extensively examined the existence and stability of solutions, less attention has been given to their oscillatory behavior. The oscillation of solutions to such equations, however, remains an active area of research, motivated by the need to establish precise conditions under which solutions oscillate or converge. In particular, the interplay between nonlinear terms and deviating arguments presents unique challenges that require refined criteria and novel analytical approaches, see ^[10,11,12].

In recent years, the study of the oscillatory and exponential behavior of DEs with delays and neutral terms across different orders has seen increasing interest, as illustrated by the work of Han et al. ^[13], Baculíková ^[14], Džurina et al. ^[15], Jadlovská et al. ^[16], Bazighifan et al. ^[17], Moaaz et al. ^[18], and Aldiaiji et al ^[19,20]. This broad interest has led to major advances in the understanding of complex periodic solutions ranging from simple harmonic motion to chaotic oscillations and has enabled accurate analyses of critical properties such as amplitude, frequency, and stability. Here is a comprehensive review of the foundational studies that have contributed significantly to this field: Baculíková ^[21] investigated the second-order delay differential equations (DDEs) oscillatory characteristics:

under the case (1.2). However, both Sun and Meng ^[22], and Kusano et al. ^[23] noted that NDEs had the following characteristics:

and the linear form that corresponds to them

Sufficient criteria have been established by Agarwal et al. ^[24] to guarantee the oscillatory behavior of second-order DEs with a neutral term:

under the conditions:

and

Han et al. ^[25] reviewed oscillations in second-order linear NDEs (1.6) where $\alpha = 1,$ and introduced criteria under the condition $0\leq u \left(\mathrm{s}\right) \leq u _{0} < \infty.$ This analysis was expanded upon by Grace and Lalli ^[26] to the equation

where $f\left(\mathrm{y}\right) /\mathrm{y}\geq k > 0$ and $\int_{\mathrm{s} _{0}}^{\infty }1/\kappa \left(\ell \right) \mathrm{d}\ell = \infty.$

Bohner et al. ^[27] also investigated the oscillations of the second-order quasi-linear NDEs

under the condition (1.2).

In similar studies, Zhang et al. ^[28] considered a particular type of second-order NDEs

where $\omega \left(\mathrm{s}\right) = \mathrm{y}\left(\mathrm{s}\right) +\sum_{i = 1}^{m} u _{i}\left(\mathrm{s}\right) \mathrm{y}\left(\tau _{i}\left(\mathrm{s}\right) \right),$ which helps simplify the analysis of these equations.

In the same context, Sun ^[29] established oscillation criteria for second-order nonlinear NDEs

they relied on a new variational principle to extract these criteria.

Finally, Moaaz et al. ^[30] presented a study on the oscillation properties of NDEs

They proposed new properties characterized by a recursive nature, and extracted oscillation conditions that guarantee the oscillation of all solutions. Alemam et al. ^[31] also made an in-depth study of the oscillatory properties of the second-order NDEs:

by using the Riccati transformation method to establish oscillation criteria.

While much of the previous research has concentrated on the oscillatory properties of linear and quasi-linear second-order NDEs, resulting in significant advancements in the understanding of their behavior, the oscillatory characteristics of nonlinear second-order NDEs have not received the same level of attention, leaving a notable gap in the literature. This study aims to address this gap by extending the investigation of oscillatory behavior to encompass nonlinear second-order equations. Building on the work of ^[30], which explored the oscillatory properties of quasi-linear second-order equations, this paper adapts and extends the approach to include nonlinear terms. Through this extension, new oscillation criteria are introduced, tailored to the distinctive features of nonlinear equations, thereby offering a more comprehensive and nuanced understanding of their oscillatory dynamics.

2.   Preliminary results

Let us define

and

for $\mathrm{s}\in \lbrack \mathrm{s}_{0}, \infty)$.

Lemma 2.1. ^[32] Assume that $\mathrm{y}(\mathrm{s})$ is an eventually positive solution of (1.1), then the corresponding function $\omega (\mathrm{s})$ satisfies one of two cases eventually:

for $\mathrm{s}\geqslant \mathrm{s}_{1}\geqslant \mathrm{s}_{0}.$

The subsequent considerations aim to demonstrate that the class $(\mathrm{C} _{2})$ is fundamental.

Lemma 2.2. If

then, the positive solution $\mathrm{y}(\mathrm{s})$ of (1.1) satisfies ($\mathrm{C}_{2}$) in Lemma 2.1 and, moreover

$\left(\mathrm{A}_{1, 1}\right)$ $\kappa ^{1/\alpha }(\mathrm{s})\omega ^{\prime }(\mathrm{s})\xi (\mathrm{s})+\omega (\mathrm{s})\geqslant 0;$

$\left(\mathrm{A}_{1, 2}\right)$ $\omega (\mathrm{s})/\xi (\mathrm{s})$ is increasing$;$

$\left(\mathrm{A}_{1, 3}\right)$ $\omega ^{\beta /\alpha -1}\left(\mathrm{s }\right) \geq \gamma;$

$\left(\mathrm{A}_{1, 4}\right)$ $\left(\kappa (\mathrm{s})\left(\omega ^{\prime }(\mathrm{s})\right) ^{\alpha }\right) ^{\prime }\leq -\omega ^{\beta }\left(\sigma \left(\mathrm{s}, b\right) \right) \widehat{\mathrm{h} }\left(\mathrm{s}\right);$

$\left(\mathrm{A}_{1, 5}\right)$ $\lim_{\mathrm{s}\rightarrow \infty }\omega (\mathrm{s}) = 0.$

Proof. Suppose on the contrary that $\mathrm{y}$ is a positive solution to (1.1) that meets case $(\mathrm{C}_{1})$ in Lemma 2.1 for $\mathrm{s}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$. Then there exists a constant $c_{0} > 0$ such that $\omega (\mathrm{s})\geq c_{0}$ and $\omega \left(\sigma (\mathrm{ s}, \ell)\right) \geq c_{0}$ eventually. Using the definition of $\omega$, we deduce that

Then (1.1) becomes

Since $\xi ^{\prime }\left(\mathrm{s}\right) < 0$ and $\tau (\mathrm{s})\leq \mathrm{s, }$ we get

and then

By combining (2.3) and (2.4) and integrating the resulting inequality from $\mathrm{s}_{1}$ to $\infty$, we conclude that

It follows from (2.2) and $\left(\text{hyp.5}\right)$ that $\int_{ \mathrm{s}_{1}}^{\mathrm{s}}\widehat{\mathrm{h}}\left(\varrho \right) \mathrm{d}\varrho$ must be unbounded. Furthermore, since $\xi ^{\prime }\left(\mathrm{s}\right) < 0,$ it's clear that

which with (2.5) gives a contradiction.

$\left(\mathrm{A}_{1, 1}\right)$ Based on case $\left(\mathrm{C} _{2}\right)$ of Lemma 2.1, it follows that $\omega \left(\mathrm{s} \right)$ is positive and decreases for every $\mathrm{s}\geq \mathrm{s} _{1}\geq \mathrm{s}_{0}.$ By the definition of $\omega \left(\mathrm{s} \right),$ we obtain $\omega \left(\mathrm{s}\right) \geq \mathrm{y}\left(\mathrm{s}\right)$ and

Since $\kappa \left(\mathrm{s}\right) \left(\omega ^{\prime }\left(\mathrm{s}\right) \right) ^{\alpha }$ is decreasing, we get

Dividing the resulting inequality by $\kappa ^{1/\alpha }\left(l\right)$ and then integrating from $\mathrm{s}$ to $\infty$, we get

$\left(\mathrm{A}_{1, 2}\right)$ From (2.8), we obtain

$\left(\mathrm{A}_{1, 3}\right)$ In the case where $\alpha = \beta$, it is easy to see that $\omega ^{\beta /\alpha -1}\left(\mathrm{s}\right) = 1$. Now, let $\alpha > \beta.$ Since $\omega ^{\prime }(\mathrm{s}) < 0,$ there exists a constant $l > 0,$ such that

and consequently,

$(\mathrm{A}_{1, 4})$ Since $\omega (\mathrm{s})/\xi (\mathrm{s})$ is increasing, we get

In view of the definition of $\omega$, we get

Thus, (1.1) becomes

that is,

$(\mathrm{A}_{1, 5})$ Since $\omega (\mathrm{s}) > 0,$ and $\omega ^{\prime }(\mathrm{s}) < 0$, then $\lim_{\mathrm{s}\rightarrow \infty }$ $\omega (\mathrm{s}) = c_{1}\geq 0$. We assert that $c_{1} = 0.$ If not, $\omega (\mathrm{s})\geq c_{1} > 0$ for $\mathrm{s}\geq \mathrm{s}_{2}\geq \mathrm{s} _{1}.$ Integrating (1.1) from $\mathrm{s}_{1}$ to $\mathrm{s}$ yields

and so

Integrating this inequality from $\mathrm{s}_{1}$ to $\infty$, we find

which contradicts (2.2). Therefore, $c_{1} = 0$.

As a result, the lemma has been completely proven. □

3.   Main results

In this section, we will discuss new monotonic properties for the solutions of (1.1).

Lemma 3.1. Let $\mathrm{y}(\mathrm{s})$ be a positive solution of (1.1), and assume that (2.2) holds. If $\delta _{0}\in \left(0, 1\right)$ with

then

$(\mathrm{A}_{2, 1})$ $\omega (\mathrm{s})/\xi ^{\rho _{0}}(\mathrm{s})$ is decreasing;

$(\mathrm{A}_{2, 2})$ $\lim_{\mathrm{s}\rightarrow \infty }$ $\omega (\mathrm{ s})/\xi ^{\rho _{0}}(\mathrm{s}) = 0;$

$(\mathrm{A}_{2, 3})$ $\omega (\mathrm{s})/\xi ^{1-\rho _{0}}(\mathrm{s})$ is increasing$.$

Proof. For the purposes of this discussion, let $\mathrm{y}(\mathrm{s})$ be an eventually positive solution of (1.1). From (3.1), it follows that:

From $\left(\mathrm{A}_{1, 5}\right),$ we know that $\lim_{\mathrm{s} \rightarrow \infty }\omega \left(\mathrm{s}\right) = 0.$ Then, there exists $\mathrm{s}_{1}\geq \mathrm{s}_{0}$ such that $\xi ^{-\alpha }(\mathrm{s})-\xi ^{-\alpha }(\mathrm{s}_{1})\geq \epsilon \xi ^{-\alpha }(\mathrm{s})$ where $\epsilon \in \left(0, 1\right).$ Thus, we have

Hence, from Lemma 2.2, we have that ($\mathrm{A}_{1, 1}$)–($\mathrm{A}_{1, 4}$) hold.

$(\mathrm{A}_{2, 1})$ Integrating $\left(\mathrm{A}_{1, 4}\right)$ from $\mathrm{s}_{1}$ to $\mathrm{s}$, we obtain

By using (3.1), we get

Since $\omega (\mathrm{s})\rightarrow 0$ as $t\rightarrow \infty,$ as stated in $\left(\mathrm{A}_{1, 5}\right)$, we have

and so, (3.2) becomes

and so,

Furthermore, from $\left(\mathrm{A}_{1, 3}\right),$ we see that

This results in

Consequently,

$(\mathrm{A}_{2, 2})$ Since $\omega \left(\mathrm{s}\right) /\xi ^{\rho _{0}}(\mathrm{s})$ is positive and decreasing, $\lim_{\mathrm{s}\rightarrow \infty }\omega \left(\mathrm{s}\right) /\xi ^{\rho _{0}}(\mathrm{s}) = c_{1}\geqslant 0.$ We assert that $c_{2} = 0.$ If not, eventually $\omega \left(\mathrm{s}\right) /\xi ^{\rho _{0}}(\mathrm{s})\geqslant c_{2} > 0$. We now present the function

We observe that $w(\mathrm{s}) > 0$ in context of $\left(\mathrm{A} _{1, 1}\right) \ $in Lemma 2.2, and

By using $\left(\mathrm{A}_{1, 3}\right),$ (3.1), (3.3) and (3.4), we find

Using the fact that $\omega \left(\mathrm{s}\right) /\xi ^{\rho _{0}}(\mathrm{s})\geqslant c_{2},$ we get

When we integrate the previous inequality from $\mathrm{s}_{1}$ to $\mathrm{s },$ we get

which is a contradiction. Thus, $c_{2} = 0.$

$\left(\mathrm{A}_{2, 3}\right)$ Finally, we have

When we integrate the previous inequality from $\mathrm{s}$ to $\infty,$ we get

Thus

and hence

Hence, the proof is complete. □

Theorem 3.1. Assume that (2.2) and (3.1) hold. If

then, (1.1) is oscillatory.

Proof. Assume, for the sake of contradiction, that $\mathrm{y}$ is an eventually positive solution of (1.1). Referring to the proof of Lemma 3.1, we obtain

and

By combining (3.6) and (3.7), we find

Since $\omega \left(\mathrm{s}\right) > 0,$ it must hold that $1-2\rho _{0}\geq 0,$ which implies that

which leads to a contradiction. This completes the proof. □

When $\rho _{0}\leq \frac{1}{2}$, it is possible to refine the results stated in Lemma 3.1. Since $\xi (\mathrm{s})$ is a decreasing function, there exists a constant $\lambda \geq 1$ such that

We introduce the constant $\rho _{1} > \rho _{0}$ as follows

Lemma 3.2. Assume (2.2) and (3.1) hold. If $\mathrm{y}(\mathrm{s})$ is a positive solution of (1.1), then

$\left(\mathrm{A}_{3, 1}\right)$ $\omega (\mathrm{s})/\xi ^{\rho _{1}}(\mathrm{s})$ is decreasing;

$\left(\mathrm{A}_{3, 2}\right)$ $\lim_{\mathrm{s}\rightarrow \infty }\omega (\mathrm{s})/\xi ^{\rho _{1}}(\mathrm{s}) = 0;$

$\left(\mathrm{A}_{3, 3}\right)$ $\omega (\mathrm{s})/\xi ^{1-\rho _{1}}(\mathrm{s})$ is increasing$.$

Proof. Assume that $\mathrm{y}(\mathrm{s})$ is an eventually positive solution of (1.1) satisfying condition ($\mathrm{C}_{2}$) in Lemma 2.1 for $\mathrm{s}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$. From Lemma 2.2, we have that ($\mathrm{A}_{1, 1}$)–($\mathrm{A}_{1, 5}$) hold. Additionally, Lemma 3.1 implies that conditions ($\mathrm{A}_{2, 1}$)-($\mathrm{A}_{2, 3}$) are satisfied.

$\left(\mathrm{A}_{3, 1}\right)$ Integrating $\left(\mathrm{A} _{1, 4}\right)$ from $\mathrm{s}_{1}$ to $\mathrm{s}$, we get

By using the fact $\omega (\mathrm{s})/\xi ^{\rho _{0}}(\mathrm{s})$ is decreasing, we have

By using (3.1) and (3.8), we get

Since $\frac{\omega (\mathrm{s})}{\xi ^{\rho _{0}}(\mathrm{s})}\rightarrow 0$ as $t\rightarrow \infty,$ as stated in ($\mathrm{A}_{2, 2}$), we have

and hence

This implies that

which is equivalent to

Consequently,

So $\omega (\mathrm{s})/\xi ^{\rho _{1}}(\mathrm{s})$ is decreasing.

The same procedures as in the Lemma 3.1 proof can be used to verify that conditions ($\mathrm{A}_{3, 2}$) and $(\mathrm{A}_{3, 3}$) are satisfied. □

If $\rho _{1} < 1/2$, we can repeat the previous process and deduce that $\delta _{2} > \delta _{1}$ as follows

More generally, if $\rho _{i} < 1/2$ for $i = 1, 2, ..., n-1,$ it is possible to describe

Additionally, by taking the identical actions as in the Lemma 3.2 proof, we may verify the following:

$(\mathrm{A}_{n, 1})\ \omega (\mathrm{s})/\xi ^{\rho _{n}}(\mathrm{s})$ is decreasing;

$(\mathrm{A}_{n, 2})$ $\lim_{\mathrm{s}\rightarrow \infty }\omega (\mathrm{s})/\xi ^{\rho _{n}}(\mathrm{s}) = 0;$

$(\mathrm{A}_{n, 3})$ $\omega (\mathrm{s})/\xi ^{1-\rho _{n}}(\mathrm{s})$ is increasing$.$

Theorem 3.2. Let (2.2) and (3.1) hold. If there exists a $n\in \mathbb{N}$ such that

then (1.1) is oscillatory.

Theorem 3.3. Let (2.2) and (3.1) hold. If there exists $n\in \mathbb{N}$ such that

then (1.1) is oscillatory.

Proof. Assume, for the sake of contradiction, that $\mathrm{y}(\mathrm{s})$ is an eventually positive solution of (1.1)$.$ Condition (2.2) guarantees that $\mathrm{y}(\mathrm{s})$ satisfies ($\mathrm{C}_{2}$). From Lemma 2.2, we have that ($\mathrm{A}_{1, 1}$)–($\mathrm{A}_{1, 4}$) hold. We generate the sequence $\left\{ \rho _{n}\right\}$ using (3.11).

We now define the function:

Based on $\left(\mathrm{A}_{1, 1}\right) \ $in Lemma 2.2, we can conclude that $\Psi (\mathrm{s})\geq 0.$ Furthermore, from ($\mathrm{A} _{n, 1}$), we can derive

Next, based on the definition of $\Psi \left(\mathrm{s}\right),$ we get

From (3.14), $\left(\mathrm{A}_{1, 4}\right)$ and (3.4), we deduce that

We observe that $\omega \left(\mathrm{s}\right) /\xi \left(\mathrm{s} \right)$ is increasing from $\left(\mathrm{A}_{1, 2}\right)$ in Lemma 2.2, then

Considering that $0 < \alpha \leq 1,$ then

Hence, (3.16) yields

From $\left(\mathrm{A}_{1, 3}\right)$ we know that $\omega ^{\beta -\alpha }\left(\sigma \left(\mathrm{s}, b\right) \right) \geq \gamma ^{\alpha }.$ Therefore the above inequality leads to

By using (3.15) we see that $w\left(\mathrm{s}\right)$ is a positive solution of

This results in a contradiction, as Theorem 2.1.1 in ^[33] ensures that condition (3.13) implies (3.17) has no positive solution. This contradiction concludes the proof of the theorem. □

4.   Examples

We provide examples to demonstrate the significance of the obtained results.

Example 4.1. Consider

where $\alpha \geq \beta,$ $0\leq \mathrm{u}_{0} < 1,$ $\tau _{0},$ $\sigma _{0}\in \left(0, 1\right),$ $\sigma _{0}\ell \leq 1,$ and $\mathrm{h} _{0} > 0.$ When comparing (1.1) and (4.1), we can see that $\kappa \left(\mathrm{s}\right) = \mathrm{s}^{2\alpha },$ $\mathrm{h}\left(\mathrm{s }, \ell \right) = \mathrm{h}_{0}\mathrm{s}^{\alpha -1},$ $\mathrm{u}\left(\mathrm{s}\right) = \mathrm{u}_{0},$ $\sigma \left(\mathrm{s}, \ell \right) = \sigma _{0}\mathrm{s}\ell$, and $\tau \left(\mathrm{s}\right) = \tau _{0} \mathrm{s}.$ It is easy to find that

and

For (3.1), we set

Applying (3.8), we obtain $\lambda = \frac{1}{\sigma _{0}}.$ Now, we define the sequence $\left\{ \rho _{n}\right\} _{n = 1}^{m}$ as

with

Then, condition (3.5) reduces to

and condition (3.13) becomes

which leads to

Theorems 3.1 and 3.3 show that the solution of (4.1) is oscillatory if either (4.2) or (4.3) holds.

Example 4.2. Consider the NDE

Clearly:

$a = 1/2,$ $b = 1,$ $\alpha = 1/3,$ $\beta = 1/5,$ $\kappa \left(\mathrm{s} \right) = \mathrm{s}^{2/3},$ $\mathrm{h}\left(\mathrm{s}, \ell \right) = \mathrm{h}_{0}\mathrm{s}^{-2/3},$ $\mathrm{u}\left(\mathrm{s}\right) = 1/4,$ $\sigma \left(\mathrm{s}, \ell \right) = \frac{\ell }{3}\mathrm{s}$ and $\tau \left(\mathrm{s}\right) = \frac{1}{2}\mathrm{s}.$ It is easy to find that

and

For (3.1), we set

Using (3.8), we have $\lambda = 3.$ Here, we define the sequence $\left\{ \rho _{n}\right\} _{n = 1}^{m}$ as

with

Then, condition (3.12) reduces to

and condition (3.13) becomes

which leads to

Theorems 3.1 and 3.3 show that the solution of (4.4) is oscillatory if either (4.5) or (4.6) holds.

Example 4.3. Consider

Clearly:

$\alpha = 1,$ $\beta = 1/3,$ $\kappa \left(\mathrm{s}\right) = \mathrm{s}^{2},$ $\mathrm{h}\left(\mathrm{s}, \ell \right) = \mathrm{h}_{0},$ $\mathrm{u} \left(\mathrm{s}\right) = 1/16,$ $\sigma \left(\mathrm{s}, \ell \right) = \frac{\ell }{4}\mathrm{s}$ and $\tau \left(\mathrm{s}\right) = \frac{1}{2} \mathrm{s}.$ It can be easily verified that

and

For (3.1), we set

From (3.8), we obtain $\lambda = 4.$ The sequence $\left\{ \rho _{n}\right\} _{n = 1}^{m}$ is then defined as

with

On the other hand, if we choose $\mathrm{h}_{0} = 0.8$ and $\gamma = 0.7$, then $\rho _{0} = 0.535\, 62$ and condition (3.5) is satisfied, which implise that (4.7) is oscillatory.

For $\mathrm{h}_{0} = 0.7$ and $\gamma = 0.5$, we compute

and (3.12) holds for $n = 4$, which implies that for $\mathrm{h}_{0} = 0.7$ and $\gamma = 0.5$ (4.7) is oscillatory.

5.   Conclusions

This research has established sufficient conditions to ensure the oscillatory behavior of all solutions within a certain class of second-order nonlinear NDEs. By focusing on the noncanonical forms of these equations, we have revealed new monotonic properties of positive solutions and proposed novel oscillation criteria, which expand the scope of current research in the field of second-order quasilinear NDEs. The contributions made in this study are an important step toward building a more comprehensive theoretical framework for understanding the oscillatory nature of these systems and paving the way for future research. Applying these analytical methods to higher-order nonlinear NDEs represents a promising path that may reveal more complex dynamics and novel oscillatory behaviors, greatly enhancing the understanding of this complex field and deepening theoretical and experimental studies in it.

Author contributions

Fahd Masood: Methodology, investigation, Writing—original draft preparation, Writing—review and editing; Salma Aljawi: Methodology, investigation, Writing—original draft preparation, Writing—review and editing; Omar Bazighifan: Methodology, investigation, Writing—review and editing, Supervision. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to acknowledge the support received from Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R514), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of Interest

There are no competing interests.