Nonlinear neutral differential equations of second-order: Oscillatory properties

1.   Introduction

Oscillation theory is a vital field of mathematics dedicated to exploring the dynamics of both oscillatory and nonoscillatory systems. This theory investigates how systems consistently fluctuate around a central value or shift between various states. Such oscillatory behavior is widespread in the natural world and in technological applications, manifesting itself in various areas such as mechanical, electrical, and biological systems.

For example, in mechanical systems, oscillation theory is essential for analyzing the stability of electrical circuits, where periodic fluctuations can affect performance. Similarly, in ecological models, it provides insight into population dynamics, helping to understand how species interact and change over time. This makes oscillation theory a crucial tool for both theoretical analysis and practical applications in multiple disciplines; see ^[1,2]. In this work, we focus our attention on the oscillation of the second-order nonlinear differential equation with the form

where the first and second terms mean the $p-$Laplace type operator $(1 < p < \infty)$ and $u\left(s\right) = w\left(s\right) +b_{1}\left(s\right) w\left(a_{1}\left(s\right) \right) +b_{2}\left(s\right) w\left(a_{2}\left(s\right) \right)$. Throughout, we assume that:

$(G_{1})$ $b_{1}, b_{2}, q\in C\left(\left[ s_{0}, \infty \right), \left[ 0, \infty \right) \right)$, $f\in C\left(\left[ s_{0}, \infty \right), \left(0, \infty \right) \right)$, $\eta \left(s_{0}\right) = \int_{s_{0}}^{\infty }f^{-1/\left(p-1\right) }\left(\upsilon \right) \mathrm{d}\upsilon < \infty.$

$(G_{2})$ $a_{1}, a_{2}, h\in C\left(\left[ s_{0}, \infty \right), \mathbb{R} \right), a_{1}\left(s\right)\leq s, a_{2}\left(s\right) \geq s, h\left(s\right) \leq s, \lim_{s\rightarrow \infty }a_{1}\left(s\right) = \lim_{s\rightarrow \infty }a_{2}\left(s\right) = \lim_{s\rightarrow \infty }h\left(s\right) = \infty \ $ and $q\left(s\right)$ is not identically zero for large $s.$

By a solution of (1.1), we mean a nontrivial function $w\left(s\right) \in C^{1}\left(\left[ s_{0}, \infty \right), \mathbb{R} \right)$, which has the properties $f\left(s\right) \left(u^{\prime }\left(s\right) \right) ^{\left(p-1\right) }\in C^{1}\left(\left[ s_{0}, \infty \right), \mathbb{R} \right)$, and $w\left(s\right)$ satisfies (1.1) on $[s_{w}, \infty)$, then u is a solution of (1.1). In our study, we focus on the solutions that satisfy $\sup \left\{ \left\vert w\left(s\right) \right\vert :s_{w}\leq s < \infty \right\} > 0$, for every $s_{w}\geq s_{0}$. If the solution to (1.1) is neither eventually positive nor eventually negative in the end, it is said to be oscillatory. Otherwise, this solution is called non-oscillatory. Therefore, we say that this equation is oscillatory if all of its solutions are oscillatory.

The $p$-Laplace equations of the oscillatory behavior of differential delay equation issues have advanced significantly over the past few decades; see ^[3,4]. It has proven to be quite helpful in the engineering and applied sciences fields in building a range of applied mathematical models that faithfully represent real-world events. neutral-type DDEs are involved in second-order equation delay systems issues, which were studied by some authors; see ^[5,6,7].

The theory of oscillatory conditions generally appears from a large adequate instance for use in drug organization, physics, engineering, trash hold hypothesis in biology, and airship control; see ^[8,9]. The impulsive conditions play an important role in modeling phenomena, principally in relating the dynamics of the population area under discussion to unexpected modification. Exploring Laplace differential equations yields a multitude of essential utilities in mechanical systems, electrical circuits, and the regulation of chemical processes; see ^[10,11,12]. Furthermore, their utility extends to ecological systems, epidemiology, and the modeling of population dynamics, as detailed in references ^{[13,14,15,16]}.

Investigations by some authors in ^[16,17,18] have yielded techniques and methodologies aimed at enhancing the oscillatory attributes of these equations. Furthermore, the work ^[19,20] has expanded this inquiry to include differential equations of neutral variety. In recent years, there has also been a significant exploration of oscillation behaviors in fourth-order delay differential equations with the $p-$ Laplace type operator, as evidenced by studies such as ^[21,22,23]. Now, in some detail, Liu et al. ^[24] introduced good conditions concerning solutions to even-order differential equations that feature a mixed term in the canonical case.

where

Graef et al. ^[7] improved some results for equations with mixed types.

where

The authors in ^[21,25] established new conditions to improve and extend some of the results for the oscillation of the equation of fourth-order with the $p-$ Laplace type operator

where $\int_{s_{0}}^{\infty }f^{-1/\left(p-1\right) }\left(\upsilon \right) \mathrm{d}\upsilon = \infty.$

This manuscript aims to broaden the scope of inquiry by incorporating second-order differential equations with mixed neutral terms under condition $\eta \left(s_{0}\right) < \infty,$ using some methods. In this context, the paper introduces innovative criteria for analyzing oscillatory solutions of equation (1.1). Three examples are shown to illustrate the results that we obtained.

2.   Basic lemmas

In this section, we first introduce some important lemmas and notation.

Lemma 2.1. ^[11] The equation

holds, if $\eta \left(\upsilon \right) : = Q\upsilon -N\left(\upsilon -L\right) ^{p/\left(p-1\right) }$, where $Q, N$, and $L$ are real constants.

Lemma 2.2. ^[24] Let $F$ and $\varpi$ be constants. Then

The following notation will be used in the remaining sections of this work:

and

3.   Main results

In this section, we discuss our main results about (1.1).

Lemma 3.1. The function $w$ is identified as the ultimate positive solution to (1.1). Then, either

or

Proof. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1). Obviously, for all $s\geq s_{1}$, $u\left(s\right) \geq w\left(s\right) > 0$ and $f\left(s\right) \left(u^{\prime }\left(s\right) \right) ^{\left(p-1\right) }$ it is not increasing. Since

therefore, $u^{\prime }$ is either eventually negative or eventually positive.

Assume first that $u^{\prime } < 0$ on $\left[ s, \infty \right).$ Since

and so

Assume now that $u^{\prime } > 0$ on $\left[ s_{1}, s\right)$, we obtain

it follows that

Thus, the proof is complete. □

Lemma 3.2. The function $w$ is identified as the ultimate positive solution to (1.1) and $u^{\prime }\left(s\right) < 0$. Then

Proof. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1). Suppose that $u^{\prime }\left(s\right) < 0$ on $\left[ s_{1}, \infty \right)$. From Lemma 3.1, we have

based on the fact that $a_{1}\left(s\right) \leq s$. Therefore,

Hence, (1.1) becomes

Since $\left(f\left(s\right) \left(u^{\prime }\left(s\right) \right) ^{\left(p-1\right) }\right) ^{\prime }\leq 0$, we have

for all $s\geq s_{1},$ from (3.3) and (3.6), we have

Combining (3.5) with (3.7) yields

for all $s\geq s_{1}$. This completes the proof. □

Theorem 3.3. Let $x_{2}\left(s\right) \geq x_{1}\left(s\right) > 0.$ If

then, (1.1) is oscillatory.

Proof. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1). Then $w\left(a_{1}\left(s\right) \right)$, $w\left(a_{2}\left(s\right) \right)$, and $w\left(h\left(s\right) \right)$ are positive. From (1.1) and $u\left(s\right) = w\left(s\right) +b_{1}\left(s\right) w\left(a_{1}\left(s\right) \right) +b_{2}\left(s\right) w\left(a_{2}\left(s\right) \right)$, we see that $u\left(s\right) \geq w\left(s\right) > 0$ and $f\left(s\right) \left(u^{\prime }\left(s\right) \right) ^{\left(p-1\right) }$ do not increase. Therefore, $u^{\prime }$ is either eventually negative or eventually positive.

Suppose first that $u^{\prime }\left(s\right) > 0$ on $\left[ s_{1}, \infty \right).\ $From Lemma 3.2 and integrating (3.4) from $s_{1}$ to $s$, we obtain

Integrating the last inequality from $s_{1}$ to $s$, we obtain

At $s\rightarrow \infty$, we arrive at a contradiction with (3.8).

Assume now that $u^{\prime }\left(s\right) < 0$ on $\left[ s_{1}, \infty \right).$ From Lemma 3.4 and integrating (3.14) from $s_{2}$ to $s$, we obtain

Since $x_{2}\left(s\right) > x_{1}\left(s\right)$, we obtain

we have shown that, according to Eq (3.10), the positivity of $u^{\prime }\left(s\right)$ as $s\rightarrow \infty$ is contradicted. Therefore, the proof is finished. □

Lemma 3.4. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1) and $u^{\prime }\left(s\right) > 0$. Then

Proof. The function $w$ is identified as the ultimate positive solution to (1.1). Suppose that $u^{\prime }\left(s\right) > 0$ on $\left[ s_{1}, \infty \right)$. From Lemma 3.1, we arrive at

From the definition of $u$, we obtain

Using that (3.11) and $u\left(a_{1}\left(s\right) \right) \leq u\left(s\right)$ where $a_{1}\left(s\right) < s$ in (3.12), we obtain

Thus, (1.1) becomes

On the other hand, it is follows from (3.8) that $\int_{s_{1}}^{s}M^{\left(p-1\right) }\left(\upsilon, \infty \right) q\left(\upsilon \right) x_{1}^{\left(p-1\right) }\left(h\left(\upsilon \right) \right) \mathrm{d}\upsilon$ must be unbounded. Further, since $M^{\prime }\left(s, \infty \right) < 0$, it is easy to see that

This completes the proof. □

Lemma 3.5. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1) and $u^{\prime }\left(s\right) < 0$. If

Then

where $\xi \in C^{1}\left(\left[ s_{0}, \infty \right), \left(0, \infty \right) \right).$

Proof. The function $w$ is identified as the ultimate positive solution to (1.1). Suppose that $u^{\prime }\left(s\right) < 0$ on $\left[ s_{1}, \infty \right)$, we can notice that $E\geq 0$ on $\left[ s_{1}, \infty \right).$ By computing the derivative of (3.15), we can conclude at

Combining (3.5) and (3.17), we have

Using Lemma 2.1 with $Q: = \xi ^{\prime }\left(s\right) /\xi \left(s\right),$ $N: = \left(p-1\right) \left(\xi \left(s\right) f\left(s\right) \right) ^{-1/\left(p-1\right) },$ $L: = \xi \left(s\right) /M^{\left(p-1\right) }\left(s, \infty \right)$ and $\upsilon : = E$, we get

which, in view of (3.18), we have

This completes the proof. □

Lemma 3.6. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1) and $u^{\prime }\left(s\right) > 0$. If

then

where $\varsigma \in C^{1}\left(\left[ s_{0}, \infty \right), \left(0, \infty \right) \right).$

Proof. The function $w$ is identified as the ultimate positive solution to (1.1). Suppose that $u^{\prime }\left(s\right) > 0$ on $\left[ s_{1}, \infty \right)$. From (3.19), we see that $D\geq 0$ on $\left[ s_{1}, \infty \right).$ Differentiating (3.19), we arrive at

Combining (3.14) and (3.21), we have

Since $\left(f\left(s\right) \left(u^{\prime }\left(s\right) \right) ^{\left(p-1\right) }\right) ^{\prime } < 0$ and $h\left(s\right) \leq s,$ we arrive at

This completes the proof. □

Now, we discuss a numerical example to highlight the significance of the conditions we obtained in Theorem 3.3.

Example 3.7. Consider the neutral equation

Let $p = 2, \; f\left(s\right) = s^{2}, \; b_{1}\left(s\right) = 1/3, \; b_{2}\left(s\right) = 1/2, \; a_{1}\left(s\right) = s/2, \; a_{2}\left(s\right) = 3s, h\left(s\right) = s/3$, $q\left(s\right) = t_{0}/s$ and

Moreover, we find

From Theorem 3.3, the Eq (3.22) is oscillatory.

Theorem 3.8. Assume that $x_{2}\left(s\right) \geq x_{1}\left(s\right) > 0$. If

then, every solutions of (1.1) is oscillatory.

Proof. The function $w$ is identified as the ultimate positive solution to (1.1). Then there exists $s_{1}\geq s_{0}$ such that $w\left(a_{1}\left(s\right) \right) > 0$, $w\left(a_{2}\left(s\right) \right) > 0$, and $w\left(h\left(s\right) \right) > 0$ for all $s\geq s_{1}.$ The same way we prove Theorem 3.3, the sign of $u^{\prime }$ becomes consistently positive or negative eventually. First, let $u^{\prime } < 0$. Integrating (3.5) from $s_{1}$ to $s$, we see that

Using (3.3) in (3.24), we obtain

Divide both sides of inequality (3.25) by $-f\left(s\right) \left(u^{\prime }\left(s\right) \right) ^{\left(p-1\right) }$ and taking the limsup, we obtain

we arrive at a contradiction with (3.23).

Let $u^{\prime } > 0$ on $\left[ s_{1}, \infty \right).$ From (3.23) and $M\left(s, \infty \right) < \infty$, we have that (3.10) holds. We notice that this part of the proof is exactly like the part of Theorem 3.3, so the proof is complete. □

Theorem 3.9. If $x_{2}\left(s\right) > 0,$ $x_{1}\left(s\right) > 0$, and $f^{\prime } > 0$ such that

and

where the functions $\varsigma, \xi \in C^{1}\left(\left[ s_{0}, \infty \right), \left(0, \infty \right) \right)$ and $s\in \left[ s_{0}, \infty \right).\ $ Then, (1.1) is oscillatory.

Proof. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1) on $\left[ s_{0}, \infty \right)$. Then $w\left(a_{1}\left(s\right) \right) > 0$, $w\left(a_{2}\left(s\right) \right) > 0$, and $w\left(h\left(s\right) \right) > 0$ for all $s\geq s_{1}.$ From Theorem 3.3, it follows that $u^{\prime }$ eventually takes on a consistent sign, either remaining negative or remaining positive, beyond a certain point. First, assuming that $u^{\prime } < 0$ on $\left[ s_{1}, \infty \right).$ By following the approach used in the proof of Theorem 3.3, we can obtain that $u$ is a solution of the inequality (3.5). From Lemma 3.5 and due to (3.3). Integrating (3.16) from $s_{2}$ to $s,$ we arrive at

From (3.3), we have

which, in view of (3.28), implies

Applying the limit superior to both sides, we are led to a contradiction with (3.26).

Now, let $u^{\prime }\left(s\right) > 0$ on $\left[ s_{1}, \infty \right).$ From Lemma 3.5 and from (3.19) and (3.20), we have

Using Lemma 2.1, with $F = \varsigma ^{\prime }\left(s\right) /\varsigma \left(s\right),$ $\varpi = \left(p-1\right) h^{\prime }\left(s\right) /\left(\varsigma ^{1/\left(p-1\right) }\left(s\right) f^{1/\left(p-1\right) }\left(s\right) \right)$ and $v = D,$ we have

Integrating (3.29) from $s_{2}$ to $s,$ we arrive at

We are taking lim sup on both sides of this inequality; we find a contradiction with (3.27), and therefore the proof is finished. □

Now, we include the following numerical example to highlight the importance of the criteria we obtained in Theorem 3.9.

Example 3.10. Let the equation

where $\xi > 1, s\geq 1$ and $\sigma > 1$. Let $p = 2, \; f\left(s\right) = s^{2}, \; b_{1}\left(s\right) = 1/3, \; b_{2}\left(s\right) = 1/3, \; a_{1}\left(s\right) = s/3, \; a_{2}\left(s\right) = \sigma s, h\left(s\right) = s/2$, and $q\left(s\right) = \xi s$. So, we find

If we set $\varsigma \left(\upsilon \right) = 1$, we obtain

By Theorem 3.9, every solution of (3.30) is oscillatory.

4.   Conclusions

In this paper, we investigate the oscillatory criteria of a non-canonical second-order differential equation featuring mixed neutral terms, driven by a $p$-Laplace differential operator. Our approach begins with transforming the equation into a canonical form, which facilitates the application of the Riccati technique to derive new oscillatory properties. The results we present not only extend but also simplify existing criteria found in the literature.

To demonstrate the practical applicability of our findings, we include several illustrative examples that highlight the effectiveness of our main results.

Looking to the future, our aim is to explore the oscillatory characteristics inherent in third-order differential equations with neutral terms, as well as to investigate fractional-order equations governed by a $p$-Laplace differential operator. We are already making progress in the study of these specific types of equations, which promise to enhance our understanding of oscillatory behavior in various contexts.

Use of Generative-AI tools declaration

The author declares that the Artificial Intelligence (AI) tools were not used in the creation of this article.

Acknowledgments

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

Conflict of interest

The author declares that there are no conflicts of interest.