On The oscillatory behavior of solutions to a class of second-order nonlinear differential equations

1.   Introduction

The study of the oscillatory behavior of delays in differential equations has advanced significantly during the last few decades; see ^[1,2,3]. It has been shown to be quite helpful in the engineering and applied sciences fields to build a range of applied mathematical models that faithfully represent real-world events, see ^[4,5,6]. DDEs of neutral type are involved in the delay system of second-order equations issues; which were studied by some authors; see ^[7,8].

The theory of oscillatory conditions generally appears from a large adequate instance of drug organization, physics, engineering, the trash hold hypothesis in biology, and air ship control. Some studies have studied the numerical aspects and computational methods of some types of equations; see ^[2,3]. Impulsive conditions play an important role in modeling phenomena, principally in relating the dynamics of the population's area under discussion to unexpected modification.

In this work, we focus our attention on the oscillation of the second-order nonlinear differential equation with the form

where $\delta_{1}, \delta_{2}, q\in C\left(\left[ s_{0}, \infty \right), \left[ 0, \infty \right) \right), f\in C\left(\left[ s_{0}, \infty \right), \left(0, \infty \right) \right),$ and $\eta \left(s_{0}\right) = \int_{s_{0}}^{\infty}f^{-1/\mu }\left(\upsilon \right) \mathrm{d}\upsilon < \infty.$ Moreover, $\mu$ is a quotient of odd positive integers, $\sigma _{1}, \sigma _{2}, h\in C\left(\left[s_{0}, \infty \right), \mathbb{R} \right), \sigma _{1}\left(s\right)\leq s, \sigma _{2}\left(s\right) \geq s, h\left(s\right) \leq s, \lim_{s\rightarrow \infty }\sigma _{1}\left(s\right) = \lim_{s\rightarrow \infty }\sigma _{2}\left(s\right) = \lim_{s\rightarrow \infty}h\left(s\right) = \infty$ 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 $f\left(s\right) \left(y^{\prime }\left(s\right) \right) ^{\mu }\in C^{1}\left(\left[ s_{0}, \infty \right), \mathbb{R} \right)$, and $w\left(s\right)$ satisfies (1.1) on $[s_{w}, \infty)$, then y 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 positive in the end nor 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 it has oscillatory solutions.

Exploring Laplace differential equations yields a multitude of essential utilities in mechanical systems, electrical circuits, and the regulation of chemical processes. In addition, their utility extends to ecological systems, epidemiology, and the modeling of population dynamics.

Investigations by some authors in ^[9,10,11] have yielded techniques and methodologies aimed at enhancing the oscillatory behavior of solutions for differential equations. For a recent review of the main results in the framework of asymptotic properties for differential equations of different order with neutral term, we refer the reader to ^[12,13,14]. Furthermore, the work ^{[15,16,17,18]} 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, as evidenced by studies such as ^{[19,20,21,22]}. Now, in some detail, Graef et al. ^[5] improved some results for mixed-type equations.

where

Li et al. ^[20,23] introduced good conditions concerning solutions to second-order differential equations featuring a mixed term in the canonical case.

where

Thandapani et al. ^[21] and Thandapani and Rama ^[22] established new conditions to improve and extend some of the results for the oscillation of the equation of second-order

where $y\left(s\right) = w\left(s\right) +\delta_{1}\left(s\right) w\left(s-h_{1}\right) +\delta_{2}\left(s\right) w\left(s+h_{2}\right).$

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 Eq (1.1). Three examples are shown to illustrate the results that we obtained.

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

and

2.   Main results

In this section, we first introduce some important lemmas and then discuss our main results about (1.1).

Lemma 2.1. ^[23] The equation

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

Lemma 2.2. Assume that $w$ eventually is a positive solution of Eq (1.1). Then either

or

Proof. Suppose that $w$ eventually is a positive solution of Eq (1.1). Obviously, for all $s\geq s_{1}$, $y\left(s\right) \geq w\left(s\right) > 0$ and $f\left(s\right) \left(y^{\prime }\left(s\right) \right) ^{\mu }$ is decreasing. Since

Then, $y^{\prime }$ is either eventually negative or eventually positive.

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

and so

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

it follows that

Thus, the proof is complete. □

Theorem 2.1. 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 $w\left(\sigma _{1}\left(s\right) \right)$, $w\left(\sigma _{2}\left(s\right) \right)$ and $w\left(h\left(s\right) \right)$ are positive. From (1.1) and $y\left(s\right) = w\left(s\right) +\delta_{1}\left(s\right) w\left(\sigma _{1}\left(s\right) \right) +\delta_{2}\left(s\right) w\left(\sigma _{2}\left(s\right) \right)$, we see $y\left(s\right) \geq w\left(s\right) > 0$ and $f\left(s\right) \left(y^{\prime }\left(s\right) \right) ^{\mu }$ is nonincreasing. Therefore, $y^{\prime }$ is either eventually negative or eventually positive.

Suppose first that $y^{\prime }\left(s\right) < 0$ on $\left[ s_{1}, \infty \right)$. From Lemma 2.2, we have

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

Hence, (1.1) becomes

Since $\left(f\left(s\right) \left(y^{\prime }\left(s\right) \right) ^{\mu }\right) ^{\prime }\leq 0$, we have

for all $s\geq s_{1},$ from (2.3) and (2.6), we have

Combining (2.5) with (2.7) yields

for all $s\geq s_{1}$. Integrating (2.8) from $s_{1}$ to $s$, we obtain

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

At $s\rightarrow \infty$, we arrive at a contradiction with (2.4).

Assume now that $y^{\prime }\left(s\right) > 0$ on $\left[ s_{1}, \infty \right).$ From Lemma 2.2, we arrive at

From the definition of $y$, we obtain

Using that (2.9) and $y\left(\sigma _{1}\left(s\right) \right) \leq y\left(s\right)$ where $\sigma _{1}\left(s\right) < s$ in (2.10), we obtain

Thus, (1.1) becomes

On the other hand, it is follows from (2.4) that $\int_{s_{1}}^{s}M^{\mu }\left(\upsilon, \infty \right) q\left(\upsilon \right) x_{1}^{\mu }\left(h\left(\upsilon \right) \right) \mathrm{d} \upsilon$ must be unbounded. Further, since $M^{\prime }\left(s, \infty \right) < 0$, it is easy to see that

Integrating (2.12) from $s_{2}$ to $s$, we get

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

we have shown that, according to Eq (2.13), the positivity of $y^{\prime }\left(s\right)$ as $s\rightarrow \infty$ is contradicted. Therefore, the proof is complete. □

Theorem 2.2. Assume that $x_{2}\left(s\right) \geq x_{1}\left(s\right) > 0$. If

then every solutions of (1.1) are 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 $w\left(\sigma _{1}\left(s\right) \right) > 0$, $w\left(\sigma _{2}\left(s\right) \right) > 0$ and $w\left(h\left(s\right) \right) > 0$ for all $s\geq s_{1}.$ The same way we prove the Theorem 2.1, the sign of $y^{\prime }$ becomes consistently positive or negative eventually. First, let $y^{\prime } < 0$. Integrating (2.5) from $s_{1}$ to $s$, we see that

Using (2.3) in (2.16), we obtain

Divide both sides of inequality (2.17) by $-f\left(s\right) \left(y^{\prime }\left(s\right) \right) ^{\mu }$ and taking the limsup, we get

we arrive at a contradiction with (2.15).

Let $y^{\prime } > 0$ on $\left[ s_{1}, \infty \right).$ From (2.15) and $M\left(s, \infty \right) < \infty$, we have that (2.13) holds. We notice that this part of the proof is exactly like the part of Theorem 2.1, so the proof is complete. □

Theorem 2.3. 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 $s\in \left[ s_{0}, \infty \right).$

Proof. Given that $w\left(s\right)$ constitutes the final positive solution of (1.1) on $\left[ s_{0}, \infty \right)$. Then $w\left(\sigma _{1}\left(s\right) \right) > 0$, $w\left(\sigma _{2}\left(s\right) \right) > 0$ and $w\left(h\left(s\right) \right) > 0$ for all $s\geq s_{1}.$ From Theorem 2.1, yields that $y^{\prime }$ is eventually takes on a consistent sign, either remaining negative or remaining positive, beyond a certain point. First, assuming that $y^{\prime } < 0$ on $\left[ s_{1}, \infty \right).$ By following the approach used in the proof of the Theorem 2.1, we can obtain that $y$ is a solution of the inequality (2.5). We now define the Riccati function as follows:

Due to (2.3), we can notice that $E\geq 0$ on $\left[ s_{1}, \infty \right).$ By Computing the derivative of (2.20), we can conclude at

Combining (2.5) and (2.21), we have

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

which, in view of (2.22), we have

Integrating (2.23) from $s_{2}$ to $s,$ we arrive at

From (2.3), we have

which, in view of (2.24), implies

Applying the limit superior to both sides, we are led to a contradiction with (2.18).

Now, let $y^{\prime }\left(s\right) > 0$ on $\left[ s_{1}, \infty \right).$ Let the function

we see that $E\geq 0$ on $\left[ s_{1}, \infty \right).$ Differentiating (2.25), we arrive at

Combining (2.12) and (2.26), we have

Since $\left(f\left(s\right) \left(y^{\prime }\left(s\right) \right) ^{\mu }\right) ^{\prime } < 0$ and $h\left(s\right) \leq s,$ we arrive at

from (2.25), we have

Using the inequality

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

Integrating (2.28) from $s_{2}$ to $s,$ we arrive at

We are taking lim sup on both sides of this inequality, we find a contradiction with (2.19), and therefore the proof is finished. □

3.   Numerical examples

Some examples are provided to demonstrate the significance of our results.

Example 3.1. Consider the neutral equation

Let $\mu = 1, \; f\left(s\right) = s^{2}, \; \delta_{1}\left(s\right) = 1/3, \; \delta_{2}\left(s\right) = 1/2, \; \sigma _{1}\left(s\right) = s/2, \; \sigma _{2}\left(s\right) = 3s, h\left(s\right) = s/3$, $q\left(s\right) = t_{0}/s$ and

Moreover, we find

From Theorem 2.1, Eq (3.1) oscillatory.

Example 3.2. Let the equation

where $\xi > 1, s\geq 1, \mu \in \left(0, 1\right)$ and $\sigma > 1$. Let $\mu = 1, \; f\left(s\right) = s^{2}, \; \delta_{1}\left(s\right) = 1/3, \; \delta_{2}\left(s\right) = 1/3, \; \sigma _{1}\left(s\right) = s/3, \; \sigma _{2}\left(s\right) = \sigma s, h\left(s\right) = s/2$ and $q\left(s\right) = \xi s$. So, we find

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

By Theorem 2.3, every solution of (3.2) is oscillatory.

4.   Conclusions

The aim of this paper is to investigate the oscillatory characteristics inherent in second-order differential equations featuring a mixed neutral term. This investigation is conducted through the application of Riccati transformations and a comparison analysis with first-order equations, ultimately leading to the derivation of oscillation criteria. The study culminates in the establishment of a central theorem related to the oscillation behavior of (1.1). In addition, three examples of the effectiveness of these criteria were discussed. As part of our future research, we will study some of the oscillatory characteristics inherent in third-order DE with neutral terms, as well as to delve into fractional-order equations. Progress is already underway in investigating these particular equation types.

Use of Generative-AI tools declaration

The author declares that no Artificial Intelligence (AI) tools were 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-2024).

Conflict of interest

The author declares that there is no conflicts of interest.