Some new oscillation results for second-order differential equations with neutral term

1.   Introduction

In this paper, we will study the oscillation behavior of the following differential equations (DEs):

We assume the following:

$(I)$ $\mathcal{P}\left(s\right) = x\left(s\right) +\mu \left(s\right) x\left(\alpha \left(s\right) \right),$ $\mu \in C\left(\left[s_{0}, \infty \right), \left[0, 1\right) \right)$, $\alpha \in C^{1}\left([s_{0}, \infty), \left(0, \infty \right) \right)$, $\alpha \left(s\right) \leq s,$ $\alpha ^{\prime }\left(s\right) > 0$ and $\lim_{s\rightarrow \infty }\alpha \left(s\right) = \infty$;

$(II)$ $\beta \in C^{1}\left([s_{0}, \infty), \left(0, \infty \right) \right),$ $\beta \left(s\right) \leq s,$ $\beta ^{\prime }\left(s\right) > 0,$ and $\lim_{s\rightarrow \infty }\beta \left(s\right) = \infty$ and

$(III)$ $q\in C^{1}\left(\left[s_{0}, \infty \right) \right)$ and $q\left(s\right) > 0$.

By a solution of (1.1), we mean a function $x$ $\in C[s_{u}, \infty), \ $ with $s_{u}: = \min \left\{ \beta \left(s_{b}\right), \alpha \left(s_{b}\right) \right\}$ for some $s_{b}\geq s_{0},$ which has the property $\mathcal{P}^{\prime }\left(s\right) \in C^{1}\left([s_{u}, \infty), \mathbb{R} \right), \ $and satisfies (1.1) on $[s_{b}, \infty)$. We only consider those nontrivial solutions $x$ of (1.1), which are defined on some half-line $[s_{b}, \infty).$ As is customary, a solution $x$ of (1.1) is said to be oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. The equation itself is called oscillatory if all its solutions oscillate.

Recently, there has been a lot of interest in the oscillatory properties of solutions to diverse types of functional DEs. The oscillation of neutral DEs was a special focus of several authors. We direct the reader to the articles ^[1,2,3], monographs ^[4,5,6], and the references listed within. The fact that neutral DEs appear in several practical issues in the natural sciences, control, and engineering ^[7] explains one of the primary causes of this interest.

For second-order equations, Agarwal et al. ^[8], Kusano et al. ^[9], Sun and Meng ^[10], and Dzurina and Stavroulakis ^[11] derived oscillation criteria for the DE

where $\lambda \left(s\right) \in C^{1}\left([s_{0}, \infty), \mathbb{R} \right),$ $\lambda \left(s\right) > 0,$ and $\psi > 0$. Baculikova ^[12] established some oscillatory properties of the DE

Their results complement and improve on the results of ^[13,14].

We briefly discuss the relevant findings that inspired our work in the remaining portion of this section.

Some oscillation criteria for the functional DE of neutral type

where $\lambda \left(s\right) \in C^{1}\left([s_{0}, \infty), \mathbb{R} \right),$ $\lambda \left(s\right) > 0,$ $\lambda ^{\prime }\left(s\right) \geq 0,$ $\psi > 0$, and $\phi > 0$ are established by Wu et al. ^[15]. They proved that the DE (1.2) is oscillatory if

where $\kappa = \min \left\{ \psi, \phi \right\},$

and

Using the Riccati substitution technique and comparing with first-order delay equations, Moaaz ^[16] derived oscillation criteria for the DE

where $\left\vert f\left(s, x\right) \right\vert \geq q\left(s\right) \left\vert x\right\vert ^{\phi },$ $\psi$ and $\phi$ are quotients of odd positive integers, $\lambda \left(s\right) \in C\left([s_{0}, \infty), \mathbb{R} \right),$ $\lambda \left(s\right) > 0$, and $\int^{\infty }\lambda ^{-1/\psi }\left(\varsigma \right) \mathrm{d}\varsigma = \infty.$ They proved that the DE (1.4) is oscillatory if $\psi \geq \phi$ and

where

and $C$ is a positive real constant.

Comparing the DE with either first-order delay DEs or inequalities, whose oscillatory behavior is known beforehand, is one of the fundamental methods in the oscillation theory for acquiring the criteria. Using the comparison method, this study aims to develop some oscillation criteria for the (1.1) solutions. The results obtained here improve some of the established results in the literature.

2.   Main results

We begin by mentioning an important lemma that we will need to prove our results.

Lemma 2.1. ^[17] Let $x$ be a positive solution of (1.1). Then,

for all sufficiently large $s$.

Theorem 2.1. If

and

then (1.1) is oscillatory.

Proof. Let (1.1) have a positive solution. It is clear that

Integrating (2.4) from $s_{0}$ to $\infty,$ we obtain

where $k = \mathcal{P}\left(s_{0}\right) -s_{0}\mathcal{P}^{\prime }\left(s_{0}\right)$. Since $\mathcal{P}\left(s\right) = x\left(s\right) +\mu \left(s\right) x\left(\alpha \left(s\right) \right)$, we obtain

Substituting (2.6) in (2.5), we find

since $\mathcal{P}^{\prime }\left(s\right) > 0$, and using (2.2), we see that

Hence

which also implies that $k > 0$ for a large enough $s_{0}$.

Therefore, we expect

If $\lim_{s\rightarrow \infty }\mathcal{P}\left(s\right) /s = n > 0.$ From (2.7), we get

Integrating (2.11) from $s_{0}$ to $\infty,$ we obtain

which contradicts (2.2); therefore, we find that $n = 0$.

Now, from (2.11), we find

Since $z^{\prime }\left(s\right) < 0$ and $\lim_{s\rightarrow \infty }z\left(s\right) = 0$, we find that

has a positive solution $z\left(s\right),$ which contradicts (2.3). End of proof. □

Lemma 2.2. Let (2.2) hold. Then

where

Proof. Let (1.1) have a positive solution. From (2.12), we get

Therefore,

Using (2.14), we see that

End of proof. □

Theorem 2.2. Let (2.2) hold. If

then (1.1) is oscillatory, where $\psi _{\varrho }\left(s\right)$ is defined as in Lemma 2.2.

Proof. Let (1.1) have a positive solution. From (2.11), we find

Using (2.13), we have

which a positive solution $z\left(s\right),$ and contradicts (2.15). End of proof. □

Corollary 2.1. Let (2.2) hold and $\lim_{s\rightarrow \infty }\psi _{\varrho }\left(\beta \left(s\right) \right) /s = 0$. If

then (1.1) is oscillatory, where $\psi _{\varrho }\left(s\right)$ is defined as in Lemma 2.2.

Proof. This is similar to the proof of Theorem 2.1, and thus we omit it. □

Lemma 2.3. Let (2.2) hold. Then

where

and

Proof. Let (1.1) have a positive solution. From (1.1) and (2.6), we obtain

Differentiating (2.19), we find

Since $\mathcal{P}\left(s\right) \geq x\left(s\right),$ we see that (2.20) becomes

Integrating (2.19) from $\beta \left(s\right)$ to $\infty,$ we have

where we used (2.1) and (2.9). Since $\mathcal{P}\left(s\right) \geq x\left(s\right),$ we see that (2.22) becomes

By using (2.21) and (2.23), we obtain

From (1.1) and the above inequality, we see that

Consequently,

hence, $\mathrm{e}^{-\phi _{2}\left(s\right) }\mathcal{P}^{\prime \prime }\left(s\right)$ is decreasing. In addition, we see that

End of proof. □

Lemma 2.4. Let (2.2) hold. Then,

Proof. The proof is similar to that of [18, Lemma 3.2]. Therefore, it has been omitted. □

Theorem 2.3. Let (2.2) hold. If

then (1.1) is oscillatory, where $\psi _{\varrho }\left(s\right)$ and $\psi _{2}\left(s\right)$ are defined as in Lemmas 2.2 and 2.3, respectively.

Proof. Let (1.1) have a positive solution. From (2.24), we have

Using (2.6) in (2.26), we get

Since $\beta \left(s\right) \leq s$, $\beta ^{\prime }\left(s\right) > 0$, and using (2.18) and (2.13), we find

Using (2.6), we get

which contradicts (2.26). End of proof. □

Example 2.1. Consider the following equation:

where $q\left(s\right) = q_{0}/s^{2},$ $q_{0} > 0,$ $\mu \left(s\right) = \mu _{0},$ $\mu _{0}\in \left[0, 1\right),$ $\beta \left(s\right) = \gamma s,$ and $\alpha \left(s\right) = \delta s$. Now, we see that the condition (2.2) is satisfied,

and

Thus, we find $\lim_{s\rightarrow \infty }\psi _{\varrho }\left(\beta \left(s\right) \right) /s = 0.$

From Corollary 2.1, we find that (2.27) is oscillatory if

From Theorem 2.3, we find that (2.27) is oscillatory if

Figure 1 shows some numerical oscillatory solutions to (2.27).

Remark 2.1. Let

be a special case of the Eq (2.27), where $q_{0} = \left(1.05\right),$ $\mu _{0} = 1/10,$ $\delta = 1/2,$ and $\gamma = 1/5.$

Let $\rho \left(s\right) = s.$\ By applying Theorem 1 in ^[15], we find that the condition (1.3) is not satisfied, where

Thus, Theorem 1 in ^[15] fails to study the oscillation of Eq (2.30).

By applying Corollary 2.2 in ^[16], we find that the condition (1.5) is not satisfied, where

Thus, Corollary 2.2 in ^[16] fails to study the oscillation of Eq (2.30).

Now, we see that the condition (2.28) becomes

By using Corollary 2.1, we find that (2.30) is oscillatory.

Additionally, condition (2.29) becomes

By using Theorem 2.3, we find that (2.30) is oscillatory.

Hence, and through the above, we find that the criteria we obtained produces results for the oscillation of Eq (2.30), while previous studies failed to study the oscillation of Eq (2.30).

3.   Conclusions

In this paper, the oscillatory behavior of (1.1) was studied. We succeeded in establishing new monotonic properties for the positive solutions of (1.1); from them, we obtained new oscillation criteria for (1.1). In addition, we provided an example and compared the results we obtained with some previous studies to show that the results we obtained improved these studies. As future work, we will try to extend our proposed results to third-order DEs.

Author contributions

Abdullah Mohammed Alomair and Ali Muhib: Conceptualization, Methodology, Validation, Writing-original draft, Writing-review & editing. The authors contributed equally to this work.

Use of Generative-AI tools declaration

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

Conflict of interest

The authors declare no conflicts of interest.