On the oscillation of fourth-order neutral differential equations with multiple delays

1.   Introduction

A key area of mathematics, differential equations (DEs) define the relationships between variables and the rates at which they change. As such, they are a vital tool for comprehending intricate biological, engineering, and physical models. Equations with complex nonlinear characteristics and higher orders are exciting in applied sciences and mathematical analysis. Due to their capacity to characterize intricate oscillatory behavior, fourth-order differential equations are especially significant in a wide range of scientific and engineering applications. Because they offer a comprehensive theoretical foundation for comprehending and managing intricate dynamic events, this class of equations continues to draw researchers (see ^[1,2]).

Neutral differential equations (NDEs) play an important role in the mathematical modeling of systems that depend on the values of present, past, or future variables. These equations are characterized by the existence of the highest-order derivative of the unknown function at the present time $\mathrm{s}$ and also at a later or future time $\mathrm{s} -\tau$. In addition to their theoretical importance, these equations have wide applications, such as the analysis of networks with lossless transmission lines, which are used in high-speed computers to connect switching circuits ^[3,4]. Therefore, the study of these equations is essential in many engineering and physical fields.

This paper examines fourth-order NDEs, which are denoted by the following form:

where

This study operates under the following hypotheses:

$(\mathcal{A}_{1})$ $\alpha,$ $\beta,$ $\gamma,$ and $\delta$ are quotients of positive odd integers, with $\gamma < 1$ and $\delta > 1$;

$(\mathcal{A}_{2})$ $\mathfrak{a}_{1}, \mathfrak{a}_{2}, \kappa _{i}\in C\left(\left[ \mathrm{s}_{0}, \infty \right), \left[ 0, \infty \right) \right),$ $i = 1, 2, ..., m$;

$(\mathcal{A}_{3})$ $\varsigma, \sigma _{i}\in C^{1}([\mathrm{s} _{0}, \infty), \mathbb{R})$ satisfies $\sigma _{i}(\mathrm{s})\leq \mathrm{s},$ $\varsigma (\mathrm{s})\leq \mathrm{s},$ $\sigma _{i}^{\prime }(\mathrm{s}) > 0$ and $\lim_{\mathrm{ s}\rightarrow \infty }\varsigma \left(\mathrm{s}\right) = \lim_{\mathrm{s} \rightarrow \infty }\sigma _{i}(\mathrm{s}) = \infty,$ $i = 1, 2, ..., m;$

$(\mathcal{A}_{4})$ $r\in C^{1}\left(\left[ \mathrm{s}_{0}, \infty \right), \left(0, \infty \right) \right)$ satisfies

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

A solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. Equation (1.1) is said to be oscillatory if all of its solutions are oscillatory.

The area of studying oscillatory processes in differential equations has a long history and is well-established, thanks to the work of early pioneers like Alexander Lyapunov and Henri Poincaré. A particular family of differential equations has attracted the interest of several scholars due to its oscillating nature (see ^[5,6,7]).

Numerous scholars have devoted their attention to comprehending the oscillatory nature of neutral differential equations in various orders. In the second order, using sophisticated methods for examining nonlinear neutral effects, Baculíková and Džurina ^[8], and Aldiaij et al. ^[9] presented significant results on the oscillation of delay equations. For the oscillation of equations with infinite neutral coefficients, Chatzarakis et al. ^[10] gave exact conditions at the third order. Using novel methodologies, Grace et al. ^[11] demonstrated sophisticated findings on the oscillation of delay equations at the fourth order. Graef et al. ^[12], Xing et al. ^[13], Alnafisah et al. ^[14], and Alqahtan et al. ^[15] expanded on the knowledge of the oscillation of delay neutral equations for higher orders.

Special instances of these equations have been the subject of the majority of research. The most well-known researchers in this area are Jadlowska et al. ^[16], who examined the linear delay equations

Using methods to enhance the special qualities of non-oscillatory solutions, they were able to offer accurate oscillation criteria for a linear differential equation with a delay.

Zhang et al. ^[17] studied the oscillatory properties of the quasi-linear delay differential equation

Their analyses focused on developing precise oscillation criteria through rigorous analytical and comparison techniques. These studies contributed significantly to the theory by clarifying how the delay and nonlinear structure affect the qualitative properties of solutions.

Subsequently, Kusano et al. ^[18] and Kamo et al. ^[19] considered a more general nonlinear form, namely

allowing for distinct exponents in the delayed term. Their work provided broader oscillation criteria under more flexible structural assumptions, relying on refined comparison techniques and inequalities to ensure that all solutions oscillate under specific parameter regimes.

Bazighifan and Cesarano ^[20] and Alatwi et al. ^[21] created specific criteria to assure oscillation in neutral nonlinear equations of the following form:

Masood et al. ^[22] studied nonlinear differential equations with a sublinear neutral term, given by the following form:

Based on prior research, the current study broadens the scope to include fourth-order neutral differential equations with mixed neutral terms, concentrating on the oscillatory behavior of all solutions. This inclusion of mixed neutral terms represents a significant extension of previous models. The complexity introduced by the higher-order nature of the equations, combined with neutral terms and multiple delays, reflects real-world dynamics found in applications such as engineering, biological systems, and control theory. Therefore, studying such equations is not only of theoretical interest but also of practical relevance. This research improves our present understanding and allows for the incorporation of new criteria that ensure the oscillations of these equations under various conditions.

2.   Preliminary results

Initially, we offer numerous helpful lemmas about the monotonic characteristics of the non-oscillatory solutions to the examined equations. In what follows, we let

and

Definition 2.1. ^[23] A function $\chi \left(\mathrm{s}\right)$ is said to be eventually positive (or eventually negative) if there exist $\mathrm{s} _{2}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$ such that $\chi \left(\mathrm{s} \right)$ is a solution on the interval $\left[ \mathrm{s}_{2}, \infty \right),$ and satisfies $\chi \left(\mathrm{s}\right) > 0$ (or $\chi \left(\mathrm{s}\right) < 0$) for all $\mathrm{s}\geq \mathrm{s}_{2}.$

Lemma 2.1. ^[24] Suppose that $y\in C^{n}\left([\mathrm{s} _{0}, \infty), \mathbb{R} ^{+}\right),$ $y^{\left(n\right) }\left(\mathrm{s}\right)$ is of fixed sign and not identically zero on $[\mathrm{s}_{0}, \infty)$ and that there exists $\mathrm{s}_{1}\geq \mathrm{s}_{0}$ such that $y^{\left(n-1\right) }\left(\mathrm{s}\right) y^{\left(n\right) }\left(\mathrm{s}\right) \leq 0$ for all $\mathrm{s}_{1}\geq \mathrm{s}_{0}.$ If $\lim_{\mathrm{s} \rightarrow \infty }y\left(\mathrm{s}\right) \neq 0,$ then, for every $\delta \in \left(0, 1\right),$ there exists $\mathrm{s}_{\epsilon }\in \lbrack \mathrm{s}_{1}, \infty)$ such that

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

Lemma 2.2. ^[25] If $A$ and $B$ are nonnegative, then

where the equality holds if and only if $A = B.$

Lemma 2.3. ^[26] Let $\alpha$ be a ratio of two odd positive integers; $A$ and $B$ are constants. Then

Lemma 2.4. ^[27] Let $y\in \mathbf{C}^{n}\left([\mathrm{s}_{0}, \infty), \left(0, \infty \right) \right)$, $y^{\left(i\right) }\left(\mathrm{s} \right) > 0$ for $i = 1, 2, ..., n$, and $y^{\left(n+1\right) }\left(\mathrm{s} \right) \leq 0$, eventually. Then, eventually,

for every $\epsilon \in \left(0, 1\right)$.

Lemma 2.5. ^[22] Assume that $\chi \left(\mathrm{s}\right)$ is an eventually positive solution of (1.1). Then, for sufficiently large $\mathrm{s},$ $\mathcal{G}\left(\mathrm{s}\right)$ satisfies one of the following cases:

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

Lemma 2.6. Let $\chi$ be an eventually positive solution of (1.1), and assume $\mathfrak{a}\in C\left(\left[ \mathrm{s}_{0}, \infty \right), \left(0, \infty \right) \right)$ such that $\mathfrak{a}_{2}\left(\mathrm{s} \right) \neq 0$ is bounded and

Then:

(ⅰ) $\chi \left(\mathrm{s}\right) \geq c\mathcal{G} \left(\mathrm{s}\right)$ for some $c\in \left(0, 1\right);$

(ⅱ) $\left(r(\mathrm{s})\left[ \mathcal{G}^{\prime \prime \prime }\left(\mathrm{s}\right) \right] ^{\alpha }\right) ^{\prime }+\kappa \left(\mathrm{s}\right) \mathcal{G}^{\beta }(\sigma (\mathrm{s}))\leq 0,$

for sufficiently large $\mathrm{s}$

Proof. From the definition of $\mathcal{G},$ it is obvious that

or

If we apply the inequality (2.1) with $\lambda = \delta > 1,$ $A = \mathfrak{a}_{2}^{1/\delta }\left(\mathrm{s}\right) \chi \left(\varsigma \left(\mathrm{s}\right) \right),$ and $B = \left(\frac{1}{\delta }\mathfrak{ a}\left(\mathrm{s}\right) \mathfrak{a}_{2}^{-1/\delta }\left(\mathrm{s} \right) \right) ^{\frac{1}{\delta -1}},$ we get

Similarly, if we apply (2.2) with $\lambda = \gamma < 1,$ $A = \mathfrak{a} _{1}^{1/\gamma }\left(\mathrm{s}\right) \chi \left(\varsigma \left(\mathrm{s}\right) \right),$ and $B = \left(\frac{1}{\gamma }\mathfrak{a} \left(\mathrm{s}\right) \mathfrak{a}_{1}^{-1/\gamma }\left(\mathrm{s} \right) \right) ^{\frac{1}{\gamma -1}},$ we get

By substituting (2.6) and (2.7) into (2.5), we obtain

Since $\mathcal{G}^{\prime }(\mathrm{s}) > 0$, we find $\mathcal{G}\left(\mathrm{s}\right) \geq c_{0}$ for some $c_{0} > 0.$ Therefore, (2.8) leads to

In light of (2.4), we can identify a constant $c\in \left(0, 1\right)$ such that

By substituting (2.9) into (1.1), we obtain

Since $\mathcal{G}^{\prime }\left(\mathrm{s}\right) > 0,$ and $\sigma (\mathrm{s})\leq \sigma _{i}(\mathrm{s})$ for all $i = 1, 2, ..., m,$ then

This completes the proof. □

3.   Main results

This section introduces additional conditions that ensure the oscillatory behavior of solutions to Eq (1.1) using the Riccati technique with varied substitutions. These requirements are based on a rigorous study that takes into consideration the equation's unique structure, eventually focusing on positive solutions. We assume that the functional inequalities hold for all large enough $s$, which simplifies the proof without losing generality.

3.1.   The scenario $\left(\mathrm{C}_{1}\right)$

This subsection discusses the criteria that exclude the existence of positive solutions to Eq (1.1) within $\left(\mathrm{C} _{1}\right).$

Lemma 3.1. Let $\beta \geq \alpha.$ If there is a nondecreasing function $\mu \in C^{1}([\mathrm{s}_{0}, \infty), \left(0, \infty \right))$ such that

holds for every $\epsilon, \epsilon _{1}\in \left(0, 1\right),$ $L > 0$, then $\mathrm{C}_{1} = \varnothing$.

Proof. Let $\chi \in \mathrm{C}_{1}.$ Then, there exists a $\mathrm{s}_{1}\geq \mathrm{s}_{0}$, such that $\chi \left(\mathrm{s}\right) > 0,$ $\chi \left(\varsigma \left(\mathrm{s}\right) \right) > 0,$ and $\chi \left(\sigma _{i}\left(\mathrm{s}\right) \right) > 0$ for $\mathrm{s}\geq \mathrm{s} _{1}\geq \mathrm{s}_{0}$ and $i = 1, 2, ..., m.$ Let us now definet

So

Using Lemma 2.6 (ⅱ), (3.2), and (3.3), we deduce that

From Lemma 2.4, we have that

and hence,

It follows from Lemma 2.1 that

for all $\epsilon _{1}\in \left(0, 1\right)$ and every sufficiently large $\mathrm{s}$. Thus, by (3.4)-(3.6), we have

By using (3.2) and (3.5), we obtain

Since $\mathcal{G}^{\prime }\left(\mathrm{s}\right) > 0$, and $\beta \geq \alpha,$ then there exist a $\mathrm{s}_{1}\geq \mathrm{s}_{0}$ and a constant $L > 0$ such that

Thus, the inequality (3.7) gives

Using Lemma 2.3, where we define $B = \mu ^{\prime }\left(\mathrm{s} \right) /\mu \left(\mathrm{s}\right),$ $A = \epsilon _{1}\alpha \mathrm{s} ^{2}/2\left(r\left(\mathrm{s}\right) \mu \left(\mathrm{s}\right) \right) ^{1/\alpha },$ and $u\left(\mathrm{s}\right) = w\left(\mathrm{s}\right),$ we derive

Integrating (3.10) from $\mathrm{s}_{2}\geq \mathrm{s}_{1}$ to $\mathrm{s },$ one arrives at

this contradicts (3.11) as $\mathrm{s}\rightarrow \infty$. This completes the proof. □

Lemma 3.2. Let $\beta \geq \alpha.$ If there is a nondecreasing functions $\mu _{1}\in C^{1}([\mathrm{s}_{0}, \infty), \left(0, \infty \right))$ such that

holds for every $\epsilon _{1}\in \left(0, 1\right),$ $L_{1} > 0$, then $\mathrm{C}_{1} = \varnothing$.

Proof. Let $\chi \left(\mathrm{s}\right) \in \mathrm{C}_{1}.$ Then there exists a $\mathrm{s}_{1}\geq \mathrm{s}_{0}$, such that $\chi \left(\mathrm{s}\right) > 0,$ $\chi \left(\varsigma \left(\mathrm{s}\right) \right) > 0,$ and $\chi \left(\sigma _{i}\left(\mathrm{s}\right) \right) > 0$ for $\mathrm{s}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$ and $i = 1, 2, ..., m.$ Now, define a function $w_{1}\left(\mathrm{s}\right)$ by

Thus

We see from Lemma 2.6 (ⅱ), (3.12), and (3.13) that

Using (3.6), we obtain

for any $\epsilon _{1}\in \left(0, 1\right)$ and sufficiently large $\mathrm{s}$. When we replace (3.15) with (3.14), we see that

Since $\left(r(\mathrm{s})\left(\mathcal{G}^{\prime \prime \prime }(\mathrm{s})\right) ^{\alpha }\right) ^{\prime } < 0,$ then

Then

Since $\beta \geq \alpha$ and $\mathcal{G}^{\prime } > 0$, there are constants $L_{1} > 0$ and $\mathrm{s}_{2}\geq \mathrm{s}_{1}$ such that

Thus, the inequality (3.16) gives

Using Lemma 2.3, where we define $B = \mu _{1}^{\prime }\left(\mathrm{s} \right) /\mu _{1}\left(\mathrm{s}\right),$ $A = \epsilon _{1}\alpha L_{1}\sigma ^{2}\left(\mathrm{s}\right) \sigma ^{\prime }\left(\mathrm{s} \right) /2\left[ \mu _{1}\left(\mathrm{s}\right) r\left(\sigma \left(\mathrm{s}\right) \right) \right] ^{1/\alpha },$ and $u\left(\mathrm{s} \right) = w_{1}\left(\mathrm{s}\right),$ we derive

Integrating (3.19) from $\mathrm{s}_{3}\geq \mathrm{s}_{2}$ to $\mathrm{ s},$ one arrives at

this contradicts (3.11) as $\mathrm{s}\rightarrow \infty$.

Thus, the proof is finished. □

Lemma 3.3. Let $0 < \beta < \alpha.$ If there is a nondecreasing function $\mu _{1}\in C^{1}([\mathrm{s}_{0}, \infty), \left(0, \infty \right))$ such that

holds for every $\epsilon _{1}\in \left(0, 1\right),$ $L_{2} > 0$, then $\mathrm{C}_{1} = \varnothing$.

Proof. Let $\chi \left(\mathrm{s}\right) \in \mathrm{C}_{1}.$ Then there exists a $\mathrm{s}_{1}\geq \mathrm{s}_{0}$, such that $\chi \left(\mathrm{s}\right) > 0,$ $\chi \left(\varsigma \left(\mathrm{s}\right) \right) > 0,$ and $\chi \left(\sigma _{i}\left(\mathrm{s}\right) \right) > 0$ for $\mathrm{s}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$ and $i = 1, 2, ..., m.$ As in the proof of (3.11) in Lemma 3.2. The function $w_{1}\left(\mathrm{s}\right)$ is defined in (3.12), then (3.13) holds. By using Lemma 2.6 (ⅱ), (3.12), and (3.13), we conclude that

By using (3.15), we see that

Note that $0 < \beta < \alpha$ and ($\mathrm{C}_{1}$) hold. Since $r^{\prime }\left(\mathrm{s}\right) \geq 0$, we deduce that $\mathcal{G}^{\left(4\right) }\left(\mathrm{s}\right) \leq 0$; this readily infers that $\mathcal{G}^{\prime \prime \prime }\left(\mathrm{s}\right)$ is nonincreasing. Then there are $L_{2} > 0$ and $\mathrm{s}_{2}\geq \mathrm{s} _{1}$ such that

From (3.21) and (3.22), it follows that

By applying Lemma 2.3, where $B = \mu _{1}^{\prime }\left(\mathrm{s} \right) /\mu _{1}\left(\mathrm{s}\right),$ $r = \epsilon _{1}\beta L_{2}\sigma ^{2}\left(\mathrm{s}\right) \sigma ^{\prime }\left(\mathrm{s} \right) /2\left[ \mu _{1}\left(\mathrm{s}\right) r\left(\mathrm{s}\right) \right] ^{1/\beta },$ and $u\left(\mathrm{s}\right) = w_{1}\left(\mathrm{s} \right),$ we drive

Integrating (3.24) throughout the interval $\left[ \mathrm{s}_{3}, \mathrm{s}\right]$ allows us to conclude

This contradicts (3.20) when $\mathrm{s}\rightarrow \infty$.

Thus, the proof is finished. □

3.2.   The scenario $\left(\mathrm{C}_{2}\right)$

This subsection discusses the criteria that exclude the existence of positive solutions to Eq (1.1) within $\left(\mathrm{C} _{2}\right).$

Lemma 3.4. Let $\beta \geq \alpha.$ If there is a nondecreasing function $\mu _{2}\in C^{1}([\mathrm{s}_{0}, \infty), \left(0, \infty \right))$ such that

holds for every $\epsilon _{2}\in \left(0, 1\right),$ $L_{3} > 0$, then $\mathrm{C}_{2} = \varnothing$.

Proof. Let $\chi \left(\mathrm{s}\right) \in \mathrm{C}_{2}.$ Then, there exists a $\mathrm{s}_{1}\geq \mathrm{s}_{0}$, such that $\chi \left(\mathrm{s}\right) > 0,$ $\chi \left(\varsigma \left(\mathrm{s}\right) \right) > 0,$ and $\chi \left(\sigma _{i}\left(\mathrm{s}\right) \right) > 0$ for $\mathrm{s}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$ and $i = 1, 2, ..., m.$ Integrating (ⅱ) from $\mathrm{s}$ to $\infty$ and applying the fact that $\left(r\left(\mathcal{G}^{\prime \prime \prime }\right) ^{\alpha }\right) ^{\prime }\leq 0$, we deduce

As $\mathcal{G}\left(\mathrm{s}\right) > 0,$ $\mathcal{G}^{\prime }\left(\mathrm{s}\right) > 0,$ and $\mathcal{G}^{\prime \prime }\left(\mathrm{s} \right) < 0,$ Lemma 2.4 implies that

Integrating (3.27) from $\sigma \left(\mathrm{s}\right)$ to $\mathrm{s},$ we obtain

Therefore, (3.26) becomes

Since $\mathcal{G}^{\prime }\left(\mathrm{s}\right) > 0,$ then

or equivalently

Integrating this inequality from $\mathrm{s}$ to $\infty$, we have

Now, define

Then, $F\left(\mathrm{s}\right) \geq 0$ for $\mathrm{s}\geq \mathrm{s} _{1}\geq \mathrm{s}_{0}$ and

Hence, by (3.28), we obtain

Because $\mathcal{G}^{\prime }\left(\mathrm{s}\right) > 0$ and $\beta \geq \alpha$, there are constants $L_{3} > 0$ and $\mathrm{s}_{2}\geq \mathrm{s} _{1}$ such that

Substituting (3.31) into (3.30), we have

Using Lemma 2.3 with $B = \mu _{2}^{\prime }\left(\mathrm{s}\right) /\mu _{2}\left(\mathrm{s}\right),$ $A = 1/\mu _{2}\left(\mathrm{s}\right)$, and $u\left(\mathrm{s}\right) = F\left(\mathrm{s}\right),$ we obtain

Consequently, (3.32) leads to

When we integrate this inequality between $\mathrm{s}_{2}$ to $\mathrm{s}$, we obtain

which contradicts (3.25) as $\mathrm{s}\rightarrow \infty$. This completes the proof. □

The following corollary is obtained by setting $\mu _{2}\left(\mathrm{s} \right) = 1$ in Lemma 3.4.

Corollary 3.1. Let $\beta \geq \alpha.$ If

holds for every $\epsilon _{2}\in \left(0, 1\right)$, then $\mathrm{C} _{2} = \varnothing$.

Lemma 3.5. Let $0 < \beta < \alpha.$ If there are nondecreasing functions $\mu _{2}\in C^{1}([\mathrm{s}_{0}, \infty), \left(0, \infty \right))$ such thatand

holds for every $\epsilon _{2}\in \left(0, 1\right),$ $L_{4} > 0,$ then $\mathrm{C}_{2} = \varnothing$.

Proof. Let $\chi \left(\mathrm{s}\right) \in \mathrm{C}_{2}.$ Then there exists a $\mathrm{s}_{1}\geq \mathrm{s}_{0}$, such that $\chi \left(\mathrm{s}\right) > 0,$ $\chi \left(\varsigma \left(\mathrm{s}\right) \right) > 0,$ and $\chi \left(\sigma _{i}\left(\mathrm{s}\right) \right) > 0$ for $\mathrm{s}\geq \mathrm{s}_{1}\geq \mathrm{s}_{0}$ and $i = 1, 2, ..., m.$ As demonstrated in the proof of (3.25) in Lemma 3.4. The function $F\left(\mathrm{s} \right)$ is defined in (3.29), and so (3.30) holds, which can be written as follows:

Since $\mathcal{G}^{\prime \prime } < 0,$ it follows that $\mathcal{G}^{\prime }$ is decreasing. Consequently, we have

Since $0 < \beta < \alpha,$ we obtain $0 < \beta /\alpha < 1,$ which, together with (3.36), leads to

Hence, the inequality (3.35) becomes

Using Lemma 2.3 with $B = \mu _{2}^{\prime }\left(\mathrm{s}\right) /\mu _{2}\left(\mathrm{s}\right),$ $A = 1/\mu _{2}\left(\mathrm{s}\right)$, and $u\left(\mathrm{s}\right) = F\left(\mathrm{s}\right),$ we can deduce that

Consequently, (3.38) leads to

When we integrate this inequality between $\mathrm{s}_{2}$ to $\mathrm{s}$, we obtain

which contradicts (3.34) as $\mathrm{s}\rightarrow \infty$. Thus, we have completed the proof. □

The following corollary is obtained by setting $\mu _{2}\left(\mathrm{s} \right) = 1$ in Lemma 3.5.

Corollary 3.2. Let $0 < \beta < \alpha.$ If

hold for some $\epsilon _{2}\in \left(0, 1\right).$ Then (1.1) is oscillatory.

4.   Oscillation theorem

In this part, we develop oscillatory criteria for the studied equation by combining the findings from the preceding sections.

Theorem 4.1. If condition (2.4) is satisfied, then Eq (1.1) will exhibit oscillatory behavior provided that at least one of the following conditions is met:

$\left(\mathrm{1}\right)$ Both conditions (3.1) and (3.25);

$\left(\mathrm{2}\right)$ Both conditions (3.11) and (3.25);

$\left(\mathrm{3}\right)$ Both conditions (3.1) and (3.33);

$\left(\mathrm{4}\right)$ Both conditions (3.11) and (3.33);

$\left(\mathrm{5}\right)$ Both conditions (3.20) and (3.34);

$\left(\mathrm{6}\right)$ Both conditions (3.20) and (3.39).

Proof. We prove the first case $\left(\mathrm{1}\right)$ and apply the same method to the rest of the cases. Let us assume that $\chi$ is eventually a positive solution to the Eq (1.1). According to Lemma 2.5, there are two possible cases of behavior: $\left(\mathrm{C}_{1}\right)$ and $\left(\mathrm{C}_{2}\right)$. Using Lemmas 3.1 and 3.2, we find that the conditions (3.1) and (3.25) exclude the existence of solutions that satisfy both of the mentioned cases, which leads to a contradiction with the initial assumption, and therefore, the solutions cannot be positive eventually, which proves the oscillation. This approach can be generalized to the other cases by the same method. Thus, the proof is complete. □

Example 4.1. Consider the NDE given by:

for $\mathrm{s}\geq 1$ and $\kappa _{0} > 0.$ Here we have

Now, we calculate

and

Therefore

which means that condition (2.4) is satisfied.

Now, consider the test function $\mu \left(\mathrm{s}\right) = \mathrm{s} ^{4}.$ Applying condition (3.1), we obtain:

which is satisfied provided that

Similarly, consider the test function $\mu _{1}\left(\mathrm{s}\right) = \mathrm{s}^{4}.$ Applying condition (3.11), we derive:

which holds provided that

Now, let us consider the test function $\mu _{2}\left(\mathrm{s}\right) = \mathrm{s}.$ Applying condition (3.25), we obtain:

which holds if

Hence, in accordance with conditions $\left(\mathrm{1}\right) -\left(\mathrm{4}\right)$ of Theorem 4.1, the satisfaction of (4.2)–(4.4) ensures that Eq (4.1) exhibits oscillatory behavior.

Example 4.2. Consider the NDE given by

for $\mathrm{s}\geq 1,$ $\kappa _{0} > 0$ and $,$ $\sigma _{i}\in \left(0, 1\right),$ $i = 1, 2, ..., m.$ Clearly, $\alpha = \beta = 1/3,$ $\gamma = \frac{1 }{3},$ $\delta = 3,$ $\varsigma \left(\mathrm{s}\right) = 0.5\mathrm{s},$ $\sigma \left(\mathrm{s}\right) = \sigma \mathrm{s, }$ where $\sigma = \min \left \{ \sigma _{i}\mathrm{, }\text{ }i = 1, 2, ..., m\right \}.$ Additionally, $r\left(\mathrm{s}\right) = \mathrm{s}^{-1/3},$ $\mathfrak{a}\left(\mathrm{s} \right) = \mathfrak{a}_{1}\left(\mathrm{s}\right) = 1/\mathrm{s},$ and $\mathfrak{a}_{2}\left(\mathrm{s}\right) = 1.$ We first evaluate the integral:

and

Moreover,

Hence,

which implies that condition (2.4) is satisfied.

Now, consider the test function $\mu \left(\mathrm{s}\right) = \mathrm{s} ^{4/3}.$ Applying condition (3.1), we obtain:

which holds provided that

Similarly, consider the test function $\mu _{1}\left(\mathrm{s}\right) = \mathrm{s}^{4/3}.$ Applying condition (3.11), we derive:

which holds if

Now, take $\mu _{2}\left(\mathrm{s}\right) = \mathrm{s}.$ Applying condition (3.25), we obtain:

which is satisfied provided that

Therefore, in accordance with conditions $\left(\mathrm{1}\right)$ and $\left(\mathrm{2}\right)$ of Theorem 4.1, if inequalities (4.6)–(4.8) are satisfied, then Eq (4.1) exhibits oscillatory behavior.

5.   Conclusions

This study constitutes a pivotal step in exploring the oscillatory behavior of solutions associated with fourth-order differential equations, which contain mixed neutral terms and multiple delays. Using the Riccati methodology, we were able to derive precise criteria that guarantee the oscillation of solutions for this class of equations. Our results are not only a natural extension of previous research but also contribute to laying new foundations for a deeper and more comprehensive understanding. The research horizon opened by this study goes beyond the fourth order, as the results can be generalized in the future to include differential equations of higher order, where $n\geq 4$, adding a new mathematical dimension to the study. Moreover, applying the proposed methodology to the study of uncanonical cases provides a promising framework for expanding the theoretical understanding of these equations. Another interesting aspect is the exploration of equations containing the matching function of the form $\mathcal{G}\left(\mathrm{s}\right) = \chi \left(\mathrm{s}\right) + \mathfrak{a}_{1}\left(\mathrm{s}\right) \chi ^{\gamma }\left(\varsigma _{1}\left(\mathrm{s}\right) \right) +\mathfrak{a}_{2}\left(\mathrm{s} \right) \chi ^{\delta }\left(\varsigma _{2}\left(\mathrm{s}\right) \right)$, as this approach opens up new horizons for research creativity and application development. Thus, this study paves the way for more in-depth analyses aimed at enhancing theoretical understanding while laying the foundations for expanding practical applications in different fields.

Author contributions

S.A., F.M., and O.B.: Methodology, investigation. S.A. and F.M.: Writing—original draft preparation. S.A., F.M., and O.B.: Writing—review and editing. O.B.: 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

All authors declare no conflicts of interest in this paper.