Functional differential equations of the neutral type: Oscillatory features of solutions

1.   Introduction

The qualitative theory of differential equations deals with the properties of solutions to these equations. The focus of this theory is to examine traits such as stability, oscillation, bifurcation, periodicity, synchronization, symmetry, and more. Researchers established this theory to gather adequate information on nonlinear models that arise while modeling physical, biological, and other processes, as seen in ^[1,2,3]. Oscillation theory is a branch of qualitative theory that establishes criteria for both oscillatory and non-oscillatory solutions to differential equations. It is a crucial mathematical tool used in many cutting-edge disciplines and technologies. In recent decades, there has been a lot of activity in studying oscillation conditions for particular functional differential equations, as seen in ^{[4,5,6,7,8,9]}.

1.1.   NDDEs with several delays

Functional differential equations (FDEs) are equations that express the derivative of an unknown function at a particular point in time in terms of the function's values at earlier points in time. A specific type of FDE is a delay differential equation (DDE), which has numerous applications in physiology, electrical engineering, and biology. For instance, in a predator-prey model, the birth rate of predators may be influenced by past predator or prey numbers rather than just the present. Neutral delay differential equations (NDDEs) are a type of DDE that have the highest derivative in the solution both with and without delay. In addition to its theoretical significance, the qualitative analysis of these equations holds great practical significance. This is because neutral differential equations are involved in a number of phenomena, such as the study of vibrating masses attached to elastic bars, the solution of variational problems with time delays, and problems involving electric networks with lossless transmission lines (such as those found in high-speed computers, where these lines are used to connect switching circuits), see ^[10]. Recently, many researchers have been interested in studying the stability of some practical models of neutral differential equations, see ^[11,12,13].

In this study, we establish novel standards for evaluating the oscillatory characteristics of nonlinear second-order NDDEs with several delays. Namely, we consider the nonlinear NDDE:

where $\nu \geq \nu _{0}\ $and $\mathcal{Z}: = u+p\cdot \left(u\circ \tau \right)$. During the study, we made the following assumptions:

(B1) $r\in {\mathbf{C}}^{1}\left(\left[ \nu _{0}, \infty \right), \mathbb{R} ^{+}\right)$, $\kappa \in \mathbb{Z} ^{+}$, and $\alpha \in \mathbb{Q} ^{+}$ is a quotient of odd numbers.

(B2) $p,$ $q\in {\mathbf{C}}\left(\left[ \nu _{0}, \infty \right), \left[ 0, \infty \right) \right)$, and $p\left(\nu \right) \leq p_{0} < 1$.

(B3) $\tau,$ $\sigma _{j}\in {\mathbf{C}}\left(\left[ \nu _{0}, \infty \right), \mathbb{R} \right)$, $\tau \left(\nu \right) \leq \nu$, $\sigma _{j}\left(\nu \right) \leq \nu$, $\lim_{\nu \rightarrow \infty }\tau \left(\nu \right) = \infty$, and $\lim_{\nu \rightarrow \infty }\sigma _{j}\left(\nu \right) = \infty$, for $j = 1, 2, ..., \kappa$.

(B4) $\psi \in {\mathbf{C}}^{1}\left(\mathbb{R}, \left(l, L\right] \right)$, where $l$ and $L$ are positive constants, and $\delta = \sqrt[\alpha ]{\left(L/l\right) }$.

(B5) $F\in {\mathbf{C}}\left(\mathbb{R}, \mathbb{R} \right)$, and $F\left(u\right) /u^{\alpha }\geq k$ for $u\neq 0$, where $k$ is a positive constant.

By a solution of NDDE (1.1), we mean a real-valued function $u\in {\mathbf{C}}^{1}\left(\left[ \nu _{u}, \infty \right), \mathbb{R} \right),$ $\nu _{u}\geq \nu _{0},$ which satisfies (1.1) on $\left[ \nu _{u}, \infty \right)$, and has the properties $r\psi \left(u\right) \left[ \mathcal{Z}^{\prime }\right] ^{\alpha }\in {\mathbf{C}}^{1}\left(\left[ \nu _{u}, \infty \right), \mathbb{R} \right)$ and $\sup \left\{ \left\vert u\left(\nu \right) \right\vert :\nu \geq \nu _{\ast }\right\} > 0$ for all $\nu _{\ast }\geq \nu _{u}$. A solution $u$ of FDE (1.1) is said to be oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Furthermore, we study the behavior of (1.1) in the canonical case, that is,

The paper is structured into three main sections: the introduction, main results, and conclusion. In the introduction section, we introduce the equation under study along with the fundamental assumptions, and then we review the previous work in the relevant literature. The main results section is further divided into three subsections: improved properties of positive solutions, oscillation theorems, and examples and discussion.

1.2.   The function class $\mathbb{H}$

In 1987, Philos ^[14] defined a new class of functions $\mathbb{H}$ to extend the results of Kamenev ^[15]. Assume that

A function $\mathcal{P}\in {\mathbf{C}}\left(\mathcal{D}, \mathbb{R}\right)$ is said to belong to the function class $\mathbb{H}$, written by $\mathcal{P}\in \mathbb{H}$, if

(ⅰ) $\mathcal{P}\left(\nu, \nu \right) = 0$ for $\nu \geq \nu _{0}$, $\mathcal{P}\left(\nu, \mu \right) > 0$ on $\mathcal{D}_{0}$.

(ⅱ) $\mathcal{P}\left(\nu, \mu \right)$ has a continuous and nonpositive partial derivative $\partial \mathcal{P}/\partial \mu$ on $\mathcal{D}_{0}$ such that the condition

for all $(\nu, \mu)\in \mathcal{D}_{0}$, is satisfied for some $\rho \in {\mathbf{C}}\left(\mathcal{D}, \mathbb{R} \right)$.

Since then, Philos class $\mathbb{H}$ has been used to study the oscillations of differential equations of different orders and types, see for example, ^[16,17,18].

1.3.   Literature review

In the first part of the related work, we will discuss the findings related to the oscillation of NDDE (1.1) or a similar case. In the second part, we will present some work that helped to enhance the relationships and inequalities used in the study of oscillation.

In 2000, Manojlovic´ ^[19] presented an oscillation condition for the nonlinear ordinary differential equation

using Philos class $\mathbb{H}$, where $F$ satisfies the condition

Džurina and Lacková ^[20] and Şahiner ^[21] studied the oscillatory behavior of NDDE

In ^[20], Džurina and Lacková used the condition

for all $N > 0$, where

and presented the following criterion for oscillation:

Theorem 1.1. (^[20], Corollary 2.4) Suppose that $F\left(u\right) = u$ and $\sigma ^{\prime }\left(\nu \right) \geq 0$. If

then every solution of (1.5) is oscillatory.

On the other hand, Şahiner ^[21] used Philos class $\mathbb{H}$ and three different forms of restrictions on $F$, namely (B5) with $\alpha = 1$, (1.4) with $\alpha = 1$, and $F^{\prime }\left(u\right) \geq A > 0$. Next, we present one of Şahiner's results in which he used condition (B5).

Theorem 1.2. (^[21], Theorem 2.3) Suppose that $\sigma ^{\prime }\left(\nu \right) \geq 0$ and there are $\xi \in {\mathbf{C}}\left(\left[ \nu _{0}, \infty \right), \mathbb{R}^{+}\right)$ and $\mathcal{P}\in \mathbb{H}$ such that

where $Q\left(\nu \right) : = q\left(\nu \right) \left[ 1-p\left(\sigma \left(\nu \right) \right) \right],$ and

Then, every solution of (1.5) is oscillatory.

Using integral average conditions of Philos-type, Xu and Weng ^[22] investigated the oscillatory properties of NDDE with distributed deviating argument

where $\tau \left(\nu \right) = \nu -\tau _{0},$ $\tau _{0} > 0$. For the more general equation

Ye and Xu ^[23] established oscillation criteria under the condition (B5), and considered the canonical and noncanonical cases. The following theorem presents the oscillation criteria for the canonical case.

Theorem 1.3. (^[23], Theorem 2.1) Suppose that $\sigma ^{\prime }\left(\nu \right) \geq 0$ and there is $\xi \in {\mathbf{C}}\left(\left[ \nu _{0}, \infty \right), \mathbb{R} ^{+}\right)$ such that

Then, every solution of (1.8) is oscillatory.

Li et al. ^[24] obtained oscillation criteria for NDDE

where $0\leq p\left(v\right) \leq p_{0} < 1$. Arul and Shobha ^[25] enhanced the findings, achieving improved oscillation results for the solutions of Eq (1.10).

In ^[26], numerous findings on oscillation were derived for the second order differential equation

where $\gamma$ and $\beta$ are ratios of positive odd integers, $\gamma \geq \beta$. As a more generalized equation than (1.11), Grace et al. ^[27] established some results for oscillation of solutions of NDDE

where

Baculikova et al. ^[28] studied the oscillatory behavior of solutions of NDDE

where $0 < \alpha \leq 1$.

In the past ten years, there has been a significant amount of research aimed at developing methods and relationships to study the oscillations of functional differential equations. Grace et al. ^[29] and Moaaz et al. ^[30,31,32] have studied the oscillation of NDDE

and enhanced the well-known results published in the literature.

By improving the relationships between $u$ and $\mathcal{Z}$, Hassan et al. ^[33] and Moaaz et al. ^[34,35] improved the oscillation results for several classes of functional differential equations.

Lemma 1.1. (^[35], Lemma 1) Suppose that $u$ is a solution of (1.1) and $u\left(\nu \right) > 0$ for $\nu \geq \nu _{1}$. Then, eventually,

for any integer $m\geq 0$, where

Oscillation theory is an important area of study that offers fascinating analytical problems. Recently, researchers have been working on developing oscillation criteria for various types of functional differential equations. This research can be found in sources such as ^{[13,36,37,38,39]}.

We investigated the oscillatory behavior of NDDE (1.1) in this study. We extended the results in ^[35] and showed a new relationship between $u$ and $\mathcal{Z}$. We established many criteria that ensure the oscillation of every solution to the studied equation. The motivations and novelty of the results are summarized as follows:

$–$ Expanding previous results in the literature to include the half-linear case as well as the case of multiple delays

$–$ Removing some monotonic constraints on delay functions

$–$ Provide criteria that take into account the effects of delay functions

$–$ Providing more efficient and accurate oscillation parameters than previous related results

2.   Main results

For convenience, we define the following: $U^{+}$ represents the class of all eventually positive solutions to NDDE (1.1), $\widetilde{\alpha } : = \alpha ^{\alpha }/\left(\alpha +1\right) ^{\alpha +1}$, $\sigma _{\min }\left(\nu \right) : = \min \left\{ \sigma _{j}\left(\nu \right) :j = 1, 2, ..., \kappa \right\}$, and

To prove the main results, we will need the following lemma:

Lemma 2.1. (^[40], Lemma 2.3) Assume that $G\left(\theta \right) = c_{1}\theta -c_{2}\theta ^{1+1/\alpha }$, where $c_{1},$ $c_{2} > 0$. Then, $G$ has the maximum value at $\theta _{_{\max }}: = \left(\alpha c_{1}/\left(\left(\alpha +1\right) c_{2}\right) \right) ^{\alpha }$ and $G\left(\theta \right) \leq G\left(\theta _{_{\max }}\right) = \widetilde{\alpha } c_{1}^{\alpha +1}c_{2}^{-\alpha }$ for $\theta \in \mathbb{R}$.

2.1.   Improved properties of positive solutions

In this section, we deduce some new and improved relationships for positive solutions to the studied equation.

Lemma 2.2. Suppose that $u\in U^{+}$. Then,

P1 $\mathcal{Z}$ and $\mathcal{Z}^{\prime }\ $are positive, and $\left(r\cdot \left(\psi \circ u\right) \cdot \left[ \mathcal{Z}^{\prime } \right] ^{\alpha }\right) ^{\prime } < 0$, eventually.

P2 $\mathcal{Z}\left(\nu \right) \geq \frac{1}{\delta }r^{1/\alpha }\left(\nu \right) \mathcal{Z}^{\prime }\left(\nu \right) \eta _{\nu _{1}}\left(\nu \right)$ and $\left(\mathcal{Z}/\eta _{\nu _{1}}^{\delta }\right) ^{\prime }\leq 0$.

Proof. Let $u\in U^{+}$. Then, there is a $\nu _{1}\geq \nu _{0}$ such that $u\left(\tau \left(\nu \right) \right) > 0$ and $u\left(\sigma _{j}\left(\nu \right) \right) > 0$ for $\nu \geq \nu _{1}$ and $j = 1, 2, ..., \kappa$. From the definition of $\mathcal{Z}$, we get that $\mathcal{Z} \left(\nu \right) > 0$ for $\nu \geq \nu _{1}$. It follows from (1.1), (B2), and (B5) that

Thus, $\left(r\cdot \left(\psi \circ u\right) \left[ \mathcal{Z}^{\prime } \right] ^{\alpha }\right) ^{\prime }\leq 0$ and $\mathcal{Z}^{\prime }$ is of a constant sign.

Suppose that $\mathcal{Z}^{\prime }\left(\nu \right) < 0$ for $\nu \geq \nu _{2}$. So,

Since $\psi \left(u\right) \leq L$, we find

Integrating this inequality leads to

Using (1.2), we obtain $\lim_{\nu \rightarrow \infty }\mathcal{Z}\left(\nu \right) = -\infty$, which contradicts the fact that $\mathcal{Z} \left(\nu \right) > 0$.

Now, we have

Integrating this inequality leads to

Using the properties in (P1), we arrive at

Then,

This completes the proof. □

Lemma 2.3. Suppose that $u\in U^{+}$. Then,

for $\nu \geq \nu _{1}$ and any integer $m\geq 0$.

Proof. Let $u\in U^{+}$. Then, there is a $\nu _{1}\geq \nu _{0}$ such that $u\left(\tau \left(\nu \right) \right) > 0$ and $u\left(\sigma _{j}\left(\nu \right) \right) > 0$ for $\nu \geq \nu _{1}$ and $j = 1, 2, ..., \kappa$. From Lemma 2.2, we have the properties in (P1) and (P2) satisfied. It follows from these properties that

and

for $\ell = 0, 1, ..., m$. Thus, from Lemma 1.1, we arrive at

This completes the proof. □

2.2.   Oscillation theorems

Theorem 2.1. Suppose that $\sigma _{\min }^{\prime }\left(\nu \right) \geq 0$, and there is a $\xi \in {\mathbf{C}}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ such that

where

and

Then, every solution of (1.1) is oscillatory.

Proof. Assume the contrary that Eq (1.1) has a nonoscillatory solution. Thus, there is a solution $u$ of (1.1) such that $u\in U^{+}$. Moreover, there is a $\nu _{1}\geq \nu _{0}$ such that $u\left(\tau \left(\nu \right) \right) > 0$ and $u\left(\sigma _{j}\left(\nu \right) \right) > 0$ for $\nu \geq \nu _{1}$ and $j = 1, 2, ..., \kappa$. Using (B5), Eq (1.1) becomes

From Lemma 2.3, we have that (1.12) holds. Combining (2.1) and (2.3), we obtain

and so

Now, we define the function

Then,

which with (2.4) gives

Since $\left(r\cdot \left(\psi \circ u\right) \cdot \left[ \mathcal{Z} ^{\prime }\right] ^{\alpha }\right) ^{\prime }\leq 0$, we get that

Substituting $\left(\mathcal{Z}^{\prime }\circ \sigma _{\min }\right)$ in (2.6), we conclude that

By multiplying the inequality

by $\mathcal{P}\left(\nu, \mu \right)$ and then integrating it from $\nu _{2}\geq \nu _{1}$ to $\nu$, we arrive at

and so

Using Lemma 2.1 with $\theta = w$, $c_{1} = \left[ \mathcal{P}\left(\nu, \mu \right) \right] ^{\alpha /\left(\alpha +1\right) }\Psi \left(\nu, \mu \right)$, and

we obtain

or

which contradicts assumption (2.2). This completes the proof. □

Theorem 2.2. Suppose that $\sigma _{\min }^{\prime }\left(\nu \right) \geq 0$, and there is a $\xi \in {\mathbf{C}}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ such that

where $\Phi _{m}$ is defined as in Theorem 2.1. Then, every solution of (1.1) is oscillatory.

Proof. Assume the contrary that Eq (1.1) has a nonoscillatory solution. Thus, there is a solution $u$ of (1.1) such that $u\in U^{+}$. Moreover, there is a $\nu _{1}\geq \nu _{0}$ such that $u\left(\tau \left(\nu \right) \right) > 0$ and $u\left(\sigma _{j}\left(\nu \right) \right) > 0$ for $\nu \geq \nu _{1}$ and $j = 1, 2, ..., \kappa$.

Now, we define $w$ as in (2.5). Proceeding as in the proof of Theorem 2.1, we arrive at (2.7). Using Lemma 2.1 with $\theta = w$, $c_{1} = \xi ^{\prime }/\xi$, and

we obtain

Integrating this inequality leads to

which contradicts assumption (2.8). This completes the proof. □

In the following theorem, we present a new criterion for the oscillation of (1.1), but unlike previous theorems, it does not need the constraint $\sigma _{\min }^{\prime }\left(\nu \right) \geq 0$.

Theorem 2.3. Suppose that there is a $\xi \in {\mathbf{C}}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ such that

for $\nu _{1}\geq \nu _{0}$, where $\Phi _{m}$ is defined as in Theorem 2.1. Then, every solution of (1.1) is oscillatory.

Proof. Assume the contrary that Eq (1.1) has a nonoscillatory solution. Thus, there is a solution $u$ of (1.1) such that $u\in U^{+}$. Moreover, there is a $\nu _{1}\geq \nu _{0}$ such that $u\left(\tau \left(\nu \right) \right) > 0$ and $u\left(\sigma _{j}\left(\nu \right) \right) > 0$ for $\nu \geq \nu _{1}$ and $j = 1, 2, ..., \kappa$.

Proceeding as in the proof of Theorem 2.1, we arrive at (2.4). Now, we define the function

Then,

which with (2.4) gives

It follows from Lemma 2.2 that $\left(\mathcal{Z}/\eta _{\nu _{1}}^{\delta }\right) ^{\prime }\leq 0$. Then,

So, (2.10) reduces to

Using Lemma 2.1 with $\theta = w$, $c_{1} = \xi ^{\prime }/\xi$, and

we obtain

Integrating this inequality leads to

which contradicts assumption (2.9). This completes the proof. □

By replacing $\mathcal{P}\left(\nu, \mu \right)$ with $\left(\nu -\mu \right) ^{n},$ $n\in \mathbb{Z} ^{+},$ in Theorem 2.1, we obtain an oscillation criterion for Eq (1.1) of the Kamenev type.

Corollary 2.1. Suppose that $\sigma _{\min }^{\prime }\left(\nu \right) \geq 0$, and there are $\xi \in {\mathbf{C}}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ and $n\in \mathbb{Z} ^{+}$ such that

where $\Phi _{m}$ and $\Psi$ are defined as in Theorem 2.1. Then, every solution of (1.1) is oscillatory.

In Theorem 2.2, by choosing $\xi \left(\mu \right) = \eta _{\nu _{1}}^{\alpha }\left(\sigma _{\min }\left(\mu \right) \right),$ we get the following oscillation criterion for Eq (1.1).

Corollary 2.2. Suppose that $\sigma _{\min }^{\prime }\left(\nu \right) \geq 0$, and

where $\Phi _{m}$ is defined as in Theorem 2.1. Then, every solution of (1.1) is oscillatory.

For the linear special case of Eq (1.1), which is:

we present the following oscillation criterion.

Corollary 2.3. Suppose that there is a $\xi \in {\mathbf{C}}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ such that

for $\nu _{1}\geq \nu _{0}$, where $\Phi _{m}$ is defined as in Theorem 2.1. Then, every solution of (2.12) is oscillatory.

2.3.   Examples and discussion

In this section, we confirm with examples and remarks the novelty of the results and their efficiency in the oscillation test compared to previous relevant results.

Remark 2.1. Let $\kappa = 1$ and $\alpha = 1$. Then, Eq (1.1) reduces to (1.5). It is simple to verify that

where $Q\left(\nu \right)$ is defined in Theorem 1.2. Moreover,

Therefore, if $m = 0$, we note that Theorems 2.1 and 2.2 turns into Theorem 1.2 and 1.3, respectively. Also, Corollary 2.2 turn into Theorems 1.1.

In general, based on (2.13), Theorems 2.1 and 2.2 generalize and improve Theorems 1.1–1.3.

Remark 2.2. Theorems 1.1–1.3, 2.1, and 2.2 require that constraint $\sigma _{\min }^{\prime }\left(\nu \right) \geq 0$. However, Theorem 2.3 does not need this constraint.

Example 2.1. Consider the NDDE

where $\alpha > 0$ is an odd integer, $\tau _{0},$ $\lambda _{j}\in \left(0, 1 \right]$ for $j = 1, 2, ..., \kappa$, $p_{0}\in \left[ 0, 1\right)$, and $a_{j}\geq 0$. We see that $r\left(\nu \right) = 1$, $p\left(\nu \right) = p_{0}$, $\tau \left(\nu \right) = \tau _{0}\nu$, $\sigma _{j}\left(\nu \right) = \lambda _{j}\nu$, $q_{j}\left(\nu \right) = a_{j}/\nu ^{\alpha +1}$, $F\left(u\right) = u^{\alpha }$, and $\psi \left(u\right) = 1/\left(1+\sin ^{2}u\right)$. It is simple to confirm that $k = 1,$ $l = 1/2,$ $L = 1,$ $\delta = \sqrt[\alpha ]{2}$, and $\tau _{\ell }\left(\nu \right) = \tau _{0}^{\ell }\nu$ for $\ell = 0, 1, ...$. Then,

where

By choosing $\mathcal{\xi }\left(\mu \right) = \mu ^{\alpha }$ and $\mathcal{P}\left(\nu, \mu \right) = \left(\nu -\mu \right) ^{\alpha +1}$, condition (2.2) reduces to

where $\lambda _{0}: =$ $\underset{1\leq j\leq \kappa }{\min }\lambda _{j}$, which is achieved if

Moreover, condition (2.8) reduces to (2.15). On the other hand, condition (2.9) becomes

which is achieved if

Therefore, using Theorems 2.1–2.3, every solution of (2.14) is oscillatory if one of the conditions (2.15) and (2.16) holds.

In order to determine which conditions are most efficient in the oscillation test, it is easy to note that

so criterion (2.15) provides the sharpest results in the oscillation test.

Remark 2.3. To compare our results with previous results in the literature, we consider the special case

where $\lambda \in \left(0, 1\right]$ and $a > 0$. Applying Theorems 1.1 –1.3, we find that every solution of (2.17) is oscillatory if $a > 1$. However, condition (2.15) becomes $a > 0.83598$, which is better for oscillation. Moreover, Figure 1 shows one of the oscillatory solutions of Eq (2.14) when $p_{0} = 0,$ $\alpha = 1,$ $k = 1,$ $a_{1} = 10$, and $\lambda _{1} = 1$. We note that the solution has an infinite number of arbitrary zeros that converge to infinity. We also notice that the distance between the zeros of this oscillatory solution increases constantly. Also, oscillation waves rise with increasing $t$.

Example 2.2. Consider the delay equation

where $\tau _{0},$ $p_{0},$ and $\lambda _{j}$ are positive, for $j = 1, 2, ..., \kappa$. We see that $\alpha = 1$, $r\left(\nu \right) = {\mathbf{e}} ^{-\nu }$, $p\left(\nu \right) = p_{0}$, $\tau \left(\nu \right) = \nu -\tau _{0}$, $\sigma _{j}\left(\nu \right) = \nu -\lambda _{j}$, $q\left(\nu \right) = {\mathbf{e}}^{a_{j}-\nu }$, $F\left(u\right) = u$, and $\psi \left(u\right) = 1+{\mathbf{e}}^{-u^{2}}$. It is simple to confirm that $l = 1$, $L = 2$, $\delta = 2$, and

where

By choosing $\mathcal{\xi }\left(\mu \right) = {\mathbf{e}}^{\mu }$ and $\mathcal{P}\left(\nu, \nu _{0}\right) = \left(\nu -\mu \right) ^{2}$, condition (2.2) becomes

where $\lambda _{0}: =$ $\underset{1\leq j\leq \kappa }{\max }\lambda _{j}$, which is achieved if

Using Theorem 2.1, every solution of (2.18) is oscillatory if (2.19) holds.

Remark 2.4. In 2018, Grace et al. ^[29] presented improved criteria for the oscillation of NDDE

Applying the results in ^[29], we find that every solution of

is oscillatory if

(Theorem 3, ^[29]), or

(Theorem 6, ^[29]), where $c = 1/\left(1+a\lambda \left(1-p_{0}\right) \right)$. For (2.20), condition (1.9) and (2.8) reduce to

and

respectively. Figure 2 compares the lower bounds of $a$-values with respect to $\lambda \in \left(0, 1\right]$ for conditions (2.21)–(2.23), when $p_{0} = 0.5$ and $\tau _{0} = 0.9$. We note that condition (2.21) provides the best results for oscillation in $\left(0, 0.214722\right)$, while condition (2.23) is the best in $\left(0, 0.214722\right]$.

Remark 2.5. Consider the special case of (2.20) in which

Table 1 compares the various criteria that confirm the oscillation of the solutions of Eq (2.24). Theorem 2.2 provides the best results for oscillation.

3.   Conclusions

The precision of the relationships and inequalities utilized has an impact on the investigation of the oscillatory behavior of FDEs. In this paper, we examined the oscillatory behavior of solutions for NDDE (1.1). We presented a novel relationship between $u$ and $\mathcal{Z}$ as an extension of the findings in ^[35]. We set more than one standard that guarantees the oscillation of all solutions of the equation under study. Through comparisons and examples, we found the following:

$–$ Our theorems extend the results in ^[20,21,23] to the half-linear case as well as to the case of multiple delays.

$–$ Theorem 2.2 does not need monotonic constraints on the delay functions, while previous results required $\sigma ^{\prime }\left(\nu \right) \geq 0$.

$–$ Based on (2.13), our results not only generalize previous results but also improve upon them. This improvement can be seen in Remarks 2.3–2.5.

$–$ The previous results do not take into account the effect of $\tau \left(\nu \right)$, while the new relationship (2.1) makes our criteria affected by $\tau \left(\nu \right)$.

It would be interesting for future studies to obtain an improved relationship between the solution and the corresponding function without having to require that $\psi \left(u\right) \geq l > 0$ and $p_{0} < 1$. Recently, there are many studies that have developed oscillation criteria for higher-order differential equations, see for example ^{[41,42,43,44,45]}. We also propose to extend our results to include the non-canonical case as well as higher-order equations.

Author contributions

O. M. developed the conceptualization and methodology. A. A. wrote the original draft. O. M. and A. A. provided examples, figures, and discussion. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

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

Acknowledgements

The authors present their sincere thanks to the editors and the anonymous reviewers for their comments that helped improve the manuscript. The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

All authors declare that there are no conflicts of interest in this paper.