Iterative oscillation criteria of third-order nonlinear damped neutral differential equations

1.   Introduction

A neutral delay differential equation contains the highest-order derivative of the unknown function both with and without delays. Because of this, the theory of neutral delay differential equations is more difficult to understand than the theory of non-neutral equations. There has been an increase in interest in the theory of neutral differential equations in recent years. Studying these equations is essential for both theory and applications, as neutral equations are used to explain a wide range of real-world phenomena, including the motion of radiating electrons, population growth, the spread of epidemics, networks incorporating lossless transmission lines, etc., see ^[2,17,19,20]. Researchers have focused a great deal of attention on the oscillation problem of functional differential equations in the recent few decades; see, for example, ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]}. For third-order delay equations, see ^{[1,2,3,4,5,6,7,12,25,26,29]}. For neutral equations, see ^{[21,22,32,33,34]} and ^{[8,15,24,35,39]} for the equations with damping. Using a generalized Riccati transformation and an integral averaging technique the authors ^[38] obtained certain necessary conditions for oscillation for the third-order nonlinear differential equation

where $\rho ^{\prime }(s) > 0$ and $\frac{f(u)}{u}\geq k > 0,$ for all $u\neq 0.$ Also, ^[11] improves and unifies the results of ^[38], reducing the third-order equations to the first and second ones. In this work, we focus our attention on the oscillation of the third-order nonlinear neutral differential equation with the form

where $s\geq s_{0}\geq 0$, $z(s): = x(s)+p(s)x(\mu (s)),$ $\varphi _{\beta }(u): = \left\vert u\right\vert ^{\beta -1}u$, $\beta > 0$; $\eta _{1}, \eta _{2} > 0$, and $m_{i}, p, q, \rho, \mu \in C(\left[s_{0}, \infty \right), \mathbb{ R}),$ $i = 1, 2, 3.$ It should be noted that the oscillation of many special cases of Eq (1.1) has been studied by many authors; see, for examples, ^{[9,10,11,12,15,16]}.

In this paper, we suppose that

$({\rm{i}})$ $0\leq p(s) < p < 1$, $q(s)\geq 0$, $m_{i}(s) > 0$, $i = 1, 2$ and $m_{3}(s)\geq 0$;

$({\rm{ii}})$ $f\in C(\mathbb{R}, \mathbb{R})$ such that $xf(x) > 0$ and $\frac{f\left(x\right)}{\varphi _{\eta }(x)} \geq k > 0$, for all $x\neq 0$, $\eta : = \eta _{1}\eta _{2}$;

$({\rm{iii}})$ $\rho (s)\leq s, \ \mu (s)\leq s,$ and $\lim\limits_{s \rightarrow \infty }\rho (s) = \lim\limits_{s\rightarrow \infty }\mu (s) = \infty;$

$({\rm{iv}})$ $\int_{\mathcal{S}}^{\infty }\left(\dfrac{1}{m_{1}(t)}\right) ^{1/\eta _{1}} \; \mathrm{d} t = \infty$ and $\int_{\mathcal{S}}^{\infty }\left(\dfrac{1}{M\left(t\right) }\right) ^{1/\eta _{2}} \mathrm{d} t = \infty,$

where $M\left(s\right) : = m_{2}\left(s\right) \exp \left(\int_{\mathcal{S}}^{s}\dfrac{m_{3}(r)}{m_{2}(r)}\; \mathrm{d}r\right),$ $\mathcal{S}\in \lbrack s_{0}, \infty).$

A function $x\left(s\right)$ is a solution of (1.1) if it satisfies Eq (1.1) for all $s\in \lbrack s_{x}, \infty)$ and satisfying $\sup \{|x(s)|:s\geq \mathcal{S}\} > 0$ for any $\mathcal{S}\geq s_{x}$ with $x(s), m_{1}(s)\varphi _{\eta _{1}}\left(z^{\prime }(s)\right),$ and $m_{2}(s)\varphi _{\eta _{2}}\left(\left[m_{1}(s)\varphi _{\eta _{1}}\left(z^{\prime }(s)\right) \right] ^{\prime }\right)$ are continuously differentiable for all $s\in \lbrack s_{x}, \infty).$ The solution on $[s_{x}, \infty)$ with arbitrary large zeros is said to be an oscillatory solution. In this paper, we investigate the oscillatory and asymptotic behavior of Eq (1.1) by a reduction in order and comparison with the oscillation of first-order delay differential equations.

2.   Main results

Throughout this paper, we define

Also, the sequences $\left\{ P_{n}(s)\right\} _{n = 1}^{\infty }$ and $\left\{ Q_{n}(s, t)\right\} _{n = 1}^{\infty }$ are defined as follows:

for $\mathcal{S}\in \lbrack s_{0}, \infty)$ and $s\in \lbrack \mathcal{S}, \infty),$ with

and

for $s\in \left[t, \infty \right) \subseteq \left[\mathcal{S}, \infty \right),$ with

for some $N > 0$ and $\mathcal{S}\in \lbrack s_{0}, \infty)$.

The subsequent lemmas will be introduced and utilized in the main result.

Lemma 2.1. Assume that $x$ is an eventually positive solution of Eq (1.1). Then there exists $\mathcal{S}\geq s_{0}$ such that either

(I) $\mathcal{L}_{1}(z(s)) > 0, \ \mathcal{L}_{2}(z(s)) > 0$,

or

(II) $\mathcal{L}_{1}(z(s)) < 0, \ \mathcal{L}_{2}(z(s)) > 0,$

for all $s\geq \mathcal{S}$.

Proof. Since $x$ is a positive solution of Eq (1.1) on $[s_{1}, \infty),$ $s_{1}\geq s_{0}$ such that $x(\rho (s)) > 0$ and $x(\mu (s)) > 0$ for $s\geq s_{1}$. From Eq (1.1), we have for all $s\geq s_{1},$

which implies that

where $M\left(s\right) = m_{2}\left(s\right) \exp \left(\int_{s_{1}}^{s} \dfrac{m_{3}(r)}{m_{2}(r)}\; \mathrm{d}r\right).$ That demonstrates that $\mathcal{L}_{1}(z(s))$ and $\mathcal{L}_{2}(z(s))$ are of one sign eventually. We claim that

If not, consider the following two cases:

Case 1. There exists $s_{2}\geq s_{1},$ sufficiently large, such that

Since $\left(M\left(s\right) \mathcal{L}_{2}(z(s))\right) ^{\prime }\leq 0,$ then there exists a negative constant $\mathcal{M}$ such that

It follows that

Integrating from $s_{2}$ to $s$, we obtain

Letting $s\rightarrow \infty$ and using (ⅳ), then $\mathcal{L}_{1}\left(z\left(s\right) \right) \rightarrow -\infty,$ which contradicts that $\mathcal{L}_{1}\left(z\left(s\right) \right) > 0.$

Case 2. There exists $s_{2}$ $\geq s_{1},$ sufficiently large, such that

which implies that $\left(m_{1}(s)\mathcal{L}_{1}(z(s))\right) ^{\prime } < 0$ and therefore,

Dividing by $m_{1}(s)$ and integrating from $s_{2}$ to $s,$ we obtain

Letting $s\rightarrow \infty,$ then (iv) yields $z(s)\rightarrow -\infty,$ which contradicts the fact that $z(s) > 0.$ This completes the proof. □

Lemma 2.2. Assume that $x$ is a positive solution of Eq (1.1) and the corresponding function $z$ satisfies (I) of Lemma 2.1. Then

Proof. Since $x$ is a positive solution of Eq (1.1) on $\left[s_{1}, \infty\right)$, then there exists $s_{2}\geq s_{1}$ such that the corresponding function $z$ satisfies (Ⅰ) of Lemma 2.1 on $\left[s_{1}, \infty \right)$. It is easy to see that Eq (1.1) can be written in the form

for all $s\geq s_{1}$. Then

Therefore,

Also, we have

Since $z^{\prime } > 0,$ we get

Substituting (2.4) into (2.3), we have

This completes the proof. □

Lemma 2.3. If $x$ is an eventually positive solution of Eq (1.1) and the corresponding function $z$ satisfies Case (I) of Lemma 2.1, then for $n\in \mathbb{N},$

Proof. Since $x$ is a positive solution of Eq (1.1) on $\left[s_{1}, \infty \right)$, then there exists $s_{2}\geq s_{1}$ such that the corresponding function $z$ satisfies (Ⅰ) of Lemma 2.1 on $\left[s_{1}, \infty \right)$. Then

Then,

Integrating the above inequality from $s_{1}$ to $s\in \left[s_{1}, \infty \right)$, we obtain

This shows that (2.5) holds for $n = 1$. Consequently,

From (2.2) and (2.7), we obtain

Using the nonicreasing nature of ${M}\left(s\right) \mathcal{L}_{2}(z(s))$ and $\rho (s)\leq s$, we obtain

Integrating the above inequality from $t$ to $s\in \left[t, \infty \right)$ implies that

Using (2.8) in (2.6), we obtain

It follows that

Again, integrating from $s_{1}$ to $s$, we obtain

This shows that (2.5) holds for $n = 2$. If this process is repeated $n$ times, we obtain (2.5). □

The asymptotic behavior of all solutions to Eq (1.1) is discussed in the results that follow.

Theorem 2.1. Let $n\in \mathbb{N}$. Assume that the first-order delay differential equation

is oscillatory. If $x(s)$ is a solution of Eq (1.1), then $x(s)$ is either oscillatory or bounded.

Proof. Assume that $x(s)$ is a nonoscillatory solution of Eq (1.1). Without loss of generality, let $x(s) > 0$, $x(\rho (s)) > 0,$ and $x(\mu (s)) > 0$ on$\ {[}$$s$$_{1}, \infty)$, $s$$_{1} \geq s_0$. It follows from Lemma 2.1 that there exists $s_{2}\geq s_{1}$ such that either (Ⅰ) or (Ⅱ) holds on ${[}$$s$$_{2}, \infty)$. Assume (Ⅰ) is valid. From (2.5), we have

Combining (2.2) and (2.10), we obtain

where $w(s): = M\left(s\right) \mathcal{L}_{2}(z(s))$. Due to [37, Theorem 1], the associated delay differential equation also has a positive solution. This is a contradiction. Now, to complete the proof, we consider (Ⅱ) valid. Since $z(s) > 0,$ and $z^{\prime }(s) < 0$ then $z(s)$ is bounded, and therefore $x\left(s\right)$ is bounded. The proof is complete. □

Theorem 2.2. Let $n\in \mathbb{N}$. Assume that the first-order delay differential equation (2.9) is oscillatory and

If $x(s)$ is a solution of Eq (1.1), then $x(s)$ is either oscillatory or tends to zero eventually.

Proof. Assume that $x(s)$ is a nonoscillatory solution of Eq (1.1). Without loss of generality, let $x(s) > 0$, $x(\rho (s)) > 0,$ and $x(\mu (s)) > 0$ on$\ {[}$$s$$_{1}, \infty)$, $s$$_{1} \geq s_0$. It follows from Lemma 2.1 that there exists $s_{2}\geq s_{1}$ such that either (Ⅰ) or (Ⅱ) holds on ${[}$$s$$_{2}, \infty)$. The proof of Case (Ⅰ) is identical to the proof of Theorem 2.1, Case (Ⅰ), and so it has been omitted. Assume (Ⅱ) is valid. It is obvious that Eq (1.1) can be written as

Since $z(s) > 0$ and $z^{\prime }(s) < 0,$ there exists a constant $l\geq 0$ such that $\lim_{t\rightarrow \infty }z(s) = l$. We claim $l = 0$. If not, then for sufficiently small $\epsilon > 0$, there exists $s_{3}\geq s_{2}$ such that $l-p(l+\epsilon) > 0$ and $l < z(s) < l+\epsilon$ for all $s > s_{3}$. Then

$N: = \frac{l-p(l+\epsilon)}{l+\epsilon } > 0$. From (2.12) and (2.13), we obtain

Then

where $K: = kN^{\eta }l^{\eta } > 0.$ Integrating the above inequality from $s\in \left[s_{3}, \infty \right)$ to $\infty,$ we obtain

It follows that

Integrating the above inequality from $s$ to $\infty,$ we obtain

Again, integrating the above inequality from $s_{2}$ to $\infty,$ we obtain

which is a contradiction to (2.11), then $\lim\limits_{s\rightarrow \infty }z(s) = 0.$ Since $0 < x(s)\leq z(s),$ then $\lim\limits_{s\rightarrow \infty }x(s) = 0$. The proof is complete. □

Lemma 2.4. If $x$ is an eventually positive solution of Eq (1.1) and the corresponding function $z$ satisfies Case (II) of Lemma 2.1, then for $n\in \mathbb{N}$ and $s\in \left[t, \infty \right),$

Proof. Let $x$ be a positive solution of Eq (1.1) such that the Case (Ⅱ) of Lemma 2.1 is satisfied on $\left[s_{1}, \infty \right)$, for some $s_{1}\geq s_{0}$. Then, for $s\geq v\geq s_{1},$

Then

Integrating the above inequality from $t$ to $s\in \left[t, \infty \right)$ with respect to $v$, we obtain

This shows that (2.15) holds for $n = 1$. Consequently,

From (2.14) and (2.17), we obtain

where $\overline{q}_{\ast }(s) = kN^{\eta }q(s)\exp \left(\int_{s_{1}}^{s} \frac{m_{3}(r)}{m_{2}(r)}\; \mathrm{d}r\right).$ Integrating the latter inequality from $u$ to $s\in \left[u, \infty \right)$ gives

From (2.16) and (2.19), we obtain

It follows that

Therefore,

Then,

This shows that (2.15) holds for $n = 2$. To obtain (2.15) for arbitrary $n\in \mathbb{N}$, this procedure can be done $n$ times. □

Theorem 2.3. Let $\rho (s)$ be nondecreasing on ${[}$$s$$_{0}, \infty)$. Suppose there exists $n\in \mathbb{N}$ such that one of the following first-order delay differential equations (2.9) is oscillatory and

Then Eq (1.1) is oscillatory.

Proof. Assume that $x(s)$ is a nonoscillatory solution of Eq (1.1). Without loss of generality, let $x(s) > 0$, $x(\rho (s)) > 0,$ and $x(\mu (s)) > 0$ on ${[}$$s$$_{1}, \infty)$, $s$$_{1}\geq$$s$$_{0}$. It follows from Lemma 2.1 that there exists $s_{2}\geq s_{1}$ such that either (Ⅰ) or (Ⅱ) holds on ${[}$$s$$_{1}, \infty)$. The proof of Case (Ⅰ) is identical to the proof of Theorem 2.1, Case (Ⅰ), and so it has been omitted. Assume (Ⅱ) is valid. As in the proof of Theorem 2.2, Case (Ⅱ), we have

Integrating the above inequality from $\rho (s)$ to $s$, we obtain

In view of the nondecreasing nature of $\rho$ and (2.15), we obtain

This is a contradiction with (2.20). The proof is complete. □

Applying the results of ^{[27,28,30,31]} in Theorems 2.1–2.3, we get the asymptotic behavior of the solutions to Eq (1.1).

Corollary 2.1. Let $\rho (s)$ be nondecreasing on ${[}$$s$$_{0}, \infty)$. Suppose there exists $n\in \mathbb{N}$ such that one of the following conditions is satisfied:

$\mathit{\pmb{(a)}}$ $\liminf_{s\rightarrow \infty }\int_{\rho (s)}^{s} \overline{q}(t)P_{n}^{\eta }(\rho (t))\; \mathrm{d}t > \dfrac{1}{\mathrm{e}};$

$\mathit{\pmb{(b)}}$ $\limsup_{s\rightarrow \infty }\int_{\rho (s)}^{s} \overline{q}(t)P_{n}^{\eta }(\rho (t))\; \mathrm{d}t > 1;$

$\mathit{\pmb{(c)}}$ $\liminf_{s\rightarrow \infty }\int_{\rho (s)}^{s} \overline{q}(t)P_{n}^{\eta }(\rho (t))\; \mathrm{d}t > \alpha$ and $\limsup_{s\rightarrow \infty }\int_{\rho (s)}^{s}\overline{q}(t)P_{n}^{\eta}(\rho (t))\; \mathrm{d}t > 1-\left(1-\sqrt{1-\alpha }\right) ^{2}.$

If $x(s)$ is a solution of Eq (1.1), then

(I) $x(s)$ is either oscillatory or bounded;

(II) $x(s)$ is either oscillatory or tends to zero eventually if (2.11) holds;

(III) $x(s)$ is oscillatory if (2.20) holds.

Remark 2.1. We note that Theorems 2.2 and 2.3 are reduced to [10, Theorems 1 and 2] when $\eta_1 = \eta_2 = 1$, $m_3(s) = 0$, $k = 1$, and $p(s) = 0.$

3.   Numerical examples

Examples are provided to demonstrate the significance of our results.

Example 3.1. Consider the third-order nonlinear neutral differential equation with a damping term of the form

where $m_{1}(s) = s-1, \ m_{2}(s) = \frac{1}{s}, \ m_{3}(s) = \frac{1}{s^{2}}, \ q(s) = 6s, \ \mu (s) = \frac{s}{2}, \ \rho (s) = s-1, \ p(s) = \frac{1}{2}, \ \eta _{1} = 3,$ and $\eta _{2} = 1$. Using Maple software, we see that condition $(iv)$ holds and $\overline{q}(s) = \frac{3}{4}s^{2}$.

Also, we have, for $n = 1$

Then, according to Corollary 2.1, every solution to Eq (3.1) is either oscillatory or tends to zero as $s\rightarrow \infty$.

Example 3.2. Consider the third-order nonlinear neutral differential equation with a damping term of the form

where $m_{1}(s) = \frac{1}{9s^2}, \ m_{2}(s) = \frac{1}{s^2}, \ m_{3}(s) = \frac{2}{s^{3}}, \ q(s) = \frac{42}{s^{7}}, \ \mu (s) = s-1, \ \rho (s) = \frac{s}{2}, \ p(s) = \frac{1}{3}, \eta _{1} = \eta _{2} = 1$, $k = \frac{1}{3},$ and $N = \frac{1}{2}$. Using Maple software, we see that condition $(iv)$ holds and $\overline{q}(s) = \frac{28}{3s^5}, q_{\ast }(s) = \frac{7}{s^5}$. Also, we have, for $n = 1,$

and

Thus, $(a)$ is satisfied for $n = 1,$ and (2.20) is satisfied for $n = 2$. Then, according to Corollary 2.1, every solution to Eq (3.2) is oscillatory.

Author contributions

Taher S. Hassan: Supervision, Writing-review editing, Software and Investigation; Emad R. Attia: Supervision, Writing-review editing, Software and Investigation; Bassant M. ElMatary: Supervision, Writing-original draft, Writing-review editing, Software and Investigation. All authors have read and agreed to the published version of the manuscript

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools 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-2024-9/1). This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this article.