Fourth-order neutral dynamic equations oscillate on timescales with different arguments

1.   Introduction

The study of neutral dynamic systems that describe oscillatory processes is currently one of the recognized areas of modern nonlinear dynamics. Analysis of the connectivity of systems based on their time series is one of the current areas of modern science. The classical approach for this purpose is the Granger causality method ^[1]. The main idea of this method is to build a predictive model and if data from the first time series helps to more accurately predict the behavior of the second, then the first system is considered to influence the second. For a review of various variants of the Granger causality method and similar measures, see ^[2]. Hilger ^[3] proposed the timescale hypothesis to connect, broaden and summarize ideas from discrete math, quantum analytics, and persistent math to self-assured timescale analytics. Bohner and Peterson's books on timescale, or measure chain ^[4,5], summarize and compile a lot of timescale mathematics. The applied hypothesis of dynamic conditions associated with the study of oscillatory miracles in creativity, normal behavior, and sociologies includes the hypothesis of movements as a key component. An essential component of the connected hypothesis of energetic conditions associated with the inquiry of oscillatory wonders in innovation, normal science and social science is the notion of oscillations, see ^[6,7,8]. In recent years, the evolution of different energy situations over timescales has attracted a lot of interest.

Let $t\in \mathcal{I}_{t_{0}} = \left[t_{0}, +\infty \right) \cap \mathbb{T}$. We explore the dynamic equation of the type

where

and $\nu \geq 1$ is the ratio of two odd positive integers. The Eq (1.1) will be studied under the following assumptions:

$\left(\mathcal{C}_{1}\right)$ The function $\Phi \in \mathcal{C}\left(\mathbb{R }, \mathbb{R}\right)$ and there exists $r\in \mathcal{C}^{1}\left(\mathcal{I}_{t_{0}}, \left[0, +\infty \right) \right)$, such that

$\left(\mathcal{C}_{2}\right)$ $\tau, \upsilon \in \mathcal{C} _{rd}^{1}\left(\mathcal{I}_{t_{0}}, \mathcal{I}_{t_{0}}\right),$ $p\in \mathcal{C}^{1}\left(\mathcal{I}_{t_{0}}, \left[0, +\infty \right) \right)$ such that

and

By the solution of (1.1), we mean a nontrivial real-valued function

Note that solutions which disappear in a neighborhood of infinity will be excluded from our consideration. A solution $\varphi$ of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory. We are interested in this type of research because there has recently been an increasing interest in obtaining suitable conditions for the oscillation and non-oscillation of combinations of various energy conditions on timescales. See, for example, the references ^[9,10,11].

J. Džurina et al. ^[9] studied oscillation for the second-order noncanonical delay differential equations

under the condition

There have been any recent occurrences on the oscillatory of (1.1). The purpose of this research is to offer a few ambiguous criteria for this condition. Zhang et al. ^[10] studied a class of second-order nonlinear delay dynamic equations of neutral theory that was investigated by

where $\nu \geq 1$ is a ratio of odd integers and $\varphi \left(t\right) = u\left(t\right) -p\left(t\right) u\left(\tau \left(t\right) \right)$.

Grace et al. ^[11] studied oscillation of fourth-order delay differential equations

under the assumption

Some of the most common examples of calculus on timescales are differential calculus, difference calculus, and quantum calculus. Timescale dynamical equations have enormous potential for applications such as population dynamics. It can model insect populations that persist during a season, die in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population. There are applications of dynamical equations on timescales in quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. Notice that the model of problem (1.1) is more general than all the problems considered before. For example, if we consider problem $\left(1\right)$, $\mathbb{T} = \mathbb{R}$, and $\Phi \left(t, u\left(t\right) \right) = q\left(t\right) u\left(t\right)$, we obtain the problem which is considered by a recent article by J. Džurina et al. in ^[9]; it discusses several possible applications.

2.   Main results

In this section, we establish some criteria that guarantee that $u$ of (1.1) oscillates on $\mathcal{I}_{t_{0}}$. Before stating the main results, we begin with the following lemma.

Lemma 2.1. ^[12] Assume that $u$ is an eventually positive solution of (1.1). Then, there are only the following three possible scenarios for $\mathcal{I}_{t_{1}}$, where $t_{1}\in \mathcal{I}_{t_{0}}$ is sufficiently large:

(1) $\varphi ^{\Delta ^{3}}\left(t\right) \geq 0,$ $\varphi ^{\Delta ^{2}}\left(t\right) \leq 0,$ $\varphi ^{\Delta }\left(t\right) \geq 0,$

(2) $\varphi ^{\Delta ^{3}}\left(t\right) \leq 0,$ $\varphi ^{\Delta ^{2}}\left(t\right) \geq 0,$ $\varphi ^{\Delta }\left(t\right) \geq 0,$

(3) $\varphi ^{\Delta ^{3}}\left(t\right) \leq 0,$ $\varphi ^{\Delta ^{2}}\left(t\right) \geq 0,$ $\varphi ^{\Delta }\left(t\right) \leq 0.$

Here,

Theorem 2.1. Assume that there exist three functions $\zeta _{1}, \zeta _{2}, \gamma \in \mathcal{C}_{rd}^{1}\left(\mathcal{I}_{t_{0}}, \left(0, \infty \right) \right)$ such that

and

where

Then every solution $u$ of (1.1) is oscillatory.

Proof. Suppose, on the contrary, that $u$ is a positive solution of (1.1) on $\left[t_{0}, +\infty \right)$. Then, there exists $t_{1}\in \mathcal{I}_{t_{0}}$, such that $u\left(\tau \left(t\right) \right) > 0,$ $u\left(\upsilon \left(t\right) \right) > 0$, and $u\left(t\right) > 0$, for all $t\geq t$. Assume that $\varphi$ fulfills (2.1) i in the claims of Lemma 2.1. Let

Then, $\Xi \left(t\right) > 0$, for $t\in \mathcal{I}_{t_{1}}$ and

Since $\tau \in \mathcal{C}^{1}\left(\mathcal{I}_{t_{0}}, \mathcal{I} _{t_{0}}\right)$, then $u\circ \tau \in \mathcal{C}^{1}\left(\mathcal{I} _{t_{0}}, \mathcal{I}_{t_{0}}\right)$, and we have

Then

Since $p^{\Delta }\left(t\right) \leq 0$ and $\tau ^{\Delta }\left(t\right) \geq 0$, for all $t\in \mathcal{I}_{t_{1}}$, then

Since $\tau$ is a bijective function, we obtain

Integrate (1.1) from $t$ to $s$, we get

When $s$ tends to $\infty$ in the above inequality, we obtain

which implies that

Integrate from $t$ to $\infty$ to get

Since $u^{\Delta }\left(t\right) \geq 0$ and $\tau \left(t\right) \leq t$, for all $t\in \mathcal{I}_{t_{1}}$, then

Substituting (2.6) in (2.5), we get

Substituting (2.7) in (2.4), we get

Since $\varphi ^{\Delta ^{2}}\left(t\right) \leq 0$, for all $t\in \mathcal{ I}_{t_{1}}$, we have

Therefore,

is nonincreasing on $\mathcal{I}_{t_{1}}$, then

Substituting (2.9) in (2.8), we have

As in ^[12], we obtain

Integrating from $t_{1}$ to $t$, we have

which contradicts (2.1).

Suppose that $\varphi$ satisfies 2.1 of Lemma 2.1. Integrating (1.1) from $t$ to $\infty$, we get

Since $\varphi ^{\Delta }\left(t\right) \geq 0,$ for all $t\in \mathcal{I} _{t_{1}},$ $u$ is decreasing on $\mathcal{I}_{t_{1}}$, and by (2.6), we have

Let

Then,

Then, by $\varphi ^{\Delta ^{3}}\left(t\right) \leq 0, \ \forall t\in \mathcal{I}_{t_{1}}$, we have

Therefore, $\frac{\varphi ^{\Delta }}{t-t_{1}}$ is a nonincreasing function $\mathcal{I}_{t_{1}}$, then

Thus $\frac{\varphi }{h_{2}\left(., t_{1}\right) }$is a nonincreasing function $\mathcal{I}_{t_{1}}$, and we get

Substituting (2.12) in (2.10), we get

As in ^[12], we obtain

Integrate from $t_{1}$ to $t$, we have

which contradicts (2.2).

Suppose that $u$ satisfies claim 2.1 of Lemma 2.1. By the definition of $\varphi$, we have $\varphi \left(t\right) \geq u\left(t\right)$, and from (1.1), we have

Thus,

is decreasing on $\mathcal{I}_{t_{1}}$, then, for any $s\geq \mathcal{I}_{t_{1}}$, we have

Then,

Integrating from $t$ to $v$,

When $v$ tends to $\infty$ in the above inequality, we obtain

thus

Therefore,

is an increasing function on $\mathcal{I}_{t_{1}}$. Thus, we get

Then,

Thus,

is a nonincreasing function on $\mathcal{I}_{t_{1}}$, and we have

then

Thus,

is an increasing function on $\mathcal{I}_{t_{1}}$, and we have

Moreover, by the definition of $\varphi$ and $\left(2.18\right)$, we have

From (2.19), and (2.13), we get

From (2.14)–(2.16), we obtain

Let

From (2.21), we have $\omega \left(t\right) \geq 0$, for all $t\in \mathcal{I}_{t_{1}}$ and

By Pöotzsche's chain rule [4, Theorem 1.90], we get

and

Substituting (2.20), (2.23), and (2.24) in (2.22), we have

Since $\varphi ^{\Delta }\left(t\right) < 0$ and $\left(\pi _{2}^{-1}\left(t\right) \varphi \left(t\right) \right) ^{\Delta } > 0$, for all $t\in \mathcal{I}_{t_{1}}$, we obtain

By (2.14) and (2.15), we get

Substituting (2.26) and (2.27) in (2.25), we have

We conclude that

Thus,

Then,

Integrating from $t_{1}$ to $t$, we have

which contradicts (2.3). □

Corollary 2.1. If

then every solution $u$ of (1.1) is oscillatory.

Proof. Similar to the proof of Theorem 2.1, we put

in Eqs (2.1)–(2.3), and we find (2.28) □

Theorem 2.2. Assume that there exist three functions $\zeta \in \mathcal{C} _{rd}^{1}\left(\mathcal{I}_{t_{0}}, \left(0, \infty \right) \right)$ such that (2.1) holds and

Then, every solution $u$ of (1.1) is oscillatory.

Proof. Suppose, on the contrary, that $u$ is a positive solution of $\left(1.1 \right)$ on $\mathcal{I}_{t_{0}}$. Then, there exists $t_{1}\in \mathcal{I} _{t_{0}}$, such that

From (1.1), we have that $q\left(t\right) \left(\varphi ^{\Delta ^{3}}\left(t\right) \right) ^{ \nu }$ is decreasing for all $t\in \mathcal{I}_{t_{1}}$. We claim that $q\left(t\right) \left(\varphi ^{\Delta ^{3}}\left(t\right) \right) ^{ \nu } > 0$, for all $t\in \mathcal{I }_{t_{1}}$. If not, then there exist a $t_{2}\in \mathcal{I}_{t_{1}}$ such that

where $m > 0$. Integrating from $t_{2}$ to $t$, we obtain

This gives

then

which is a contradiction. □

As some applications of the main results, we present the following example.

Example 2.1. Consider a fourth-order half-linear delay dynamic equation

Here, $\mathbb{T} = \mathbb{R}$, $\nu = 1$, $q\left(t\right) = p\left(t\right) = 1$, $r\left(t\right) = t^{-3}$, $\tau \left(t\right) = \frac{t}{2}$, and $\upsilon \left(t\right) = 2t$.

Set $\zeta \left(t\right) = 1$, then

Thus, (2.1) holds. By Theorem 2.2, Eq (2.29) is oscillatory.

Remark 2.1. If fourth-order hybrid nonlinear functional dynamic equations with damping on timescale are considered

on an arbitrary timescale $\mathbb{T}$ with $\sup \mathbb{T} = \infty$, with the function

such that

and there exists $\psi \in \mathcal{C}\left(\mathcal{I}_{t_{0}}, \left[0, +\infty \right) \right)$, so that

Thus, (1.1) becomes a special case of (2.30) in a case $h\equiv 1$. Using the same method presented in this research, we can obtain some oscillation criteria for (2.30). It means obtaining generalizations of Theorems 2.1 and 2.2.

3.   Conclusions

The investigation of sufficient circumstances ensures oscillation of all fourth-order neutral functional differential equation delay solutions. However, this problem remains largely open for future research. We used the generalized Riccati transformations to present a few new theorems for the oscillation of (1.1). Our results here support some well-known results that have recently been published in the literature. A certain number of results reported in the literature are supplemented by the results provided; for more details, see ^[13,14,15].

Furthermore, our results can be extended to study a class of systems of higher order hybrid advanced differential equations, for example,

where

and $\nu \geq 1$ is the ratio of two odd positive integers, see ^{[16,17,18,19]}. It will be our next work.

Author contributions

Abdelkader Moumen: Methodology, Formal analysis; Amin Benaissa Cherif: Writing-original draft preparation; Fatima Zohra Ladrani: Writing-original draft preparation; Keltoum Bouhali: Supervision; Mohamed Bouye: Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large group research project under grant number RGP2/281/45.

Conflict of interest

The authors declare that there is no conflict of interest.