Oscillation Theorems of Fourth-Order Differential Equations with a Variable Argument Using the Comparison Technique

Abstract

In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation $(a$$\varphi \left(u\right)$$u^{‴})$+q$(u\circ h)=0$, which is of a functional type with a delay. The oscillation behavior of solutions of fourth-order delay equations has been studied using many techniques, but previous results did not take into account the existence of the function $\varphi$ except in second-order studies. The existence of $\varphi$ increases the difficulty of obtaining monotonic and asymptotic properties of the solutions and also increases the possibility of applying the results to a larger area of special cases. We present two criteria to ensure the oscillation of the solutions of the studied equation for two different cases of $\varphi$. Our approach is based on using the comparison principle with equations of the first or second order to benefit from recent developments in studying the oscillation of these orders. We also provide several examples and compare our results with previous ones to illustrate the novelty and effectiveness.

1. Introduction

Because of their symmetry and structural complexity, higher-order differential equations, especially even ones, are frequently used to simulate dynamic systems. Stability and periodicity are important characteristics of dynamic and periodic systems, where these qualitative solutions are commonly found. Since it is difficult, if not impossible, to derive closed-form solutions for these complicated systems, it is imperative to investigate their properties, particularly their oscillatory behavior. Refer to [1,2,3] for more details. Gaining knowledge of the circumstances that cause the system to oscillation can help one regulate and anticipate how the system will react. This offers adequate oscillation criteria for even-order equation solutions, which is of great theoretical and practical significance for specialized study.

In order to improve material design, analyze beam and slab vibrations, and guarantee structural stability, higher-order equations are essential. They are employed in electrical engineering to design signal filters and oscillators, which are essential parts of electronic and communication systems [4]. Furthermore, these equations are used in biomedical engineering to model intricate processes like nerve signal propagation and blood flow dynamics, yielding data for improving medical technologies [5]. When taken as a whole, these applications demonstrate how crucial it is to conduct more research in this area in order to meet changing scientific and technological problems.

In this work, we consider the delay differential equation (DDE)

$${\left(a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right)\right)}^{\prime }+p\left(\nu \right)y\left(h\left(\nu \right)\right)=0,$$

(1)

where $\nu \ge {\nu }_0$. During this study, we assume the following:

(H1)

$a,$$p\in \mathbf{C}\left(\left[{\nu }_0,\infty \right),\left[0,\infty \right)\right)$, $a\left(\nu \right)>0$, and

$${\int }_{{\nu }_1}^{\nu }{a}^{-1/r}\left(w\right)\mathrm{d}w\to \infty \mathrm{as}\nu \to \infty ;$$

(H2)

$h\in \mathbf{C}\left(\left[{\nu }_0,\infty \right),\mathbb{R}\right)$, $h\left(\nu \right)\le \nu$, $h^{\prime }\left(\nu \right)\ge 0$, and ${lim}_{\nu \to \infty }h\left(\nu \right)=\infty ;$

(H3)

$\varphi \in {\mathbf{C}}^1\left(\mathbb{R},\mathbb{R}\right)$, $\varphi \left(y\right)>0$ for $y\ne 0$, and $\varphi \left(y\right)\le M$ where $M\in {\mathbb{R}}^{+}$ (positive real numbers).

A function $y\in {\mathbf{C}}^3\left(\left[{\nu }_y,\infty \right),\mathbb{R}\right),$${\nu }_y\ge {\nu }_0$ is said to be a proper solution of (1) if $\left(a·\varphi \left(y\right)·y^{‴}\right)\in {\mathbf{C}}^1\left(\left[{\nu }_y,\infty \right),\mathbb{R}\right)$, y satisfies (1) on $\left[{\nu }_y,\infty \right)$, and $sup\left\{\left|y\left(\nu \right)\right|:\nu \ge {\nu }_1\right\}>0$ for every ${\nu }_1\ge {\nu }_y$. If a solution y of (1) has arbitrarily large zeros, it is considered oscillatory; if not, it is called non-oscillatory. If every solution to (1) oscillates, the equation is said to be oscillatory.

The study of the oscillatory behavior of even-order DDE has developed through several stages. In 1973, Dahiya and Bhagat [6] discussed the issue of oscillation of the DDE

$$y^{\left(n\right)}+p{\left(y\left(\nu -h\left(\nu \right)\right)\right)}^{\beta }=0,$$

where $n\in {\mathbb{Z}}^{+}$ is even. Later, Grace and Lalli [7] investigated the oscillatory behavior of the DDEs

$$y^{\left(n\right)}+p\left(F\circ y\circ h\right)=0$$

(2)

and

$$y^{\left(n\right)}+\rho {\left|y^{\left(n-1\right)}\right|}^{\beta }{y}^{\left(n-1\right)}+p\left(F\circ y\circ h\right)=0,$$

using the weighted integral approach. These results were improved in [8]. This was followed by a study by Zafer [9] who dealt with the oscillation of the DDE

$${\left[y+\rho \left(y\circ g\right)\right]}^{\left(n\right)}+F\left(\nu ,y,\left(y\circ g\right)\right)=0.$$

(3)

Moreover, Agarwal et al. [10] and Zhang and Yan [11] studied the oscillation of (3) and improved the results in [9].

Agarwal and Grace [12] investigated the oscillatory behavior of the DDEs

$${\left[{\left(y^{\left(n-1\right)}\right)}^{\alpha }\right]}^{\prime }+p\left(F\circ y\circ h\right)=0,$$

using comparison with first-order equations.

In this work, we present a new comparison criterion that tests the oscillation of Equation (1) by comparing it with second-order equations. To the best of our knowledge, previous work has studied the oscillatory behavior of fourth-order equations in the case where $\varphi \left(y\right)=1$; see [6,7,8,9,10,11,12]. The new criterion develops and extends previous results in the literature. The examples and comparisons in the last section illustrate the novelty and effectiveness of our results.

2. Main Results

Here, we suppose that $\mathbb{Y}$ is the class of all eventually positive solutions to Equation (1). Moreover, we show whether the function is increasing or decreasing by using the symbols ↑ and ↓, respectively. For convenience, we define, for $\nu \ge {\nu }_1$,

$${\mathcal{A}}_0\left(\nu \right):={\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{1}{a\left(l\right)}}\mathrm{d}l,$$

$${\mathcal{A}}_1\left(\nu \right):={\int }_{{\nu }_1}^{\nu }{\mathcal{A}}_0\left(l\right)\mathrm{d}l,{\widehat{\mathcal{A}}}_1\left(\nu \right):={\int }_{{\nu }_1}^{\nu }{\mathcal{A}}_0^{\kappa }\left(l\right)\mathrm{d}l$$

$${\mathcal{A}}_2\left(\nu \right)={\int }_{{\nu }_1}^{\nu }{\mathcal{A}}_1\left(l\right)\mathrm{d}l,\mathrm{and}{\widehat{\mathcal{A}}}_2\left(\nu \right)={\int }_{{\nu }_1}^{\nu }{\widehat{\mathcal{A}}}_1\left(l\right)\mathrm{d}l,$$

where ${\nu }_1\ge {\nu }_0$ is large enough.

We will consider the following two cases for the function $\varphi$:

(N1)

$\varphi \left(y\right)\ge m$ and $\kappa :=M/m$, where $m\in {\mathbb{R}}^{+}$.

(N2)

$y{\varphi }^{\prime }\left(y\right)>0$ for $y\ne 0$.

2.1. Auxiliary Lemmas

Based on Lemma 2.2.1 in [13], the solutions in class $\mathbb{Y}$ have the following monotonic properties.

Lemma 1

([13], Lemma 2.2.1). Suppose that $y\in \mathbb{Y}$. Then, $y\left(\nu \right)↑$, $y^{″}\left(\nu \right)↑$, $\left(a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right)\right)↓$, and $y^{\prime }\left(\nu \right)$ is either ↑ or ↓.

Lemma 2.

Suppose that $y\in \mathbb{Y}$, and $y^{\prime }\left(\nu \right)↓$. Then, the DDE

$$y^{″}\left(\nu \right)+\left({\displaystyle \frac{1}{M}}{\int }_{\nu }^{\infty }{\int }_w^{\infty }{\displaystyle \frac{1}{a\left(w\right)}}p\left(l\right)\mathrm{d}l\mathrm{d}w\right)y\left(h\left(\nu \right)\right)=0$$

(4)

has a positive solution.

Proof. 

Let $y\in \mathbb{Y}$ and $y^{\prime }\left(\nu \right)↓$. Integrating (1) from $\nu$ to ∞, we obtain

$$a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right)\ge {\int }_{\nu }^{\infty }p\left(l\right)y\left(h\left(l\right)\right)\mathrm{d}l.$$

Since $y\left(\nu \right)↑$ and $h\left(\nu \right)↑$, we have $y\left(h\left(l\right)\right)\ge y\left(h\left(v\right)\right)$ for $l\ge v$, and so

$$a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right)\ge y\left(h\left(\nu \right)\right){\int }_{\nu }^{\infty }p\left(l\right)\mathrm{d}l.$$

Hence,

$$y^{‴}\left(\nu \right)\ge {\displaystyle \frac{1}{M}}{\displaystyle \frac{1}{a\left(\nu \right)}}y\left(h\left(\nu \right)\right){\int }_{\nu }^{\infty }p\left(l\right)\mathrm{d}l.$$

(5)

By integration again, we conclude that

$$y^{″}\left(\nu \right)\le -{\displaystyle \frac{1}{M}}{\int }_{\nu }^{\infty }{\int }_w^{\infty }{\displaystyle \frac{1}{a\left(w\right)}}p\left(l\right)y\left(h\left(w\right)\right)\mathrm{d}l\mathrm{d}w,$$

which implies that

$$y^{″}\left(\nu \right)+\left({\displaystyle \frac{1}{M}}{\int }_{\nu }^{\infty }{\int }_w^{\infty }{\displaystyle \frac{1}{a\left(w\right)}}p\left(l\right)\mathrm{d}l\mathrm{d}w\right)y\left(h\left(\nu \right)\right)\le 0.$$

According to Corollary 1 in [14], DDE (4) also has a positive solution, which completes the proof. □

Lemma 3.

Suppose that $y\in \mathbb{Y}$, $y^{\prime }\left(\nu \right)↑$, and (N1) holds. Then,

$$\left({\displaystyle \frac{y^{″}}{{\mathcal{A}}_0^{\kappa }}}\right)↓$$

(6)

and

$$\left({\displaystyle \frac{y^{\left(i\right)}}{{\widehat{\mathcal{A}}}_{2-i}}}\right)↓\mathrm{for}i=0,1.$$

(7)

Proof. 

Let $y\in \mathbb{Y}$ and $y^{\prime }\left(\nu \right)↑$. Using the monotonic properties of y (Lemma 1), we have

$$\begin{array}{ccc}\hfill y^{″}\left(\nu \right)& \ge & {\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{a\left(l\right)\varphi \left(y\left(l\right)\right)y^{‴}\left(l\right)}{a\left(l\right)\varphi \left(y\left(l\right)\right)}}\mathrm{d}l\hfill \\ & \ge & a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right){\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{1}{a\left(l\right)\varphi \left(y\left(l\right)\right)}}\mathrm{d}l\hfill \\ & \ge & {\displaystyle \frac{1}{\kappa }}a\left(\nu \right)y^{‴}\left(\nu \right){\mathcal{A}}_0\left(\nu \right).\hfill \end{array}$$

So,

$$\left({\displaystyle \frac{y^{″}}{{\mathcal{A}}_0^{\kappa }}}\right)↓.$$

This implies that

$$y^{\prime }\left(\nu \right)\ge {\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{y^{″}\left(l\right)}{{\mathcal{A}}_0^{\kappa }\left(l\right)}}{\mathcal{A}}_0^{\kappa }\left(l\right)\mathrm{d}l\ge {\displaystyle \frac{{\widehat{\mathcal{A}}}_1\left(\nu \right)}{{\mathcal{A}}_0^{\kappa }\left(\nu \right)}}{y}^{″}\left(\nu \right).$$

It follows that

$$\left({\displaystyle \frac{y^{\prime }}{{\widehat{\mathcal{A}}}_1}}\right)↓.$$

By repeating this procedure one last time, we find that ${\widehat{\mathcal{A}}}_1\left(\nu \right)y\left(\nu \right)\ge {\widehat{\mathcal{A}}}_2\left(\nu \right)y^{\prime }\left(\nu \right)$ and

$$\left({\displaystyle \frac{y}{{\widehat{\mathcal{A}}}_2}}\right)↓.$$

This completes the proof. □

Lemma 4.

Suppose that $y\in \mathbb{Y}$, $y^{\prime }\left(\nu \right)↑$, and (N2) holds. Then,

$$\left({\displaystyle \frac{y^{″}}{{\mathcal{A}}_0}}\right)↓$$

(8)

and

$$\left({\displaystyle \frac{y^{\left(i\right)}}{{\mathcal{A}}_{2-i}}}\right)↓\mathrm{for}i=0,1.$$

(9)

Proof. 

Let $y\in \mathbb{Y}$ and $y^{\prime }\left(\nu \right)↑$. Since ${\varphi }^{\prime }\left(y\right)>0$, we have that

$$0\ge {\left(a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right)\right)}^{\prime }\ge \varphi \left(y\left(\nu \right)\right){\left(a\left(\nu \right)y^{‴}\left(\nu \right)\right)}^{\prime }.$$

(10)

So, $\left(a{y}^{‴}\right)↓$. Thus,

$$y^{″}\left(\nu \right)\ge {\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{a\left(l\right)y^{‴}\left(l\right)}{a\left(l\right)}}\mathrm{d}l\ge a\left(\nu \right)y^{‴}\left(\nu \right){\mathcal{A}}_0\left(\nu \right).$$

Proceeding as in the proof of Lemma 3, we obtain that (8) and (9) hold. □

2.2. Oscillation Criterion

Theorem 1.

Assume that (N1) holds. If DDEs (4) and

$${\left({\displaystyle \frac{{\mathcal{A}}_0^{\kappa -1}\left(\nu \right)}{{\widehat{\mathcal{A}}}_1\left(\nu \right)}}{\phi }^{\prime }\left(\nu \right)\right)}^{\prime }+{\displaystyle \frac{1}{M\kappa }}{\displaystyle \frac{{\widehat{\mathcal{A}}}_2\left(h\left(\nu \right)\right)}{{\widehat{\mathcal{A}}}_2\left(\nu \right)}}p\left(\nu \right)\phi \left(\nu \right)=0$$

(11)

are oscillatory, then (1) is oscillatory.

Proof. 

Assume the contrary that $y\in \mathbb{Y}$. According to Lemma 2, the oscillation of Equation (4) necessarily excludes the case where $y^{\prime }\left(\nu \right)↓$. So, $y^{\prime }\left(\nu \right)↑$. From Lemma 3, we have

$$y^{\prime }\ge {\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{{\widehat{\mathcal{A}}}_1}{{\mathcal{A}}_0^{\kappa -1}}}a{y}^{‴}.$$

Now, we define

$$\omega :={\displaystyle \frac{a\varphi \left(y\right)y^{‴}}{y}}>0.$$

Then,

$$\begin{array}{ccc}\hfill {\omega }^{\prime }& =& {\displaystyle \frac{{\left(a\varphi \left(y\right)y^{‴}\right)}^{\prime }}{y}}-{\displaystyle \frac{a\varphi \left(y\right)y^{‴}}{y^2}}{y}^{\prime }\hfill \\ & \le & -p{\displaystyle \frac{y\left(h\right)}{y}}-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{a\varphi \left(y\right)y^{‴}}{y^2}}{\displaystyle \frac{{\widehat{\mathcal{A}}}_1}{{\mathcal{A}}_0^{\kappa -1}}}a{y}^{‴}\hfill \\ & \le & -p{\displaystyle \frac{y\left(h\right)}{y}}-{\displaystyle \frac{1}{M\kappa }}{\displaystyle \frac{{\widehat{\mathcal{A}}}_1}{{\mathcal{A}}_0^{\kappa -1}}}{\left({\displaystyle \frac{a\varphi \left(y\right)y^{‴}}{y}}\right)}^2,\hfill \end{array}$$

which with (7) yields

$${\omega }^{\prime }+{\displaystyle \frac{1}{M\kappa }}{\displaystyle \frac{{\widehat{\mathcal{A}}}_1}{{\mathcal{A}}_0^{\kappa -1}}}{\omega }^2+p{\displaystyle \frac{{\widehat{\mathcal{A}}}_2\left(h\right)}{{\widehat{\mathcal{A}}}_2}}\le 0.$$

(12)

From [15,16], every solution of (11) is non-oscillatory if and only if there exists a function $\omega \in C\left(\left[{\nu }_1,\infty \right),\mathbb{R}\right)$ satisfying (12), which is a contradiction.

This completes the proof. □

Theorem 2.

Assume that (N2) holds. If DDEs (4) and

$${\left(a\left(\nu \right){\upsilon }^{\prime }\left(\nu \right)\right)}^{\prime }+{\displaystyle \frac{1}{M}}{\displaystyle \frac{{\mathcal{A}}_2\left(h\left(\nu \right)\right)}{{\mathcal{A}}_0\left(h\left(\nu \right)\right)}}p\left(\nu \right)\upsilon \left(h\left(\nu \right)\right)=0$$

(13)

are oscillatory, then (1) is oscillatory.

Proof. 

Assume the contrary that $y\in \mathbb{Y}$. According to Lemma 2, the oscillation of Equation (4) necessarily excludes the case where $y^{\prime }\left(\nu \right)↓$. So, $y^{\prime }\left(\nu \right)↑$. From Lemma 4, we have

$$\begin{array}{ccc}\hfill M{\left(a\left(\nu \right)y^{‴}\left(\nu \right)\right)}^{\prime }& \le & \varphi \left(y\left(\nu \right)\right){\left(a\left(\nu \right)y^{‴}\left(\nu \right)\right)}^{\prime }\hfill \\ & \le & {\left(a\left(\nu \right)\varphi \left(y\left(\nu \right)\right)y^{‴}\left(\nu \right)\right)}^{\prime }\hfill \\ & =& -p\left(\nu \right)y\left(h\left(\nu \right)\right).\hfill \end{array}$$

(14)

Using (9), we find

$${\left(a\left(\nu \right)y^{‴}\left(\nu \right)\right)}^{\prime }+{\displaystyle \frac{1}{M}}{\displaystyle \frac{{\mathcal{A}}_2\left(h\left(\nu \right)\right)}{{\mathcal{A}}_0\left(h\left(\nu \right)\right)}}p\left(\nu \right)y^{″}\left(h\left(\nu \right)\right)\le 0.$$

(15)

Therefore, $\upsilon :=y^{″}$ is a positive solution to (15). According to Corollary 1 in [14], DDE (13) also has a positive solution, a contradiction.

This completes the proof. □

According to the precise oscillation criterion for second-order DDEs in [17], we can derive the following corollary for the oscillation of DDE (1). We will need the following conditions:

$$\begin{array}{c}{\lambda }_1:=\underset{\nu \to \infty }{liminf}{\displaystyle \frac{\nu }{h\left(\nu \right)}}<\infty ,\hfill \\ {\lambda }_2:=\underset{\nu \to \infty }{liminf}{\displaystyle \frac{{\mathcal{A}}_0\left(\nu \right)}{{\mathcal{A}}_0\left(h\left(\nu \right)\right)}}<\infty .\hfill \end{array}$$

(16)

and

$${\lambda }_1={\lambda }_2=\infty .$$

(17)

Corollary 1.

Equation (1) is oscillatory if

$$\underset{\nu \to \infty }{liminf}\left(\nu h\left(\nu \right){\int }_{\nu }^{\infty }{\int }_w^{\infty }{\displaystyle \frac{1}{a\left(w\right)}}p\left(l\right)\mathrm{d}l\mathrm{d}w\right)>M{\mu }_1$$

(18)

and one of the following cases is satisfied:

(i)

(N1) holds and

$$\underset{\nu \to \infty }{liminf}\left({\displaystyle \frac{{\mathcal{A}}_0^{\kappa -1}\left(\nu \right)}{{\widehat{\mathcal{A}}}_1\left(\nu \right)}}{\displaystyle \frac{{\widehat{\mathcal{A}}}_2\left(h\left(\nu \right)\right)}{{\widehat{\mathcal{A}}}_2\left(\nu \right)}}{\mathcal{R}}^2\left(\nu \right)p\left(\nu \right)\right)>{\displaystyle \frac{M\kappa }{4}};$$

(ii)

(N2) holds and

$$\underset{\nu \to \infty }{liminf}\left(a\left(\nu \right){\mathcal{A}}_0\left(\nu \right){\mathcal{A}}_2\left(h\left(\nu \right)\right)p\left(\nu \right)\right)>M{\mu }_2,$$

where

$${\mu }_i=\left\{\begin{array}{cc}\underset{k\in \left(0,1\right)}{max}\left\{k\left(1-k\right){\lambda }_i^{-k}\right\},\hfill & \mathrm{if}\left(16\right)\mathrm{holds};\\ 0\hfill & \mathrm{if}\left(17\right)\mathrm{holds},\end{array}\right.$$

for $i=1,2$, and

$$\mathcal{R}\left(\nu \right)={\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{{\widehat{\mathcal{A}}}_1\left(l\right)}{{\mathcal{A}}_0^{\kappa -1}\left(l\right)}}\mathrm{d}l.$$

2.3. Further Oscillation Results

In this section, we seek to obtain new monotonic and asymptotic properties of the function

$$z:={\displaystyle \frac{y^{″}}{{\mathcal{A}}_0}},$$

and employ these properties to obtain an oscillation criterion for (1) using comparison with a first-order equation.

Lemma 5.

Suppose that $y\in \mathbb{Y}$, $y^{\prime }\left(\nu \right)↑$, (N2) holds, and

$${\int }_{{\nu }_1}^{\infty }{\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l=\infty .$$

(19)

Then, ${lim}_{\nu \to \infty }z\left(\nu \right)=0$ and

$$\left(\rho \left(\nu \right)z\left(\nu \right)\right)↓,$$

(20)

where

$$\eta \left(\nu \right):={\displaystyle \frac{1}{M}}{\displaystyle \frac{1}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}{\int }_0^{\nu }{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l$$

and

$$\rho \left(\nu \right):=exp\left({\int }_{{\nu }_1}^{\nu }\eta \left(l\right)\mathrm{d}l\right).$$

Proof. 

Let $y\in \mathbb{Y}$ and $y^{\prime }\left(\nu \right)↑$. As in the proof of Theorem 2, we obtain that (14) holds. Now, we have

$$\begin{array}{ccc}\hfill {\left(a{\mathcal{A}}_0^2{z}^{\prime }\right)}^{\prime }& =& {\left(a{\mathcal{A}}_0^2{\left[{\displaystyle \frac{y^{″}}{{\mathcal{A}}_0}}\right]}^{\prime }\right)}^{\prime }\hfill \\ & =& {\left({\mathcal{A}}_{0}a{y}^{‴}-y^{″}\right)}^{\prime }\hfill \\ & =& {\mathcal{A}}_0{\left(a{y}^{‴}\right)}^{\prime }\hfill \\ & \le & -{\displaystyle \frac{1}{M}}{\mathcal{A}}_{0}py\left(h\right),\hfill \end{array}$$

which with (7) gives

$${\left(a{\mathcal{A}}_0^2{z}^{\prime }\right)}^{\prime }\le -{\displaystyle \frac{1}{M}}{\mathcal{A}}_0{\mathcal{A}}_2\left(h\right)pz\left(h\right)$$

(21)

Since $z>0$ and $z↓$, we have that ${lim}_{\nu \to \infty }z\left(\nu \right)=z_0\ge 0$. Supposing $z_0>0$, there is ${\nu }_1\ge {\nu }_0$ such that $z\left(\nu \right)\ge z_0$. An integration of (21) from ${\nu }_1$ to $\nu$ yields

$$\begin{array}{ccc}& & z^{\prime }\left(\nu \right)+{\displaystyle \frac{1}{M}}{\displaystyle \frac{1}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}{\int }_{{\nu }_1}^{\nu }{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)z\left(h\left(l\right)\right)\mathrm{d}l\hfill \\ & \le & -{\displaystyle \frac{L}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}<0.\hfill \end{array}$$

(22)

where $L:=-a\left({\nu }_1\right){\mathcal{A}}_0^2\left({\nu }_1\right)z^{\prime }\left({\nu }_1\right)>0$. An integration of (22) from ${\nu }_1$ to ∞ leads to

$$\begin{array}{ccc}& & z_0-z\left({\nu }_1\right)\hfill \\ & <& -{\displaystyle \frac{z_0}{M}}{\int }_{{\nu }_1}^{\infty }{\displaystyle \frac{1}{a\left(w\right){\mathcal{A}}_0^2\left(w\right)}}{\int }_{{\nu }_1}^w{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l\mathrm{d}w\hfill \\ & =& -{\displaystyle \frac{z_0}{M}}{\int }_{{\nu }_1}^{\infty }{\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l,\hfill \end{array}$$

which contradicts (19). Thus, $z_0=0$.

Next, from (22), we obtain

$$\begin{array}{ccc}& & z^{\prime }\left(\nu \right)\hfill \\ & \le & -{\displaystyle \frac{\frac{1}{M}}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}\left[L+{\int }_{{\nu }_1}^{\nu }{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)z\left(h\left(l\right)\right)\mathrm{d}l\right]\hfill \\ & \le & -{\displaystyle \frac{\frac{1}{M}}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}\left[L+z\left(h\left(\nu \right)\right){\int }_{{\nu }_1}^{\nu }{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l\right]\hfill \\ & \le & -{\displaystyle \frac{\frac{1}{M}}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}\left[L-z\left(h\left(\nu \right)\right){\int }_0^{{\nu }_1}{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l\right.\hfill \\ & & +\left.z\left(h\left(\nu \right)\right){\int }_0^{\nu }{\mathcal{A}}_0\left(l\right){\mathcal{A}}_2\left(h\left(l\right)\right)p\left(l\right)\mathrm{d}l\right].\hfill \end{array}$$

Since $z\left(\nu \right)\to 0$ as $\nu \to \infty$, we get that

$$z^{\prime }\left(\nu \right)+\eta \left(\nu \right)z\left(\nu \right)\le 0,$$

or

$${\left[z\left(\nu \right)exp\left({\int }_{{\nu }_1}^{\nu }\eta \left(l\right)\mathrm{d}l\right)\right]}^{\prime }\le 0.$$

This completes the proof. □

Theorem 3.

Assume that (N2) and (19) hold. If DDEs (4) and

$$z^{\prime }\left(\nu \right)+{\displaystyle \frac{1}{M}}{\displaystyle \frac{\rho \left(h\left(\nu \right)\right)}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}z\left(h\left(\nu \right)\right){\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{{\mathcal{A}}_2\left(h\left(l\right)\right)}{\rho \left(h\left(l\right)\right)}}{\mathcal{A}}_0\left(l\right)p\left(l\right)\mathrm{d}l=0$$

(23)

are oscillatory, then (1) is oscillatory.

Proof. 

Assume the contrary that $y\in \mathbb{Y}$. According to Lemma 2, the oscillation of Equation (4) necessarily excludes the case where $y^{\prime }\left(\nu \right)↓$. So, $y^{\prime }\left(\nu \right)↑$. As in the proof of Lemma 5, we have that (22) holds. Using (20) in (22), we find that z is a positive solution of

$$z^{\prime }\left(\nu \right)+{\displaystyle \frac{1}{M}}{\displaystyle \frac{\rho \left(h\left(\nu \right)\right)}{a\left(\nu \right){\mathcal{A}}_0^2\left(\nu \right)}}z\left(h\left(\nu \right)\right){\int }_{{\nu }_1}^{\nu }{\displaystyle \frac{{\mathcal{A}}_2\left(h\left(l\right)\right)}{\rho \left(h\left(l\right)\right)}}{\mathcal{A}}_0\left(l\right)p\left(l\right)\mathrm{d}l<0.$$

(24)

In view of Theorem 1 in [18], the DDE (23) also has a positive solution. □

Corollary 2.

Assume that (N2), (19), and (18) hold. If

$$\begin{array}{ccc}& & \underset{\nu \to \infty }{liminf}\left({\int }_{h\left(\nu \right)}^{\nu }{\displaystyle \frac{\rho \left(h\left(w\right)\right)}{a\left(w\right){\mathcal{A}}_0^2\left(w\right)}}{\int }_{{\nu }_1}^w{\displaystyle \frac{{\mathcal{A}}_2\left(h\left(l\right)\right)}{\rho \left(h\left(l\right)\right)}}{\mathcal{A}}_0\left(l\right)p\left(l\right)\mathrm{d}l\mathrm{d}w\right)\hfill \\ & >& {\displaystyle \frac{M}{\mathrm{e}}}\hfill \end{array}$$

(25)

then (1) is oscillatory.

According to Theorem 2.1.1 in [19], the existence of a positive solution to (23) contradicts the condition (25). Therefore, the proof of the previous corollary is straightforward.

2.4. Examples and Comparison

Example 1.

Consider the Euler DDE

$$y^{\left(4\right)}\left(\nu \right)+{\displaystyle \frac{p_0}{{\nu }^4}}y\left(\beta \nu \right)=0,$$

(26)

where $p_0>0$ and $\beta \in \left(0,1\right]$. Based on Corollary 1, the oscillation criterion for this equation is

$$p_0>max\left\{{\displaystyle \frac{6{\mu }_1}{\beta }},{\displaystyle \frac{6{\mu }_2}{{\beta }^3}}\right\}.$$

(27)

Using Corollary 2, Equation (26) is oscillatory

$$p_0>{\displaystyle \frac{6{\mu }_1}{\beta }}$$

and

$${\displaystyle \frac{1}{6\left(1-\frac{1}{6}{p}_0{\beta }^3\right)}}{p}_0{\beta }^{3}ln\left(1/\beta \right)>{\displaystyle \frac{1}{\mathrm{e}}}.$$

Figure 1 shows an approximate solution to Equation (26) when $p_0=10$ and $\beta =0.17$.

Remark 1.

Consider Euler DDE (26) when $\beta =1/2$.

Table 1. Comparison of the oscillation criteria of Equation (26) when $\beta =1/2$.

Source	[20]	[9]	[8]	[21]	[22]	[27]
Criterion	$p_0>13,824$	$p_0>815.22$	$p_0>28.360$	$p_0>25.472$	$p_0>13$	$p_0>8.7403$

shows the oscillation criteria for the Equation (26) in this case.

Example 2.

Consider the DDE

$${\left({\displaystyle \frac{1}{1+{sin}^2\left(y\left(\nu \right)\right)}}{y}^{‴}\left(\nu \right)\right)}^{\prime }+{\displaystyle \frac{p_0}{{\nu }^4}}y\left(\beta \nu \right)=0,$$

(28)

where $p_0>0$ and $\beta \in \left(0,1\right]$. Observe that while $\varphi \left(y\right)={\left(1+{sin}^2\left(y\left(\nu \right)\right)\right)}^{-1}$ does not satisfy condition (N2), it does satisfy the condition (N1), since $1/2\le \varphi \left(y\right)\le 1$. Not that $\kappa =2$, ${\mathcal{A}}_0\left(\nu \right)=\nu$, ${\widehat{\mathcal{A}}}_1\left(\nu \right)={\displaystyle \frac{1}{3}}{\nu }^3$, ${\widehat{\mathcal{A}}}_2\left(\nu \right)={\displaystyle \frac{1}{12}}{\nu }^4$, and $\mathcal{R}\left(\nu \right)={\displaystyle \frac{1}{9}}{\nu }^3$. Based on Corollary 1, the oscillation criterion for this equation is

$$p_0>max\left\{{\displaystyle \frac{6{\mu }_1}{\beta }},{\displaystyle \frac{27}{2{\beta }^4}}\right\}.$$

Example 3.

Consider the DDE

$${\left({\nu }^{\alpha }\left(1-{\mathrm{e}}^{-y^2}\right)y^{‴}\left(\nu \right)\right)}^{\prime }+{\displaystyle \frac{p_0}{{\nu }^{4-\alpha }}}y\left(\beta \nu \right)=0$$

(29)

where $p_0>0$, $\alpha \in \left(0,1\right)$, and $\beta \in \left(0,1\right]$. Observe that while $\varphi \left(y\right)=1-{\mathrm{e}}^{-y^2}$ does not satisfy condition (N1), it does satisfy condition (N2), since $y{\varphi }^{\prime }\left(y\right)=$$2y^2{e}^{-y^2}>0$ for $y\ne 0$. We have

$${\mathcal{A}}_i\left(\nu \right)={\displaystyle \frac{1}{{\prod }_{k=0}^i\left(1+k-\alpha \right)}}{\nu }^{1+i-\alpha },$$

for $i=0,1,2$. Based on Corollary 1, the oscillation criterion for this equation is

$$p_0>max\left\{{\displaystyle \frac{6{\mu }_1}{\beta }},{\displaystyle \frac{{\left(1-\alpha \right)}^2\left(2-\alpha \right)\left(3-\alpha \right){\mu }_2}{{\beta }^{3-\alpha }}}\right\}.$$

Remark 2.

To our knowledge, there are no previous results using the approach taken in this study to test the oscillation of higher-order equations with the function $\varphi \left(y\right)$, such as (28) and (29). Therefore, we can say that this is a new finding, which does not appear to have been studied previously.

3. Conclusions

This work established novel oscillation criteria for solutions of the DDE (1). Numerous methods have been used to study the oscillation behavior of fourth-order DDEs; however, prior findings did not consider the existence of the function $\varphi$, with the exception of second-order studies. We considered two different cases of the function $\varphi$, one of which restricts it to bounds from below and the other of which imposes monotonic constraints on it. For two distinct situations of $\varphi$, we provided two conditions to guarantee the oscillation of the solutions of the examined equation. Our method is based on applying the comparison principle to first- or second-order equations in order to benefit from new developments in understanding the oscillation of higher-order equations. We illustrated the importance of the new results through examples, comparisons and remarks.

As there are no studies that have addressed the oscillatory behavior of higher-order differential equations with the function $\varphi$, our findings can be extended to scenarios like neutral, advanced, half-linear, and other equations through a variety of future research avenues.