**New Oscillation Criteria for Advanced Differential Equations of Fourth Order**

#### **Omar Bazighifan 1,2,† , Hijaz Ahmad 3,† and Shao-Wen Yao 4,\* ,†**


Received: 8 April 2020; Accepted: 2 May 2020; Published: 6 May 2020

**Abstract:** The main objective of this paper is to establish new oscillation results of solutions to a class of fourth-order advanced differential equations with delayed arguments. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Four examples are provided to illustrate the main results.

**Keywords:** advanced differential equations; oscillations; Riccati transformations; fourth-order delay equations

#### **1. Introduction**

In the last decades, many researchers have devoted their attention to introducing more sophisticated analytical and numerical techniques to solve mathematical models arising in all fields of science, technology and engineering. Fourth-order advanced differential equations naturally appear in models concerning physical, biological and chemical phenomena, having applications in dynamical systems such as mathematics of networks and optimization, and applications in the mathematical modeling of engineering problems, such as electrical power systems, materials and energy, also, problems of elasticity, deformation of structures, or soil settlement, see [1].

The present paper deals with the investigation of the oscillatory behavior of the fourth order advanced differential equation of the following form

$$\left(a\left(v\right)\left(y'''\left(v\right)\right)^{\beta}\right)' + \sum\_{i=1}^{j} q\_i\left(v\right)g\left(y\left(\eta\_i\left(v\right)\right)\right) = 0, \ v \ge v\_{0\prime} \tag{1}$$

where *j* ≥ 1 and *β* is a quotient of odd positive integers. Throughout the paper, we suppose the following assumptions:

*a* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)), *a* ′ (*υ*) ≥ 0, *q<sup>i</sup>* , *<sup>η</sup><sup>i</sup>* <sup>∈</sup> *<sup>C</sup>* ([*υ*0, <sup>∞</sup>), <sup>R</sup>), *<sup>q</sup><sup>i</sup>* (*υ*) <sup>≥</sup> 0, *<sup>η</sup><sup>i</sup>* (*υ*) <sup>≥</sup> *<sup>υ</sup>*, *<sup>i</sup>* <sup>=</sup> 1, 2, .., *<sup>j</sup>*, *<sup>g</sup>* <sup>∈</sup> *C* (R, R) such that *g* (*x*) /*x <sup>β</sup>* <sup>≥</sup> ℓ > 0, for *<sup>x</sup>* <sup>6</sup><sup>=</sup> 0 and under the condition

$$\int\_{v\_0}^{\infty} \frac{1}{a^{1/\beta} \left(s\right)} ds = \infty. \tag{2}$$

During this decade, several works have been accomplished in the development of the oscillation theory of higher order advanced equations by using the Riccati transformation and the theory of comparison between first and second-order delay equations. Further, the oscillation theory of fourth and second order delay equations has been studied and developed by using an integral averaging technique and the Riccati transformation, see [2–23].

In this paper, we are aimed to complement the results reported in [24–26], therefore we discuss their findings and results below.

Moaaz et al. [27] considered the fourth-order differential equation

$$\left( a\left(\upsilon\right) \left( y^{\prime\prime\prime}(\upsilon) \right)^{\gamma} \right)' + q\left(\upsilon\right) y^{\alpha}\left(\eta\left(\upsilon\right)\right) = 0, \alpha$$

where *γ*, *α* are quotients of odd positive integers.

Grace et al. [28] considered the equation

$$\left(a\left(\upsilon\right)\left(y''\left(\upsilon\right)\right)^{\gamma}\right)'' + q\left(\upsilon\right)g\left(y\left(\eta\left(\upsilon\right)\right)\right) = 0,\tag{3}$$

where *η* (*υ*) ≤ *υ*.

Zhang et al. in [29] studied qualitative behavior of the fourth-order differential equation

$$\left( a\left(\upsilon\right) \left(w^{\prime\prime\prime}\left(\upsilon\right)\right)^{\beta} \right)' + q\left(\upsilon\right)w\left(\sigma\left(\upsilon\right)\right) = 0,$$

where *σ* (*υ*) ≤ *υ*, *β* is a quotient of odd positive integers and they used the Riccati transformation. Agarwal and Grace [24] considered the equation

$$\left( \left( y^{\left( \mathbf{x} - 1 \right)} \left( \upsilon \right) \right)^{\beta} \right)' + q \left( \upsilon \right) y^{\beta} \left( \eta \left( \upsilon \right) \right) = \mathbf{0}, \tag{4}$$

where *κ* is even, and they established some new oscillation criteria by using the comparison technique. Among others, they proved it oscillatory if

$$\liminf\_{\upsilon \to \infty} \int\_{\upsilon}^{\eta(\upsilon)} \left( \eta \left( s \right) - s \right)^{\chi - 2} \left( \int\_{\eta(\upsilon)}^{\infty} q \left( \upsilon \right) d\upsilon \right)^{1/\beta} ds > \frac{(\kappa - 2)!}{\varepsilon}. \tag{5}$$

Agarwal et al. in [25] extended the Riccati transformation to obtain new oscillatory criteria for ODE (4) under the condition

$$\limsup\_{\upsilon \to \infty} \upsilon^{\beta(\kappa - 1)} \int\_{\upsilon}^{\infty} q\left(s\right) ds > \left( (\kappa - 1)! \right)^{\beta}. \tag{6}$$

Authors in [26] studied oscillatory behavior of Equation (4) where *β* = 1 and if there exists a function *τ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)), also, they proved oscillatory by using the Riccati transformation if

$$\int\_{\upsilon}^{\infty} \left( \tau \left( s \right) q \left( s \right) - \frac{\left( \kappa - 2 \right)! \left( \tau' \left( s \right) \right)^{2}}{2^{3 - 2\kappa} s^{\kappa - 2} \tau \left( s \right)} \right) ds = \infty. \tag{7}$$

To compare the conditions, we apply the previous results to the equation

$$y^{(4)}\left(\upsilon\right) + \frac{q\_0}{\upsilon^4}y\left(3\upsilon\right) = 0, \ \upsilon \ge 1,\tag{8}$$

1. By applying Condition (5) in [24], we get

*q*<sup>0</sup> > 13.6

2. By applying Condition (6) in [25], we get

*q*<sup>0</sup> > 18.

3. By applying Condition (7) in [26], we get

$$q\_0 > 576.$$

The main aim of this paper is to establish new oscillation results of solutions to a class of fourth-order differential equations with delayed arguments and they essentially complement the results reported in [24–26].

The rest of the paper is organized as follows. In Section 2, four lemmas are given to prove the main results. In Section 3, we establish new oscillation results for Equation (1), comparisons are carried out with oscillations of first and second-order delay differential equations and some examples are presented to illustrate the main results. Some conclusions are discussed in Section 4.

#### **2. Some Auxiliary Lemmas**

In this section, the following some auxiliary lemmas are provided

**Lemma 1** ([23])**.** *Suppose that y* ∈ *C κ* ([*υ*0, ∞),(0, ∞)), *y* (*κ*) *is of a fixed sign on* [*υ*0, ∞), *y* (*κ*) *not identically zero and there exists a υ*<sup>1</sup> ≥ *υ*<sup>0</sup> *such that*

$$y^{(\kappa -1)}\left(v\right)y^{(\kappa)}\left(v\right) \le 0\_{\kappa}$$

*for all υ* ≥ *υ*1*. If we have* lim*υ*→<sup>∞</sup> *y* (*υ*) 6= 0*, then there exists υ<sup>θ</sup>* ≥ *υ*<sup>1</sup> *such that*

$$y\left(\upsilon\right) \ge \frac{\theta}{\left(\kappa - 1\right)!} \upsilon^{\kappa - 1} \left| y^{\left(\kappa - 1\right)}\left(\upsilon\right) \right| \nu$$

*for every θ* ∈ (0, 1) *and υ* ≥ *υ<sup>θ</sup> .*

**Lemma 2** ([30])**.** *Let β be a ratio of two odd numbers, V* > 0 *and U are constants. Then*

$$\mathcal{U}\mathfrak{x} - V\mathfrak{x}^{(\mathfrak{f}+1)/\beta} \le \frac{\mathfrak{f}^{\mathfrak{f}}}{(\mathfrak{f}+1)^{\mathfrak{f}+1}} \frac{\mathcal{U}^{\mathfrak{f}+1}}{V^{\beta}}.$$

*for all positive x.*

**Lemma 3** ([9])**.** *If y*(*i*) (*υ*) > 0, *i* = 0, 1, ..., *κ*, *and y*(*κ*+1) (*υ*) < 0, *then*

$$\frac{y\left(\upsilon\right)}{\upsilon^{\kappa}/\kappa!} \ge \frac{y'\left(\upsilon\right)}{\upsilon^{\kappa-1}/\left(\kappa-1\right)!}.$$

**Lemma 4** ([7])**.** *Suppose that y is an eventually positive solution of Equation (1). Then, there exist two possible cases:*

$$\begin{array}{llll} (\mathbf{S\_1}) & y \ (\upsilon) > 0, \ y' \ (\upsilon) > 0, \ y'' \ (\upsilon) > 0, \ y''' \ (\upsilon) > 0, \ y^{(4)} \ (\upsilon) < 0, \\\ (\mathbf{S\_2}) & y \ (\upsilon) > 0, \ y' \ (\upsilon) > 0, \ y'' \ (\upsilon) < 0, \ y''' \ (\upsilon) > 0, \ y^{(4)} \ (\upsilon) < 0, \end{array}$$

*for υ* ≥ *υ*1, *where υ*<sup>1</sup> ≥ *υ*<sup>0</sup> *is sufficiently large.*

#### **3. Oscillation Criteria**

In this section, we shall establish some oscillation criteria for fourth order advanced differential Equation (1).

**Remark 1.** *It is well known (see [31]), the differential equation*

$$\left[a\left(\upsilon\right)\left(y'\left(\upsilon\right)\right)^{\beta}\right]' + q\left(\upsilon\right)y^{\beta}\left(g\left(\upsilon\right)\right) = 0, \quad \upsilon \ge \upsilon\_{0},\tag{9}$$

*where <sup>β</sup>* <sup>&</sup>gt; <sup>0</sup> *is the ratio of odd positive integers, <sup>a</sup> , <sup>q</sup>* <sup>∈</sup> *<sup>C</sup>* ([*υ*0, <sup>∞</sup>), <sup>R</sup>+) *is nonoscillatory if and only if there exists a number υ* ≥ *υ*0*, and a function ς* ∈ *C* 1 ([*υ*, ∞), R), *satisfying the following inequality*

$$
\left(\boldsymbol{\varrho}'\left(\upsilon\right) + \gamma \boldsymbol{a}^{-1/\beta}\left(\upsilon\right)\left(\boldsymbol{\varrho}\left(\upsilon\right)\right)^{(1+\beta)/\beta} + q\left(\upsilon\right) \leq 0, \quad \text{on } [\upsilon, \infty).
$$

In what follows, we compare the oscillatory behavior of Equation (1) with the second-order half-linear equations of the type in Equation (9). There are numerous results concerning the oscillation of (9), which included Hille and Nehari types, Philos type, etc.

**Theorem 1.** *Assume that Equation (2) holds. If the differential equations*

$$\left(\frac{2a^{\frac{1}{\beta}}\left(\upsilon\right)}{\left(\theta\upsilon^{2}\right)^{\beta}}\left(y'\left(\upsilon\right)\right)^{\beta}\right)' + \sum\_{i=1}^{j} q\_{i}\left(\upsilon\right)y^{\beta}\left(\upsilon\right) = 0\tag{10}$$

*and*

$$\int y''(v) + y\left(v\right) \int\_{v}^{\infty} \left(\frac{1}{a\left(\xi\right)} \int\_{\xi}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \mathrm{d}s\right)^{1/\beta} \mathrm{d}\xi = 0\tag{11}$$

*are oscillatory for some constant θ* ∈ (0, 1)*, then every solution of Equation (1) is oscillatory.*

**Proof.** By contradiction, assume that *y* is a positive solution of Equation (1). Then, we can suppose that *y* (*υ*) and *y* (*η<sup>i</sup>* (*υ*)) are positive for all *υ* ≥ *υ*<sup>1</sup> sufficiently large. From Lemma 4, we have two possible cases (**S**1) and (**S**2).

Let case (**S**1) holds, then with the help of Lemma 1, we get

$$y'(v) \ge \frac{\theta}{2} v^2 y''''(v) \,. \tag{12}$$

for every *θ* ∈ (0, 1) and for all large *υ*.

Define

$$\varphi\left(\upsilon\right) := \tau\left(\upsilon\right) \left(\frac{a\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{\beta}}{y^{\beta}\left(\upsilon\right)}\right),\tag{13}$$

we see that *ϕ* (*υ*) > 0 for *υ* ≥ *υ*1, where there exists a positive function *τ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)) and

$$\begin{array}{rcl} \varphi'\left(\upsilon\right) &=& \tau'\left(\upsilon\right) \frac{a\left(\upsilon\right)\left(\mathcal{Y}^{\prime\prime}\left(\upsilon\right)\right)^{\beta}}{\mathcal{Y}^{\beta}\left(\upsilon\right)} + \tau\left(\upsilon\right) \frac{\left(a\left(\mathcal{Y}^{\prime\prime}\right)^{\beta}\right)^{\prime}\left(\upsilon\right)}{\mathcal{Y}^{\beta}\left(\upsilon\right)}\\ & & - \beta\tau\left(\upsilon\right) \frac{\mathcal{Y}^{\beta-1}\left(\upsilon\right)\mathcal{Y}\left(\upsilon\right)a\left(\upsilon\right)\left(\mathcal{Y}^{\prime\prime}\left(\upsilon\right)\right)^{\beta}}{\mathcal{Y}^{2\beta}\left(\upsilon\right)}. \end{array}$$

Using Equations (12) and (13), we obtain

$$\begin{split} \left| \boldsymbol{\varrho}' \left( \upsilon \right) \right| &\leq \quad \frac{\tau'\_{+} \left( \upsilon \right)}{\tau \left( \upsilon \right)} \boldsymbol{\varrho} \left( \upsilon \right) + \tau \left( \upsilon \right) \frac{\left( a \left( \upsilon \right) \left( \boldsymbol{y}''' \left( \upsilon \right) \right)^{\beta} \right)'}{\boldsymbol{y}^{\beta} \left( \upsilon \right)} \\ &\quad \quad - \beta \tau \left( \upsilon \right) \frac{\theta}{2} \upsilon^{\kappa - 2} \frac{a \left( \upsilon \right) \left( \boldsymbol{y}''' \left( \upsilon \right) \right)^{\beta + 1}}{\boldsymbol{y}^{\beta + 1} \left( \upsilon \right)} \\ &\leq \quad \frac{\tau' \left( \upsilon \right)}{\tau \left( \upsilon \right)} \boldsymbol{\varrho} \left( \upsilon \right) + \tau \left( \upsilon \right) \frac{\left( a \left( \upsilon \right) \left( \boldsymbol{y}''' \left( \upsilon \right) \right)^{\beta} \right)'}{\boldsymbol{y}^{\beta} \left( \upsilon \right)} \\ &\quad \quad - \frac{\beta \theta \upsilon^{2}}{2 \left( \tau \left( \upsilon \right) a \left( \upsilon \right) \right)^{\frac{1}{\beta}}} \boldsymbol{\varrho} \left( \upsilon \right)^{\frac{\beta + 1}{\delta}} . \end{split} \tag{14}$$

From Equations (1) and (14), we obtain

$$\log'(v) \le \frac{\tau'(v)}{\tau(v)} \varphi\left(v\right) - \ell \tau\left(v\right) \frac{\sum\_{i=1}^j q\_i\left(v\right) y^{\oint}\left(\eta\_i\left(v\right)\right)}{y^{\oint}\left(v\right)} - \frac{\beta \theta v^2}{2\left(\tau\left(v\right)a\left(v\right)\right)^{\frac{1}{\overline{\beta}}}} \varphi\left(v\right)^{\frac{\beta+1}{\overline{\beta}}}.$$

Note that *y* ′ (*υ*) > 0 and *η<sup>i</sup>* (*υ*) ≥ *υ*, thus, we get

$$\varphi'(v) \le \frac{\tau'(v)}{\tau(v)} \varphi(v) - \ell \tau\left(v\right) \sum\_{i=1}^{j} q\_i\left(v\right) - \frac{\beta \theta v^2}{2\left(\tau\left(v\right)a\left(v\right)\right)^{\frac{1}{\beta}}} \varphi\left(v\right)^{\frac{\beta+1}{\beta}}.\tag{15}$$

If we set *τ* (*υ*) = ℓ = 1 in Equations (15), then we find

$$\left(\boldsymbol{\varrho}'\left(\boldsymbol{\upsilon}\right) + \frac{\beta\theta\boldsymbol{\upsilon}^2}{2a^{\frac{1}{\beta}}\left(\boldsymbol{\upsilon}\right)}\boldsymbol{\varrho}\left(\boldsymbol{\upsilon}\right)^{\frac{\beta+1}{\beta}} + \sum\_{i=1}^{j} q\_i\left(\boldsymbol{\upsilon}\right) \leq \mathbf{0}.\right)$$

Thus, we can see that Equation (10) is a nonoscillatory, which is a contradiction. Let suppose the case (**S**2) holds. Define

$$
\psi\left(v\right) := \theta\left(v\right) \frac{y'\left(v\right)}{y\left(v\right)},
$$

we see that *ψ* (*υ*) > 0 for *υ* ≥ *υ*1, where there exist a positive function *ϑ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)). By differentiating *ψ* (*υ*), we obtain

$$\psi'(\upsilon) = \frac{\vartheta'(\upsilon)}{\vartheta(\upsilon)}\psi(\upsilon) + \vartheta\left(\upsilon\right)\frac{y''(\upsilon)}{y(\upsilon)} - \frac{1}{\vartheta(\upsilon)}\psi\left(\upsilon\right)^2. \tag{16}$$

Now, integrating Equation (1) from *υ* to *m* and using *y* ′ (*υ*) > 0, we obtain

$$a\left(m\right)\left(y^{\prime\prime\prime}\left(m\right)\right)^{\notin} - a\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{\notin} = -\int\_{\upsilon}^{m} \sum\_{i=1}^{j} q\_{i}\left(s\right)g\left(y\left(\eta\_{i}\left(s\right)\right)\right)ds.$$

By virtue of *y* ′ (*υ*) > 0 and *η<sup>i</sup>* (*υ*) ≥ *υ*, we get

$$a\left(m\right)\left(y^{\prime\prime\prime}\left(m\right)\right)^{\beta} - a\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{\beta} \leq -\ell y^{\beta}\left(\upsilon\right) \int\_{\upsilon}^{m} \sum\_{i=1}^{j} q\_{i}\left(s\right) \, ds.$$

Letting *<sup>m</sup>* → <sup>∞</sup> , we see that

$$a\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{\beta} \geq \ell y^{\beta}\left(\upsilon\right) \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \mathrm{d}s$$

and hence

$$y^{\prime\prime\prime}(\upsilon) \ge y(\upsilon) \left( \frac{\ell}{a\left(\upsilon\right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \, \mathrm{d}s \right)^{1/\beta} \, \mathrm{d}s$$

Integrating again from *υ* to ∞, we get

$$(y''(v) + y(v) \int\_{v}^{\infty} \left(\frac{\ell}{a\left(\underline{\varsigma}\right)} \int\_{\mathfrak{s}}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \mathrm{d}s\right)^{1/\beta} \mathrm{d}\underline{\varsigma} \le 0. \tag{17}$$

From Equations (16) and (17), we obtain

$$\psi'(\upsilon) \le \frac{\vartheta'(\upsilon)}{\vartheta(\upsilon)} \psi(\upsilon) - \vartheta(\upsilon) \int\_{\upsilon}^{\infty} \left( \frac{\ell}{a(\emptyset)} \int\_{\xi}^{\infty} \sum\_{i=1}^{j} q\_i(s) \, \mathrm{d}s \right)^{1/\beta} \, \mathrm{d}\xi - \frac{1}{\vartheta(\upsilon)} \psi\left(\upsilon\right)^2 . \tag{18}$$

If we now set *ϑ* (*υ*) = ℓ = 1 in Equation (18), then we obtain

$$
\psi'(\upsilon) + \psi^2(\upsilon) + \int\_{\upsilon}^{\infty} \left( \frac{1}{a\left(\underline{\varsigma}\right)} \int\_{\underline{\varsigma}}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \, \mathrm{d}s \right)^{1/\beta} \, \mathrm{d}\underline{\varsigma} \le 0.
$$

Thus, it can be seen that Equation (11) is non oscillatory, which is a contradiction. Hence, Theorem 1 is proved.

**Remark 2.** *It is well known (see [19]) that if*

$$\int\_{v\_0}^{\infty} \frac{1}{a\left(v\right)} \mathrm{d}v = \infty, \text{ and } \liminf\_{v \to \infty} \left(\int\_{v\_0}^{v} \frac{1}{a\left(s\right)} \mathrm{d}s\right) \int\_{v}^{\infty} q\left(s\right) \mathrm{d}s > \frac{1}{4}.$$

*then Equation (9) with β* = 1 *is oscillatory.*

Based on the above results and Theorem 1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with *β* = 1.

**Theorem 2.** *Let β* = ℓ = 1, *and assuming that Equation (2) holds, if*

$$\int\_{v\_0}^{\infty} \frac{\theta v^2}{2a\left(v\right)} \mathrm{d}v = \infty$$

*and*

$$\liminf\_{v \to \infty} \left( \int\_{v\_0}^v \frac{\theta s^2}{2a\left(s\right)} \mathrm{d}s \right) \int\_v^\infty \sum\_{i=1}^j q\_i\left(s\right) \mathrm{d}s > \frac{1}{4},\tag{19}$$

*also, if*

$$\liminf\_{\upsilon \to \infty} \upsilon \int\_{\upsilon\_0}^{\upsilon} \int\_{\upsilon}^{\infty} \left( \frac{1}{a \left( \underline{\zeta} \right)} \int\_{\xi}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) \, \mathrm{d}s \right) \mathrm{d}\zeta \mathrm{d}\upsilon > \frac{1}{4} \,\tag{20}$$

*for some constant θ* ∈ (0, 1)*, then all solutions of Equation (1) are oscillatory.*

In the following theorem, we compare the oscillatory behavior of Equation (1) with the first-order differential equations:

**Theorem 3.** *Assume that Equation (2) holds, if the differential equations*

$$\mathbf{x}'\left(\upsilon\right) + \ell \sum\_{i=1}^{j} q\_i\left(\upsilon\right) \left(\frac{\theta \upsilon^2}{2a^{1/\beta}\left(\upsilon\right)}\right)^{\beta} \mathbf{x}\left(\eta\left(\upsilon\right)\right) = \mathbf{0} \tag{21}$$

*and*

$$z'(v) + vz(v) \int\_{v}^{\infty} \left( \frac{\ell}{a(\emptyset)} \int\_{\emptyset}^{\infty} \sum\_{i=1}^{j} q\_i(s) \, \mathrm{d}s \right)^{1/\beta} \, \mathrm{d}\emptyset = 0 \tag{22}$$

*are oscillatory for some constant θ* ∈ (0, 1)*, then every solutions of Equation (1) is oscillatory.*

**Proof.** We prove this theorem by contradiction again, assume that *y* is a positive solution of Equation (1). Then, we can suppose that *y* (*υ*) and *y* (*η<sup>i</sup>* (*υ*)) are positive for all *υ* ≥ *υ*<sup>1</sup> sufficiently large. From Lemma 4, we have two possible cases (**S**1) and (**S**2). In the case where (**S**1) holds, from Lemma 1, we see

$$y\left(\upsilon\right) \ge \frac{\theta \upsilon^2}{2a^{1/\beta}\left(\upsilon\right)} \left(a^{1/\beta}\left(\upsilon\right)y^{\prime\prime\prime}\left(\upsilon\right)\right)\prime$$

for every *θ* ∈ (0, 1) and for all large *υ*. Thus, if we set

$$\mathfrak{x}\left(\upsilon\right) = a\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{\beta} > 0\,,$$

then we see that *ψ* is a positive solution of the inequality

$$\left(\mathbf{x}'\left(\upsilon\right) + \ell \sum\_{i=1}^{j} q\_i\left(\upsilon\right) \left(\frac{\theta \upsilon^2}{2a^{1/\beta}\left(\upsilon\right)}\right)^{\beta} \mathbf{x}\left(\eta\left(\upsilon\right)\right) \leq 0. \tag{23}$$

From [20] [Theorem 1], we conclude that the corresponding Equation (21) has a positive solution, which is a contradiction. In the case where (**S**2) holds. From Lemma 3, we get

$$y\left(\upsilon\right) \ge \upsilon y'\left(\upsilon\right),\tag{24}$$

From Equations (17) and (24), we get

$$\int y'''\left(\upsilon\right) + \upsilon y'\left(\upsilon\right) \int\_{\upsilon}^{\infty} \left(\frac{\ell}{a\left(\emptyset\right)} \int\_{\xi}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \mathrm{d}s\right)^{1/\beta} \mathrm{d}\xi \le 0.1$$

Now, we set

$$z\left(v\right) = y'\left(v\right).$$

Thus, we find *ψ* is a positive solution of the inequality

$$z'\left(\upsilon\right) + \upsilon z\left(\upsilon\right) \int\_{\upsilon}^{\infty} \left(\frac{\ell}{a\left(\varsigma\right)} \int\_{\xi}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \mathrm{d}s\right)^{1/\beta} \mathrm{d}\xi \leq 0. \tag{25}$$

From ([20], Theorem 1), we conclude that the corresponding Equation (22) has a positive solution, which is a contradiction again. Thus the proof is completed.

**Corollary 1.** *Let Equation (2) hold, if*

$$\liminf\_{\upsilon \to \infty} \int\_{\upsilon}^{\eta\_i(\upsilon)} \ell \sum\_{i=1}^{j} q\_i \left( s \right) \left( \frac{\theta s^2}{2a^{1/\beta} \left( s \right)} \right)^{\beta} \mathrm{d}s > \frac{\theta^{\beta}}{\mathsf{e}} \tag{26}$$

*and*

$$\liminf\_{\upsilon \to \infty} \int\_{\upsilon}^{\eta\_i(\upsilon)} s \int\_{\upsilon}^{\infty} \left( \frac{\ell}{a \left( \emptyset \right)} \int\_{\xi}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) \, \mathrm{ds} \right)^{1/\beta} \mathrm{d}\xi \, \mathrm{ds} > \frac{1}{\mathrm{e}} \tag{27}$$

*for some constant θ* ∈ (0, 1)*, then every solutions of Equation (1) is oscillatory.*

**Example 1.** *Consider a differential equation*

$$\left(v^{\mathfrak{z}} \left(w^{\prime\prime\prime}(v)\right)^{\mathfrak{z}}\right)' + \frac{q\_0}{v^6} w^{\mathfrak{z}} \left(\mathfrak{z}v\right) = 0, \; v \ge 1,\tag{28}$$

*where q*<sup>0</sup> *is a constant. Let β* = 3, *a* (*υ*) = *υ* 3 , *q* (*υ*) = *q*0/*υ* 6 *and η* (*υ*) = 2*υ*. *If we set* ℓ = 1, *then Condition (26) becomes*

$$\begin{aligned} \liminf\_{v \to \infty} \int\_{v}^{\eta\_{l}(v)} \ell \sum\_{i=1}^{j} q\_{i} \left( s \right) \left( \frac{\theta s^{2}}{2a^{1/\beta} \left( s \right)} \right)^{\beta} ds &= \liminf\_{v \to \infty} \int\_{v}^{2v} \frac{q\_{0}}{s^{6}} \left( \frac{\theta s^{2}}{2s^{1/3}} \right)^{3} ds \\ &= \liminf\_{v \to \infty} \left( \frac{q\_{0} \theta^{3}}{8} \right)^{3} \int\_{v}^{2v} \frac{q\_{0}}{s} ds \\ &= \frac{q\_{0} \theta^{3} \ln 2}{8} > \frac{\theta^{3}}{8} \end{aligned}$$

*and Condition (27) holds. Therefore, from Corollary 1, all solutions of Equation (28) are oscillatory if q*<sup>0</sup> > 1728/ *θ* 3 e ln 2 *for some constant θ* ∈ (0, 1).

**Example 2.** *Let the equation*

$$y^{(4)}\left(\upsilon\right) + \frac{q\_0}{\upsilon^4}y\left(2\upsilon\right) = 0, \ \upsilon \ge 1,\tag{29}$$

*where q*<sup>0</sup> > 0 *is a constant. Let β* = 1, *a* (*υ*) = 1, *q* (*υ*) = *q*0/*υ* 4 *and η* (*υ*) = 2*υ. If we set* ℓ = 1, *then Condition (19) becomes*

$$\begin{aligned} \liminf\_{\upsilon \to \infty} \left( \int\_{\upsilon\_0}^{\upsilon} \frac{\theta s^2}{2a\left(s\right)} \mathrm{d}s \right) \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \mathrm{d}s &= \liminf\_{\upsilon \to \infty} \left( \frac{\upsilon^3}{3} \right) \int\_{\upsilon}^{\infty} \frac{q\_0}{s^4} \mathrm{d}s \\ &= \quad \frac{q\_0}{9} > \frac{1}{4} \end{aligned}$$

*and Condition (20) becomes*

$$\begin{aligned} \liminf\_{\upsilon \to \infty} \upsilon \int\_{\upsilon\_0}^{\upsilon} \int\_{\upsilon}^{\infty} \left( \frac{1}{a \left( \underline{\zeta} \right)} \int\_{\zeta}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) \mathrm{d}s \right)^{1/\beta} \mathrm{d}\zeta \mathrm{d}\upsilon &= \liminf\_{\upsilon \to \infty} \upsilon \left( \frac{q\_0}{\delta \upsilon} \right) \\ &= \frac{q\_0}{6} > \frac{1}{4}. \end{aligned}$$

*Therefore, from Theorem 2, all solutions of Equation (29) are oscillatory if q*<sup>0</sup> > 2.25*.*

**Remark 3.** *We compare our result with the known related criteria*


**Example 3.** *Consider a differential Equation (8) where q*<sup>0</sup> > 0 *is a constant. Note that β* = 1, *κ* = 4, *a* (*υ*) = 1, *q* (*υ*) = *q*0/*υ* 4 *and η* (*υ*) = 3*υ. If we set* ℓ = 1, *then Condition (19) becomes*

$$\frac{q\_0}{9} > \frac{1}{4}.$$

*Therefore, from Theorem 2, all the solutions of Equation (8) are oscillatory if q*<sup>0</sup> > 2.25*.*

**Remark 4.** *We compare our result with the known related criteria*


**Example 4.** *Let the equation*

$$y^{(4)}\left(\upsilon\right) + \frac{q\_0}{\upsilon^2}y\left(\upsilon\upsilon\right) = 0, \ \upsilon > 1,\tag{30}$$

*where q*<sup>0</sup> > 0, *c* > 1 *are constants. Note that β* = 1, *a* (*υ*) = 1, *q* (*υ*) = *q*0/*υ* 2 *and η* (*υ*) = *cυ*. *From ([14], Corollary 2.4), we have that the equation*

> *υ* 2

*y* ′′ (*υ*) + *q*0 *y* (*cυ*) = 0, *c* > 1, *q*<sup>0</sup> > 0,

*is oscillatory if*

$$q\_0 \left( 1 + q\_0 \ln c \right) > \frac{1}{4}.$$

*Therefore, from Theorem 1, all the solutions of Equation (30) are oscillatory if q*<sup>0</sup> (1 + *q*<sup>0</sup> ln *c*) > 1/4.

#### **4. Conclusions**

In this paper, the main aim to provide a study of asymptotic behavior of the fourth order advanced differential equation has been achieved. We used the theory of comparison with first and second-order delay equations and the Riccati substitution to ensure that every solution of this equation is oscillatory. The presented results complement a number of results reported in the literature. Furthermore, the findings of this paper can be extended to study a class of systems of higher order advanced differential equations.

**Author Contributions:** Writing—original draft preparation, O.B. and H.A.; writing—review and editing, O.B., H.A. and S.-W.Y.; formal analysis, O.B., H.A. and S.-W.Y.; funding acquisition, O.B., H.A., S.-W.Y.; supervision, O.B., H.A. and S.-W.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This work was supported by the National Natural Science Foundation of China (No. 71601072) and Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006). The authors thank the reviewers for their useful comments, which led to the improvement of the content of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article*

### **New Results for Kneser Solutions of Third-Order Nonlinear Neutral Differential Equations**

### **Osama Moaaz 1,† , Belgees Qaraad 1,2,†, Rami Ahmad El-Nabulsi 3,**∗**,† and Omar Bazighifan 4,5,†**


Received: 24 March 2020; Accepted: 22 April 2020; Published: 1 May 2020

**Abstract:** In this paper, we consider a certain class of third-order nonlinear delay differential equations *r* (*w* ′′) *α* ′ (*v*) + *q* (*v*) *x β* (*ς* (*v*)) = 0, for *v* ≥ *v*0, where *w* (*v*) = *x* (*v*) + *p* (*v*) *x* (*ϑ* (*v*)). We obtain new criteria for oscillation of all solutions of this nonlinear equation. Our results complement and improve some previous results in the literature. An example is considered to illustrate our main results.

**Keywords:** oscillation criteria; thrid-order; delay differential equations

#### **1. Introduction**

The continuous development in various sciences is accompanied by the continued emergence of new models of difference and differential equations that describe this development. Studying the qualitative properties of differential equations helps to understand and analyze many life phenomena and problems; see [1]. Recently, the study of the oscillatory properties of differential equations has evolved significantly; see [2–10]. However, third-order differential equations attract less attention compared to first and second-order equations; see [11–20].

In this paper, we consider the third-order neutral nonlinear differential equation of the form

$$\left(r\left(w^{\prime\prime}\right)^{a}\right)^{\prime}(v) + q\left(v\right)x^{\beta}\left(\boldsymbol{\zeta}\left(v\right)\right) = 0,\text{ for }v \ge v\_{0\prime} \tag{1}$$

where *w* (*v*) = *x* (*v*) + *p* (*v*) *x* (*ϑ* (*v*)), *α* and *β* are ratios of odd positive integers. In this work, we assume the following conditions:


A solution of (1) means *<sup>x</sup>* ∈ *<sup>C</sup>* ([*v*0, <sup>∞</sup>)) with *<sup>v</sup>*<sup>∗</sup> ≥ *<sup>v</sup>*0, which satisfies the properties *<sup>w</sup>* ∈ *<sup>C</sup>* 2 ([*v*∗, <sup>∞</sup>)), *r* (*w* ′′) *<sup>α</sup>* <sup>∈</sup> *<sup>C</sup>* 1 ([*v*∗, <sup>∞</sup>)) and satisfies (1) on [*v*∗, <sup>∞</sup>). We consider the nontrivial solutions of (1) which exist on some half-line [*v*∗, <sup>∞</sup>) and satisfy the condition sup{|*<sup>x</sup>* (*v*)| : *<sup>v</sup>*<sup>1</sup> ≤ *<sup>v</sup>* < <sup>∞</sup>} > 0 for any *<sup>v</sup>*<sup>1</sup> ≥ *<sup>v</sup>*∗.

**Definition 1.** *The class S*<sup>1</sup> *is a set of all solutions x of Equation (1) such that their corresponding function w satisfies*

$$\mathbf{Case \ (i):} \quad w\left(v\right) > 0, w'\left(v\right) > 0, w''\left(v\right) > 0;$$

*and the class S*<sup>2</sup> *is a set of all solutions of Equation (1) such that their corresponding function w satisfies*

$$\mathbf{Case (ii)}: \quad w\left(v\right) > 0, w'\left(v\right) < 0, w''\left(v\right) > 0.$$

**Definition 2.** *If the nontrivial solution x is neither positive nor negative eventually, then x is called an oscillatory solution. Otherwise, it is a non-oscillatory solution.*

When studying the oscillating properties of neutral differential equations with odd-order, most of the previous studies have been concerned with creating a sufficient condition to ensure that the solutions are oscillatory or tend to zero; see [11–20]. For example, Baculikova and Dzurina [11,12], Candan [13], Dzurina et al. [15], Li et al. [18] and Su et al. [19] studied the oscillatory properties of (1) in the case where *α* = *β* and 0 ≤ *p* (*v*) ≤ *p*<sup>0</sup> < 1. Elabbasy et al. [16] studied the oscillatory behavior of general differential equation

$$\left(r\_2\left(\left(r\_1\left(w'\right)^a\right)'\right)^{\beta}\right)'(v) + q\left(v\right)f\left(\mathbf{x}\left(\emptyset\left(v\right)\right)\right) = 0, \text{ for } v \ge v\_{0\prime}$$

For an odd-order, Karpuz at al. [17] and Xing at al. [20] established several oscillation theorems for equation

$$\left(r\_2 \left(w^{(n-1)}\right)'\right)'(v) + q\left(v\right)x^{\mathfrak{a}}\left(\mathfrak{g}\left(v\right)\right) = 0, \text{ for } v \ge v\_0.$$

As an improvement and completion of the previous studies, Dzurina et al. [14], established standards to ensure that all solutions of linear equation

$$\left(r\_2 \left(r\_1 w'\right)'\right)'(v) + q\left(v\right) \ge \left(\xi\left(v\right)\right) = 0\_\prime$$

by comparison with first-order delay equations.

The main objective of this paper is to obtain new criteria for oscillation of all solution of nonlinear Equation (1). Our results complement and improve the results in [11–19] which only ensure that non-oscillating solutions tend to zero.

Next, we state the following lemmas, which will be useful in the proof of our results.

**Lemma 1.** *Assume that c*1, *<sup>c</sup>*<sup>2</sup> ∈ [0, <sup>∞</sup>) *and <sup>γ</sup>* > <sup>0</sup>*. Then*

$$(\mathbf{c}\_1 + \mathbf{c}\_2)^\gamma \le \mu \left(\mathbf{c}\_1^\gamma + \mathbf{c}\_2^\gamma\right),\tag{2}$$

*where*

$$\mu := \begin{cases} 1 & \text{if } \gamma \le 1 \\\ 2^{\gamma - 1} & \text{if } \gamma > 1. \end{cases}$$

**Lemma 2.** *Let <sup>u</sup>*, *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>* ([*v*0, <sup>∞</sup>), <sup>R</sup>), *<sup>u</sup>* (*v*) <sup>=</sup> *<sup>g</sup>* (*v*) <sup>+</sup> *ag* (*<sup>v</sup>* <sup>−</sup> *<sup>b</sup>*) *for <sup>v</sup>* <sup>≥</sup> *<sup>v</sup>*<sup>0</sup> <sup>+</sup> max{0, *<sup>c</sup>*}, *where <sup>a</sup>* <sup>6</sup><sup>=</sup> <sup>1</sup>*, b are constants. Suppose that there exists a constant l* <sup>∈</sup> <sup>R</sup> *such that* lim*v*→<sup>∞</sup> *<sup>u</sup>* (*v*) <sup>=</sup> *l.*

> (**H**<sup>1</sup> ) : *If* lim inf*v*→<sup>∞</sup> *<sup>g</sup>* (*v*) <sup>=</sup> *<sup>g</sup>*<sup>∗</sup> <sup>∈</sup> <sup>R</sup>, *then g*<sup>∗</sup> <sup>=</sup> *<sup>l</sup>*/ (<sup>1</sup> <sup>+</sup> *<sup>a</sup>*); (**H**<sup>2</sup> ) : *If* lim sup*v*→<sup>∞</sup> *<sup>g</sup>* (*v*) <sup>=</sup> *<sup>g</sup>* <sup>∗</sup> <sup>∈</sup> <sup>R</sup>, *then g*<sup>∗</sup> <sup>=</sup> *<sup>l</sup>*/ (<sup>1</sup> <sup>+</sup> *<sup>a</sup>*).

**Lemma 3.** *Let x* ∈ *C n* ([*v*0, ∞),(0, ∞)). *Assume that x* (*n*) (*v*) *is of fixed sign and not identically zero on* [*v*0, ∞) *and that there exists a v*<sup>1</sup> ≥ *v*<sup>0</sup> *such that x* (*n*−1) (*v*) *x* (*n*) (*v*) ≤ 0 *for all v* ≥ *v*1*. If* lim*v*→<sup>∞</sup> *x* (*v*) 6= 0, *then for every µ* ∈ (0, 1) *there exists v<sup>µ</sup>* ≥ *v*<sup>1</sup> *such that*

$$\mathbf{x}^\*(v) \ge \frac{\mu}{(n-1)!} v^{n-1} \left| \mathbf{x}^{(n-1)}(v) \right| \text{ for } v \ge v\_\mu.$$

#### **2. Criteria for Nonexistence of Decreasing Solutions**

Through this paper, we will be using the following notation:

$$\begin{aligned} \mathcal{E}w\left(v\right) &:= r\left(w^{\prime\prime}\right)^{\alpha}\left(v\right),\\ \widetilde{q}\left(v\right) &:= \min\left\{q\left(v\right), q\left(\theta\left(v\right)\right)\right\}, \end{aligned}$$

and

$$\eta\left(v, u\right) := \int\_{\mu}^{v} \frac{1}{r\_{\pi}^{\frac{1}{a}}\left(s\right)} \text{ds and } \widetilde{\eta}\left(v, u\right) = \int\_{\mu}^{v} \left(\int\_{\mu}^{s} \frac{1}{r\_{\pi}^{\frac{1}{a}}\left(\zeta\right)} \text{d}\zeta\right) \text{ds}\,\omega$$

where *<sup>v</sup>* ∈ [*v*0, <sup>∞</sup>).

**Lemma 4.** *Assume that x* ∈ *S*2*. Then*

$$w\left(\mu\right) \ge \tilde{\eta}\left(\left.\sigma,\mu\right) \mathcal{E}^{1/\mathfrak{a}} w\left(\left.\sigma\right),\right.\tag{3}$$

*for u* ≤ *̟, and*

$$\left(\varepsilon w\left(v\right) + \frac{(p\_0)^{\beta}}{\vartheta\_0} \varepsilon w\left(\vartheta\left(v\right)\right)\right)' \le -\frac{1}{\mu} \tilde{q}\left(v\right) w^{\beta}\left(\emptyset\left(v\right)\right). \tag{4}$$

**Proof.** Let *x* be an eventually positive solution of (1). Then, we can assume that *x* (*v*) > 0, *x* (*ϑ* (*v*)) > 0 and *x* (*ς* (*v*)) > 0 for *v* ≥ *v*1, where *v*<sup>1</sup> is sufficiently large. From Lemma 1, (1) and (**I**2), we obtain

$$w^{\mathfrak{G}}\left(\upsilon\right) \le \mu\left(\mathbf{x}^{\mathfrak{G}}\left(\upsilon\right) + p\_0^{\mathfrak{G}}\mathbf{x}^{\mathfrak{G}}\left(\mathfrak{G}\left(\upsilon\right)\right)\right). \tag{5}$$

Since *£w* (*v*) is non-increasing, we have

$$-w'(u) \ge \int\_{u}^{\mathcal{O}} \frac{1}{r^{1/a}(s)} \mathcal{E}^{1/a} w\left(s\right) ds \ge \mathcal{E}^{1/a} w\left(\mathcal{o}\right) \int\_{u}^{\mathcal{O}} \frac{1}{r^{1/a}\left(s\right)} ds,\text{ for } u \le \mathcal{o}.\tag{6}$$

Integrating this inequality from *u* to *̟*, we get

$$w\left(\mu\right) - w\left(\mathcal{a}\right) \ge \varepsilon^{1/\alpha} w\left(\mathcal{a}\right) \int\_{\mu}^{\mathcal{O}} \left(\int\_{\mu}^{\sigma} \frac{1}{r^{1/\alpha}\left(s\right)} \mathrm{d}s\right) \mathrm{d}\sigma.$$

Thus,

$$w\left(u\right) \ge \tilde{\eta}\left(\left.\sigma\right|u\right)\mathcal{E}^{1/a}w\left(\left.\sigma\right|\right).\tag{7}$$

Now, from (1) and (**I**3), we obtain

$$\left(\left(\varepsilon w\left(\vartheta\left(v\right)\right)\right)' \frac{1}{\vartheta'\left(v\right)} + \eta\left(\vartheta\left(v\right)\right)x^{\theta}\left(\zeta\left(\vartheta\left(v\right)\right)\right) = 0. \tag{8}$$

Using (1), (5) and (8), we have

$$\begin{split} 0 &\geq \quad \left(\mathsf{E}w\left(\boldsymbol{v}\right)\right)' + q\left(\boldsymbol{v}\right)\mathbf{x}^{\boldsymbol{\theta}}\left(\boldsymbol{\zeta}\left(\boldsymbol{v}\right)\right) + p\_{0}^{\boldsymbol{\theta}}\left(\frac{1}{\mathsf{\boldsymbol{\theta}}\_{0}}\left(\mathsf{\boldsymbol{\varepsilon}}\boldsymbol{w}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\right)' + q\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\mathbf{x}^{\boldsymbol{\theta}}\left(\boldsymbol{\varrho}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\right)\right) \\ &\geq \quad \left(\mathsf{\boldsymbol{\varepsilon}}{\mathsf{w}\left(\boldsymbol{v}\right)}\right)' + \frac{1}{\mathsf{\boldsymbol{\theta}}{\boldsymbol{\theta}}\_{0}}p\_{0}^{\boldsymbol{\theta}}\left(\mathsf{\boldsymbol{\varepsilon}}{\mathsf{w}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)}\right)' + \tilde{q}\left(\boldsymbol{v}\right)\left(\mathsf{x}^{\boldsymbol{\theta}}\left(\boldsymbol{\varsigma}\left(\boldsymbol{v}\right)\right) + p\_{0}^{\boldsymbol{\theta}}\mathsf{x}^{\boldsymbol{\theta}}\left(\boldsymbol{\varsigma}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\right)\right). \end{split}$$

Thus,

$$\left(\xi w\left(v\right) + \frac{1}{\theta\_0} p\_0^{\theta} \xi w\left(\theta\left(v\right)\right)\right)' + \frac{1}{\mu} \tilde{q}\left(v\right) w^{\theta}\left(\xi\left(v\right)\right) \le 0. \tag{9}$$

The proof of the lemma is complete.

**Theorem 1.** *If there exists a function <sup>δ</sup>* ∈ *<sup>C</sup>* ([*v*0, <sup>∞</sup>),(0, <sup>∞</sup>)) *such that <sup>ϑ</sup>* (*v*) ≤ *<sup>δ</sup>* (*v*), *<sup>ς</sup>* −1 (*δ* (*v*)) < *v and the delay differential equation*

$$\phi'(v) + \frac{1}{\mu} \left(\frac{\varsigma\_0}{\varsigma\_0 + p\_0^{\beta}}\right)^{\not p/a} \tilde{q}\left(v\right) \left(\tilde{\eta}\left(\theta\left(v\right), \delta\left(v\right)\right)\right)^{\not p} \phi^{\beta/a}\left(\varepsilon^{-1}\left(\delta\left(v\right)\right)\right) = 0 \tag{10}$$

*is oscillatory, then S*<sup>2</sup> *is an empty set.*

**Proof.** Assume the contrary that *x* is a positive solution of (1) and which satisfies case **(ii)**. Then, we assume that *x* (*v*) > 0, *x* (*ς* (*v*)) > 0 and *x* (*ϑ* (*v*)) > 0 for *v* ≥ *v*1, where *v*<sup>1</sup> is sufficiently large. Thus, from (1), we get *r* (*w* ′′) *α* ′ (*v*) ≤ 0 for *v* ≥ *v*1. Using Lemma 4, we get (3) and (4). Combining (4) and (3) with [*u* = *ϑ* (*v*) and *̟* = *δ* (*v*)], we find

$$\left(\varepsilon w\left(v\right) + \frac{1}{\xi\_0} p\_0^{\beta} \varepsilon w\left(\boldsymbol{\varsigma}\left(v\right)\right)\right)' + \frac{1}{\mu} \widetilde{q}\left(v\right) \left(\widetilde{\boldsymbol{\eta}}\left(\boldsymbol{\theta}, \delta\right)\right)^{\beta} \mathcal{E}^{\boldsymbol{\beta}/a} w\left(\boldsymbol{\delta}\left(v\right)\right) \leq 0. \tag{11}$$

Since *£w* (*v*) is non-increasing, we see that *£w* (*v*) ≤ *£w* (*ς* (*v*)), and hence

$$
\varepsilon \varepsilon w\left(v\right) + \frac{1}{\xi\_0} p\_0^{\beta} \varepsilon w\left(\boldsymbol{\xi}\left(v\right)\right) \le \left(1 + \frac{1}{\xi\_0} p\_0^{\beta}\right) \varepsilon w\left(\boldsymbol{\xi}\left(v\right)\right). \tag{12}
$$

Using (11) along with (12), we have that *φ* (*v*) := *£w* (*v*) + <sup>1</sup> *ς*0 *p β* 0 *£w* (*ς* (*v*)) is a positive solution of the differential inequality

$$\left(\phi'\left(v\right) + \frac{1}{\mu}\widetilde{q}\left(v\right)\left(\widetilde{\eta}\left(\theta,\delta\right)\right)^{\beta}\left(\frac{\xi\,0}{\xi\_0 + p\_0^{\beta}}\right)^{\beta/a}\phi^{\beta/a}\left(\xi^{-1}\left(\delta\left(v\right)\right)\right) \le 0.1$$

By Theorem 1 [21], the associated delay Equation (10) also has a positive solution, which is a contradiction. The proof is complete.

**Theorem 2.** *Assume that <sup>β</sup>* ≥ *<sup>α</sup>. If there exists a function <sup>θ</sup>* ∈ *<sup>C</sup>* ([*v*0, <sup>∞</sup>),(0, <sup>∞</sup>)) *such that <sup>θ</sup>* (*v*) ≤ *<sup>v</sup>*, *ϑ* (*v*) ≤ *ς* (*θ* (*v*)) *and*

$$\limsup\_{\upsilon \to \infty} M^{\mathfrak{H}-n} \eta^a \left( \theta, \varsigma \left( \theta \right) \right) \int\_{\theta(\upsilon)}^{\upsilon} \tilde{q} \left( s \right) \, \mathrm{d}s > \mu \left( 1 + \frac{1}{\mathfrak{H}\_0} p\_0^{\mathfrak{H}} \right), \tag{13}$$

*then S*<sup>2</sup> *is an empty set.*

**Proof.** As in the proof of Theorem 1, we obtain (12). Using Lemma 4, we get (3) and (4). Integrating (4) from *θ* (*v*) to *v*, we get

$$0 < \varepsilon w\left(v\right) + \frac{1}{\mathfrak{H}} p\_0^{\mathfrak{H}} \varepsilon w\left(\mathfrak{z}\left(v\right)\right) \le \varepsilon w\left(\mathfrak{e}\left(v\right)\right) + \frac{1}{\mathfrak{H}} p\_0^{\mathfrak{H}} \varepsilon w\left(\mathfrak{z}\left(\mathfrak{e}\left(\mathfrak{z}\left(v\right)\right)\right) - \frac{1}{\mu} \int\_{\mathfrak{H}\left(v\right)}^{v} \widetilde{q}\left(\mathfrak{s}\right) w^{\mathfrak{G}}\left(\mathfrak{e}\left(\mathfrak{s}\right)\right) d\mathfrak{s} \omega$$

which together with (12) gives

$$
\left(1 + \frac{1}{\varsigma\_0} p\_0^{\beta}\right) \sharp w\left(\boldsymbol{\varrho}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\right) \geq \frac{1}{\mu} w^{\beta}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right) \int\_{\boldsymbol{\theta}\left(\boldsymbol{v}\right)}^{\boldsymbol{v}} \widetilde{q}\left(\boldsymbol{s}\right) \operatorname{d}\mathbf{s}.\tag{14}
$$

Since *w* ′ (*v*) < 0, there exists a constant *M* > 0 such that *w* (*v*) ≥ *M* for *v* ≥ *v*2, and hence (14) becomes

$$\left(1+\frac{1}{\varsigma\_0}p\_0^{\beta}\right)\operatorname{\boldsymbol{\varepsilon}}w\left(\boldsymbol{\varsigma}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\right) \geq \frac{M^{\beta-\alpha}}{\mu}w^{\mu}\left(\boldsymbol{\theta}\left(\boldsymbol{v}\right)\right)\int\_{\boldsymbol{\theta}\left(\boldsymbol{v}\right)}^{\upsilon}\tilde{q}\left(\boldsymbol{s}\right)\operatorname{\boldsymbol{\mathsf{d}}}\boldsymbol{\mathsf{s}}.$$

From (3) [*u* = *ϑ* (*v*) and *̟* = *ς* (*θ* (*v*))], we find

$$\left(1+\frac{1}{\xi\_0}p\_0^{\beta}\right) \ge \frac{M^{\beta-\alpha}}{\mu} \eta^{\alpha} \left(\theta \iota \xi\left(\theta\right)\right) \int\_{\theta\left(v\right)}^{v} \widetilde{q}\left(s\right) \,\mathrm{d}s.$$

From above inequality, taking the lim sup on both sides, we obtain a contradiction to (13). The proof is complete.

**Corollary 1.** *Assume that there exists a function <sup>δ</sup>* ∈ *<sup>C</sup>* ([*v*0, <sup>∞</sup>),(0, <sup>∞</sup>)) *such that <sup>ϑ</sup>* (*v*) ≤ *<sup>δ</sup>* (*v*), *ς* −1 (*δ* (*v*)) < *v. Then S*<sup>2</sup> *is an empty set, if one of the statements is hold:* (**b**1) *α* = *β and*

$$\lim\_{v \to \infty} \inf \int\_{\theta^{-1}(\delta(v))}^v \tilde{q}\left(s\right) \tilde{\eta}\left(\emptyset\left(s\right), \delta\left(s\right)\right) ds > \frac{\theta\_0 + p\_0^{\beta}}{\theta\_0 \mu e};\tag{15}$$

(**b**2) *α* < *β*, *there exists a function ξ* (*v*) ∈ *C* 1 ([*v*0, ∞)) *such that ξ* ′ (*v*) > <sup>0</sup>*,* lim*v*→<sup>∞</sup> *<sup>ξ</sup>* (*v*) = <sup>∞</sup>,

$$\limsup\_{v \to \infty} \frac{\beta \tilde{\varsigma}' \left( \theta^{-1} \left( \delta \left( v \right) \right) \right) \left( \theta^{-1} \left( \delta \left( v \right) \right) \right)'}{a \tilde{\varsigma}' \left( v \right)} < 1 \tag{16}$$

*and*

$$\liminf\_{v \to \infty} \left[ \frac{1}{\mu\_{\mathfrak{F}}^{\mathfrak{F}'}(v)} \left( \frac{\mathfrak{e}\_0}{\mathfrak{e}\_0 + p\_0^{\mathfrak{E}}} \right)^{\mathfrak{F}/a} \tilde{q}(v) \not\subset (\mathfrak{e}, \delta) \, e^{-\mathfrak{F}(v)} \right] > 0. \tag{17}$$

**Proof.** It is well-known from [22,23] that conditions (15)–(17) imply the oscillation of (10).

#### **3. Criteria for Nonexistence of Increasing Solutions**

**Theorem 3.** *Assume that ϑ* (*v*) ≤ *ς* (*v*) *and ς* ′ (*v*) > 0. *If there exist a function σ* (*v*) *and v*<sup>1</sup> ≥ *v*<sup>0</sup> *such that*

$$\limsup\_{\mathbb{D}\to\infty} \int\_{\upsilon\_1}^{\upsilon} \left[ \frac{1}{\mu} \sigma \left( s \right) \tilde{q} \left( s \right) - \frac{\left( \sigma' \left( s \right) \right)^{a+1}}{\left( a+1 \right)^{a+1} \left( \sigma \left( s \right) \eta \left( \xi \left( s \right), s\_1 \right) \xi' \left( s \right) \right)^{a}} \left( 1 + \frac{\sigma\_0^{\delta}}{\theta\_0} \right) \right] \, \mathrm{d}s = \infty,\tag{18}$$

*then S*<sup>1</sup> *is an empty set.*

**Proof.** Let *x* be a positive solution of (1) and which satisfies case **(i)**. In view of case **(i)**, we can define a positive function by

$$
\psi\left(\upsilon\right) = \sigma\left(\upsilon\right) \frac{\varepsilon w\left(\upsilon\right)}{w^{\alpha}\left(\emptyset\left(\upsilon\right)\right)}.\tag{19}
$$

Hence, by differentiating (19), we get

$$\psi'(v) = \sigma'(v) \frac{\varepsilon w(v)}{w^{\mathfrak{a}}\left(\boldsymbol{\xi}\left(\boldsymbol{v}\right)\right)} + \sigma\left(v\right) \frac{\left(\varepsilon w\left(v\right)\right)'}{w^{\mathfrak{a}}\left(\boldsymbol{\xi}\left(\boldsymbol{v}\right)\right)} - \frac{a\sigma\left(v\right)\varepsilon w\left(v\right)w^{\mathfrak{a}-1}\left(\boldsymbol{\xi}\left(v\right)\right)w'\left(\boldsymbol{\xi}\left(v\right)\right)\boldsymbol{\xi}'\left(v\right)}{w^{2\mathfrak{a}}\left(\boldsymbol{\xi}\left(v\right)\right)}.\tag{20}$$

Substituting (19) into (20), we have

$$
\psi'(v) = \sigma'(v) \frac{(\xi w(v))'}{w^a(\xi(v))} + \frac{\sigma'(v)}{\sigma'(v)} \psi(v) - \frac{a \eta\left(\xi\left(v\right), v\_1\right) \xi'(v)}{\sigma^{\frac{1}{a}}(v)} \psi^{\frac{a+1}{a}}(v) \,. \tag{21}
$$

Now, define another positive function by

$$
\omega \left( v \right) = \sigma \left( v \right) \frac{\varepsilon w \left( \theta \left( v \right) \right)}{w^a \left( \emptyset \left( v \right) \right)}. \tag{22}
$$

By differentiating (22), we get

$$
\varpi'(v) = \left. \sigma'(v) \frac{\xi w\left(\theta\left(v\right)\right)}{w^a\left(\xi\left(v\right)\right)} + \sigma\left(v\right) \frac{\left(\xi w\left(\theta\left(v\right)\right)\right)'}{w^a\left(\xi\left(v\right)\right)}\right| \tag{23}
$$

$$-\frac{a\sigma\left(v\right)\left\|zw\left(\theta\left(v\right)\right)w^{\alpha-1}\left(\emptyset\left(v\right)\right)w^{\prime}\left(\emptyset\left(v\right)\right)\xi^{\prime}\left(v\right)}{w^{2\alpha}\left(\emptyset\left(v\right)\right)}.\tag{24}$$

Substituting (22) into (23) implies

$$
\varpi'(v) = \sigma'(v) \frac{\left(\xi w\left(\theta\left(v\right)\right)\right)'}{w^a\left(\xi\left(v\right)\right)} + \frac{\sigma'(v)}{\sigma\left(v\right)} \sigma\left(v\right) - \frac{a\eta\left(\xi\left(v\right), v\_1\right)\xi'\left(v\right)}{\sigma^{\frac{1}{a}}\left(v\right)} \sigma^{\frac{a+1}{a}}\left(v\right) \,. \tag{25}
$$

We can write the inequalities (21) and (25) in the form

$$\begin{split} \psi'(v) + \frac{\sigma\_0^\beta}{\theta\_0} \phi'(v) &\leq \quad \sigma(v) \frac{(\operatorname{Ew}(v))' + \frac{\sigma\_0^\beta}{\theta\_0} (\operatorname{Ew}(\varPhi(v)))'}{w^\kappa \left(\operatorname{\zeta}(v)\right)} \\ &+ \frac{\sigma'(v)}{\sigma(v)} \psi(v) - \frac{a\eta \left(\varPhi\left(\operatorname{v}\right), \nu\_1\right) \operatorname{\zeta}'(v)}{\sigma^\frac{1}{\varPhi}\left(v\right)} \psi^{\frac{a+1}{a}}(v) \\ &+ \frac{\sigma\_0^\beta}{\theta\_0} \left(\frac{\sigma'(v)}{\sigma(v)} \varPhi\left(v\right) - \frac{a\eta \left(\varPhi\left(\operatorname{v}\right), \nu\_1\right) \operatorname{\zeta}'(v)}{\sigma^\frac{1}{\varPhi}\left(v\right)} \phi^{\frac{a+1}{a}}(v)\right). \end{split} \tag{26}$$

Taking into account Lemma 1, (4) and (26), we obtain

$$\begin{split} \psi'(v) + \frac{\sigma\_0^\beta}{\theta\_0} \phi'(v) &\leq \ -\sigma(v) \left( \frac{\tilde{q}\left(v\right)}{\mu} \right) \\ &+ \frac{\sigma'(v)}{\sigma\left(v\right)} \psi\left(v\right) - \frac{a\eta\left(\xi\left(v\right), v\_1\right) \xi'\left(v\right)}{\sigma^{\frac{1}{a}}\left(v\right)} \psi^{\frac{a+1}{a}}\left(v\right) \\ &+ \frac{\sigma\_0^\beta}{\theta\_0} \left( \frac{\sigma'\left(v\right)}{\sigma\left(v\right)} \phi\left(v\right) - \frac{a\eta\left(\xi\left(v\right), v\_1\right) \xi'\left(v\right)}{\sigma^{\frac{1}{a}}\left(v\right)} \phi^{\frac{a+1}{a}}\left(v\right) \right). \end{split}$$

Applying the following inequality

$$B\mu - A\mu^{\frac{\alpha+1}{\alpha}} \le \frac{\alpha^{\alpha}B^{\alpha+1}}{\left(\alpha+1\right)^{\alpha+1}A^{\alpha}}, \ A > 0,$$

with

$$A = \frac{\mathfrak{a}\eta\left(\mathfrak{g}\left(\boldsymbol{v}\right), \boldsymbol{v}\_1\right)\mathfrak{g}'\left(\boldsymbol{v}\right)}{\sigma^{\frac{1}{\mathfrak{a}}}\left(\boldsymbol{v}\right)} \text{ and } B = \frac{\sigma'\left(\boldsymbol{v}\right)}{\sigma\left(\boldsymbol{v}\right)}.$$

we get

$$\begin{split} \left(\boldsymbol{\eta}^{\prime}\left(\boldsymbol{v}\right) + \frac{\boldsymbol{\sigma}\_{0}^{\beta}}{\theta\_{0}}\boldsymbol{\sigma}^{\prime}\left(\boldsymbol{v}\right) &\leq \ -\boldsymbol{\sigma}\left(\boldsymbol{v}\right) \frac{\widetilde{\boldsymbol{q}}\left(\boldsymbol{v}\right)}{\mu} + \frac{\left(\boldsymbol{\sigma}^{\prime}\left(\boldsymbol{v}\right)\right)^{\boldsymbol{\alpha}+1}}{\left(\boldsymbol{\alpha}+1\right)^{\boldsymbol{\alpha}+1}\left(\boldsymbol{\sigma}\left(\boldsymbol{v}\right)\right)\left(\boldsymbol{\xi}\left(\boldsymbol{v}\right),\boldsymbol{\upsilon}\_{1}\right)\boldsymbol{\xi}^{\prime}\left(\boldsymbol{v}\right)\boldsymbol{\epsilon}^{\boldsymbol{\alpha}}} \\ &\quad + \frac{\frac{\boldsymbol{\sigma}\_{0}^{\beta}}{\theta\_{0}}\left(\boldsymbol{\sigma}^{\prime}\left(\boldsymbol{v}\right)\right)^{\boldsymbol{\alpha}+1}}{\left(\boldsymbol{\alpha}+1\right)^{\boldsymbol{\alpha}+1}\left(\boldsymbol{\sigma}\left(\boldsymbol{v}\right)\boldsymbol{\eta}\left(\boldsymbol{\xi}\left(\boldsymbol{v}\right),\boldsymbol{\upsilon}\_{1}\right)\boldsymbol{\xi}^{\prime}\left(\boldsymbol{v}\right)\right)^{\boldsymbol{\alpha}}}. \end{split}$$

Integrating last inequality from *v*<sup>1</sup> to *v*, we arrive at

$$\int\_{\upsilon\_1}^{\upsilon} \left[ \sigma \left( s \right) \frac{\widetilde{q} \left( s \right)}{\mu} - \frac{\left( \sigma' \left( s \right) \right)^{a+1}}{\left( \mathfrak{a} + 1 \right)^{a+1} \left( \sigma \left( s \right) \mathfrak{a} \left( \mathfrak{s} \left( s \right), s\_1 \right) \mathfrak{s}' \left( s \right) \right)^{a}} \left( 1 + \frac{\sigma\_0^{\beta}}{\theta\_0} \right) \right] \mathrm{d}s \leq \psi \left( \upsilon\_2 \right) + \frac{\sigma\_0^{\beta}}{\theta\_0} \mathfrak{a} \left( \upsilon\_2 \right) \dots$$

The proof is complete.

**Theorem 4.** *Assume that there exist continuously differentiable functions σ* (*v*) *and ξ* (*v*) *and ϑ* −1 (*δ* (*v*)) *such that ϑ* −1 (*δ* (*v*))′ > 0, *ξ* ′ (*v*) > 0 *and if (3) and one of the conditions (16), (17) or (15) holds, then Equation (1) is oscillatory.*

**Theorem 5.** *Assume that <sup>x</sup> is a positive solution of (1). If there exist <sup>θ</sup>* ∈ *<sup>C</sup>* ([*v*0, <sup>∞</sup>),(0, <sup>∞</sup>)) *such that <sup>θ</sup>* (*v*) < *<sup>v</sup>*, *ς* (*v*) < *ϑ* (*θ* (*v*)) *and if conditions (3) and (13) hold, then Equation (1) is oscillatory.*

In this section we state and prove some results by considering

$$
\zeta\left(v\right) = v - \delta\_0 \text{ for } \delta\_0 \ge 0,\\
p\left(v\right) = p\_0 \ne 1.
$$

**Lemma 5.** *Let x* (*v*) *be positive solution of Equation (1), eventually. Assume that w* (*v*) *satisfies case* (**ii**). *If*

$$\int\_{\upsilon\_0}^{\infty} \int\_{\phi}^{\infty} \left( \frac{1}{r \left( u \right)} \int\_{u}^{\infty} q \left( s \right) \, \mathrm{ds} \right)^{1/a} \, \mathrm{d}u \, \mathrm{d}\phi = \infty,\tag{27}$$

*then*

$$\lim\_{\upsilon \to \infty} \mathbf{x}\left(\upsilon\right) = 0.\tag{28}$$

**Proof.** Since *w* (*v*) is a non-increasing positive function, there exists a constant *w*<sup>0</sup> ≥ 0 such that lim*v*→<sup>∞</sup> *w* (*v*) = *w*<sup>0</sup> ≥ 0. We claim that *w*<sup>0</sup> = 0. Otherwise, using Lemma 2, we conclude that lim*v*→<sup>∞</sup> *w* (*v*) = *w*0/ (1 + *p*0) > 0. Therefore, there exists a *v*<sup>2</sup> ≥ *v*<sup>0</sup> such that, for all *v* ≥ *v*<sup>2</sup>

$$
\ln \left( \zeta \left( v \right) \right) > \frac{w\_0}{2 \left( 1 + p\_0 \right)} > 0. \tag{29}
$$

From (1) and (29), we see that

$$(\varepsilon w\left(\left(v\right)\right))' \le -q\left(v\right)\left(\frac{w\_0}{2\left(1+p\_0\right)}\right)^\beta.$$

Integrating above inequality from *v* to ∞, we have

$$
\mathcal{E}w\left(\left(v\right)\right) \ge \left(\frac{w\_0}{2\left(1+P\_0\right)}\right)^{\beta} \int\_v^{\infty} q\left(s\right) \,\mathrm{d}s.
$$

It follows that

$$w''(v) \ge \left(\frac{w\_0}{2\left(1+P\_0\right)}\right)^{\frac{\beta}{a}} \left(\frac{1}{r\left(v\right)} \int\_v^{\infty} q\left(s\right) \,\mathrm{ds}\right)^{\frac{1}{a}}.\tag{30}$$

Integrating (30) from *v* to ∞, yields

$$-w'\left(v\right) \ge \left(\frac{w\_0}{2\left(1+P\_0\right)}\right)^{\frac{\beta}{\alpha}} \int\_v^\infty \left(\frac{1}{r\left(u\right)} \int\_v^\infty \mathbf{q}\left(s\right) \,\mathrm{ds}\right)^{1/\alpha} \,\mathrm{d}u.$$

Integrating again from *v*<sup>2</sup> to ∞, we obtain

$$w\left(v\_{2}\right) \geq \left(\frac{w\_{0}}{2\left(1+P\_{0}\right)}\right)^{\frac{\beta}{\alpha}} \int\_{v\_{2}}^{\infty} \int\_{\phi}^{\infty} \left(\frac{1}{r\left(u\right)} \int\_{u}^{\infty} \mathbf{q}\left(s\right) \,\mathrm{ds}\right)^{1/\alpha} \,\mathrm{d}u \,\mathrm{d}\phi\_{s}$$

which contradicts with (27). Therefore, lim*v*→<sup>∞</sup> *w* (*v*) = 0, and from the inequality 0 < *x* (*v*) ≤ *w* (*v*), we have property (28). The proof is complete.

**Theorem 6.** *Let condition (27) be satisfied and suppose that there exists a function ̺* <sup>∈</sup> *<sup>C</sup>* (*I*, <sup>R</sup>) *such that ̺* (*v*) ≤ *<sup>ς</sup>* (*v*), *̺* (*v*) < *v and* lim*v*→<sup>∞</sup> *̺* (*v*) = <sup>∞</sup>. *If the first-order delay differential equation*

$$\left(y'\left(\upsilon\right) + \frac{q\left(\upsilon\right)}{\left(1+p\_0\right)^{\beta}} \left(\int\_{\upsilon\_1}^{\varrho\left(\upsilon\right)} \int\_{\iota\_1}^{\Phi} a^{-1/\gamma}\left(s\right) \mathsf{d}s \mathsf{d}u\right)^{\beta} y^{\frac{\beta}{\alpha}} \left(\varrho\left(\upsilon\right)\right) = 0$$

*is oscillatory, then every solution x* (*v*) *of Equation (1) is either oscillatory or satisfies (28).*

**Proof.** Assume that *x* (*v*) is positive solution of (1), eventually. This implies that there exists *v*<sup>1</sup> ≥ *v<sup>o</sup>* such that either (**i**) or (**ii**) hold for all *v* ≥ *v*1.

For (**ii**), by lemma 5, we see that (28) holds.

For (**i**), since *w* ′ (*v*) is a non-decreasing positive function, there exists a constant *c*<sup>0</sup> such that lim*v*→<sup>∞</sup> *w* ′ (*v*) = *c*<sup>0</sup> > 0 (or *c*<sup>0</sup> = ∞). By Lemma 2, we have

$$\lim\_{v \to \infty} \mathfrak{a}'(v) = c\_0/(1+p\_0) > 0\_\prime$$

which implies that *x* (*v*) is a non-decreasing function and taking into account *δ*<sup>0</sup> ≥ 0, we get

$$w\left(v\right) = \mathbf{x}\left(v\right) + p\_0 \mathbf{x}\left(v - \delta\_0\right) \le \left(1 + p\_0\right) \mathbf{x}\left(v\right) \mathbf{y}$$

therefore

$$\mathbf{x}\left(v\right) \ge \frac{1}{1+p\_0} w\left(v\right).$$

for *̺* (*v*) ≤ *ς* (*v*), and

$$\text{tr}\left(\boldsymbol{\varrho}\left(\boldsymbol{v}\right)\right) \ge \text{tr}\left(\boldsymbol{\varrho}\left(\boldsymbol{v}\right)\right) \ge \frac{1}{1+p\_0} w\left(\boldsymbol{\varrho}\left(\boldsymbol{v}\right)\right) \dots$$

By substitution in (1), we have

$$\left(\left(\varepsilon w\left(v\right)\right)' + \frac{q\left(v\right)}{\left(1+p\_0\right)^{\beta}} w^{\beta}\left(\varrho\left(v\right)\right) \le 0. \tag{31}$$

Using (7) and (31), we get

$$\left(\left(\varepsilon w\left(v\right)\right)' + \frac{q\left(v\right)}{\left(1+p\_0\right)^{\beta}} \left(\int\_{v\_2}^{\varrho(v)} \int\_{u\_1}^{\phi} a^{-1/\gamma}\left(s\right) \operatorname{dsdu}\right)^{\beta} \left(\left\varepsilon w\left(\varrho\left(v\right)\right)\right)^{\frac{\beta}{\alpha}} \leq 0.\right)$$

Therefore, we have *y* = *£w* (*v*) is positive solution of a the first order delay equation

$$y'(v) + \frac{q\left(v\right)}{\left(1+p\_0\right)^{\beta}} \left(\int\_{v\_1}^{\varrho(v)} \int\_{u\_1}^{\phi} a^{-1/\gamma}\left(s\right) \operatorname{dsd}\mu\right)^{\beta} y^{\frac{\beta}{\pi}}\left(\varrho\left(v\right)\right) \le 0.$$

The proof is complete.

**Theorem 7.** *If the first-order delay differential equation*

$$w'(v) + \frac{1}{\mu} \left(\frac{\theta\_0}{\theta\_0 + p\_0^{\delta}}\right) \tilde{q}\left(v\right) \frac{\lambda^{\beta} \xi^{2\beta}\left(v\right)}{2^{\delta} r^{\beta/a}\left(\xi\left(v\right)\right)} w^{\delta/a}\left(\xi\left(v\right)\right) = 0 \tag{32}$$

*is oscillatory, eventually*. *Then, every solution x* (*v*) *of Equation (1) is either oscillatory or satisfies (28).*

**Proof.** As in the proof of Lemma 1, we get, from (1), (5) and (8), that (9) holds. Now, by using Lemma 3, we have

$$w\left(v\right) > \frac{\lambda}{2}v^2w''\left(v\right). \tag{33}$$

Since <sup>d</sup> d*v £w* (*v*) ≤ 0 and *ϑ* (*v*) ≤ *v*, we obtain *£w* (*ϑ* (*v*)) ≥ *£w* (*v*), and so

$$
\varepsilon \varepsilon w \left( v \right) + \frac{1}{\mathfrak{d}\_0} p\_0^{\beta} \varepsilon w \left( \mathfrak{d} \left( v \right) \right) \le \left( 1 + \frac{1}{\mathfrak{d}\_0} p\_0^{\beta} \right) \varepsilon w \left( v \right),
$$

which with (9) gives

$$(\left(\varepsilon w\left(v\right)\right)' + \frac{1}{\mu} \left(\frac{\theta\_0}{\theta\_0 + p\_0^{\beta}}\right) \tilde{q}\left(v\right)w^{\beta}\left(\boldsymbol{\zeta}\left(v\right)\right) \leq 0.$$

Thus, from (33), we find

$$\left(\left(\varepsilon w\left(v\right)\right)' + \frac{1}{\mu} \left(\frac{\mathfrak{d}\_0}{\mathfrak{d}\_0 + p\_0^{\mathfrak{f}}}\right) \widetilde{q}\left(v\right) \frac{\lambda^{\mathfrak{f}}}{2^{\mathfrak{f}}} \xi^{2\mathfrak{f}}\left(v\right) \left(w''\left(\xi\left(v\right)\right)\right)^{\mathfrak{f}} \leq 0.$$

If we set *w* := *£w* (*v*) = *r* (*w* ′′) *α* , then we have that *w* > 0 is a solution of delay inequality

$$w'(v) + \frac{1}{\mu} \left(\frac{\mathfrak{d}\_0}{\mathfrak{d}\_0 + p\_0^{\beta}}\right) \tilde{q}(v) \frac{\lambda^{\beta} \mathfrak{g}^{2\beta}(v)}{2^{\beta} r^{\beta/\alpha} \left(\mathfrak{g}\left(v\right)\right)} w^{\beta/\alpha} \left(\mathfrak{g}\left(v\right)\right) \le 0.$$

By Theorem 1 [21] the associated delay differential Equation (32) also has a positive solution. The proof is complete.

**Example 1.** *Consider the third order delay differential equation*

$$\left[\left(\left[\mathbf{x}\left(\upsilon\right) + p\mathbf{x}\left(\lambda\upsilon\right)\right]^{\prime\prime}\right)^{a}\right]^{\prime} + \frac{q\_{0}}{\upsilon^{a(n-1)+1}}\mathbf{x}^{a}\left(\gamma\upsilon\right) = 0,\tag{34}$$

.

*where <sup>γ</sup>*, *<sup>λ</sup>* <sup>∈</sup> (0, 1). *Then <sup>q</sup>*e(*v*) <sup>=</sup> *q*0 *v* 2*α*+1 *, ς* (*v*) = *γv*, *ϑ* (*v*) = *λv, set σ* (*v*) = *v* 2 , *ζ* (*v*) = (*γ*+*λ*)*v* 2 *. It is easy to get <sup>η</sup>* (*v*, *<sup>u</sup>*) <sup>=</sup> (*<sup>v</sup>* <sup>−</sup> *<sup>u</sup>*), *<sup>η</sup>*e(*v*, *<sup>u</sup>*) <sup>=</sup> (*v*−*u*) 2 2 *and ϑ* −1 (*v*) = *<sup>v</sup> γ* . *By Theorem 3, (18) imply*

$$q\_0 > \frac{(2)^{\beta - 1} \left(2\alpha\right)^{\alpha + 1}}{\gamma^{2\alpha} \left(\alpha + 1\right)^{\alpha + 1}} \left(1 + \frac{\sigma\_0^{\beta}}{\theta\_0}\right) \sqrt{}$$

*also, by (15) with α* = 1, *we get*

$$\frac{q\_0}{8} \left(\gamma - \lambda\right)^2 \ln \frac{2\gamma}{\lambda + \gamma} > \frac{\theta\_0 + p\_0}{\theta\_0 e} \lambda$$

*By Theorem 4 with α* = 1, *the Equation (34) is oscillatory if*

$$q\_0 > \max\left\{ \frac{1}{\gamma^2} \left( 1 + \frac{\sigma\_0}{\theta\_0} \right), \frac{8\left(\theta\_0 + p\_0\right)}{\left(\gamma - \lambda\right)^2 \left( \ln \frac{2\gamma}{\lambda + \gamma} \right) \theta\_0 e} \right\}$$

**Remark 1.** *The results in [11–19] only ensure that the non-oscillating solutions to Equation (34) tend to zero, so our method improves the previous results.*

**Remark 2.** *For interested researchers, there is a good problem which is finding new results for non existence of Kneser solutions for (1) without requiring*

$$
\theta \circ \mathfrak{g} = \mathfrak{g} \circ \mathfrak{G} \text{ or } \left(\theta^{-1}\left(v\right)\right)' \ge \mathfrak{G}\_0.
$$

**Author Contributions:** Writing original draft, formal analysis, writing review and editing, O.M., B.Q. and O.B.; writing review and editing, funding and supervision, R.A.E.-N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors thank the reviewers for for their useful comments, which led to the improvement of the content of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**



© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

## **Improved Approach for Studying Oscillatory Properties of Fourth-Order Advanced Differential Equations with** *p***-Laplacian Like Operator**

**Omar Bazighifan 1,2,\* ,† and Thabet Abdeljawad 3,4,5,\* ,†**


Received: 16 April 2020; Accepted: 25 April 2020; Published: 26 April 2020

**Abstract:** This paper aims to study the oscillatory properties of fourth-order advanced differential equations with *p*-Laplacian like operator. By using the technique of Riccati transformation and the theory of comparison with first-order delay equations, we will establish some new oscillation criteria for this equation. Some examples are considered to illustrate the main results.

**Keywords:** oscillation; advanced differential equations; *p*-Laplacian equations; comparison theorem

#### **1. Introduction**

In the last decades, many researchers from all fields of science, technology and engineering have devoted their attention to introducing more sophisticated analytical and numerical techniques to solve and analyze mathematical models arising in their fields.

Fourth-order advanced differential equations naturally appear in models concerning physical, biological, chemical phenomena applications in dynamical systems, mathematics of networks,and optimization. They also appear in the mathematical modeling of engineering problems to study electrical power systems, materials and energy, elasticity, deformation of structures, and soil settlement [1]. The *p*-Laplace equations have some applications in continuum mechanics, see for example [2–4].

An active and essential research area in the above investigations is to study the sufficient criterion for oscillation of delay differential equations. In fact, during this decade, Several works have been accomplished in the development of the oscillation theory of higher order delay and advanced equations by using the Riccati transformation and the theory of comparison between first and second-order delay equations, (see [5–12]). Further, the oscillation theory of fourth and second order delay equations has been studied and developed by using integral averaging technique and the Riccati transformation, (see [13–27]). The study of oscillation has been carried to fractional equations in the setting of fractional operators with singular and nonsingular kernels, as well (see [28,29] and the references therein).

We provide oscillation properties of the fourth order advanced differential equation with a *p*-Laplacian like operator

$$\left( b\left( \upsilon\right) \left| y^{\prime\prime\prime}\left( \upsilon\right) \right|^{p-2} y^{\prime\prime\prime}\left( \upsilon\right) \right)^{\prime} + \sum\_{i=1}^{j} q\_i\left( \upsilon\right) g\left( y\left( \eta\_i\left( \upsilon\right) \right) \right) = 0,\tag{1}$$

where *υ* ≥ *υ*<sup>0</sup> and *j* ≥ 1. Throughout this paper, we assume that:

(*D*1) *p* > 1 is a real number, (*D*2) *q<sup>i</sup>* , *<sup>η</sup><sup>i</sup>* <sup>∈</sup> *<sup>C</sup>* ([*υ*0, <sup>∞</sup>), <sup>R</sup>), *<sup>q</sup><sup>i</sup>* (*υ*) <sup>≥</sup> 0, (*D*3) *<sup>η</sup><sup>i</sup>* (*υ*) <sup>≥</sup> *<sup>υ</sup>*, lim*υ*→<sup>∞</sup> *η<sup>i</sup>* (*υ*) = ∞, *i* = 1, 2, .., *j*, (*D*4) *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>* (R, <sup>R</sup>) such that *g* (*x*) / |*x*| *p*−2 *x* ≥ *k* > 0, for *x* 6= 0. (2)

(*D*5) *b* ∈ *C* 1 ([*υ*0, ∞), R), *b* (*υ*) > 0, *b* ′ (*υ*) ≥ 0 and under the condition

$$\int\_{v\_0}^{\infty} \frac{1}{b^{1/(p-1)}\left(s\right)} ds = \infty. \tag{3}$$

In fact, our aim in this paper is complete and improves the results in [5–7]. For the sake of completeness, we first recall and discuss these results. Li et al. [3] examined the oscillation of equation

$$\left( a\left(\upsilon\right) \left| z^{\prime\prime\prime}\left(\upsilon\right) \right|^{p-2} z^{\prime\prime\prime}\left(\upsilon\right) \right)^{\prime} + \sum\_{i=1}^{j} q\_i\left(\upsilon\right) \left| w\left(\delta\_i\left(\upsilon\right)\right) \right|^{p-2} w\left(\delta\_i\left(\upsilon\right)\right) = 0,$$

where *p* > 1 is a real number. The authors used the Riccati transformation and integral averaging technique. Park et al. [8] used Riccati technique to obtain necessary and sufficient conditions for oscillation of

$$\left( a\left( \upsilon \right) \left| w^{\left( \kappa -1 \right)} \left( \upsilon \right) \right|^{p-2} w^{\left( \kappa -1 \right)} \left( \upsilon \right) \right)' + q\left( \upsilon \right) g\left( w\left( \delta \left( \upsilon \right) \right) \right) = 0,$$

where *κ* is even and under the condition

$$\int\_{v\_0}^{\infty} \frac{1}{a^{1/(p-1)}\,(s)} \mathrm{d}s = \infty.$$

Agarwal and Grace [5] considered the equation

$$\left( \left( y^{\left( \kappa - 1 \right)} \left( v \right) \right)^{\gamma} \right)' + q \left( v \right) y^{\gamma} \left( \eta \left( v \right) \right) = 0,\tag{4}$$

where *κ* is even and they proved it oscillatory if

$$\liminf\_{\upsilon \to \infty} \int\_{\upsilon}^{\eta(\upsilon)} (\eta \left(s\right) - s)^{\kappa - 2} \left( \int\_{s}^{\infty} q\left(\upsilon\right) d\upsilon \right)^{1/\gamma} ds > \frac{(\kappa - 2)!}{e}. \tag{5}$$

Agarwal et al. in [6] studied Equation (4) and obtained the criterion of oscillation

$$\limsup\_{\upsilon \to \infty} v^{\gamma(\kappa - 1)} \int\_{\upsilon}^{\infty} q\left(s\right) ds > ((\kappa - 1)!)^{\gamma}.\tag{6}$$

Authors in [7] studied oscillatory behavior of (4) where *γ* = 1, *κ* is even and if there exists a function *δ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)), also, they proved it oscillatory by using the Riccati transformation if

$$\int\_{\nu\_0}^{\infty} \left( \delta \left( s \right) q \left( s \right) - \frac{\left( \kappa - 2 \right)! \left( \delta' \left( s \right) \right)^2}{2^{3 - 2\kappa} s^{\kappa - 2} \delta \left( s \right)} \right) ds = \infty. \tag{7}$$

To compare the conditions, we apply the previous results to the equation

$$y^{(4)}\left(\upsilon\right) + \frac{q\_0}{\upsilon^4}y\left(\mathfrak{B}\upsilon\right) = 0, \ \upsilon \ge 1,\tag{8}$$

1. By applying condition (5) on Equation (8), we get

$$q\_0 > 13.6.$$

2. By applying condition (6) on Equation (8), we get

*q*<sup>0</sup> > 18.

3. By applying condition (7) on Equation (8), we get

*q*<sup>0</sup> > 576.

From the above we find the results in [6] improves results [7]. Moreover, the results in [5] improves results [6,7], we see this clearly in the Section 3. Thus, the motivation in studying this paper is complement and improve results [5–7].

We will need the following lemmas.

**Lemma 1** ([18])**.** *If the function y satisfies y*(*i*) (*υ*) > 0, *i* = 0, 1, ..., *n*, *and y*(*n*+1) (*υ*) < 0, *then*

$$\frac{y\left(\upsilon\right)}{\upsilon^n/n!} \ge \frac{y'\left(\upsilon\right)}{\upsilon^{n-1}/\left(n-1\right)!}.$$

**Lemma 2** ([10])**.** *Suppose that y* ∈ *C n* ([*υ*0, ∞),(0, ∞)), *y* (*n*) *is of a fixed sign on* [*υ*0, ∞), *y* (*n*) *not identically zero and there exists a υ*<sup>1</sup> ≥ *υ*<sup>0</sup> *such that*

$$y^{(n-1)}\left(\upsilon\right)y^{(n)}\left(\upsilon\right) \le 0\_{\prime}$$

*for all υ* ≥ *υ*1*. If we have* lim*υ*→<sup>∞</sup> *y* (*υ*) 6= 0*, then there exists υ<sup>λ</sup>* ≥ *υ*<sup>1</sup> *such that*

$$y\left(v\right) \ge \frac{\lambda}{(n-1)!} v^{n-1} \left| y^{(n-1)}\left(v\right) \right| \cdot $$

*for every λ* ∈ (0, 1) *and υ* ≥ *υλ.*

**Lemma 3** ([21])**.** *Let γ be a ratio of two odd numbers, V* > 0 *and U are constants. Then*

$$\mathcal{U}\mathbf{x} - V\mathbf{x}^{(\gamma+1)/\gamma} \le \frac{\gamma^{\gamma}}{(\gamma+1)^{\gamma+1}} \frac{\mathcal{U}^{\gamma+1}}{V^{\gamma}}, \; V > 0.$$

**Lemma 4** ([15])**.** *Assume that y is an eventually positive solution of (1). Then, there exist two possible cases:*

$$\begin{array}{ll} \left(\mathbf{S}\_1\right) & y\left(\upsilon\right) > 0, \ y'\left(\upsilon\right) > 0, \ y''\left(\upsilon\right) > 0, \ y^{\prime\prime\prime}\left(\upsilon\right) > 0, \ y^{\prime(4)}\left(\upsilon\right) \leq 0, \\\left(\mathbf{S}\_2\right) & y\left(\upsilon\right) > 0, \ y'\left(\upsilon\right) > 0, \ y^{\prime\prime}\left(\upsilon\right) < 0, \ y^{\prime\prime\prime}\left(\upsilon\right) > 0, \ y^{\prime(4)}\left(\upsilon\right) \leq 0, \end{array}$$

*for υ* ≥ *υ*1, *where υ*<sup>1</sup> ≥ *υ*<sup>0</sup> *is sufficiently large.*

#### **2. Oscillation Criteria**

In this section, we shall establish some oscillation criteria for equation (1).

**Lemma 5.** *Assume that y be an eventually positive solution of (1) and* (**S**1) *holds. If*

$$\pi \left( \upsilon \right) := \delta \left( \upsilon \right) \left( \frac{b \left( \upsilon \right) \left( y^{\prime \prime \prime} \left( \upsilon \right) \right)^{p-1}}{y^{p-1} \left( \upsilon \right)} \right) , \tag{9}$$

*where δ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)), *then*

$$
\pi'\left(\upsilon\right) \le \frac{\delta'\left(\upsilon\right)}{\delta\left(\upsilon\right)}\pi\left(\upsilon\right) - k\delta\left(\upsilon\right)\sum\_{i=1}^{j}q\_{i}\left(\upsilon\right) - \frac{\left(p-1\right)\varepsilon\upsilon^{2}}{2\left(\delta\left(\upsilon\right)b\left(\upsilon\right)\right)^{\frac{1}{\left(p-1\right)}}}\pi\left(\upsilon\right)^{\frac{p}{\left(p-1\right)}}\right.\tag{10}
$$

*for all υ* > *υ*1*, where υ*<sup>1</sup> *large enough*.

**Proof.** Let *y* is an eventually positive solution of (1) and (**S**1) holds. Thus, from Lemma 2, we get

$$y'\left(\upsilon\right) \ge \frac{\varepsilon}{2}\upsilon^2 y'''\left(\upsilon\right),\tag{11}$$

for every *ε* ∈ (0, 1) and for all large *υ*. From (9), we see that *π* (*υ*) > 0 for *υ* ≥ *υ*1, and

$$\begin{aligned} \pi'(\upsilon) &= \, \_\delta'(\upsilon) \, \frac{b\left(\upsilon\right) \left(\boldsymbol{y}^{\prime\prime\prime}(\upsilon)\right)^{p-1}}{\boldsymbol{y}^{p-1}\left(\upsilon\right)} + \delta\left(\upsilon\right) \, \frac{\left(b\left(\boldsymbol{y}^{\prime\prime\prime}\right)^{p-1}\right)^{\prime}\left(\upsilon\right)}{\boldsymbol{y}^{p-1}\left(\upsilon\right)} \\ &- \left(p-1\right)\delta\left(\upsilon\right) \, \frac{\boldsymbol{y}^{p-2}\left(\upsilon\right)\boldsymbol{y}^{\prime}\left(\upsilon\right)b\left(\upsilon\right)\left(\boldsymbol{y}^{\prime\prime\prime}\left(\upsilon\right)\right)^{p-1}}{\boldsymbol{y}^{2\left(p-1\right)}\left(\upsilon\right)}. \end{aligned}$$

Using (11) and (9), we obtain

$$\begin{split} \pi'(\boldsymbol{v}) &\leq \quad \frac{\delta'(\boldsymbol{v})}{\delta\left(\boldsymbol{v}\right)}\pi(\boldsymbol{v}) + \delta\left(\boldsymbol{v}\right) \frac{\left(\boldsymbol{b}\left(\boldsymbol{v}\right)\left(\boldsymbol{y}'''\left(\boldsymbol{v}\right)\right)^{p-1}\right)'}{\boldsymbol{y}^{p-1}\left(\boldsymbol{v}\right)} \\ &- \left(p-1\right)\delta\left(\boldsymbol{v}\right)\frac{\varepsilon}{2}\boldsymbol{v}^{2}\frac{\boldsymbol{b}\left(\boldsymbol{v}\right)\left(\boldsymbol{y}'''\left(\boldsymbol{v}\right)\right)^{p}}{\boldsymbol{y}^{p}\left(\boldsymbol{v}\right)} \\ &\leq \quad \frac{\delta'\left(\boldsymbol{v}\right)}{\delta\left(\boldsymbol{v}\right)}\pi\left(\boldsymbol{v}\right) + \delta\left(\boldsymbol{v}\right) \frac{\left(\boldsymbol{b}\left(\boldsymbol{v}\right)\left(\boldsymbol{y}'''\left(\boldsymbol{v}\right)\right)^{p-1}\right)'}{\boldsymbol{y}^{p-1}\left(\boldsymbol{v}\right)} \\ &- \frac{\left(p-1\right)\epsilon\boldsymbol{v}^{2}}{2\left(\delta\left(\boldsymbol{v}\right)\boldsymbol{b}\left(\boldsymbol{v}\right)\right)^{\frac{1}{\left(p-1\right)}}}\pi\left(\boldsymbol{v}\right)^{\frac{p}{\left(p-1\right)}}. \end{split} \tag{12}$$

From (1) and (12), we get

$$\pi'(v) \le \frac{\delta'(v)}{\delta(v)}\pi(v) - k\delta'(v) \frac{\sum\_{i=1}^j q\_i(v) \, y^{p-1} \left(\eta\_i(v)\right)}{y^{p-1} \left(v\right)} - \frac{(p-1)\varepsilon v^2}{2 \left(\delta\left(v\right) b\left(v\right)\right)^{\frac{1}{\left(p-1\right)}}} \pi(v)^{\frac{p}{\left(p-1\right)}} \,.$$

Note that *y* ′ (*υ*) > 0 and *η<sup>i</sup>* (*υ*) ≥ *υ*, thus, we find

$$\pi'\left(\upsilon\right) \le \frac{\delta'\left(\upsilon\right)}{\delta\left(\upsilon\right)}\pi\left(\upsilon\right) - k\delta\left(\upsilon\right)\sum\_{i=1}^j q\_i\left(\upsilon\right) - \frac{\left(p-1\right)\varepsilon\upsilon^2}{2\left(\delta\left(\upsilon\right)b\left(\upsilon\right)\right)^{\frac{1}{\left(p-1\right)}}}\pi\left(\upsilon\right)^{\frac{p}{\left(p-1\right)}}.$$

The proof is complete.

**Lemma 6.** *Assume that y be an eventually positive solution of (1) and* (**S**2) *holds. If*

$$
\zeta'(\upsilon) := \sigma\left(\upsilon\right) \frac{y'\left(\upsilon\right)}{y\left(\upsilon\right)}.\tag{13}
$$

*where σ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)), *then*

$$\xi'(v) \le \frac{\sigma'(v)}{\sigma(v)}\xi(v) - \sigma(v) \int\_{v}^{\infty} \left(\frac{k}{b'(v)} \int\_{v}^{\infty} \sum\_{i=1}^{j} q\_i(s) \, \mathrm{ds}\right)^{1/(p-1)} \, \mathrm{d}v - \frac{1}{\sigma'(v)}\xi'(v)^2 \, \mathrm{.}\tag{14}$$

*for all υ* > *υ*1*, where υ*<sup>1</sup> *large enough*.

**Proof.** Let *y* is an eventually positive solution of (1) and (**S**2) holds. Integrating (1) from *υ* to *m* and using *y* ′ (*υ*) > 0, we obtain

$$b\left(m\right)\left(y^{\prime\prime\prime}\left(m\right)\right)^{p-1} - b\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{p-1} = -\int\_{\upsilon}^{m} \sum\_{i=1}^{j} q\_i\left(s\right)g\left(y\left(\eta\_i\left(s\right)\right)\right)ds.$$

By virtue of *y* ′ (*υ*) > 0 and *η<sup>i</sup>* (*υ*) ≥ *υ*, we get

$$b\left(m\right)\left(y^{\prime\prime\prime}\left(m\right)\right)^{p-1} - b\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{p-1} \le -ky^{p-1}\left(\upsilon\right)\int\_{\upsilon}^{\mu} \sum\_{i=1}^{j} q\_i\left(s\right)ds.$$

Letting *<sup>m</sup>* → <sup>∞</sup> , we see that

$$b\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{p-1} \geq k y^{p-1}\left(\upsilon\right) \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \,\mathrm{d}s$$

and so

$$y^{\prime\prime\prime}(\upsilon) \ge y^{\prime}(\upsilon) \left( \frac{k}{b^{\prime}(\upsilon)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i^{\prime}(s) \, \mathrm{d}s \right)^{1/(p-1)}$$

Integrating again from *υ* to ∞, we get

$$y''(v) + y\left(v\right) \int\_{v}^{\infty} \left(\frac{k}{b\left(v\right)} \int\_{v}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \mathrm{d}s\right)^{1/\left(p-1\right)} \mathrm{d}v \le 0. \tag{15}$$

.

From the definition of *ξ* (*υ*), we see that *ξ* (*υ*) > 0 for *υ* ≥ *υ*1. By differentiating, we find

$$
\xi'\left(\upsilon\right) = \frac{\sigma'\left(\upsilon\right)}{\sigma\left(\upsilon\right)}\xi\left(\upsilon\right) + \sigma\left(\upsilon\right)\frac{y''\left(\upsilon\right)}{y\left(\upsilon\right)} - \frac{1}{\sigma\left(\upsilon\right)}\xi\left(\upsilon\right)^2. \tag{16}
$$

From (15) and (16), we obtain

$$\xi'(v) \le \frac{\sigma'(v)}{\sigma(v)}\xi(v) - \sigma(v) \int\_{v}^{\infty} \left(\frac{k}{b'(v)} \int\_{v}^{\infty} \sum\_{i=1}^{j} q\_i(s) \, \mathrm{d}s\right)^{1/(p-1)} \, \mathrm{d}v - \frac{1}{\sigma'(v)}\xi(v)^2 \, \mathrm{d}s$$

The proof is complete.

**Theorem 1.** *Assume that there exist positive functions δ*, *σ* ∈ *C* 1 ([*υ*0, ∞),(0, ∞)) *such that*

$$\int\_{\upsilon\_0}^{\infty} \left( k \delta \left( s \right) \sum\_{i=1}^{j} q\_i \left( s \right) - \frac{2^{p-1} b \left( s \right) \left( \delta' \left( s \right) \right)^p}{p^p \left( s^2 \varepsilon \delta \left( s \right) \right)^{p-1}} \right) \mathrm{d}s = \infty,\tag{17}$$

*for some ε* ∈ (0, 1)*, and either*

$$\int\_{v\_0}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) ds = \infty \tag{18}$$

*or*

$$\int\_{\nu\_0}^{\infty} \left( \sigma \left( s \right) \int\_{\upsilon}^{\infty} \left( \frac{k}{b \left( v \right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) \mathrm{d}s \right)^{1/(p-1)} \mathrm{d}v - \frac{1}{4 \sigma \left( s \right)} \left( \sigma' \left( s \right) \right)^2 \right) \mathrm{d}s = \infty. \tag{19}$$

*Then every solution of (1) is oscillatory.*

**Proof.** Assume that *y* is eventually positive solution of (1). Then, we can suppose that *y* (*υ*)and *y* (*η<sup>i</sup>* (*υ*)) are positive for all *υ* ≥ *υ*<sup>1</sup> sufficiently large. From Lemma 4, we have two possible cases (**S**1) and (**S**2).

Assume that case (**S**1) holds. From Lemma 5, we get that (10) holds. Using Lemma 3 with

$$\mathcal{U} = \delta'(v) / \delta(v) \, , \, V = (p - 1) \, \varepsilon v^2 / \left( 2 \left( \delta \left( v \right) b \left( v \right) \right)^{\frac{1}{(p - 1)}} \right) \text{ and } \mathfrak{x} = \pi \left( v \right) \, \mathfrak{x}$$

we get

$$\frac{\delta'(\upsilon)}{\delta\left(\upsilon\right)}\pi\left(\upsilon\right) - \frac{(p-1)\epsilon\upsilon^2}{2\left(\delta\left(\upsilon\right)b\left(\upsilon\right)\right)^{\frac{1}{\left(p-1\right)}}}\pi\left(\upsilon\right)^{\frac{p}{\left(p-1\right)}} \leq -\frac{2^{p-1}b\left(\upsilon\right)\left(\delta'\left(\upsilon\right)\right)^p}{p^p \left(\upsilon^2\varepsilon\delta\left(\upsilon\right)\right)^{p-1}}.\tag{20}$$

From (10) and (20), we obtain

$$\pi'\left(\upsilon\right) \le -k\delta\left(\upsilon\right)\sum\_{i=1}^j q\_i\left(\upsilon\right) + \frac{2^{p-1}b\left(\upsilon\right)\left(\delta'\left(\upsilon\right)\right)^p}{p^p \left(\upsilon^2\varepsilon\delta\left(\upsilon\right)\right)^{p-1}}.$$

Integrating from *υ*<sup>1</sup> to *υ*, we get

$$\int\_{\upsilon\_1}^{\upsilon} \left( k \delta \left( s \right) \sum\_{i=1}^{j} q\_i \left( s \right) - \frac{2^{p-1} b \left( s \right) \left( \delta' \left( s \right) \right)^p}{p^p \left( s^2 \varepsilon \delta \left( s \right) \right)^{p-1}} \right) \mathrm{d}s \le \pi \left( \upsilon\_1 \right) \iota$$

for every *ε* ∈ (0, 1), which contradicts (17).

Let case (**S**2) holds. Integrating (1) from *m* to *υ*, we conclude that

$$-b\left(m\right)\left(y^{\prime\prime\prime}\left(m\right)\right)^{p-1} = -\int\_{\mathfrak{m}}^{\upsilon} \sum\_{i=1}^{j} q\_i\left(s\right)g\left(y\left(\eta\_i\left(s\right)\right)\right)ds.$$

By virtue of *y* ′ (*υ*) > 0 and *η<sup>i</sup>* (*υ*) ≥ *υ*, we get

$$\int\_{m}^{\upsilon} \sum\_{i=1}^{j} q\_i\left(s\right) ds \le \frac{b\left(m\right) \left(y^{\prime\prime\prime}\left(m\right)\right)^{p-1}}{ky^{p-1}\left(m\right)}\lambda$$

which contradicts (18).

From Lemma 6, we get that (14) holds. Using Lemma 3 with

$$\mathcal{U} = \sigma'\left(\upsilon\right)/\sigma\left(\upsilon\right), \; V = 1/\sigma\left(\upsilon\right) \text{ and } \mathfrak{x} = \mathfrak{f}\left(\upsilon\right), \; \mathfrak{x}$$

we get

$$\frac{\sigma'(v)}{\sigma(v)}\mathfrak{f}\left(v\right) - \frac{1}{\sigma\left(v\right)}\mathfrak{f}^2\left(v\right) \le -\frac{1}{4\sigma\left(v\right)}\left(\sigma'\left(v\right)\right)^2. \tag{21}$$

From (14) and (21), we obtain

$$\xi'(v) \le -\sigma\left(v\right) \int\_{v}^{\infty} \left(\frac{k}{b\left(v\right)} \int\_{v}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \, \mathrm{ds}\right)^{1/\left(p-1\right)} \, \mathrm{d}v + \frac{1}{4\sigma\left(v\right)} \left(\sigma'\left(v\right)\right)^{2} \, . \tag{22}$$

Integrating from *υ*<sup>1</sup> to *υ*, we get

$$\int\_{\nu\_1}^{v} \left( \sigma \begin{pmatrix} s \\ \end{pmatrix} \int\_{\upsilon}^{\infty} \left( \frac{k}{b \left( \upsilon \right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) \, \mathrm{ds} \right)^{1/\gamma} \, \mathrm{d}\upsilon - \frac{1}{4\sigma \left( s \right)} \left( \sigma' \left( s \right) \right)^2 \right) \, \mathrm{ds} \leq \xi \left( v\_1 \right) \, \mathrm{s}$$

which contradicts (19). The proof is complete.

When putting *δ* (*υ*) = *υ* <sup>3</sup> and *σ* (*υ*) = *υ* into Theorem 1, we get the following oscillation criteria:

**Corollary 1.** *Let (3) hold. Assume that*

$$\int\_{\upsilon\_0}^{\infty} \left( s^3 \sum\_{i=1}^j q\_i(s) - \frac{2^{p-1} 3^p s^{-3(p-1)+2} b(s)}{p^p \varepsilon^{p-1}} \right) ds = \infty,\tag{23}$$

*or some ε* ∈ (0, 1). *If (18) holds and*

$$\int\_{\nu\_0}^{\infty} \left( s \int\_{\upsilon}^{\infty} \left( \frac{k}{b \left( \upsilon \right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) ds \right)^{1/(p-1)} \mathrm{d}\upsilon - \frac{1}{4s} \right) \mathrm{d}s = \infty,\tag{24}$$

*then every solution of (1) is oscillatory.*

In the next theorem, we compare the oscillatory behavior of (1) with the first-order differential equations:

**Theorem 2.** *Assume that (3) holds. If the differential equations*

$$\theta'\left(\upsilon\right) + k \sum\_{i=1}^{j} q\_i\left(\upsilon\right) \left(\frac{\varepsilon \upsilon^2}{2b^{1/\gamma}\left(\upsilon\right)}\right)^{p-1} \theta\left(\eta\left(\upsilon\right)\right) = 0\tag{25}$$

*and*

$$\left(\phi'\left(\upsilon\right) + \upsilon\phi\left(\upsilon\right)\int\_{\upsilon}^{\infty} \left(\frac{k}{b\left(\upsilon\right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) \mathrm{d}s\right)^{1/(p-1)} \mathrm{d}\upsilon = 0\tag{26}$$

*are oscillatory, then every solution of (1) is oscillatory.*

**Proof.** Assume the contrary that *y* is a positive solution of (1). Then, we can suppose that *y* (*υ*)and *y* (*η<sup>i</sup>* (*υ*)) are positive for all *υ* ≥ *υ*<sup>1</sup> sufficiently large. From Lemma 4, we have two possible cases (**S**1) and (**S**2).

In the case where (**S**1) holds, from Lemma 2, we get

$$y\left(\upsilon\right) \ge \frac{\varepsilon \upsilon^2}{2b^{1/(p-1)}\left(\upsilon\right)} \left(b^{1/\left(p-1\right)}\left(\upsilon\right) y^{\prime\prime\prime}\left(\upsilon\right)\right),$$

for every *ε* ∈ (0, 1) and for all large *υ*. Thus, if we set

$$\theta\left(\upsilon\right) = b\left(\upsilon\right)\left(y^{\prime\prime\prime}\left(\upsilon\right)\right)^{p-1} > 0\_{\prime\prime}$$

then we see that *ξ* is a positive solution of the inequality

$$
\theta'\left(\upsilon\right) + k \sum\_{i=1}^{j} q\_i\left(\upsilon\right) \left(\frac{\varepsilon \upsilon^2}{2b^{1/(p-1)}\left(\upsilon\right)}\right)^{p-1} \theta\left(\eta\left(\upsilon\right)\right) \le 0. \tag{27}
$$

From ([27], Theorem 1), we conclude that the corresponding Equation (25) also has a positive solution, which is a contradiction.

In the case where (**S**2) holds, from Lemma 1, we get

$$y\left(\upsilon\right) \ge \upsilon y'\left(\upsilon\right),\tag{28}$$

From (28) and (15), we get

$$\left(y''\left(v\right) + vy'\left(v\right)\int\_{v}^{\infty} \left(\frac{k}{b\left(v\right)} \int\_{v}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \mathrm{d}s\right)^{1/\left(p-1\right)} \mathrm{d}v \le 0.$$

Thus, if we set

$$
\phi\left(v\right) = y'\left(v\right)'
$$

then we see that *ξ* is a positive solution of the inequality

$$\left(\phi'\left(\upsilon\right) + \upsilon\phi\left(\upsilon\right)\right) \int\_{\upsilon}^{\infty} \left(\frac{k}{b\left(\upsilon\right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_{i}\left(s\right) \mathrm{d}s\right)^{1/\left(p-1\right)} \mathrm{d}\upsilon \le 0. \tag{29}$$

It is well known (see ([27], Theorem 1)) that the corresponding Equation (26) also has a positive solution, which is a contradiction. The proof is complete.

**Corollary 2.** *Assume that (3) holds. If*

$$\liminf\_{v \to \infty} \int\_{\eta\_i(v)}^v \sum\_{i=1}^j q\_i \left( s \right) \left( \frac{\varepsilon \mathbf{s}^2}{2b^{1/(p-1)} \left( s \right)} \right)^{p-1} \mathbf{ds} > \frac{((n-1)!)^{p-1}}{\mathbf{e}} \tag{30}$$

*and*

$$\liminf\_{\upsilon \to \infty} \int\_{\eta\_i(\upsilon)}^{\upsilon} v \int\_{\upsilon}^{\infty} \left( \frac{k}{b\left(\upsilon\right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i\left(s\right) ds \right)^{1/(p-1)} \,\mathrm{d}\upsilon \mathrm{ds} > \frac{1}{\mathbf{e}'} \tag{31}$$

*then every solution of (1) is oscillatory.*

#### **3. Examples**

For an application of Corollary 1, we give the following example:

**Example 1.** *Consider a differential equation*

$$y^{(4)}\left(\upsilon\right) + \frac{q\_0}{\upsilon^4}y\left(2\upsilon\right) = 0, \ \upsilon \ge 1,\tag{32}$$

*where q*<sup>0</sup> > 0 *is a constant. Note that p* = 2, *b* (*υ*) = 1, *q* (*υ*) = *q*0/*υ* 4 *and η* (*υ*) = 2*υ. If we set k* = 1, *then condition (23) becomes*

$$\begin{aligned} \int\_{v\_0}^{\infty} \left( s^3 \sum\_{i=1}^j q\_i \left( s \right) - \frac{2^{p-1} 3^p s^{-3(p-1)+2} b \left( s \right)}{p^p \varepsilon^{p-1}} \right) ds &= \quad \int\_{v\_0}^{\infty} \left( \frac{q\_0}{s} - \frac{9}{2\varepsilon s} \right) ds \\ &= \quad \left( q\_0 - \frac{9}{2\varepsilon} \right) \int\_{v\_0}^{\infty} \frac{1}{s} ds \\ &= \quad \text{or} \quad \text{if} \quad q\_0 > 4.5 \end{aligned}$$

*and condition (24) becomes*

$$\begin{aligned} \int\_{\upsilon\_0}^{\infty} \left( s \int\_{\upsilon}^{\infty} \left( \frac{k}{b \,(\upsilon)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i \,(s) \, \mathrm{ds} \right)^{1/(p-1)} \mathrm{d}\upsilon - \frac{1}{4s} \right) \mathrm{ds} &= \quad \int\_{\upsilon\_0}^{\infty} \left( \frac{q\_0}{6s} - \frac{1}{4s} \right) \mathrm{ds} \\ &= \quad \quad \quad \quad \quad \quad \quad q\_0 > \frac{3}{2}. \end{aligned}$$

*Therefore, from Corollary 1, all solution Equation (32) is oscillatory if q*<sup>0</sup> > 4.5*.*

**Remark 1.** *We compare our result with the known related criteria for oscillation of this equation are as follows:*


*Therefore, it is clear that we see our result improves results [5–7].*

For an application of Theorem 1, we give the following example.

**Example 2.** *Consider a differential equation*

$$\left(\upsilon\left(y^{\prime\prime\prime}(\upsilon)\right)\right)' + \frac{a}{\upsilon^3} y\left(\upsilon\upsilon\right) = 0, \upsilon \ge 1,\tag{33}$$

*where c* > 0 *and a* > 1 *is a constant. Note that p* = 2, *b* (*υ*) = *υ*, *q* (*υ*) = *a*/*υ* 3 . *If we set k* = 1, *δ* (*s*) = *σ* (*s*) = *s* 2 , *then conditions (17) and (19) become*

Z ∞ *υ*0 *kδ* (*s*) *j* ∑ *i*=1 *q<sup>i</sup>* (*s*) − 2 *p*−1 *b* (*s*) (*δ* ′ (*s*))*<sup>p</sup> p p* (*s* 2 *εδ* (*s*))*p*−<sup>1</sup> ! d*s* = Z ∞ *υ*0 *a s* − 2 *sε* d*s* = *a* − 2 *ε* Z ∞ *υ*0 1 *s* d*s* = ∞ *if a* > 2 *ε*

*and*

$$\begin{split} \int\_{v\_0}^{\infty} \left( \sigma \left( s \right) \int\_{v}^{\infty} \left( \frac{k}{b \left( v \right)} \int\_{\upsilon}^{\infty} \sum\_{i=1}^{j} q\_i \left( s \right) \mathrm{d}s \right)^{1/\left(p-1\right)} \mathrm{d}\upsilon - \frac{1}{4\sigma \left( s \right)} \left( \sigma' \left( s \right) \right)^{2} \right) \mathrm{d}s &= \int\_{v\_0}^{\infty} \left( \frac{a}{4} - \frac{1}{4} \right) \mathrm{d}s \\ &= \quad \infty, \quad \text{if} \quad q\_0 > 1. \end{split}$$

*for some constant ε* ∈ (0, 1)*. Hence, by Theorem 1, every solution of Equation (33) is oscillatory if*

$$a > \frac{2}{\varepsilon}.$$

**Remark 2.** *By applying condition (23) in Equation (8), we find*

*q*<sup>0</sup> > 4.5,

*while the conditions that we obtained in the introduction as follows:*


*Therefore, our result improves results [5–7].*

#### **4. Conclusions**

This paper is concerned with the oscillatory properties of the fourth-order differential equations with *p*-Laplacian like operators. New oscillation criteria are established, and they essentially improves the related contributions to the subject. In this paper the following methods were used:


Further, in the future work we get some oscillation criteria of (1) under the condition R <sup>∞</sup> *υ*0 1 *b* 1/(*p*−1) (*s*) d*s* < ∞.

**Author Contributions:** O.B.: Writing original draft, writing review and editing. T.A.: Formal analysis, writing review and editing, funding and supervision. All authors have read and agreed to the published version of the manuscript.

**Funding:** The second author would like to thank Prince Sultan University for the support through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

**Acknowledgments:** The authors thank the reviewers for their useful comments, which led to the improvement of the content of the paper.

**Conflicts of Interest:** There are no competing interests between the authors.

#### **References**


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