**1. Introduction**

The theory of differential equations is an adequate mathematical apparatus for the simulation of processes and phenomena observed in biotechnology, neural networks, physics etc, see [1]. One area of active research in recent times is to study the sufficient criterion for oscillation of delay differential equations, see [1–28].

In this work, we establish the asymptotic behavior of fourth-order neutral differential equation of the form

$$\left(r\left(t\right)\left(\mathcal{N}\_{x}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)^{\prime} + \int\_{a}^{b} q\left(t,\theta\right) \ge^{6} \left(\delta\left(t,\theta\right)\right)d\theta = 0,\tag{1}$$

where *t* ≥ *t*0 and *Nx* (*t*) := *x* (*t*) + *p* (*t*) *x* (*ϕ* (*t*)). In this paper, we assume that:


**Definition 1.** *The function x* ∈ *<sup>C</sup>*<sup>3</sup>[*ty*, <sup>∞</sup>), *ty* ≥ *t*0, *is called a solution of (1), if r* (*t*) ( *N*--- *x* (*t*))*<sup>α</sup>* ∈ *<sup>C</sup>*<sup>1</sup>[*ty*, <sup>∞</sup>), *and x* (*t*) *satisfies (1) on* [*ty*, <sup>∞</sup>)*.*

**Definition 2.** *A solution of (1) is called oscillatory if it has arbitrarily large zeros on* [*tx*, <sup>∞</sup>), *and otherwise is called to be nonoscillatory.*

**Definition 3.** *The Equation (1) is called oscillatory if every its solutions are oscillatory.*

In the following, we discuss some important papers: Chatzarakisetal.[9]provedthe equation(1)where*α*=*β*,isoscillatory,

$$\int\_{t\_0}^{\infty} \left( \mathcal{O}\left(s\right) - \frac{2^{\alpha}r\left(s\right)}{\mu^{a\_{\mathrm{s}}} \rho^{\alpha}\left(s\right)} \left(\frac{\rho'\left(s\right)}{\alpha+1}\right)^{\alpha+1} \right) \mathrm{d}s = \infty,$$

 if

for some *μ* ∈ (0, 1) and

$$\int\_{t\_0}^{\infty} \left( \vartheta \left( s \right) \left( \int\_t^{\infty} (Q \left( \upsilon \right))^{\frac{1}{\alpha}} r^{\frac{-1}{\alpha}} \left( \upsilon \right) d\upsilon \right) - \frac{\theta\_+^{\prime 2} \left( s \right)}{4 \theta \left( s \right)} \right) \mathrm{d}s = \infty, 1$$

where (*t*) := *kρ* (*t*) *Q* (*t*) (1 − *p* (*δ* (*t*, *a*)))*<sup>α</sup>* (*δ* (*t*, *a*) \*t*)<sup>3</sup>*<sup>α</sup>*and *ρ*, *θ* ∈ *C*<sup>1</sup> ([*<sup>υ</sup>*0, <sup>∞</sup>),(0, <sup>∞</sup>)).

Moaaz et al. in [19] extended the Riccati transformation to obtain new oscillatory criteria for (1) as condition 

$$\int\_{t\_0}^{\infty} \left[ \theta \left( s \right) Q \left( s \right) - \frac{1}{\lambda 4} \left( \frac{\theta' \left( s \right)}{\theta \left( s \right)} \right)^2 \right] \mathrm{d}s = \infty.$$

where *λ* ∈ (0, 1) and a function *θ* ∈ *C*<sup>1</sup> ([*<sup>υ</sup>*0, <sup>∞</sup>),(0, <sup>∞</sup>)).

Authors in [24] studied oscillatory behavior of equation

$$N\_x^{(n)}\left(t\right) + q\left(t\right) \ge \left(\delta\left(t\right)\right) = 0,\tag{2}$$

where *n* is even, they proved it oscillatory by using the Riccati transformation if either

$$\liminf\_{t \to \infty} \int\_{\varrho(t)}^t Q\left(s\right) ds > \frac{(n-1)!}{\mathbf{e}},\tag{3}$$

or

$$\limsup\_{t \to \infty} \int\_{\varrho(t)}^t Q\left(s\right) ds > (n-1)!\_{\prime}$$

where *Q* (*t*) := *ϕ<sup>n</sup>*−<sup>1</sup> (*t*) (1 − *p* (*ϕ* (*t*))) *q* (*t*).

Xing et al. [22] proved that the even-order differential equation

$$\left(r\left(t\right)\left(N\_{\mathbf{x}}^{\left(n-1\right)}\left(t\right)\right)^{\alpha}\right)' + q\left(t\right)\mathbf{x}^{\left\{\boldsymbol{\delta}\right\}}\left(\boldsymbol{\delta}\left(t\right)\right) = 0\,\mathrm{s}$$

is oscillatory, if

$$\left(\delta^{-1}\left(t\right)\right)' \ge \delta\_0 > 0,\ \left.\rho'\left(t\right)\right| \ge \left.\rho\_0 > 0,\ \left.\rho^{-1}\left(\delta\left(t\right)\right) < t$$

and

$$\lim\inf\_{t\to\infty} \int\_{\varphi^{-1}(\delta(t))}^{t} \frac{\widehat{q}\left(s\right)}{r\left(s\right)} \left(s^{n-1}\right)^{a} ds > \frac{\left(\frac{1}{\delta\_{0}} + \frac{p\_{0}^{a}}{\delta\_{0}\rho\_{0}}\right)}{\mathbf{e}\left((n-1)!\right)^{-a}}\tag{4}$$

where *q*%(*t*) := min &*q <sup>δ</sup>*−<sup>1</sup> (*t*) , *q <sup>δ</sup>*−<sup>1</sup> (*ϕ* (*t*))' and *n* is even.

To prove this, we apply the previous results to the equation

$$\left(\mathbf{x}\left(t\right) + p\mathbf{x}\left(qt\right)\right)^{(n)} + b\mathbf{x}\left(\delta t\right) = \mathbf{0}, \ t \ge 1,\tag{5}$$

where *n* = 4, *p* = 7/8, *ϕ* = 1/e, *δ* = 1/e<sup>2</sup> and *b* = *q*0/*υ*4, we find: 1. By applying condition (3) in (5), we find

$$q\_0 > 3561.9.$$

2. By applying condition (4) in (5), we ge<sup>t</sup>

> *q*0 > 3008.5.

Hence, [22] improved the results in [24].

Thus, the motivation in studying this paper is complement results in [9] and improve results [22,24].

By using the Riccati transformations, we establish a new oscillation criterion for a class of fourth-order neutral differential equations (1). An example is provided to illustrate the main results.

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

We shall employ the following lemmas

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

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

**Lemma 2** ([16])**.** *Let the function x satisfies x*(*i*) (*t*) > 0, *i* = 0, 1, ..., *n*, *and x*(*n*+<sup>1</sup>) (*t*) < 0, *then*

$$\frac{\mathbf{x}\left(t\right)}{t^n/n!} \ge \frac{\mathbf{x}'\left(t\right)}{t^{n-1}/\left(n-1\right)!}.$$

**Lemma 3** ([4])**.** *Assume that x*, *v* ≥ 0 *and α* ≥ 1 *is a positive real number. Then*

$$(\mathfrak{x} + v)^{\alpha} \le \mathfrak{z}^{\alpha - 1} \left( \mathfrak{x}^{\alpha} + v^{\alpha} \right)$$

*and*

$$(\mathbf{x} + \mathbf{v})^{\beta} \le \mathbf{x}^{\beta} + \mathbf{v}^{\beta}, \text{ for } \beta \le 1.$$

**Lemma 4** ([9])**.** *Assume that x is an eventually positive solution of (1). Then, there exist two possible cases:*

$$\begin{aligned} \left( \left( \mathbf{S}\_1 \right) \ N\_x^{(\kappa)} \left( t \right) > 0 \text{ for } \kappa = 0, 1, 2, 3; \\ \left( \left( \mathbf{S}\_2 \right) \ N\_x \left( t \right) > 0, \ N\_x^{\prime} \left( t \right) > 0, \ N\_x^{\prime \prime} \left( t \right) < 0 \text{ and } N\_x^{\prime \prime \prime} \left( t \right) > 0, \end{aligned} $$

*for t* ≥ *t*1, *where t*1 ≥ *t*0 *is sufficiently large.* **Notation 1.** *We consider the following notations:*

$$\begin{array}{rcl}p\_{1}\left(t\right)&=&\frac{1}{p\left(q^{-1}\left(t\right)\right)}\left(1-\frac{\left(q^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)^{3}}{\left(q^{-1}\left(t\right)\right)^{3}p\left(q^{-1}\left(q^{-1}\left(t\right)\right)\right)}\right),\\p\_{2}\left(t\right)&=&\frac{1}{p\left(q^{-1}\left(t\right)\right)}\left(1-\frac{\left(q^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}{\left(q^{-1}\left(t\right)\right)p\left(q^{-1}\left(q^{-1}\left(t\right)\right)\right)}\right)\\\Psi\left(t\right)&=&M\_{1}^{\theta-a}\theta\left(t\right)\int\_{a}^{b}q\left(t,\theta\right)p\_{1}^{\theta}\left(\delta\left(t,\theta\right)\right)d\theta\\\mathcal{R}\left(t\right)&=&\int\_{a}^{b}\left(\frac{\mu\left(q^{-1}\left(\eta\left(t,\theta\right)\right)\right)^{3}}{6}\right)^{\theta}q\left(t,\theta\right)p\_{1}^{\theta}\left(\eta\left(t,\theta\right)\right)r^{-\hat{\beta}/a}\left(\boldsymbol{\varrho}^{-1}\left(\eta\left(t,\theta\right)\right)\right)\,\mathrm{d}\theta\\\mathcal{R}\left(t\right)&=&\int\_{t}^{\infty}\left(\frac{1}{r\left(\boldsymbol{\varrho}\right)}\int\_{\boldsymbol{\varrho}}^{\infty}\left(\int\_{a}^{b}q\left(s,\theta\right)\left(\frac{\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}\left(s,\theta\right)\right)}{\boldsymbol{s}}\right)^{\theta}\,\mathrm{d}\theta\right)\,\mathrm{d}\theta\right)^{1/a}\,\mathrm{d}\phi,\end{array}$$

*and*

$$\Phi(t) := p\_2^{\beta/a} \theta\_1\left(t\right) M\_2^{(\beta-a)/a} \int\_t^{\infty} \left(\frac{1}{r\left(\varrho\right)} \int\_{\varrho}^{\infty} \left(\int\_a^b q\left(s, \vartheta\right) \left(\frac{\varrho^{-1}\left(\delta\left(s, \vartheta\right)\right)}{s}\right)^{\beta} \mathrm{d}\vartheta\right) \mathrm{ds}\right)^{1/a} \mathrm{d}\varrho.$$

#### **3. Main Results**

In this part, we will discuss some oscillation criteria for Equation (1).

**Lemma 5.** *Assume that x is an eventually positive solution of (1) and*

$$\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)^{3} < \left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)^{3} p \left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right). \tag{6}$$

.

.

*Then*

$$\text{var}\left(t\right) \ge \frac{1}{p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)} \left(\mathcal{N}\_{\text{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right) - \frac{1}{p\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}\mathcal{N}\_{\text{x}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)\right). \tag{7}$$

**Proof.** Let *x* be an eventually positive solution of (1) on [*<sup>t</sup>*0, <sup>∞</sup>). From the definition of *z* (*t*), we see that

$$p\left(t\right) \ge \left(\boldsymbol{\varrho}\left(\boldsymbol{t}\right)\right) = \boldsymbol{N}\_{\boldsymbol{x}}\left(\boldsymbol{t}\right) - \boldsymbol{x}\left(\boldsymbol{t}\right),$$

and so

$$p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\mathbf{x}\left(t\right) = \mathbf{N}\_{\mathbf{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right) - \mathbf{x}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)$$

Repeating the same process, we obtain

$$\exp\left(t\right) = \frac{1}{p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)} \left(\mathcal{N}\_{\boldsymbol{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right) - \left(\frac{\mathcal{N}\_{\boldsymbol{x}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}{p\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)} - \frac{\mathcal{x}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}{p\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}\right)\right) \cdot \mathcal{N}\_{\boldsymbol{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)$$

which yields

$$\propto(t) \ge \frac{N\_{\mathbf{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)}{p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)} - \frac{1}{p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)} \frac{N\_{\mathbf{x}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}{p\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}$$

Thus, (7) holds. This completes the proof. **Theorem 1.** *Let δ* (*t*) ≤ *ϕ* (*t*) *and (6) holds. If there exist positive functions θ*, *θ*1 ∈ *C*<sup>1</sup> ([*<sup>t</sup>*0, <sup>∞</sup>), R) *such that*

$$\int\_{t\_0}^{\infty} \left( \Psi \left( s \right) - \frac{2^a}{\left( a+1 \right)^{a+1}} \frac{r \left( \varrho^{-1} \left( \delta \left( s, a \right) \right) \right) \left( \theta' \left( s \right) \right)^{a+1}}{\left( \mu\_1 \theta \left( s \right) \left( \varrho^{-1} \left( \delta \left( s, a \right) \right) \right)' \left( \delta \left( s, a \right) \right)' \left( \varrho^{-1} \left( \delta \left( s, a \right) \right) \right)^2 \right)^a} \right) \, ds = \infty \tag{8}$$

*and*

$$\int\_{t\_0}^{\infty} \left( \Phi \left( s \right) - \frac{\left( \theta\_1' \left( s \right) \right)^2}{4 \theta\_1 \left( s \right)} \right) \mathrm{d}s = \infty,\tag{9}$$

*for some μ*1 ∈ (0, 1) *and every M*1, *M*2 > 0*, then (1) is oscillatory.*

**Proof.** Let *x* be a non-oscillatory solution of (1) on [*<sup>t</sup>*0, <sup>∞</sup>). Without loss of generality, we can assume that *x* is eventually positive. It follows from Lemma 4 that there exist two possible cases (**<sup>S</sup>**1) and (**<sup>S</sup>**2).

Let (**<sup>S</sup>**1) holds. From Lemma 2, we obtain *Nx* (*t*) ≥ 13 *tNx* (*t*) and hence the function *<sup>t</sup>*−<sup>3</sup>*Nx* (*t*) is nonincreasing, which with the fact that *ϕ*<sup>−</sup><sup>1</sup> (*t*) ≤ *ϕ*<sup>−</sup><sup>1</sup> *ϕ*<sup>−</sup><sup>1</sup> (*t*) gives

$$\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)^{\mathfrak{J}}\mathcal{N}\_{\mathfrak{x}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right) \leq \left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)^{\mathfrak{J}}\mathcal{N}\_{\mathfrak{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right).\tag{10}$$

From (7) and (10), we ge<sup>t</sup> that

$$\begin{aligned} \text{tr}\begin{pmatrix} t \end{pmatrix} &\geq \quad \frac{N\_{\text{X}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)}{p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)} \left(1 - \frac{\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)^{n-1}}{\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)^{n-1} p\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}\right) \\ &\geq \quad p\_{1}\left(t\right)N\_{\text{X}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right). \end{aligned} \tag{11}$$

From (1) and (11), we obtain

$$\left(r\left(t\right)\left(\mathcal{N}\_{\mathbf{x}}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)^{\prime} + \int\_{a}^{b} q\left(t,\theta\right) p\_{1}^{\theta}\left(\delta\left(t,\theta\right)\right) \mathcal{N}\_{\mathbf{x}}^{\theta}\left(\boldsymbol{\varrho}^{-1}\left(\delta\left(t,\theta\right)\right)\right) d\theta \leq 0. \tag{12}$$

Since *δ* (*t*, *ξ*) is nondecreasing with respect tos, we ge<sup>t</sup> *δ* (*t*, *ϑ*) ≥ *δ* (*t*, *a*) for *ξ* ∈ (*a*, *b*) and so

$$\left(r\left(t\right)\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)^{\prime} + \mathcal{N}\_{\boldsymbol{x}}^{\boldsymbol{\theta}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}\left(t,\boldsymbol{a}\right)\right)\right)\int\_{\boldsymbol{a}}^{b} q\left(t,\boldsymbol{\vartheta}\right) \boldsymbol{p}\_{1}^{\boldsymbol{\theta}}\left(\boldsymbol{\delta}\left(t,\boldsymbol{\vartheta}\right)\right) d\boldsymbol{\vartheta} \leq 0.$$

Next, we define a function *ω* by

$$
\omega\left(t\right) := \theta\left(t\right) \frac{r\left(t\right) \left(N\_x^{\prime\prime\prime}\left(t\right)\right)^{\alpha}}{N\_x^{\alpha}\left(\rho^{-1}\left(\delta\left(t,a\right)\right)\right)} > 0.
$$

Differentiating and using (12), we obtain

$$\begin{split} \omega^{\prime}\left(t\right) &\leq \quad \frac{\theta^{\prime}\left(t\right)}{\theta\left(t\right)}\omega\left(t\right) - \theta\left(t\right)N\_{\text{x}}^{\mathbb{R}-\mathfrak{a}}\left(\mathfrak{g}^{-1}\left(\delta\left(t,a\right)\right)\right)\int\_{a}^{b}q\left(t,\theta\right)p\_{1}^{\mathbb{R}}\left(\delta\left(t,\theta\right)\right)d\theta\\ &\quad -a\theta\left(t\right)\frac{r\left(t\right)\left(N\_{\text{x}}^{\prime\prime\mathfrak{a}}\left(t\right)\right)^{\mathfrak{a}}\left(\mathfrak{g}^{-1}\left(\delta\left(t,a\right)\right)\right)'\left(\delta\left(t,a\right)\right)'N\_{\text{x}}^{\prime\mathfrak{a}}\left(\mathfrak{g}^{-1}\left(\delta\left(t,a\right)\right)\right)}{N\_{\text{x}}^{\mathfrak{a}+1}\left(\mathfrak{g}^{-1}\left(\delta\left(t,a\right)\right)\right)}.\end{split} \tag{13}$$

Recalling that *r* (*t*) (*N*--- *x* (*t*))*<sup>α</sup>* is decreasing, we ge<sup>t</sup>

> *r ϕ*<sup>−</sup><sup>1</sup> (*δ* (*t*, *a*)) *N*--- *x ϕ*<sup>−</sup><sup>1</sup> (*δ* (*t*, *a*))*<sup>α</sup>* ≥ *r* (*t*) *N*--- *x* (*t*)*<sup>α</sup>* .

> This yields

$$\left(N\_x^{\prime\prime\prime}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)\right)^{\boldsymbol{a}} \geq \frac{r\left(t\right)}{r\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)}\left(N\_x^{\prime\prime\prime}\left(t\right)\right)^{\boldsymbol{a}}.\tag{14}$$

It follows from Lemma 1 that

$$N\_x^{\prime} \left( \boldsymbol{\varrho}^{-1} \left( \boldsymbol{\delta} \left( t, \boldsymbol{a} \right) \right) \right) \geq \frac{\mu\_1}{2} \left( \boldsymbol{\varrho}^{-1} \left( \boldsymbol{\delta} \left( t, \boldsymbol{a} \right) \right) \right)^2 N\_x^{\prime \prime \prime} \left( \boldsymbol{\varrho}^{-1} \left( \boldsymbol{\delta} \left( t, \boldsymbol{a} \right) \right) \right), \tag{15}$$

for all *μ*1 ∈ (0, <sup>1</sup>). Thus, by (13)–(15), we ge<sup>t</sup>

$$\begin{split} \omega^{\prime}\left(t\right) &\leq \quad \frac{\theta^{\prime}\left(t\right)}{\theta\left(t\right)}\omega\left(t\right) - \theta\left(t\right)\operatorname{N}\_{x}^{\theta-a}\left(\operatorname{\boldsymbol{\rho}}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)\int\_{a}^{b}\boldsymbol{q}\left(t,\theta\right)\operatorname{p}\_{1}^{\mathsf{f}}\left(\boldsymbol{\delta}\left(t,\theta\right)\right)d\theta\\ &\quad -a\theta\left(t\right)\frac{\mu\_{1}}{2}\left(\frac{r\left(t\right)}{r\left(\boldsymbol{\rho}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)}\right)^{1/a}\frac{r\left(t\right)\left(\operatorname{N}\_{x}^{\mathsf{f}\mathsf{m}}\left(t\right)\right)^{a+1}\left(\operatorname{\boldsymbol{\rho}}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)^{\prime}\left(\operatorname{\boldsymbol{\delta}}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)^{\prime}\left(\operatorname{\boldsymbol{\rho}}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)^{2}}{\mathrm{N}\_{x}^{\mathsf{f}+1}\left(\boldsymbol{\rho}^{-1}\left(\boldsymbol{\delta}\left(t,a\right)\right)\right)} \end{split}$$

Hence,

$$\begin{split} \omega^{\prime}(t) &\leq \quad \frac{\theta^{\prime}(t)}{\theta(t)} \omega^{\prime}(t) - \theta \left(t \right) \operatorname{N}\_{x}^{\theta^{-\mathrm{ad}}} \left(\operatorname{\boldsymbol{\varrho}}^{-1} \left(\boldsymbol{\delta}^{\prime}(t, \boldsymbol{a})\right)\right) \int\_{a}^{b} q \left(t, \theta\right) p\_{1}^{\theta} \left(\boldsymbol{\delta}^{\prime}(t, \boldsymbol{\theta})\right) d\theta \\ &- a \frac{\mu\_{1}}{2} \left(\frac{r\left(t\right)}{r\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}^{\prime}(t, \boldsymbol{a})\right)\right)}\right)^{1/a} \frac{\left(\operatorname{\boldsymbol{\varrho}}^{-1} \left(\boldsymbol{\delta}^{\prime}(t, \boldsymbol{a})\right)\right)^{\prime} \left(\boldsymbol{\delta}^{\prime}(t, \boldsymbol{a})\right)^{\prime} \left(\boldsymbol{\varrho}^{-1} \left(\boldsymbol{\delta}^{\prime}(t, \boldsymbol{a})\right)\right)^{2}}{\left(r\boldsymbol{\theta}\right)^{1/a} \left(t\right)} \omega^{\prime \frac{a+1}{a}}(t) . \end{split}$$

Since *Nx* (*t*) > 0, there exist a *t*2 ≥ *t*1 and a constant *M* > 0 such that

$$N\_{\mathbf{x}}\left(t\right) > M\_{\mathbf{\cdot}}\tag{16}$$

for all *t* ≥ *t*2. Using the inequality

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

with

$$\mathcal{U} = \frac{\theta'(t)}{\theta(t)},\\\ V = a \frac{\mu\_1}{2} \left( \frac{r\left(t\right)}{r\left(\varrho^{-1}\left(\delta\left(t,a\right)\right)\right)} \right)^{1/a} \frac{\left(\varrho^{-1}\left(\delta\left(t,a\right)\right)\right)'\left(\delta\left(t,a\right)\right)'\left(\varrho^{-1}\left(\delta\left(t,a\right)\right)\right)^2}{\left(r\theta\right)^{1/a}\left(t\right)}$$

and *x* = *ω*, we ge<sup>t</sup>

$$
\omega'\left(t\right) \le -\Psi\left(t\right) + \frac{2^a}{\left(a+1\right)^{a+1}} \frac{r\left(\varrho^{-1}\left(\delta\left(t,a\right)\right)\right)\left(\theta'\left(t\right)\right)^{a+1}}{\left(\mu\_1\theta\left(t\right)\left(\varrho^{-1}\left(\delta\left(t,a\right)\right)\right)'\left(\delta\left(t,a\right)\right)'\left(\varrho^{-1}\left(\delta\left(t,a\right)\right)\right)^2\right)^a}.
$$

This implies that

$$\int\_{t\_1}^t \left( \Psi \left( s \right) - \frac{2^a}{\left( a+1 \right)^{a+1}} \frac{r \left( \varrho^{-1} \left( \delta \left( t, a \right) \right) \right) \left( \theta' \left( t \right) \right)^{a+1}}{\left( \mu\_1 \theta \left( t \right) \left( \varrho^{-1} \left( \delta \left( t, a \right) \right) \right)' \left( \delta \left( t, a \right) \right)' \left( \varrho^{-1} \left( \delta \left( t, a \right) \right) \right)^2 \right)^a} \right) \, ds \le \omega \left( t\_1 \right) \, ds$$

which contradicts (8).

> In the case where (**<sup>S</sup>**2) satisfies, by using Lemma 2, we find that

$$N\_{\mathbf{x}}\left(t\right) \ge tN\_{\mathbf{x}}'\left(t\right) \tag{17}$$

and hence *<sup>t</sup>*−<sup>1</sup>*Nx* (*t*)- ≤ 0. Therefore,

$$
\varphi^{-1}\left(t\right)\mathcal{N}\_{\mathbf{x}}\left(\varphi^{-1}\left(\varphi^{-1}\left(t\right)\right)\right) \le \varphi^{-1}\left(\varphi^{-1}\left(t\right)\right)\mathcal{N}\_{\mathbf{x}}\left(\varphi^{-1}\left(t\right)\right).\tag{18}
$$

From (7) and (18), we have

$$\begin{aligned} \text{tr}\begin{pmatrix} t \end{pmatrix} &\geq \quad \frac{1}{p\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)} \left(1 - \frac{\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}{\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)p\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right)\right)}\right)N\_{\boldsymbol{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right) \\ &= \quad p\_{2}\left(t\right)N\_{\boldsymbol{x}}\left(\boldsymbol{\varrho}^{-1}\left(t\right)\right), \end{aligned}$$

which with (1) gives

$$\left(r\left(t\right)\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(t\right)\right)^{\boldsymbol{a}}\right)^{\prime} \leq -\int\_{\boldsymbol{a}}^{\boldsymbol{b}} q\left(t,\theta\right) p\_{2}^{\boldsymbol{\beta}}\left(\boldsymbol{\delta}\left(t,\theta\right)\right) \mathcal{N}\_{\boldsymbol{x}}^{\boldsymbol{\beta}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}\left(t,\theta\right)\right)\right) d\theta.$$

Integrating this inequality from *t* to , we obtain

$$\left(r\left(\boldsymbol{\varrho}\right)\left(\boldsymbol{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(\boldsymbol{\varrho}\right)\right)^{a} - r\left(t\right)\left(\boldsymbol{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(t\right)\right)^{a}\right\\ \leq -\int\_{t}^{\varrho} \left(\int\_{a}^{b} q\left(t,\theta\right) \boldsymbol{p}\_{2}^{\boldsymbol{\theta}}\left(\boldsymbol{\delta}\left(t,\theta\right)\right) \boldsymbol{N}\_{\boldsymbol{x}}^{\boldsymbol{\theta}}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}\left(t,\theta\right)\right)\right) \mathrm{d}\theta\right) \mathrm{d}\mathbf{x}.\tag{19}$$

From (17), we ge<sup>t</sup> that

$$N\_x \left( \varphi^{-1} \left( \delta \left( t, \theta \right) \right) \right) \ge \frac{\varphi^{-1} \left( \delta \left( t, \theta \right) \right)}{t} N\_x \left( t \right) \,. \tag{20}$$

Letting → ∞ in (19) and using (20), we obtain

$$\Pr\left(t\right) \left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(t\right)\right)^{a} \geq p\_{2}^{\mathcal{S}}\left(\boldsymbol{\delta}\left(t,\boldsymbol{a}\right)\right) \mathcal{N}\_{\boldsymbol{x}}^{\mathcal{S}}\left(t\right) \int\_{t}^{\infty} \left(\int\_{a}^{b} q\left(s,\boldsymbol{\vartheta}\right) \left(\frac{\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\delta}\left(s,\boldsymbol{\vartheta}\right)\right)}{s}\right)^{\beta} \mathrm{d}\boldsymbol{\vartheta}\right) \mathrm{d}s.$$

Integrating this inequality again from *t* to <sup>∞</sup>, we ge<sup>t</sup>

$$N\_x^{\prime\prime} \left( t \right) \le -p\_2^{\delta/a} N\_x^{\delta/a} \left( t \right) \int\_t^{\infty} \left( \frac{1}{r \left( \varrho \right)} \int\_{\varrho}^{\infty} \left( \int\_a^b q \left( s, \vartheta \right) \left( \frac{\varrho^{-1} \left( \delta \left( s, \vartheta \right) \right)}{s} \right)^{\beta} d\vartheta \right) ds \right)^{1/a} d\varrho,\tag{21}$$

for all *μ*2 ∈ (0, <sup>1</sup>).

> Now, we define

$$w\left(t\right) = \theta\_1\left(t\right) \frac{N\_{\chi}'\left(t\right)}{N\_{\chi}\left(t\right)}.$$

Then *w* (*t*) > 0 for *t* ≥ *t*1. By differentiating *w* and using (21), we find

$$\begin{split} \boldsymbol{w}^{\prime}(t) &= \quad \frac{\theta\_{1}^{\prime}(t)}{\theta\_{1}(t)} \boldsymbol{w}(t) + \boldsymbol{\theta}\_{1} \left( t \right) \frac{\boldsymbol{N}\_{\boldsymbol{x}}^{\prime\prime}(t)}{\boldsymbol{N}\_{\boldsymbol{x}}\left( t \right)} - \boldsymbol{\theta}\_{1} \left( t \right) \left( \frac{\boldsymbol{N}\_{\boldsymbol{x}}^{\prime}(t)}{\boldsymbol{N}\_{\boldsymbol{x}}\left( t \right)} \right)^{2} \\ &\leq \quad \frac{\theta\_{1}^{\prime}(t)}{\theta\_{1}(t)} \boldsymbol{w}(t) - \frac{1}{\theta\_{1}\left( t \right)} \boldsymbol{w}^{2}(t) \\ &\quad - p\_{2}^{\beta/a} \theta\_{1}\left( t \right) \boldsymbol{N}\_{\boldsymbol{x}}^{\beta/a-1}(t) \int\_{t}^{\infty} \left( \frac{1}{r\left( \boldsymbol{\varrho} \right)} \int\_{\boldsymbol{\vartheta}}^{\infty} \left( \int\_{a}^{b} q\left( s, \boldsymbol{\vartheta} \right) \left( \frac{\boldsymbol{\varrho}^{-1}\left( \boldsymbol{\delta}\left( s, \boldsymbol{\vartheta} \right) \right)}{s} \right)^{\beta} d\boldsymbol{\vartheta} \right) \operatorname{d}\boldsymbol{\vartheta} \right)^{1/a} \operatorname{d}\boldsymbol{\varrho}. \end{split}$$

Thus, we obtain

$$w'(t) \le -\Phi\left(t\right) + \frac{\theta\_1'\left(t\right)}{\theta\_1\left(t\right)}w\left(t\right) - \frac{1}{\theta\_1\left(t\right)}w^2\left(t\right),$$

and so

$$w'(t) \le -\Phi(t) + \frac{\left(\theta\_1'(t)\right)^2}{4\theta\_1(t)}.$$

Then, we ge<sup>t</sup>

$$\int\_{t\_1}^{t} \left(\Phi\left(s\right) - \frac{\left(\theta'\left(t\right)\right)^2}{4\theta\left(t\right)}\right) ds \le w\left(t\_1\right),$$

which contradicts (9). This completes the proof.

**Theorem 2.** *Let*

$$\frac{\left(\varrho^{-1}\left(\varrho^{-1}\left(t\right)\right)\right)^{n-1}}{\left(\varrho^{-1}\left(t\right)\right)^{n-1}\left(\varrho^{-1}\left(\varrho^{-1}\left(t\right)\right)\right)} \le 1. \tag{22}$$

*Suppose that there exist positive functions η*, *σ* ∈ *p*1 ([*<sup>t</sup>*0, <sup>∞</sup>), R) *satisfying*

$$\eta\left(t\right) \le \delta\left(t\right), \; \eta\left(t\right) < \phi\left(t\right), \; \sigma\left(t\right) \le \delta\left(t\right), \; \sigma\left(t\right) < \phi\left(t\right), \; \sigma'\left(t\right) \ge 0 \; and \; \lim\_{t \to \infty} \eta\left(t\right) = \lim\_{t \to \infty} \sigma\left(t\right) = \infty. \tag{23}$$

*If the equations*

$$
\psi'(t) + \mathcal{R}\left(t\right)\psi^{\otimes/a}\left(\varphi^{-1}\left(\eta\left(t,a\right)\right)\right) = 0\tag{24}
$$

*and*

$$\left(\boldsymbol{\phi}'\left(t\right) + p\_2^{\boldsymbol{\beta}/\boldsymbol{a}}\left(\boldsymbol{\varphi}\left(t\right,\boldsymbol{a}\right)\right)\right)^{\boldsymbol{\beta}/\boldsymbol{a}} \boldsymbol{R}\left(t\right) \boldsymbol{\phi}^{\boldsymbol{\beta}/\boldsymbol{a}}\left(\boldsymbol{\varphi}\left(t\right,\boldsymbol{a}\right)\right) = 0\tag{25}$$

*are oscillatory, then (1) is oscillatory.*

**Proof.** Let *x* be a non-oscillatory solution of (1) on [*<sup>t</sup>*0, <sup>∞</sup>). Without loss of generality, we suppose that *x* > 0. From Lemma 4, we find there exist two possible cases (**<sup>S</sup>**1) and (**<sup>S</sup>**2).

Assume that Case (**<sup>S</sup>**1) holds. From Theorem 1, we ge<sup>t</sup> that (12) holds. Since *η* (*t*) ≤ *δ* (*t*) and *z*- (*t*) > 0, we obtain

$$\left(r\left(t\right)\left(\mathcal{N}\_{\mathbf{x}}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)^{\prime} \leq -\int\_{a}^{b} q\left(t,\theta\right) p\_{1}^{\beta}\left(\eta\left(t,\theta\right)\right) \mathcal{N}\_{\mathbf{x}}^{\beta}\left(\boldsymbol{\uprho}^{-1}\left(\eta\left(t,\theta\right)\right)\right) \mathrm{d}\boldsymbol{\uprho}.\tag{26}$$

Now, by using Lemma 1, we have

$$N\_{\mathbf{x}}\left(t\right) \geq \frac{\mu}{6} t^3 N\_{\mathbf{x}}^{\prime\prime\prime}\left(t\right). \tag{27}$$

for some *μ* ∈ (0, <sup>1</sup>). It follows from (26) and (27) that, for all *μ* ∈ (0, <sup>1</sup>),

$$\left(r\left(t\right)\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)^{\prime} + \int\_{\boldsymbol{t}}^{\boldsymbol{b}} \left(\frac{\mu\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\eta}\left(t,\boldsymbol{\vartheta}\right)\right)\right)^{3}}{6}\right)^{\delta} q\left(t,\boldsymbol{\vartheta}\right) p\_{1}^{\delta}\left(\boldsymbol{\eta}\left(t,\boldsymbol{\vartheta}\right)\right) \left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\eta}\left(t,\boldsymbol{\vartheta}\right)\right)\right)\right)^{\delta} \mathrm{d}\boldsymbol{\theta} \leq 0.5\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(\boldsymbol{\varrho}\right)\right)\right)^{\delta} q\left(t,\boldsymbol{\vartheta}\right) \left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime}\right)\right)\right)\right)^{\delta} \mathrm{d}\boldsymbol{\theta} \leq 0.5\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime\prime}\left(\mathcal{N}\_{\boldsymbol{x}}^{\prime}\right)\right)\right)^{\delta}\right)^{\delta} q\left(t,\boldsymbol{\vartheta}\right)$$

Thus, we choose

$$
\psi\left(t\right) = r\left(t\right)\left(N\_x^{\prime\prime\prime}\left(t\right)\right)^{\alpha}.
$$

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

$$
\psi'(t) + \mathcal{R}\left(t\right)\psi^{\otimes/a}\left(\boldsymbol{\varrho}^{-1}\left(\boldsymbol{\eta}\left(t,\boldsymbol{a}\right)\right)\right) \leq 0.
$$

Using (see ([15] Theorem 1)), we see (24) also has a positive solution, a contradiction.

Suppose that Case (**<sup>S</sup>**2) holds. From Theorem 1, we ge<sup>t</sup> that (21) holds. Since *σ* (*t*) ≤ *δ* (*t*) and *Nx* (*t*) > 0, we have that

$$N\_x''(t) \le -p\_2^{\theta/a} N\_x^{\theta/a} \left( \boldsymbol{\varrho}^{-1} \left( \boldsymbol{\sigma} \left( t, \boldsymbol{a} \right) \right) \right) \int\_t^\infty \left( \frac{1}{r \left( \boldsymbol{\varrho} \right)} \int\_{\boldsymbol{\theta}}^\infty \left( \int\_a^b q \left( s, \boldsymbol{\theta} \right) \left( \frac{\boldsymbol{\varrho}^{-1} \left( \boldsymbol{\sigma} \left( s, \boldsymbol{\theta} \right) \right)}{s} \right)^{\boldsymbol{\beta}} d\boldsymbol{\theta} \right) ds \right)^{1/a} d\boldsymbol{\varrho}, \tag{28}$$

Using Lemma 2, we ge<sup>t</sup> that

$$N\_{\mathbf{x}}\left(t\right) \ge tN\_{\mathbf{x}}'\left(t\right). \tag{29}$$

From (18) and (29), we obtain

$$N\_{\boldsymbol{x}}^{\prime\prime}(t) \leq -p\_2^{\boldsymbol{\beta}/a} \left( N\_{\boldsymbol{x}}^{\prime} \left( \boldsymbol{\varrho}^{-1} \left( \boldsymbol{\sigma} \left( t, \boldsymbol{a} \right) \right) \right) \right)^{\boldsymbol{\beta}/a} \left( \boldsymbol{\varrho}^{-1} \left( \boldsymbol{\sigma} \left( t, \boldsymbol{a} \right) \right) \right)^{\boldsymbol{\beta}/a} \boldsymbol{R} \left( t \right) \dots$$

Now, we choose *φ* (*t*) := *Nx* (*t*), thus, we find that *φ* is a positive solution of

$$\left(\boldsymbol{\phi}'\left(t\right) + p\_2^{\otimes/a}\left(\boldsymbol{\varphi}^{-1}\left(\boldsymbol{\sigma}\left(t,\boldsymbol{a}\right)\right)\right)^{\otimes/a}\boldsymbol{R}\left(t\right)\boldsymbol{\phi}^{\otimes/a}\left(\boldsymbol{\varphi}^{-1}\left(\boldsymbol{\sigma}\left(t,\boldsymbol{a}\right)\right)\right) \leq 0. \tag{30}$$

Using (see ([15] Theorem 1)), we see (25) also has a positive solution, a contradiction. The proof is complete.

#### **Example 1.** *Consider the differential equation*

$$
\int \left( \mathbf{x}\left(t\right) + \frac{1}{2}\mathbf{x}\left(\frac{t}{3}\right) \right)^{\prime\prime\prime} \right)^{\prime} + \int\_{0}^{1} \left(\frac{q\_{0}}{t^{4}}\right) \theta \mathbf{x}\left(\frac{t-\frac{\pi}{4}}{2}\right) d\theta = 0,\tag{31}
$$

*where q*0 > 0 *is a constant. Let α* = *β* = 1, *r* (*t*) = 1, *p* (*t*) = 1/2, *ϕ* (*t*) = *t*/3, *ϕ*<sup>−</sup><sup>1</sup> (*t*) = 3*t*, *δ* (*t*, *a*) = *t*/2, *q* (*t*, *ϑ*) = *<sup>q</sup>*0\*t*<sup>4</sup> *ϑ*.

*Thus, by using Theorem 1, then Equation (31) is oscillatory.*

**Remark 1.** *By applying our results in (5), we see that our results improve [22,24].*

**Remark 2.** *One can easily see that the results obtained in [24] cannot be applied to conditions in Theorem 1, so our results are new.*

## **4. Conclusions**

In this work, our method is based on using the Riccati transformations to ge<sup>t</sup> some oscillation criteria of (1). There are numerous results concerning the oscillation criteria of fourth order equations, which include various forms of criteria as Hille/Nehari, Philos, etc. This allows us to obtain also various criteria for the oscillation of (1). Further, we can try to ge<sup>t</sup> some oscillation criteria of (1) if *Nx* (*t*) := *x* (*t*) − *p* (*t*) *x* (*ϕ* (*t*)) in the future work.

**Author Contributions:** The authors claim to have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors received no direct funding for this work.

**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:** There are no competing interests between the authors.
