*Article* **Oscillation Results for Higher Order Differential Equations**

#### **Choonkil Park 1,***∗***,†, Osama Moaaz 2,† and Omar Bazighifan 2,†**


Received: 22 December 2019; Accepted: 25 January 2020; Published: 3 February 2020

**Abstract:** The objective of our research was to study asymptotic properties of the class of higher order differential equations with a *p*-Laplacian-like operator. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided.

**Keywords:** ocillation; higher-order; differential equations; *p*-Laplacian equations

#### **1. Introduction**

In this work, we are concerned with oscillations of higher-order differential equations with a *p*-Laplacian-like operator of the form

$$\left(r\left(t\right)\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|^{p-2}y^{\left(n-1\right)}\left(t\right)\right)' + q\left(t\right)\left|y\left(\tau\left(t\right)\right)\right|^{p-2}y\left(\tau\left(t\right)\right) = 0.\tag{1}$$

We assume that *<sup>p</sup>* <sup>&</sup>gt; 1 is a constant, *<sup>r</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> ([*t*0, <sup>∞</sup>), <sup>R</sup>), *<sup>r</sup>* (*t*) <sup>&</sup>gt; 0, *<sup>q</sup>*, *<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>* ([*t*0, <sup>∞</sup>), <sup>R</sup>), *<sup>q</sup>* <sup>&</sup>gt; 0, *τ* (*t*) ≤ *t*, lim*t*→<sup>∞</sup> *τ* (*t*) = ∞ and the condition

$$
\eta\left(t\_0\right) = \infty,\tag{2}
$$

where

$$\eta\left(t\right) := \int\_{t}^{\infty} \frac{ds}{r^{1/\left(p-1\right)}\left(s\right)}.$$

By a solution of (1) we mean a function *<sup>y</sup>* <sup>∈</sup> *<sup>C</sup>n*−1[*Ty*, <sup>∞</sup>), *Ty* <sup>≥</sup> *<sup>t</sup>*0, which has the property *r* (*t*) *y*(*n*−1) (*t*) *p*−2 *<sup>y</sup>*(*n*−1) (*t*) <sup>∈</sup> *<sup>C</sup>*1[*Ty*, <sup>∞</sup>), and satisfies (1) on [*Ty*, <sup>∞</sup>). We consider only those solutions *y* of (1) which satisfy sup{|*y* (*t*)| : *t* ≥ *T*} > 0, for all *T* > *Ty*. A solution of (1) is called oscillatory if it has arbitrarily large number of zeros on [*Ty*, ∞), and otherwise it is called to be nonoscillatory; (1) is said to be oscillatory if all its solutions are oscillatory.

In recent decades, there has been a lot of research concerning the oscillation of solutions of various classes of differential equations; see [1–24].

It is interesting to study Equation (1) since the *p*-Laplace differential equations have applications in continuum mechanics [14,25]. In the following, we briefly review some important oscillation criteria obtained for higher-order equations, which can be seen as a motivation for this paper.

Elabbasy et al. [26] proved that the equation

$$\left| \left( r\left( t \right) \left| \left( y^{\left(n-1\right)}\left( t\right) \right) \right|^{p-2} y^{\left(n-1\right)}\left( t\right) \right)' + q\left( t\right) f\left( y\left( \tau\left( t\right) \right) \right) = 0, \quad \forall t$$

is oscillatory, under the conditions

$$\int\_{t\_0}^{\infty} \frac{1}{r^{r^{-1}}(t)} dt = \infty;$$

additionally,

$$\int\_{\ell\_0}^{\infty} \left( \psi \left( s \right) - \frac{1}{p^p} \phi^p \left( s \right) \frac{\left( (n-1)! \right)^{p-1} \rho \left( s \right) a \left( s \right)}{\left( (p-1) \left\mu s^{n-1} \right)^{p-1}} - \frac{(p-1) \rho \left( s \right)}{a^{1/\left(p-1\right)} \left( s \right) \eta^p \left( s \right)} \right) ds = +\infty, \quad \mu \in \mathbb{R}$$

for some constant *μ* ∈ (0, 1) and

$$\int\_{\ell\_0}^{\infty} kq \left( \mathbf{s} \right) \frac{\pi \left( \mathbf{s} \right)^{p-1}}{\mathbf{s}^{p-1}} d\mathbf{s} = \infty.$$

Agarwal et al. [2] studied the oscillation of the higher-order nonlinear delay differential equation

$$\left[ \left| y^{(n-1)} \left( t \right) \right|^{a-1} y^{(n-1)} \left( t \right) \right]' + q \left( t \right) \left| y \left( \tau \left( t \right) \right) \right|^{a-1} y \left( \tau \left( t \right) \right) = 0.$$

where *α* is a positive real number. In [27], Zhang et al. studied the asymptotic properties of the solutions of equation

$$\left[r\left(t\right)\left(y^{\left(n-1\right)}\left(t\right)\right)^{a}\right]' + q\left(t\right)y^{\beta}\left(\tau\left(t\right)\right) = 0, \quad t \ge t\_{0}.$$

where *α* and *β* are ratios of odd positive integers, *β* ≤ *α* and

$$\int\_{t\_0}^{\infty} r^{-1/a} \left( s \right) ds < \infty. \tag{3}$$

In this work, by using the Riccati transformations, the integral averaging technique and comparison principles, we establish a new oscillation criterion for a class of higher-order neutral delay differential Equations (1). This theorem complements and improves results reported in [26]. An illustrative example is provided.

In the sequel, all occurring functional inequalities are assumed to hold eventually; that is, they are satisfied for all *t* large enough.

#### **2. Main Results**

In this section, we establish some oscillation criteria for Equation (1). For convenience, we denote that *F*<sup>+</sup> (*t*) := max {0, *F* (*t*)} ,

$$B\left(t\right) := \frac{1}{(n-4)!} \int\_{t}^{\infty} (\theta - t)^{n-4} \left(\frac{\int\_{\theta}^{\infty} q\left(s\right) \left(\frac{\tau(s)}{s}\right)^{p-1} ds}{r\left(\theta\right)}\right)^{1/(p-1)} d\theta$$

and

$$D\left(s\right) := \frac{r\left(s\right)\delta\left(s\right)\left|h\left(t,s\right)\right|^p}{p^p \left[H\left(t,s\right)A\left(s\right)\mu \frac{s^{n-2}}{\left(n-2\right)!}\right]^{p-1}}.$$

We begin with the following lemmas.

**Lemma 1** (Agarwal **[1]**)**.** *Let <sup>y</sup>*(*t*) <sup>∈</sup> *<sup>C</sup><sup>m</sup>* [*t*0, <sup>∞</sup>) *be of constant sign and <sup>y</sup>*(*m*) (*t*) <sup>=</sup> <sup>0</sup> *on* [*t*0, <sup>∞</sup>) *which satisfies y* (*t*) *<sup>y</sup>*(*m*) (*t*) <sup>≤</sup> 0. *Then,*

(**I**) *There exists a t*<sup>1</sup> <sup>≥</sup> *<sup>t</sup>*<sup>0</sup> *such that the functions y*(*i*) (*t*), *<sup>i</sup>* <sup>=</sup> 1, 2, ..., *<sup>m</sup>* <sup>−</sup> <sup>1</sup> *are of constant sign on* [*t*0, <sup>∞</sup>);

(**II**) *There exists a number k* ∈ {1, 3, 5, ..., *m* − 1} *when m is even, k* ∈ {0, 2, 4, ..., *m* − 1} *when m is odd, such that, for t* ≥ *t*1*,*

$$y\left(t\right)y^{\left(i\right)}\left(t\right) > 0\_{\prime\prime}$$

*for all i* = 0, 1, ..., *k and*

$$(-1)^{m+i+1}y\left(t\right)y^{(i)}\left(t\right) > 0,$$

*for all i* = *k* + 1, ..., *m*.

**Lemma 2** (Kiguradze **[15]**)**.** *If the function y satisfies y*(*j*) > 0 *for all j* = 0, 1, ..., *m*, *and y*(*m*+1) < 0, *then*

$$\frac{m!}{t^m}y\left(t\right) - \frac{(m-1)!}{t^{m-1}}y'\left(t\right) \ge 0.$$

**Lemma 3** (Bazighifan **[7]**)**.** *Let <sup>h</sup>* <sup>∈</sup> *<sup>C</sup><sup>m</sup>* ([*t*0, <sup>∞</sup>),(0, <sup>∞</sup>)). *Suppose that <sup>h</sup>*(*m*) (*t*) *is of a fixed sign, on* [*t*0, <sup>∞</sup>)*, <sup>h</sup>*(*m*) (*t*) *not identically zero, and that there exists a t*<sup>1</sup> <sup>≥</sup> *<sup>t</sup>*<sup>0</sup> *such that, for all t* <sup>≥</sup> *<sup>t</sup>*1,

$$h^{(m-1)}\left(t\right)h^{(m)}\left(t\right) \le 0.$$

*If we have* lim *<sup>t</sup>*→∞*<sup>h</sup>* (*t*) <sup>=</sup> 0, *then there exists t<sup>λ</sup>* <sup>≥</sup> *<sup>t</sup>*<sup>0</sup> *such that*

$$h\left(t\right) \ge \frac{\lambda}{(m-1)!} t^{m-1} \left| h^{\left(m-1\right)}\left(t\right) \right| \cdot \lambda$$

*for every λ* ∈ (0, 1) *and t* ≥ *tλ.*

**Lemma 4.** *Let n* ≥ 4 *be even, and assume that y is an eventually positive solution of Equation (1). If (2) holds, then there exists two possible cases for t* ≥ *t*1, *where t*<sup>1</sup> ≥ *t*<sup>0</sup> *is sufficiently large:*

> (*C*1) *y* (*t*) > 0, *y* (*t*) > 0, *y*(*n*−1) (*t*) > 0, *y*(*n*) (*t*) < 0, (*C*2) *y*(*j*)(*t*) > 0, *y*(*j*+1)(*t*) < 0 *for all odd integer <sup>j</sup>* ∈ {1, 2, ..., *<sup>n</sup>* <sup>−</sup> <sup>3</sup>}, *<sup>y</sup>*(*n*−1) (*t*) <sup>&</sup>gt; 0, *<sup>y</sup>*(*n*) (*t*) <sup>&</sup>lt; 0.

**Proof.** Let *y* be an eventually positive solution of Equation (1). By virtue of (1), we get

$$\left(r\left(t\right)\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|^{p-2}y^{\left(n-1\right)}\left(t\right)\right)'<0.\tag{4}$$

From ([11] Lemma 4), we have that *y*(*n*−1) (*t*) > 0 eventually. Then, we can write (4) in the from

$$\left(r\left(t\right)\left(y^{\left(n-1\right)}\left(t\right)\right)^{p-1}\right)'<0,$$

which gives

$$r'\left(t\right)\left(y^{\left(n-1\right)}\left(t\right)\right)^{p-1} + r\left(t\right)\left(p-1\right)\left(y^{\left(n-1\right)}\left(t\right)\right)^{p-2}y^{\left(n\right)}\left(t\right) < 0.$$

Thus, *y*(*n*) (*t*) < 0 eventually. Thus, by Lemma 1, we have two possible cases (*C*1) and (*C*2). This completes the proof.

**Lemma 5.** *Let y be an eventually positive solution of Equation (1) and assume that Case* (*C*1) *holds. If*

$$\omega\left(t\right) := \delta\left(t\right) \left(\frac{r\left(t\right) \left|\left(y^{(n-1)}\left(t\right)\right)\right|^{p-1}}{y^{p-1}\left(t\right)}\right),\tag{5}$$

*where <sup>δ</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> ([*t*0, <sup>∞</sup>),(0, <sup>∞</sup>)), *then*

$$
\omega'(t) \le \frac{\delta\_+'(t)}{\delta(t)} \omega\left(t\right) - \delta\left(t\right) q\left(t\right) \left(\frac{\tau^{n-1}\left(t\right)}{t^{n-1}}\right)^{p-1} - \frac{(p-1)\,\mu t^{n-2}}{(n-2)!\left(\delta\left(t\right)\,r\left(t\right)\right)^{1/(p-1)}}\omega^{p/(p-1)}\left(t\right)\,. \tag{6}
$$

**Proof.** Let *y* be an eventually positive solution of Equation (1) and assume that Case (*C*1) holds. From the definition of *ω*, we see that *ω* (*t*) > 0 for *t* ≥ *t*1, and

$$\begin{aligned} \omega'\left(t\right) &\leq \left.\delta'\left(t\right) \frac{r\left(t\right) \left|\left(y^{(n-1)}\left(t\right)\right)\right|^{p-1}}{y^{p-1}\left(t\right)} + \delta\left(t\right) \frac{\left(r\left(t\right) \left|\left(y^{(n-1)}\left(t\right)\right)\right|^{p-1}\right)'}{y^{p-1}\left(t\right)}\right| \\ &\quad - \delta\left(t\right) \frac{\left(p-1\right)y'\left(t\right)r\left(t\right) \left|\left(y^{(n-1)}\left(t\right)\right)\right|^{p-1}}{y^p\left(t\right)} .\end{aligned}$$

Using Lemma 3 with *m* = *n* − 1, *h* (*t*) = *y* (*t*), we get

$$y'(t) \ge \frac{\mu}{(n-2)!} t^{n-2} y^{(n-1)}\left(t\right),\tag{7}$$

for every constant *μ* ∈ (0, 1). From (5) and (7), we obtain

$$\begin{array}{rcl} \omega^{\prime}\begin{pmatrix} t \end{pmatrix} & \leq & \delta^{\prime}\begin{pmatrix} t \end{pmatrix} \frac{r(t)\big|\left(y^{(n-1)}(t)\right)\big|^{p-1}}{y^{p-1}(t)} + \delta^{\prime}\begin{pmatrix} t \end{pmatrix} \frac{\left(r(t)\big|\left(y^{(n-1)}(t)\right)\big|^{p-1}\right)^{\prime}}{y^{p-1}(t)}\\ & - \delta^{\prime}\begin{pmatrix} t \end{pmatrix} \frac{(p-1)\mu^{n-2}}{(n-2)!} \frac{r(t)\big|\left(y^{(n-1)}(t)\right)\big|^{p}}{y^{p}(t)}. \end{array} \tag{8}$$

.

By Lemma 2, we have

$$\frac{y\left(t\right)}{y'\left(t\right)} \ge \frac{t}{n-1}$$

Integrating this inequality from *τ* (*t*) to *t*, we obtain

$$\frac{y\left(\tau\left(t\right)\right)}{y\left(t\right)} \ge \frac{\tau^{n-1}\left(t\right)}{t^{n-1}}.\tag{9}$$

Combining (1) and (8), we get

$$\begin{array}{ll} \omega' \begin{pmatrix} t \end{pmatrix} & \leq & \delta' \begin{pmatrix} t \end{pmatrix} \frac{r(t) \big| \left( y^{(n-1)}(t) \right) \big|^{p-1}}{y^{p-1}(t)} - \delta \begin{pmatrix} t \end{pmatrix} \frac{q(t) \big| \left( y^{(p-1)}(\tau(t)) \right)}{y^{p-1}(t)} \\ & - \delta \begin{pmatrix} t \end{pmatrix} \frac{(p-1)\mu t^{n-2}}{\left( n-2 \right)!} \frac{r(t) \big| \left( y^{(n-1)}(t) \right) \big|^{p}}{y^{p}(t)} . \end{array} \tag{10}$$

From (9) and (10), we obtain

$$
\omega'(t) \le \frac{\delta'\_+(t)}{\delta(t)} \omega'(t) - \delta\left(t\right) q\left(t\right) \left(\frac{\tau^{n-1}\left(t\right)}{t^{n-1}}\right)^{p-1} - \frac{(p-1)\,\mu t^{n-2}}{(n-2)!\left(\delta\left(t\right)\,r\left(t\right)\right)^{1/\left(p-1\right)}} \omega^{p/\left(p-1\right)}\left(t\right). \tag{11}
$$

It follows from (11) that

$$\left(\delta\left(t\right)q\left(t\right)\left(\frac{\tau^{n-1}\left(t\right)}{t^{n-1}}\right)^{p-1} \leq \frac{\delta'\_+\left(t\right)}{\delta\left(t\right)}\omega\left(t\right) - \omega'\left(t\right) - \frac{\left(p-1\right)\mu t^{n-2}}{\left(n-2\right)!\left(\delta\left(t\right)r\left(t\right)\right)^{1/\left(p-1\right)}}\omega^{p/\left(p-1\right)}\left(t\right)\dots$$

This completes the proof.

**Lemma 6.** *Let y be an eventually positive solution of Equation (1) and assume that Case* (*C*2) *holds. If*

$$
\psi\left(t\right) := \sigma\left(t\right) \frac{y'\left(t\right)}{y\left(t\right)},\tag{12}
$$

*where <sup>σ</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> ([*t*0, <sup>∞</sup>),(0, <sup>∞</sup>)), *then*

$$
\sigma\left(t\right)\mathcal{B}\left(t\right) \le -\psi'\left(t\right) + \frac{\sigma'\left(t\right)}{\sigma\left(t\right)}\psi\left(t\right) - \frac{1}{\sigma\left(t\right)}\psi^2\left(t\right). \tag{13}
$$

**Proof.** Let *y* be an eventually positive solution of Equation (1) and assume that Case (*C*2) holds. Using Lemma 2, we obtain

$$y\left(t\right) \ge t y'\left(t\right).$$

Thus we find that *y*/*t* is nonincreasing, and hence

$$y\left(\tau\left(t\right)\right) \ge y\left(t\right) \frac{\tau\left(t\right)}{t}.\tag{14}$$

Since *y* > 0, (1) becomes

$$\left(r\left(t\right)\left(y^{\left(n-1\right)}\left(t\right)\right)^{p-1}\right)' + q\left(t\right)y^{p-1}\left(\tau\left(t\right)\right) = 0.$$

Integrating that equation from *t* to ∞, we see that

$$\lim\_{t \to \infty} \left( r\left( t \right) \left( y^{(n-1)} \left( t \right) \right)^{p-1} \right) - r\left( t \right) \left( y^{(n-1)} \left( t \right) \right)^{p-1} + \int\_{t}^{\infty} q\left( s \right) y^{p-2} \left( \tau\left( s \right) \right) = 0. \tag{15}$$

Since the function *r y*(*n*−1) *<sup>p</sup>*−<sup>1</sup> is positive *r* > 0 and *y*(*n*−1) > 0 . and nonincreasing *r y*(*n*−1) *<sup>p</sup>*−<sup>1</sup> < 0 , there exists a *t*<sup>2</sup> ≥ *t*<sup>0</sup> such that *r y*(*n*−1) *<sup>p</sup>*−<sup>1</sup> is bounded above for all *t* ≥ *t*2, and so lim*<sup>t</sup>*→<sup>∞</sup> *r* (*t*) *y*(*n*−1) (*t*) *<sup>p</sup>*−<sup>1</sup> = *c* ≥ 0. Then, from (15), we obtain

$$-r\left(t\right)\left(y^{\left(n-1\right)}\left(t\right)\right)^{p-1} + \int\_{t}^{\infty} q\left(s\right)y^{p-2}\left(\tau\left(s\right)\right) \leq -c \leq 0.$$

From (14), we obtain

$$-r\left(t\right)\left(\mathcal{Y}^{(n-1)}\left(t\right)\right)^{p-1} + \int\_{t}^{\infty} q\left(s\right)\left(s\right)^{p-1} \frac{\tau\left(s\right)^{p-1}}{s^{p-1}}ds \le 0.1$$

It follows from *y* (*t*) > 0 that

$$-y^{(n-1)}\left(t\right) + \frac{y\left(t\right)}{r^{1/\left(p-1\right)}\left(t\right)} \left(\int\_t^{\infty} q\left(s\right) \left(\frac{\tau\left(s\right)}{s}\right)^{p-1} ds\right)^{1/\left(p-1\right)} \le 0.$$

Integrating the above inequality from *t* to ∞ for a total of (*n* − 3) times, we get

$$y^{\prime\prime}\_{\ }(t) + \frac{\int\_{t}^{\infty} \left(\theta - t\right)^{n-4} \left(\frac{\int\_{\theta}^{\infty} q(s) \left(\frac{r(s)}{s}\right)^{p-1} ds}{r(\theta)}\right)^{1/(p-1)} d\theta}{(n-4)!} y^{\prime}(t) \le 0. \tag{16}$$

From the definition of *ψ* (*t*), we see that *ψ* (*t*) > 0 for *t* ≥ *t*1, and

$$
\psi'(t) = \sigma'(t) \frac{y'(t)}{y'(t)} + \sigma'(t) \frac{y''(t) \, y'(t) - (y'(t))^2}{y^2(t)}.\tag{17}
$$

It follows from (16) and (17) that

$$
\sigma'(t) \, ^\flat \boldsymbol{\beta} \, (t) \le -\psi'(t) + \frac{\sigma'(t)}{\sigma'(t)} \boldsymbol{\psi} \, (t) - \frac{1}{\sigma'(t)} \boldsymbol{\psi}^2 \, (t) \, .
$$

This completes the proof.

#### **Definition 1.** *Let*

$$D = \{(t, s) \in \mathbb{R}^2 : t \ge s \ge t\_0\} \text{ and } D\_0 = \{(t, s) \in \mathbb{R}^2 : t > s \ge t\_0\}.$$

*We say that a function H* ∈ *C* (*D*, R) *belongs to the class if*

(**i**1) *H* (*t*, *t*) = 0 *for t* ≥ *t*0, *H* (*t*,*s*) > 0, (*t*,*s*) ∈ *D*0.

(**i**2) *H has a nonpositive continuous partial derivative ∂H*/*∂s on D*<sup>0</sup> *with respect to the second variable.*

**Theorem 1.** *Let n* ≥ 4 *be even. Assume that there exist functions H*, *H*<sup>∗</sup> ∈ , *δ*, *A*, *σ*, *A*<sup>∗</sup> ∈ *<sup>C</sup>*<sup>1</sup> ([*t*0, <sup>∞</sup>),(0, <sup>∞</sup>)) *and h*, *<sup>h</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>C</sup>* (*D*0, <sup>R</sup>) *such that*

$$-\frac{\partial}{\partial s}\left(H\left(t,s\right)A\left(s\right)\right) = H\left(t,s\right)A\left(s\right)\frac{\delta'\left(t\right)}{\delta\left(t\right)} + h\left(t,s\right). \tag{18}$$

*and*

$$-\frac{\partial}{\partial s} \left( H\_\*\left(t, \mathbf{s}\right) A\_\*\left(\mathbf{s}\right) \right) = H\_\*\left(t, \mathbf{s}\right) A\_\*\left(\mathbf{s}\right) \frac{\sigma'\left(t\right)}{\sigma\left(t\right)} + h\_\*\left(t, \mathbf{s}\right) \,. \tag{19}$$

*If*

$$\limsup\_{t \to \infty} \frac{1}{H\left(t, t\_0\right)} \int\_{t\_0}^{t} \left[ H\left(t, s\right) A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\tau^{n-1}\left(s\right)}{s^{n-1}}\right)^{p-1} - D\left(s\right) \right] ds = \infty,\tag{20}$$

*for some constant μ* ∈ (0, 1) *and*

$$\limsup\_{t \to \infty} \frac{1}{H\_\*\left(t, t\_0\right)} \int\_{t\_0}^t \left( H\_\*\left(t, s\right) A\_\*\left(s\right) \sigma\left(s\right) B\left(s\right) - \frac{\sigma\left(s\right) \left|h\_\*\left(t, s\right)\right|^2}{4H\_\*\left(t, s\right) A\_\*\left(s\right)}\right) ds = \infty,\tag{21}$$

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

**Proof.** Let *y* be a nonoscillatory solution of Equation (1) on the interval [*t*0, ∞). Without loss of generality, we can assume that *y* is an eventually positive. By Lemma 4, there exist two possible cases for *t* ≥ *t*1, where *t*<sup>1</sup> ≥ *t*<sup>0</sup> is sufficiently large.

Assume that (*C*1) holds. From Lemma 5, we get that (6) holds. Multiplying (6) by *H* (*t*,*s*) *A* (*s*) and integrating the resulting inequality from *t*<sup>1</sup> to *t*, we have

$$\begin{aligned} \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\tau^{n-1}\left(s\right)}{s^{n-1}}\right)^{p-1} ds \\ \leq & - \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \omega'\left(s\right) ds + \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \frac{\delta'\left(s\right)}{\delta\left(s\right)} \omega\left(s\right) ds \\ = & - \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \frac{(p-1)\,\mu s^{n-2}}{(n-2)! \left(\delta\left(s\right) \operatorname{r}\left(s\right)\right)^{1/\left(p-1\right)}} \omega^{p/\left(p-1\right)}\left(s\right) ds \end{aligned}$$

Thus

$$\begin{aligned} &\int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\mathfrak{r}^{n-1}\left(s\right)}{\mathfrak{s}^{n-1}}\right)^{p-1} ds \\ &\leq H\left(t,t\_1\right) A\left(t\_1\right) \omega\left(t\_1\right) - \int\_{t\_1}^t \left(-\frac{\partial}{\partial s}\left(H\left(t,s\right)A\left(s\right)\right) - H\left(t,s\right)A\left(s\right)\frac{\delta'\left(t\right)}{\delta\left(t\right)}\right) \omega\left(s\right) ds \\ &\quad - \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \frac{\left(p-1\right)\mu s^{n-2}}{\left(n-2\right)! \left(\delta\left(s\right)r\left(s\right)\right)^{1/\left(p-1\right)}} \omega^{p/\left(p-1\right)}\left(s\right) ds \end{aligned}$$

This implies

$$\begin{split} \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\tau^{n-1}\left(s\right)}{s^{n-1}}\right)^{p-1} ds \\ \leq & H\left(t,t\_1\right) A\left(t\_1\right) \omega\left(t\_1\right) + \int\_{t\_1}^t \left| h\left(t,s\right) \right| \omega\left(s\right) d\left(s\right) \\ \quad - \int\_{t\_1}^t H\left(t,s\right) A\left(s\right) \frac{\left(p-1\right)\mu s^{n-2}}{\left(n-2\right)! \left(\delta\left(s\right)r\left(s\right)\right)^{1/\left(p-1\right)}} \omega^{p/\left(p-1\right)}\left(s\right) ds. \end{split} \tag{22}$$

Using the inequality

$$
\beta l l V^{\beta -1} - l l^{\beta} \le (\beta - 1) \, V^{\beta}, \quad \beta > 1, \,\, lI \ge 0 \text{ and } \, V \ge 0,\tag{23}
$$

,

with *β* = *p*/ (*p* − 1),

$$M = \left( \left( p - 1 \right) H \left( t, s \right) A \left( s \right) \frac{\mu s^{n-2}}{\left( n - 2 \right)!} \right)^{(p-1)/p} \frac{\omega \left( s \right)}{\left( \delta \left( s \right) r \left( s \right) \right)^{1/p}}$$

and

$$V = \left(\frac{p-1}{p}\right)^{p-1} |h\left(t,s\right)|^{p-1} \left(\frac{\delta\left(s\right)r\left(s\right)}{\left(\left(p-1\right)H\left(t,s\right)A\left(s\right)\frac{\mu s^{n-2}}{\left(n-2\right)!}\right)^{p-1}}\right)^{(p-1)/p}$$

we get

$$\begin{aligned} \left| h\left(t,s\right) \right| \left\| \left(s\right) - H\left(t,s\right)A\left(s\right) \frac{\left(p-1\right) \mu s^{n-2}}{\left(n-2\right)! \left(\delta\left(s\right)r\left(s\right)\right)^{1/\left(p-1\right)}} \omega^{p^{f}\left(p-1\right)} \right. \\ \left. \left. \begin{aligned} \left(s\right)r\left(s\right) \left(s\right) \\ \left(H\left(t,s\right)A\left(s\right) \frac{\mu s^{n-2}}{\left(n-2\right)!} \right)^{p-1} \end{aligned} \right| \frac{\left| h\left(t,s\right) \right|}{p} \end{aligned}$$

which with (23) gives

$$\begin{aligned} \int\_{t\_1}^{t} \left( H\left(t,s\right) A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\mathfrak{r}^{n-1}\left(s\right)}{s^{n-1}}\right)^{p-1} - D\left(s\right) \right) ds & \leq & H\left(t, t\_1\right) A\left(t\_1\right) \omega\left(t\_1\right) \\ & \leq & H\left(t, t\_0\right) A\left(t\_1\right) \omega\left(t\_1\right) \dots \omega\left(t\_{n-1}\right) \end{aligned}$$

Then

$$\begin{aligned} \left\| \frac{1}{H\left(t, t\_0\right)} \int\_{t\_0}^t \left( H\left(t, s\right) A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\mathsf{T}^{n-1}\left(s\right)}{\mathsf{s}^{n-1}}\right)^{p-1} - D\left(s\right) \right) ds \right\| \\ \leq & A\left(t\_1\right) \omega\left(t\_1\right) + \int\_{t\_0}^{t\_1} A\left(s\right) \delta\left(s\right) q\left(s\right) \left(\frac{\mathsf{T}^{n-1}\left(s\right)}{\mathsf{s}^{n-1}}\right)^{p-1} ds \\ < & \infty, \end{aligned}$$

for some *μ* ∈ (0, 1), which contradicts (20).

Assume that Case (*C*2) holds. From Lemma 6, we get that (13) holds. Multiplying (13) by *H*<sup>∗</sup> (*t*,*s*) *A*<sup>∗</sup> (*s*), and integrating the resulting inequality from *t*<sup>1</sup> to *t*, we have

$$\begin{split} \int\_{t\_1}^{t} H\_\*\left(t,s\right) A\_\*\left(s\right) \sigma\left(s\right) B\left(s\right) ds &\leq \ -\int\_{t\_1}^{t} H\_\*\left(t,s\right) A\_\*\left(s\right) \mathfrak{p}'\left(s\right) ds + \int\_{t\_1}^{t} H\_\*\left(t,s\right) A\_\*\left(s\right) \frac{\sigma'\left(s\right)}{\sigma\left(s\right)} \mathfrak{p}\left(s\right) ds \\ &\quad -\int\_{t\_1}^{t} \frac{H\_\*\left(t,s\right) A\_\*\left(s\right)}{\sigma\left(s\right)} \mathfrak{p}^2\left(s\right) ds \\ &= \ \ \left. H\_\*\left(t,t\_1\right) A\_\*\left(t\_1\right) \mathfrak{p}\left(t\_1\right) - \int\_{t\_1}^{t} \frac{H\_\*\left(t,s\right) A\_\*\left(s\right)}{\sigma\left(s\right)} \mathfrak{p}^2\left(s\right) ds \\ &\quad -\int\_{t\_1}^{t} \left( -\frac{\partial}{\partial s} \left( H\_\*\left(t,s\right) A\_\*\left(s\right)\right) - H\_\*\left(t,s\right) A\_\*\left(s\right) \frac{\sigma'\left(t\right)}{\sigma\left(t\right)} \right) \Psi\left(s\right) ds. \end{split}$$

Then

$$\begin{aligned} \left| \int\_{t\_1}^{t} H\_\*\left(t, s\right) A\_\*\left(s\right) \sigma\left(s\right) B\left(s\right) ds \right| & \leq \left. H\_\*\left(t, t\_1\right) A\_\*\left(t\_1\right) \psi\left(t\_1\right) + \int\_{t\_1}^{t} \left| h\_\*\left(t, s\right) \right| \psi\left(s\right) d\left(s\right) \right| \\ & \quad - \int\_{t\_1}^{t} \frac{H\_\*\left(t, s\right) A\_\*\left(s\right)}{\sigma\left(s\right)} \psi^2\left(s\right) ds. \end{aligned}$$

Hence we have

$$\begin{aligned} \int\_{t\_1}^{t} \left( H\_\*\left(t,s\right) A\_\*\left(s\right) \sigma\left(s\right) B\left(s\right) - \frac{\sigma\left(s\right) \left|h\_\*\left(t,s\right)\right|^2}{4H\_\*\left(t,s\right)A\_\*}\right) ds & \leq & H\_\*\left(t,t\_1\right) A\_\*\left(t\_1\right) \Psi\left(t\_1\right) \\ & \leq & H\_\*\left(t,t\_0\right) A\_\*\left(t\_1\right) \Psi\left(t\_1\right) \dots \end{aligned}$$

This implies

$$\begin{aligned} \frac{1}{H\_\*\left(t, t\_0\right)} \int\_{t\_0}^t \left( H\_\*\left(t, s\right) A\_\*\left(s\right) \sigma\left(s\right) B\left(s\right) - \frac{\sigma\left(s\right) \left|h\_\*\left(t, s\right)\right|^2}{4H\_\*\left(t, s\right) A\_\*}\right) ds \\ \leq A\_\*\left(t\_1\right) \Psi\left(t\_1\right) + \int\_{t\_0}^t A\_\*\left(s\right) \sigma\left(s\right) B\left(s\right) ds < \infty \end{aligned}$$

which contradicts (21). Therefore, every solution of (1) is oscillatory.

In the next theorem, we establish new oscillation results for Equation (1) by using the comparison technique with the first-order differential inequality:

**Theorem 2.** *Let n* ≥ 2 *be even and r* (*t*) > 0*. Assume that for some constant λ* ∈ (0, 1)*, the differential equation*

$$\log'\left(t\right) + \frac{q\left(t\right)}{r\left(\pi\left(t\right)\right)} \left(\frac{\lambda\pi^{n-1}\left(t\right)}{(n-1)!}\right)^{p-1} \wp\left(\pi\left(t\right)\right) = 0\tag{24}$$

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

**Proof.** Let (1) have a nonoscillatory solution *y*. Without loss of generality, we can assume that *y* (*t*) > 0 for *t* ≥ *t*1, where *t*<sup>1</sup> ≥ *t*<sup>0</sup> is sufficiently large. Since *r* (*t*) > 0, we have

$$y'(t) > 0, \ y^{(n-1)}\left(t\right) > 0 \text{ and } y^{(n)}\left(t\right) < 0. \tag{25}$$

From Lemma 3, we get

$$y\left(t\right) \ge \frac{\lambda t^{n-1}}{(n-1)! r^{1/p-1}\left(t\right)} r^{1/p-1}\left(t\right) y^{\left(n-1\right)}\left(t\right),\tag{26}$$

for every *λ* ∈ (0, 1). Thus, if we set

$$\varphi\left(t\right) = r\left(t\right)\left[y^{\left(n-1\right)}\left(t\right)\right]^{p-1} > 0\_{\prime\prime}$$

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

$$
\varphi'\left(t\right) + \frac{q\left(t\right)}{r\left(\pi\left(t\right)\right)} \left(\frac{\lambda\pi^{n-1}\left(t\right)}{(n-1)!}\right)^{p-1} \varphi\left(\pi\left(t\right)\right) \le 0. \tag{27}
$$

From [22] (Theorem 1), we conclude that the corresponding Equation (24) also has a positive solution, which is a contradiction.

Theorem 2 is proved.

**Corollary 1.** *Assume that (2) holds and let n* ≥ 2 *be even. If*

$$\lim\_{t \to \infty} \inf \int\_{\tau(t)}^t \frac{q\left(s\right)}{r\left(\tau\left(s\right)\right)} \left(\tau^{n-1}\left(s\right)\right)^{p-1} ds > \frac{\left((n-1)!\right)^{p-1}}{\varepsilon},\tag{28}$$

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

Next, we give the following example to illustrate our main results.

**Example 1.** *Consider the equation*

$$y^{(4)}\left(t\right) + \frac{\gamma}{t^4}y\left(\frac{9}{10}t\right) = 0, \ t \ge 1,\tag{29}$$

*where γ* > 0 *is a constant. We note that n* = 4, *r* (*t*) = 1, *p* = 2, *τ* (*t*) = 9*t*/10 *and q* (*t*) = *γ*/*t* <sup>4</sup>*. If we set H* (*t*,*s*) = *H*<sup>∗</sup> (*t*,*s*) = (*t* − *s*) <sup>2</sup> , *<sup>A</sup>* (*s*) <sup>=</sup> *<sup>A</sup>*<sup>∗</sup> (*s*) <sup>=</sup> 1, *<sup>δ</sup>* (*s*) <sup>=</sup> *<sup>t</sup>* 3, *<sup>σ</sup>* (*s*) <sup>=</sup> *<sup>t</sup>*, *<sup>h</sup>* (*t*,*s*) <sup>=</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*) <sup>5</sup> <sup>−</sup> <sup>3</sup>*ts*−<sup>1</sup> *and h*<sup>∗</sup> (*t*,*s*) = (*t* − *s*) <sup>3</sup> <sup>−</sup> *ts*−<sup>1</sup> *then we get*

$$\eta\left(s\right) = \int\_{t\_0}^{\infty} \frac{1}{r^{1/\left(p-1\right)}\left(s\right)} ds = \infty$$

*and*

$$\begin{aligned} \mathcal{B}\left(t\right) &=& \frac{1}{\left(n-4\right)!} \int\_{t}^{\infty} \left(\theta - t\right)^{n-4} \left(\frac{\int\_{\theta}^{\infty} q\left(s\right) \left(\frac{\tau(s)}{s}\right)^{p-1} ds}{r\left(\theta\right)}\right)^{1/\left(p-1\right)} d\theta \\ &=& \Im \gamma / \left(20t^{2}\right) .\end{aligned}$$

*Hence conditions (20) and (21) become*

$$\begin{split} &\limsup\_{t\to\infty} \frac{1}{H\left(t,t\_{0}\right)} \int\_{t\_{0}}^{t} \Big( H\left(t,s\right)A\left(s\right)\delta\left(s\right)q\left(s\right) \left(\frac{\mathsf{r}^{n-1}\left(s\right)}{\mathsf{s}^{n-1}}\right)^{p-1} - D\left(s\right) \Big) ds \\ &= \limsup\_{t\to\infty} \frac{1}{\left(t-1\right)^{2}} \int\_{1}^{t} \Big[ \frac{729\gamma}{1000} \mathsf{i}^{2} \mathsf{s}^{-1} + \frac{729\gamma}{1000} \mathsf{s} - \frac{729\gamma}{500} t - \frac{\mathsf{s}}{2\mu} \left(25 + 9t^{2} \mathsf{s}^{-2} - 30t \mathsf{s}^{-1}\right) \Big] ds \\ &= \infty \quad \left(\mathcal{Y}\gamma > 500/81\right) \end{split}$$

*and*

$$\begin{split} \limsup\_{t \to \infty} &\frac{1}{H\_{\ast}\left(t, t\_{0}\right)} \int\_{t\_{0}}^{t} \Big( H\_{\ast}\left(t, s\right) A\_{\ast}\left(s\right) \sigma\left(s\right) B\left(s\right) - \frac{\sigma\left(s\right) \left|h\_{\ast}\left(t, s\right)\right|^{2}}{4H\_{\ast}\left(t, s\right) A\_{\ast}\left(s\right)} \Big) ds \\ &= \limsup\_{t \to \infty} \frac{1}{\left(t - 1\right)^{2}} \int\_{1}^{t} \left[\frac{3\gamma}{20} t^{2} s^{-1} + \frac{3\gamma}{20} s - \frac{3\gamma}{10} t - \frac{s}{4} \left(9 - 630t s^{-1} + t^{2} s^{-2}\right)\right] ds \\ &= \infty \quad \left(\text{if } \gamma > 5/3\right). \end{split}$$

*Thus, by Theorem 1, every solution of Equation (29) is oscillatory if γ* > 500/81*.*

#### **3. Conclusions**

In this work, we have discussed the oscillation of the higher-order differential equation with a p-Laplacian-like operator and we proved that Equation (1) is oscillatory by using the following methods:


Additionally, in future work we could try to get some oscillation criteria of Equation (1) under the condition <sup>∞</sup> *t*0 1 *r*1/(*p*−<sup>1</sup>)(*t*) *dt* < ∞. Thus, we would discuss the following two cases:

$$\begin{array}{llll} \left(\mathbb{C}\_{1}\right) & y\left(t\right) > 0, \ y^{\left(n-1\right)}\left(t\right) > 0, \ y^{\left(n\right)}\left(t\right) < 0, \\\left(\mathbb{C}\_{2}\right) & y\left(t\right) > 0, \ y^{\left(n-2\right)}\left(t\right) > 0, \ y^{\left(n-1\right)}\left(t\right) < 0. \end{array}$$

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

#### **References**


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