**Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order**

#### **Osama Moaaz 1,† , Ioannis Dassios 2,\* ,†, Omar Bazighifan 3,4,† and Ali Muhib 5,†**


Received: 6 March 2020; Accepted: 28 March 2020; Published: 3 April 2020

**Abstract:** We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results.

**Keywords:** deviating argument; fourth order; differential equation; oscillation

#### **1. Introduction**

In this paper, we are concerned with the oscillation and the asymptotic behavior of solutions of the following two fourth-order differential equations. The nonlinear differential equation:

$$\left(r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)' + q\left(t\right)\mathbf{x}^{\beta}\left(\sigma\left(t\right)\right) = \mathbf{0},\tag{1}$$

and the differential equation with the middle term of the form:

$$\left(r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)' + p\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a} + q\left(t\right)\mathbf{x}^{\beta}\left(\sigma\left(t\right)\right) = \mathbf{0},\tag{2}$$

where *<sup>α</sup>* and *<sup>β</sup>* are quotient of odd positive integers, *<sup>r</sup>*, *<sup>q</sup>* ∈ *<sup>C</sup>* ([*t*0, <sup>∞</sup>), [0, <sup>∞</sup>)), *<sup>r</sup>* (*t*) > 0, *<sup>q</sup>* (*t*) > 0, *<sup>σ</sup>* (*t*) ∈ *<sup>C</sup>* ([*t*0, <sup>∞</sup>), <sup>R</sup>), *<sup>σ</sup>* (*t*) <sup>≤</sup> *<sup>t</sup>*, lim*t*→<sup>∞</sup> *<sup>σ</sup>* (*t*) <sup>=</sup> <sup>∞</sup>. Moreover, we study Equation (1) under the condition

$$\int\_{t\_0}^{\infty} \frac{1}{r^{1/\alpha}(s)} ds = \infty \tag{3}$$

and Equation (2) under the conditions *<sup>p</sup>* ∈ *<sup>C</sup>* ([*t*0, <sup>∞</sup>), [0, <sup>∞</sup>)), *<sup>r</sup>* ′ (*t*) + *p* (*t*) ≥ 0 and

$$\int\_{t\_0}^{\infty} \left[ \frac{1}{r(s)} \exp \left( - \int\_{t\_0}^{s} \frac{p\left(u\right)}{r\left(u\right)} du \right) \right]^{1/a} ds = \infty. \tag{4}$$

We aim for a solution of Equation (1) or Equation (2) as a function *<sup>x</sup>*(*t*) : [*tx*, <sup>∞</sup>) <sup>→</sup> <sup>R</sup>, *<sup>t</sup><sup>x</sup>* <sup>≥</sup> *<sup>t</sup>*<sup>0</sup> such that *x*(*t*) and *r* (*t*) (*x* ′′′ (*t*))*<sup>α</sup>* are continuously differentiable for all *<sup>t</sup>* ∈ [*tx*, <sup>∞</sup>) and sup{|*x*(*t*)| : *<sup>t</sup>* ≥ *<sup>T</sup>*} > 0 for any *T* ≥ *tx*. We assume that Equation (1) or Equation (2) possesses such a solution. A solution of

Equation (1) or Equation (2) is called oscillatory if it has arbitrarily large zeros on [*tx*' ∞). Otherwise, it is called non-oscillatory. Equation (1) or Equation (2) is said to be oscillatory if all its solutions are oscillatory. The equation itself is called oscillatory if all of its solutions are oscillatory.

In mechanical and engineering problems, questions related to the existence of oscillatory and non-oscillatory solutions play an important role. As a result, there has been much activity concerning oscillatory and asymptotic behavior of various classes of differential and difference equations (see, e.g., [1–34], and the references cited therein).

Zhang et al. [30] considered Equation (1) where *α* = *β* and obtained some oscillation criteria. Baculikova et al. [5] proved that the equation

$$\left[r\left(t\right)\left(\mathbf{x}^{\left(n-1\right)}\left(t\right)\right)^{\alpha}\right]' + q\left(t\right)f\left(\mathbf{x}\left(\mathbf{r}\left(t\right)\right)\right) = 0$$

is oscillatory if the delay differential equations

$$y'(t) + q(t)f\left(\frac{\delta \tau^{n-1}(t)}{(n-1)!r^{\frac{1}{a}}(\tau(t))}\right)f\left(y^{\frac{1}{a}}(\tau(t))\right) = 0$$

is oscillatory and under the assumption that Equation (3) holds, and obtained some comparison theorems. In [15], El-Nabulsi et al. studied the asymptotic properties of the solutions of equation

$$\left(r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a}\right)' + q\left(t\right)\mathbf{x}^{a}\left(\sigma\left(t\right)\right) = \mathbf{0},\tag{5}$$

where *α* is ratios of odd positive integers and under the condition (3).

Elabbasy et al. [14] proved that Equation (2) where *α* = *β* = 1 is oscillatory if

$$\int\_{t\_0}^{\infty} \left( \rho \left( s \right) q \left( s \right) \frac{\mu}{2} r^2 \left( s \right) - \frac{1}{4 \rho \left( s \right) r \left( s \right)} \left[ \frac{\rho\_+' \left( s \right)}{\rho \left( s \right)} - \frac{p \left( s \right)}{r \left( s \right)} \right]^2 \right) ds = \infty \mu$$

for some *µ* ∈ (0, 1), and

$$\int\_{t\_0}^{\infty} \left[ \theta \left( s \right) \int\_s^{\infty} \left[ \frac{1}{r \left( \upsilon \right)} \int\_{\upsilon}^{\infty} q \left( \upsilon \right) \left( \frac{\tau^2 \left( \upsilon \right)}{\upsilon^2} \right) d\upsilon \right] d\upsilon - \frac{\left( \theta' \left( s \right) \right)^2}{4 \theta \left( s \right)} \right] ds = \infty$$

where positive functions *ρ*, *ϑ* ∈ *C* 1 ([*ν*0, ∞), R) and under the condition in Equation (4).

The motivation in studying this paper improves results in [15]. An example is presented in the last section to illustrate our main results.

We firstly provide the following lemma, which is used as a tool in the proofs our theorems.

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

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

*If we have* lim*t*→<sup>∞</sup> *h* (*t*) 6= 0, *then there exists t<sup>λ</sup>* ≥ *t*<sup>0</sup> *such that*

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

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

**Lemma 2** ([26])**.** *If the function x satisfies x*(*i*) (*t*) > 0, *i* = 0, 1, ..., *n*, *and x*(*n*+1) (*t*) < 0, *then*

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

**Lemma 3** ([27] Lemma 1.2)**.** *Assume that α is a quotient of odd positive integers, V* > 0 *and U are constants. Then,*

$$My - Vy^{(a+1)/a} \le \frac{a^a}{(a+1)^{a+1}} \mathcal{U}^{a+1} V^{-a}.\tag{6}$$

#### **2. Oscillation Results**

Firstly we establish oscillation results for Equation (1). For convenience, we denote

$$\begin{aligned} \mathcal{G}\left(t\right) &:= \frac{\lambda^{\beta}}{6^{\beta}} \frac{q\left(t\right) \sigma^{3\beta}\left(t\right)}{r^{\beta/\alpha}\left(\sigma\left(t\right)\right)}, \\\\ \mathcal{R}\left(t\right) &:= \int\_{t}^{\infty} \left(\frac{1}{r\left(u\right)} \int\_{u}^{\infty} q\left(s\right) \, \mathrm{d}s\right)^{1/\alpha} \, \mathrm{d}u \end{aligned}$$

and

$$\widetilde{\mathcal{R}}\left(t\right) := \mu^{\beta/\alpha} \int\_{t}^{\infty} \left( \frac{1}{r\left(u\right)} \int\_{u}^{\infty} q\left(s\right) \left(\frac{\sigma\left(s\right)}{s}\right)^{\beta} \mathrm{d}s \right)^{1/\alpha} \mathrm{d}u \,\rho$$

where *λ*, *µ* ∈ (0, 1).

**Lemma 4.** *Assume that Equation (3) holds. If x is an eventually positive solution of Equation (1); then, x* ′ > 0 *and x*′′′ > 0*.*

**Proof.** Assume that *x* is an eventually positive solution of Equation (1); then, *x* (*t*) > 0 and *x* (*σ* (*t*)) > 0 for *t* ≥ *t*1. From Equation (1), we get

$$\left(r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha}\right)' = -q\left(t\right)\mathbf{x}^{\beta}\left(\sigma\left(t\right)\right) < 0.$$

Hence, *r* (*t*) (*x* ′′′ (*t*))*<sup>α</sup>* is decreasing of one sign. Thus, we see that

$$x'''(t) > 0.$$

From Equation (1), we obtain

$$\left(r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha}\right)' = r^{\prime}\left(t\right) + ar\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha -1}\mathbf{x}^{(4)}\left(t\right) \le 0, \alpha$$

from which it follows that *x* (4) (*t*) ≤ 0, hence *x* ′ (*t*) > 0, *t* ≥ *t*1. The proof is complete.

**Theorem 1.** *Assume that Equation (3) holds. If the differential equation*

$$
\mu'\left(t\right) + G\left(t\right)\mu^{\otimes/\mathfrak{a}}\left(\sigma\left(t\right)\right) = 0\tag{7}
$$

*is oscillatory for some λ* ∈ (0, 1), *then Equation (1) is oscillatory.*

**Proof.** Assume to the contrary that Equation (1) has a nonoscillatory solution in [*t*0, ∞). Without loss of generality, we only need to be concerned with positive solutions of Equation (1). Then, there exists a *t*<sup>1</sup> ≥ *t*<sup>0</sup> such that *x* (*t*) > 0 and *x* (*σ* (*t*)) > 0 for *t* ≥ *t*1. Let

$$\mu\left(t\right) := r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha} > 0 \text{ [from Lemma 4]}.$$

which with Equation (1) gives

$$\mu'\left(t\right) + q\left(t\right)x^{\mathcal{G}}\left(\sigma\left(t\right)\right) = 0. \tag{8}$$

Since *x* is positive and increasing, we have lim*t*→<sup>∞</sup> *x* (*t*) 6= 0. Thus, from Lemma 1, we get

$$\mathbf{x}^{\mathfrak{G}}\left(\sigma\left(t\right)\right) \geq \frac{\lambda^{\mathfrak{G}}}{6^{\mathfrak{G}}} \sigma^{\mathfrak{G}\mathfrak{G}}\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}\left(\sigma\left(t\right)\right)\right)^{\mathfrak{G}},\tag{9}$$

for all *λ* ∈ (0, 1). By Equations (8) and (9), we see that

$$
\mu'\left(t\right) + \frac{\lambda^{\beta}}{6^{\beta}} q\left(t\right) \sigma^{\Im\beta}\left(t\right) \left(\mathfrak{x}^{\prime\prime\prime}\left(\sigma\left(t\right)\right)\right)^{\beta} \leq 0.1
$$

Thus, we note that *u* is positive solution of the differential inequality

$$
\mu'\left(t\right) + G\left(t\right)\mu^{\beta/\alpha}\left(\sigma\left(t\right)\right) \le 0.
$$

In view of [25] (Theorem 1), the associated Equation (7) also has a positive solution, which is a contradiction. The theorem is proved.

**Corollary 1.** *Assume that α* = *β and Equation (3) holds. If*

$$\liminf\_{t \to \infty} \int\_{\sigma(t)}^t \mathcal{G}\left(s\right) \, \mathrm{d}s > \frac{1}{\mathrm{e}'} \,\tag{10}$$

*for some λ* ∈ (0, 1), *then Equation (1) is oscillatory.*

**Proof.** It is well-known (see [28] (Theorem 2.1.1)) that Equation (10) implies the oscillation of Equation (11).

**Lemma 5.** *Assume that Equation (3) holds and x is an eventually positive solution of Equation (1). If*

$$\int\_{t\_0}^{\infty} \left( M^{\mathfrak{f}-a} \rho \left( t \right) q \left( t \right) \frac{\sigma^{3a} \left( t \right)}{t^{3a}} - \frac{2^a}{\left( a+1 \right)^{a+1}} \frac{r \left( t \right) \left( \rho' \left( t \right) \right)^{a+1}}{\mu^a t^{2a} \rho^a \left( t \right)} \right) \mathrm{d}s = \infty,\tag{11}$$

*for some µ* ∈ (0, 1), *then x*′′ < 0.

**Proof.** Assume to the contrary that *x* ′′ (*t*) > 0. Using Lemmas 2 and 1, we obtain

$$\frac{\mathfrak{x}\left(\sigma\left(t\right)\right)}{\mathfrak{x}\left(t\right)} \geq \frac{\sigma^3\left(t\right)}{t^3} \tag{12}$$

and

$$\mathbf{x}'\left(t\right) \ge \frac{\mu}{2}t^2\mathbf{x}''\left(t\right),\tag{13}$$

for all *µ* ∈ (0, 1) and every sufficiently large *t*. Now, we define a function *ψ* by

$$
\psi\left(t\right) := \rho\left(t\right) \frac{r\left(t\right) \left(\mathfrak{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha}}{\mathfrak{x}^{\alpha}\left(t\right)} > 0.
$$

By differentiating and using Equations (12) and (13), we obtain

$$\psi'(t) \le \frac{\rho'(t)}{\rho(t)} \omega\left(t\right) - \rho\left(t\right)q\left(t\right) \frac{\sigma^{3\alpha}\left(t\right)}{t^{3\alpha}} x^{\theta - a}\left(\sigma\left(t\right)\right) - \frac{a\mu}{2} \frac{t^2}{\rho^{1/a}\left(t\right)r^{1/a}\left(t\right)} \psi^{1+1/a}\left(t\right). \tag{14}$$

Since *x* ′ (*t*) > 0, there exist a *t*<sup>2</sup> ≥ *t*<sup>1</sup> and a constant *M* > 0 such that *x* (*t*) > *M*, for all *t* ≥ *t*2. Using the inequality in Equation (6) with *U* = *ρ* ′/*ρ*, *V* = *αµt* 2/ 2*r* 1/*α* (*t*) *ρ* 1/*α* (*t*) and *y* = *ψ*, we get

$$\psi'(t) \le -M^{\beta-a} \rho\left(t\right) q\left(t\right) \frac{\sigma^{3a}\left(t\right)}{t^{3a}} + \frac{2^a}{\left(a+1\right)^{a+1}} \frac{r\left(t\right)\left(\rho'\left(t\right)\right)^{a+1}}{\mu^a t^{2a} \rho^a\left(t\right)}.$$

This implies that

$$\int\_{t\_1}^t \left( M^{\mathfrak{H}-a} \rho \begin{pmatrix} t \end{pmatrix} q \begin{pmatrix} t \end{pmatrix} - \frac{\sigma^{3a} \begin{pmatrix} t \end{pmatrix}}{t^{3a}} - \frac{2^a}{(a+1)^{a+1}} \frac{r \begin{pmatrix} t \end{pmatrix} (\rho' \begin{pmatrix} t \end{pmatrix})^{a+1}}{\mu^a t^{2a} \rho^a \begin{pmatrix} t \end{pmatrix}} \right) \mathrm{d}s \le \psi \left( t\_1 \right) \,\_2\theta\_1 \left( \begin{pmatrix} t \end{pmatrix} \right)^{a+1} \left( \begin{pmatrix} t \end{pmatrix} - \frac{\rho^a}{(a+1)^{a+1}} \frac{r \begin{pmatrix} t \end{pmatrix} (\rho' \begin{pmatrix} t \end{pmatrix} )}{t^{2a+1}} \right) \mathrm{d}s \right) = \psi \left( t \right) \,\_2\theta\_1 \left( \begin{pmatrix} t \end{pmatrix} \right) \,\_2\theta\_1 \left( \begin{pmatrix} t \end{pmatrix} \right)$$

which contradicts Equation (11). The proof is complete.

**Theorem 2.** *Assume that β* ≥ *α and Equations (3) and (11) hold, for some µ* ∈ (0, 1)*. If*

$$y''\left(t\right) + M^{\mathfrak{f}-\mathfrak{a}}\widetilde{\mathcal{R}}\left(t\right)y\left(t\right) = 0\tag{15}$$

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

**Proof.** Assume to the contrary that Equation (1) has a nonoscillatory solution in [*t*0, ∞). Without loss of generality, we only need to be concerned with positive solutions of Equation (1). Then, there exists a *t*<sup>1</sup> ≥ *t*<sup>0</sup> such that *x* (*t*) > 0 and *x* (*σ* (*t*)) > 0 for *t* ≥ *t*1. From Lemmas 4 and 1, we have that

$$\mathbf{x}'(t) > \mathbf{0}, \ x''(t) < \mathbf{0} \text{ and } \mathbf{x}''''(t) > \mathbf{0},\tag{16}$$

for *t* ≥ *t*2, where *t*<sup>2</sup> is sufficiently large. Now, integrating Equation (1) from *t* to *l*, we have

$$r\left(l\right)\left(\mathbf{x}^{\prime\prime\prime}\left(l\right)\right)^{a} = r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a} - \int\_{t}^{l} q\left(s\right)\mathbf{x}^{\emptyset}\left(\sigma\left(s\right)\right)\,\mathrm{d}s.\tag{17}$$

Using Lemma 3 from [29] with Equation (16), we get

$$\frac{\mathfrak{x}\left(\sigma\left(t\right)\right)}{\mathfrak{x}\left(t\right)} \geq \lambda \frac{\sigma\left(t\right)}{t} \lambda$$

for all *λ* ∈ (0, 1), which with Equation (17) gives

$$r\left(l\right)\left(\mathbf{x}^{\prime\prime\prime}\left(l\right)\right)^{a} - r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a} + \lambda^{\beta} \int\_{t}^{l} q\left(s\right) \left(\frac{\sigma\left(s\right)}{s}\right)^{\beta} \mathbf{x}^{\beta}\left(s\right) \,\mathrm{d}s \le 0.1$$

It follows by *x* ′ > 0 that

$$r\left(l\right)\left(\mathbf{x}^{\prime\prime\prime}\left(l\right)\right)^{a} - r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{a} + \lambda^{\beta}\mathbf{x}^{\beta}\left(t\right)\int\_{t}^{l} q\left(\mathbf{s}\right)\left(\frac{\sigma\left(\mathbf{s}\right)}{\mathbf{s}}\right)^{\beta}\mathbf{ds} \leq \mathbf{0}.\tag{18}$$

.

Taking *<sup>l</sup>* → <sup>∞</sup>, we have

$$-r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha} + \lambda^{\beta}\mathbf{x}^{\beta}\left(t\right)\int\_{t}^{\infty} q\left(s\right)\left(\frac{\sigma\left(s\right)}{s}\right)^{\beta}ds \le 0\_{\lambda}$$

that is

$$\mathbf{x}^{\prime\prime\prime}(t) \ge \frac{\lambda^{\beta/\alpha}}{r^{1/\alpha}(t)} \mathbf{x}^{\beta/\alpha}(t) \left( \int\_{t}^{\infty} q\left(s\right) \left(\frac{\sigma\left(s\right)}{s}\right)^{\beta} \mathrm{d}s \right)^{1/\alpha}$$

Integrating the above inequality from *t* to ∞, we obtain

$$-\mathfrak{x}''(t) \ge \lambda^{\beta/a} \mathfrak{x}^{\beta/a}(t) \int\_{t}^{\infty} \left( \frac{1}{r(u)} \int\_{u}^{\infty} q\left(s\right) \left(\frac{\sigma\left(s\right)}{s}\right)^{\beta} \mathrm{ds} \right)^{1/a} \mathrm{d}u,$$

hence

$$\mathbf{x}^{\prime\prime}(t) \le -\widetilde{\mathbf{R}}\left(t\right)\mathbf{x}^{\beta/\mathfrak{a}}\left(t\right). \tag{19}$$

Now, if we define *ω* by

$$
\omega\left(t\right) = \frac{\mathbf{x}'\left(t\right)}{\mathbf{x}\left(t\right)}\mathbf{y}'
$$

then *ω* (*t*) > 0 for *t* ≥ *t*1, and

$$
\omega'(t) = \frac{\mathfrak{x}''(t)}{\mathfrak{x}(t)} - \left(\frac{\mathfrak{x}'(t)}{\mathfrak{x}(t)}\right)^2.
$$

By using Equation (19) and definition of *ω* (*t*), we see that

$$
\omega'\left(t\right) \le -\tilde{\mathcal{R}}\left(t\right) \frac{\mathfrak{x}^{\beta/a}\left(t\right)}{\mathfrak{x}\left(t\right)} - \omega^2\left(t\right). \tag{20}
$$

Since *x* ′ (*t*) > 0, there exists a constant *M* > 0 such that *x* (*t*) ≥ *M*, for all *t* ≥ *t*2, where *t*<sup>2</sup> is sufficiently large. Then, Equation (20) becomes

$$
\omega'\left(t\right) + \omega^2\left(t\right) + M^{\beta - \alpha}\tilde{\mathcal{R}}\left(t\right) \le 0. \tag{21}
$$

It is well known (see [3]) that the differential equation in Equation (15) is nonoscillatory if and only if there exists *t*<sup>3</sup> > max {*t*1, *t*2} such that Equation (21) holds, which is a contradiction. Theorem is proved.

**Theorem 3.** *Assume that β* ≥ *α and σ* ′ (*t*) > 1 *and Equations (3) and (11) hold, for some µ* ∈ (0, 1)*. If*

$$\left(\frac{1}{\sigma'\left(t\right)}y'\left(t\right)\right)' + M^{\theta/a - 1}\mathbb{R}\left(t\right)y\left(t\right) = 0\tag{22}$$

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

**Proof.** Proceeding as in the proof of Theorem 2, we obtain Equation (17). Thus, it follows from *σ* ′ (*t*) ≥ 0 and *x* ′ (*t*) ≥ 0 that

$$r\left(l\right)\left(\mathbf{x}^{\prime\prime\prime}\left(l\right)\right)^{\mathfrak{a}} - r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\mathfrak{a}} + \mathbf{x}^{\mathfrak{f}}\left(\sigma\left(t\right)\right)\int\_{\mathfrak{f}}^{\mathfrak{l}} q\left(s\right) \mathbf{ds} \leq \mathbf{0}.\tag{23}$$

Thus, Equation (16) becomes

$$\mathbf{x}^{\prime\prime}(t) \le -\mathsf{R}\left(t\right)\mathbf{x}^{\beta/\mathfrak{a}}\left(\boldsymbol{\sigma}\left(t\right)\right).\tag{24}$$

Now, if we define *w* by

$$w\left(t\right) = \frac{x'\left(t\right)}{\mathfrak{x}\left(\sigma\left(t\right)\right)'}$$

then *w* (*t*) > 0 for *t* ≥ *t*1, and

$$\begin{aligned} w'(t) &= \frac{\mathfrak{x}''(t)}{\mathfrak{x}\left(\sigma\left(t\right)\right)} - \frac{\mathfrak{x}'(t)}{\mathfrak{x}^2\left(\sigma\left(t\right)\right)} \mathfrak{x}'\left(\sigma\left(t\right)\right) \sigma'\left(t\right) \\ &\leq \quad \frac{\mathfrak{x}''(t)}{\mathfrak{x}\left(\sigma\left(t\right)\right)} - \sigma'\left(t\right) \left(\frac{\mathfrak{x}'(t)}{\mathfrak{x}\left(\sigma\left(t\right)\right)}\right)^2 . \end{aligned}$$

By using Equation (24) and definition of *w* (*t*), we see that

$$w'\left(t\right) + M^{\beta/\alpha - 1}R\left(t\right) + \sigma'\left(t\right)w^2\left(t\right) \le 0. \tag{25}$$

It is well known (see [3]) that the differential equation in Equation (22) is nonoscillatory if and only if there exists *t*<sup>3</sup> > max {*t*1, *t*2} such that Equation (25) holds, which is a contradiction. Theorem is proved.

There are many results concerning the oscillation of Equations (15) and (22), which include Hille–Nehari types, Philos type, etc. On the basis of [33,34], we have the following corollary, respectively.

**Corollary 2.** *Assume that β* = *α and Equations (3) and (11) hold, for some µ* ∈ (0, 1)*. If*

$$\lim\_{t \to \infty} \frac{1}{H\left(t, t\_0\right)} \int\_{t\_0}^{t} \left(H\left(t, s\right)\widetilde{\mathcal{R}}\left(s\right) - \frac{1}{4}h^2\left(t, s\right)\right)ds = \infty$$

$$\liminf\_{t \to \infty} \int\_{t}^{\infty} \widetilde{\mathcal{R}}\left(s\right)ds > \frac{1}{4},\tag{26}$$

*or*

*then Equation (1) is oscillatory.*

**Corollary 3.** *Assume that β* = *α and Equations (3) and (11) hold, for some µ* ∈ (0, 1)*. If there exists a constant κ* ∈ (0, 1/4] *such that*

$$t^2 \tilde{\mathcal{R}}\left(s\right) \ge \kappa$$

*and*

$$\limsup\_{t \to \infty} \left( t^{\kappa - 1} \int\_{t\_0}^t s^{2 - \kappa} \widetilde{\mathcal{R}}\left( s \right) \, \mathrm{d}s + t^{1 - \tilde{\kappa}} \int\_t^{\infty} s^{\tilde{\kappa}} \widetilde{\mathcal{R}}\left( s \right) \, \mathrm{d}s \right) > 1\_{\kappa}$$

*where <sup>κ</sup>*<sup>e</sup> <sup>=</sup> <sup>1</sup> 2 1 − √ 1 − 4*κ , then Equation (1) is oscillatory.*

We will now define the following notation:

$$\eta\_{t\_0}(t) := \exp\left(\int\_{t\_0}^t \frac{p\left(u\right)}{r\left(u\right)} \mathrm{d}u\right),$$

and

$$\widehat{\mathcal{R}}\left(t\right) := \mu\_1^{\otimes/a} \int\_t^{\infty} \left( \frac{1}{r\left(u\right)\eta\_{t\_0}\left(t\right)} \int\_u^{\infty} \eta\_{t\_0}\left(t\right) q\left(s\right) \left(\frac{\sigma\left(s\right)}{s}\right)^{\beta} \mathrm{d}s \right)^{1/a} \mathrm{d}u \,\rho$$

where *µ*<sup>1</sup> ∈ (0, 1). We establish oscillation results for Equation (2) by converting into the form of Equation (1). It is not difficult to see that

$$\begin{split} \frac{1}{\eta\_{l\_{0}}(t)} \frac{d}{dt} \left( \mu\left(t\right) r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a} \right) &=& \frac{1}{\eta\_{l\_{0}}(t)} \left[ \eta\_{l\_{0}}\left(t\right) \left(r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a}\right)' + \eta\_{l\_{0}}'\left(t\right) r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a} \right] \\ &=& \left(r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a}\right)' + \frac{\eta\_{l\_{0}}'(t)}{\eta\_{l\_{0}}(t)} r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a} \\ &=& \left(r\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a}\right)' + p\left(t\right) \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{a}, \end{split}$$

which with Equation (2) gives

$$\left(\eta\_{t\_0}\left(t\right)r\left(t\right)\left(\mathbf{x}^{\prime\prime\prime}\left(t\right)\right)^{\alpha}\right)' + \eta\_{t\_0}\left(t\right)q\left(t\right)\mathbf{x}^{\beta}\left(\sigma\left(t\right)\right) = \mathbf{0}.$$

**Corollary 4.** *Assume that α* = *β and Equation (4) holds. If*

$$\liminf\_{t \to \infty} \int\_{\sigma(t)}^t \widehat{G}\left(s\right) \, \mathrm{d}s > \frac{1}{\mathrm{e}'} $$

*for some λ* ∈ (0, 1), *where*

$$
\widehat{G}\left(t\right) := \frac{\lambda^{\beta}}{6^{\beta}} \frac{\eta\_{t\_0}\left(t\right) q\left(t\right) \sigma^{3\beta}\left(t\right)}{\eta\_{t\_0}^{\beta/\alpha}\left(\sigma\left(t\right)\right) r^{\beta/\alpha}\left(\sigma\left(t\right)\right)},
$$

*then Equation (2) is oscillatory.*

**Corollary 5.** *Assume that β* = *α, Equation (4) and*

$$\int\_{t\_0}^{\infty} \left( M^{6-a} \rho \left( t \right) \eta\_{l\_0} \left( t \right) \rho \left( t \right) \frac{\sigma^{3a} \left( t \right)}{t^{3a}} - \frac{2^a}{(a+1)^{a+1}} \frac{r \left( t \right) \eta\_{l\_0} \left( t \right) \left( \rho' \left( t \right) \right)^{a+1}}{\mu^a t^{2a} \rho^a \left( t \right)} \right) \mathrm{d}s = \infty,\tag{27}$$

*hold, for some µ* ∈ (0, 1)*. If*

$$\lim\_{t \to \infty} \frac{1}{H\left(t, t\_0\right)} \int\_{t\_0}^{t} \left( H\left(t, s\right) \hat{\mathcal{R}}\left(s\right) - \frac{1}{4} h^2\left(t, s\right) \right) ds = \infty$$

*or*

$$\liminf\_{t \to \infty} \int\_{t}^{\infty} \widehat{R}\left(s\right) \,\mathrm{d}s > \frac{1}{4}\prime$$

*then Equation (2) is oscillatory.*

**Corollary 6.** *Assume that β* = *α and Equations (4) and (27) hold, for some µ* ∈ (0, 1)*. If there exists a constant κ* ∈ (0, 1/4] *such that*

$$t^2 \widehat{\mathbb{R}}\left(s\right) \ge \kappa$$

*and*

$$\limsup\_{t \to \infty} \left( t^{\kappa - 1} \int\_{t\_0}^t \mathbf{s}^{2 - \kappa} \widehat{\mathcal{R}}\left( \mathbf{s} \right) \mathbf{ds} + t^{1 - \tilde{\kappa}} \int\_t^{\infty} \mathbf{s}^{\tilde{\kappa}} \widehat{\mathcal{R}}\left( \mathbf{s} \right) \mathbf{ds} \right) > 1/2$$

*where <sup>κ</sup>*<sup>e</sup> *is defined as Corollary 3, then Equation (2) is oscillatory.*

#### **3. Example**

In this section, we give the following example to illustrate our main results.

**Example 1.** *For t* ≥ 1, *consider a differential equation:*

$$\left(t^{\mathfrak{z}} \left(\mathbf{x}^{\prime\prime\prime}(t)\right)^{\mathfrak{z}}\right)' + \frac{q\_0}{t^{\mathfrak{z}}} \mathbf{x}^{\mathfrak{z}} \left(\gamma t\right) = \mathbf{0},\tag{28}$$

*where γ* ∈ (0, 1] *and q*<sup>0</sup> > 0*. We note that α* = *β* = 3, *r* (*t*) = *t* 3 , *σ* (*t*) = *γt and q* (*t*) = *q*0/*t* 7 *. Thus, it is easy to verify that*

$$\mathcal{G}\left(t\right) = \frac{\lambda^3 \gamma^6}{6^3} \frac{q\_0}{t} \; \text{and} \; \tilde{\mathcal{R}}\left(t\right) = \lambda \left(\frac{q\_0}{6}\right)^{1/3} \gamma \frac{1}{2t^2}.$$

*By using Corollary 1, we see that Equation (28) is oscillatory if*

$$q\_0 > \frac{6^3}{e\left(\ln\frac{1}{\gamma}\right)\gamma^6}.\tag{29}$$

*This result can be obtained from [5].*

*For using Corollary 2, we see that the conditions in Equations (11) and (26) become*

$$q\_0 > \left(\frac{3^4}{2}\right) \frac{1}{\gamma^9}$$

*and*

$$q\_0 > 6\left(\frac{1}{4\gamma}\right)^3$$

*respectively. Thus, Equation (28) is oscillatory if*

$$q\_0 > \max\left\{ \left(\frac{3^4}{2}\right) \frac{1}{\gamma^9}, 6\left(\frac{1}{4\gamma}\right)^3 \right\} = \left(\frac{3^4}{2}\right) \frac{1}{\gamma^9}.\tag{30}$$

**Remark 1.** *By applying equation Equation (30) on the work in [15] where γ* = 1/2*, we find*

*q*<sup>0</sup> > 20736.

*Therefore, our result improves results [15].*

#### **4. Conclusions**

In this article, we study the oscillatory behavior of a class of non-linear fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. The outcome of this article extends a number of related results reported in the literature.

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

**Funding:** This work was supported by Science Foundation Ireland (SFI), by funding Ioannis Dassios, under Investigator Programme Grant No. SFI/15 /IA/3074.

**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.

#### **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/).

**Osama Moaaz 1,\* , Ali Muhib 1,2 and Shyam S. Santra <sup>3</sup>**


**Abstract:** It is easy to notice the great recent development in the oscillation theory of neutral differential equations. The primary aim of this work is to extend this development to neutral differential equations of mixed type (including both delay and advanced terms). In this work, we consider the second-order non-canonical neutral differential equations of mixed type and establish a new single-condition criterion for the oscillation of all solutions. By using a different approach and many techniques, we obtain improved oscillation criteria that are easy to apply on different models of equations.

**Keywords:** non-canonical differential equations; second-order; neutral delay; mixed type; oscillation criteria

#### **1. Introduction**

This paper discusses the oscillatory behavior of solutions of second-order neutral differential equations of mixed type:

$$\begin{aligned} \left( r(s) \left( (\mathbf{x}(s) + p\_1(s)\mathbf{x}(\varrho\_1(s)) + p\_2(s)\mathbf{x}(\varrho\_2(s)))' \right)' \right)' \\ + q\_1(s)\mathbf{x}^a(\theta\_1(s)) + q\_2(s)\mathbf{x}^a(\theta\_2(s)) = 0, \end{aligned} \tag{1}$$

where *s* ≥ *s*0. Throughout this paper, we assume the following:

**(C1)** *α* ∈ *Q* + *odd* :<sup>=</sup> {*a*/*<sup>b</sup>* : *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> <sup>Z</sup> <sup>+</sup> are odd};


**(C4)** *<sup>p</sup>*1, *<sup>p</sup>*2, *<sup>q</sup>*1, *<sup>q</sup>*<sup>2</sup> ∈ *<sup>C</sup>*([*s*0, <sup>∞</sup>), [0, <sup>∞</sup>)) and *<sup>q</sup>*1, *<sup>q</sup>*<sup>2</sup> are not identically zero for large *<sup>s</sup>*.

Let *<sup>x</sup>* be a real-valued function defined for all *<sup>s</sup>* in a real interval [*sx*, <sup>∞</sup>), *<sup>s</sup><sup>x</sup>* ≥ *<sup>s</sup>*0, which has the properties

$$\mathfrak{x} + p\_1 \cdot \mathfrak{x} \circ \varrho\_1 + p\_2 \cdot \mathfrak{x} \circ \varrho\_2 \in \mathbb{C}^1([s\_{\mathfrak{x}}, \infty), \mathbb{R})$$

and

$$r \cdot (\mathfrak{x} + p\_1 \cdot \mathfrak{x} \circ \varrho\_1 + p\_2 \cdot \mathfrak{x} \circ \varrho\_2)' \in \mathbb{C}^1([\mathfrak{s}\_{\mathfrak{X}} \infty), \mathbb{R}).$$

Then, *<sup>x</sup>* is called a solution of (1) on [*sx*, <sup>∞</sup>) if *<sup>x</sup>* satisfies (1) for all *<sup>s</sup>* ≥ *<sup>s</sup>x*. We will consider only the solutions of (1) that exist on some half-line [*sx*, <sup>∞</sup>) for *<sup>s</sup><sup>x</sup>* ≥ *<sup>s</sup>*<sup>0</sup> and satisfy the condition

sup{|*x*(*s*)| : *<sup>s</sup><sup>c</sup>* ≤ *<sup>s</sup>* < <sup>∞</sup>} > 0 for any *<sup>s</sup><sup>c</sup>* ≥ *<sup>s</sup>x*.

A nontrivial solution *x* of any differential equation is said to be oscillatory if it has arbitrary large zeros; otherwise, it is said to be non-oscillatory.


**Citation:** Moaaz, O.; Muhib; A.; Santra S.S. An Oscillation Test for Solutions of Second-Order Neutral Differential Equations of Mixed Type. *Mathematics* **2021**, *9*, 1634. https://doi.org/10.3390/math9141634

Academic Editors: Eva Kaslik and Christopher Goodrich

Received: 20 January 2021 Accepted: 19 February 2021 Published: 11 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

The oscillation and asymptotic behavior of solutions to various classes of delay and advanced differential equations have been widely discussed in the literature. For second-order delay equations, the studies found in [1–5] were concerned with studying the oscillatory behavior of the equation:

$$\left(r(s)\left(\left(\mathbf{x}(s) + p\_1(s)\mathbf{x}(\varrho\_1(s))\right)'\right)^a\right)' + q\_1(s)\mathbf{x}^a(\theta\_1(s)) = 0,\tag{2}$$

with the canonical operator *π*(*s*0) = ∞, where

$$
\pi(s) := \int\_{s\_0}^s r^{-1/\alpha}(\xi) \, \mathbf{d}\tilde{\xi} \, .
$$

One can find developments and comparisons of the oscillation criteria of (2) in the recently published paper by Moaaz et al. [4] for a non-canonical case, that is,

$$\int\_{\mathbf{s}\_0}^{\infty} r^{-1/\alpha}(\boldsymbol{\xi}) \mathbf{d}\boldsymbol{\xi} < \infty.$$

Bohner et al. [6] simplified and improved the previous results found by Agarwal et al. [7] and Han et al. [8]. For more general equations and more accurate results, see [9,10].

For second-order advanced equations, Chatzarakis et al. [11,12] studied the asymptotic behavior of the equation:

$$\left(r(s)\left(\mathfrak{x}(s)'\right)'\right)' + q\_2(s)\mathfrak{x}^a(\theta\_2(s)) = 0,$$

in the non-canonical case, and improved a number of pre-existing results.

Although there are many results of studies of the oscillation of solutions of delay differential equations, the results that concern the study of mixed equations are few—see, for example [13–24]. By using the Riccati transformation technique, Arul and Shobha [13] obtained some sufficient conditions for oscillation of the equation:

$$\left(r(s)(\mathfrak{x}(s) + a(s)\mathfrak{x}(s-\varrho) + b(s)\mathfrak{x}(s+\delta))'\right)' + q(s)f(\mathfrak{x}(\theta(s))) = 0\_{\kappa}$$

where 0 ≤ *<sup>a</sup>*(*s*) ≤ *<sup>a</sup>* < <sup>∞</sup>, 0 ≤ *<sup>b</sup>*(*s*) ≤ *<sup>b</sup>* < <sup>∞</sup>, and *<sup>f</sup>*(*u*)/*<sup>u</sup>* ≥ *<sup>k</sup>* > 0. Dzurina et al. [22] established some criteria for the oscillation of the equation

$$\left(\mathbf{x}(\mathbf{s}) + p\_1 \mathbf{x}(\mathbf{s} - \varrho\_1) + p\_2 \mathbf{x}(\mathbf{s} + \varrho\_2)\right)^{\prime\prime} = q\_1(\mathbf{s})\mathbf{x}(\mathbf{s} - \theta\_1) + q\_2(\mathbf{s})\mathbf{x}(\mathbf{s} + \theta\_2)\mathbf{x}$$

where *̺<sup>i</sup>* , *θ<sup>i</sup>* ≥ 0 are constants, *q<sup>i</sup>* is nonnegative, and *i* = 1, 2. Tunc et al. [24] studied the oscillatory behavior of solutions of the equation:

$$\left(r(s)\left(\left(\mathbf{x}(s) + p\_1(s)\mathbf{x}(\varrho\_1(s)) + p\_2(s)\mathbf{x}(\varrho\_2(s))\right)'\right)^a\right)' + q(s)\mathbf{x}^a(\theta(s)) = 0, \quad s = 0$$

in the canonical case *π*(*s*0) = ∞, and considered the cases:

$$p\_1(\mathbf{i}) \not\supset p\_1(\mathbf{s}) \ge \mathbf{0}, \ p\_2(\mathbf{s}) \ge \mathbf{1} \text{ and } p\_2(\mathbf{s}) \ne \mathbf{1} \text{ eventually}$$

and

$$p\_1(\text{ii}) \ p\_2(s) \ge 0, \ p\_1(s) \ge 1 \text{ and } p\_2(s) \ne 1 \text{ eventually.}$$

Thandapani et al. [23] considered the equation

$$\left( \left( \mathbf{x}(s) + p\_1 \mathbf{x}(s - \varrho\_1) + p\_2 \mathbf{x}(s + \varrho\_2) \right)^a \right)^{\prime \prime} + q\_1(s) \mathbf{x}^{\notin}(s - \theta\_1) + q\_2(s) \mathbf{x}^{\uparrow}(s + \theta\_2) = \mathbf{0}\_n$$

where *α*, *β*, and *γ* are the ratios of odd positive integers, and established some sufficient conditions for the oscillation of all of the solutions. For more results, techniques, and approaches that deal with the oscillation of delay differential equations of higher orders, see [25–33].

The objective of this paper is to study the oscillatory and asymptotic properties of a class of delay differential equations of mixed neutral type with the non-canonical operator. The oscillation criteria are obtained via only one condition, and hence, they are easy to apply. Moreover, by using generalized Riccati substitution, we get new criteria that improve some of the results reported in the literature. An example is provided to illustrate the significance of the main results.

#### **2. Preliminary Results**

In the following, we present the notations used in this study:


$$\kappa(\mu, v) := \int\_{\mathfrak{u}}^{v} r^{-1/\alpha}(\delta) \mathbf{d} \delta \mathbf{y}$$


$$\upsilon(s) := \mathbf{x}(s) + p\_1(s)\mathbf{x}(\varrho\_1(s)) + p\_2(s)\mathbf{x}(\varrho\_2(s)), \text{ for } s \ge s\_0.$$


$$\begin{array}{rcl} \mathcal{B}\_{1}(s) &:& = 1 - p\_{1}(s) \frac{\kappa(\varrho\_{1}(s),\infty)}{\kappa(s,\infty)} - p\_{2}(s), \\\ H(s) &:& = q\_{1}(s) \mathcal{B}\_{2}^{\mathfrak{a}}(\theta\_{1}(s)) + q\_{2}(s) \mathcal{B}\_{2}^{\mathfrak{a}}(\theta\_{2}(s)), \\\ G(s) &:& = q\_{1}(s) \mathcal{B}\_{1}^{\mathfrak{a}}(\theta\_{1}(s)) + q\_{2}(s) \mathcal{B}\_{1}^{\mathfrak{a}}(\theta\_{2}(s)) \end{array}$$

and

$$B\_2(s) := 1 - p\_1(s) - p\_2(s) \frac{\kappa(s\_{1\prime} \varrho\_2(s))}{\kappa(s\_{1\prime}s)}, \text{ for } s \ge s\_1 \ge s\_0.$$

**Lemma 1** ([6], Lemma 2.6)**.** *Assume that* <sup>Θ</sup>(*v*) := *Av* − *<sup>B</sup>*(*<sup>v</sup>* − *<sup>C</sup>*) (*α*+1)/*α , where A*, *B, and C are real constants, B* > 0*, and α* ∈ *Q* + *odd. Then, the maximum value of* <sup>Θ</sup> *on* <sup>R</sup> *at <sup>v</sup>* ∗ = *C* + (*αA*/((*α* + 1)*B*))*<sup>α</sup> is*

$$
\Theta(v^\*) \le \max\_{v \in \mathbb{R}} \Theta(v) = A\mathbb{C} + \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}} A^{\alpha+1} B^{-\alpha}.
$$

**Lemma 2.** *Let x be a positive solution of (1). If υ is decreasing, then*

$$\left(\frac{v(s)}{\kappa(s,\infty)}\right)' \ge 0,\tag{3}$$

*eventually. Further, if υ is increasing, then*

$$\left(\frac{v(s)}{\kappa(s\_{1\prime}s)}\right)' \le 0,\tag{4}$$

*for all s* ≥ *s*<sup>1</sup> ≥ *s*0.

**Proof.** Suppose that (1) has a positive solution *<sup>x</sup>* on [*s*0, <sup>∞</sup>). Obviously, *<sup>υ</sup>*(*s*) ≥ *<sup>x</sup>*(*s*) > 0 for all *s* ≥ *s*<sup>1</sup> ≥ *s*0. Thus, from (1), we get

$$\left(r(s)\left(\upsilon'(s)\right)^{\mathfrak{a}}\right)' = -q\_1(s)\mathfrak{x}^{\mathfrak{a}}(\theta\_1(s)) - q\_2(s)\mathfrak{x}^{\mathfrak{a}}(\theta\_2(s)) \le 0.1$$

Hence, *r*(*s*)(*υ* ′ (*s*))*<sup>α</sup>* is non-increasing, and so *υ* ′ (*s*) has a constant sign for *s* ≥ *s*1. Assume that *υ* ′ (*s*) < 0 on [*s*1, ∞). Then,

$$v(s) \ge -\int\_s^\infty r^{-1/a}(\xi) r^{1/a}(\xi) v'(\xi) d\xi \ge -\kappa(s,\infty) r^{1/a}(s) v'(s),\tag{5}$$

and so,

$$\left(\frac{\upsilon(s)}{\kappa(s,\infty)}\right)' = \frac{\kappa(s,\infty)\upsilon'(s) + r^{-1/\alpha}(s)\upsilon(s)}{\left(\kappa(s,\infty)\right)^2} \ge 0.$$

Next, assume that *υ* ′ (*s*) > 0 on [*s*1, ∞). Hence, we obtain

$$\upsilon(s) \ge \int\_{s\_1}^s r^{-1/a}(\xi) r^{1/a}(\xi) \upsilon'(\xi) d\xi \ge \kappa(s\_1, s) r^{1/a}(s) \upsilon'(s).$$

and it follows that

$$\left(\frac{\upsilon(s)}{\kappa(s\_1, s)}\right)' = \frac{\kappa(s\_1, s)\upsilon'(s) - r^{-1/\kappa}(s)\upsilon(s)}{\kappa^2(s\_1, s)} \le 0.1$$

Thus, the proof is complete.

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

**Theorem 1.** *Assume that H*(*s*) ≥ *G*(*s*) > 0*. If*

$$\limsup\_{s \to \infty} \int\_{s\_1}^{s} \frac{1}{r^{1/a}(u)} \left( \int\_{s\_1}^{u} G(\xi) \kappa^a(\theta\_2(\xi), \infty) d\xi \right)^{1/a} du = \infty,\tag{6}$$

*for s*<sup>1</sup> ≥ *s*0*, then all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary: that (1) has a non-oscillatory solution *x* on [*s*0, ∞). Without loss of generality (since the substitution *y* = −*x* transforms (1) into an equation of the same form), we suppose that *x* is an eventually positive solution. Then, there exists *s*<sup>1</sup> ≥ *s*<sup>0</sup> such that *x*(*̺*1(*s*)), *x*(*̺*2(*s*)), *x*(*θ*1(*s*)), and *x*(*θ*2(*s*)) are positive for all *s* ≥ *s*1. Thus, from (1) and the definition of *υ*, we note that *υ*(*s*) ≥ *x*(*s*) > 0 and *r*(*s*)(*υ* ′ (*s*))*<sup>α</sup>* is non-increasing. Hence, *υ* ′ > 0 or *υ* ′ < 0 eventually.

Assume that *υ* ′ (*s*) < 0 on [*s*1, ∞). By using Lemma 2, we have

$$\upsilon(\varrho\_1(s)) \le \frac{\kappa(\varrho\_1(s), \infty)}{\kappa(s, \infty)} \upsilon(s),$$

based on the fact that *̺*1(*s*) ≤ *s*. Therefore,

$$\begin{aligned} \varkappa(s) &= \begin{aligned} \upsilon(s) - p\_1(s)\varkappa(\varrho\_1(s)) - p\_2(s)\varkappa(\varrho\_2(s)) \\ &\ge \quad \upsilon(s) - p\_1(s)\upsilon(\varrho\_1(s)) - p\_2(s)\upsilon(\varrho\_2(s)) \\ &\ge \quad \left(1 - p\_1(s)\frac{\varkappa(\varrho\_1(s),\infty)}{\varkappa(s,\infty)} - p\_2(s)\right)\upsilon(s) \\ &= \quad B\_1(s)\upsilon(s). \end{aligned} \end{aligned}$$

Hence, (1) becomes

$$\left(r(s)\left(v'(s)\right)^a\right)' \le -q\_1(s)B\_1^a(\theta\_1(s))v^a(\theta\_1(s)) - q\_2(s)B\_1^a(\theta\_2(s))v^a(\theta\_2(s)).$$

and since *θ*1(*s*) ≤ *θ*2(*s*), we have

$$\begin{aligned} \left(r(s)\left(v'(s)\right)^{a}\right)' &\leq \ -q\_1(s)\mathcal{B}\_1^a(\theta\_1(s))v''(\theta\_2(s)) - q\_2(s)\mathcal{B}\_1^a(\theta\_2(s))v''(\theta\_2(s))\\ &\leq \ -(q\_1(s)\mathcal{B}\_1^a(\theta\_1(s)) + q\_2(s)\mathcal{B}\_1^a(\theta\_2(s)))v''(\theta\_2(s))\\ &= \ -G(s)v''(\theta\_2(s)). \end{aligned} \tag{7}$$

Since *r*(*s*)(*υ* ′ (*s*))*<sup>α</sup>* ′ <sup>≤</sup> 0, we have

$$r(s)\left(v'(s)\right)^a \le r(s\_1)\left(v'(s\_1)\right)^a := -L < 0,\tag{8}$$

for all *s* ≥ *s*1, and from (5) and (8), we have

$$
\upsilon^{\mathfrak{a}}(s) \ge L\kappa^{\mathfrak{a}}(s,\infty) \text{ for all } s \ge s\_1. \tag{9}
$$

Combining (7) with (9) yields

$$\left(r(s)\left(v'(s)\right)^a\right)' \le -G(s)L\kappa^a(\theta\_2(s),\infty),\tag{10}$$

for all *s* ≥ *s*1. Integrating (10) from *s*<sup>1</sup> to *s*, we obtain

$$\begin{aligned} r(s) \left(v'(s)\right)^{\alpha} &\leq \quad r(s\_1) \left(v'(s\_1)\right)^{\alpha} - L \int\_{s\_1}^{s} G(\xi) \kappa^{\alpha}(\theta\_2(\xi), \infty) d\xi \\ &\leq \quad -L \int\_{s\_1}^{s} G(\xi) \kappa^{\alpha}(\theta\_2(\xi), \infty) d\xi. \end{aligned}$$

Integrating the last inequality from *s*<sup>1</sup> to *s*, we get

$$\upsilon(s) \le \upsilon(s\_1) - L^{1/\alpha} \int\_{s\_1}^{s} \frac{1}{r^{1/\alpha}(u)} \left( \int\_{s\_1}^{u} G(\xi) \kappa^{\kappa}(\theta\_2(\xi), \infty) d\xi \right)^{1/\alpha} d\mu.$$

Passing to the limit as *<sup>s</sup>* → <sup>∞</sup>, we arrive at a contradiction with (6). Now, assume that *υ* ′ (*s*) > 0 on [*s*1, ∞). From Lemma 2, we arrive at

$$
\upsilon(\varrho\_2(s)) \le \frac{\kappa(s\_1, \varrho\_2(s))}{\kappa(s\_1, s)} \upsilon(s). \tag{11}
$$

From the definition of *υ*, we obtain

$$\begin{aligned} x(s) &=& v(s) - p\_1(s)x(\varrho\_1(s)) - p\_2(s)x(\varrho\_2(s)) \\ &\ge& v(s) - p\_1(s)v(\varrho\_1(s)) - p\_2(s)v(\varrho\_2(s)). \end{aligned} \tag{12}$$

Using that (11) and *υ*(*̺*1(*s*)) ≤ *υ*(*s*), where *̺*1(*s*) < *s* in (12), we obtain

$$\begin{aligned} \varkappa(s) &\geq & v(s) \Big( 1 - p\_1(s) - p\_2(s) \frac{\kappa(s\_1, \varrho\_2(s))}{\kappa(s\_1, s)} \Big) \\ &\geq & B\_2(s) v(s). \end{aligned} \tag{13}$$

Hence, (1) becomes

$$\left(r(s)\left(v'(s)\right)^a\right)' \le -q\_1(s)B\_2^a(\theta\_1(s))v^a(\theta\_1(s)) - q\_2(s)B\_2^a(\theta\_2(s))v^a(\theta\_2(s)),$$

and since *θ*1(*s*) ≤ *θ*2(*s*), we have

$$\begin{split} \left( r(s) \left( v'(s) \right)^{a} \right)' &\leq \ -q\_1(s) \mathcal{B}\_2^a(\theta\_1(s)) v^a(\theta\_1(s)) - q\_2(s) \mathcal{B}\_2^a(\theta\_2(s)) v^a(\theta\_1(s)) \\ &\leq \ -(q\_1(s) \mathcal{B}\_2^a(\theta\_1(s)) + q\_2(s) \mathcal{B}\_2^a(\theta\_2(s))) v^a(\theta\_1(s)) \\ &= \ -H(s) v^a(\theta\_1(s)). \end{split} \tag{14}$$

On the other hand, it follows from (6) and (C2) that R *<sup>s</sup> s*1 *G*(*ξ*)*κ α* (*θ*2(*ξ*), ∞)d*ξ* must be unbounded. Further, since *κ* ′ (*s*, ∞) < 0, it is easy to see that

$$\int\_{s\_1}^{s} \mathbf{G}(\xi) \mathbf{d}\xi \to \infty \text{ as } s \to \infty. \tag{15}$$

Integrating (14) from *s*<sup>2</sup> to *s*, we get

$$\begin{aligned} r(s) \left(v'(s)\right)^{\mathfrak{a}} &\leq \quad r(s\_2) \left(v'(s\_2)\right)^{\mathfrak{a}} - \int\_{s\_2}^{s} H(\xi) v^{\mathfrak{a}}(\theta\_1(\xi)) \mathbf{d}\xi \\ &\leq \quad r(s\_2) \left(v'(s\_2)\right)^{\mathfrak{a}} - v^{\mathfrak{a}}(\theta\_1(s\_2)) \int\_{s\_2}^{s} H(\xi) \mathbf{d}\xi. \end{aligned}$$

Since *H*(*s*) > *G*(*s*), we get

$$r(s)\left(v'(s)\right)^a \le r(s\_2)\left(v'(s\_2)\right)^a - v^a(\theta\_1(s\_2))\int\_{s\_2}^s \mathcal{G}(\xi)d\xi.\tag{16}$$

which, with (15), contradicts the fact that *υ* ′ (*s*) > 0. The proof is complete.

**Theorem 2.** *Assume that H*(*s*) ≥ *G*(*s*) > 0*. If*

$$\limsup\_{s \to \infty} \mathbb{x}^{\mathfrak{a}}(\theta\_2(s), \infty) \int\_{s\_1}^{s} \mathbb{G}(\xi) \, \mathrm{d}\xi > 1,\tag{17}$$

*then all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary: that (1) has a non-oscillatory solution *x* on [*s*0, ∞). Without loss of generality (since the substitution *y* = −*x* transforms (1) into an equation of the same form), we suppose that *x* is an eventually positive solution. Then, there exists *s*<sup>1</sup> ≥ *s*<sup>0</sup> such that *x*(*̺*1(*s*)) > 0, *x*(*̺*2(*s*)) > 0, *x*(*θ*1(*s*)) > 0, and *x*(*θ*2(*s*)) > 0 for all *s* ≥ *s*1. As in the proof of Theorem 1, *υ* ′ > 0 or *υ* ′ < 0 eventually.

Assume that *υ* ′ < 0 on [*s*1, ∞). Integrating (7) from *s*<sup>1</sup> to *s*, we get

$$r(s)\left(v'(s)\right)^{\alpha} \leq \left.r(s\_1)\left(v'(s\_1)\right)^{\alpha} - \int\_{s\_1}^{s} G(\xi)v^{\alpha}(\theta\_2(\xi))d\xi\right|$$

$$\leq \left.-v''(\theta\_2(s))\int\_{s\_1}^{s} G(\xi)d\xi. \tag{18}$$

Since *θ*2(*s*) ≥ *s*, then from (3), we have

$$
\upsilon(\theta\_2(s)) \ge \frac{\kappa(\theta\_2(s), \infty)}{\kappa(s, \infty)} \upsilon(s), \tag{19}
$$

and using (19) and (5) in (18), we obtain

$$r(s)\left(\upsilon'(s)\right)^a \le r(s)\left(\upsilon'(s)\right)^a \kappa^a(\theta\_2(s),\infty) \int\_{s\_1}^s \mathcal{G}(\xi)d\xi.\tag{20}$$

Dividing both sides of inequality (20) by *r*(*s*)(*υ* ′ (*s*))*<sup>α</sup>* and taking the *limsup*, we get

$$\limsup\_{s \to \infty} \kappa^{\mathfrak{a}}(\theta\_2(s), \infty) \int\_{s\_1}^{s} G(\xi) \mathbf{d}\xi \le 1,$$

we obtain a contradiction with the condition (17).

Now, assume that *υ* ′ > 0 on [*s*1, ∞). From (17) and the fact that *κ*(*θ*2(*s*), ∞) < ∞, we have that (15) holds. Then, this part of the proof is similar to that of Theorem 1. Therefore, the proof is complete.

**Theorem 3.** *Assume that H*(*s*) ≥ *G*(*s*) > 0 *and (15) hold. Further, if the differential equation*

$$\left(\upsilon'(s) + \frac{1}{r^{1/a}(s)} \frac{\kappa(\theta\_2(s), \infty)}{\kappa(\theta\_1(s), \infty)}\right) \left(\int\_{s\_1}^s G(\xi) d\xi\right)^{1/a} \upsilon(\theta\_1(s)) = 0 \tag{21}$$

*is oscillatory, then all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary: that (1) has a non-oscillatory solution *x* on [*s*0, ∞). Without loss of generality (since the substitution *y* = −*x* transforms (1) into an equation of the same form), we suppose that *x* is an eventually positive solution. Then, there exists *s*<sup>1</sup> ≥ *s*<sup>0</sup> such that *x*(*̺*1(*s*)) > 0, *x*(*̺*2(*s*)) > 0, *x*(*θ*1(*s*)) > 0, and *x*(*θ*2(*s*)) > 0 for all *s* ≥ *s*1. As in the proof of Theorem 1, *υ* ′ > 0 or *υ* ′ < 0 eventually.

Assume that *υ* ′ < 0 on [*s*1, <sup>∞</sup>). Since *<sup>θ</sup>*2(*s*) ≥ *<sup>θ</sup>*1(*s*), we get, from (3), that

$$\upsilon(\theta\_2(s)) \ge \frac{\kappa(\theta\_2(s), \infty)}{\kappa(\theta\_1(s), \infty)} \upsilon(\theta\_1(s))\nu$$

which, with (18), gives

$$\left(r(s)\left(\upsilon'(s)\right)^{\alpha}\leq \frac{\kappa^{\alpha}(\theta\_{2}(s),\infty)}{\kappa^{\alpha}(\theta\_{1}(s),\infty)}\upsilon^{\alpha}(\theta\_{1}(s))\int\_{s\_{1}}^{s}G(\boldsymbol{\xi})\,\mathrm{d}\boldsymbol{\xi}\right)$$

Now, we see that *υ* > 0 is a solution of the inequality

$$\left(\upsilon'(s) + \frac{1}{r^{1/a}(s)} \frac{\kappa(\theta\_2(s), \infty)}{\kappa(\theta\_1(s), \infty)} \left(\int\_{s\_1}^s G(\xi) d\xi\right)^{1/a} \upsilon(\theta\_1(s)) \le 0.\right)$$

Using [34], we find that (21) also has a positive solution—a contradiction. By proceeding as in the proof of Theorem 1, the proof of this theory is completed.

**Corollary 1.** *Assume that H*(*s*) ≥ *G*(*s*) > 0 *and (15) hold. If*

$$\liminf\_{s \to \infty} \int\_{\theta\_1(s)}^s \frac{1}{r^{1/a}(u)} \frac{\kappa(\theta\_2(u), \infty)}{\kappa(\theta\_1(u), \infty)} \left( \int\_{s\_1}^u G(\xi) d\xi \right)^{1/a} du > \frac{1}{\mathbf{e}'} \tag{22}$$

*then all solutions of (1) are oscillatory.*

**Proof.** Using ([35], Theorem 2), we have that (22) implies the oscillation of (21). From Theorem 3, we have that (1) is oscillatory.

**Theorem 4.** *Assume that H*(*s*) > 0, *G*(*s*) > 0*. If there exist functions ψ*, *δ* ∈ *C* 1 ([*s*0, ∞),(0, ∞))*, and s*<sup>1</sup> ∈ [*s*0, <sup>∞</sup>) *such that*

$$\limsup\_{s \to \infty} \left\{ \frac{\kappa^a(s, \infty)}{\delta(s)} \int\_{s\_1}^s \left( \delta(\tilde{\xi}) G(\tilde{\xi}) \frac{\kappa^a(\theta\_2(\tilde{\xi}), \infty)}{\kappa^a(\tilde{\xi}, \infty)} - \frac{r(\tilde{\xi}) (\delta'(\tilde{\xi}))^{a+1}}{(a+1)^{a+1} (\delta(\tilde{\xi}))^a} \right) d\xi \right\} > 1 \tag{23}$$

*and*

$$\limsup\_{s \to \infty} \int\_{s\_1}^{s} \left( \psi(\xi) H(\xi) - \frac{1}{(a+1)^{a+1}} \frac{r(\xi) \left( \psi'(\xi) \right)^{a+1}}{\psi^a(\xi) \left( \theta\_1'(\xi) \right)^a} \right) d\xi = \infty,\tag{24}$$

*then all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary: that (1) has a non-oscillatory solution *x* on [*s*0, ∞). Without loss of generality (since the substitution *y* = −*x* transforms (1) into an equation of the same form), we suppose that *x* is an eventually positive solution. Then, there exists *s*<sup>1</sup> ≥ *s*<sup>0</sup> such that *x*(*̺*1(*s*)) > 0, *x*(*̺*2(*s*)) > 0, *x*(*θ*1(*s*)) > 0, and *x*(*θ*2(*s*)) > 0 for all *s* ≥ *s*1. From Theorem 1, *υ* ′ > 0 or *υ* ′ < 0 eventually.

Assume that *υ* ′ < 0 on [*s*1, ∞). As in the proof of Theorem 1, we arrive at (7). Now, we define the function

$$
\omega(s) = \delta(s) \left( \frac{r(s) \left(v'(s)\right)^a}{v^a(s)} + \frac{1}{\kappa^a(s,\infty)} \right) \text{ on } [s\_1,\infty). \tag{25}
$$

From (5), we get that *<sup>ω</sup>* ≥ 0 on [*s*1, <sup>∞</sup>). Differentiating (25), we obtain

$$\begin{split} \omega'(s) &= \quad \frac{\delta'(s)}{\delta(s)}\omega(s) + \delta(s) \frac{\left(r(s)\left(v'(s)\right)^{\kappa}\right)'}{v^{\mu}(s)} - a\delta(s)r(s) \left(\frac{v'(s)}{v(s)}\right)^{\kappa+1} \\ &+ \frac{a\delta(s)}{r^{1/\kappa}(s)\kappa^{\alpha+1}(s,\infty)} \\ &\leq \quad \frac{\delta'(s)}{\delta(s)}\omega(s) + \delta(s) \frac{\left(r(s)\left(v'(s)\right)^{\kappa}\right)'}{v^{\mu}(s)} - \frac{a}{\left(\delta(s)r(s)\right)^{1/\kappa}} \left(\omega(s) - \frac{\delta(s)}{\kappa^{\alpha}(s,\infty)}\right)^{(\kappa+1)/\kappa} \\ &+ \frac{a\delta(s)}{r^{1/\kappa}(s)\kappa^{\alpha+1}(s,\infty)}. \end{split} \tag{26}$$

Combining (7) and (26), we have

$$\begin{split} \omega'(s) &\leq \ -\frac{\mathfrak{a}}{\left(\delta(s)r(s)\right)^{1/a}} \Big(\omega(s) - \frac{\delta(s)}{\kappa^a(s,\infty)}\Big)^{(a+1)/a} - \delta(s)G(s)\frac{v^a(\theta\_2(s))}{v^a(s)} \\ &+ \frac{\mathfrak{a}\delta(s)}{r^{1/a}(s)\kappa^{a+1}(s,\infty)} + \frac{\delta'(s)}{\delta(s)}\omega(s). \end{split} \tag{27}$$

Using Lemma 1 with *A* := *δ* ′ (*s*)/*δ*(*s*), *B* := *α*(*δ*(*s*)*r*(*s*))−1/*<sup>α</sup>* , *C* := *δ*(*s*)/*κ α* (*s*, ∞) and *ξ* := *ω*, we get

$$\begin{split} \frac{\delta'(s)}{\delta(s)}\omega(s) - \frac{a}{\left(\delta(s)r(s)\right)^{1/a}} \left(\omega(s) - \frac{\delta(s)}{\kappa^a(s,\infty)}\right)^{(a+1)/a} &\leq \frac{r(s)}{\left(a+1\right)^{a+1}} \frac{\left(\delta'(s)\right)^{a+1}}{\left(\delta(s)\right)^a} \\ &+ \frac{\delta'(s)}{\kappa^a(s,\infty)}, \end{split} \tag{28}$$

and since *s* ≤ *θ*2(*s*), we arrive at

$$
v(\theta\_2(s)) \ge \frac{\kappa(\theta\_2(s), \infty)}{\kappa(s, \infty)} v(s),\tag{29}$$

which, in view of (27), (28), and (29), gives

$$\begin{split} \omega^{\prime}(s) &\leq \quad \frac{\delta^{\prime}(s)}{\kappa^{a}(s,\infty)} + \frac{1}{(\mathfrak{a}+1)^{a+1}} r(s) \frac{(\delta^{\prime}(s))^{a+1}}{(\delta(s))^{a}} - \delta(s) G(s) \frac{v^{a}(\theta\_{2}(s))}{v^{a}(s)} \\ &+ \frac{a\delta(s)}{r^{1/a}(s)\kappa^{a+1}(s,\infty)} \\ &\leq \quad -\delta(s) G(s) \frac{\kappa^{a}(\theta\_{2}(s),\infty)}{\kappa^{a}(s,\infty)} + \left(\frac{\delta(s)}{\kappa^{a}(s,\infty)}\right)^{\prime} + \frac{r(s)(\delta^{\prime}(s))^{a+1}}{(a+1)^{a+1}(\delta(s))^{a}}. \end{split}$$

Integrating (30) from *s*<sup>2</sup> to *s*, we arrive at

$$\begin{split} \left| \int\_{s\_2}^{s} \Big( \delta(\xi) G(\xi) \frac{\kappa^a(\theta\_2(\xi), \infty)}{\kappa^a(\underline{\xi}, \infty)} - \frac{r(\xi) (\delta'(\underline{\xi}))^{a+1}}{(a+1)^{a+1} (\delta(\underline{\xi}))^a} \Big) d\underline{\xi} \right| & \leq \left. \left( \frac{\delta(s)}{\kappa^a(s, \infty)} - \omega(s) \right) \right|\_{s\_2}^{s} \\ & \leq \left. - \left( \delta(s) \frac{r(s) (\upsilon'(s))^a}{\upsilon^a(s)} \right) \right|\_{s\_2}^{s} (\mathbf{31}) . \end{split} \tag{31}$$

From (5), we have

$$-\frac{r^{1/\kappa}(s)v'(s)}{v(s)} \le \frac{1}{\kappa(s,\infty)'} $$

which, in view of (31), implies

$$\frac{\kappa^{\mathfrak{a}}(s,\infty)}{\delta(s)} \int\_{s\_2}^{s} \Big( \delta(\xi) G(\xi) \frac{\kappa^{\mathfrak{a}}(\theta\_2(\xi),\infty)}{\kappa^{\mathfrak{a}}(\xi,\infty)} - \frac{r(\xi)(\delta'(\xi))^{\mathfrak{a}+1}}{(\mathfrak{a}+1)^{\mathfrak{a}+1}(\delta(\xi))^{\mathfrak{a}}} \Big) d\xi \le 1.$$

Thus, we get a contradiction with (23).

Now, assume that *υ* ′ (*s*) > 0 on [*s*1, ∞). Let us define the Riccati function

$$\varphi(s) = \psi(s) \frac{r(s)(\upsilon'(s))^{\alpha}}{\upsilon^{\alpha}(\theta\_1(s))}, \text{ on } [s\_1, \infty). \tag{32}$$

We find that *<sup>ϕ</sup>* ≥ 0 on [*s*1, <sup>∞</sup>). Differentiating (32), we get

$$\varphi'(s) = \frac{\psi'(s)}{\psi(s)}\varphi(s) + \psi(s)\frac{\left(r(s)(\upsilon'(s))^\kappa\right)'}{\upsilon^a(\theta\_1(s))} - a\psi(s)r(s)\frac{(\upsilon'(s))^\kappa\upsilon'(\theta\_1(s))\theta\_1'(s)}{\upsilon^{a+1}(\theta\_1(s))}.\tag{33}$$

Combining (14) and (33), we have

$$
\varphi'(s) \le \frac{\psi'(s)}{\psi(s)} \varphi(s) - \psi(s)H(s) - \alpha \psi(s) r(s) \frac{(\upsilon'(s))^\kappa \upsilon'(\theta\_1(s)) \theta\_1'(s)}{\upsilon^{\kappa+1}(\theta\_1(s))}.
$$

Since *r*(*s*)(*υ* ′ (*s*))*<sup>α</sup>* ′ <sup>&</sup>lt; 0 and *<sup>θ</sup>*1(*s*) <sup>≤</sup> *<sup>s</sup>*, we arrive at

$$
\varphi'(s) \le \frac{\psi'(s)}{\psi(s)} \varphi(s) - \psi(s)H(s) - \kappa \psi(s)r(s)\theta\_1'(s) \frac{\left(\upsilon'(s)\right)^{a+1}}{\upsilon^{a+1} \left(\theta\_1(s)\right)}.
$$

and from (32), we have

$$
\varphi'(s) \le \frac{\psi'(s)}{\psi(s)} \varphi(s) - \psi(s) H(s) - \frac{a \theta\_1'(s)}{\psi^{1/a}(s) r^{1/a}(s)} \phi^{(a+1)/a}(s).
$$

Using the inequality

$$Kv - sv^{(a+1)/a} \le \frac{\mathfrak{a}^a}{(\mathfrak{a}+1)^{a+1}} \frac{K^{a+1}}{s^a}, s > 0,\tag{34}$$

with *K* = *ψ* ′ (*s*)/*ψ*(*s*), *s* = *αθ*′ 1 (*s*)/*ψ* 1/*α* (*s*)*r* 1/*α* (*s*), and *v* = *ϕ*, we have

$$
\varphi'(s) \le -\psi(s)H(s) + \frac{1}{(\alpha+1)^{\alpha+1}} \frac{r(s) \left(\psi'(s)\right)^{\alpha+1}}{\psi^{\alpha}(s) \left(\theta\_1'(s)\right)^{\alpha}}.\tag{35}
$$

Integrating (35) from *s*<sup>2</sup> to *s*, we arrive at

$$\int\_{s\_2}^{s} \left( \psi(\xi) H(\xi) - \frac{1}{\left(\alpha + 1\right)^{\alpha + 1}} \frac{r(\xi) \left(\psi'(\xi)\right)^{\alpha + 1}}{\psi^{\alpha}(\xi) \left(\theta\_1'(\xi)\right)^{\alpha}} \right) d\xi \le \varphi(s\_2).$$

Taking the limsup on both sides of this inequality, we have a contradiction with (24). The proof of the theorem is complete.

**Theorem 5.** *Assume that H*(*s*) > 0 *and G*(*s*) > 0*. If there exist the functions δ* ∈ *C* 1 ([*s*0, ∞),(0, ∞)) *and s*<sup>1</sup> ∈ [*s*0, <sup>∞</sup>) *such that (23) and*

$$\liminf\_{s \to \infty} \frac{a}{\Psi(s)} \int\_s^\infty \frac{\theta\_1'(\xi)}{r^{1/a}(\xi)} \Psi^{(a+1)/a}(\xi) d\xi > \frac{a}{(a+1)^{(a+1)/a}} \tag{36}$$

*hold, where*

$$\Psi(s) = \int\_s^\infty H(\mathfrak{f}) \, \mathrm{d}\mathfrak{f}\_\prime$$

*then all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary: that (1) has a non-oscillatory solution *x* on [*s*0, ∞). Without loss of generality (since the substitution *y* = −*x* transforms (1) into an equation of the same form), we suppose that *x* is an eventually positive solution. Then, there exists *s*<sup>1</sup> ≥ *s*<sup>0</sup> such that *x*(*̺*1(*s*)) > 0, *x*(*̺*2(*s*)) > 0, *x*(*θ*1(*s*)) > 0, and *x*(*θ*2(*s*)) > 0 for all *s* ≥ *s*1. Theorem 1 yields that *υ* ′ eventually has one sign.

Assume that *υ* ′ (*s*) < 0 on [*s*1, ∞). The proof is similar to that of Theorem 4. Now, assume that *υ* ′ (*s*) > 0 on [*s*1, ∞).Let us define the Riccati function

$$\varphi(s) = \frac{r(s)(v'(s))^\alpha}{v^\alpha(\theta\_1(s))}.\tag{37}$$

We see that *<sup>ϕ</sup>* ≥ 0 on [*s*1, <sup>∞</sup>). Differentiating (37), we arrive at

$$\varphi'(s) = \frac{\left(r(s)(v'(s))^a\right)'}{v^a(\theta\_1(s))} - ar(s)\frac{(v'(s))^a v'(\theta\_1(s))\theta\_1'(s)}{v^{a+1}(\theta\_1(s))}.\tag{38}$$

Combining (14) and (38), we have

$$
\varrho'(s) \le -H(s) - \alpha r(s) \frac{(\upsilon'(s))^\alpha \upsilon'(\theta\_1(s)) \theta\_1'(s)}{\upsilon^{a+1}(\theta\_1(s))}.
$$

Since *r*(*s*)(*υ* ′ (*s*))*<sup>α</sup>* ′ <sup>&</sup>lt; 0 and *<sup>θ</sup>*1(*s*) <sup>≤</sup> *<sup>s</sup>*, we arrive at

$$
\phi'(s) \le -H(s) - \mathfrak{a}r(s)\theta\_1'(s)\frac{\left(\upsilon'(s)\right)^{\mathfrak{a}+1}}{\upsilon^{\mathfrak{a}+1}(\theta\_1(s))}.
$$

which, with (37), gives

$$
\varphi'(s) \le -H(s) - \frac{a\theta\_1'(s)}{r^{1/\alpha}(s)} \varphi^{(a+1)/\alpha}(s). \tag{39}
$$

Integrating (39) from *s* to ∞, and using the fact that *ϕ*(*s*) > 0 and *ϕ* ′ (*s*) < 0, we get

$$-\varphi(s) \le -\int\_s^\infty H(\xi) \mathrm{d}\xi - \int\_s^\infty \frac{a \theta\_1'(\xi)}{r^{1/\alpha}(\xi)} \varphi^{(\alpha+1)/\alpha}(\xi) \,\mathrm{d}\xi.$$

Hence, we have

$$\frac{\varrho(s)}{\Psi(s)} \ge 1 + \frac{1}{\Psi(s)} \int\_s^\infty \frac{a \theta\_1'(\xi)}{r^{1/a}(\xi)} \Psi^{(a+1)/a}(\xi) \left(\frac{\varrho(\xi)}{\Psi(\xi)}\right)^{(a+1)/a} d\xi. \tag{40}$$

(*α*+1)/*<sup>α</sup>*

,

Let *<sup>ϑ</sup>* = inf*s*≥*<sup>s</sup> <sup>ϕ</sup>*(*s*)/Ψ(*s*);then, obviously, *<sup>ϑ</sup>* ≥ 1. Hence, it follows from (40) and (36) that

> *ϑ α* + 1

or

$$\frac{\vartheta}{\alpha+1} \ge \frac{1}{\alpha+1} + \frac{\alpha}{\alpha+1} \left(\frac{\vartheta}{\alpha+1}\right)^{(\alpha+1)/\alpha}$$

which contradicts the admissible value of *ϑ* and *α*. Therefore, the proof is complete.

**Corollary 2.** *Assume that H*(*s*) > 0 *and G*(*s*) > 0. *If (36) and either*

*ϑ* ≥ 1 + *α*

$$\limsup\_{s \to \infty} \int\_{s}^{s} \left( G(\xi) \kappa^{a}(\theta\_{2}(\xi), \infty) - \frac{\kappa^{a+1}}{(a+1)^{a+1} r^{1/a}(\xi) \kappa(\xi, \infty)} \right) d\xi > 1,\tag{41}$$

$$\limsup\_{s\to\infty} \kappa^{a-1}(s,\infty) \int\_{s}^{s} \left( G(\xi) \frac{\kappa^{a}(\theta\_{2}(\xi),\infty)}{\kappa^{a-1}(\xi,\infty)} - \frac{1}{(a+1)^{a+1}r^{1/a}(\xi)\kappa^{a}(\xi,\infty)} \right) d\xi > 1,\tag{42}$$

*or*

$$\limsup\_{s \to \infty} \kappa^{\mathfrak{a}}(s, \infty) \int\_{s}^{s} G(\boldsymbol{\xi}) \frac{\kappa^{\mathfrak{a}}(\theta\_{2}(\boldsymbol{\xi}), \infty)}{\kappa^{\mathfrak{a}}(\boldsymbol{\xi}, \infty)} \mathrm{d}\boldsymbol{\xi} > 1,\tag{43}$$

*hold, then all solutions of (1) are oscillatory.*

**Proof.** By choosing *δ*(*s*) = *κ α* (*s*, ∞), *δ*(*s*) = *κ*(*s*, ∞), or *δ*(*s*) = 1, the condition (23) reduces to one of the conditions (41)–(43), respectively.

**Example 1.** *Consider the second-order neutral differential equation*

$$\left(s^2\left(\mathbf{x}(s) + p\_1^\*\mathbf{x}\left(\frac{s}{\lambda}\right) + p\_2^\*\mathbf{x}(\lambda s)\right)'\right)' + q\_1^\*\mathbf{x}\left(\frac{s}{\mu}\right) + q\_2^\*\mathbf{x}(\mu s) = 0,\tag{44}$$

*where s* ≥ 1, *λ* ≥ 1, *µ* ≥ 1, *p* ∗ <sup>1</sup> > *p* ∗ 2 *, and λ p* ∗ <sup>1</sup> + *p* ∗ 2 ∈ (0, 1)*. Now, we note that r*(*s*) = *s* 2 , *p*1(*s*) = *p* ∗ 1 , *p*2(*s*) = *p* ∗ 2 , *̺*1(*s*) = *s*/*λ*, *̺*2(*s*) = *λs*, *q*1(*s*) = *q* ∗ 1 , *q*2(*s*) = *q* ∗ 2 , *θ*1(*s*) = *s*/*µ*, *and θ*2(*s*) = *µs.Thus, we have that*

$$B\_1(s) = 1 - \lambda p\_1^\* - p\_{2'}^\* \quad B\_2(s) = 1 - p\_1^\* - p\_2^\* \left(\frac{s - \frac{1}{\lambda}}{s - 1}\right)$$

*and*

$$G(s) = (q\_1^\* + q\_2^\*)(1 - \lambda p\_1^\* - p\_2^\*).$$

*Set W*(*s*) = *<sup>s</sup>* <sup>−</sup> <sup>1</sup> *λ* /(*s* − 1)*. Since* lim*s*→<sup>∞</sup> *W*(*s*) = 1*, there exists s<sup>ǫ</sup>* > *s*<sup>0</sup> *such that W*(*s*) < 1 + *ǫ for all ǫ* > 0 *and every s* ≥ *sǫ. By choosing ǫ* = *λ* − 1, *we obtain W*(*s*) < *λ for all* *s* ≥ *s*∗*. Thus, and taking into account the fact that p* ∗ <sup>1</sup> > *p* ∗ 2 *and λp* ∗ <sup>1</sup> + *p* ∗ <sup>2</sup> ∈ (0, 1)*, we get that B*<sup>2</sup> ≥ *B*<sup>1</sup> > 0*. Now, from Theorem 2, we have that equation (44) is oscillatory if*

$$q\_1^\* + q\_2^\* > \frac{\mu}{1 - \lambda p\_1^\* - p\_2^\*}. \tag{C1}$$

*On the other hand, using Corollary 1, we see that (44) is oscillatory if*

$$q\_1^\* + q\_2^\* > \frac{\mu}{\left(1 - \lambda p\_1^\* - p\_2^\*\right)} \frac{\mu}{\text{e} \ln \mu}.\tag{C2}$$

*Next, since W*(*s*) < *λ for all s* ≥ *s*∗*, we find that B*2(*s*) > 1 − *p* ∗ <sup>1</sup> + *λp* ∗ 2 *, and so, H*(*s*) > *q* ∗ <sup>1</sup> + *q* ∗ 2 <sup>1</sup> <sup>−</sup> *p* ∗ <sup>1</sup> + *λp* ∗ 2 *.Hence, by choosing ψ*(*s*) = 1*, condition (24) holds, directly. Using Theorem 4, we see that (44) is oscillatory if*

$$q\_1^\* + q\_2^\* > \frac{1}{4} \frac{\mu}{\left(1 - \lambda \left. p\_1^\* - p\_2^\* \right)}. \tag{C3}$$

**Remark 1.** *Taking the fact that µ* > *e*ln *µ into account, it is easy to notice that condition (C3) supports the most efficient condition for oscillation of (44). Figures 1 and 2 display a comparison of the criteria (C1)–(C3).*

**Figure 1.** Comparison of the criteria (C1)–(C3) when *λ* = 2, *p* ∗ <sup>1</sup> = 0.25, and *p* ∗ <sup>2</sup> = 0

**Figure 2.** Comparison of the criteria (C1)–(C3) when *µ* = 2, *p* ∗ <sup>1</sup> = 0.5, and *p* ∗ <sup>2</sup> = 0

**Remark 2.** *In the special case of (44), p*∗ <sup>2</sup> = *q* ∗ <sup>2</sup> = 0*, that is,*

$$\left(s^2 \left(\mathfrak{x}(s) + p\_1^\* \mathfrak{x}\left(\frac{s}{\lambda}\right)\right)'\right)' + q\_1^\* \mathfrak{x}\left(\frac{s}{\mu}\right) = 0.$$

*The oscillation criterion (C3) reduces to*

$$q\_1^\* > \frac{1}{4} \frac{\mu}{\left(1 - \lambda p\_1^\*\right)} \,\tag{45}$$

*which is the exact criterion that was obtained in Example 3.1 in [7]. Moreover, if p* ∗ <sup>1</sup> = 0 *and µ* = 1*, then condition (45) reduces so that q* ∗ <sup>1</sup> > 1/4*, which is a sharp condition for oscillation of the second-order Euler equation.*

#### **4. Conclusions**

Most works that studied the oscillatory behavior of mixed equations regarded the canonical case *π*(*l*0) = ∞. Likewise, works that were concerned with the non-canonical case of neutral equations obtained two conditions for testing the oscillation. In this paper, we focused on studying the non-canonical case, and we created criteria with only one condition that is easy to verify. Therefore, our results are an extension, complement, and improvement to previous results in the literature. It is interesting to extend the results of this paper to higher-order equations.

**Author Contributions:** Formal analysis, O.M., A.M. and S.S.S.; Investigation, O.M., A.M. and S.S.S.; Methodology, O.M.; Writing—original draft, A.M. and S.S.S.; Writing—review and editing, A.M. and O.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** There was no external funding for this article.

**Acknowledgments:** The authors present their sincere thanks to the editors and two anonymous referees.

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

#### **References**


### *Article* **Hyperbolic Center of Mass for a System of Particles in a Two-Dimensional Space with Constant Negative Curvature: An Application to the Curved 2-Body Problem**

**Pedro Pablo Ortega Palencia <sup>1</sup> , Ruben Dario Ortiz Ortiz 2,\* and Ana Magnolia Marin Ramirez <sup>2</sup>**


**Abstract:** In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere L 1 *R* .

**Keywords:** center of mass; conformal metric; geodesic; hyperbolic lever law

**Citation:** Ortega Palencia, P.P.; Ortiz Ortiz, R.D.; Marín Ramírez, A.M. Hyperbolic Center of Mass for a System of Particles in a Two-Dimensional Space with Constant Negative Curvature: An Application to the Curved 2-Body Problem. *Mathematics* **2021**, *9*, 531. https://doi.org/10.3390/math9050531

Academic Editor: Ioannis Dassios

Received: 17 December 2020 Accepted: 27 February 2021 Published: 3 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

The center of mass (center of gravity or centroid) is a fundamental concept, and its geometrical and mechanical properties are for understanding many physical phenomena. Its definition for Euclidean spaces is elemental; nevertheless, in spaces with the non-zero curvature, it is rare. In [1], the author gives an extensive explanation showing that the possibility of the concept can be correctly defined in more general spaces, and he signalizes the difficulties in defining spaces of non-zero curvature concerning the lack of the linear structure of ones. While it is true that the author synthesizes the basic properties of the center of mass, in his approach there are some entities without physical meaning, such as the non-conservation of the total mass of the system or the presence of unbounded speeds under normal conditions. In [2], there is a definition of center of mass for two particles in hyperbolic space, in the same direction to the one presented here, but the authors do not give an expression for calculating it. In [3], the author mentions the difficulty of defining the center of mass in curved spaces. He provides a class of orbits in the curved *n*-body problem for which "no point that could play the role of the center of mass is fixed or moves uniformly along a geodesic". This proves that the equations of motion lack center-of-mass and linear momentum integrals. Nevertheless, he does not provide a way to calculate or determine this element. In [4], the center of the mass problem on two-point homogeneous spaces and the connection of existing mass center concepts with the two-body Hamiltonian functions are considered. We discussed different possibilities for defining a center of the mass in spaces of constant and non-zero curvature, and it was established that a natural way of defining a concept of center of mass for two particles on a Riemannian space as the point on the shortest geodesic interval joining these particles that divides the interval in the ratio of the masses of particles and this is denoted by *R*1.

This last approach is followed in the present work.

In this article, the problem of finding a mathematical expression for computing the center of mass of a system of *n* particles sited on the two-dimensional hyperbolic sphere L 2 *<sup>R</sup>* = {(*x*, *y*, *z*) : *x* <sup>2</sup> + *y* <sup>2</sup> <sup>−</sup> *<sup>z</sup>* <sup>2</sup> <sup>=</sup> <sup>−</sup>*<sup>R</sup>* <sup>2</sup>} is considered. The stereographic projection of the upper sheet of L 2 *R* on the Poincaré disk D<sup>2</sup> *<sup>R</sup>* = {(*x*, *y*) : *x* <sup>2</sup> + *y* <sup>2</sup> < *R* <sup>2</sup>} <sup>=</sup> {*<sup>w</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*w*<sup>|</sup> <sup>&</sup>lt; *<sup>R</sup>*}, endowed with the conformal metric (see [5]).

$$ds^2 = \frac{4\mathcal{R}^4 \, dwd\overline{w}}{(\mathcal{R}^2 - |w|^2)^2} \tag{1}$$

Both D<sup>2</sup> *<sup>R</sup>* with the metric (1) and <sup>L</sup> 2 *<sup>R</sup>* with the Euclidean metric have the same Gaussian curvature *k* = −1/*R* 2 , and for the Minding's Theorem they belong to the isometric differentiable class (see [6,7], chapter 2). In [8], we use the lever law, an explicit formula that allows us to calculate the center of mass of a system of *n* particles with masses *m*1, *m*2, ..., *m<sup>n</sup>* > 0, located on the superior half plane of Lobachevsky *H*<sup>2</sup> , endowed with a conformal metric which induces a constant and negative Gaussian curvature.

Following the basic geometry methods, we obtain the expression for the center of mass for a system of *n* particles sited in the hyperbolic sphere L 2 *<sup>R</sup>* with arbitrary *R*.

We organized this article as follows: In Section 1 we introduced some concepts relative to the center of mass in the Euclidean spaces. In Section 2, some properties of stereographical projection are remembered, and we proceeded to deduce the expression for the center of mass, for two particles on the upper branch of hyperbola, from the "hyperbolic rule of the lever" (see [1,4]) extended to the surface of L 2 *R* . After obtaining the expression for the center of mass for two particles in L 1 *R* , we naturally extended to a system of *n* particles in L 1 *R* , and in the same way, to a system of *n* particles in L 2 *R* .

#### **2. One-Dimensional Euclidean Case**

Let us consider two particles with positive masses sited in the real line at points *x*<sup>1</sup> and *x*2. The point defines the center of mass of the system

$$\mathbf{x}\_{\mathcal{C}} = \frac{m\_1 \mathbf{x}\_1 + m\_2 \mathbf{x}\_2}{m\_1 + m\_2},\tag{2}$$

A direct calculation shows that *m*1|*x<sup>c</sup>* − *x*1| = *m*2|*x<sup>c</sup>* − *x*2| (Euclidean rule of the lever). It is easy to prove that *x<sup>c</sup>* is the unique point in a segment (geodesic) joining *x*<sup>1</sup> and *x*<sup>2</sup> with this property. We extended this definition to more dimensions in Euclidean spaces. Nevertheless, this definition cannot be extended to spaces in general because it is possible in such cases that it is not defined as a linear structure. However, with the "rule of the lever" in mind is possible to have this definition to Riemannian surfaces, as we shall see later.

#### **3. Center of Masses in a Two-Dimensional Hyperbolic Space**

*3.1. Some Observations about the Stereographic Projection of Hyperbolic Sphere on the Poincaré Disk*

Let D<sup>2</sup> *<sup>R</sup>* <sup>=</sup> {*<sup>w</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*w*<sup>|</sup> <sup>&</sup>lt; *<sup>R</sup>*} and *<sup>P</sup>* : <sup>L</sup> 2 *<sup>R</sup>* <sup>→</sup> <sup>D</sup><sup>2</sup> *R* be the stereographic projection; then, for (*x*, *<sup>y</sup>*, *<sup>z</sup>*) <sup>∈</sup> <sup>L</sup> 2 *R* , we have *P*(*x*, *y*, *z*) = *w* = *u* + *iv*, where *u* = *Rx R*+*z* , *v* = *Ry R*+*z* and the inverse projection is *P* −1 : D<sup>2</sup> *<sup>R</sup>* <sup>→</sup> <sup>L</sup> 2 *R* , where (see [5]),

$$P^{-1}(\mu + iv) = \left(\frac{2R^2u}{R^2 - u^2 - v^2}, \frac{2R^2v}{R^2 - u^2 - v^2}, \frac{R(u^2 + v^2 + R^2)}{R^2 - u^2 - v^2}\right).$$

*P* −1 transforms lines through the origin in meridians (hyperbolas through the point (0, 0, *<sup>R</sup>*), with the axis being the *<sup>z</sup>*-axis) and circles with center at the origin, {*<sup>w</sup>* <sup>∈</sup> <sup>D</sup><sup>2</sup> *R* : |*w*| = *const*. < *R*} in parallels (horizontal concentric circles on the upper sheet of hyperboloid).

If we consider the stereographic projection of the one-dimensional hyperbolic sphere L 1 *R* on the real line, we reduce the above equation to

*P*(*x*, *y*) = *u* where *u* = *Rx R*+*y* , *u* ∈ (−*R*, *R*), and the inverse projection is

$$P^{-1}(\mu) = \left(\frac{2\mathcal{R}^2\mu}{\mathcal{R}^2 - \mu^2}, \frac{\mathcal{R}(\mathcal{R}^2 + \mu^2)}{\mathcal{R}^2 - \mu^2}\right).$$

**Theorem 1.** *Let us consider two masses m*1, *m*<sup>2</sup> *sited at the points Q*1, *Q*2*, respectively, and let Qc*(*xc*, *yc*) *be the coordinates of the hyperbolic center of mass, and s*<sup>1</sup> *the length of the arc from Q*<sup>1</sup> *to Q<sup>c</sup> and s*<sup>2</sup> *the length of the arc from Q<sup>c</sup> to Q*2*; then, from the relation (hyperbolic rule of the lever) m*1*r*<sup>1</sup> = *m*2*r*<sup>2</sup> *it follows that:*

$$\left(\frac{\mathcal{R} + \boldsymbol{\mu}\_{\mathcal{L}}}{\mathcal{R} - \boldsymbol{\mu}\_{\mathcal{L}}}\right)^{m} = \left(\frac{\mathcal{R} + \boldsymbol{\mu}\_{1}}{\mathcal{R} - \boldsymbol{\mu}\_{1}}\right)^{m\_{1}} \left(\frac{\mathcal{R} + \boldsymbol{\mu}\_{2}}{\mathcal{R} - \boldsymbol{\mu}\_{2}}\right)^{m\_{2}}.\tag{3}$$

*where m* = *m*<sup>1</sup> + *m*<sup>2</sup> *is the total mass of the system.*

**Proof.** In this case, the length of the arc from the South Pole *P<sup>s</sup>* to the arbitrary point (*x*, *y*) is

$$r = \int\_0^{\mu} \frac{2R^2 dt}{R^2 - t^2} = R \ln \left( \frac{R+\mu}{R-\mu} \right).$$

More generally, the length of the arc *s* from point *Q*1(*x*1, *y*1) to *Q*2(*x*2, *y*2) in the same parallel, if their stereographical projections are *u*<sup>1</sup> and *u*2, is

$$r = R\left(\ln\left(\frac{R+\mu\_2}{R-\mu\_2}\right) - \ln\left(\frac{R+\mu\_1}{R-\mu\_1}\right)\right).$$

Thus,

$$r\_1 = R\left(\ln\left(\frac{R+u\_1}{R-u\_1}\right) - \ln\left(\frac{R+u\_c}{R-u\_c}\right)\right),\tag{4}$$

and

$$r\_2 = R\left(\ln\left(\frac{R+\mu\_c}{R-\mu\_c}\right) - \ln\left(\frac{R+\mu\_2}{R-\mu\_2}\right)\right),\tag{5}$$

$$\operatorname{Rm}\_1\left(\ln\left(\frac{R+\iota\_{\mathcal{C}}}{R-\iota\_{\mathcal{C}}}\right)-\ln\left(\frac{R+\iota\_{1}}{R-\iota\_{1}}\right)\right) = \operatorname{Rm}\_2\left(\ln\left(\frac{R+\iota\_{2}}{R-\iota\_{2}}\right)-\ln\left(\frac{R+\iota\_{\mathcal{C}}}{R-\iota\_{\mathcal{C}}}\right)\right).$$

Therefore,

$$\ln\left(\frac{R+\nu\_c}{R-\nu\_c}\right) = \frac{1}{m\_1+m\_2}\left(m\_1\ln\left(\frac{R+\nu\_1}{R-\nu\_1}\right) + m\_2\ln\left(\frac{R+\nu\_2}{R-\nu\_2}\right)\right).\tag{6}$$

and from there the result follows.

Inductively, we can extend the last argument to *n* particles with masses *m*1,*m*2, . . . ,*m<sup>n</sup>* sited on L 1 *R* as expressed by the following.

**Corollary 1.** *Let m*1*,m*2*, . . . ,m<sup>n</sup> be positive masses sited on the points* (*x*1, *y*1)*,*(*x*2, *y*2)*, . . . ,*(*xn*, *yn*) *of* L 1 *<sup>R</sup> with stereographical projections u*1*,u*2*, . . . ,un, respectively. Then there is a unique point u<sup>c</sup>* ∈ (−*R*, *R*) *such that such that the following expression is fulfilled:*

$$\left(\frac{\mathcal{R} + \boldsymbol{\mu}\_c}{\mathcal{R} - \boldsymbol{\mu}\_c}\right)^m = \prod\_{k=1}^n \left(\frac{\mathcal{R} + \boldsymbol{\mu}\_k}{\mathcal{R} - \boldsymbol{\mu}\_k}\right)^{m\_k} \tag{7}$$

*where m* = ∑ *n <sup>k</sup>*=<sup>1</sup> *m<sup>k</sup> is the total mass of the system.*

#### *3.2. Center of Mass for a System of Two Particles in* L 2 *R*

Now, we extend the "rule of the lever" to a more general context:

Consider a Riemannian surface *T* and two particles with masses *m*1, *m*<sup>2</sup> sited in the points *x*1, *x*<sup>2</sup> ∈ *T*, respectively. Then we define the *T*-center of mass as the point *x<sup>c</sup>* in the geodesic joining *x*<sup>1</sup> to *x*<sup>2</sup> such that the following relation is verified:

$$m\_1 d(\mathbf{x}\_1, \mathbf{x}\_c) = m\_2 d(\mathbf{x}\_2, \mathbf{x}\_c)$$

where *d* is the metric in *T*, and *d*(*x*1, *xc*) + *d*(*xc*, *x*2) = *d*(*x*1, *x*2). For the case of L 2 *R* , geodesics are hyperbolas determined for the intersection of the upper sheet of the hyperboloid with the plane drawn for the pair of points and the origin (0, 0, 0).

Now we can calculate the hyperbolic center of mass for a system with a finite number of particles on L 2 *R*] . Following exactly the same reasoning as in the previous section, we obtain the following.

**Corollary 2.** *Let m*1*,m*2*, . . . ,m<sup>n</sup> be n masses of particles sited, respectively, in the points* (*x*1, *y*1, *z*1)*,* (*x*2, *y*2, *z*2)*, . . . ,* (*xn*, *yn*, *zn*) *on the same geodesic of* L 2 *<sup>R</sup> with stereographical projections w*1*, w*2*, . . . , wn, in the Poincaré disk. And let w<sup>c</sup> be their hyperbolic center of mass; then, the next relation is fulfilled:*

$$\left(\frac{\mathcal{R} + w\_c}{\mathcal{R} - w\_c}\right)^m = \prod\_{k=1}^n m\_k \left(\frac{\mathcal{R} + w\_k}{\mathcal{R} - w\_k}\right)^{m\_k},\tag{8}$$

*where m* = ∑ *n <sup>k</sup>*=<sup>1</sup> *m<sup>k</sup> .*

**Remark 1.** *If each fraction is divided, their numerator and denominator for R and both sides rise to the power R, when R* → <sup>∞</sup>*, is obtained,*

$$\exp(2mw\_{\mathfrak{c}}) = \exp\left(2\sum\_{k=1}^{n} m\_k w\_k\right).$$

*Or equivalently,*

$$w\_{\mathcal{C}} = \frac{1}{m} \sum\_{k=1}^{n} m\_k w\_k. \tag{9}$$

*This corresponds to the equation for the center of mass in the Euclidean complex plane, that is, the complex plane (or* R<sup>2</sup> *), with Euclidean metric and zero curvature.*

#### **4. An Application to the Curved 2-Body Problem**

In [5], the curved *n*-body problem in a two-dimensional space with constant negative curvature is studied, and the model L 2 *R* is considered, in which there are systems. Let *<sup>z</sup>* = (*z*1, *<sup>z</sup>*2, . . . , *<sup>z</sup>n*) <sup>∈</sup> (*D*<sup>2</sup> *R* ) *<sup>n</sup>* be the configuration of *n* point particles with masses *<sup>m</sup>*1, *<sup>m</sup>*2, . . . , *<sup>m</sup><sup>n</sup>* <sup>&</sup>gt; <sup>0</sup> <sup>∈</sup> <sup>D</sup><sup>2</sup> *R* .

$$m\_k \ddot{z}\_k = -\frac{2m\_k \overline{z}\_k \dot{z}\_k^2}{\mathcal{R}^2 - |z\_k|^2} + \frac{2}{\lambda(z\_k, \overline{z}\_k)} \frac{\partial \mathcal{U}\_\mathcal{R}}{\partial \overline{z}\_k}, \quad k = 1, \dots, n \tag{10}$$

where

$$
\lambda(z\_k, \overline{z}\_k) = \frac{4R^4}{(R^2 - |z\_k^2|)^2}
$$

is the conformal function of the Riemannian metric,

$$\frac{\partial \mathcal{U}\_R}{\partial \boldsymbol{z}\_k} = \sum\_{j=1, j \neq k}^n \frac{2m\_k m\_j \mathcal{R} \mathcal{P}\_{2,(k,j)}(\boldsymbol{z}, \boldsymbol{\bar{z}}\_k)}{(\Theta\_{2,(k,j)}(\boldsymbol{z}, \boldsymbol{\bar{z}}\_k))^{3/2}},$$

$$\mathcal{P}\_{2,(k,j)}(\boldsymbol{z}, \boldsymbol{\bar{z}}\_k) = (\mathcal{R}^2 - |\boldsymbol{z}\_k|^2)(\mathcal{R}^2 - |\boldsymbol{z}\_j|^2)^2(\boldsymbol{z}\_j - \boldsymbol{z}\_k)(\mathcal{R}^2 - \boldsymbol{z}\_k \boldsymbol{\bar{z}}\_j).$$

$$\Theta\_{2,(k,\mathfrak{j})}(z,\bar{z}\_{k}) = [2(z\_{k}\bar{z}\_{\mathfrak{j}} + z\_{\mathfrak{j}}\bar{z}\_{\mathfrak{k}})\mathbb{R}^{2} - (|z\_{k}|^{2} + \mathbb{R}^{2})(|z\_{\mathfrak{j}}|^{2} + \mathbb{R}^{2})]^{2} - (\mathbb{R}^{2} - |z\_{k}|^{2})^{2}(\mathbb{R}^{2} - |z\_{\mathfrak{j}}|^{2})^{2},$$

$$k\_{\mathfrak{i}}j \in \{1, \dots, n\}, k \neq j.$$

We consider functions of the form

$$w\_k(t) = e^{\dot{t}t} z\_k(t)\_{\prime}$$

where *z* = (*z*1, . . . , *zn*) is a solution of Equation (10). Straightforward computations show that

$$\begin{cases} \dot{w}\_k = (\dot{z}\_k + \dot{z}\_k)e^{it} \\ \ddot{w}\_k = (\dot{z}\_k + 2\dot{e}\_k - z\_k)e^{it} \\ \frac{d\mathcal{Z}\_k}{d\mathcal{D}\_k} = e^{it} \quad k = 1, \dots, n. \end{cases}$$

For the configurations called relative equilibrium, concerning the 2-body problem, the next result is established.

**Theorem 2.** *Consider two point particles of masses m*1, *m*<sup>2</sup> > 0 *moving on the Poincaré disk* D<sup>2</sup> *R , whose center is the origin,* 0*, of the coordinate system. Then z* = (*z*1, *z*2) *is an elliptic relative equilibrium of system* (10) *with n* = 2 *if and only if, for every circle centered at* 0 *of radius α, with* 0 < *α* < *R, along which m*<sup>1</sup> *moves, there is a unique circle centered at* 0 *of radius r, which satisfies* 0 < *r* < *R, along which m*<sup>2</sup> *moves, such that, at every time instant, m*<sup>1</sup> *and m*<sup>2</sup> *are on some diameter of* D<sup>2</sup> *R , with* 0 *between them. Moreover,*

*1. if m*<sup>2</sup> > *m*<sup>1</sup> > 0 *and α are given, then r* < *α;*

*2. if m*<sup>1</sup> = *m*<sup>2</sup> > 0 *and α are given, then r* = *α;*

*3. if m*<sup>1</sup> > *m*<sup>2</sup> > 0 *and α are given, then r* > *α.*

This result was reformulated in a more precise form, using the expression for the hyperbolic center of mass taking into account that in a configuration corresponding to a relative equilibrium is invariant with the time, because the distance and angles between particles do not change. This is sufficient, considering the initial configuration on the *x*-axis, and *α* corresponds to the length measure over the Poincaré disk of the projection of arc *r*<sup>1</sup> over the hyperbolic sphere L 2 *R* , and *r* is the projection length in disk one of the arc *r*<sup>2</sup> over the hyperbolic sphere. Then we have the next relations:

> *R* + *α R* − *α*

$$\text{The first-order coupling between the two-dimensional } \mathcal{N} \text{-matrices is the only possible } \mathcal{N} \text{-matrices with } \mathcal{N} = \{0, 1, 2, \dots, N\} \text{ and } \mathcal{N} = \{0, 1, 2, \dots, N\}.$$

and

*<sup>r</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*ln R* + *r R* − *r* .

*<sup>r</sup>*<sup>1</sup> <sup>=</sup> *ln*

Substituting in the hyperbolic rule *m*1*r*<sup>1</sup> = *m*2*r*<sup>2</sup> and the expression for the center of mass, we obtain

$$\left(\frac{R+\alpha}{R-\alpha}\right)^{m\_1} = \left(\frac{R+r}{R-r}\right)^{-m\_2}.$$

Thus it follows from Equation (6) that

$$\frac{\mathcal{R} - w\_c}{\mathcal{R} + w\_c} = 1$$

It follows that *w<sup>c</sup>* = 0 and so the center of mass is fixed for every time in the South Pole of the hyperbolic sphere (0, 0, *R*), and Theorem 2 can be expressed in the following form.

**Theorem 3.** *For every configuration of elliptic relative equilibrium for the* 2*-body problem with masses m*1, *m*<sup>2</sup> *sited in the points P*1(*x*1, *y*1, *z*1) *and P*2(*x*2, *y*2, *z*2) *on the hyperbolic sphere of radius R. If r*<sup>1</sup> *and r*<sup>2</sup> *are the lengths of arcs measured from the South Pole* (0, 0, *R*) *to the points P*<sup>1</sup> *and P*2*, respectively, then it satisfies the relation m*1*r*<sup>1</sup> = *m*2*r*2*, and the center of mass of the system is fixed in* (0, 0, *R*) *for every time.*

The hyperbolic spaces are very special in relativity. A hyperbolic (i.e., Lobachevskian) space can be represented upon one sheet of a two-sheeted cylindrical hyperboloid in Minkowski space–time. According to works that recently appeared in literature (see [9]), in hyperbolic spaces, the expression for the center of mass obtained by adopting the relativistic rule of lever reads

$$m\_1 \sinh \sqrt{-k} r\_1 = m\_2 \sinh \sqrt{-k} r\_2 \tag{11}$$

with *r<sup>i</sup>* , *i* = 1, 2, denoting the Riemannian distance of *m<sup>i</sup>* to the center of mass and *k* the (negative) Gaussian curvature, respectively. Using the stereographic projection of a hyperbolic sphere on the Poincaré disk.

For both the Euclidean and the hyperbolic spaces, the center of mass for the system particles plays a central role in the conserved momentum principle. Adoption of the conserved momentum principle for 2-body is expressed in spaces with negative Gaussian curvature is along the following lines.

**Theorem 4.** *Consider two masses m*1, *m*<sup>2</sup> *sited in the points Q*1*, Q*2*, respectively, and r*1*, r*<sup>2</sup> *the length or arc from <sup>Q</sup><sup>c</sup> to <sup>Q</sup>*2*; then, from the relation (hyperbolic rule of the lever) <sup>m</sup>*<sup>1</sup> sinh <sup>√</sup> −*kr*<sup>1</sup> = *<sup>m</sup>*<sup>2</sup> sinh <sup>√</sup> −*kr*<sup>2</sup> *and using the stereographic projection of a hyperbolic sphere on the Poincaré disk the conserved momentum principle for 2-body expressed in spaces with negative Gaussian curvature is*

$$m\_1 \sinh\left(R \ln\left(\frac{R+w\_c}{R-w\_c}\frac{R-w\_1}{R+w\_1}\right)\right) = m\_2 \sinh\left(R \ln\left(\frac{R-w\_c}{R+w\_c}\frac{R+w\_2}{R-w\_2}\right)\right) \tag{12}$$

**Proof.** Following the ideas from [1] on relativistic momentum, we have

$$p = m \frac{v}{\sqrt{1 - v^2}}$$

where *p* is the momentum, *v* is the velocity of the particle with respect to a frame of reference and the velocity of the light is *c* = 1. We take

$$r = \frac{1}{2} \ln \left( \frac{1+v}{1-v} \right).$$

the distance of the particle with respect to the center of the reference frame. This solution with respect to *v* yields

$$v = \tanh r.$$

Evaluating this in the momentum gives *p* = *m* sinh *r*. From the energy of the particle *E* = *m* √ 1 1−*v* 2 and replacing the velocity of the particle *v* = tanh *r* gets *E* = *m* cosh *r*. From the hyperbolic identity cosh *x* <sup>2</sup> <sup>−</sup> sinh *<sup>x</sup>* <sup>2</sup> = 1, the constant *E* <sup>2</sup> <sup>−</sup> *<sup>p</sup>* <sup>2</sup> = *m*<sup>2</sup> can be obtained. Let *r*<sup>1</sup> be the distance between the particle with mass *m*<sup>1</sup> and the mass center, and *r*<sup>2</sup> the distance between the particle with mass *m*<sup>2</sup> and the mass center; then, *v*<sup>1</sup> = tanh *r*1, and *v*<sup>1</sup> = tanh *r*<sup>1</sup> are the velocities of particles with mass *m*<sup>1</sup> and *m*2, respectively. In consequence, the relativistic momentum is constant,

$$m\_1 \sinh r\_1 = m\_2 \sinh r\_2$$

The expression of the center of mass for a system of two particles in *L* 2 R is presented in the following result.

**Theorem 5.** *Consider two masses m*1, *m*<sup>2</sup> *sited in the points Q*1*, Q*2*, respectively, and r*1*, r*<sup>2</sup> *the length or arc from <sup>Q</sup><sup>c</sup> to <sup>Q</sup>*2*; then, from the relation (hyperbolyc rule of the lever) <sup>m</sup>*<sup>1</sup> sinh <sup>√</sup> −*kr*<sup>1</sup> = *<sup>m</sup>*<sup>2</sup> sinh <sup>√</sup> −*kr*<sup>2</sup> *and using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the center of mass for a system of two particles in L*<sup>2</sup> R *is given by*

$$m = m\_1 \cosh\left(\sqrt{-k}\ln\left(\frac{(\mathcal{R} + w\_\mathcal{E})(\mathcal{R} - w\_1)}{(\mathcal{R} - w\_\mathcal{E})(\mathcal{R} + w\_1)}\right)\right) + m\_2 \cosh\left(\sqrt{-k}\ln\left(\frac{(\mathcal{R} - w\_\mathcal{E})}{(\mathcal{R} + w\_\mathcal{E})}\frac{(\mathcal{R} + w\_2)}{(\mathcal{R} - w\_2)}\right)\right) \tag{13}$$

*where k is the (negative) Gaussian curvature*

**Proof.** From the principle of conservation of relativistic momentum and total energy we obtain the desired result. Following the ideas from [1],

$$E = \mathfrak{m} = \mathfrak{m}\_1 \cosh r\_1 + \mathfrak{m}\_2 \cosh r\_2$$

with *E* = *m* cosh 0 = *m*.

One of the most interesting aspects concerning the determination of the center of mass of a particle system lies in its physical applications. As known, for Euclidean space, Equations (2) may be derived from the Lever rule. If we suppose that the particles are under the influence of an attractive potential force, depending only on their mutual distance, this equation may be derived from the other two different characteristics of the center of mass: (a) Collision point and (b) center of steady rotation.

In situation (a), for a collisional point, if the particles are initially at rest they will collide at the center of mass; the expression of the centre of mass is along the following lines.

**Theorem 6.** *Consider two masses m*1, *m*<sup>2</sup> *sited in the points Q*1*, Q*2*, respectively, and r*1*, r*<sup>2</sup> *the length or arc from Q<sup>c</sup> to Q*2*; then, from the relation (hyperbolyc rule of the lever) m*1*r*<sup>1</sup> = *m*2*r*<sup>2</sup> *and using the stereographic projection of a hyperbolic sphere on the Poincaré disk the center of mass for a system of two particles in L*<sup>2</sup> R *is given by*

$$\left(\frac{1+\sqrt{-k}w\_{\varepsilon}}{1-\sqrt{-k}w\_{\varepsilon}}\right)^{m} = \left(\frac{1+\sqrt{-k}w\_{1}}{1-\sqrt{-k}w\_{1}}\right)^{m\_{1}}\left(\frac{1+\sqrt{-k}w\_{2}}{1-\sqrt{-k}w\_{2}}\right)^{m\_{2}}.\tag{14}$$

**Proof.** Following the ideas from [9] we obtain our result.

In situation (b), for the center of steady rotation, if the particles rotate uniformly along with concentric circles, maintaining a constant distance over time, then the center of mass coincides with the circle's center, the expression of the center of mass is given by the following.

**Theorem 7.** *Consider two masses m*1, *m*<sup>2</sup> *sited in the points Q*1*, Q*2*, respectively, and r*1*, r*<sup>2</sup> *the length or arc from <sup>Q</sup><sup>c</sup> to <sup>Q</sup>*2*; then, from the relation (hyperbolyc rule of the lever) <sup>m</sup>*<sup>1</sup> sinh 2<sup>√</sup> −*kr*<sup>1</sup> = *<sup>m</sup>*<sup>2</sup> sinh 2<sup>√</sup> −*kr*<sup>2</sup> *and using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the center of mass for a system of two particles in L*<sup>2</sup> R *is given by*

$$m = m\_1 \cosh\left(2\sqrt{-k}\ln\left(\frac{(R+w\_\varepsilon)(R-w\_1)}{(R-w\_\varepsilon)(R+w\_1)}\right)\right) + m\_2 \cosh\left(2\sqrt{-k}\ln\left(\frac{(R-w\_\varepsilon)}{(R+w\_\varepsilon)}\frac{(R+w\_2)}{(R-w\_2)}\right)\right) \tag{15}$$

**Proof.** Following the ideas of [2,9], we can generalize, using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the center of mass for a system of two particles in *L* 2 R our idea and so obtain the result.

**Remark 2.** *It has been established that if the particles have distinct masses, then the above definitions of the center of mass are not equivalent for hyperbolic spaces. Similarly, using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the three meanings for the center of mass (lever rule, collision point and center of steady rotation) are not equivalent. We consider that, from the physical point of view, the most appropriate definition is the definition present here, because it inherits two properties of the Euclidean center of mass (lever rule and collision point), while the relativistic definition only preserves one (conservation of angular momentum).*

#### **5. Conclusions**

In the present work, an analytical formula is obtained that allows the exact calculation of the coordinates of the center of mass for a system of particles with positive masses located on a two-dimensional Riemannian manifold with constant and negative Gaussian curvature. The model of such a variety is taken as a model, Poincaré's disk *D*2*R*, with the conformal metric resulting from the stereographic projection of the hyperbolic sphere *L* <sup>2</sup>*R*. The formula obtained is derived using the hyperbolic lever law in this context, as one of the possibilities that is referenced in [4], and with it a previously obtained result is established more precisely that allows characterizing the relative equilibria for a 2-body problem on Poincaré's disk.

**Author Contributions:** P.P.O.P., A.M.M.R., and R.D.O.O. supervised the entire article. All authors performed the formal analysis, and participated in the writing and revising of the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by UNIVERSIDAD DE CARTAGENA grant number 062-2019.

**Acknowledgments:** The authors wishes to thank Joaquín Luna Torres and José Guadalupe Reyes Victoria, for their valuable suggestions.

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

#### **References**


### *Article* **Some Important Criteria for Oscillation of Non-Linear Differential Equations with Middle Term**

**Saad Althobati 1,†, Omar Bazighifan 2,3,\* ,† and Mehmet Yavuz 4,\* ,†**


**Abstract:** In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with first-order equations. The obtained results extend and simplify known conditions in the literature. Furthermore, examining the validity of the proposed criteria is demonstrated via particular examples.

**Keywords:** higher-order; neutral delay; oscillation

#### **1. Introduction**

Neutral equations contribute to many applications in physics, engineering, biology, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. Also, oscillation of neutral equations contribute to many applications of problems dealing with vibrating masses attached to an elastic bar, see [1].

In this paper, we investigate the oscillatory properties of solutions of the higher-order neutral differential equation

$$\left(\mathfrak{a}\_1(\mathbf{x})\left(\mathfrak{a}^{(\ell-1)}(\mathbf{x})\right)^{(p-1)}\right)' + \mathfrak{a}\_2(\mathbf{x})\left(\mathfrak{a}^{(\ell-1)}(\mathbf{x})\right)^{(p-1)} + \mathfrak{f}(\mathbf{x})\delta^{(p-1)}(\mathfrak{f}\_2(\mathbf{x})) = \mathfrak{0}, \ \mathbf{x} \ge \mathbf{x}\_0,\tag{1}$$

where

$$
\mathfrak{a}(\mathfrak{x}) := \delta(\mathfrak{x}) + \mathfrak{c}(\mathfrak{x})\delta(\mathfrak{z}\_1(\mathfrak{x})).\tag{2}
$$

The main results are obtained under the following conditions:

 *α*<sup>1</sup> ∈ *C* 1 ([*x*0, ∞)), *α*1(*x*) > 0, *α* ′ 1 (*x*) ≥ 0, 1 < *<sup>p</sup>* < <sup>∞</sup>, *<sup>c</sup>*, *<sup>α</sup>*2, *<sup>ζ</sup>* ∈ *<sup>C</sup>*([*x*0, <sup>∞</sup>)), *<sup>α</sup>*2(*x*) > 0, *<sup>ζ</sup>*(*x*) > 0, 0 ≤ *<sup>c</sup>*(*x*) < *<sup>c</sup>*<sup>0</sup> < 1, *β*<sup>1</sup> ∈ *C* 1 ([*x*0, <sup>∞</sup>)), *<sup>β</sup>*<sup>2</sup> ∈ *<sup>C</sup>*([*x*0, <sup>∞</sup>)), *<sup>β</sup>* ′ 1 (*x*) > 0, *<sup>β</sup>*1(*x*) ≤ *<sup>x</sup>*, lim*x*→<sup>∞</sup> *<sup>β</sup>*1(*x*) = lim*x*→<sup>∞</sup> *<sup>β</sup>*2(*x*) = <sup>∞</sup>, ℓ ≥ 4 is an even natural number, *ζ* is not identically zero for large *x*.

Moreover, we establish the oscillatory behavior of (1) under the conditions

$$
\beta\_2(\mathbf{x}) < \beta\_1(\mathbf{x}), \ \beta\_2'(\mathbf{x}) \ge 0 \text{ and } \left(\beta\_1^{-1}(\mathbf{x})\right)' > 0 \tag{3}
$$

and

$$\int\_{x\_0}^{\infty} \left( \frac{1}{a\_1(s)} \exp \left( - \int\_{x\_0}^s \frac{a\_2(\mathcal{a})}{a\_1(\mathcal{a})} d\mathcal{a} \right) \right)^{1/(p-1)} ds = \infty. \tag{4}$$


**Citation:** Althobati, S.; Bazighifan, O.; Yavuz, M. Some Important Criteria for Oscillation of Non-Linear Differential Equations with Middle Term. *Mathematics* **2021**, *9*, 346. https://doi.org/10.3390/math9040346

Academic Editor: Alberto Cabada Received: 20 January 2021 Accepted: 8 February 2021 Published: 9 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

Over the past few years, there has been much research activity concerning the oscillation and asymptotic behavior of various classes of differential equations; see [2–11]. In particular, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see [12–23].

Bazighifan et al. [2] examined the oscillation of higher-order delay differential equations with damping of the form

$$\begin{cases} \left(a\_1(\mathbf{x})\varPhi\_p[\mathcal{O}^{(\ell-1)}(\mathbf{x})]\right)' + a\_2(\mathbf{x})\varPhi\_p[f\left(\mathcal{O}^{(\ell-1)}(\mathbf{x})\right)] + \sum\_{i=1}^j \zeta\_i(\mathbf{x})\varPhi\_p[g\left(\mathcal{O}(\mathcal{S}\_i(\mathbf{x}))\right)] = 0, \\ \varPhi\_p[\mathbf{s}] = |\mathbf{s}|^{p-2}\mathbf{s}, \; j \ge 1, \; \mathbf{x} \ge \mathbf{x}\_0 > 0. \end{cases}$$

This time, the authors used the Riccati technique.

Zhang et al. in [3] considered a higher-order differential equation

$$\begin{cases} L\_{\mathcal{Q}}' + a\_2(\mathbf{x}) \Big| \mathcal{o}^{(\ell-1)}(\mathbf{x}) \Big|^{p-2} \mathcal{o}^{(\ell-1)}(\mathbf{x}) + \zeta(\mathbf{x}) |\delta(\mathcal{\beta}\_2(\mathbf{x}))|^{p-2} \delta(\mathcal{\beta}\_2(\mathbf{x})) = 0, \\ 1 < p < \infty, \ \mathbf{x} \ge \mathbf{x}\_0 > 0, \ \mathbf{o}(\mathbf{x}) = \delta(\mathbf{x}) + c(\mathbf{x}) \delta(\mathcal{\beta}\_1(\mathbf{x})), \end{cases}$$

where

$$L\_{\mathcal{O}} = \left| \mathcal{O}^{(\ell-1)}(\mathfrak{x}) \right|^{p-2} \mathcal{O}^{(\ell-1)}(\mathfrak{x}).$$

Bazighifan and Ramos [4] considered the oscillation of the even-order nonlinear differential equation with middle term of the form

$$\begin{cases} \left(a\_1(\mathbf{x}) \left(\mathcal{O}^{(\ell-1)}(\mathbf{x})\right)^{p-1}\right)' + a\_2(\mathbf{x}) \left(\mathcal{O}^{(\ell-1)}(\mathbf{x})\right)^{p-1} + \zeta(\mathbf{x})\mathcal{O}(\beta(\mathbf{x})) = 0, \\ \mathbf{x} \ge \mathbf{x}\_0 > 0, \end{cases}$$

where 1 < *p* < ∞.

Liu et al. [5] investigated the higher-order differential equations

$$\begin{cases} \left(a\_1(\mathbf{x})\Phi\Big(\boldsymbol{\phi}^{(\ell-1)}(\mathbf{x})\Big)\right)' + a\_2(\mathbf{x})\Phi\Big(\boldsymbol{\phi}^{(\ell-1)}(\mathbf{x})\Big) + \zeta(\mathbf{x})\Phi(\boldsymbol{\phi}(\boldsymbol{\beta}(\mathbf{x}))) = \mathbf{0},\\ \Phi = |\mathbf{s}|^{p-2}\mathbf{s}, \; \mathbf{x} \ge \mathbf{x}\_0 > \mathbf{0}, \; \ell \text{ is even}, \end{cases}$$

where *n* is even and used integral averaging technique.

The authors in [6,7] discussed oscillation criteria for the equations

$$\begin{cases} \left(\mathfrak{a}\_1(\mathfrak{x}) \Big| \mathcal{o}^{(\ell-1)}(\mathfrak{x}) \Big| \stackrel{p-2}{\cdot} \mathcal{o}^{(\ell-1)}(\mathfrak{x}) \right)' + \sum\_{i=1}^j \zeta\_i(\mathfrak{x}) \mathcal{g}(\mathcal{o}(\beta\_i(\mathfrak{x}))) = 0, \\ j \ge 1, \ \mathfrak{x} \ge \mathfrak{x}\_0 > 0, \end{cases}$$

where ℓ is even and *p* > 1 is a real number, the authors used comparison method with first and second-order equations.

Li et al. [8] studied the oscillation of fourth-order neutral differential equations

$$\begin{cases} \left(\alpha\_1(\mathbf{x})|\boldsymbol{\alpha}^{\prime\prime\prime}(\mathbf{x})|^{p-2}\boldsymbol{\alpha}^{\prime\prime\prime}(\mathbf{x})\right)' + \zeta(\mathbf{x})|\delta(\beta\_2(\mathbf{x}))|^{p-2}\delta(\beta\_2(\mathbf{x})) = 0, \\ 1 < p < \infty, \ \mathbf{x} \ge \mathbf{x}\_0 > 0, \end{cases}$$

where *̟*(*x*) = *δ*(*x*) + *c*(*x*)*δ*(*β*1(*x*)).

In [9,10], the authors considered the equation

$$
\omega^{(\ell)}(\mathbf{x}) + \zeta(\mathbf{x})\delta(\beta\_2(\mathbf{x})) = \mathbf{0} \tag{5}
$$

by using the Riccati method, they proved that this equation is oscillatory if

$$\liminf\_{x \to \infty} \int\_{\beta\_2(x)}^x z(s)ds > \frac{(\ell - 1)2^{(\ell - 1)(\ell - 2)}}{\mathbf{e}} \tag{6}$$

and

$$\lim\limits\_{x\to\infty} \inf \int\_{\beta\_2(x)}^x z(s)ds > \frac{(\ell - 1)!}{\mathbf{e}}, \text{ respectively.}\tag{7}$$

where *z*(*x*) := *β* ℓ−1 2 (*x*)(1 − *α*2(*β*2(*x*)))*ζ*(*x*).

We can easily apply conditions (6) and (7) to the equation

$$
\left(\delta(\mathbf{x}) + \frac{1}{2}\delta\left(\frac{1}{2}\mathbf{x}\right)\right)^{(4)} + \frac{\zeta\_0}{\mathbf{x}^4}\delta\left(\frac{9}{10}\mathbf{x}\right) = \mathbf{0}, \; \mathbf{x} \ge 1,\tag{8}
$$

then we get that (8) is oscillatory if


Hence, [10] improved the results in [9].

Thus, the main purpose of this article is to extend the results in [9,10,23]. An example is considered to illustrate the main results.

We mention some important lemmas:

$$\textbf{Lemma 1 ([11]). Let } \delta \in \mathcal{C}^{\ell}([\mathtt{x}\_{0}, \infty), (0, \infty)), \delta^{(\ell-1)}(\mathtt{x})\delta^{(\ell)}(\mathtt{x}) \le 0 \, and \lim\_{\mathbf{x} \to \infty} \delta(\mathbf{x}) \ne 0 \,, \textbf{then}$$

$$\delta(\mathbf{x}) \ge \frac{\mu}{(\ell - 1)!} \mathbf{x}^{\ell - 1} \Big| \delta^{(\ell - 1)}(\mathbf{x}) \Big| \text{ for } \mathbf{x} \ge \mathbf{x}\_{\mu}, \ \mu \in (0, 1).$$

**Lemma 2** ([16])**.** *If δ* (*i*) (*x*) > 0, *i* = 0, 1, ..., ℓ, *and δ* (ℓ+1) (*x*) < 0, *then*

$$\frac{\delta(\mathbf{x})}{\mathbf{x}^{\ell}/\ell!} \ge \frac{\delta'(\mathbf{x})}{\mathbf{x}^{\ell-1}/(\ell-1)!}.$$

**Lemma 3** ([13])**.** *Let (30) hold and*

*δ be an eventually positive solution of (1).* (9)

*Then, we have these cases:*

$$\begin{array}{ll} (\mathbf{I}\_{1}): & \mathcal{a}(\mathbf{x}) > 0, \; \mathcal{a}'(\mathbf{x}) > 0, \; \mathcal{a}''(\mathbf{x}) > 0, \; \mathcal{a}^{(\ell-1)}(\mathbf{x}) > 0 \; \text{and} \; \mathcal{a}^{(\ell)}(\mathbf{x}) < 0, \\\ (\mathbf{I}\_{2}): & \mathcal{a}(\mathbf{x}) > 0, \; \mathcal{a}^{(j)}(\mathbf{x}) > 0, \; \mathcal{a}^{(j+1)}(\mathbf{x}) < 0 \; \text{for all odd integer} \\\ & j \in \{1, 2, \ldots, \ell - 3\}, \; \mathcal{a}^{(\ell-1)}(\mathbf{x}) > 0 \; \text{and} \; \mathcal{a}^{(\ell)}(\mathbf{x}) < 0, \end{array}$$

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

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

**Theorem 1.** *If the differential equation*

$$\phi'(\mathbf{x}) + (1 - \mathfrak{c}(\beta\_2(\mathbf{x})))^{(p-1)} \zeta(\mathbf{x}) \frac{y\_{\mathbf{x}\_0}(\mathbf{x})}{y\_{\mathbf{x}\_0}(\beta\_2(\mathbf{x}))} \left(\frac{\mu \beta\_2^{\ell-1}(\mathbf{x})}{(\ell-1)! a\_1^{1/(p-1)}(\beta\_2(\mathbf{x}))}\right)^{(p-1)} \phi(\beta\_2(\mathbf{x})) = 0 \tag{10}$$

*is oscillatory for some constant µ* ∈ (0, 1)*, where*

$$y\_{\mathbf{x}\_0}(\mathbf{x}) := \exp\left(\int\_{\mathbf{x}\_0}^{\mathbf{x}} \frac{\mathfrak{a}\_2(t)}{t\_1(t)} \mathbf{d}t\right).$$

*then (1) is oscillatory.*

**Proof.** Let (9) hold. Then, we see that *δ*(*x*), *δ*(*β*1(*x*)) and *δ*(*β*2(*x*)) are positive for all *x* ≥ *x*<sup>1</sup> sufficiently large. It is not difficult to see that

$$\begin{split} &\frac{1}{y\_{x\_{0}}(\mathbf{x})}\frac{d}{dx}\Big(y\_{x\_{0}}(\mathbf{x})a\_{1}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}\Big) \\ & \qquad \qquad = \frac{1}{y\_{x\_{0}}(\mathbf{x})}\Big(y\_{x\_{0}}(\mathbf{x})\Big(a\_{1}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}\Big)' + y\_{x\_{0}}'(\mathbf{x})a\_{1}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}\Big) \\ & \qquad = \Big(a\_{1}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}\Big)' + \frac{y\_{x\_{0}}'(\mathbf{x})}{y\_{x\_{0}}(\mathbf{x})}a\_{1}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)} \\ & \qquad = \Big(a\_{1}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}\Big)' + a\_{2}(\mathbf{x})\Big(\boldsymbol{\mathcal{O}}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}. \end{split} \tag{11}$$

Taking into account (2) and *̟*′ (*x*) > 0, we get that *δ*(*x*) ≥ (1 − *c*(*x*))*̟*(*x*). Thus, from (1) and (11), we have that

$$\left(y\_{\ge 0}(\mathbf{x})a\_1(\mathbf{x})\left(\mathcal{O}^{(\ell-1)}(\mathbf{x})\right)^{(p-1)}\right)' + y\_{\ge 0}(\mathbf{x})\zeta(\mathbf{x})(1-c(\beta\_2(\mathbf{x})))^{(p-1)}\mathcal{O}^{(p-1)}(\beta\_2(\mathbf{x})) \le 0,\tag{12}$$

for *c*<sup>0</sup> < 1.

Using Lemma 1, we get that

$$
\mathfrak{w}(\mathfrak{x}) \ge \frac{\mu}{(\ell - 1)!} \mathfrak{x}^{\ell - 1} \mathfrak{w}^{(\ell - 1)}(\mathfrak{x}), \tag{13}
$$

for some *µ* ∈ (0, 1). From (1), (12) and (13), we see that

$$\left(y\_{30}(\mathbf{x})a\_1(\mathbf{x})\Big(\mathcal{O}^{(\ell-1)}(\mathbf{x})\Big)^{(p-1)}\right)' + y\_{30}(\mathbf{x})\zeta(\mathbf{x})(1-c(\mathfrak{f}\_2(\mathbf{x})))^{(p-1)}\Big(\frac{\mu\mathfrak{f}\_2^{\ell-1}(\mathbf{x})}{(\ell-1)!}\Big)^{(p-1)}\left(a^{(\ell-1)}(\mathfrak{f}\_2(\mathbf{x}))\right)^{(p-1)} \le 0. \quad \forall \mathbf{x} \in \mathcal{X}\_{\mathbf{x}}$$

Then, if we set *<sup>φ</sup>*(*x*) = *<sup>y</sup>x*<sup>0</sup> (*x*)*α*1(*x*) *̟*(ℓ−1) (*x*) (*p*−1) , then we have that *φ* > 0 is a solution of the delay inequality

$$\left(\mathfrak{g}'(\mathbf{x}) + (1 - \mathfrak{c}(\mathfrak{gz}(\mathbf{x})))^{(p-1)}\mathfrak{z}(\mathbf{x}) \frac{\mathfrak{y}\_{\mathfrak{x}\_0}(\mathbf{x})}{\mathfrak{y}\_{\mathfrak{x}\_0}(\mathfrak{z}\_2(\mathbf{x}))} \left(\frac{\mu \mathfrak{f}\_2^{\ell-1}(\mathbf{x})}{(\ell - 1)! a\_1^{1/(p-1)}(\mathfrak{f}\_2(\mathbf{x}))}\right)^{(p-1)} \mathfrak{e}(\mathfrak{gz}(\mathbf{x})) \le 0.$$

It is clear that the equation (10) has a positive solution (see [17], Theorem 1), this is a contradiction. The proof is complete.

**Theorem 2.** *Assume that (3) and (30) hold. If the differential equations*

$$z'(\mathbf{x}) + \zeta(\mathbf{x}) \frac{y\_{\mathbf{x}\_0}(\mathbf{x})}{y\_{\mathbf{x}\_0}\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right)} \left(\frac{\mu\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right)^{\ell-1} c\_\ell(\boldsymbol{\beta}\_2(\mathbf{x}))}{(\ell-1)! a\_1^{1/(p-1)}\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right)}\right)^{(p-1)} z\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right) = 0 \tag{14}$$

*and*

$$
\omega'(\mathbf{x}) + \beta\_1^{-1}(\beta\_2(\mathbf{x})) \tilde{y}\_{\ell-3}(\mathbf{x}) \omega\left(\beta\_1^{-1}(\beta\_2(\mathbf{x}))\right) = 0 \tag{15}
$$

*are oscillatory, where*

$$\begin{aligned} \widetilde{y}\_0(\mathbf{x}) &:= \left( \frac{1}{y\_{\mathbf{x}\_1}(\mathbf{x}) \mathbf{a}\_1(\mathbf{x})} \int\_{\mathbf{x}}^{\infty} \zeta(s) y\_{\mathbf{x}\_1}(s) c\_2^{(p-1)}(\beta\_2(s)) \mathbf{ds} \right)^{1/(p-1)}, \\\widetilde{y}\_k(\mathbf{x}) &:= \int\_{\mathbf{x}}^{\infty} \widetilde{y}\_{k-1}(s) \mathbf{ds}, \ k = 1, 2, \dots, \ell - 2 \end{aligned}$$

*and*

$$c\_m(\mathbf{x}) := \frac{1}{c\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\alpha})\right)} \left(1 - \frac{\left(\boldsymbol{\beta}\_1^{-1}\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\alpha})\right)\right)^{m-1}}{\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\alpha})\right)^{m-1}c\left(\boldsymbol{\beta}\_1^{-1}\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\alpha})\right)\right)}\right), m = 2, \ell, \ldots$$

*then (1) is oscillatory.*

**Proof.** Let (9) hold. Then, we see that *δ*(*x*), *δ*(*β*1(*x*)) and *δ*(*β*2(*x*)) are positive.

Let (**I**1) hold, from Lemma 2, we find *̟*(*x*) <sup>≥</sup> <sup>1</sup> (ℓ−1) *x̟*′ (*x*) and then *x* <sup>1</sup>−ℓ*̟*(*x*) ′ ≤ 0. Hence, since *β* −1 1 (*x*) ≤ *β* −1 1 *β* −1 1 (*x*) , we obtain

$$
\sigma\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right) \leq \frac{\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right)^{\ell-1}}{\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)^{\ell-1}}\sigma\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right). \tag{16}
$$

From (2), we obtain

$$\begin{split} c\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\delta(\boldsymbol{x}) &=& \mathcal{O}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right) - \delta\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right) \\ &=& \mathcal{O}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right) - \left(\frac{\mathcal{O}\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right)}{c\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right)} - \frac{\delta\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right)}{c\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right)}\right) \\ &\geq& \mathcal{O}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right) - \frac{1}{c\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right)}\mathcal{O}\left(\boldsymbol{\beta}\_{1}^{-1}\left(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{x})\right)\right), \end{split} \tag{17}$$

which with (1), (11) and (17) give

$$\begin{split} & \left( y\_{\mathbf{x}\_{0}}(\mathbf{x}) a\_{1}(\mathbf{x}) \left( \sigma^{(\ell-1)}(\mathbf{x}) \right)^{(p-1)} \right)' \\ & \quad + \frac{y\_{\mathbf{x}\_{0}} \xi(\mathbf{x})}{c^{(p-1)} \left( \boldsymbol{\beta}\_{1}^{-1} (\boldsymbol{\beta}\_{2}(\mathbf{x})) \right)} \left( \sigma \big( \boldsymbol{\beta}\_{1}^{-1} (\boldsymbol{\beta}\_{2}(\mathbf{x})) \big) - \frac{\sigma \big( \boldsymbol{\beta}\_{1}^{-1} \left( \boldsymbol{\beta}\_{1}^{-1} (\boldsymbol{\beta}\_{2}(\mathbf{x})) \right) \big)}{c \big( \boldsymbol{\beta}\_{1}^{-1} \left( \boldsymbol{\beta}\_{1}^{-1} (\boldsymbol{\beta}\_{2}(\mathbf{x})) \right)} \right)^{(p-1)} \leq 0. \end{split} \tag{18}$$

We have that (18), which (16) gives

$$\left(y\_{\mathbf{x}\_{1}}(\mathbf{x})a\_{1}(\mathbf{x})\left(\mathcal{O}^{(\ell-1)}(\mathbf{x})\right)^{(p-1)}\right)' + y\_{\mathbf{x}\_{1}}(\mathbf{x})\zeta(\mathbf{x})c\_{\ell}^{(p-1)}(\beta\_{2}(\mathbf{x}))\mathcal{O}^{(p-1)}\left(\beta\_{1}^{-1}(\beta\_{2}(\mathbf{x}))\right) \le 0. \tag{19}$$

From Lemma 1, we get (13). Therefore, from (19), we obtain

$$\begin{aligned} & \left( y\_{\mathbf{x}\_1}(\mathbf{x}) a\_1(\mathbf{x}) \left( \mathcal{O}^{(\ell-1)}(\mathbf{x}) \right)^{(p-1)} \right)' \\ & \le \ -y\_{\mathbf{x}\_1}(\mathbf{x}) \zeta(\mathbf{x}) \left( \frac{\mu c\_\ell(\boldsymbol{\beta}\_2(\mathbf{x}))}{(\ell-1)!} \left( \boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x})) \right)^{\ell-1} \right)^{(p-1)} \left( \mathcal{O}^{(\ell-1)} \left( \boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x})) \right) \right)^{(p-1)} . \end{aligned}$$

Then, if we set *z*(*x*) = *yx*<sup>0</sup> (*x*)*α*1(*x*) *̟*(ℓ−1) (*x*) (*p*−1) , then we have that *z* > 0 is a solution of the delay inequality

$$z'(\mathbf{x}) + \zeta(\mathbf{x}) \frac{y\_{\mathbf{x}\_1}(\mathbf{x})}{y\_{\mathbf{x}\_1}\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right)} \left(\frac{\mu\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right)^{\ell-1}c\_\ell(\boldsymbol{\beta}\_2(\mathbf{x}))}{(\ell-1)!\boldsymbol{\alpha}\_1^{1/(p-1)}\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right)}\right)^{(p-1)} z\left(\boldsymbol{\beta}\_1^{-1}(\boldsymbol{\beta}\_2(\mathbf{x}))\right) \leq 0.$$

It is clear (see [17] Theorem 1) that the Equation (14) also has a positive solution, this is a contradiction.

Let (**I**2) hold, from Lemma 2, we obtain

$$
\omega \sigma(\mathbf{x}) \ge \mathbf{x} \sigma'(\mathbf{x}) \tag{20}
$$

and then *x* <sup>−</sup>1*̟*(*x*) ′ <sup>≤</sup> 0. Hence, since *<sup>β</sup>* −1 1 (*x*) ≤ *β* −1 1 *β* −1 1 (*x*) , we get

$$\left(\mathscr{O}\_{1}^{-1}\left(\mathscr{S}\_{1}^{-1}(\mathbbm{x})\right)\right) \leq \frac{\mathscr{S}\_{1}^{-1}\left(\mathscr{S}\_{1}^{-1}(\mathbbm{x})\right)}{\mathscr{S}\_{1}^{-1}(\mathbbm{x})} \varpi\left(\mathscr{S}\_{1}^{-1}(\mathbbm{x})\right),\tag{21}$$

which with (18) yields

$$\left(y\_{\mathbf{x}\_{1}}(\mathbf{x})a\_{1}(\mathbf{x})\left(\mathcal{a}^{(\ell-1)}(\mathbf{x})\right)^{(p-1)}\right)' + \zeta(\mathbf{x})y\_{\mathbf{x}\_{1}}(\mathbf{x})c\_{2}^{(p-1)}(\beta\_{2}(\mathbf{x}))\mathcal{a}^{(p-1)}\left(\beta\_{1}^{-1}(\beta\_{2}(\mathbf{x}))\right) \le 0. \tag{22}$$

Integrating (22) from *x* to ∞, we obtain

$$\begin{split} -\mathcal{a}^{(\ell-1)}(\mathbf{x}) &\leq \ -\Big(\frac{1}{y\_{\mathbf{1}}(\mathbf{x})\mathbf{a}\_{1}(\mathbf{x})}\int\_{\mathbf{x}}^{\infty}\zeta(s)y\_{\mathbf{1}\_{\mathbf{1}}}(s)c\_{2}^{(p-1)}(\boldsymbol{\beta}\_{2}(\mathbf{s}))\boldsymbol{\alpha}^{(p-1)}\Big(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{\beta}\_{2}(\mathbf{s}))\Big)ds\Big)^{1/(p-1)} \\ &\leq \ -\tilde{y}\_{0}(\mathbf{x})\boldsymbol{\omega}\Big(\boldsymbol{\beta}\_{1}^{-1}(\boldsymbol{\beta}\_{2}(\mathbf{x}))\Big). \end{split}$$

Integrating this inequality ℓ − 3 times from *<sup>x</sup>* to <sup>∞</sup>, we find

$$
\varpi''(\mathbf{x}) + \tilde{y}\_{\ell-3}(\mathbf{x})\varpi\left(\beta\_1^{-1}(\beta\_2(\mathbf{x}))\right) \le 0,\tag{23}
$$

which with (20) gives

$$
\omega''(\mathfrak{x}) + \beta\_1^{-1}(\beta\_2(\mathfrak{x})) \widetilde{y}\_{\ell-3}(\mathfrak{x}) \mathfrak{o}' \left(\beta\_1^{-1}(\beta\_2(\mathfrak{x}))\right) \le 0.
$$

Thus, if we put *ω*(*x*) := *̟*′ (*x*), then we conclude that *ω* > 0 is a solution of

$$
\omega'(\mathbf{x}) + \beta\_1^{-1}(\beta\_2(\mathbf{x})) \tilde{y}\_{\ell-3}(\mathbf{x}) \omega\left(\beta\_1^{-1}(\beta\_2(\mathbf{x}))\right) \le 0. \tag{24}
$$

It is clear (see [17] Theorem 1) that the equation (15) also has a positive solution, this is a contradiction. The proof is complete.

Next, we establish new oscillation conditions for Equation (1) according to the results obtained some related contributions to the subject.

**Corollary 1.** *Assume that c*<sup>0</sup> < 1 *and (30) hold. If*

$$\liminf\_{\chi \to \infty} \int\_{\beta\_2(x)}^{\chi} (1 - c(\beta\_2(s)))^{(p-1)} \zeta(s) \frac{y\_{\mathfrak{n}\_0}(s)}{y\_{\mathfrak{n}\_0}(\beta\_2(s))} \left( \frac{\mu \beta\_2^{\ell-1}(s)}{a\_1^{1/(p-1)}(\beta\_2(s))} \right)^{(p-1)} ds > \frac{((\ell-1)!)^{(p-1)}}{\mathfrak{e}} \tag{25}$$

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

**Corollary 2.** *Let (3) and (30) hold. If*

$$\liminf\_{x \to \infty} \int\_{\beta\_1^{-1}(\beta\_2(x))}^x \zeta(s) \frac{y\_{x\_0}(s)}{y\_{x\_0}\left(\beta\_1^{-1}(\beta\_2(s))\right)} \left(\frac{\mu\left(\beta\_1^{-1}(\beta\_2(s))\right)^{\ell-1} c\_\ell(\beta\_2(s))}{a\_1^{1/(p-1)}\left(\beta\_1^{-1}(\beta\_2(s))\right)}\right)^{(p-1)} ds > \frac{((\ell-1)!)^{(p-1)}}{\mathsf{e}} \tag{26}$$

*and*

$$\liminf\_{x \to \infty} \int\_{\beta\_1^{-1}(\beta\_2(x))}^{x} \beta\_1^{-1}(\beta\_2(s)) \tilde{y}\_{\ell-3}(s) ds > \frac{1}{\mathbf{e}} \tag{27}$$

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

#### **3. Applications**

This section presents some interesting application which are addressed based on above hypothesis to show some interesting results in this paper.

**Example 1.** *Let the equation*

$$\left(\delta(\mathbf{x}) + \frac{1}{2}\delta\left(\frac{\mathbf{x}}{3}\right)\right)^{(4)}(\mathbf{x}) + \frac{1}{\mathbf{x}}\boldsymbol{\sigma}^{(3)}(\mathbf{x}) + \frac{\zeta\_0}{\mathbf{x}^4}\delta\left(\frac{\mathbf{x}}{2}\right) = \mathbf{0},\tag{28}$$

*where ζ*<sup>0</sup> > 0 *is a constant. Let p* = 2, ℓ = 4, *α*1(*x*) = 1, *α*2(*x*) = 1/*x*, *ζ*(*x*) = *ζ*0/*x* 4 , *β*2(*x*) = *x*/2 *and β*1(*x*) = *x*/3*. So, we get*

$$y\_{\mathbf{x}\_0}(\mathbf{x}) = \mathbf{x}\_\prime \ y\_{\mathbf{x}\_0}(\beta\_2(\mathbf{x})) = \mathbf{x}/2.$$

*Thus, we find*

$$\begin{split} &\liminf\_{\mathbf{x}\to\mathbf{0}}\int\_{\beta\_{2}(\mathbf{x})}^{\mathbf{x}} (1-c(\beta\_{2}(\mathbf{s})))^{(p-1)} \zeta(s) \frac{y\_{\mathbf{x}\mathbf{0}}(s)}{y\_{\mathbf{x}\mathbf{0}}(\beta\_{2}(\mathbf{s}))} \left(\frac{\mu\beta\_{2}^{\ell-1}(\mathbf{s})}{a\_{1}^{1/(p-1)}(\beta\_{2}(\mathbf{s}))}\right)^{(p-1)} \mathbf{ds} \\ &= \ \liminf\_{\mathbf{x}\to\mathbf{0}}\int\_{\mathbf{x}/2}^{\mathbf{x}} \frac{\zeta\_{0}}{\mathbf{x}^{4}} \left(\frac{\mathbf{x}^{3}}{8}\right) \mathbf{ds} = \frac{\zeta\_{0}}{8} \ln 2. \end{split}$$

*Hence, the condition becomes*

$$
\zeta\_0 > \frac{48}{\text{e} \ln 2}.\tag{29}
$$

*Therefore, by Corollary 1, every solution of (28) is oscillatory if ζ*<sup>0</sup> > 25.5*.*

**Remark 1.** *Consider the equation (8), by Corollary 1, all solution of (8) is oscillatory if ζ*<sup>0</sup> > 57.5*. Whereas, the criterion obtained from the results of [9,10] are ζ*<sup>0</sup> > 1839.2 *and ζ*<sup>0</sup> > 59.5*. So, our results extend the results in [9].*

#### **4. Conclusions**

In this paper, we obtain sufficient criteria for oscillation of solutions of higher-order differential equation with middle term. We discussed the oscillation behavior of solutions for Equation (1). We obtain some oscillation criteria by comparison method with first order equations. Our results unify and improve some known results for differential equations with middle term. In future work, we will discuss the oscillatory behavior of these equations using integral averaging method and under condition

$$\int\_{x\_0}^{\infty} \left( \frac{1}{\mathfrak{a}\_1(s)} \exp \left( - \int\_{x\_0}^s \frac{\mathfrak{a}\_2(\mathcal{O})}{\mathfrak{a}\_1(\mathcal{O})} d\mathcal{O} \right) \right)^{1/(p-1)} ds < \infty. \tag{30}$$

For researchers interested in this field, and as part of our future research, there is a nice open problem which is finding new results in the following cases:

$$\begin{array}{llll} (\mathbf{F}\_1) & \mathcal{o}(\mathbf{x}) > 0, & \mathcal{o}'(\mathbf{x}) > 0, & \mathcal{o}''(\mathbf{x}) > 0, & \mathcal{o}^{(\ell-1)}(\mathbf{x}) > 0, & \mathcal{o}^{(\ell)}(\mathbf{x}) < 0, \\\ (\mathbf{F}\_2) & \mathcal{o}(\mathbf{x}) > 0, & \mathcal{o}^{(j)}(\mathbf{x}) > 0, & \mathcal{o}^{(j+1)}(\mathbf{x}) < 0 \text{ for all odd integers} \end{array}$$

$$j \in \{1, 3, \dots, \ell - 3\}, \mathcal{o}^{(\ell - 1)}(\mathfrak{x}) > 0, \mathcal{o}^{(\ell)}(\mathfrak{x}) < 0.$$

**Author Contributions:** Conceptualization, S.A. Formal analysis, O.B.; Methodology, S.A., O.B. and M.Y.; Software, O.B.; Writing—original draft, S.A., O.B. and M.Y.; Writing—review and editing, S.A. and M.Y. All authors have read and agreed to the published version of the manuscript.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

#### **References**


### *Article* **On the Stability of la Cierva's Autogiro**

#### **M. Fernández-Martínez 1,‡, Juan L.G. Guirao 2,3,\* ,‡**


Received: 17 October 2020; Accepted: 13 November 2020; Published: 15 November 2020 -

**Abstract:** In this paper, we rediscover in detail a series of unknown attempts that some Spanish mathematicians carried out in the 1930s to address a challenge posed by Mr. la Cierva in 1934, which consisted of mathematically justifying the stability of la Cierva's autogiro, the first practical use of the direct-lift rotary wing and one of the first helicopter type aircraft.

**Keywords:** la Cierva's autogiro; la Cierva's equation; stability; differential equation with periodic coefficients

#### **1. Introduction**

The autogiro was the first practical use of the direct-lift rotary wing, where a windmilling rotor replaces the wing of the airplane, and the propulsive force is generated by a propeller. Interestingly, the autogiro allows a very slow flight and also behaves like an airplane in cruise. This kind of aircraft was developed by the Spanish aeronautical engineer Mr. Juan de la Cierva y Codorníu (Murcia (Spain), 1895–Croydon (UK), 1936), who also coined the term "autogiro". The origins of the autogiro come back to 1919, when an airplane that had been designed by Mr. la Cierva crashed due to stall near the ground. This fact encouraged him to design an aircraft with both a low landing speed and take-off.

Mr. la Cierva evolved the autogiro over the years. Firstly, the C-3 autogiro, which included a five-bladed rigid rotor, was built in 1922. The use of articulated rotor blades on the autogiro was suggested later, and Mr. la Cierva was the first to successfully apply a flap hinge in a rotary-wing aircraft. The C-4 autogiro (1923), which equipped a four-bladed rotor with flap hinges on the blades, was proved to fly with success. Thereafter, in 1924, it was built the C-6 autogiro with a rotor consisting of four flapping blades. This type, which is considered to be the first successful model of la Cierva's autogiro, took part in a demonstration at the Royal Aircraft Establishment the next year (c.f. [1]).

The Cierva Autogiro Company was founded in 1925 in UK by Mr. la Cierva, and about 500 autogiros were built in the next decade, many of them under license of the Cierva Company. In this regard, and for illustration purposes, Figure 1 depicts an autogiro constructed under license by Pitcairn in the United States (c.f. [2]). In those times, the autogiro was described as an easy to handle and fast aircraft, ahead of its time, which could land almost without rolling and take off in less than 30 m, and being able to stop off in the air, just to name some of its features. Certainly, the autogiro developments had an effect on the subsequent helicopter developments. Presently, however, the aircraft design seems to have evolved differently from the times of la Cierva's autogiro. In fact, novel settings consisting of combinations of four or more electric motors driving blades of carbon fiber will allow for less pollution and noise, and also lead to higher efficient aircraft. From a

mathematical viewpoint, several problems related to modern aviation have been addressed by means of Fractional Calculus (c.f. [3]), path planning algorithms (c.f. [4]), or non-linear hyperbolic partial differential equations (c.f. [5]), to name some groundbreaking techniques.

**Figure 1.** The picture above (public domain) shows a PCA-2 autogiro built in the United States by Pitcairn under license of the Cierva Company. This unit was used by the National Advisory Committee for Aeronautics (NACA) for research purposes on rotor systems (c.f. [2]).

One of the first versions of la Cierva's aircraft, the C-3 autogiro, exhibited a certain tendency to fall over side-ways [1]. This issue made him to pay special attention to several aspects related to the stability of the autogiro. In this regard, in 1934, he attended a lecture at the Escuela Superior Aerotécnica (Madrid - Spain), and posed the following linear differential equation with periodic coefficients [6]:

$$m\frac{\mathrm{d}^2\Theta}{\mathrm{d}\,\varrho^2} + \left(\frac{3}{4} + \lambda\,\sin\varrho\right)\frac{\mathrm{d}\,\Theta}{\mathrm{d}\,\varrho} + \left(m + \lambda\,\cos\varrho + \frac{3}{4}\lambda^2\,\sin(2\varrho)\right)\Theta = 0,\tag{1}$$

where *ϕ* is the azimuthal angle of the autogiro's blade, Θ is a function of *ϕ* that measures the angle of deviation of the blade with respect to its position of dynamic equilibrium when rotating, *λ* is a parameter that provides a relationship between the forward speed of the aircraft and the peripheral speed, and *m* is the ratio of the mass of the air volume (assumed to be contained in a rectangular parallelepiped with sides equal to the radius of the rotor and the width of the blade, twice) to the mass of the blade. The periodic nature of the coefficients of that equation is clear due to the autogiro's blade movement.

Following [7], we shall refer to Equation (1) as la Cierva's equation hereafter. It is worth mentioning that Mr. la Cierva appeared interested in mathematically determine whether the expression that bears his name admits convergent solutions since it could imply positive consequences concerning the stability of the autogiro. However, that expression resisted the attempts by Spanish and British mathematicians to that date, and in fact, some articles requiring the attention of mathematicians to address that equation can be found in the press of the time (c.f., e.g., [6]).

Next, let us provide some further comments regarding the parameters *λ* and *m* that are involved in Equation (1). Firstly, notice that *λ* increases as the speed does. In this way, Mr. la Cierva posed *λ* = 1 as an appropriate limit value, thus taking into account future evolutions of the autogiro, the so-called ultrarrapid autogiro. On the other hand, Mr. la Cierva suggested the parameter *m* to vary in the

range [0.15, 1], depending on the aircraft. However, for a given autogiro, that parameter remains constant except in the case of large variations concerning the air density. As such, *m* = 0.5 was then considered to be an acceptable average value.

As stated in [7], Mr. la Cierva was especially interested in mathematically justifying the stability of the movement of the blades of the autogiro rather than quantitatively integrating Equation (1) for certain initial conditions. It is worth mentioning that such a stability had been fully verified in all the autogiros that had been assembled until then, and was also expected for higher speeds of values of the parameter *λ*. As such, the problem regarding the stability of la Cierva's autogiro could be mathematically stated in the following terms: does Θ go to zero as *ϕ* is increased regardless of the initial conditions? Regarding the latter, the reader may think of possible gusts of wind that could affect the movement of the blasts of the aircraft.

The main goal of this paper is to unveil the unknown attempts that some Spanish mathematicians carried out in the 1930s to solve the problem of the stability of la Cierva's autogiro. As such, the structure of this paper is as follows. Section 2 contains some preliminaries regarding differential equations with periodic coefficients. In this way, the concepts of characteristic exponent, characteristic number, and characteristic equation will be introduced. Section 3 describes in detail the first attempt of Prof. Orts y Aracil to analytically integrate Equation (1). Section 4 develops the calculations made by Prof. Orts y Aracil leading to sufficient conditions to guarantee that Equation (1) possesses convergent solutions. Shortly thereafter, the renowned Spanish engineer and mathematician Pedro Puig Adam (Barcelona (Spain), 1900–Madrid (Spain), 1960), Ph.D. in mathematics in 1921, published a qualitative approach regarding the stability of la Cierva's autogiro. Their calculations, which we have described in detail, have been included in Section 5 together with numerical calculations we have carried out in Mathematica. On the other hand, Section 6 contains some results that Puig-Adam obtained in regard to the reduced la Cierva's equation. Finally, Section 7 presents some additional remarks to complete the present study.

#### **2. Preliminaries**

In this section, we recall the basics on differential equations with periodic coefficients, thus paying special attention to the key concepts of characteristic exponent, characteristic number, and characteristic equation associated with a differential equation with periodic coefficients.

Firstly, it is clear that the so-called la Cierva's equation (c.f. Equation (1)) stands as a particular case of the following expression:

$$\frac{\mathrm{d}^2 y(\mathbf{x})}{\mathrm{d}\,\mathrm{x}^2} + p\_1(\mathbf{x})\frac{\mathrm{d}\,y(\mathbf{x})}{\mathrm{d}\,\mathrm{x}} + p\_2(\mathbf{x})\,y(\mathbf{x}) = \mathbf{0},\tag{2}$$

where *p*1(*x*) and *p*2(*x*) are continuous and *ω*−periodic functions (with *ω* = 2*π* in the case of la Cierva's equation). Furthermore, if *y*(*x*) is a solution of Equation (2), then *y*(*x* + *ω*) also is.

Let *y*1(*x*) and *y*2(*x*) be two linearly independent solutions of Equation (2). Hence, *y*1(*x* + *ω*) and *y*2(*x* + *ω*) also are. Thus, we can write

$$\begin{aligned} y\_1(\mathbf{x} + \omega) &= a\_{11} y\_1(\mathbf{x}) + a\_{12} y\_2(\mathbf{x}) \\ y\_2(\mathbf{x} + \omega) &= a\_{21} y\_1(\mathbf{x}) + a\_{22} y\_2(\mathbf{x}) .\end{aligned} \tag{3}$$

Moreover, the coefficients *aij* : *i*, *j* = 1, 2 in Equation (3) could be calculated just by assigning particular values to the independent variable *x*.

Let *<sup>a</sup>* <sup>∈</sup> <sup>R</sup> and *<sup>ϕ</sup>*(*x*) be a *<sup>ω</sup>*−periodic function. Then the logarithmic derivative of the function *η*(*x*) := *e ax ϕ*(*x*) (i.e., *<sup>η</sup>* ′ (*x*) *η*(*x*) ) is also *ω*−periodic, though *η*(*x*) is not. In fact, it holds that

$$
\eta(\mathbf{x} + \omega) = e^{a(\mathbf{x} + \omega)} \,\, \boldsymbol{\varrho}(\mathbf{x} + \omega) = e^{a\omega} \,\, e^{a\mathbf{x}} \,\, \boldsymbol{\varrho}(\mathbf{x}) = e^{a\omega} \,\, \eta(\mathbf{x})\tag{4}$$

for all *x* ∈ dom (*η*). Thus, if the variable *x* is increased in *ω* units, then the image of *x* + *ω* by *η* coincides with *η*(*x*) multiplied by a factor equal to *s* := *e <sup>a</sup>ω*. In this context, *a* is named the characteristic exponent, whereas the factor *s* is known as the characteristic number. Notice that either the characteristic number or the characteristic exponent provides information about whether *η*(*x*) goes to zero as *<sup>x</sup>* → <sup>∞</sup>. In particular, if |*s*| < 1, then *<sup>µ</sup>*(*x*) → 0 as *<sup>x</sup>* → <sup>∞</sup>, which means that the oscillations of the movement of the autogiro blade would get dampened. In fact, the amplitude of the oscillations of that blade would be multiplied by a factor less than the unit each new rotation. As such, we are interested in the calculation of those characteristic numbers, *s*.

Let *ϕ*(*x*) be a *ω*−periodic solution of Equation (2). Then we can write *η*(*x*) as a linear combination of both *y*1(*x*) and *y*2(*x*), namely

$$
\eta(\mathbf{x}) = \mathbb{C}\_1 \, y\_1(\mathbf{x}) + \mathbb{C}\_2 \, y\_2(\mathbf{x}).\tag{5}
$$

Hence, we have that

$$\begin{split} \eta(\mathbf{x} + \boldsymbol{\omega}) &= \mathbb{C}\_1 \, y\_1(\mathbf{x} + \boldsymbol{\omega}) + \mathbb{C}\_2 \, y\_2(\mathbf{x} + \boldsymbol{\omega}) \\ &= \mathbb{C}\_1 \left( a\_{11} \, y\_1(\mathbf{x}) + a\_{12} \, y\_2(\mathbf{x}) \right) + \mathbb{C}\_2 \left( a\_{21} \, y\_1(\mathbf{x}) + a\_{22} \, y\_2(\mathbf{x}) \right) \\ &= \left( \mathbb{C}\_1 \, a\_{11} + \mathbb{C}\_2 \, a\_{21} \right) y\_1(\mathbf{x}) + \left( \mathbb{C}\_1 \, a\_{12} + \mathbb{C}\_2 \, a\_{22} \right) y\_2(\mathbf{x}) \\ &= s \, \eta(\mathbf{x}) = s \, \mathbb{C}\_1 y\_1(\mathbf{x}) + s \, \mathbb{C}\_2 y\_2(\mathbf{x}), \end{split} \tag{6}$$

where the identity at Equation (5) has been used in the first equality, Equation (3) has been applied in the second identity, the fourth one is a consequence of *η*(*x*) assumed to be *ω*−periodic and Equation (4), and the last identity is due to *η*(*x*) being a particular solution of Equation (2) (c.f. Equation (5)). By identifying coefficients between the expressions at both the third and fifth equalities of Equation (6), it holds that

$$\begin{aligned} \mathbb{C}\_{1} \left( a\_{11} - s \right) + \mathbb{C}\_{2} a\_{21} &= 0 \\ \mathbb{C}\_{1} a\_{12} + \mathbb{C}\_{2} \left( a\_{22} - s \right) &= 0. \end{aligned} \tag{7}$$

Therefore, the so-called characteristic equation stands from the following expression:

$$
\begin{vmatrix} a\_{11} - s & a\_{21} \\ a\_{12} & a\_{22} - s \end{vmatrix} = 0,\tag{8}
$$

which is equivalent to

$$s^2 - \left(a\_{11} +\_{22}\right)s + \left[a\_{11}a\_{22} - a\_{12}a\_{21}\right] = 0.\tag{9}$$

Assume that the polynomial in Equation (9) possesses two distinct roots, *s*<sup>1</sup> and *s*2. If both of them are introduced in Equation (7), then a pair of specific values for each constant *C*<sup>1</sup> and *C*<sup>2</sup> will be obtained, thus leading to a pair of functions, *η*1(*x*) and *η*2(*x*) (c.f. Equation (5)) satisfying the condition at Equation (4). Accordingly, each solution of Equation (2) could be written as a linear combination of the functions *ηi*(*x*) : *i* = 1, 2. Following the above, the next result holds.

**Theorem 1.** *If the polynomial in Equation (8) has two distinct roots being less than the unit in absolute value, then ηi*(*x*) : *i* = 1, 2 *go to zero as x goes to infinity. More generally, any solution of Equation (2) would go to zero as x goes to infinity.*

A consequence of Theorem 1 is that the movement of the blade of the autogiro will be in equilibrium regardless the initial conditions.

We conclude this section by providing the statement of a known result concerning harmonic combinations of periodic functions. In fact,

**Theorem 2.** *Let <sup>α</sup>*, *<sup>β</sup>* <sup>∈</sup> <sup>R</sup> *with <sup>α</sup>* <sup>6</sup><sup>=</sup> <sup>0</sup>*. Then*

$$
\mu \sin \mathfrak{x} + \beta \cos \mathfrak{x} = A \sin(\mathfrak{x} + \gamma)\mathfrak{y}
$$

*.*

$$where \ \gamma = \arctan\left(\frac{\beta}{a}\right) \text{ and } A = \text{sgn}(a)\sqrt{a^2 + \beta^2}$$

The proof of that result becomes straightforward by using that sin(*x* + *γ*) = sin *x* cos *γ* + cos *x* sin *γ*, and identiying coefficients with those from *α* sin *x* + *β* cos *x*. This result will be applied in forthcoming Section 6.

#### **3. Towards a Particular Solution of la Cierva's Equation**

In this section, we revisit in detail a first approach that Prof. José Mª Orts y Aracil (Paterna, Valencia (Spain), 1891–Barcelona (Spain), 1968) contributed in [8] to mathematically determine a particular solution to Equation (1). First, it is clear that la Cierva's equation can be rewritten as follows:

$$\frac{\mathrm{d}^2\Theta}{\mathrm{d}\,\varrho^2} + \frac{1}{m} \left( \frac{3}{4} + \lambda \sin\varrho \right) \frac{\mathrm{d}\,\Theta}{\mathrm{d}\,\varrho} + \frac{1}{m} \left( m + \lambda \cos\varrho + \frac{3}{4} \lambda^2 \sin(2\varrho) \right) \Theta = 0. \tag{10}$$

Let Θ = *u e<sup>v</sup>* , where both *u* and *v* are functions of *ϕ*. Then it is clear that

$$
\Theta' = e^v \left( u' + v' u \right) \qquad \Theta'' = e^v \left( u'' + 2v' u' + (v'^2 + v'') u \right). \tag{11}
$$

If we replace the expressions at Equation (11) in Equation (10), then we have

$$\begin{aligned} e^{\nu} \left( u^{\prime \prime} + 2 \, \nu^{\prime} u^{\prime} + \left( v^{\prime 2} + v^{\prime \prime} \right) u \right) + \frac{1}{m} \, \left( \frac{3}{4} + \lambda \sin \varphi \right) \left( u^{\prime} + v^{\prime} u \right) e^{\upsilon} \\ + \frac{1}{m} \left( m + \lambda \cos \varphi + \frac{3}{4} \lambda^{2} \sin(2\varphi) \right) u \, e^{\upsilon} = 0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} u'' + \left( 2v' + \frac{1}{m} \left( \frac{3}{4} + \lambda \sin \varphi \right) \right) u' \\ + \left( v'' + v'^2 + \frac{1}{m} \left( \frac{3}{4} + \lambda \sin \varphi \right) v' + 1 + \frac{\lambda}{m} \cos \varphi + \frac{3\lambda^2}{4m} \sin(2\varphi) \right) u &= 0. \end{aligned} \tag{12}$$

Next, we cancel the coefficient of *u* ′ in Equation (12). In fact,

$$2v' + \frac{1}{m} \left(\frac{3}{4} + \lambda \sin \varphi\right) = 0 \Leftrightarrow v' = -\frac{1}{2m} \left(\frac{3}{4} + \lambda \sin \varphi\right). \tag{13}$$

Hence, it is clear that

$$v = \frac{1}{2m} \left(\lambda \cos \varphi - \frac{3}{4} \varphi \right) \text{ and } v'' = -\frac{\lambda}{2m} \cos \varphi. \tag{14}$$

As such, Equation (12) can be reduced as follows:

$$
u'' + p(\boldsymbol{\varrho})\,\boldsymbol{u} = 0,\\
$$

where

$$p(\varphi) = v^{\prime\prime} + v^{\prime 2} + \frac{1}{m} \left(\frac{3}{4} + \lambda \sin \varphi\right) v^{\prime} + 1 + \frac{\lambda}{m} \cos \varphi + \frac{3\lambda^2}{4m} \sin(2\varphi). \tag{15}$$

If we replace the expressions in both Equations (13) and (14) into Equation (15), then

$$p(\varphi) = -\frac{1}{4m^2} \left( \frac{9}{16} + \lambda^2 \sin^2 \varphi + \frac{3\lambda}{2} \sin \varphi \right) + \frac{\lambda}{2m} \cos \varphi + 1 + \frac{3\lambda^2}{4m} \sin(2\varphi). \tag{16}$$

Moreover, if we replace sin<sup>2</sup> *ϕ* by <sup>1</sup> 2 (1 − cos(2*ϕ*)) in Equation (16), then we have

$$\begin{aligned} p(\boldsymbol{\varrho}) &= 1 - \frac{9}{64m^2} - \frac{\lambda^2}{8m^2} + \frac{\lambda}{2m} \cos \boldsymbol{\varrho} + \frac{\lambda^2}{8m^2} \cos(2\boldsymbol{\varrho}) \\ &- \frac{3\lambda}{8m^2} \sin \boldsymbol{\varrho} + \frac{3\lambda^2}{4m} \sin(2\boldsymbol{\varrho}). \end{aligned}$$

As such, Equation (12) can be expressed in the following terms:

$$\frac{\text{d}^2 \,\mu}{\text{d} \,\rho^2} = \left( a\_0 + a\_1 \cos \varphi + a\_2 \cos(2\varphi) + b\_1 \sin \varphi + b\_2 \sin(2\varphi) \right) u\_\prime \tag{17}$$

where

$$\begin{aligned} a\_0 &= \frac{9 + 8\,\lambda^2}{64\,m^2} - 1, \quad a\_1 = -\frac{\lambda}{2\,m'}, \quad a\_2 = -\frac{\lambda^2}{8\,m^2} \\ b\_1 &= \frac{3}{8}\,\frac{\lambda}{m^2}, \quad b\_2 = -\frac{3}{4}\,\frac{\lambda^2}{m}. \end{aligned} \tag{18}$$

Additionally, by writing *u* = *e* R *z* d *ϕ* , the expression in Equation (17) can be rewritten as follows:

$$\frac{\mathrm{d}\,z}{\mathrm{d}\,\varphi} + z^2 = a\_0 + a\_1 \cos\varphi + a\_2 \cos(2\varphi) + b\_1 \sin\varphi + b\_2 \sin(2\varphi),\tag{19}$$

which leads to a Ricatti type equation. The next expression was suggested by Prof. Orts y Aracil as a potential solution of Equation (19):

$$z\_1 = \mathfrak{a} + \beta \sin \mathfrak{q} + \gamma \cos \mathfrak{q},\tag{20}$$

where *α*, *β*, and *γ* are three constants that can be determined by introducing Equation (20) in the former Equation (19) and identifying coefficients in both sides of that expression. As such, we obtain that

$$\begin{array}{llll} a\_0 = a^2 + \frac{1}{2} \left( \beta^2 + \gamma^2 \right), & a\_1 = \beta + 2 \text{ a } \gamma, \quad a\_2 = \frac{1}{2} \left( \gamma^2 - \beta^2 \right) \\\ b\_1 = 2 \text{ a } \beta - \gamma, & b\_2 = \beta \text{ } \gamma. \end{array} \tag{21}$$

Next, we observe that

$$a\_2^2 + b\_2^2 = \left(\frac{1}{2}\left(\gamma^2 + \beta^2\right)\right)^2 \ge 0\_\prime$$

so *α* <sup>2</sup> + q *a* 2 <sup>2</sup> + *b* 2 <sup>2</sup> = *α* <sup>2</sup> + <sup>1</sup> 2 (*γ* <sup>2</sup> + *β* 2 ) = *a*0. Therefore,

$$a\_0 \ge \frac{1}{2} \left(\gamma^2 + \beta^2\right) = \sqrt{a\_2^2 + b\_2^2}.$$

On the other hand, it is clear that

$$
\sqrt{a\_2^2 + b\_2^2} - a\_2 = \frac{1}{2} \left( \gamma^2 + \beta^2 \right) - \frac{1}{2} \left( \gamma^2 - \beta^2 \right) = \beta^2 \ge 0.
$$

Furthermore, it holds that *a*<sup>2</sup> + q *a* 2 <sup>2</sup> + *b* 2 <sup>2</sup> <sup>=</sup> <sup>1</sup> 2 (*γ* <sup>2</sup> <sup>−</sup> *<sup>β</sup>* 2 ) + <sup>1</sup> 2 (*γ* <sup>2</sup> + *β* 2 ) = *γ* <sup>2</sup> <sup>≥</sup> 0. All the calculations above lead to the following values of the parameters *α*, *β*, and *γ* of the particular solution at Equation (19):

$$\alpha = \sqrt{a\_0 - \sqrt{a\_2^2 + b\_2^2}}, \quad \beta = \sqrt{\sqrt{a\_2^2 + b\_2^2} - a\_2}, \quad \gamma = \sqrt{a\_2 + \sqrt{a\_2^2 + b\_2^2}}.$$

Going beyond, it is possible to reduce the parameters *α*, *β*, and *γ* in Equation (21), thus leading to a pair of relationships among the coefficients *a<sup>i</sup>* and *b<sup>j</sup>* for *i* = 0, 1, 2 and *j* = 1, 2. Recall that *a<sup>i</sup>* and *b<sup>j</sup>* can be expressed, in turn, in terms of *λ* and *m* (c.f. Equation (18)). In fact, the following expressions hold.

$$1327104\,m^8 + 359424\,m^6 + 35712\,m^4 - 324\,m^2 = 0$$

$$\lambda^2 = \frac{9 + 16\,m^2 - (9 - 48\,m^2)\sqrt{1 + 36\,m^2}}{8\sqrt{1 + 36\,m^2}\left(1 - \sqrt{1 + 36\,m^2}\right)}.\tag{22}$$

If the eight order polynomial in *m* at Equation (22) is divided by 12 *m*<sup>2</sup> (under the assumption that *<sup>m</sup>* <sup>6</sup><sup>=</sup> 0), and the change of variable *<sup>t</sup>* <sup>=</sup> <sup>48</sup> *<sup>m</sup>*<sup>2</sup> is considered, then the following third order polynomial stands:

$$t^3 + 13t^2 + 62t - 27 = 0.\tag{23}$$

By Bolzano's Theorem, it is clear that the polynomial in Equation (23) possesses a root, say *t*1, in the subinterval [0.4007, 0.4008]. That root could be approximated by some numerical method, though in [8], *t*<sup>1</sup> was considered merely as the middle point of that subinterval, i.e., *t*<sup>1</sup> = 0.40075. Since *t*<sup>1</sup> = 48 *m*<sup>2</sup> 1 , then we have *m*<sup>1</sup> ≃ 0.0914. Hence, the second expression in Equation (22) leads to *λ*<sup>1</sup> ≃ 0.7249. With the values of both parameters *m* and *λ* estimated, the coefficients *α*, *β*, and *γ* of the particular solution of Equation (19) given by Equation (20) can be calculated by Equation (18). In fact, that particular solution remains as follows:

$$z\_1 = 3.8391 + 4.1036 \sin \varphi + 1.0511 \cos \varphi. \tag{24}$$

Also, we have *u*<sup>1</sup> = exp (3.8391 *ϕ* + 1.0511 sin *ϕ* − 4.1036 cos *ϕ*), and hence,

$$\begin{aligned} \Theta\_1 &= \exp(3.8391 \,\varrho + 1.0511 \,\sin \varphi) \\ &+ 0.6898 \left(0.0914 \,\cos \varphi - \frac{3}{4} \,\varphi \right) - 4.1036 \,\cos \varphi), \end{aligned}$$

stands as a particular solution of Equation (10), the differential equation which models the equilibrium of the blade of la Cierva's autogiro.

As Prof. Orts y Aracil commented, the approach contributed in this section threw a value of *λ*<sup>1</sup> = 0.7249 lying within the range suggested by Mr. Herrera in [6], i.e., the subinterval [0, 1], though the value of *m*<sup>1</sup> = 0.091 appears out of its corresponding range, the subinterval [0.15, 1]. In this regard, it was argued that the problem of the equilibrium of la Cierva's autogiro had been addressed from a mathematical (and not an Saee) viewpoint.

#### **4. Sufficient Conditions on the Existence of Convergent Solutions**

In this section, sufficient conditions are provided to guarantee the existence of convergent solutions for la Cierva's equation, an issue that was further addressed by Prof. Orts y Aracil in [9]. With this aim, we start by sketching an alternative approach to that one described in Section 3 with the aim to integrate the expression at Equation (10).

First of all, let us denote by *p*(*ϕ*) the continuous and periodic function that appears at the right term of Equation (19), i.e.,

$$p(\varphi) = a\_0 + a\_1 \cos \varphi + a\_2 \cos(2\varphi) + b\_1 \sin \varphi + b\_2 \sin(2\varphi),\tag{25}$$

which allows rewriting Equation (17) as follows:

$$\frac{\text{d}^2 \,\mu}{\text{d} \,\rho^2} = p(\rho) \,\text{u.}\tag{26}$$

Such a kind of differential equations can be integrated by means of a characteristic equation of the form

$$s^2 - As + 1 = 0,\text{ where}\tag{27}$$

$$\begin{aligned} A &= 2 + \sum\_{n=1}^{+\infty} \left[ F\_n(2\pi) + f\_n'(2\pi) \right], & F\_n(\varphi) &= \int\_0^{\varphi} \mathbf{d} \, \boldsymbol{\varrho} \int\_0^{\varphi} p(\boldsymbol{\varrho}) \, F\_{n-1}(\boldsymbol{\varrho}) \, \mathbf{d} \, \boldsymbol{\varrho}, \\\ f\_n(\boldsymbol{\varrho}) &= \int\_0^{\varphi} \mathbf{d} \, \boldsymbol{\varrho} \int\_0^{\varphi} p(\boldsymbol{\varrho}) \, f\_{n-1}(\boldsymbol{\varrho}) \, \mathbf{d} \, \boldsymbol{\varrho}, \quad F\_0(\boldsymbol{\varrho}) = 1, \text{ and } f\_0(\boldsymbol{\varrho}) &= \boldsymbol{\varrho} \end{aligned} \tag{28}$$

(c.f. ([10] [item 49, p. 402]) and ([11] [Chapter 3, Section 55])). Moreover, if *p*(*ϕ*) ≥ 0 for all *ϕ* > 0, then it holds that *Fn*(*ϕ*), *fn*(*ϕ*), *f* ′ *n* (*ϕ*) <sup>&</sup>gt; 0 for all *<sup>ϕ</sup>* <sup>&</sup>gt; 0 and all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Hence, *<sup>A</sup>* <sup>&</sup>gt; 2 and the expression at Equation (27) possesses two positive roots, say *s*<sup>1</sup> and *s*2, with one of them being greater (resp., smaller) than the unit and the other being smaller (resp., greater) than the unit.

A fundamental system of solutions for Equation (26) is provided by the functions

$$u\_1 = e^{\frac{\varrho}{2\pi} \int\_{S\_1}} \cdot \mathfrak{a}(\varrho), \qquad u\_2 = e^{\frac{\varrho}{2\pi} \int\_{S\_2}} \cdot \mathfrak{f}(\varrho), \tag{29}$$

where *α*(*ϕ*) and *β*(*ϕ*) are 2*π*−periodic continuous functions.

Hence, one of the integrals at Equation (29), say *<sup>u</sup>*1, goes to zero as *<sup>ϕ</sup>* → <sup>∞</sup>, and so does <sup>Θ</sup>. In this way, the so-called Liapounov's condition can be stated as follows (c.f. [10]).

**Theorem 3** (Liapounov's condition)**.** *The second order differential equation in Equation (26) admits a convergent integral as <sup>ϕ</sup>* → <sup>∞</sup>*, if and only if, p*(*ϕ*) ≥ <sup>0</sup> *for all <sup>ϕ</sup>* > <sup>0</sup>*.*

Following the above, our next goal is to verify that sufficient condition. To deal with, let us apply the change of variable *x* = tan( *ϕ* 2 ) to the periodic function at Equation (25). As such, we have

$$\begin{split} p(\boldsymbol{\varphi}) &= a\_0 + a\_1 \cos \boldsymbol{\varphi} + a\_2 \frac{1 - \tan^2 \boldsymbol{\varphi}}{1 + \tan^2 \boldsymbol{\varphi}} + b\_1 \sin \boldsymbol{\varphi} + b\_2 \frac{2 \tan \boldsymbol{\varphi}}{1 + \tan^2 \boldsymbol{\varphi}} \\ &= a\_0 + a\_1 \frac{1 - \mathbf{x}^2}{1 + \mathbf{x}^2} + a\_2 \frac{1 - 6 \mathbf{x}^2 + \mathbf{x}^4}{(1 + \mathbf{x}^2)^2} + b\_1 \frac{2 \times}{1 + \mathbf{x}^2} + b\_2 \frac{4 \times (1 - \mathbf{x}^2)}{(1 + \mathbf{x}^2)^2} \end{split} \tag{30}$$

and hence, we can write *p*(*ϕ*) = <sup>1</sup> (1+*x* 2) <sup>2</sup> *q*(*x*), where

$$\begin{split} q(\mathbf{x}) &= a\_0 \left( 1 + \mathbf{x}^2 \right)^2 + a\_1 \left( 1 - \mathbf{x}^2 \right) \left( 1 + \mathbf{x}^2 \right) + a\_2 \left( 1 - 6\mathbf{x}^2 + \mathbf{x}^4 \right) \\ &+ 2 \, b\_1 \ge \left( 1 + \mathbf{x}^2 \right) + 4 \, b\_2 \ge \left( 1 - \mathbf{x}^2 \right) \\ &= \left( a\_0 + a\_1 + a\_2 \right) + 2 \left( b\_1 + 2 \, b\_2 \right) \ge + 2 \left( a\_0 - 3 \, a\_2 \right) \mathbf{x}^2 \\ &+ 2 \left( b\_1 - 2 \, b\_2 \right) \mathbf{x}^3 + \left( a\_0 - a\_1 + a\_2 \right) \mathbf{x}^4. \end{split} \tag{31}$$

Notice that the first equality at Equation (30) has been applied that

$$\sin(2\varphi) = \frac{2\tan\varphi}{1+\tan^2\varphi}, \quad \cos(2\varphi) = \frac{1-\tan^2\varphi}{1+\tan^2\varphi}.$$

whereas the second identity at that expression holds since that change of variable implies that

$$\cos\varphi = \frac{1-\mathfrak{x}^2}{1+\mathfrak{x}^2}, \quad \sin\varphi = \frac{2\mathfrak{x}}{1+\mathfrak{x}^2}, \quad \tan\varphi = \frac{2\tan(\frac{\varphi}{2})}{1-\tan^2(\frac{\varphi}{2})} = \frac{2\mathfrak{x}}{1-\mathfrak{x}^2}.$$

Moreover, by writing

$$q(\mathbf{x}) = c\_4 + c\_3 \mathbf{x} + c\_2 \mathbf{x}^2 + c\_1 \mathbf{x}^3 + c\_0 \mathbf{x}^4,\tag{32}$$

we can identify coefficients with those ones at the right side of Equation (31). In fact,

$$\begin{aligned} c\_0 &= a\_0 - a\_1 + a\_2 = \frac{9}{64 \, m^2} - 1 + \frac{\lambda}{2 \, m} \\ c\_1 &= 2 \left( b\_1 - 2 \, b\_2 \right) = \frac{3}{4} \, \frac{\lambda}{m^2} + 3 \, \frac{\lambda^2}{m} \\ c\_2 &= 2 \left( a\_0 - 3 \, a\_2 \right) = \frac{9}{32 \, m^2} - 2 + \frac{\lambda^2}{m^2} \\ c\_3 &= 2 \left( b\_1 + 2 \, b\_2 \right) = \frac{3}{4} \, \frac{\lambda}{m^2} - 3 \, \frac{\lambda^2}{m} \\ c\_4 &= a\_0 + a\_1 + a\_2 = \frac{9}{64 \, m^2} - \frac{\lambda}{2 \, m} - 1 \end{aligned} \tag{33}$$

where Equation (18) allows writing the *c<sup>i</sup>* 's in terms of the parameters *λ* and *m*.

On the other hand, a necessary condition to get *p*(*ϕ*) ≥ 0 for all *ϕ* > 0 consists of both coefficients *c*<sup>0</sup> and *c*<sup>4</sup> of the polynomial at Equation (32) being positive. In this way, Equation (33) implies that

$$\begin{aligned} \alpha\_4 > 0 &\Leftrightarrow 32m \left(\lambda + 2m\right) < 9 \Leftrightarrow X^2 - Y^2 < 9. \\ \alpha\_0 > 0 &\Leftrightarrow 32m \left(2m - \lambda\right) < 9 \Leftrightarrow m < \frac{3}{8} \Leftrightarrow X - Y < 3, \end{aligned} \tag{34}$$

where *X* := 8*m* + 2*λ* and *Y* := 2*λ*. Observe that *X*,*Y* > 0 since both parameters *m* and *λ* are positive. In fact, regarding the second equivalence at the first line of Equation (34), just observe that we can write

$$\begin{aligned} 9 &> 32 \, m \, (\lambda + 2m) = 64 \, m^2 + 32 \, m \lambda \\ &= 8^2 m^2 + 2 \times 16 \, m \lambda + (2\lambda)^2 - (2\lambda)^2 \\ &= (8m + 2\lambda)^2 - (2\lambda)^2 = X^2 - Y^2. \end{aligned}$$

Thus, the condition *c*<sup>4</sup> > 0 is equivalent to a point at the first quadrant, (*X*,*Y*), located above the hyperbola *X* <sup>2</sup> <sup>−</sup> *<sup>Y</sup>* <sup>2</sup> = 9.

#### **5. Puig-Adam's Qualitative Approach**

In this section, we revisit in detail the approach contributed by Puig-Adam in [7] to approach the solutions of la Cierva's equation from a qualitative viewpoint.

According to the contents of Section 2, we are interested in obtaining two particular solutions of la Cierva's equation (c.f. Equation (10)), say *y*1(*x*) and *y*2(*x*). Let them be given by the following initial conditions:

$$\begin{aligned} y\_1(0) &= 1, \quad & y\_1'(0) = 0 \\ y\_2(0) &= 0, \quad & y\_2'(0) = 1. \end{aligned} \tag{35}$$

It is clear that *y*1(*x*) and *y*2(*x*) would be independent since their Wronskian at 0 is distinct from zero, *W*(*y*1, *y*2)(0) = 1. Moreover, from Equation (3), we have that

$$\begin{aligned} y\_1'(\mathbf{x} + \omega) &= a\_{11} y\_1'(\mathbf{x}) + a\_{12} y\_2'(\mathbf{x}) \\ y\_2'(\mathbf{x} + \omega) &= a\_{21} y\_1'(\mathbf{x}) + a\_{22} y\_2'(\mathbf{x}) \end{aligned} \tag{36}$$

for all *x*. If we particularize both Equations (3) and (36) in *x* = 0, then

$$\begin{aligned} y\_1(\omega) &= a\_{11\prime} & y\_1'(\omega) &= a\_{12} \\ y\_2(\omega) &= a\_{21\prime} & y\_2'(\omega) &= a\_{22\prime} \end{aligned}$$

Moreover, from Equation (9), the characteristic equation holds from the following expression:

$$s^2 - \left(y\_1(\omega) + y\_2'(\omega)\right)s + \left[y\_1(\omega)\,y\_2'(\omega) - y\_1'(\omega)\,y\_2(\omega)\right] = 0. \tag{37}$$

As such, Equation (37) would be fully determined once *yi*(*ω*) and their derivatives, *y* ′ *i* (*ω*) for *i* = 1, 2, have been calculated. It is also worth mentioning that the coefficients of the characteristic polynomial are independent from the initial conditions that were selected, i.e., such coefficients only depend on the coefficients of the given differential equation. In particular, notice that the independent term of Equation (37), which coincides with *W*(*y*1, *y*2)(*ω*), can be calculated in terms of *p*1(*x*) (recall Equation (2)), by means of the following expression (c.f., e.g., [11]):

$$\mathcal{W}(y\_1, y\_2)(\mathbf{x}) = \mathcal{W}(y\_1, y\_2)(\mathbf{x}\_0) \cdot \exp\left[-\int\_{\mathbf{x}\_0}^{\mathbf{x}} p\_1(\mathbf{x}) \, \mathbf{d} \, \mathbf{x}\right].\tag{38}$$

In fact, observe that the former expression can be justified just by identifying the differential equation in Equation (2) with the next one:

$$W(y\_1 y\_1, y\_2)(\mathbf{x}) = \mathbf{0}.\tag{39}$$

In fact, Equation (39) is equivalent to

$$\begin{aligned} \left( \left( y\_1(\mathbf{x}) \, y\_2'(\mathbf{x}) - y\_1'(\mathbf{x}) \, y\_2(\mathbf{x}) \right) y\_1''(\mathbf{x}) + \left( y\_1''(\mathbf{x}) \, y\_2(\mathbf{x}) - y\_1(\mathbf{x}) \, y\_2''(\mathbf{x}) \right) y\_1'(\mathbf{x}) \right) \\ + \left( y\_1'(\mathbf{x}) \, y\_2''(\mathbf{x}) - y\_1''(\mathbf{x}) \, y\_2'(\mathbf{x}) \right) y(\mathbf{x}) = 0, \end{aligned}$$

which leads to

$$y''(\mathbf{x}) + \frac{y\_1''(\mathbf{x}) \, y\_2(\mathbf{x}) - y\_1(\mathbf{x}) \, y\_2''(\mathbf{x})}{y\_1(\mathbf{x}) \, y\_2'(\mathbf{x}) - y\_1'(\mathbf{x}) \, y\_2(\mathbf{x})} y'(\mathbf{x}) + \frac{y\_1'(\mathbf{x}) \, y\_2''(\mathbf{x}) - y\_1''(\mathbf{x}) \, y\_2'(\mathbf{x})}{y\_1(\mathbf{x}) \, y\_2'(\mathbf{x}) - y\_1'(\mathbf{x}) \, y\_2(\mathbf{x})} y(\mathbf{x}) = 0 \tag{40}$$

since *y*1(*x*) and *y*2(*x*) have been assumed to be independent solutions (and hence, *W*(*y*1, *y*2)(*x*) 6= 0 for all *x*). Thus, if the expressions in both Equations (2) and (40) coincide term by term, then it holds that

$$p\_1(\mathbf{x}) = \frac{y\_1'''(\mathbf{x}) \, y\_2(\mathbf{x}) - y\_1(\mathbf{x}) \, y\_2''(\mathbf{x})}{y\_1(\mathbf{x}) \, y\_2'(\mathbf{x}) - y\_1'(\mathbf{x}) \, y\_2(\mathbf{x})} = -\frac{\mathcal{W}'(y\_1, y\_2)(\mathbf{x})}{\mathcal{W}(y\_1, y\_2)(\mathbf{x})}.$$

Following the above, it holds that the independent term of Equation (37) can be obtained just by applying Equation (38) in the open interval (*x*0, *x*) = (0, *ω*). Since *W*(*y*1, *y*2)(0) = 1, then we have that

$$\begin{split} \mathcal{W}(y\_1, y\_2)(2\pi) &= \exp\left[-\int\_0^{2\pi} p\_1(\mathbf{x}) \, \mathbf{d} \, \mathbf{x}\right] \\ &= \exp\left[-\int\_0^{2\pi} \frac{1}{m} \left(\frac{3}{4} + \lambda \sin \mathbf{x}\right) \mathbf{d} \, \mathbf{x}\right] = e^{-\frac{3}{2m}\pi} \end{split} \tag{41}$$

where it has been used that *p*1(*x*) = <sup>1</sup> *m* 3 <sup>4</sup> + *λ* sin *x* and *ω* = 2*π* in the case of la Cierva's equation (c.f. Equation (10)). Hence, the characteristic polynomial associated to la Cierva's equation remains as follows:

$$s^2 - \left(y\_1(2\pi) + y\_2'(2\pi)\right)s + e^{-\frac{3}{2\pi}\cdot\pi} = 0.\tag{42}$$

Interestingly, it holds that the independent term, *e* <sup>−</sup> <sup>3</sup> 2*m <sup>π</sup>*, does not depend on the forward speed.

However, to fully determine the characteristic polynomial at Equation (42), it becomes necessary to know the values of both functions *y*1(*x*) and *y* ′ 2 (*x*) at *x* = 2*π*. With this aim, Puig-Adam, instead of carrying out a power series expansion in regard to the periodic coefficients of the starting equation, for instance, preferred to apply a (second order) Runge-Kutta numerical approach to each particular solution, *y*1(*x*) and *y*2(*x*), of la Cierva's equation in the closed bounded interval [0, 2*π*] with parameters *m* = 0.5 and *λ* = 1, that according to Puig-Adam, had been suggested by Mr. la Cierva. In [7], it was stated that the trapezoidal method had been applied. In this paper, though, we shall apply a explicit midpoint method (also known as modified Euler method), which appears implemented in Mathematica. In any case, both of them are second-order approaches.

In this way, and similarly to [7], Figures 1 and 2 depicts our approximations to each particular solution of Mr. la Cierva's equation, *y*1(*x*) with initial conditions *y*1(0) = 1, *y* ′ 1 (0) = 0, and *y*2(*x*) with initial conditions *y*2(0) = 0, *y* ′ 2 (0) = 1 (c.f. Equation (35)), as provided by the second-order (Runge-Kutta explicit) midpoint approach on the interval [0, 2*π*], which corresponds to a turn of the blade of the autogiro.

**Figure 2.** Second order Runge-Kutta approximations (obtained by the explicit midpoint method) to each particular solution of la Cierva's equation according to the procedure described in Section 5, i.e., *y*1(*ϕ*) (blue line) and *y*2(*ϕ*), where *ϕ* varies in the range [0, 2*π*], which means a turn of the blade of the autogiro, and the choice of parameters was as suggested by Mr. la Cierva, i.e., *m* = 0.5 and *λ* = 1.

According to our numerical calculations, it holds that

*y*1(2*π*) = −0.0222528, *y* ′ 2 (2*π*) = 0.0230689,

and hence, the characteristic polynomial at Equation (42) remains as follows:

$$s^2 - 0.000816093 \, s + e^{-\frac{3}{20} \cdot \pi} = 0.\tag{43}$$

As such, it holds that the polynomial at Equation (43) possesses two complex (conjugated) roots, namely *s*<sup>1</sup> = 0.000408046 − 0.00897402 *i* and *s*<sup>2</sup> = 0.000408046 + 0.00897402 *i*. Since |*s<sup>i</sup>* | = 0.00898329 ≪ 1 for

*i* = 1, 2, it can be guaranteed that the blade movement of la Cierva's autogiro behaves quite stably for that choice of parameters.

**Remark 1.** *It is worth mentioning that, in the original study carried out by Puig-Adam, the following values were obtained by the numerical approach carried out therein: y*1(2*π*) = −0.013 *and y* ′ 2 (2*π*) = 0.04197*, which led to the next characteristic equation:*

$$s^2 - 0.02897s + e^{-\frac{3}{2m}\pi} = 0\_\prime$$

*whose real roots are t*<sup>1</sup> = 0.00312209 *and t*<sup>2</sup> = 0.0258479*. Such results mainly differ from ours in the nature of the roots of the characteristic polynomial. That issue was mainly caused by the approximation errors made due to the limitations of the calculation systems available in the 1930s. It is also true that we have approximated the coefficients y*1(2*π*) *and y* ′ 2 (2*π*) *by the midpoint method instead of the trapezoidal approach used by Puig-Adam. However, both are second-order approaches, so they should lead to close results.*

Recall also that *W*(*y*1, *y*2)(2*π*) = *e* <sup>−</sup>3*<sup>π</sup>* <sup>≃</sup> 8.0699518 <sup>×</sup> <sup>10</sup>−<sup>5</sup> (c.f. Equation (41)). Alternatively, if we calculate an approximation to that Wronskian by means of the expression appeared at Equation (37) and the values of the coefficients provided by the numerical approach used by Puig-Adam, then we have

$$\begin{split} W(y\_1, y\_2)(2\pi) &= y\_1(2\pi) \, y\_2'(2\pi) - y\_1'(2\pi) \, y\_2(2\pi) \\ &\simeq -0.013 \times 0.04197 + 0.00398 \times 0.18509 \\ &\simeq 1.910482 \times 10^{-4} = W\_{\text{PA}}(y\_1, y\_2)(2\pi) \end{split} \tag{44}$$

where *W*PA(*y*1, *y*2)(2*π*) denotes the Puig-Adam's numerical approximation to that quantity. As such, the absolute error obtained when comparing the theoretical value of that Wronskian with respect to *<sup>W</sup>*PA(*y*1, *<sup>y</sup>*2)(2*π*), (c.f. Equation (44)) was found to be approximately equal to 1.10349 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , quite close to zero. Going beyond, our midpoint-based approach, which approximated *W*(*y*1, *y*2)(2*π*) by the quantity 8.0699523 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , threw an absolute error approximately equal to 5.33809 <sup>×</sup> <sup>10</sup>−<sup>12</sup> .

Furthermore, it is possible to provide a qualitative viewpoint in regard to the stability of the oscillations of the blade of the autogiro in its upcoming turns. In fact, let *ω* = 2*π* and consider Equation (35). Applying such initial conditions to both Equations (3) and (36), it holds that the former turns into the following expression:

$$\begin{aligned} y\_1(\mathbf{x} + 2\pi) &= y\_1(2\pi) \, y\_1(\mathbf{x}) + y\_1'(2\pi) \, y\_2(\mathbf{x}) \\ y\_2(\mathbf{x} + 2\pi) &= y\_2(2\pi) \, y\_1(\mathbf{x}) + y\_2'(2\pi) \, y\_2(\mathbf{x}) . \end{aligned} \tag{45}$$

By recursively applying Equation (45), we have

$$\begin{aligned} y\_1(\mathbf{x} + 4\pi) &= y\_1(2\pi)y\_1(\mathbf{x} + 2\pi) + y\_1'(2\pi) \, y\_2(\mathbf{x} + 2\pi) \\ &= \left( y\_1^2(2\pi) + y\_1'(2\pi) \, y\_2(2\pi) \right) y\_1(\mathbf{x}) \\ &+ y\_1'(2\pi) \left( y\_1(2\pi) + y\_2'(2\pi) \right) y\_2(\mathbf{x}) \\ y\_2(\mathbf{x} + 4\pi) &= y\_2(2\pi) \, y\_1(\mathbf{x} + 2\pi) + y\_2'(2\pi) \, y\_2(\mathbf{x} + 2\pi) \\ &= y\_2(2\pi) \left( y\_1(2\pi) + y\_2'(2\pi) \right) y\_1(\mathbf{x}) \\ &+ \left( y\_2'(2\pi) + y\_2(2\pi) \, y\_1'(2\pi) \right) \, y\_2(\mathbf{x}) \end{aligned} \tag{46}$$

which corresponds to the second turn of the blade. Figure 3 depicts (numerical approximations of) both solutions after two turns of the autogiro's blade. Also, regarding the third turn of the blade, the following expression holds:

$$\begin{aligned} y\_1(\mathbf{x} + 6\pi) &= a\_1 y\_1(\mathbf{x}) + a\_2 y\_2(\mathbf{x}) \\ y\_2(\mathbf{x} + 6\pi) &= \beta\_1 y\_1(\mathbf{x}) + \beta\_2 y\_2(\mathbf{x}) \end{aligned} \tag{47}$$

where

$$\begin{aligned} a\_1 &= \left( y\_1^2(2\pi) + y\_1'(2\pi) \, y\_2(2\pi) \right) \, y\_1(2\pi) + y\_1'(2\pi) \left( y\_1(2\pi) + y\_2'(2\pi) \right) \, y\_2(2\pi) \\ a\_2 &= \left( y\_1^2(2\pi) + y\_1'(2\pi) \, y\_2(2\pi) \right) \, y\_1'(2\pi) + y\_1'(2\pi) \left( y\_1(2\pi) + y\_2'(2\pi) \right) \, y\_2'(2\pi) \\ \beta\_1 &= y\_2(2\pi) \, \left( y\_1(2\pi) + y\_2'(2\pi) \right) \, y\_1(2\pi) + \left( y\_2^2(2\pi) + y\_2(2\pi) \, y\_1'(2\pi) \right) \, y\_2(2\pi) \\ \beta\_2 &= y\_2(2\pi) \, \left( y\_1(2\pi) + y\_2'(2\pi) \right) \, y\_1'(2\pi) + \left( y\_2^2(2\pi) + y\_2(2\pi) \, y\_1'(2\pi) \right) \, y\_2'(2\pi) . \end{aligned} \tag{48}$$

As with Figure 3, (numerical approximations) of the particular solutions of la Cierva's equation (for parameters *m* = 0.5 and *λ* = 1) after three turns of the autogiro's blade are illustrated at Figure 4. It can be seen that for angles beyond <sup>5</sup>*<sup>π</sup>* 2 , the graph of the first particular solution of la Cierva's equation at the second turn of the blade becomes indistinguishable from the *x*−axis, as it is the case of the plot of *y*2(*x*) as of the third turn of the blade.

Notice that, as Puig-Adam pointed out, the initial conditions *y*1(0) = 1, *y* ′ 2 (0) = 1 (c.f. Equation (35)) are quite extreme. Nevertheless, for *k* small enough, particular solutions of the form *ky*<sup>1</sup> and *ky*2, which exhibit smaller oscillations than those from *y*<sup>1</sup> and *y*2, and whose graphs can be depicted by a *y*−axis rescaling of those appeared in Figure 2, are possible.

**Figure 3.** (Numerical approximations to the) particular solutions of la Cierva's equation after two turns of the autogiro's blade (c.f. Equation (46)). In this occassion, the blue lines have been used to distinguish the curves of both particular solutions in regard to the first turn to their prolongations to the second turn of the blade. In addition, notice that the dotted line corresponds to *y*2(*ϕ*).

**Figure 4.** (Numerical approximations to the) particular solutions of la Cierva's equation after the first three turns of the autogiro's blade (c.f. Equations (47) and (48)). The blue lines represent the curves of both particular solutions at the first turn, the orange lines correspond to their prolongations to the second turn of the blade, and the green lines depict the extensions of such solutions to the third turn. As with Figure 3, the dotted line corresponds to *y*2(*ϕ*).

#### **6. La Cierva's Reduced Equation**

The aim of this section is to calculate a pair of particular solutions to la Cierva's equation by means of the so-called reduced la Cierva's equation. Furthermore, a comparison of such solutions with those solutions obtained in Section 5 is carried out.

First, recall that in Section 5, it was provided a numerical criterion to determine whether the solutions of la Cierva's equation (c.f. Equation (1)) behave stably for a choice of parameters (*λ*, *m*). Specifically, let *y*<sup>1</sup> and *y*<sup>2</sup> be the particular solutions of that equation (as provided by a Runge-Kutta method, in this case) in the interval [0, 2*π*], and calculate *y*1(2*π*) + *y* ′ 2 (2*π*). If that quantity stands <1 in absolute value, then the behavior of the oscillations of the autogiro's blade is stable for such parameters.

Firstly, we recall the original expression of la Cierva's equation (c.f. Equation (1)):

$$m\Theta'' + \left(\frac{3}{4} + \lambda\sin\varphi\right)\Theta' + \left(m + \lambda\cos\varphi + \frac{3}{4}\lambda^2\sin(2\varphi)\right)\Theta = 0,\tag{49}$$

where *ϕ* is the azimuthal angle of the autogiro's blade and Θ is a function of *ϕ* that measures the angle of deviation of the blade with respect to its position of dynamic equilibrium when rotating.

Let Θ = *uv*. Then it is clear that Θ′ = *u* ′*v* + *uv*′ and Θ′′ = *u* ′′*v* + 2 *u* ′*v* ′ + *uv*′′. If we apply that change of variable to Equation (49), then that expression turns into the next one:

$$\begin{aligned} m\left(u''v + 2\,u'v' + uv''\right) + \left(\frac{3}{4} + \lambda\,\sin\varphi\right)\left(u'v + uv'\right) \\ + \left(m + \lambda\,\cos\varphi + \frac{3}{4}\,\lambda^2\,\sin(2\varphi)\right)uv &= 0, \end{aligned} \tag{50}$$

which is equivalent to

$$\begin{aligned} m v u'' + \left( 2m v' + \left( \frac{3}{4} + \lambda \sin \varphi \right) v \right) u' \\ + \left( m v'' + \left( \frac{3}{4} + \lambda \sin \varphi \right) v' + \left( m + \lambda \cos \varphi + \frac{3}{4} \lambda^2 \sin(2\varphi) \right) v \right) u \\ = 0. \end{aligned} \tag{51}$$

The next goal is to cancel the coefficient of *u* ′ in Equation (51). In fact,

$$\left(2mv' + \left(\frac{3}{4} + \lambda\sin\varphi\right)v = 0 \Leftrightarrow \frac{v'}{v} = -\frac{1}{2m}\left(\frac{3}{4} + \lambda\sin\varphi\right).\tag{52}$$

The integration of the expression in Equation (52) leads to

$$v = \exp\left[\frac{\lambda}{2m}\cos\varphi - \frac{3}{8m}\varphi\right].\tag{53}$$

As such, Equation (50) has been reduced to the next one:

$$m\,u'' + \left(m\,\frac{v''}{v} + \left(\frac{3}{4} + \lambda\sin\varphi\right)\frac{v'}{v} + m + \lambda\cos\varphi + \frac{3}{4}\lambda^2\sin(2\varphi)\right)u = 0.\tag{54}$$

Since

$$\frac{\mathbf{d}}{\mathbf{d}\,\,\phi} \left( \frac{v'}{v} \right) = \frac{v''}{v} - \left( \frac{v'}{v} \right)^2,\tag{55}$$

then it is clear that

$$\begin{split} \frac{v''}{v} &= \left(\frac{v'}{v}\right)^2 + \frac{\mathbf{d}}{\mathbf{d}\,\varrho} \left(\frac{v'}{v}\right) \\ &= \left[ -\frac{1}{2m} \left( \frac{3}{4} + \lambda \sin\varrho \right) \right]^2 - \frac{\lambda}{2m} \cos\varrho. \end{split}$$

Hence, Equation (54) can be rewritten as *u* ′′ = −*q*(*ϕ*) *u*, where

$$\begin{aligned} q(\boldsymbol{\varrho}) &= 1 + \frac{\lambda}{2m} \cos \boldsymbol{\varrho} + \frac{3}{4m} \lambda^2 \sin(2\boldsymbol{\varrho}) - \frac{1}{4m^2} \left( \frac{3}{4} + \lambda \sin \boldsymbol{\varrho} \right)^2 \\ &= 1 + \frac{\lambda}{2m} \cos \boldsymbol{\varrho} + \frac{3}{4m} \lambda^2 \sin(2\boldsymbol{\varrho}) - \frac{9}{64} m^2 - \frac{\lambda^2}{4m^2} \sin^2 \boldsymbol{\varrho} - \frac{3\lambda}{8m^2} \sin \boldsymbol{\varrho}. \end{aligned}$$

Firstly, notice that we can write

$$\frac{3\lambda}{8m^2}\sin\varphi - \frac{\lambda}{2m}\cos\varphi = A\sin(\varphi + \varphi\_1)\lambda$$

where *A* = *<sup>λ</sup>* 2*m* q 1 + 3 4*m* 2 and *<sup>ϕ</sup>*<sup>1</sup> <sup>=</sup> arctan(−<sup>4</sup> <sup>3</sup> *<sup>m</sup>*). In fact, just apply Theorem 2 for *<sup>α</sup>* <sup>=</sup> <sup>3</sup>*<sup>λ</sup>* <sup>8</sup>*m*<sup>2</sup> > 0 and *<sup>β</sup>* <sup>=</sup> <sup>−</sup> *<sup>λ</sup>* 2*m* .

On the other hand, we also affirm that

$$\frac{\lambda^2}{4m^2}\sin^2\varphi - \frac{3}{4m}\lambda^2\sin(2\varphi) = B\sin(2\varphi + \varphi\_2)\_{\prime\prime}$$

where *<sup>B</sup>* <sup>=</sup> <sup>−</sup> *<sup>λ</sup>* 2 4*m* q 9 + <sup>1</sup> <sup>4</sup>*m*<sup>2</sup> and *<sup>ϕ</sup>*<sup>2</sup> <sup>=</sup> arctan 1 6*m* . In this case, it has been used that sin<sup>2</sup> *ϕ* = 1 2 (<sup>1</sup> <sup>−</sup> cos(2*ϕ*)), and applied Theorem 2 again for *<sup>α</sup>* <sup>=</sup> <sup>−</sup> <sup>3</sup> 4*m λ* <sup>2</sup> and *<sup>β</sup>* <sup>=</sup> <sup>−</sup> *<sup>λ</sup>* 2 8*m*<sup>2</sup> . Following the above, it holds that Equation (54) is equivalent to the next one, that we shall name as la Cierva's reduced equation, hereafter:

$$u^{\prime\prime} = \left(a + b\sin(\varphi + \varphi\_1) + c\sin(2\varphi + \varphi\_2)\right)u,\tag{56}$$

where

$$\begin{aligned} a &= \frac{9}{64}m^2 + \frac{\lambda^2}{8\,m^2} - 1, \quad b = \frac{\lambda}{2m}\sqrt{1 + \left(\frac{3}{4\,m}\right)^2}, \quad c = -\frac{\lambda^2}{4m}\sqrt{9 + \frac{1}{4\,m^2}}\\ \varphi\_1 &= \arctan\left(-\frac{4}{3}m\right), \quad \varphi\_2 = \arctan\left(\frac{1}{6\,m}\right). \end{aligned} \tag{57}$$

Going beyond, it is possible to turn la Cierva's reduced equation into a Riccati type one. In fact, similarly to Equation (55), we have that

$$\frac{u''}{u} = \frac{\mathbf{d}}{\mathbf{d}\,\,\rho} \left(\frac{u'}{u}\right) + \left(\frac{u'}{u}\right)^2 = \frac{\mathbf{d}\,\,\eta}{\mathbf{d}\,\,\rho} + \eta^2 \,\,\rho$$

where the second equality has been denoted *η* := *<sup>u</sup>* ′ *u* . Hence, Equation (56) can be even rewritten in terms of a first order Ricatti type equation:

$$
\eta' = a + b\sin(\varphi + \varphi\_1) + c\sin(2\varphi + \varphi\_2) - \eta^2,\tag{58}
$$

where the coefficients *a*, *b*, *c* appear in Equation (57). In this regard, in [7], Puig-Adam realized that a particular solution to la Cierva's equation had been obtained previously by Prof. Aracil in [8]. Despite the form of that particular solution was similar to the one provided in Equation (58) (c.f. Equation (24)), it is worth pointing out that it was obtained for the choice of parameters *λ* = 0.7249 and *m* = 0.0914 /∈ [0.15, 1], the range proposed by Mr. la Cierva.

As with the numerical analysis carried out in Section 5 regarding la Cierva's equation, next we shall apply the midpoint method approach to a pair of (independent) particular solutions of the reduced la Cierva's equation (c.f. Equation (56)), namely *u*1(*ϕ*) and *u*2(*ϕ*), with initial conditions *u*1(0) = 1, *u* ′ 1 (0) = 0, and *u*2(0) = 0, *u* ′ 2 (0) = 1. Also, the same parameters as in Section 5 will be used, i.e., *λ* = 1 and *m* = 0.5, and both solutions will be numerically approximated in the subinterval [0, 2*π*] (a turn of the autogiro's blade). In this case, the values of the coefficients and angles in Equation (57) are as follows: *a* ≃ 0.0625, *b* ≃ 80278, *c* ≃ −1.58114, *ϕ*<sup>1</sup> ≃ −0.588003, and *ϕ*<sup>2</sup> ≃ 0.321751. Figure 5 depicts both particular solutions.

Our next goal is to compare the particular solutions (obtained by the midpoint method) of la Cierva's equation (c.f. Figure 2) to the ones of the reduced la Cierva's one. Since {*u*1(*ϕ*), *u*2(*ϕ*)} is a fundamental system of solutions of the reduced la Cierva's equation, then {*u*1(*ϕ*) *v*(*ϕ*), *u*2(*ϕ*) *v*(*ϕ*)} is a fundamental system of solutions of la Cierva's equation, where *<sup>v</sup>*(*ϕ*) = exp *λ* 2*m* cos *<sup>ϕ</sup>* <sup>−</sup> <sup>3</sup> 8*m ϕ* (c.f. Equation (53)). Hence, each solution of la Cierva's equation, *y*(*ϕ*), can be expressed in the following terms:

$$\begin{split} \boldsymbol{y}(\boldsymbol{\varrho}) &= \boldsymbol{v}(\boldsymbol{\varrho}) \left( \mathbb{C} \, \boldsymbol{u}\_{1}(\boldsymbol{\varrho}) + \boldsymbol{D} \, \boldsymbol{u}\_{2}(\boldsymbol{\varrho}) \right) \\ &= \boldsymbol{\varepsilon}^{\cos \boldsymbol{\varrho} - \frac{3}{4} \boldsymbol{\varrho}} \left( \mathbb{C} \, \boldsymbol{u}\_{1}(\boldsymbol{\varrho}) + \boldsymbol{D} \, \boldsymbol{u}\_{2}(\boldsymbol{\varrho}) \right) : \mathbb{C}, \boldsymbol{D} \in \mathbb{R}, \end{split} \tag{59}$$

where the last identity has been used that *λ* = 1 and *m* = 0.5. Also, it is clear that

$$y'(\varphi) = e^{\cos\varphi - \frac{3}{4}\varphi} \left[ \mathcal{C}u'\_1 + Du'\_2 - \left(\frac{3}{4} + \sin\varphi \right) \left(\mathcal{C}u\_1 + Du\_2\right) \right]. \tag{60}$$

Since *y*1(0) = 1, then Equation (59) leads to *C* = <sup>1</sup> *e* . Moreover, the condition *y* ′ 1 (0) = 0 applied to Equation (60) implies that *D* = <sup>3</sup> 4*e* . As such, we have

$$\Psi\_1(\varphi) = e^{\cos\varphi - \frac{3}{4}\varphi - 1} \left( u\_1(\varphi) + \frac{3}{4} u\_2(\varphi) \right). \tag{61}$$

On the other hand, the initial condition *y*2(0) = 0 applied to Equation (59) gives *C* = 0. Furthermore, *y* ′ 2 (0) = 1 implies that *D* = <sup>1</sup> *e* . Thus,

$$y\_2(\varphi) = e^{\cos\varphi - \frac{3}{4}\varphi - 1}u\_2(\varphi). \tag{62}$$

Upcoming Figure 6 displays a graphical comparison involving the particular solutions of la Cierva's equation from both Sections 5 and 6. Specifically, the first particular solutions of that equation are depicted in blue (the dotted line corresponds to the expression in Equation (61)), and the second particular solutions appear in orange (the dashed line corresponds to the expression in Equation (62)). Observe that all the curves behave similarly, especially for angles *<sup>ϕ</sup>* <sup>≥</sup> *<sup>π</sup>* 2 .

**Figure 5.** Second order Runge-Kutta approximations (obtained by the explicit midpoint method) to each particular solution from the reduced la Cierva's equation, where the first particular solution, *y*1(*ϕ*) (c.f. Equation (61)), is depicted by a blue line, *ϕ* varies in the range [0, 2*π*], and the choice of parameters was as suggested by Mr. la Cierva, i.e., *m* = 0.5 and *λ* = 1.

**Figure 6.** Second order Runge-Kutta approximations (obtained by the explicit midpoint method) to those pairs of particular solutions of la Cierva's equation that were obtained in Sections 5 and 6, respectively. The blue dotted line depicts the first particular solution of that equation as it appears in Equation (61), whereas the orange dashed line illustrates the second particular solution of la Cierva's equation (c.f. Equation (62)). On the other hand, the continuous curves correspond to the particular solutions of la Cierva's equation as they were obtained in Section 5. As with Figure 5, *ϕ* varies in the range [0, 2*π*], which means a turn of the blade of the autogiro, and the choice of parameters has been as suggested by Mr. la Cierva, i.e., *m* = 0.5 and *λ* = 1. Notice that the *y*-axis has been labeled as Θ(*ϕ*) to denote an approximation to each particular solution of la Cierva's equation for *ϕ* ∈ [0, 2*π*].

#### **7. Final Remarks**

Next, we provide some additional remarks allowing us to complete our study on the stability of la Cierva's autogiro.

1. We recall that the conditions provided in Section 4 to guarantee the existence of convergent solutions for la Cierva's equation are sufficient but not necessary. In fact, let us consider the reduced la Cierva's equation (c.f. Equation (56)), and define

$$q(\varphi) = a + b\sin(\varphi + \varphi\_1) + c\sin(2\varphi + \varphi\_2),\tag{63}$$

where the coefficients *a*, *b*, and *c* are given as in Equation (57). Then for *λ* = 1 and *m* = 0.5, i.e., the choice of parameters used in both Sections 5 and 6, it holds that the function *q*(*ϕ*) is not positive in the whole interval [0, 2*π*] (c.f., e.g., Figure 7). As such, the Liapounov's condition (c.f. Theorem 3) cannot guarantee the existence of convergent solutions in regard to the reduced la Cierva's equation for that choice of parameters. However, as proved in Section 5, la Cierva's equation behaves stably for such parameters.

**Figure 7.** Graph of the function *q*(*ϕ*) (as defined in Equation (63)) in the interval [0, 2*π*].

2. Let

$$y''(\mathbf{x}) + p\_2(\mathbf{x})\,y(\mathbf{x}) = \mathbf{0} \tag{64}$$

be a second order differential equation with *p*1(*x*) = 0, as it is the case of the la Cierva's reduced equation (c.f. Equation (56)). Then its associated characteristic equation can be expressed in the following terms (c.f. Equation (27)):

$$s^2 - As + 1 = 0,\tag{65}$$

.

where the roots of Equation (65) are of the form

$$s\_1 = e^{2\pi \mathfrak{a}} \qquad s\_2 = e^{-2\pi \mathfrak{a}}$$

Hence, it is clear that *A* = *s*<sup>1</sup> + *s*<sup>2</sup> = 2 cosh(2*πα*), which leads to

$$
\alpha = \frac{1}{2\pi} \operatorname{arcosh}\left(\frac{A}{2}\right).
$$

Let Θ = *uv*, where *u* = *e* <sup>±</sup>*α<sup>x</sup>* and *v* being as in Equation (53). Then the aperiodic part of Θ is given by the next expression:

$$\exp\left[\left(-\frac{3}{8m} \pm \alpha\right)|x|\right]$$

Since *α* > 0, then it is clear that

$$-\frac{3}{8m} - \alpha < \alpha - \frac{3}{8m}.$$

As such, *<sup>α</sup>* <sup>−</sup> <sup>3</sup> <sup>8</sup>*<sup>m</sup>* <sup>&</sup>lt; 0 implies <sup>−</sup> <sup>3</sup> <sup>8</sup>*<sup>m</sup>* − *α* < 0. Observe that the stability condition consists of *<sup>α</sup>* <sup>−</sup> <sup>3</sup> <sup>8</sup>*<sup>m</sup>* < 0, which is satisfied whether *A* < 2 cosh( 3*π* 4*m* ). On the other hand, the condition *A* < 2 is fulfilled provided that the characteristic exponent *<sup>α</sup>* <sup>∈</sup> *<sup>i</sup>* <sup>R</sup>. In that case, the aperiodic part of <sup>Θ</sup> is of the form exp <sup>−</sup> <sup>3</sup>*<sup>x</sup>* 8*m* , which goes to 0 as *<sup>x</sup>* → +∞.

Notice that *A* could be approximated by the quantity *u*1(2*π*) + *u* ′ 2 (2*π*) through the midpoint approach, for instance, as carried out in both Sections 5 and 6.

3. In Section 4, it was provided a method, first proposed by Liapounov in [10], which allows calculating the coefficient *A* that appears in characteristic equations of the form Equation (65) that are associated to the next kind of differential equations (c.f. Equation (64)):

$$\frac{\mathrm{d}^2 \, y(\mathfrak{x})}{\mathrm{d}\, \mathfrak{x}^2} = \varepsilon \, p(\mathfrak{x}) \, y(\mathfrak{x}).$$

In fact, it holds that

$$A = 1 + \frac{1}{2} \sum\_{n=1}^{+\infty} \left[ F\_n(\omega) + f\_n' \omega \right] \text{ } \varepsilon^n \text{.} \tag{66}$$

where *ε* ∈ (0, 1), and *Fn*(*ω*) and *fn*(*ω*) being as in Equation (28). On the other hand, in [11], Goursat applied that method for *ε* = 1, thus leading to the expressions contained in Equation (28). However, even under the assumption that the series in Equation (66) is convergent, it holds that such a convergence would be quite slow, especially as the period *ω* increases. As a consequence, that particular expression becomes quite limited to deal with practical applications regarding the calculation of the coefficient *A*.

4. The reader may think, at least at a first glance, that the form of the reduced la Cierva's equation is similar to the one of the generalized Hill's type equation, whose origins go back to the study of the movement of the Moon under the influence of the gravitational field of the system Earth-Sun. That equation admits the following expression:

$$\frac{\mathbf{d}^2 \ y(\mathbf{x})}{\mathbf{d} \, \mathbf{x}^2} + \left[\lambda + \gamma \, \Phi(\mathbf{x})\right] y(\mathbf{x}) = 0 : \lambda \, \gamma \in \mathbb{R}.$$

However, notice that the parameters at the reduced la Cierva's equation, *λ* and *m*, do not appear linearly in Equation (56) (c.f. Equation (57)). As such, the reduced la Cierva's equation cannot be understood as a particular case of the generalized Hill's equation.


almost all la Cierva's surface. On the other hand, Figures 10 and 11 depict a neighborhood of Puig-Adam's choice of parameters where la Cierva's surface behaves stably, as stated in remark (5).

**Figure 8.** La Cierva's surface, *S* = {(*λ*, *m*, *kλ*,*m*) : (*λ*, *m*) ∈ *R*}, where *R* = [0, 1] × [0.15, 1] (above). The plane {(*λ*, *m*, 1) : (*λ*, *m*) ∈ *R*} has been graphically displayed as a benchmark regarding the limit of the stability zone for la Cierva's surface (below).

**Figure 9.** Contours of la Cierva's surface. Observe that the Puig-Adam's choice of parameters, *λ* = 1, *m* = 0.5 is indeed surrounded by a region of points with low *kλ*,*<sup>m</sup>* numbers. Notice that almost all the whole surface behaves stably.

**Figure 10.** A neighborhood of the Puig-Adam's choice of parameters where la Cierva's surface is stable.

**Figure 11.** Contours of a neighborhood of the Puig-Adam's choice of parameters where la Cierva's surface behaves stably.

**Author Contributions:** Conceptualization, M.F.-M. and J.L.G.G.; methodology, M.F.-M. and J.L.G.G.; software, M.F.-M. and J.L.G.G.; validation, M.F.-M. and J.L.G.G.; formal analysis, M.F.-M. and J.L.G.G.; investigation, M.F.-M. and J.L.G.G.; resources, M.F.-M. and J.L.G.G.; data curation, M.F.-M. and J.L.G.G.; writing–original draft preparation, M.F.-M. and J.L.G.G.; writing–review and editing, M.F.-M. and J.L.G.G.; visualization, M.F.-M. and J.L.G.G.; supervision, M.F.-M. and J.L.G.G.; project administration, M.F.-M. and J.L.G.G.; funding acquisition, M.F.-M. and J.L.G.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** Both authors were partially funded by Ministerio de Ciencia, Innovación y Universidades, grant number PGC2018-097198-B-I00, and by Fundación Séneca of Región de Murcia, grant number 20783/PI/18.

**Acknowledgments:** The authors would also like to express their gratitude to the anonymous reviewers whose suggestions, comments, and remarks have allowed them to enhance the quality of this paper. M.F.M. would like to dedicate this work to the memory of Susi, who passed away while writing this article. The authors sincerely appreciate some insightful comments that were made to this manuscript by Prof. Tareq Saeed from King Abdulaziz University at Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


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### *Article* **Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains**

### **Zhen Yang and Junjie Ma \***

School of Mathematics and Statistics, Guizhou University, Guiyang 550025, Guizhou, China **\*** Correspondence: jjma@gzu.edu.cn

Received: 31 July 2020; Accepted: 22 October 2020; Published: 2 November 2020

**Abstract:** In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.

**Keywords:** highly oscillatory integral; Chebyshev polynomial; nearly singular; Levin quadrature rule; adaptive mesh refinement

#### **1. Introduction**

Highly oscillatory integrals frequently arise in acoustic scattering [1], computational physical optics [2], computational electromagnetics [3], and related fields. Generally, dramatically changing integrands make classical approximations perform poor. Therefore, studies on numerical calculation of highly oscillatory integrals have attracted much attention during the past few decades, and a variety of contributions has been made, for example, Filon-type quadrature [4,5], numerical steepest descent method [6], Levin method [7], and so on.

When the phase is nonlinear, researchers usually resort to Levin-type quadrature, which originates from David Levin's pioneering work in [7]. By transforming the oscillatory integration problem into a special ordinary differential equation, one could get an efficient approximation to the generalized Fourier transform with the help of collocation methods. Afterwards, Levin analyzed the convergence rate of the innovative approach in [8]. Analogous to Filon-type quadrature, the Levin-type method based on Hermite interpolation was developed by Olver in [9]. Application of Hermite interpolation definitely increased the convergence rate of the numerical method with respect to the frequency. In [10], Li et al. proposed a stable and high-order Levin quadrature rule by employing the spectral Chebyshev collocation method and truncated singular value decomposition technique. Multiquadric radial basis functions were applied to Levin's equation and an innovative composite Levin method was presented in [11]. Numerical tests manifested that such kind of algorithms was able to deal with stationary problems. Sparse solvers for Levin's equation in one-dimension were constructed by employing recurrence of Chebyshev polynomials in [12,13]. Meanwhile, a class of preconditioners was proposed by the second author to deal with the ill-conditioned linear system in [13]. Molabahrami studied the Galerkin method for Levin's equation and developed the Galerkin–Levin method for oscillatory integrals in [14].

Levin-type quadrature rules were extended to solving singular problems in the past several years. In [15], Wang and Xiang employed the technique of singularity separation and transformed Levin's equation into coupled non-singular ordinary differential equations. By solving the transformed equations numerically, they obtained an efficient Levin quadrature rule for weakly singular integrals with highly oscillatory Fourier kernels. Recently, the second author proposed the fractional Jacobi-Galerkin–Levin quadrature by investigating fractional Jacobi approximations in [16]. Through properly choosing weighted Jacobi polynomials, the discretized Levin's equation was turned into a sparse linear system. It had been verified that the convergence rate of this kind of Levin quadrature rules could be analyzed by studying coefficients of the fractional Jacobi expansion of the error function. In [17], a multi-resolution quadrature rule was applied to deal with the singularity, and the modified Levin quadrature rule coupled with the multi-quadric radial basis function was developed to calculate oscillatory integrals with Bessel and Airy kernels.

Levin's quadrature rule also plays an important role in solving multi-dimensional problems. By introducing a multivariate ordinary differential equation, Levin found a non-oscillatory approximation to the integrand in [7], which led to an efficient algorithm for computing oscillatory integrals over rectangular regions. In [18], Li et al. devised a class of spectral Levin methods for multi-dimensional integrals by utilizing the Chebyshev differential matrix and delaminating quadrature rule. An innovative procedure for multivariate highly oscillatory integrals was devised by employing multi-resolution analysis in [19]. Meanwhile, the meshless approximation was obtained by truncated singular value decomposition. In [20], the second author studied a fast algorithm for Hermite differential matrix by the barycentric formula. With the help of delaminating quadrature, the spectral Levin-type method for calculation of highly oscillatory integrals over rectangular regions was constructed.

Although researchers have made much contribution to numerical calculation of highly oscillatory integrals, little attention has been paid to the computation of nearly singular and highly oscillatory integrals, for example,

$$I[\mathcal{F}, \mathcal{G}\_1, \mathcal{G}\_2, \omega] = \int\_{-1}^{1} \int\_{-1}^{1} \frac{F(\mathbf{x}, y)}{(\mathbf{x} - a)^2 + (\mathbf{x} - b)^2 + \epsilon^2} \mathbf{e}^{\mathbf{i}\omega(\mathcal{G}\_1(\mathbf{x}) + \mathcal{G}\_2(y))} d\mathbf{x} dy. \tag{1}$$

In this paper, we are concerned with efficient computation of Integral (1), and partly fill in the gap in this field. Moreover, we suppose that *F*(*x*, *y*) is analytic with respect to both variables, *G*1(*x*) and *G*2(*y*) are sufficiently smooth functions without stationary points, and the frequency parameter *ω* ≫ 1, (*a*, *<sup>b</sup>*) <sup>∈</sup> <sup>R</sup><sup>2</sup> , and |*ǫ*| ≪ 1.

A large frequency parameter *ω* implies that integrands of Integral (1) are highly oscillatory, and classical quadrature rules suffer from the computational cost. In Table 1, we list numerical results computed by the classical delaminating quadrature rule coupled with Clenshaw–Curtis quadrature (CCQ), where the quadrature nodes are fixed 16. Referenced values are computed by CHEBFUN toolbox (see [21]). CHEBFUN, which approximate functions by Chebyshev interplant, was firstly developed in 2004. Due to the fast and high-order approximation to the integrand, numerical integration methods in CHEBFUN usually provide efficient numerical approaches for univariate and multivariate integrals. Hence, the 2D quadrature method in CHEBFUN is chosen as a benchmark. It can be seen from Table 1 that, as *ω* goes to infinity, CCQ diverges from the referenced values when we do not add quadrature nodes.


**Table 1.** Numerical results of classical delaminating quadrature rules for highly oscillatory multivariate integrals <sup>Z</sup> <sup>1</sup> −1 Z 1 −1 cos(*x* + *y*)e i*ω*(*x*+*y*) *dxdy*.

In contrast to oscillatory integrals arising in existing studies, when the point (*a*, *b*) in Integral (1) is close to or falls in the integration domain and *ǫ* is particularly small, the integrand attains its peak value around (*a*, *b*) and decays dramatically away from such a critical point. In general, the point (*a*, *b*) is called the nearly singular point. Plenty of additional quadrature nodes have to be used if we want to make the numerical formula retain a tolerance error.

There also exist several contributions to tackle the nearly singular problem. The sinh transformation is deemed one of the most important tools. For nearly singular moments arising in Laplace's equation, Johnston et al. proposed the sinh transformation in [22]. Occorsio and Serafini considered two kinds of cubature rules for nearly singular and highly oscillatory integrals in [23]. With the help of 2D-dilation technique, Occorsio and Serafini were able to relax the fast changing integrand and applied Gauss–Jacobi quadrature to the transformed integral. Numerical experiments verified that such an approximation procedure greatly increased the numerical performance of Gauss quadrature.

The remaining parts are organized as follows. In the second section, we review some results with regard to the calculation of Chebyshev series and present the convergence property of Chebyshev interplant and series. In Section 3, we first extend the idea in [13] to two-dimensional oscillatory integrals. Compared with existing Levin quadrature, the new approach has an advantage in computational time. Then, noting that there is little convergence analysis of 2D Levin quadrature rules, we try to fill the gap through examining the modified Levin equation. Finally, we present an innovative composite Levin quadrature rule for solving nearly singular and highly oscillatory problems. In contrast to existing numerical integration methods, the proposed composite method does not suffer from high oscillation and nearly singular amplitudes. Numerical tests included in Section 4 are conducted to verify the efficiency of the proposed approach, and some remarks are concluded in Section 5.

#### **2. Auxillary Tools**

In this section, we first revisit auxillary operators with regard to Chebyshev series, which help develop numerical algorithms for computation of two-dimensional oscillatory integrals. Then, error bounds for coefficients of Chebyshev series and Clenshaw–Curtis interplant are introduced.

When the given function *f*(*x*) is analytic in a sufficiently large domain containing [−1, 1], one can compute its Chebyshev series by (see [24])

$$f(\mathbf{x}) = \sum\_{n=0}^{\infty} f\_n T\_n(\mathbf{x}),$$

with

$$f\_0 = \frac{1}{\pi} \int\_{-1}^{1} \frac{f(s)}{\sqrt{1 - s^2}} ds,\\ f\_n = \frac{2}{\pi} \int\_{-1}^{1} \frac{f(s)T\_n(s)}{\sqrt{1 - s^2}} ds, \ n = 1, 2, \dots, \dots$$

Here, *Tn*(*x*) denotes the first-kind Chebyshev polynomial of order *n*. Noting the relation between the first- and second-kind Chebyshev polynomials *Tn*(*x*), *Un*(*x*) (see [24])

$$\frac{d}{d\mathfrak{x}}T\_n(\mathfrak{x}) = \begin{cases} \begin{array}{c} n!L\_{n-1}(\mathfrak{x}), & n \geq 1, \\ 0, & n = 0, \end{array} \end{cases} \tag{2}$$

and

$$T\_n(\mathbf{x}) = \begin{cases} \mathcal{U}\_0(\mathbf{x}), & n = 0, \\ \frac{1}{2} \mathcal{U}\_1(\mathbf{x}), & n = 1, \\ \frac{1}{2} (\mathcal{U}\_n(\mathbf{x}) - \mathcal{U}\_{n-2}(\mathbf{x})), & n \ge 2, \end{cases} \tag{3}$$

we are able to compute

$$f'(\mathbf{x}) = \sum\_{n=0}^{\infty} f'\_n T\_n(\mathbf{x}) = \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{\infty} (2n+4k+2) f\_{n+2k+1} \right) T\_n(\mathbf{x})\_n$$

which implies Chebyshev coefficients of the derivative can be represented by

$$
\begin{pmatrix} f'\_0 \\ f'\_1 \\ f'\_2 \\ f'\_3 \\ \vdots \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 & 3 & 0 & 5 & \cdots \\ 0 & 0 & 4 & 0 & 8 & 0 & \ddots \\ 0 & 0 & 0 & 6 & 0 & 10 & \ddots \\ 0 & 0 & 0 & 0 & 8 & 0 & \ddots \\ 0 & 0 & 0 & 0 & 8 & 0 & \ddots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \end{pmatrix} \begin{pmatrix} f\_0 \\ f\_1 \\ f\_2 \\ f\_3 \\ \vdots \end{pmatrix} = \mathcal{D} \begin{pmatrix} f\_0 \\ f\_1 \\ f\_2 \\ f\_3 \\ \vdots \end{pmatrix} . \tag{4}
$$

Secondly, suppose that there exists a sufficiently smooth function

$$a(\mathbf{x}) = \sum\_{n=0}^{\infty} a\_n T\_n(\mathbf{x}).$$

Noting the identity (see [24])

$$T\_m(\mathfrak{x})T\_n(\mathfrak{x}) = \frac{1}{2}(T\_{m+n}(\mathfrak{x}) + T\_{|m-n|}(\mathfrak{x}))\_n$$

we can compute the product *a*(*x*)*f*(*x*) by

$$a(\mathfrak{x})f(\mathfrak{x}) = \sum\_{n=0}^{\infty} c\_n T\_n(\mathfrak{x}),$$

where **a** = [*a*0, *a*1, · · · ] *T* , and coefficients {*cn*} ∞ *n*=0 are defined by

$$
\begin{pmatrix} c\_0 \\ c\_1 \\ c\_2 \\ \vdots \end{pmatrix} = \frac{1}{2} \left( \begin{pmatrix} 2a\_0 & a\_1 & a\_2 & a\_3 & \cdots \\ a\_1 & 2a\_0 & a\_1 & a\_2 & \ddots \\ & a\_2 & a\_1 & 2a\_0 & a\_1 & \ddots \\ & \vdots & \ddots & \ddots & \ddots & \ddots \\ \vdots & \ddots & \ddots & \ddots & \ddots \end{pmatrix} + \begin{pmatrix} 0 & 0 & 0 & 0 & \cdots \\ a\_1 & a\_2 & a\_3 & a\_4 & \ddots \\ & a\_2 & a\_3 & a\_4 & a\_5 & \ddots \\ & \vdots & \ddots & \ddots & \ddots & \ddots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots \end{pmatrix} \right) \begin{pmatrix} f\_0 \\ f\_1 \\ f\_2 \\ \vdots \\ \vdots \end{pmatrix}
$$

$$
= \mathcal{M}[\mathbf{a}] \begin{pmatrix} f\_0 \\ f\_1 \\ f\_2 \\ \vdots \end{pmatrix}. \tag{5}
$$

Operators D,M[**a**] have been verified to be efficient tools for discretizing Levin's equation. For more details, one can refer to [13].

On the other hand, when *f*(*x*) is analytic with | *f*(*x*)| ≤ *M* in the region bounded by the Bernstein ellipse with the radius *ρ* > 1, we have for every *n* 6= 0 (see [25])

$$|f\_n| \le \mathbf{2}M\rho^{-n}.$$

Noting that

$$f'\_n = \sum\_{k=0}^{\infty} (2n + 4k + 2) f\_{n+2k+1} \rho$$

we can compute

$$|f'\_n| \le \sum\_{k=0}^{\infty} (2n + 4k + 2)|f\_{n+2k+1}| \le \sum\_{k=0}^{\infty} (2n + 4k + 2) 2M\rho^{-n-2k-1} \le 4M\rho^{-n-1} \left( (n+1) \sum\_{k=0}^{\infty} \rho^{-2k} + 2 \sum\_{k=0}^{\infty} k \rho^{-2k} \right).$$

Employing

$$\sum\_{k=0}^{\infty} \rho^{-2k} = \frac{\rho^2}{\rho^2 - 1}, \quad \sum\_{k=0}^{\infty} k \rho^{-2k} = \frac{\rho^2}{(\rho^2 - 1)^2}$$

leads to

$$|f'\_n| \le 4M\rho^{-n-1} \left( (n+1)\frac{\rho^2}{\rho^2 - 1} + 2\frac{\rho^2}{(\rho^2 - 1)^2} \right) \le 4M\rho^{-n-1} \frac{\rho(\rho - 1)(n+1) + 2}{(\rho - 1)^2} \le 4M(n+1)\frac{\rho^{-n+1}}{(\rho - 1)^2}.$$

Furthermore, according to [25, Theorems 2.1, 2.4], we have

$$\|f - p\_N\|\_{\infty} \le 4M \frac{\rho^{-N}}{\rho - 1} \tag{6}$$

and

$$\|f' - p\_N'\|\_{\infty} \le 4M(N+1)^2 \frac{\rho^{-N+2}}{(\rho-1)^3}.\tag{7}$$

Here, *pN*(*x*) denotes the interplant of *f*(*x*) at Clenshaw–Curtis nodes or the truncated Chebyshev series of *f*(*x*).

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

This section is devoted to investigating fast algorithms for calculation of Integral (1). To begin with, let us consider the computation of oscillatory integral without nearly singular integrands, that is,

$$\hat{I}[\mathcal{F}, \mathcal{G}\_1, \mathcal{G}\_2, \omega] = \int\_{-1}^{1} \int\_{-1}^{1} \mathcal{F}(\mathbf{x}, y) \mathbf{e}^{\mathrm{i}\omega(\mathcal{G}\_1(\mathbf{x}) + \mathcal{G}\_2(y))} d\mathbf{x} dy. \tag{8}$$

Here, *F*(*x*, *y*), *G*1(*x*), *G*2(*y*) are smooth functions with sufficiently large analytic regions, and *G*1(*x*), *G*2(*y*) do not have stationary points in the complex plane.

Consider the inner integral

$$H(y) = \int\_{-1}^{1} F(x, y) \mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(x)} d\mathbf{x}.\tag{9}$$

For fixed *y<sup>j</sup>* = cos *j N π*, *j* = 0, 1, · · · , *N*, we are restricted to finding a function *Pj*(*x*) satisfying

$$P\_j'(\mathbf{x}) + \mathbf{i}\omega \mathbf{G}\_1'(\mathbf{x}) P\_j(\mathbf{x}) = F(\mathbf{x}, y\_j). \tag{10}$$

Noting that *G* ′ 1 (*x*) never vanishes over the interval [−1, 1], we can get the modified Levin equation,

$$\frac{P\_j'(\mathbf{x})}{G\_1'(\mathbf{x})} + \mathbf{i}\omega P\_j(\mathbf{x}) = \frac{F(\mathbf{x}, y\_j)}{G\_1'(\mathbf{x})}.\tag{11}$$

Let

$$P\_{\dot{f}}(\mathbf{x}) = \sum\_{n=0}^{\infty} p\_n^{\dot{f}} T\_n(\mathbf{x}), \ 1/G\_1'(\mathbf{x}) = \sum\_{n=0}^{\infty} g\_n^1 T\_n(\mathbf{x}), \ F(\mathbf{x}, y\_{\dot{f}}) = \sum\_{n=0}^{\infty} f\_n^{\dot{f}} T\_n(\mathbf{x}).$$

With the help of operators D,M[**a**], we rewrite modified Levin's Equation (11) as

$$\mathcal{M}[\mathbf{G}\_1]\mathcal{D}\mathbf{P}\_{\dot{j}} + \mathrm{i}\omega \mathbf{P}\_{\dot{j}} = \mathcal{M}[\mathbf{G}\_1]\mathbf{F}\_{\dot{j}\prime} \tag{12}$$

where

$$\mathbf{x}\_{j} = \cos\frac{j}{N}\pi,\ \mathbf{P}\_{j} = \begin{pmatrix} p\_{0}^{j} \\ p\_{1}^{j} \\ \vdots \end{pmatrix},\ \mathbf{G}\_{1} = \begin{pmatrix} g\_{0}^{1} \\ g\_{1}^{1} \\ \vdots \end{pmatrix},\ \mathbf{F}\_{j} = \begin{pmatrix} f\_{0}^{j} \\ f\_{1}^{j} \\ \vdots \end{pmatrix}.$$

Solving Equation (12) by the truncation method [26] gives the unknown coefficients *p j <sup>n</sup>*, *n* = 0, 1, · · · , *N*, and we can get approximations to *Pj*(±1) by Clenshaw algorithm,

$$P\_{\hat{l}}(\pm 1) \approx \frac{b\_0(\pm 1) - b\_2(\pm 1)}{2}$$

with

$$\begin{cases} b\_{N+1}(\pm 1) = 0, b\_N(\pm 1) = p\_N^{j,N} \\\ b\_k(\pm 1) = (\pm 2) \times b\_{k+1}(\pm 1) + p\_k^{j,N} \end{cases}$$

where *p j*,*N k* denotes the approximation to *p j k* . Hence, the inner integral (9) is computed by

$$\int\_{-1}^{1} F(\mathbf{x}, y\_j) \mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(\mathbf{x})} d\mathbf{x} \approx \mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(1)} \frac{b\_0(1) - b\_2(1)}{2} - \mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(-1)} \frac{b\_0(-1) - b\_2(-1)}{2}. \tag{13}$$

Since *HN*(*yj*), the approximation to *H*(*y*) at Clenshaw–Curtis nodes, has been obtained, we are able to construct the polynomial *HN*(*y*) by

$$H\_N(y) = \sum\_{j=0}^N H\_N(y\_j) L\_j(y) = \sum\_{n=0}^N h\_n T\_n(y) \mu\_n$$

where *Lj*(*y*) denotes Lagrange basis with respect to Clenshaw–Curtis nodes and *h<sup>n</sup>* can be computed by fast Fourier transform. Letting

$$Q(y) = \sum\_{n=0}^{\infty} q\_n T\_n(y)$$

denote the function satisfying

$$\mathcal{M}[\mathbf{G}\_2]\mathcal{D}\mathbf{Q} + \mathrm{i}\omega \mathbf{Q} = \mathcal{M}[\mathbf{G}\_2]\mathbf{H}\_{N\prime} \tag{14}$$

where

$$1/G\_2'(y) = \sum\_{n=0}^{\infty} g\_n^2 T\_n(y), \\ H\_N(y) = \sum\_{n=0}^{N} h\_n T\_n(y).$$

and

$$\mathbf{Q} = \begin{pmatrix} q\_0 \\ q\_1 \\ \vdots \end{pmatrix}, \mathbf{G}\_2 = \begin{pmatrix} \mathbf{g}\_0^2 \\ \mathbf{g}\_1^2 \\ \vdots \end{pmatrix}, \mathbf{H}\_N = \begin{pmatrix} h\_0 \\ h\_1 \\ \vdots \end{pmatrix}.$$

we are able to approximate *q*0, · · · , *q<sup>N</sup>* by the truncation method again. Computing *a*0(±1), *a*2(±1) by

$$\begin{cases} a\_{N+1}(\pm 1) = 0, a\_N(\pm 1) = q\_{N'}^N\\ a\_k(\pm 1) = (\pm 2) \times a\_{k+1}(\pm 1) + q\_{k'}^N \end{cases}$$

where *q N k* denotes the approximation to *q<sup>k</sup>* , we arrive at 2D spectral coefficient Levin quadrature for Integral (8)

$$\hat{I}[\mathbf{F}, \mathbf{G}\_1, \mathbf{G}\_2, \omega] \approx \hat{I}\_\mathbf{N}[\mathbf{F}, \mathbf{G}\_1, \mathbf{G}\_2, \omega] := \mathbf{e}^{\mathrm{i}\omega \mathbf{G}\_2(1)} \frac{a\_0(1) - a\_2(1)}{2} - \mathbf{e}^{\mathrm{i}\omega \mathbf{G}\_2(-1)} \frac{a\_0(-1) - a\_2(-1)}{2}.\tag{15}$$

In [27], Xiang established the relation between Filon and Levin quadrature rules in the case of the phase *g*(*x*) = 1 and analyzed the convergence property of Levin quadrature. Instead of resorting to Filon quadrature, we consider the convergence rate of the above spectral coefficient Levin method with respect to quadrature nodes and frequency in the case of nonlinear oscillators through examining the decaying rate of the coefficients.

For any fixed *y* ∈ [−1, 1], *F*(*x*, *y*) turns to the univariate function with regard to *x*. Let *MH*, *MF*(*y*), *MG*<sup>1</sup> , *MG*<sup>2</sup> denote the maximum of *H*(*y*), *F*(*x*, *y*), 1/*G* ′ 1 (*x*), 1/*G* ′ 2 (*y*) within their corresponding Bernstein ellipse with radiuses *ρH*, *ρF*(*y*), *ρG*<sup>1</sup> , *ρG*<sup>2</sup> , respectively. Furthermore, denoting

$$\rho\_{\mathcal{F}} = \inf\_{y \in [-1, 1]} \{ \rho\_{\mathcal{F}}(y) \},\\M\_{\mathcal{F}} := \sup\_{y \in [-1, 1]} \{ M\_{\mathcal{F}}(y) \},\\M\_{\mathcal{2}} := \sup\_{y \in [-1, 1]} \{ G\_2''(y) \},\\\mathfrak{m}\_{\mathcal{2}} := \inf\_{y \in [-1, 1]} \{ G\_2'(y) \}.$$

we summarize the convergence analysis in the following theorem.

**Theorem 1.** *Suppose*


*Then, for sufficiently large ω*, *we have*

$$\begin{split} & \left| \hat{\mathbb{I}}[\mathcal{F}, \mathcal{G}\_{1}, \mathcal{G}\_{2}, \omega] - \hat{\mathbb{I}}\_{\mathrm{N}}[\mathcal{F}, \mathcal{G}\_{1}, \mathcal{G}\_{2}, \omega] \right| \\ & \leq \mathbb{C} \left( \frac{(\mathcal{N} + 1)^{2}}{\omega} \frac{\rho\_{H}^{-N+2}}{(\rho\_{H} - 1)^{3}} + \frac{(\mathcal{N} + 2)^{3} \log(\mathcal{N} + 1)}{\omega} \frac{\rho\_{F}^{-N+4}}{(\rho\_{F} - 1)^{6}} + \frac{(\mathcal{N} + 2)^{3}}{\omega} \frac{\rho\_{H}^{-N+4}}{(\rho\_{H} - 1)^{6}} \right) . \end{split}$$

*where the constant C does not depend on ω*, *N*.

**Proof.** Let *H*ˆ *<sup>N</sup>*(*y*) = *N* ∑ *j*=0 *H*(*yj*)*Lj*(*y*) denote the interplant of *H*(*y*) at Clenshaw–Curtis points. A direct calculation implies the quadrature error can be decomposed into

<sup>ˆ</sup>*I*[*F*, *<sup>G</sup>*1, *<sup>G</sup>*2, *<sup>ω</sup>*] <sup>−</sup> <sup>ˆ</sup>*IN*[*F*, *<sup>G</sup>*1, *<sup>G</sup>*2, *<sup>ω</sup>*] = Z 1 −1 *H*(*y*)e i*ωG*2(*y*) *dy* − e i*ωG*2(1) *a*0(1) − *a*2(1) 2 − e i*ωG*2(−1) *a*0(−1) − *a*2(−1) 2 = Z 1 −1 Z 1 −1 *F*(*x*, *y*)e i*ω*(*G*<sup>1</sup> (*x*)+*G*2(*y*))*dxdy* <sup>−</sup> Z 1 −1 *H*ˆ *<sup>N</sup>*(*y*)e i*ωG*2(*y*) *dy* + Z 1 −1 *N* ∑ *j*=0 *H*(*yj*)*Lj*(*y*)e i*ωG*2(*y*) *dy* − Z 1 −1 *N* ∑ *j*=0 *Pj*(1)e i*ωG*<sup>1</sup> (1) <sup>−</sup> *<sup>P</sup>j*(−1)<sup>e</sup> i*ωG*<sup>1</sup> (−1) *Lj*(*y*)e i*ωG*2(*y*) *dy* + Z 1 −1 *HN*(*y*)e i*ωG*2(*y*) *dy* − *Q*(1)e <sup>i</sup>*ωG*2(1) <sup>−</sup> *<sup>Q</sup>*(−1)<sup>e</sup> i*ωG*2(−1) =*E*<sup>1</sup> + *E*<sup>2</sup> + *E*3.

Here,

$$\begin{split} E\_1 &:= \int\_{-1}^1 \int\_{-1}^1 F(\mathbf{x}, y) \mathbf{e}^{i\omega(\mathbb{G}\_1(\mathbf{x}) + \mathbb{G}\_2(y))} dx dy - \int\_{-1}^1 \hat{H}\_N(y) \mathbf{e}^{i\omega \mathbb{G}\_2(y)} dy, \\ E\_2 &:= \int\_{-1}^1 \sum\_{j=0}^N H(y\_j) L\_j(y) \mathbf{e}^{i\omega \mathbb{G}\_2(y)} dy - \int\_{-1}^1 \sum\_{j=0}^N \left( P\_j(1) \mathbf{e}^{i\omega \mathbb{G}\_1(1)} - P\_j(-1) \mathbf{e}^{i\omega \mathbb{G}\_1(-1)} \right) L\_j(y) \mathbf{e}^{i\omega \mathbb{G}\_2(y)} dy, \\ E\_3 &:= \int\_{-1}^1 H\_N(y) \mathbf{e}^{i\omega \mathbb{G}\_2(y)} dy - \left( Q(1) \mathbf{e}^{i\omega \mathbb{G}\_2(1)} - Q(-1) \mathbf{e}^{i\omega \mathbb{G}\_2(-1)} \right). \end{split}$$

In the remaining work, we give estimates for *E*1, *E*2, *E*<sup>3</sup> with respect to the increasing truncation term *N* and frequency *ω*.

For *E*1, note that *H*(*y*) is bounded within its Bernstein ellipse by

$$|H(y)| = \left| \int\_{-1}^{1} F(\mathbf{x}, y) \mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(\mathbf{x})} d\mathbf{x} \right| \le \int\_{-1}^{1} |F(\mathbf{x}, y)| |\mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(\mathbf{x})}| d\mathbf{x} \le 2M\_F. \tag{16}$$

As a result, we have according to integration by parts


where *C*<sup>1</sup> := 16*M<sup>F</sup> m*<sup>2</sup> 1 + *M*<sup>2</sup> *m*<sup>2</sup> . For *E*2, since

$$\hat{H}\_N(y) = \sum\_{j=0}^N H(y\_j) L\_j(y), \\ H\_N(y) = \sum\_{j=0}^N H\_N(y\_j) L\_j(y), \\ H\_N(y\_j) = P\_j(1) \mathbf{e}^{i\omega \mathbf{G}\_1(1)} - P\_j(-1) \mathbf{e}^{i\omega \mathbf{G}\_1(-1)}.$$

letting

$$E\_{2,j} := H(y\_j) - H\_N(y\_j)\_{\prime}$$

we obtain

$$E\_2 = \sum\_{j=0}^{N} E\_{2,j} \int\_{-1}^{1} L\_j(y) \mathbf{e}^{\mathbf{i}\omega G\_2(y)} dy.$$

Furthermore, letting

$$\text{Err}\_{2,j}(\mathbf{x}) := \frac{F(\mathbf{x}, y\_j) - P\_j'(\mathbf{x}) - \mathbf{i}\omega G\_1'(\mathbf{x}) P\_j(\mathbf{x})}{G\_1'(\mathbf{x})} \rho$$

we have

$$E\_{2,j} = \int\_{-1}^{1} \text{Err}\_{2,j}(\mathfrak{x}) \mathbf{G}\_1'(\mathfrak{x}) \mathbf{e}^{\mathbf{i}\omega \mathbf{G}\_1(\mathfrak{x})} d\mathfrak{x}.$$

A direct calculation as is done in the estimation procedure for *E*<sup>1</sup> results in

$$|E\_{2,j}| \le \frac{2||\text{Err}\_{2,j}(\mathbf{x})||\_{\infty}}{\omega} + \frac{2||\text{Err}\_{2,j}'(\mathbf{x})||\_{\infty}}{\omega}.\tag{18}$$

Then, let us consider the decaying rate of coefficients of Chebyshev expansions of Err2,*j*(*x*), which helps to analyze kErr2,*j*(*x*)k<sup>∞</sup> and kErr′ 2,*j* (*x*)k∞. In fact, the truncation technique implies

$$\mathbf{M}\_N[\mathbf{G}\_1]\mathcal{D}\_N\mathbf{P}\_{j,N} + \mathrm{i}\omega\mathbf{P}\_{j,N} = \mathcal{M}\_N[\mathbf{G}\_1]\mathbf{F}\_{j,N} \tag{19}$$

with

$$\mathbf{P}\_{j,N} = \begin{pmatrix} p\_0^{j,N} \\ p\_1^{j,N} \\ \vdots \\ p\_N^{j,N} \end{pmatrix} \text{ } \mathbf{F}\_{j,N} = \begin{pmatrix} f\_0^j \\ f\_1^j \\ \vdots \\ f\_N^j \end{pmatrix} \text{ } \mathbf{f}$$

and *p j*,*N k* denotes the approximation to *p j k* in Equation (12). For sufficiently large *ω* > *N*, it follows that 1 *ω* kM*N*[**G**1]D*N*k<sup>∞</sup> < 1. By Neumann's lemma, we have

$$\mathbf{P}\_{\mathbf{j},N} = \frac{1}{\mathbf{i}\omega} \left( \sum\_{n=0}^{\infty} \left( -\frac{1}{\mathbf{i}\omega} \right)^{n} \mathcal{M}\_{N}^{n} [\mathbf{G}\_{1}] \mathcal{D}\_{N}^{n} \right) \mathcal{M}\_{N} [\mathbf{G}\_{1}] \mathbf{F}\_{\mathbf{j},N}. \tag{20}$$

.

Denoting the maximum of ∞ ∑ *n*=0 − 1 i*ω <sup>n</sup>* <sup>M</sup>*<sup>n</sup> <sup>N</sup>*[**G**1]D *n N* ! M*N*[**G**1] by *SN*, we notice that

$$|p\_n^{j,N}| \le \frac{2\mathcal{S}\_N M\_F}{\omega} \rho\_F^{-n} \, |d p\_n^{j,N}| \le \frac{4\mathcal{S}\_N M\_F}{\omega} (n+1) \frac{\rho\_F^{-n+1}}{(\rho\_F - 1)^2} \le 4\mathcal{S}\_N M\_F \frac{\rho\_F^{-n+1}}{(\rho\_F - 1)^2}.$$

The Chebyshev coefficients of Err2,*j*(*x*) can be computed by

$$c\_n^{2,j} = \int\_{-1}^1 \frac{\text{Errr}\_{2,j}(\mathbf{x}) T\_n(\mathbf{x})}{\sqrt{1 - \mathbf{x}^2}} d\mathbf{x}, \ n = 0, 1, \dots, \dots$$

Noting the construction technique in the modified spectral Levin coefficient method, we get *c* 2,*j <sup>n</sup>* = 0 for *n* = 0, 1, · · · , *N*. On the other hand, for *n* ≥ *N* + 1, it follows that

$$|c\_{n}^{2,j}| \leq \left| \int\_{-1}^{1} \frac{F(\mathbf{x}, y\_{j}) T\_{n}(\mathbf{x})}{G\_{1}'(\mathbf{x}) \sqrt{1 - \mathbf{x}^{2}}} d\mathbf{x} \right| + \left| \int\_{-1}^{1} \frac{P\_{j}'(\mathbf{x}) T\_{n}(\mathbf{x})}{G\_{1}'(\mathbf{x}) \sqrt{1 - \mathbf{x}^{2}}} d\mathbf{x} \right|.$$

It is noted that the first term in the right-hand side of the above equation is the coefficient of *F*(*x*, *yj*) *G*′ 1 (*x*) and the second term is that of *P* ′ *j* (*x*) *G*′ 1 (*x*) , where we denote coefficients to be *c FG n* , *c PG n* , respectively. Recalling

the product operator in Equation (5), we have

$$|c\_{n}^{FG}| \leq \frac{3}{2} \left( |g\_{n}^{1}| |f\_{0}^{j}| + |g\_{n-1}^{1}| |f\_{1}^{j}| + \dots + |g\_{0}^{1}| |f\_{n}^{j}| \right) \leq 6M\_{\mathcal{G}\_{1}}M\_{F}(n+1)\rho\_{F}^{-n}$$

and

$$|c\_{n}^{PG}| \leq \frac{3}{2} \left( |g\_{n}^{1}| |dp\_{0}^{jN}| + |g\_{n-1}^{1}| |dp\_{1}^{jN}| + \dots + |g\_{0}^{1}| |dp\_{n}^{jN}| \right) \leq 12M\_{\mathbb{G}\_{1}}M\_{\mathbb{F}}\mathbb{S}\_{N}(n+1)\frac{\rho\_{F}^{-n+1}}{(\rho\_{F}-1)^{2}}.$$

Therefore, it follows

$$\begin{split} |c\_{n}^{2^{j}}| \leq & 6M\_{\mathrm{G}\_{1}}M\_{\mathrm{F}}(n+1)\rho\_{\mathrm{F}}^{-n} + 12M\_{\mathrm{G}\_{1}}M\_{\mathrm{F}}S\_{N}(n+1)\frac{\rho\_{\mathrm{F}}^{-n+1}}{(\rho\_{\mathrm{F}}-1)^{2}} \\ \leq & 6M\_{\mathrm{G}\_{1}}M\_{\mathrm{F}}(n+1)\frac{\rho\_{\mathrm{F}}^{-n+1}}{(\rho\_{\mathrm{F}}-1)^{2}} + 12M\_{\mathrm{G}\_{1}}M\_{\mathrm{F}}S\_{N}(n+1)\frac{\rho\_{\mathrm{F}}^{-n+1}}{(\rho\_{\mathrm{F}}-1)^{2}} \\ \leq & C'(n+1)\frac{\rho\_{\mathrm{F}}^{-n+1}}{(\rho\_{\mathrm{F}}-1)^{2}}. \end{split} \tag{21}$$

Here, *C* ′ := 2 max{6*MG*<sup>1</sup> *MF*, 12*MG*<sup>1</sup> *MFSN*}. Hence, kErr2,*j*(*x*)k<sup>∞</sup> and kErr′ 2,*j* (*x*)k<sup>∞</sup> can be bounded by

$$\|\|\text{Err}\_{2,j}(\mathbf{x})\|\|\_{\infty} \le \sum\_{n=N+1}^{\infty} |c\_n^{2,j}| \le \frac{\mathcal{C}'}{(\rho\_F - 1)^2} \sum\_{n=N+1}^{\infty} (n+1)\rho\_F^{-n+1} \le \mathcal{C}' \frac{(N+2)\rho\_F^{-N+2}}{(\rho\_F - 1)^4}.$$

and

$$\|\mathrm{Err}'\_{2,j}(\mathbf{x})\|\_{\infty} \le 2\sum\_{n=N+1}^{\infty} |c\_n^{2,j}| \|T'\_n(\mathbf{x})\|\_{\infty} \le 2\sum\_{n=N+1}^{\infty} \mathcal{C}'(n+1)\frac{\rho\_F^{-n+1}}{(\rho\_F - 1)^2} n^2 \le \mathcal{C}'(N+2)^3 \frac{\rho\_F^{-N+4}}{(\rho\_F - 1)^6}.$$

As a result, it follows that

$$|E\_{2,j}| \le \frac{2}{\omega} (||\text{Errr}\_{2,j}(\mathbf{x})||\_{\infty} + ||\text{Errr}'\_{2,j}(\mathbf{x})||\_{\infty}) \le \frac{8C'}{\omega} (N+2)^3 \frac{\rho\_F^{-N+4}}{(\rho\_F - 1)^6}.$$

Now, we arrive at the fact

$$|E\_2| \le \sum\_{j=0}^{N} |E\_{2j}| \left| \int\_{-1}^{1} L\_j(y) \mathbf{e}^{i\omega \mathbf{G}\_2(y)} dy \right| \le \mathcal{C}\_2 \frac{(N+2)^3 \log(N+1)}{\omega} \frac{\rho\_F^{-N+4}}{(\rho\_F - 1)^6} \tag{22}$$

where *C*<sup>2</sup> := 64*C* ′ *π* .

The estimation procedure for *E*<sup>3</sup> is similar to that of *E*2,*<sup>j</sup>* . We ignore details and give the conclusion directly

$$|E\_3| \le C\_3 \frac{(N+2)^3}{\omega} \frac{\rho\_H^{-N+4}}{(\rho\_H - 1)^{6'}} \tag{23}$$

where *C*<sup>3</sup> does not depend on *N* and *ω*.

To sum up, we arrive at the following error bound by combining Equations (17), (22), and (23),

$$\begin{split} & \left| \hat{I}[F, G\_{1}, G\_{2}, \omega] - \hat{I}\_{N}[F, G\_{1}, G\_{2}, \omega] \right| \\ \leq & |E\_{1}| + |E\_{2}| + |E\_{3}| \\ \leq & \frac{\mathcal{C}\_{1}(N+1)^{2}}{\omega} \frac{\rho\_{H}^{-N+2}}{(\rho\_{H}-1)^{3}} + \mathcal{C}\_{2} \frac{(N+2)^{3}\log(N+1)}{\omega} \frac{\rho\_{F}^{-N+4}}{(\rho\_{F}-1)^{6}} + \mathcal{C}\_{3} \frac{(N+2)^{3}}{\omega} \frac{\rho\_{H}^{-N+4}}{(\rho\_{H}-1)^{6}} \\ \leq & \mathcal{C} \left( \frac{(N+1)^{2}}{\omega} \frac{\rho\_{F}^{-N+2}}{(\rho\_{H}-1)^{3}} + \frac{(N+2)^{3}\log(N+1)}{\omega} \frac{\rho\_{F}^{-N+4}}{(\rho\_{F}-1)^{6}} + \frac{(N+2)^{3}}{\omega} \frac{\rho\_{H}^{-N+4}}{(\rho\_{H}-1)^{6}} \right), \end{split} \tag{24}$$

with *C* = max{*C*1, *C*2, *C*3}. It is easily seen that the constant *C* does not depend on *N* and *ω*. This completes the proof.

Finally, let us turn to the construction of the composite quadrature rule for calculation of Integral (1). It is observed in the above theorem that, when the radiuses *ρF*, *ρ<sup>H</sup>* are close to 1, the error bound would expand dramatically. Therefore, an efficient quadrature rule has to guarantee the fact that the integrand has a relatively large analytic radius over the integration domain. To make this judgment be satisfied, we choose a non-uniform grid instead of partitioning the integration region uniformly.

To begin with, the singular point *z* ∗ is projected into the plane containing the integration region and we get the projection point *z*. In the case of the projected point *z* falling into the integration domain (Case I), the first box is determined by the distance between *z* ∗ and *z*. We construct a square with its center being *z* and its side length being 2k*z* <sup>∗</sup> − *z*k. Then, the side length of level-2 box's with the center *<sup>z</sup>* is set to be 22k*<sup>z</sup>* <sup>∗</sup> − *z*k. To devise the composite quadrature rule, we first select level-1 box as a subdomain. Noting that the remaining domain is not a rectangle, we partition it into four subdomains, that is, Box21, Box22, Box23, and Box24 (see Figure 1). In general, the side length of level-*l* box's with the center *<sup>z</sup>* is set to be 2*l*k*<sup>z</sup>* <sup>∗</sup> − *z*k, and the integration subdomain is constructed similarly, which finally results in a nonuniform grid (see Figure 2).

When the projected point falls out of the integration domain, for example, it is around the side (Case II) or vertex (Case III), we implement a similar partition procedure like that in Case I. The final partition grid is shown in Figures 3 and 4. Applying 2D spectral coefficient Levin quadrature rule in the subdomain leads to a class of composite 2D spectral coefficient Levin quadrature. It is noted that such kind of partition techniques guarantee the fact that the distance between the singular and the integration interval is no less than 2 when we map the integration domain into [−1, 1] × [−1, 1].

**Figure 1.** The integration subdomains for level-2 box.


**Figure 2.** The partition method in Case I (left: location of the singular point *z* ∗ and projection point *z*. right: the nonuniform grid).


**Figure 3.** The partition method in Case II (left: location of the singular point *z* ∗ and projection point *z*. right: the nonuniform grid).


**Figure 4.** The partition method in Case III (left: location of the singular point *z* ∗ and projection point *z*. right: the nonuniform grid).

#### **4. Numerical Experiments**

This section is devoted to illustrating the numerical performance of 2D spectral coefficient Levin quadrature (2DSC-Levin) and composite 2D spectral coefficient Levin quadrature (C2DSC-Levin) given in Section 3.

**Example 1.** *Let us consider the computation of the oscillatory integral*

$$\int\_{-1}^{1} \int\_{-1}^{1} \cos(x+y) \mathbf{e}^{i\omega(x+y)} dx dy.$$

*The phase x* + *y has no stationary points within the domain* [−1, 1] × [−1, 1], *and the amplitude* cos(*x* + *y*) *is analytic with respect to both variables. Therefore, it is expected that the new approach has an exponential convergence rate.*

It is noted that approximation results derived from classical cubature usually do not make sense when *ω* ≫ 1. We first employ CCQ and give the computational results in Table 2, where both quantity of quadrature nodes *N* and oscillation parameter *ω* are variables.


**Table 2.** Absolute errors of CCQ for Example 1.

Although plenty of quadrature nodes have been used in the above example, computed results are not satisfactory especially in the case of high oscillation. Now, we list approximated results of 2DSC-Levin in Table 3, where the referenced exact value is computed by the CHEBFUN toolbox again. It can be seen from Table 3 that absolute errors do not increase as the frequency *ω* enlarges, which implies that the new method is robust to high oscillation. On the other hand, when we raise the truncation term of Chebyshev series, the absolute error decays fast. In Figure 5, we give tendencies of absolute errors with respect to increasing frequencies and compare the computational time of 2DSC-Levin and referenced algorithm in CHEBFUN. It can be found that the consumed time of 2DSC-Levin does not vary as the frequency *ω* enlarges, whereas that of CHEBFUN's 2D quadrature (2D-Cheb) increases dramatically. Since 2D-Cheb is a class of self-adaptive algorithms, it has to increase quadrature nodes when the frequency goes to infinity to retain a tolerance error, which results in

the dramatically growing curve. However, approximations derived from 2DSC-Levin do not suffer from high oscillation according to Theorem 1. Hence, there is no need to raise quadrature nodes of 2DSC-Levin in high oscillation, and we do not witness an obvious change in Figure 5.


**Table 3.** Absolute errors of 2DSC-Levin for Example 1.

**Figure 5.** Comparison between 2DSC-Levin and 2D-Cheb in Example 1, *ω* is a variable (left: absolute errors, right: CPU time).

We also employ 2D-Cheb–Levin quadrature given in [18] to give a comparison. Computed results are shown in Table 4.


**Table 4.** Absolute errors of 2D-Cheb–Levin quadrature for Example 1.

Comparison between Tables 3 and 4 illustrates that the accuracy of 2DSC-Levin and 2D-Cheb–Levin quadrature is similar. However, 2DSC-Levin does a little better than 2D-Cheb–Levin quadrature when CPU time is considered. Since both approaches consist of approximations to a series of one-dimensional integrals, we show the consuming time of both approaches for computing the final highly oscillatory integrals in Table 5, where it can be seen that 2DSC-Levin is slightly faster than 2D-Cheb–Levin quadrature, which is partly due to the sparse structure of the discretizatized modified Levin equation.


**Table 5.** Comparison of CPU time for 2DSC-Levin and 2D-Cheb–Levin quadrature for fixed *ω* = 1000.

For computation of univariate oscillatory integrals, Levin quadrature does well in solving problems with complicate phases. In the following example, we consider a highly oscillatory integral with nonlinear oscillators over [0, 1] × [0, 1].

**Example 2.** *Let us consider the computation of the oscillatory integral*

$$\int\_0^1 \int\_0^1 \frac{1}{\mathfrak{x}^2 + y^2 + 15} \mathrm{e}^{\mathrm{i}\omega(\mathfrak{x}^2 + \mathfrak{x} + y^2 + y)} d\mathfrak{x} dy.$$

*The amplitude* <sup>1</sup> *x* <sup>2</sup> + *y* <sup>2</sup> + 15 *is no longer an entire function, and the inverse of the phase function x* <sup>2</sup> + *x* + *y* <sup>2</sup> + *y can not be calculated directly.*

We show absolute errors and CPU time of 2DSC-Levin in Table 6 and Figure 6, respectively. Due to the fact that the amplitude in Example 2 has a limited analytic radius, 2DSC-Levin converges to the machine precision much more slowly than that in Example 1. However, noting the decaying curve in the left part of Figure 6 manifests that 2DSC-Levin has the property that the higher the oscillation, the better the approximation, which also coincides with the theoretical estimate in Theorem 1. Hence, 2DSC-Levin is feasible for calculation of highly oscillatory integrals over rectangle regions when the oscillator *g*(*x*, *y*) is nonlinear. In addition, it is interesting that the curve of CPU time of 2DSC-Levin has a jump at about *ω* = 5500 in the right part of Figure 6. Such a phenomenon may originate from the fact that the Levin equation can be solved more efficiently as the frequency becomes larger. However, this is still a conjecture and we need more theoretical investigation in the future work.


**Table 6.** Absolute errors of 2DSC-Levin for Example 2.

**Figure 6.** Comparison between 2DSC-Levin and 2D-Cheb in Example 2, *ω* is a variable (left: absolute errors, right: CPU time).

To illustrate the effectiveness of the composite 2D spectral coefficient Levin quadrature rule (C2DSC Levin), we give a comparison among the new approach, 2D sinh transformation (JJE) in [22], and 2D dilation quadrature (2D-d) in [23].

For Integral (1), the sinh transformation is defined by

$$\mathfrak{x} = a + \mathfrak{e}\sinh(\mu\_1 u - \eta\_1), \\ \mathfrak{y} = b + \mathfrak{e}\sinh(\mu\_2 u - \eta\_2),$$

where

$$\begin{aligned} \mu\_1 &= \frac{1}{2} \left( \operatorname{arcsinh} \left( \frac{1+a}{\varepsilon} \right) + \operatorname{arcsinh} \left( \frac{1-a}{\varepsilon} \right) \right), \\ \mu\_2 &= \frac{1}{2} \left( \operatorname{arcsinh} \left( \frac{1+a}{\varepsilon} \right) - \operatorname{arcsinh} \left( \frac{1-a}{\varepsilon} \right) \right). \end{aligned}$$

Since the transformed integrand is no longer nearly singular, a direct 2D Gauss cubature can be applied in practical computation.However, it should be noted that, although JJE can efficiently deal with nearly singular problems, it generally suffers from the highly oscillatory integrands.

In [23], Occorsio and Serafini proposed 2D-d for the integral

$$\mathcal{Z}(\mathbf{F},\omega) = \int\_{D} \mathbf{F}(\mathbf{x}) \mathbf{K}(\mathbf{x},\omega) d\mathbf{x},\tag{25}$$

where *D* := [−1, 1] × [−1, 1], **x** = (*x*, *y*), and

$$F(\mathbf{x}) = \frac{1}{(\mathbf{x} - a)^2 + (y - b)^2 + \varepsilon^2}, \mathbf{K}(\mathbf{x}, \omega) = \mathbf{e}^{\mathbf{i}\omega(\mathbf{G}\_1(\mathbf{x}) + \mathbf{G}\_2(y))}.$$

Letting *ω*<sup>1</sup> = q |*ω*|, *x* = *η ω*<sup>1</sup> , *y* = *θ ω*<sup>1</sup> , we have

$$\mathcal{Z}(\mathbf{F},\omega) = \omega\_1^2 \int\_{[-\omega\_1,\omega\_1]^2} \mathbf{F}\left(\frac{\eta}{\omega\_1}, \frac{\theta}{\omega\_1}\right) \mathbf{K}\left(\frac{\eta}{\omega\_1}, \frac{\theta}{\omega\_1}, \omega\right) d\eta d\theta.$$

Properly choosing *<sup>d</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> and *<sup>S</sup>* <sup>=</sup> 2*ω*<sup>1</sup> *d* <sup>∈</sup> <sup>N</sup> results in

$$\mathcal{Z}(\mathbf{F},\omega) = \omega \sum\_{i=1}^{S} \sum\_{j=1}^{S} \int\_{D\_{i,j}} \mathbf{F}\left(\frac{\eta}{\omega\_1}, \frac{\theta}{\omega\_1}\right) \mathbf{K}\left(\frac{\eta}{\omega\_1}, \frac{\theta}{\omega\_1}, \omega\right) d\eta d\theta.$$

Here, *Di*,*<sup>j</sup>* := [−*ω*<sup>1</sup> + (*i* − 1)*d*, −*ω*<sup>1</sup> + *id*] × [−*ω*<sup>1</sup> + (*j* − 1)*d*, −*ω*<sup>1</sup> + *jd*]. Employing the transformed Gauss–Jacobi quadrature to the moment integral gives 2D-dilation quadrature.

**Example 3.** *Consider the computation of*

$$\int\_{-1}^{1} \int\_{-1}^{1} \frac{\sin(xy)}{(x+0.5)^2 + (y-0.5)^2 + 0.09} \mathbf{e}^{\mathbf{i}\omega(x+y)} d\mathbf{x} dy.$$

*It is noted that the integrand will reach its peak value at* (−0.5, 0.5), *and dramatically decrease away from such a critical point.*

We list computed absolute errors of JJE, 2D-d and C2DSC-Levin in Tables 7–9. As the number of quadrature nodes *N* increases, all of algorithms converge fast to the referenced value. When the integrand does not change rapidly, JJE provides the best numerical approximation. However, as the frequency *ω* enlarges, JJE and 2D-d suffer from high oscillation, while C2DSC-Levin is still able to maintain a relatively high-order approximation. It should be noted that the dilation parameter *d* in 2D-d is restricted to make the number of quadrature nodes coincide with the other two methods, and a slightly modified choice as is done in [23] may make 2D-d be able to deal with some highly oscillatory problems. The corresponding results are shown in Figure 7. Numerical results in this figure indicate that the absolute error derived from JJE increases dramatically when *ω* goes beyond 500, while the error of 2D-d rises slowly. It is also found that both absolute errors and computational time of the new approach do not suffer from the varying frequency *ω*. Hence, C2DSC-Levin is the most effective tool for computing oscillatory and nearly singular integrals.

**Table 7.** Absolute errors of JJE for Example 3.





**Table 9.** Absolute errors of C2DSC-Levin for Example 3.

**Figure 7.** Comparison of JJE, 2D-d and C2DSC-Levin in Example 3, *ω* is a variable (left: absolute errors, right: CPU time).

Although JJE is efficient for solving some nearly singular problems, it may fail when *b* = 0 in the integrand.

**Example 4.** *Let us consider the computation of the following integral*

$$\int\_{0}^{1} \int\_{0}^{1} \frac{1}{(\mathbf{x} + 0.02)^{2} + (y + 0.02)^{2}} \mathbf{e}^{\mathbf{i}\omega(\mathbf{x}^{3} + 3\mathbf{x} + y^{2} + 6y)} d\mathbf{x} dy \,\omega$$

*It is noted that JJE does not work in this case.*

Absolute errors derived from 2D-d and C2DSC-Levin are listed in Tables 10 and 11, respectively. It can be found that 2D-d provides more accurate approximation than that of C2DSC-Levin in the relatively low frequency. Nevertheless, the absolute error of C2DSC-Levin never increases in the high frequency while its 2D-d counterpart enlarges. Furthermore, although 2D-d will provide a more accurate approximation if we employ the choice of the dilation parameter considered in [23], it still cannot beat C2DSC-Levin in the high frequency when both absolute errors and computational time are considered (see Figure 8).


**Table 10.** Absolute errors of 2D-d for Example 4.



**Figure 8.** Comparison of 2D-d and C2DSC-Levin in Example 4, *ω* is a variable (left: absolute errors, right: CPU time).

#### **5. Conclusions**

In this paper, we have presented the modified spectral coefficient Levin quadrature for calculation of highly oscillatory integrals over rectangle regions and established its convergence rate with respect to the truncation term and oscillation parameter. Furthermore, by considering numerical calculation of moments over a non-uniform mesh, we derive the composite Levin quadrature. Numerical experiments indicate that the non-uniform partition technique greatly reduces the nearly singular problem. Recently, sharp bounds for coefficients of multivariate Gegenbauer expansion of analytic functions have been studied in [28], which definitely opens a door for our ongoing work about convergence analysis of Levin quadrature in high dimensional hypercube.

On the other hand, studies on the asymptotic and oscillatory behavior of solutions to highly oscillatory integral and differential equations have attracted much attention during the past decades [29–32]. It is noted that computation and numerical analysis of oscillatory integrals provide efficient tools for such kinds of studies and investigation of application of the proposed approaches to oscillatory equations is also necessary in the future work.

**Author Contributions:** Z.Y. and J.M. conceived and designed the experiments; Z.Y. performed the experiments; Z.Y. and J.M. analyzed the data; J.M. contributed reagents/materials/analysis tools; Z.Y. and J.M. wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (No. 11901133) and the Science and Technology Foundation of Guizhou Province (No. QKHJC[2020]1Y014).

**Acknowledgments:** The authors thank referees for their helpful suggestions.

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

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


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### *Article* **Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order**

**Taher S. Hassan 1,2,\* ,† Yuangong Sun 3,\* ,† and Amir Abdel Menaem 4,†**


Received: 12 September 2020; Accepted: 23 October 2020; Published: 31 October 2020

**Abstract:** In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

**Keywords:** time scales; functional dynamic equations; second order; oscillation criteria

#### **1. Introduction**

In order to combine continuous and discrete analysis, the theory of dynamic equations on time scales was proposed by Stefan Hilger in [1]. There are different types of time scales applied in many applications (see [2]). The cases when the time scale T as an arbitrary closed subset is equal to the reals or to the integers represent the classical theories of differential and of difference equations. The theory of dynamic equations includes the classical theories for the differential equations and difference equations cases and other cases in between these classical cases. That is, we are eligible to consider the q-difference equations when T =*q* <sup>N</sup><sup>0</sup> :<sup>=</sup> {*<sup>q</sup> k* : *<sup>k</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> for *<sup>q</sup>* <sup>&</sup>gt; <sup>1</sup>} which has significant applications in quantum theory (see [3]) and different types of time scales like T =*h*N, T = N 2 and T = T*<sup>n</sup>* (the set of the harmonic numbers) can also be applied. For more details of time scales calculus, see [2,4,5]. The study of nonlinear dynamic equations is considered in this work because these equations arise in various real-world problems like the turbulent flow of a polytrophic gas in a porous medium, non-Newtonian fluid theory, and in the study of *p*−Laplace equations. Therefore, we are interested in the oscillatory behavior of the nonlinear functional dynamic equation of second order with deviating arguments

$$\left[a(\zeta)\,\varphi\_{\gamma}\left(z^{\Delta}(\zeta)\right)\right]^{\Delta} + q(\zeta)\,\varphi\_{\beta}\left(z(\eta(\zeta))\right) = 0\tag{1}$$

on an above-unbounded time scale <sup>T</sup>, where *<sup>ϕ</sup>α*(*u*) :<sup>=</sup> <sup>|</sup>*u*<sup>|</sup> *α* sgn*u*, *α* > 0; *a* and *q* are positive rd-continuous functions on T such that

$$\int\_{}^{\infty} \frac{\Delta \varphi}{a^{\frac{1}{\gamma}}(\varphi)} = \infty;\tag{2}$$

and *<sup>η</sup>* : <sup>T</sup> <sup>→</sup> <sup>T</sup> is a rd-continuous function such that lim*ζ*→<sup>∞</sup> *<sup>η</sup>*(*ζ*) = <sup>∞</sup>.

By a solution of Equation (1) we mean a nontrivial real-valued function *z* ∈ C 1 rd[*ζz*, ∞)<sup>T</sup> for some *<sup>ζ</sup><sup>z</sup>* <sup>≥</sup> *<sup>ζ</sup>*<sup>0</sup> with *<sup>ζ</sup>*<sup>0</sup> <sup>∈</sup> <sup>T</sup> such that *<sup>z</sup>* <sup>∆</sup>, *a*(*ζ*)*ϕ<sup>γ</sup> z* <sup>∆</sup>(*ζ*) ∈ C 1 rd[*ζz*, ∞)<sup>T</sup> and *z*(*ζ*) satisfies Equation (1) on [*ζz*, ∞)T, where Crd is the space of right-dense continuous functions. It should be mentioned that in a particular case when T = R then

$$\sigma(\zeta) = \zeta,\ \mu(\zeta) = 0,\ g^{\Delta}(\zeta) = g'(\zeta),\ \int\_a^b g(\zeta) \Delta \zeta = \int\_a^b g(\zeta) d\zeta.$$

and (1) turns as the nonlinear functional differential equation

$$\left[a(\zeta)\,\varphi\_{\gamma}\left(z'(\zeta)\right)\right]' + q(\zeta)\,\varphi\_{\beta}\left(z(\eta(\zeta))\right) = 0. \tag{3}$$

The oscillation properties of Equation (3) and special cases were investigated by Nehari [6], Fite [7], Hille [8], Wong [9], Erbe [10], and Ohriska [11] as follows: The oscillatory behavior of the linear differential equation of second order

$$z''(\zeta) + q(\zeta)z(\zeta) = 0,\tag{4}$$

is investigated in Nehari [6] and showed that if

$$\liminf\_{\tilde{\zeta}\to\infty} \frac{1}{\tilde{\zeta}} \int\_{\tilde{\zeta}\_0}^{\tilde{\zeta}} \varkappa^2 q(\varkappa) \mathrm{d}\varkappa > \frac{1}{4},\tag{5}$$

then all solutions of (4) are oscillatory. Fite [7] proved that if

$$\int\_{\tilde{\zeta}\_0}^{\infty} q(\varphi) \mathrm{d}\varphi = \infty,\tag{6}$$

then all solutions of Equation (4) are oscillatory. Hille [8] developed the condition (6) and illustrated that if

$$\liminf\_{\zeta \to \infty} \zeta \int\_{\zeta}^{\infty} q(\varphi) d\varphi > \frac{1}{4},\tag{7}$$

then all solutions of Equation (4) are oscillatory. For the delay differential equation

$$z''(\zeta) + q(\zeta)z(\eta(\zeta)) = 0,\tag{8}$$

the Hille-type condition (7) is generalized by Wong [9], where *η*(*ζ*) ≥ *γζ* with 0 < *γ* < 1, and showed that if

$$\liminf\_{\zeta \to \infty} \zeta \int\_{\zeta}^{\infty} q(\varphi) \mathrm{d}\varphi > \frac{1}{4\gamma'} \tag{9}$$

then all solutions of (8) are oscillatory. Erbe [10] enhanced the condition (9) and examined that if

lim inf *<sup>ζ</sup>*→<sup>∞</sup> *ζ* Z ∞ *ζ q*(≁) *η*(≁) ≁ d≁ > 1 4 , (10)

then all solutions of (8) are oscillatory where *η*(*ζ*) ≤ *ζ*. Ohriska [11] proved that, if

$$\limsup\_{\zeta \to \infty} \zeta \int\_{\zeta}^{\infty} q(\omega) \frac{\eta(\omega)}{\omega} d\omega > 1,\tag{11}$$

then all solutions of (8) are oscillatory.

When T = Z, then

$$
\sigma(\zeta) = \zeta + 1, \; \mu(\zeta) = 1, \; g^{\Delta}(\zeta) = \Delta \mathbf{g}(\zeta), \\
\int\_a^b g(\zeta) \Delta \zeta = \sum\_{\zeta=a}^{b-1} \mathbf{g}(\zeta) \zeta
$$

and (1) turns as the nonlinear functional difference equation

$$
\Delta \left[ a(\zeta) \varrho\_{\gamma} \left( \Delta z(\zeta) \right) \right] + q(\zeta) \varrho\_{\beta} \left( z(\eta(\zeta)) \right) = 0. \tag{12}
$$

The oscillation of Equation (12) when *a*(*ζ*) = 1, *η*(*ζ*) = *ζ*, and *γ* = *β* is the quotient of odd positive integers was elaborated by Thandapani et al. [12] in which *q*(*ζ*) is a positive sequence and showed that every solution of (12) is oscillatory, if

$$\sum\_{k=k\_0}^{\infty} q(k) = \infty.$$

We will examine that our results not only unite some of the known oscillation results for differential and difference equations but they also can be applied on other cases in which the oscillatory behavior of solutions for these equations on various types of time scales was not known. Note that, if T =*h*Z, *h* > 0, then

$$
\sigma(\zeta) = \zeta + h, \,\mu(\zeta) = h, \, z^{\Delta}(\zeta) = \Delta\_h z(\zeta) = \frac{z(\zeta + h) - z(\zeta)}{h},
$$

$$
\int\_a^b g(\zeta) \Delta \zeta = \sum\_{k=0}^{\frac{b-a-h}{h}} g(a+kh)h,
$$

and (1) turns as the nonlinear functional difference equation

$$
\Delta\_h \left[ a(\zeta) \varrho\_\gamma \left( \Delta\_h z(\zeta) \right) \right] + q(\zeta) \varrho\_\beta \left( z(\eta(\zeta)) \right) = 0. \tag{13}
$$

If

$$\mathbb{T} = q^{\mathbb{N}\_0} = \{ \mathbb{zeta} : \mathbb{zeta} = q^k, \ k \in \mathbb{N}\_0, \ q > 1 \} \mathbb{z}$$

then

$$
\sigma(\zeta) = q \,\,\zeta,\,\mu(\zeta) = (q-1)\zeta,\,\,z^{\Delta}(\zeta) = \Delta\_{\overline{q}}z(\zeta) = (z(q\,\zeta) - z(\zeta)) / (q-1)\,\,\zeta,
$$

$$
\int\_{\zeta\_0}^{\infty} g(\zeta) \Delta \zeta = \sum\_{k=n\_0}^{\infty} g(q^k) \mu(q^k) .
$$

where *t*<sup>0</sup> = *q <sup>n</sup>*<sup>0</sup> , and (1) turns as the second order *<sup>q</sup>*−nonlinear difference equation

$$
\Delta\_q \left[ a(\zeta) \,\varphi\_\gamma \left( \Delta\_q z(\zeta) \right) \right] + q(\zeta) \,\varphi\_\beta \left( z(\eta(\zeta)) \right) = 0. \tag{14}
$$

If

$$\mathbb{T} = \mathbb{N}\_0^2 := \{ n^2 : n \in \mathbb{N}\_0 \}\_{\mathsf{A}}$$

then

$$\sigma(\zeta) = (\sqrt{\zeta} + 1)^2 \text{ . } \mu(\zeta) = 1 + 2\sqrt{\zeta} \text{ . } \Delta\_N z(\zeta) = \frac{z((\sqrt{\zeta} + 1)^2) - z(\zeta)}{1 + 2\sqrt{\zeta}}.$$

and (1) turns as the second order nonlinear difference equation

$$
\Delta\_N \left[ a(\zeta) \,\varphi\_\gamma \left( \Delta\_N z(\zeta) \right) \right] + q(\zeta) \,\varphi\_\beta \left( z(\eta(\zeta)) \right) = 0. \tag{15}
$$

If <sup>T</sup> <sup>=</sup> {*H<sup>n</sup>* : *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>0} where *<sup>H</sup><sup>n</sup>* is the harmonic numbers defined by

$$H\_0 = 0, \ H\_{\mathbb{N}} = \sum\_{k=1}^n \frac{1}{k}, n \in \mathbb{N}\_{\succ}$$

then

$$
\sigma(H\_n) = H\_{n+1}, \; \mu(H\_n) = \frac{1}{n+1}, \; z^\Delta(t) = \Delta\_{H\_n} z(H\_n) = (n+1)\Delta z(H\_n).
$$

and (1) turns as the second order nonlinear harmonic difference equation

$$
\Delta\_{H\_{\rm H}}\left[a(H\_{\rm n})\varphi\_{\gamma}\left(\Delta\_{H\_{\rm n}}z(H\_{\rm n})\right)\right] + q(H\_{\rm n})\varphi\_{\not p}\left(z(\eta(H\_{\rm n}))\right) = 0. \tag{16}
$$

For dynamic equations, Erbe et al. in [13,14] expanded the Hille and Nehari oscillation criteria to the half-linear delay dynamic equation of second order

$$(a(\zeta)(z^{\Delta}(\zeta))^{\gamma})^{\Delta} + q(\zeta)z^{\gamma}(\eta(\zeta)) = 0,\tag{17}$$

where *γ* is a quotient of odd positive integers,

$$\eta(\zeta) \le \zeta, \ a^{\Delta}(\zeta) \ge 0, \int\_{\zeta\_0}^{\infty} \eta^{\gamma}(\zeta) q(\zeta) d\zeta = \infty. \tag{18}$$

The authors showed that if either of the following conditions holds

$$\liminf\_{\zeta \to \infty} \zeta^{\gamma} \int\_{\sigma(\zeta)}^{\infty} q(\omega) \left( \frac{\eta(\omega)}{\sigma(\omega)} \right)^{\gamma} \Delta \omega > \frac{\gamma^{\gamma}}{l^{\gamma^2} (\gamma + 1)^{\gamma + 1}},\tag{19}$$

or

$$\liminf\_{\zeta \to \infty} \zeta^{\gamma} \int\_{\sigma(\zeta)}^{\infty} q(\gamma \omega) \left( \frac{\eta(\omega)}{\sigma(\omega)} \right)^{\gamma} \Delta \omega + \liminf\_{\zeta \to \infty} \frac{1}{\widetilde{\zeta}} \int\_{\widetilde{\zeta}\_0}^{\zeta} \omega^{\gamma + 1} q(\gamma \omega) \left( \frac{\eta(\omega)}{\sigma(\omega)} \right)^{\gamma} \Delta \omega > \frac{1}{l^{\gamma(\gamma + 1)}},$$

where *l* := lim inf*ζ*→<sup>∞</sup> *ζ σ*(*ζ*) , then all solutions of (17) are oscillatory. We refer the reader to related results [15–35] and the references cited therein.

A natural question now is: Do the oscillation criteria (5), (6), (7) and (11) for the differential equations of second order by Nehari, Fite, Hille and Ohriska extend to the nonlinear dynamic equation of second order (1) without the restrictive condition (18) in both cases *η*(*ζ*) ≤ *ζ* and *η*(*ζ*) ≥ *ζ*, and when *β* ≥ *γ* and *β* ≤ *γ*.

The aim of this paper is to propose an obvious answer to the above question. We will establish Nehari, Hille and Ohriska type oscillation criteria for (1) without imposing the restrictive condition (18), which generalize and improve the aforementioned results in the literature.

### **2. Oscillation Criteria of** (1) **when** *β* ≥ *γ*

In the subsequent results, we will use the subsequent notations

$$A\left(\zeta\right) := \int\_{\zeta\_0}^{\zeta} \frac{\Delta \varkappa}{a^{\frac{1}{\gamma}}\left(\varkappa\right)} \quad \text{and} \quad l := \liminf\_{\zeta \to \infty} \frac{A\left(\zeta\right)}{A\left(\sigma(\zeta)\right)} \le 1,$$

and

$$\phi(\zeta) := \begin{cases} 1, & \eta(\zeta) \ge \zeta. \\ \left(\frac{A(\eta(\zeta))}{A(\zeta)}\right)^{\beta}, & \eta(\zeta) \le \zeta. \end{cases}$$

Furthermore, *l* > 0 is assuming in the next results.

First, we derive Nehari type to the nonlinear dynamic equation of second order (1).

**Theorem 1.** *Let* (2) *holds, and*

$$\begin{aligned} \liminf\_{\substack{\zeta \to \infty \\ \zeta \to \infty}} \frac{1}{A(\zeta)} \int\_{\Upsilon}^{\zeta} A^{\gamma+1}(\omega) \, \phi(\omega) q(\omega) \Delta \omega &> \frac{1}{l^{\gamma}(\gamma+1)} \left( 1 - \frac{l^{\gamma}}{\gamma l^{\gamma} + 1} \right), \quad 0 < \gamma \le 1, \\\liminf\_{\substack{\zeta \to \infty \\ \zeta \to \infty}} \frac{1}{A(\zeta)} \int\_{\Upsilon}^{\zeta} A^{\gamma+1}(\omega) \, \phi(\omega) q(\omega) \Delta \omega &> \frac{\gamma}{l^{\gamma}(\gamma+1)} \frac{\gamma}{(\gamma + l^{\gamma})}, \qquad \gamma \ge 1, \end{aligned} \tag{20}$$

*for enough large T* ∈ [*ζ*0, <sup>∞</sup>)T*. Then all solutions of Equation* (1) *are oscillatory.*

**Proof.** Assume *z* (*t*) is a nonoscillatory solution of Equation (1) on [*ζ*0, ∞)T. Thus, without loss of generality, let *<sup>z</sup>*(*ζ*) <sup>&</sup>gt; 0 and *<sup>z</sup>*(*η*(*ζ*)) <sup>&</sup>gt; 0 on [*ζ*0, <sup>∞</sup>)T. Since *<sup>q</sup>* <sup>∈</sup> <sup>C</sup>rd ([*ζ*0, <sup>∞</sup>)T, <sup>R</sup>+) and then

$$\left[a(\zeta)\,\rho\_{\gamma}\left(z^{\Delta}(\zeta)\right)\right]^{\Delta} < 0 \quad \text{for } \zeta \ge \zeta\_0.$$

Hence *z* <sup>∆</sup>(*ζ*) > 0, otherwise, it leads to a contradiction. Define

$$w(\zeta) := \frac{a(\zeta)\,\varphi\_\gamma\left(z^\Delta(\zeta)\right)}{z^\gamma(\zeta)}.$$

Using the product and quotient rules, we reach

$$\begin{split} w^{\Delta}(\xi) &= \quad \left( \frac{a(\xi)\,\varphi\_{\gamma}\left(z^{\Delta}(\xi)\right)}{z^{\gamma}(\xi)} \right)^{\Delta} \\ &= \quad \frac{1}{z^{\gamma}(\xi)} \left[ a(\xi)\,\varphi\_{\gamma}\left(z^{\Delta}(\xi)\right) \right]^{\Delta} \\ &\quad + \left( \frac{1}{z^{\gamma}(\xi)} \right)^{\Delta} \left[ a(\xi)\,\varphi\_{\gamma}\left(z^{\Delta}(\xi)\right) \right]^{\sigma} \\ &= \quad \frac{\left[ a(\xi)\,\varphi\_{\gamma}\left(z^{\Delta}(\xi)\right) \right]^{\Delta}}{z^{\gamma}(\xi)} - \frac{(z^{\gamma}(\xi))^{\Delta}}{z^{\gamma}(\xi)z^{\gamma}(\sigma(\xi))} \left[ a(\xi)\,\varphi\_{\gamma}\left(z^{\Delta}(\xi)\right) \right]^{\sigma} . \end{split} \tag{21}$$

From (1) and the definition of *w*(*ζ*), we have

$$w^{\Lambda}(\zeta) = -\left(\frac{z\left(\eta(\zeta)\right)}{z(\zeta)}\right)^{\beta} z^{\beta-\gamma}\left(\zeta\right)q(\zeta) - \frac{(z^{\gamma}(\zeta))^{\Lambda}}{z^{\gamma}(\zeta)}w\left(\sigma(\zeta)\right).$$

Since *z* <sup>∆</sup> <sup>&</sup>gt; 0, then *<sup>z</sup>* (*ζ*) <sup>≥</sup> *<sup>z</sup>* (*ζ*0) for *<sup>ζ</sup>* <sup>≥</sup> *<sup>ζ</sup>*<sup>0</sup> and so

$$z^{\beta-\gamma} \left( \zeta \right) \ge z^{\beta-\gamma} \left( \zeta\_0 \right) =: k > 0 \quad \text{for } \zeta \ge \zeta\_0.$$

Therefore,

$$w^{\Delta}(\zeta) \le -k \left(\frac{z\left(\eta(\zeta)\right)}{z(\zeta)}\right)^{\beta} q(\zeta) - \frac{(z^{\gamma}(\zeta))^{\Delta}}{z^{\gamma}(\zeta)} w\left(\sigma(\zeta)\right).$$

Let *<sup>ζ</sup>* ∈ [*ζ*0, <sup>∞</sup>)<sup>T</sup> be fixed. If *<sup>η</sup>*(*ζ*) ≥ *<sup>ζ</sup>*, then *<sup>z</sup>*(*η*(*ζ*)) ≥ *<sup>z</sup>*(*ζ*) by the fact that *<sup>z</sup>* <sup>∆</sup> > 0. Now the case *<sup>η</sup>*(*ζ*) <sup>≤</sup> *<sup>ζ</sup>* is considered. Since *a ϕ<sup>γ</sup> z* ∆ <sup>∆</sup> < 0 on [*ζ*0, ∞)T, we achieve

$$\begin{aligned} z(\zeta) &\geq \quad z(\zeta) - z(\zeta\_1) = \int\_{\zeta\_0}^{\zeta} z^{\Lambda}(\varkappa) \Delta \varkappa \\ &\geq \quad a^{\frac{1}{\gamma}}(\zeta) z^{\Lambda}(\zeta) \int\_{\zeta\_0}^{\zeta} \frac{\Delta \varkappa}{a^{\frac{1}{\gamma}}(\varkappa)} \\ &= \quad a^{\frac{1}{\gamma}}(\zeta) z^{\Lambda}(\zeta) A(\zeta) .\end{aligned}$$

Therefore

$$\begin{aligned} \left[\frac{z(\zeta)}{A(\zeta)}\right]^\Delta &=& \frac{A(\zeta)z^\Delta(\zeta) - z(\zeta)a^{-\frac{1}{\gamma}}(\zeta)}{A(\zeta)A^\sigma(\zeta)}\\ &=& \frac{a^{-\frac{1}{\gamma}}(\zeta)}{A(\zeta)A^\sigma(\zeta)}\left(a^{\frac{1}{\gamma}}(\zeta)z^\Delta(\zeta)A(\zeta) - z(\zeta)\right),\\ &\leq& 0, \quad \zeta \in (\zeta\_0, \infty)\_\mathbb{T}. \end{aligned}$$

So there exists a *<sup>ζ</sup>*<sup>1</sup> ∈ (*ζ*0, <sup>∞</sup>)<sup>T</sup> such that *<sup>η</sup>*(*ζ*) ∈ (*ζ*0, <sup>∞</sup>)<sup>T</sup> for *<sup>ζ</sup>* ≥ *<sup>ζ</sup>*<sup>1</sup> and so

$$\frac{z(\eta(\zeta))}{z(\zeta)} \ge \frac{A(\eta(\zeta))}{A(\zeta)} \quad \text{ for } \zeta \in [\zeta\_{1\prime}\infty)\_{\mathbb{T}}.$$

In both cases and from the definition of *φ*(*ζ*) we have that

$$\left(\frac{z\left(\eta(\zeta)\right)}{z(\zeta)}\right)^{\beta} \ge \phi(\zeta),\tag{22}$$

and so

$$w^{\Delta}(\zeta) \le -k \, \phi(\zeta)q(\zeta) - \frac{(z^{\gamma}(\zeta))^{\Delta}}{z^{\gamma}(\zeta)} w \, (\sigma(\zeta)) \, \, \, \, \zeta \in [\zeta\_1, \infty)\_{\mathbb{T}}.\tag{23}$$

Then by using the Pötzsche chain rule ([2], Theorem 1.90), we get that

$$\begin{split} \left( z^{\gamma}(\boldsymbol{\zeta}) \right)^{\Delta} &= \quad \gamma \left( \int\_{0}^{1} \left[ z(\boldsymbol{\zeta}) + h\mu(\boldsymbol{\zeta}) z^{\Delta}(\boldsymbol{\zeta}) \right]^{\gamma - 1} \, \mathrm{d}h \right) z^{\Delta}(\boldsymbol{\zeta}) \\ &= \quad \gamma \left( \int\_{0}^{1} \left[ \left( 1 - h \right) z(\boldsymbol{\zeta}) + hz \left( \sigma(\boldsymbol{\zeta}) \right) \right]^{\gamma - 1} \, \mathrm{d}h \right) z^{\Delta}(\boldsymbol{\zeta}) \\ &\geq \quad \left\{ \begin{array}{ll} \gamma z^{\gamma - 1} \left( \sigma(\boldsymbol{\zeta}) \right) z^{\Delta}(\boldsymbol{\zeta}), & 0 < \gamma \leq 1, \\ \gamma z^{\gamma - 1}(\boldsymbol{\zeta}) z^{\Delta}(\boldsymbol{\zeta}), & \gamma \geq 1. \end{array} \right. \end{split}$$

If 0 < *γ* ≤ 1, then

$$w^{\Delta}(\zeta) < -k \, \phi(\zeta)q(\zeta) - \gamma \frac{z^{\Delta}(\zeta)}{z \, (\sigma(\zeta))} \left(\frac{z \, (\sigma(\zeta))}{z(\zeta)}\right)^{\gamma} w \, (\sigma(\zeta)) \cdot \zeta$$

and if *γ* ≥ 1, then

$$w^{\Delta}(\zeta) \le -k \, \phi(\zeta)q(\zeta) - \gamma \frac{z^{\Delta}(\zeta)}{z \, (\sigma(\zeta))} \frac{z \, (\sigma(\zeta))}{z(\zeta)} w \, (\sigma(\zeta)) \, .$$

Note that *z* <sup>∆</sup> > 0 and *a ϕ<sup>γ</sup> z* ∆ <sup>∆</sup> < 0 on [*ζ*1, ∞)T, we see for *γ* > 0,

$$\begin{split} \left| w^{\Delta}(\zeta) \right| &\leq \ -k \, \phi(\zeta)q(\zeta) - \gamma \frac{z^{\Delta}(\zeta)}{z \, (\sigma(\zeta))} w \, (\sigma(\zeta)) \\ &\leq \ -k \, \phi(\zeta)q(\zeta) - \gamma a^{-\frac{1}{\gamma}}(\zeta)w^{1+\frac{1}{\gamma}} \, (\sigma(\zeta)) \ , \quad \zeta \in [\zeta\_1, \infty)\_{\mathbb{T}}. \end{split} \tag{24}$$

Multiplying both sides of (24) by *A γ*+1 (*ζ*) and integrating from *<sup>ζ</sup>*<sup>2</sup> to *<sup>ζ</sup>* ∈ [*ζ*2, <sup>∞</sup>)T, we get

$$\begin{array}{rcl}\int\_{\tilde{\varsigma}\_{2}}^{\tilde{\varsigma}} A^{\gamma+1}(\varkappa) w^{\Delta}(\varkappa) \Delta \varkappa & \leq & -k \int\_{\tilde{\varsigma}\_{2}}^{\tilde{\varsigma}} A^{\gamma+1}(\varkappa) \phi(\varkappa) q(\varkappa) \Delta \varkappa\\ & -\gamma \int\_{\tilde{\varsigma}\_{2}}^{\tilde{\varsigma}} a^{-\frac{1}{\gamma}}(\varkappa) \left(A^{\gamma}(\varkappa) w(\sigma(\varkappa))\right)^{\frac{\gamma+1}{\gamma}} \Delta \varkappa \end{array}$$

By integration by parts, we have

$$\begin{split} A^{\gamma+1}(\boldsymbol{\zeta})w(\boldsymbol{\zeta}) &\leq \quad A^{\gamma+1}(\boldsymbol{\zeta}\_{2})w(\boldsymbol{\zeta}\_{2}) + \int\_{\boldsymbol{\zeta}\_{2}}^{\boldsymbol{\zeta}} \Big( A^{\gamma+1}(\boldsymbol{\omega}) \Big)^{\Delta} w\left(\boldsymbol{\sigma}(\boldsymbol{\omega}) \right) \Delta \boldsymbol{\omega}, \\ &\quad - k \int\_{\boldsymbol{\zeta}\_{2}}^{\boldsymbol{\zeta}} A^{\gamma+1}(\boldsymbol{\omega}) \boldsymbol{\phi}(\boldsymbol{\omega}) q(\boldsymbol{\omega}) \Delta \boldsymbol{\omega} \\ &\quad - \gamma \int\_{\boldsymbol{\zeta}\_{2}}^{\boldsymbol{\zeta}} a^{-\frac{1}{\gamma}}(\boldsymbol{\omega}) \left( A^{\gamma}(\boldsymbol{\omega}) w\left(\boldsymbol{\sigma}(\boldsymbol{\omega}) \right) \right)^{\frac{\gamma+1}{\gamma}} \Delta \boldsymbol{\omega}. \end{split}$$

Using the Pötzsche chain rule, we arrive

$$\begin{split} \left( A^{\gamma+1}(\boldsymbol{\omega}) \right)^{\Delta} &= \quad (\gamma+1) \int\_{0}^{1} [A(\boldsymbol{\omega}) + h\mu(\boldsymbol{\omega})A^{\Delta}(\boldsymbol{\omega})]^{\gamma} \mathrm{d}\boldsymbol{h} \, \frac{1}{a^{1/\gamma}(\boldsymbol{\omega})} \\ &= \quad (\gamma+1) \int\_{0}^{1} [(1-h)A(\boldsymbol{\omega}) + hA^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\omega}))]^{\gamma} \mathrm{d}\boldsymbol{h} \, \frac{1}{a^{1/\gamma}(\boldsymbol{\omega})} \\ &\leq \quad (\gamma+1) \frac{A^{\gamma}\left( \boldsymbol{\sigma}(\boldsymbol{\omega}) \right)}{a^{1/\gamma}(\boldsymbol{\omega})} . \end{split} \tag{25}$$

Hence

$$\begin{split} A^{\gamma+1}(\boldsymbol{\zeta})w(\boldsymbol{\zeta}) &\leq \quad A^{\gamma+1}(\boldsymbol{\zeta}\_{2})w(\boldsymbol{\zeta}\_{2}) - \int\_{\overline{\boldsymbol{\zeta}}\_{2}}^{\overline{\boldsymbol{\zeta}}} A^{\gamma+1}(\boldsymbol{\omega})\boldsymbol{\theta}(\boldsymbol{\omega})q(\boldsymbol{\omega})d\boldsymbol{\omega} \\ &\quad + (\gamma+1)\int\_{\overline{\boldsymbol{\zeta}}\_{2}}^{\overline{\boldsymbol{\zeta}}} \frac{1}{a^{1/\gamma}(\boldsymbol{\omega})} \left[\frac{A\left(\boldsymbol{\sigma}(\boldsymbol{\omega})\right)}{A(\boldsymbol{\omega})}\right]^{\gamma} A^{\gamma}(\boldsymbol{\omega})w\left(\boldsymbol{\sigma}(\boldsymbol{\omega})\right)d\boldsymbol{\omega} \\ &\quad - \gamma\int\_{\overline{\boldsymbol{\zeta}}\_{2}}^{\overline{\boldsymbol{\zeta}}} \frac{1}{a^{1/\gamma}(\boldsymbol{\omega})} \left(A^{\gamma}(\boldsymbol{\omega})w\left(\boldsymbol{\sigma}(\boldsymbol{\omega})\right)\right)^{\frac{\gamma+1}{\gamma}} \boldsymbol{\Delta}\boldsymbol{\omega}. \end{split}$$

It follows that *w* <sup>∆</sup>(*ζ*) <sup>≤</sup> 0 on [*ζ*1, <sup>∞</sup>)T. Let *<sup>ε</sup>* <sup>&</sup>gt; 0, then we choose *<sup>ζ</sup>*<sup>2</sup> <sup>∈</sup> [*ζ*1, <sup>∞</sup>)T, enough large, so for *<sup>ζ</sup>* ∈ [*ζ*2, <sup>∞</sup>)T,

$$A^\gamma \left( \zeta \right) w \left( \sigma \left( \zeta \right) \right) \geq a\_\* - \varepsilon,\tag{26}$$

and

$$\frac{A\left(\zeta\right)}{A\left(\sigma\left(\zeta\right)\right)} \ge l - \varepsilon,\tag{27}$$

where *a*<sup>∗</sup> is defined by

$$a\_\* := \liminf\_{\zeta \to \infty} A^\gamma(\zeta) w\left(\sigma(\zeta)\right) \le 1. \tag{28}$$

By (27), we then get that

$$\begin{aligned} A^{\gamma+1}(\zeta)w(\zeta) &\leq A^{\gamma+1}(\zeta\_2)w(\zeta\_2) - k \int\_{\zeta\_2}^{\zeta} A^{\gamma+1}(\omega)\phi(\omega)q(\omega)d\omega \\\\ \int\_{\zeta\_2}^{\zeta} \frac{1}{a^{1/\gamma}(\omega)} \left[ \frac{\gamma+1}{(l-\varepsilon)^{\gamma}} A^{\gamma}(\omega)w(\sigma(\omega)) - \gamma \left(A^{\gamma}(\omega)w(\sigma(\omega))\right)^{\frac{\gamma+1}{\gamma}} \right] d\omega \\\\ \text{which} \end{aligned}$$

Using the inequality

+

$$\|\mathbf{Y}u - \mathbf{X}u^{\frac{\gamma+1}{\gamma}}\| \le \frac{\gamma^{\gamma}}{(\gamma+1)^{\gamma+1}} \frac{Y^{\gamma+1}}{X^{\gamma}}\tag{29}$$

with *X* = *γ*, *Y* = *γ* + 1 (*l* − *ε*) *γ* and *u* = *A γ* (≁) *w* (*σ*(≁)), we get

$$\begin{split} A^{\gamma+1}(\boldsymbol{\zeta})w(\boldsymbol{\zeta}) &\leq \quad A^{\gamma+1}(\boldsymbol{\zeta}\_{2})w(\boldsymbol{\zeta}\_{2}) - k \int\_{\boldsymbol{\zeta}\_{2}}^{\boldsymbol{\zeta}} A^{\gamma+1}(\boldsymbol{\omega})\phi(\boldsymbol{\omega})q(\boldsymbol{\omega})d\boldsymbol{\omega} \\ &\quad + \frac{1}{(l-\varepsilon)^{\gamma(\gamma+1)}} \left[A(\boldsymbol{\zeta}) - A(\boldsymbol{\zeta}\_{2})\right]. \end{split}$$

Dividing both sides by *A*(*ζ*), we obtain

$$\begin{split} A^{\gamma}(\boldsymbol{\zeta})w(\boldsymbol{\zeta}) &\leq \quad \frac{A^{\gamma+1}(\boldsymbol{\zeta}\_{2})w(\boldsymbol{\zeta}\_{2})}{A(\boldsymbol{\zeta})} - \frac{k}{A(\boldsymbol{\zeta})} \int\_{\boldsymbol{\zeta}\_{2}}^{\boldsymbol{\zeta}} A^{\gamma+1}(\boldsymbol{\omega})\phi(\boldsymbol{\omega})q(\boldsymbol{\omega})d\boldsymbol{\omega} \\ &+ \frac{1}{(l-\varepsilon)^{\gamma(\gamma+1)}} \left[1 - \frac{A(\boldsymbol{\zeta}\_{2})}{A(\boldsymbol{\zeta})}\right]. \end{split}$$

Since *w σ* (*ζ*) ≤ *w*(*ζ*) we get

$$\begin{array}{rcl} A^{\gamma}(\boldsymbol{\zeta})w\left(\boldsymbol{\sigma}(\boldsymbol{\zeta})\right) & \leq & \frac{A^{\gamma+1}(\boldsymbol{\zeta}\_{2})w(\boldsymbol{\zeta}\_{2})}{A(\boldsymbol{\zeta})} - \frac{k}{A(\boldsymbol{\zeta})} \int\_{\boldsymbol{\zeta}\_{2}}^{\boldsymbol{\zeta}} A^{\gamma+1}(\boldsymbol{\omega})\phi(\boldsymbol{\omega})q(\boldsymbol{\omega})d\boldsymbol{\omega} \\ & + \frac{1}{(l-\varepsilon)^{\gamma(\gamma+1)}} \left[1 - \frac{A(\boldsymbol{\zeta}\_{2})}{A(\boldsymbol{\zeta})}\right]. \end{array}$$

Taking the lim sup of both sides as *<sup>ζ</sup>* → <sup>∞</sup> we get

$$A\_\* \le -\liminf\_{\zeta \to \infty} \frac{k}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma+1}(\omega) \phi(\omega) q(\omega) \Delta \omega + \frac{1}{(l-\varepsilon)^{\gamma(\gamma+1)}}.$$

where

$$A\_\* := \limsup\_{\zeta \to \infty} A^\gamma(\zeta)w\left(\sigma(\zeta)\right).$$

Since *k*, *ε* > 0 are arbitrary constants, we obtain

$$A\_\* \le -\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma+1}(\omega) \phi(\omega) q(\omega) \Delta \omega + \frac{1}{l^{\gamma(\gamma+1)}}.\tag{30}$$

Now, multiplying both sides of (24) by *A γ*+1 (*ζ*), we get

$$\begin{split} A^{\gamma+1}\left(\zeta\right)w^{\Lambda}\left(\zeta\right) &\leq \ -k \ A^{\gamma+1}\left(\zeta\right)\phi(\zeta)q(\zeta) - \gamma a^{-1/\gamma}(\zeta)A^{\gamma+1}\left(\zeta\right)w^{1+\frac{1}{\gamma}}\left(\sigma(\zeta)\right), \\ &= \ -A^{\gamma+1}\left(\zeta\right)\phi(\zeta)q(\zeta) \\ &\quad - \gamma a^{-1/\gamma}(\zeta)A^{\gamma}\left(\zeta\right)w\left(\sigma(\zeta)\right)A\left(\zeta\right)w^{\frac{1}{\gamma}}\left(\sigma(\zeta)\right). \end{split}$$

Using (26) gives

$$A^{\gamma+1} \left( \zeta \right) w^{\Lambda} (\zeta) \le -k \left. A^{\gamma+1} \left( \zeta \right) \phi(\zeta) \eta(\zeta) - \theta a^{-1/\gamma}(\zeta), \quad \zeta \in [\zeta\_2, \infty)\_{\mathbb{T}'} \tag{31}$$

where *ϑ* = *γ* (*a*<sup>∗</sup> − *ε*) 1+ <sup>1</sup> *<sup>γ</sup>* . Integrating the inequality (31) from *<sup>ζ</sup>*<sup>2</sup> to *<sup>ζ</sup>* ∈ [*ζ*2, <sup>∞</sup>)T, we get

$$\int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} A^{\gamma+1} \left( \leadsto \right) w^{\Delta} (\leadsto) \Delta \varphi \leq -k \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} A^{\gamma+1} \left( \leadsto \right) \phi(\leadsto) q(\leadsto) \Delta \omega - \theta \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} a^{-1/\gamma} (\leadsto) \Delta \omega$$

Using integrating by parts, we get

$$A^{\gamma+1}\left(\zeta\right)w(\zeta) \quad \le \left. A^{\gamma+1}\left(\zeta\_2\right)w^{\Delta}(\zeta\_2) + \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} \left[A^{\gamma+1}\left(\omega\right)\right]^{\Delta}w\left(\sigma\left(\omega\right)\right)\Delta\omega$$

$$-k \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} A^{\gamma+1}\left(\omega\right)\phi(\omega)q(\omega)\Delta\omega - \theta\left[A(\zeta) - A(\zeta\_2)\right].\tag{32}$$

We consider the forthcoming two cases:

(I) When 0 < *γ* ≤ 1. Using the product rule, we have

$$\left[A^{\gamma+1}\left(\varphi\right)\right]^\Delta = \left[A^\gamma\left(\varphi\right)A\left(\varphi\right)\right]^\Delta = \left[A^\gamma\left(\varphi\right)\right]^\Delta A\left(\varphi\right) + A^\gamma\left(\sigma\left(\varphi\right)\right)A^\Delta\left(\varphi\right) \dots$$

Again use the Pötzsche chain rule, we get

$$\begin{split} \left(A^{\gamma}\left(\varphi\right)\right)^{\Delta} &= \quad \gamma \left(\int\_{0}^{1} \left[A\left(\varphi\right) + h\mu\left(\varphi\right)A^{\Delta}\left(\varphi\right)\right]^{\gamma-1} \mathrm{d}h\right)A^{\Delta}\left(\varphi\right) \\ &= \quad \gamma \left(\int\_{0}^{1} \left[\left(1-h\right)A\left(\varphi\right) + hA\left(\sigma\left(\varphi\right)\right)\right]^{\gamma-1} \mathrm{d}h\right)A^{\Delta}\left(\varphi\right) \\ &\leq \quad \gamma A^{\gamma-1}\left(\varphi\right)A^{\Delta}\left(\varphi\right). \end{split}$$

Then

$$\left[A^{\gamma+1}\left(\varphi\right)\right]^\Delta \le \left(\gamma A^\gamma\left(\varphi\right) + A^\gamma\left(\sigma\left(\varphi\right)\right)\right)A^\Lambda\left(\varphi\right).$$

and so

*A*

*γ*+1 (*ζ*) *w*(*ζ*) ≤ *A γ*+1 (*ζ*2) *w* ∆ (*ζ*2) + Z *ζ ζ*2 (*γA γ* (≁) + *A γ* (*σ* (≁))) *A* ∆ (≁) *w* (*σ* (≁)) ∆≁ −*k* Z *ζ ζ*2 *A γ*+1 (≁) *<sup>φ</sup>*(≁)*q*(≁)∆<sup>≁</sup> <sup>−</sup> *<sup>ϑ</sup>* [*A*(*ζ*) <sup>−</sup> *<sup>A</sup>*(*ζ*2)] = *A γ*+1 (*ζ*2) *w* ∆ (*ζ*2) + Z *ζ ζ*2 *γ* + *A* (*σ* (≁)) *A* (≁) *γ A* ∆ (≁) *A γ* (≁) *w* (*σ* (≁)) ∆≁ −*k* Z *ζ ζ*2 *A γ*+1 (≁) *<sup>φ</sup>*(≁)*q*(≁)∆≁) <sup>−</sup> *<sup>ϑ</sup>* [*A*(*ζ*) <sup>−</sup> *<sup>A</sup>*(*ζ*2)] ≤ *A γ*+1 (*ζ*2) *w* ∆ (*ζ*2) + *γ* + 1 (*l* − *ε*) *γ* (*A*<sup>∗</sup> + *ε*) [*A*(*ζ*) − *A*(*ζ*2)] −*k* Z *ζ ζ*2 *A γ*+1 (≁) *<sup>φ</sup>*(≁)*q*(≁)∆≁) <sup>−</sup> *<sup>ϑ</sup>* [*A*(*ζ*) <sup>−</sup> *<sup>A</sup>*(*ζ*2)] .

Dividing both sides by *A*(*ζ*), we have

$$\begin{split} A^{\gamma} \left( \zeta \right) w \left( \sigma(\zeta) \right) &\leq \quad A^{\gamma} \left( \zeta \right) w(\zeta) \leq \frac{A^{\gamma+1} \left( \zeta\_{2} \right) w^{\Delta}(\zeta\_{2})}{A(\zeta)} \\ &\quad + \left[ \gamma + \frac{1}{\left( l - \varepsilon \right)^{\gamma}} \right] \left( A\_{\*} + \varepsilon \right) \left[ 1 - \frac{A(\zeta\_{2})}{A(\zeta)} \right] \\ &\quad - \frac{k}{A(\zeta)} \int\_{\tilde{\zeta}\_{2}}^{\tilde{\zeta}} A^{\gamma+1} \left( \varkappa \right) \phi(\varkappa) q(\varkappa) \Delta \varkappa - \vartheta \left[ 1 - \frac{A(\zeta\_{2})}{A(\zeta)} \right]. \end{split}$$

Taking the lim sup of both sides as *<sup>ζ</sup>* → <sup>∞</sup> and using (2), we get

$$A\_\* \le \left[\gamma + \frac{1}{\left(l - \varepsilon\right)^{\gamma}}\right] \left(A\_\* + \varepsilon\right) - \liminf\_{\tilde{\zeta} \to \infty} \frac{k}{A(\tilde{\zeta})} \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} A^{\gamma + 1} \left(\varkappa\right) \phi(\varkappa) q(\varkappa) \Delta \varphi - \vartheta.$$

Since *k* and *ε* > 0 are arbitrary constants, we achieve the demanded inequality

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma + 1} \left( \leadsto \right) \phi(\asymp) \eta(\asymp) \Delta \circ \simeq A\_\* \left[ \gamma - 1 + \frac{1}{l^{\gamma}} \right] - \gamma a\_\*^{1 + \frac{1}{\gamma}}.\tag{33}$$

From (30) and (33), we obtain

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma+1}(\omega) \phi(\omega) q(\omega) \Delta \omega \le \frac{1}{l^{\gamma(\gamma+1)}} \left( 1 - \frac{l^{\gamma}}{\gamma l^{\gamma} + 1} \right) \zeta$$

which contradicts the condition (20) if 0 < *γ* ≤ 1.

(II) When *γ* ≥ 1. Using the product rule, we have

$$\left[A^{\gamma+1}\left(\varphi\right)\right]^\Delta = \left[A^\gamma\left(\varphi\right)A\left(\varphi\right)\right]^\Delta = \left[A^\gamma\left(\varphi\right)\right]^\Delta A\left(\sigma\left(\varphi\right)\right) + A^\gamma\left(\varphi\right)A^\Delta\left(\varphi\right)\dots$$

Again by the Pötzsche chain rule we obtain

$$\begin{split} \left(A^{\gamma}\left(\varphi\right)\right)^{\Delta} &= \quad \gamma \left(\int\_{0}^{1} \left[A\left(\varphi\right) + h\mu\left(\varphi\right)A^{\Delta}\left(\varphi\right)\right]^{\gamma-1} \mathrm{d}h\right)A^{\Delta}\left(\varphi\right) \\ &= \quad \gamma \left(\int\_{0}^{1} \left[\left(1-h\right)A\left(\varphi\right) + hA\left(\sigma\left(\varphi\right)\right)\right]^{\gamma-1} \mathrm{d}h\right)A^{\Delta}\left(\varphi\right) \\ &\leq \quad \gamma A^{\gamma-1}\left(\sigma\left(\varphi\right)\right)A^{\Delta}\left(\varphi\right). \end{split}$$

Then

$$\left[A^{\gamma+1}\left(\varphi\right)\right]^\Delta \le \left(\gamma A^\gamma\left(\sigma\left(\varphi\right)\right) + A^\gamma\left(\varphi\right)\right)A^\Lambda\left(\varphi\right)\dots$$

and so

*A γ*+1 (*ζ*) *w*(*ζ*) ≤ *A γ*+1 (*ζ*2) *w* ∆ (*ζ*2) + Z *ζ ζ*2 (*γA γ* (*σ* (≁)) + *A γ* (≁)) *A* ∆ (≁) *w* (*σ* (≁)) ∆≁ −*k* Z *ζ ζ*2 *A γ*+1 (≁) *<sup>φ</sup>*(≁)*q*(≁)∆<sup>≁</sup> <sup>−</sup> *<sup>ϑ</sup>* [*A*(*ζ*) <sup>−</sup> *<sup>A</sup>*(*ζ*2)] = *A γ*+1 (*ζ*2) *w* ∆ (*ζ*2) + Z *ζ ζ*2 *γ A* (*σ* (≁)) *A* (≁) *γ* + 1 *A* ∆ (≁) *A γ* (≁) *w* (*σ* (≁)) ∆≁ −*k* Z *ζ ζ*2 *A γ*+1 (≁) *<sup>φ</sup>*(≁)*q*(≁)∆≁) <sup>−</sup> *<sup>ϑ</sup>* [*A*(*ζ*) <sup>−</sup> *<sup>A</sup>*(*ζ*2)] ≤ *A γ*+1 (*ζ*2) *w* ∆ (*ζ*2) + *γ* (*l* − *ε*) *<sup>γ</sup>* + 1 (*A*<sup>∗</sup> + *ε*) [*A*(*ζ*) − *A*(*ζ*2)] −*k* Z *ζ ζ*2 *A γ*+1 (≁) *<sup>φ</sup>*(≁)*q*(≁)∆<sup>≁</sup> <sup>−</sup> *<sup>ϑ</sup>* [*A*(*ζ*) <sup>−</sup> *<sup>A</sup>*(*ζ*2)] .

Dividing both sides by *A*(*ζ*), we have

$$\begin{split} \left| A^{\gamma} \left( \zeta \right) w(\zeta) \right| &\leq \quad \frac{A^{\gamma+1} \left( \zeta\_{2} \right) w^{\Delta}(\zeta\_{2})}{A(\zeta)} + \left( \frac{\gamma}{(l-\varepsilon)^{\gamma}} + 1 \right) \left( A\_{\*} + \varepsilon \right) \left[ 1 - \frac{A(\zeta\_{2})}{A(\zeta)} \right] \\ &\quad - \frac{k}{A(\zeta)} \int\_{\zeta\_{2}}^{\zeta} A^{\gamma+1} \left( \varkappa \right) \phi(\varkappa) q(\varkappa) \Delta \varkappa - \vartheta \left[ 1 - \frac{A(\zeta\_{2})}{A(\zeta)} \right]. \end{split}$$

Taking the lim sup of both sides as *<sup>ζ</sup>* → <sup>∞</sup> and by (2), we obtain

$$A\_\* \le \left(\frac{\gamma}{(l-\varepsilon)^\gamma} + 1\right) (A\_\* + \varepsilon) - \liminf\_{\zeta \to \infty} \frac{k}{A(\zeta)} \int\_{\tilde{\zeta}\_2}^{\zeta} A^{\gamma+1}(\omega) \,\phi(\omega) q(\omega) \Delta \omega - \theta.$$

Since *k*, *ε* > 0 are arbitrary constants, we reach the demanded inequality

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma + 1}(\simeq) \,\phi(\simeq) q(\simeq) \Delta \varphi \leq \gamma \left( \frac{A\_\*}{l^{\gamma}} - a\_\*^{1 + \frac{1}{\gamma}} \right). \tag{34}$$

,

From (30) and (34), we get

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma+1} \left( \omega \right) \phi(\omega) q(\omega) \Delta \omega \le \frac{\gamma}{l^{\gamma(\gamma+1)} \left( \gamma + l^{\gamma} \right)}$$

which is in contrast to the condition (20) if *γ* ≥ 1. The proof is accomplished.

**Theorem 2.** *Let* (2) *holds, and*

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_T^\zeta A^{\gamma+1}(\omega) \phi(\omega) q(\omega) \Delta \omega > \frac{1}{l^{\gamma(\gamma+1)}} \left( 1 - \frac{l^\gamma}{\gamma+1} \right) . \tag{35}$$

*for enough large T* ∈ [*ζ*0, <sup>∞</sup>)T*. Then all solutions of Equation* (1) *are oscillatory.*

**Proof.** Assume *z* is a nonoscillatory solution of Equation (1) on [*ζ*0, ∞)T. Thus, without loss of generality, let *z*(*ζ*) > 0 and *z*(*η*(*ζ*)) > 0 on [*ζ*0, ∞)T. As shown in the proof of Theorem 1, we obtain

$$A^{\gamma+1}\left(\zeta\right)w(\zeta) \quad \le \left. A^{\gamma+1}\left(\zeta\_2\right)w^{\Delta}(\zeta\_2) + \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} \left[ A^{\gamma+1}\left(\omega\right) \right]^{\Delta} w\left(\sigma\left(\omega\right)\right) \Delta\omega$$

$$-k \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} A^{\gamma+1}\left(\gamma\right) \phi(\omega) q(\omega) \Delta\omega - \theta\left[ A(\zeta) - A(\zeta\_2) \right],\tag{36}$$

where *ϑ* = *γ* (*a*<sup>∗</sup> − *ε*) 1+ <sup>1</sup> *<sup>γ</sup>* . In addition, we have

$$\left[A^{\gamma+1}(\varphi)\right]^\Delta \le (\gamma+1)A^\gamma\left(\sigma(\varphi)\right)a^{-1/\gamma}(\varphi). \tag{37}$$

Substituting (37) into (36) we get

$$\begin{split} A^{\gamma+1}\left(\boldsymbol{\zeta}\right)w(\boldsymbol{\zeta}) &\leq \quad A^{\gamma+1}\left(\boldsymbol{\zeta}\_{2}\right)w^{\Delta}(\boldsymbol{\zeta}\_{2})\\ &+ \left(\gamma+1\right)\int\_{\boldsymbol{\zeta}\_{2}}^{\zeta} \left[\frac{A\left(\boldsymbol{\sigma}\left(\boldsymbol{\omega}\right)\right)}{A\left(\boldsymbol{\omega}\right)}\right]^{\gamma}a^{-1/\gamma}(\boldsymbol{\omega})A^{\gamma}\left(\boldsymbol{\omega}\right)w\left(\boldsymbol{\sigma}\left(\boldsymbol{\omega}\right)\right)\Delta\boldsymbol{\omega}\\ &-k\int\_{\boldsymbol{\zeta}\_{2}}^{\zeta}A^{\gamma+1}\left(\boldsymbol{\omega}\right)\boldsymbol{\phi}(\boldsymbol{\omega})q(\boldsymbol{\omega})\Delta\boldsymbol{\omega}-\boldsymbol{\theta}\left[A(\boldsymbol{\zeta})-A(\boldsymbol{\zeta}\_{2})\right]\\ &\leq \quad A^{\gamma+1}\left(\boldsymbol{\zeta}\_{2}\right)w^{\Delta}(\boldsymbol{\zeta}\_{2})+\frac{\gamma+1}{(l-\varepsilon)^{\gamma}}\left(a\_{\ast}+\varepsilon\right)\left[A(\boldsymbol{\zeta})-A(\boldsymbol{\zeta}\_{2})\right]\\ &-k\int\_{\boldsymbol{\zeta}\_{2}}^{\zeta}A^{\gamma+1}\left(\boldsymbol{\omega}\right)\boldsymbol{\phi}(\boldsymbol{\omega})q(\boldsymbol{\omega})\Delta\boldsymbol{\omega}-\boldsymbol{\theta}\left[A(\boldsymbol{\zeta})-A(\boldsymbol{\zeta}\_{2})\right].\end{split}$$

Dividing both sides by *A*(*ζ*), we have

$$\begin{split} A^{\gamma} \left( \zeta \right) w \left( \sigma(\zeta) \right) &\leq \quad A^{\gamma} \left( \zeta \right) w(\zeta) \leq \frac{A^{\gamma+1} \left( \zeta\_{2} \right) w^{\Delta}(\zeta\_{2})}{A(\zeta)}\\ &\quad + \frac{\left( \gamma + 1 \right)}{\left( l - \varepsilon \right)^{\gamma}} \left( a\_{\*} + \varepsilon \right) \left[ 1 - \frac{A(\zeta\_{2})}{A(\zeta)} \right] \\ &\quad - \frac{k}{A(\zeta)} \int\_{\tilde{\zeta}\_{2}}^{\zeta} A^{\gamma+1} \left( \varkappa \right) \phi(\varkappa) q(\varkappa) \Delta \varkappa - \vartheta \left[ 1 - \frac{A(\zeta\_{2})}{A(\zeta)} \right] \end{split}$$

Taking the lim sup of both sides as *<sup>ζ</sup>* → <sup>∞</sup> and by (2), we obtain

$$a\_\* \le \frac{(\gamma + 1)}{(l - \varepsilon)^{\gamma}} \left( a\_\* + \varepsilon \right) - \liminf\_{\tilde{\zeta} \to \infty} \frac{1}{A(\tilde{\zeta})} \int\_{\tilde{\zeta}\_2}^{\tilde{\zeta}} a^{\gamma + 1} \left( \varkappa \right) \phi(\varkappa) q(\varkappa) \Delta \varkappa - \vartheta.$$

Since *k*, *ε* > 0 are arbitrary, we get the required inequality

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma + 1} \left( \simeq \right) \phi(\simeq) q(\simeq) \Delta \simeq \underline{a}\_\* \left[ \frac{\gamma + 1}{l^{\gamma}} - 1 \right] - \gamma a\_\*^{1 + \frac{1}{\gamma}}.\tag{38}$$

From (30) and (38), we obtain

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta\_2}^{\zeta} A^{\gamma+1}(\omega) \phi(\omega) q(\omega) \Delta \omega \le \frac{1}{l^{\gamma(\gamma+1)}} \left( 1 - \frac{l^{\gamma}}{\gamma+1} \right) \zeta$$

which is in contrast to the condition (35). The proof is accomplished.

**Example 1.** *Consider the nonlinear dynamic equation of second order*

$$\left[\zeta^{\gamma-1}\varrho\_{\gamma}\left(z^{\Delta}(\zeta)\right)\right]^{\Delta} + \frac{\delta\zeta^{\frac{1-\gamma}{\gamma}}}{\phi(\zeta)A^{\gamma+1}(\zeta)}\varrho\_{\beta}\left(z(\eta(\zeta))\right) = 0,\tag{39}$$

*where γ*, *β*, *and δ are positive constants with β* ≥ *γ*. *Here a*(*ζ*) = *ζ γ*−1 *, and <sup>q</sup>*(*ζ*) = *δζ*−(*γ*+1) *φ*(*ζ*)*Aγ*+1(*ζ*) *, then the condition* (2) *holds since*

$$\int^{\infty} \frac{\Delta \varkappa}{a^{\frac{1}{\gamma}}(\varkappa)} = \int^{\infty} \frac{\Delta \varkappa}{\varkappa^{1-\frac{1}{\gamma}}} = \infty$$

*by Example 5.60 in [5]. In addition, a straightforward computation yields that*

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta}^{\zeta} A^{\gamma+1}(\omega) \phi(\omega) q(\omega) \Delta \omega = \delta \liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\zeta}^{\zeta} \frac{\Delta \omega}{\omega^{\gamma+1}} = \delta.$$

*By Theorem 2, every solution of* (39) *is oscillatory if*

$$
\delta > \frac{1}{l^{\gamma(\gamma+1)}} \left( 1 - \frac{l^{\gamma}}{\gamma + 1} \right).
$$

We present a Fite–Wintner type oscillation criterion for (1). The proof is similar to that in [7], and hence is omitted.

**Theorem 3.** *Let* (2) *holds, and*

$$\int\_{\tilde{\zeta}\_0}^{\infty} q(\varphi) \Delta \varphi = \infty. \tag{40}$$

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

From Theorem 3, we assume without loss of generality that

$$\int\_{\zeta\_0}^{\infty} \phi(\varkappa) q(\varkappa) \Delta \varkappa < \infty.$$

Otherwise, we have that (40) holds due to *φ*(*ζ*) ≤ 1, which implies that Equation (1) is oscillatory by Theorem 3. The next theorem is generalized Hille type to the second order nonlinear dynamic Equation (1).

**Theorem 4.** *Let* (2) *holds, and*

$$\liminf\_{\zeta \to \infty} A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega > \frac{\gamma^{\gamma}}{l^{\gamma^2} (\gamma + 1)^{\gamma + 1}}.\tag{41}$$

*Then every solutions of Equation* (1) *is oscillatory.*

**Proof.** Assume *z* (*t*) be a nonoscillatory solution of Equation (1) on [*ζ*0, ∞)T. Thus, without loss of generality, let *z*(*ζ*) > 0 and *z*(*η*(*ζ*)) > 0 on [*ζ*0, ∞)T. As depicted in the proof of Theorem 1, we obtain (24) for *<sup>ζ</sup>* ≥ *<sup>ζ</sup>*1, for some *<sup>ζ</sup>*<sup>1</sup> ∈ (*ζ*0, <sup>∞</sup>)<sup>T</sup> such that *<sup>η</sup>*(*ζ*) ∈ (*ζ*0, <sup>∞</sup>)<sup>T</sup> for *<sup>ζ</sup>* ≥ *<sup>ζ</sup>*1. Also for *<sup>ε</sup>* > 0, then we can pick *<sup>ζ</sup>*<sup>2</sup> <sup>∈</sup> [*ζ*1, <sup>∞</sup>)T, sufficiently large, so that (26) and (27) for *<sup>ζ</sup>* <sup>∈</sup> [*ζ*2, <sup>∞</sup>)T. Replacing *<sup>ζ</sup>* by <sup>≁</sup> in the inequality (24) and then integrating it from *<sup>σ</sup>*(*ζ*) ≥ *<sup>ζ</sup>*<sup>2</sup> to *<sup>v</sup>* ∈ [*ζ*, <sup>∞</sup>)<sup>T</sup> and using the fact *<sup>w</sup>* > 0, we have

$$\begin{aligned} -w\left(\sigma(\zeta)\right) &\leq \quad w\left(v\right) - w\left(\sigma(\zeta)\right) \\ &\leq \quad -k \int\_{\sigma(\zeta)}^{v} \phi(\varkappa)q\left(\varkappa\right)\Delta\varkappa - \gamma \int\_{\sigma(\zeta)}^{v} a^{-\frac{1}{\gamma}}(\varkappa) \, w^{1+\frac{1}{\gamma}}\left(\sigma(\varkappa)\right) \Delta\varkappa. \end{aligned}$$

Taking *<sup>v</sup>* → <sup>∞</sup> we obtain

$$-w\left(\sigma(\zeta)\right) \le -k \int\_{\sigma(\zeta)}^{\infty} \phi(\varkappa) q\left(\varkappa\right) \Delta \varkappa - \gamma \int\_{\sigma(\zeta)}^{\infty} a^{-1/\gamma}(\varkappa) \, w^{1+1/\gamma}\left(\sigma(\varkappa)\right) \Delta \varkappa. \tag{42}$$

Multiplying both sides of (42) by *A γ* (*ζ*), we obtain

$$\begin{split} -A^{\gamma} \left( \zeta \right) w(\sigma(\zeta)) &\leq \ -k \ A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\varkappa) q \left( \varkappa \right) \Delta \varkappa \\ &\quad - \gamma A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} a^{-1/\gamma} (\varkappa) \, w^{1 + \frac{1}{\gamma}} \left( \sigma(\varkappa) \right) \Delta \varkappa \\ &= \ -k \ A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\varkappa) q \left( \varkappa \right) \Delta \varkappa \\ &\quad - \gamma A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \frac{A^{\Delta} \left( \varkappa \right)}{A^{\gamma + 1} (\varkappa)} \left[ A^{\gamma} (\varkappa) w \left( \sigma(\varkappa) \right) \right]^{1 + \frac{1}{\gamma}} \Delta \varkappa. \end{split}$$

It follows from (26) that

$$\begin{array}{rcl}-A^{\gamma}\left(\zeta\right)w(\sigma(\zeta)) & \leq & -kA^{\gamma}\left(\zeta\right)\int\_{\sigma(\zeta)}^{\infty}\phi(\omega)q(\omega)\Delta\omega\\ & & -\left(a\_{\ast}-\varepsilon\right)^{1+\frac{1}{\gamma}}A^{\gamma}\left(\zeta\right)\int\_{\sigma(\zeta)}^{\infty}\gamma\frac{A^{\Delta}\left(\omega\right)}{A^{\gamma+1}\left(\omega\right)}\Delta\omega. \end{array} \tag{43}$$

By Pötzsche chain rule, we reach

$$\left(\frac{-1}{A^{\gamma}}\right)^{\Delta} = \gamma \int\_0^1 \frac{1}{[A + h\mu(\cdot \omega)A^{\Delta}]^{\gamma + 1}} \text{d}h \, A^{\Delta} \le \gamma \frac{A^{\Delta}}{A^{\gamma + 1}}.\tag{44}$$

Then from (43) and (44), we have

$$\begin{split} -A^{\gamma} \left( \zeta \right) w(\sigma(\zeta)) &\leq \ -k \ A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega - \left( a\_{\ast} - \varepsilon \right)^{1 + \frac{1}{\gamma}} \left[ \frac{A \left( \zeta \right)}{A \left( \sigma(\zeta) \right)} \right]^{\gamma} \\ &\leq \ -k \ A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega - \left( l - \varepsilon \right)^{\gamma} \left( a\_{\ast} - \varepsilon \right)^{1 + \frac{1}{\gamma}} \end{split}$$

which yields

$$k A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega \leq A^{\gamma} \left( \zeta \right) w(\sigma(\zeta)) - (l - \varepsilon)^{\gamma} \left( a\_\* - \varepsilon \right)^{1 + \frac{1}{\gamma}}.$$

By taking the lim inf of both sides as *<sup>ζ</sup>* → <sup>∞</sup> we obtain that

$$\liminf\_{\zeta \to \infty} k \, A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega \leq a\_\* - (l - \varepsilon)^{\gamma} \left( a\_\* - \varepsilon \right)^{1 + \frac{1}{\gamma}}.$$

Since *k* and *ε* > 0 are arbitrary, we achieve the following inequality

$$\liminf\_{\zeta \to \infty} A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega \leq a\_\* - l^{\gamma} \ a\_\*^{1 + \frac{1}{\gamma}}.$$

Using the inequality (29) with *z* = *l γ* , *Y* = 1 and *u* = *a*∗, we get the desired inequality

$$\liminf\_{\zeta \to \infty} A^{\gamma}(\zeta) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) q(\omega) \Delta \omega \le \frac{\gamma^{\gamma}}{l^{\gamma^2} (\gamma + 1)^{\gamma + 1}} \zeta$$

which is in contrast to the condition (41). The proof is accomplished in Theorem 4.

**Example 2.** *Consider the nonlinear second order dynamic equation*

$$\left[\varphi\_{\gamma}\left(z^{\Delta}(\zeta)\right)\right]^{\Delta} + \frac{\kappa\gamma}{L\zeta^{\gamma+1}}\varphi\_{\beta}\left(z(\eta(\zeta))\right) = 0,\tag{45}$$

*where γ, β*, *κ are positive constants, and L* = lim inf*ζ*→<sup>∞</sup> *ζ σ* (*ζ*) *<sup>γ</sup> with β* ≥ *γ. Here a*(*ζ*) = 1*, η*(*ζ*) ≥ *ζ and q*(*ζ*) = *ηγ Lζ γ*+1 *, then the condition* (2) *holds, A*(*ζ*) = *ζ* − *ζ*<sup>0</sup> *and φ*(*ζ*) = 1*. In addition,*

$$\begin{split} \liminf\_{\zeta \to \infty} A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\varkappa) q(\varkappa) \Delta \varkappa &=& \frac{\kappa}{L} \liminf\_{\zeta \to \infty} A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \frac{\gamma \Delta \varkappa}{\varkappa^{\omega \gamma + 1}} \\ &\geq \quad \frac{\kappa}{L} \liminf\_{\zeta \to \infty} A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \left( \frac{-1}{\varpi^{\gamma}} \right)^{\Lambda} \Delta \varkappa \\ &=& \frac{\kappa}{L} \liminf\_{\zeta \to \infty} \left( \frac{\zeta}{\sigma \left( \zeta \right)} - \frac{\zeta\_0}{\sigma \left( \zeta \right)} \right)^{\gamma} = \kappa \end{split}$$

*if κ* > *γ γ l γ*2 (*γ* + 1) *γ*+1 . *Then by Theorem 4, all solutions of* (45) *are oscillatory if κ* > *γ γ l γ*2 (*γ* + 1) *γ*+1 .

**Remark 1.** *We could refer to the recent results due to [13,14] and others do not apply to Equations* (39) *and* (45)*.* **Theorem 5.** *Let* (2) *hold, and*

$$\limsup\_{\zeta \to \infty} A^{\gamma}(\zeta) \int\_{\zeta}^{\infty} \phi(\omega) q(\omega) \Delta \omega > 1. \tag{46}$$

#### *Then all solutions of Equation* (1) *oscillate.*

**Proof.** Assume *z* (*t*) is a nonoscillatory solution of Equation (1) on [*ζ*0, ∞)T. Thus, without loss of generality, let *z*(*ζ*) > 0 and *z*(*η*(*ζ*)) > 0 on [*ζ*0, ∞)T. Integrating both sides of the dynamic Equation (1) from *<sup>ζ</sup>* to *<sup>v</sup>* ∈ [*ζ*0, <sup>∞</sup>)T, we obtain

$$\int\_{\tilde{\zeta}}^{v} q(\varphi \circ) z^{\mathfrak{f}}(\eta(\varphi \circ)) \Delta \circ = a(\zeta)(z^{\Delta}(\zeta))^{\gamma} - a(v)(z^{\Delta}(v))^{\gamma} \le a(\zeta)(z^{\Delta}(\zeta))^{\gamma}. \tag{47}$$

As shown in the proof of Theorem 1, there exists *<sup>ζ</sup>*<sup>1</sup> ∈ (*ζ*0, <sup>∞</sup>)<sup>T</sup> satisfying *<sup>η</sup>*(*ζ*) ∈ (*ζ*0, <sup>∞</sup>)<sup>T</sup> for *<sup>ζ</sup>* ≥ *<sup>ζ</sup>*<sup>1</sup> such that for *ζ* ≥ *ζ*<sup>1</sup>

$$z^{\mathfrak{G}}(\eta(\zeta)) \ge k \,\phi(\zeta) z^{\gamma}(\zeta) \,\tag{48}$$

and

$$z^{\gamma}(\zeta) \ge a(\zeta) \left(z^{\Delta}(\zeta)\right)^{\gamma} A^{\gamma}(\zeta). \tag{49}$$

From (47) and (48), we obtain

$$k\int\_{\tilde{\zeta}}^{v} \phi(\varkappa)q(\varkappa)z^{\gamma}(\varkappa)\Delta\varkappa \le a(\zeta)(z^{\Delta}(\zeta))^{\gamma}.$$

Since *z* <sup>∆</sup>(*ζ*) > 0, we get that

$$k \, z^{\gamma}(\zeta) \int\_{\zeta}^{v} \phi(\omega) q(\omega) \Delta \omega \le a(\zeta) (z^{\Delta}(\zeta))^{\gamma}. \tag{50}$$

From (49) and (50), we get

$$k\,A^{\gamma}(\zeta)\int\_{\zeta}^{v}\phi(\omega)q(\omega)\Delta\omega \le 1.$$

$$\text{Taking } v \to \infty \text{, we have}$$

$$k\,A^{\gamma}(\zeta)\int\_{\zeta}^{\infty}\phi(\omega)q(\omega)\Delta\omega \le 1.$$

Since *k* > 0 is arbitrary, we have

$$A^\gamma(\zeta) \int\_{\zeta}^{\infty} \phi(\varkappa) q(\varkappa) \Delta \varkappa \le 1\_{\varkappa}$$

which gives us the contradiction

$$\limsup\_{\zeta \to \infty} A^{\gamma}(\zeta) \int\_{\zeta}^{\infty} \phi(\omega) q(\omega) \Delta \omega \le 1.$$

The proof of Theorem 5 is accomplished.

### **3. Oscillation Criteria of** (1) **when** *β* ≤ *γ*

Assume that

$$z(\zeta) > 0, \; z(\eta(\zeta)) > 0, \; z^{\Delta}(\zeta) > 0, \; \left[a(\zeta)\rho\_{\gamma}\left(z^{\Delta}(\zeta)\right)\right]^{\Delta} < 0$$

eventually. Integrating Equation (1) from *<sup>ζ</sup>* to *<sup>v</sup>* ∈ [*ζ*, <sup>∞</sup>)<sup>T</sup> and then using (22) and the fact that *<sup>z</sup>* <sup>∆</sup> > 0, we obtain

$$\begin{aligned} -a(v)\,\eta\_{\gamma}\left(z^{\Delta}(v)\right) + a(\zeta)\,\eta\_{\gamma}\left(z^{\Delta}(\zeta)\right) &= \int\_{\zeta}^{v} q\left(\omega\right)\,\eta\_{\beta}\left(z\left(\eta\left(\omega\right)\right)\right)\,\Delta\omega\\ &\geq \int\_{\zeta}^{v} \phi\left(\omega\right)q\left(\omega\right)\,\eta\_{\beta}\left(z\left(\omega\right)\right)\,\Delta\omega\\ &\geq \quad \varrho\_{\beta}\left(z\left(\zeta\right)\right)\int\_{\zeta}^{v} \phi\left(\omega\right)q\left(\omega\right)\,\Delta\omega, \end{aligned}$$

and *a*(*v*)*ϕ<sup>γ</sup> z* <sup>∆</sup>(*v*) > 0 gives

$$a(\zeta)\varrho\_{\gamma}\left(z^{\Delta}(\zeta)\right) \ge \left.\varrho\_{\beta}\left(z\left(\zeta\right)\right)\int\_{\zeta}^{v}\phi(\omega)q\left(\omega\right)\Delta\omega\right.$$

Hence by taking limits as *<sup>v</sup>* → <sup>∞</sup> we have

$$a(\zeta)\,\wp\_{\gamma}\left(z^{\Delta}(\zeta)\right) \ge \,\wp\_{\beta}\left(z\left(\zeta\right)\right)\int\_{\zeta}^{\infty}\phi(\omega)q\left(\omega\right)\,\Delta\omega.\tag{51}$$

Since - *a*(*ζ*)*ϕ<sup>γ</sup> z* <sup>∆</sup>(*ζ*) <sup>∆</sup> < 0 eventually, then

$$a(\zeta)\,\varphi\_{\gamma}\left(z^{\Delta}(\zeta)\right) \le a(\zeta\_2)\,\varphi\_{\gamma}\left(z^{\Delta}(\zeta\_2)\right) =: b \quad \text{for } \zeta \ge \zeta\_2\sqrt{\xi}$$

and hence from (51), we have

$$b \ge a(\zeta)\varrho\_{\gamma}\left(z^{\Delta}(\zeta)\right) \ge \left.\varrho\_{\beta}\left(z\left(\zeta\right)\right)\int\_{\zeta}^{\infty} \phi(\varkappa)q\left(\varkappa\right)\Delta\varkappa\_{\varkappa}\right|$$

and so

$$z^{\beta-\gamma}\left(\zeta\right) = \left[\varrho\_{\beta}\left(z\left(\zeta\right)\right)\right]^{\frac{\beta-\gamma}{\beta}} \ge c \left[\int\_{\zeta}^{\infty} \phi(\varkappa)q\left(\varkappa\right)d\varkappa\right]^{\frac{\gamma-\beta}{\beta}}\zeta$$

where *c* := *b β*−*γ <sup>β</sup>* > 0. Combining all these we see that for every arbitrary *c* > 0,

$$z^{\beta-\gamma} \left( \zeta \right) \ge c \left[ \int\_{\zeta}^{\infty} \phi(\omega) q\left(\omega \right) \Delta \omega \right]^{\frac{\gamma-\beta}{\beta}},\tag{52}$$

eventually. Let

$$Q\left(\zeta\right) := q\left(\zeta\right) \left[ \int\_{\zeta}^{\infty} \phi(\varkappa) q\left(\varkappa\right) \Delta\varkappa \right]^{\frac{\gamma-\beta}{\beta}}.$$

Therefore, by (52) and the definition of *Q* (*ζ*), as direct consequence of Theorems 1, 2, 4 and 5, we get oscillation criteria for Equation (1) with *β* ≤ *γ*.

**Theorem 6.** *Let* (2) *hold, and*

$$\begin{aligned} \liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\mathbb{T}}^{\zeta} A^{\gamma+1}(\omega) \, \phi(\omega) Q(\omega) \Delta \omega &> \frac{1}{l^{\gamma(\gamma+1)}} \left( 1 - \frac{l^{\gamma}}{\gamma l^{\gamma} + 1} \right), & 0 < \gamma \le 1, \\\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_{\mathbb{T}}^{\zeta} A^{\gamma+1}(\omega) \, \phi(\omega) Q(\omega) \Delta \omega &> \frac{\gamma}{l^{\gamma(\gamma+1)} \left( \gamma + l^{\gamma} \right)}, & \gamma \ge 1, \end{aligned} \tag{53}$$

*for enough large T* ∈ [*ζ*0, <sup>∞</sup>)T*. Then all solutions of Equation* (1) *oscillate.*

**Theorem 7.** *Let* (2) *holds, and*

$$\liminf\_{\zeta \to \infty} \frac{1}{A(\zeta)} \int\_T^{\zeta} A^{\gamma+1}(\omega) \phi(\omega) Q(\omega) \Delta \omega > \frac{1}{l^{\gamma(\gamma+1)}} \left( 1 - \frac{l^{\gamma}}{\gamma+1} \right) \zeta$$

*for enough large T* ∈ [*ζ*0, <sup>∞</sup>)T*. Then all solutions of Equation* (1) *oscillate.*

**Theorem 8.** *Let* (2) *holds, and*

$$\liminf\_{\zeta \to \infty} A^{\gamma} \left( \zeta \right) \int\_{\sigma(\zeta)}^{\infty} \phi(\omega) \mathbb{Q}(\omega) \Delta \omega > \frac{\gamma^{\gamma}}{l^{\gamma^2} (\gamma + 1)^{\gamma + 1}}.$$

*Then all solutions of Equation* (1) *oscillate.*

**Theorem 9.** *Let* (2) *holds, and*

$$\limsup\_{\zeta \to \infty} A^{\gamma}(\zeta) \int\_{\zeta}^{\infty} \phi(\omega) Q(\omega) \Delta \omega > 1.$$

*Then all solutions of Equation* (1) *oscillate.*

#### **4. Conclusions**

	- (i) Condition (41) reduces to (7) in the case if T = R, *γ* = *β* = 1, *a* (*ζ*) = 1, and *η* (*ζ*) = *ζ*;
	- (ii) Condition (41) reduces to (10) in the case when <sup>T</sup> <sup>=</sup> <sup>R</sup>, *<sup>γ</sup>* <sup>=</sup> *<sup>β</sup>* <sup>=</sup> 1, *<sup>a</sup>* (*ζ*) <sup>=</sup> 1, and *<sup>g</sup>* (*ζ*) <sup>≤</sup> *<sup>ζ</sup>*;
	- (iii) Condition (41) reduces to (19) under the assumptions that *γ* = *β*, *a* <sup>∆</sup> (*ζ*) <sup>≥</sup> 0, and *<sup>g</sup>* (*ζ*) <sup>≤</sup> *<sup>ζ</sup>*;
	- (iv) Conditions (46) reduces to (11) supposing that <sup>T</sup> <sup>=</sup> <sup>R</sup>, *<sup>γ</sup>* <sup>=</sup> *<sup>β</sup>* <sup>=</sup> 1, *<sup>a</sup>* (*ζ*) <sup>=</sup> 1, and *<sup>g</sup>* (*ζ*) <sup>≤</sup> *<sup>ζ</sup>*.

**Author Contributions:** Conceptualization, T.S.H.; Data curation, A.A.M.; Formal analysis, T.S.H. and Y.S.; Project administration, Y.S.; Writing—original draft, T.S.H.; Resources, A.A.M.; Supervision, T.S.H. and Y.S.; Investigation, A.A.M.; Validation, T.S.H., Y.S. and A.A.M.; Writing—review & editing, T.S.H., Y.S. and A.A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The reported study was supported by the National Natural Science Foundation of China under Grant 61873110 and the Foundation of Taishan Scholar of Shandong Province under Grant ts20190938.

**Conflicts of Interest:** The authors declare that they have no competing interests. There are not any non-financial competing interests (political, personal, religious, ideological, academic, intellectual, commercial, or any other) to declare in relation to this manuscript.

#### **References**


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