Extended Approach to the Asymptotic Behavior and Symmetric Solutions of Advanced Differential Equations

Abstract

We studied the asymptotic behavior of fourth-order advanced differential equations of the form ${\left(a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+q\left(\upsilon \right)g\left(w\left(\delta \left(\upsilon \right)\right)\right)=0$. New results are presented for the oscillatory behavior of these equations in the form of Philos-type and Hille–Nehari oscillation criteria. Some illustrative examples are presented.

1. Introduction

As is well known, differential equations have many real-world applications [1]. Advanced differential equations, in particular, find applications in dynamical systems, mathematics of networks, optimization, and in the mathematical modeling of engineering processes, such as those found in electrical power systems, materials, and energy [2]. In the last decade, one important area of active research is the study of the qualitative oscillation behavior of differential equations [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. In this paper, we investigate the oscillation of fourth-order nonlinear advanced differential equations of the following form:

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

(1)

where $\upsilon \ge {\upsilon }_0$. Our main aim is to complement and improve the results in [19,20,21]. To motivate that, we briefly review and put into context those and related results.

In [17,18], Zhang et al. obtain, under mild assumptions and with the help of the comparison method with first-order equations, an oscillation criterion ensuring that every solution w of equation

$${\left({\left(w^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+q\left(\upsilon \right)w^{\alpha }\left(\delta \left(\upsilon \right)\right)=0$$

(2)

with $\delta \left(\upsilon \right)\le \upsilon$, $\alpha \le \beta$, $\kappa$ even, and $\alpha ,\beta$ ratios of odd positive integers, is either oscillatory or satisfies $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$. For the special case when $\beta =\alpha$, Zhang et al. [22] obtain—under similar assumptions of those in [17,18], but now using the comparison method with second-order equations—some results on the asymptotic behavior of (2) in the case of $\kappa =4$. Agarwal and Grace [19] and Agarwal et al. [20] study canonical even-order nonlinear advanced differential equations, as follows:

$${\left({\left(w^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+q\left(\upsilon \right)w^{\beta }\left(\delta \left(\upsilon \right)\right)=0$$

(3)

by means of the Riccati transformation technique, establishing some oscillation criteria for (3) when $\delta \left(\upsilon \right)\ge \upsilon$, $\kappa$ is even, and $\beta$ is a ratio of odd positive integers. As a special case, when $\beta =1$, Equation (3) becomes

$$w^{\left(\kappa \right)}\left(\upsilon \right)+q\left(\upsilon \right)w\left(\delta \left(\upsilon \right)\right)=0.$$

(4)

Grace and Lalli [21] study the oscillation of (4), in the case when $\kappa$ is even, under the following condition:

$${\int }_{{\upsilon }_0}^{\infty }\frac{1}{a^{1/\beta }\left(s\right)}\mathrm{d}s=\infty .$$

(5)

To prove their results, they apply previous mentioned results to the following equation:

$$w^{\left(\kappa \right)}\left(\upsilon \right)+\frac{q_0}{{\upsilon }^{\kappa }}w\left(\lambda \upsilon \right)=0,\upsilon \ge 1.$$

(6)

If we set $\kappa =4$ and $\lambda =2$, then, by applying the conditions in [19,20,21] to Equation (6), we find the results in [20], which improve those of [21]. Moreover, the results in [19] improve the ones of [20,21]. Thus, our motivation here is to complement and improve the results of [19,20,21]. From them, we obtain new criteria for the oscillation of Equation (1).

The paper is organized as follows. We begin with Section 2 of preliminaries, fixing our assumptions and notations and recalling necessary definitions and results from the literature. Our results are then given in Section 3: we prove conditions assuring that every solution w of (1) is either oscillatory or satisfies $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$ (see Theorems 1 and 2 and Corollary 1). In Section 4, we give two simple examples for which previous results of the literature do not apply, while our Hille–Nehari-type oscillation criterion holds. We end with Section 5—conclusions and future works—posing an interesting and challenging open question.

2. Hypotheses and Preliminaries

Throughout the work, we assume the following assumptions to (1):

A₁:

$\beta$ is a quotient of odd positive integers;

A₂:

$a\in C^1\left([{\upsilon }_0,\infty ),\mathbb{R}\right)$, $a\left(\upsilon \right)>0$, and $a^{\prime }\left(\upsilon \right)\ge 0$ with

$${\int }_{{\upsilon }_0}^{\infty }\frac{1}{a^{1/\beta }\left(s\right)}\mathrm{d}s<\infty ;$$

(7)

A₃:

$q\in C\left([{\upsilon }_0,\infty ),\mathbb{R}\right)$ with $q\left(\upsilon \right)\ge 0$;

A₄:

$\delta \in C\left([{\upsilon }_0,\infty ),\mathbb{R}\right)$ with $\delta \left(\upsilon \right)\ge \upsilon$ and $\underset{\upsilon \to \infty }{lim}\delta \left(\upsilon \right)=\infty$;

A₅:

$g\in C\left(\mathbb{R},\mathbb{R}\right)$ such that $g\left(x\right)/x^{\beta }\ge k>0$ for $x\ne 0$.

By a solution of (1), we mean the function $w\in C^3[{\upsilon }_w,\infty )$, ${\upsilon }_w\ge {\upsilon }_0$, which has the property $a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\in C^1[{\upsilon }_w,\infty )$ and satisfies (1) on $[{\upsilon }_w,\infty )$. We consider only those solutions w of (1) that satisfy $sup\{\left|w\left(\upsilon \right)\right|:\upsilon \ge {\upsilon }_w\}>0$. To prove our results, we make use of the following methods: (i) an integral averaging technique; (ii) Riccati transformation techniques; (iii) a comparison method with second-order differential equations.

Definition 1

(See [23]). We say that the differential equation

$${\left[a\left(\upsilon \right){\left(w^{\prime }\left(\upsilon \right)\right)}^{\beta }\right]}^{\prime }+q\left(\upsilon \right)w^{\beta }\left(g\left(\upsilon \right)\right)=0\upsilon \ge {\upsilon }_{0,}$$

(8)

where $\beta >0$ is a ratio of odd positive integers and a, $q\in C\left([{\upsilon }_0,\infty ),{\mathbb{R}}^{+}\right)$, is nonoscillatory, if there exists a number $\upsilon \ge {\upsilon }_0$ and a function $\varsigma \in C^1\left([{\upsilon }_0,\infty ),\mathbb{R}\right)$ satisfying the inequality

$${\varsigma }^{\prime }\left(\upsilon \right)+\gamma a^{-1/\beta }\left(\upsilon \right){\left(\varsigma \left(\upsilon \right)\right)}^{\left(1+\beta \right)/\beta }+q\left(\upsilon \right)\le 0$$

on $[{\upsilon }_0,\infty )$.

Definition 2

(See [24]). Let $D=\{\left(\upsilon ,s\right)\in {\mathbb{R}}^2:\upsilon \ge s\ge {\upsilon }_0\}\mathrm{and}{D}_0=\{\left(\upsilon ,s\right)\in {\mathbb{R}}^2:\upsilon >s\ge {\upsilon }_0\}$. A kernel function $H_i\in C\left(D,\mathbb{R}\right),$$(i=1,2,\cdots ,n)$ is said to belong to the set ℑ, written by $H\in \Im$, if, for $i=1,2$, one has:

(i)

$H_i\left(\upsilon ,s\right)=0$ for $\upsilon \ge {\upsilon }_0$ and $H_i\left(\upsilon ,s\right)>0$ for $\left(\upsilon ,s\right)\in D_0$;

(ii)

$H_i\left(\upsilon ,s\right)$ has a continuous and non-positive partial derivative $\partial H_i/\partial s$ on $D_0$, and there exist functions $\eta ,\vartheta \in C^1\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right)$ and $h_i\in C\left(D_0,\mathbb{R}\right)$, such that

$$\frac{\partial }{\partial s}{H}_1\left(\upsilon ,s\right)+\frac{{\eta }^{\prime }\left(s\right)}{\eta \left(s\right)}{H}_1\left(\upsilon ,s\right)=h_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)$$

(9)

and

$$\frac{\partial }{\partial s}{H}_2\left(\upsilon ,s\right)+\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right)=h_2\left(\upsilon ,s\right)\sqrt{H_2\left(\upsilon ,s\right)}.$$

(10)

Notation 1.

For convenience, we denote

$$\begin{array}{ccc}\hfill \zeta \left(s\right)& :=& {\int }_{{\upsilon }_0}^{\infty }\frac{1}{a^{1/\beta }\left(s\right)}\mathrm{d}s,\hfill \\ \hfill \pi \left(s,\upsilon \right)& :=& \frac{h_1^{\beta +1}\left(\upsilon ,s\right)H_1^{\beta }\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{2^{\beta }\eta \left(s\right)a\left(s\right)}{{\left(\theta s^2\right)}^{\beta }},\hfill \\ \hfill \tilde{\pi }\left(s,\upsilon \right)& :=& \frac{{\beta }^{\beta +1}{H}_3\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{1}{a^{1/\beta }\left(s\right)\zeta \left(s\right)}\hfill \end{array}$$

where $\theta \in \left(0,1\right)$, and

$$\varpi \left(\upsilon \right):={\int }_{\upsilon }^{\infty }{\left(\frac{k}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma .$$

We shall employ the following four lemmas:

Lemma 1

(See [18]). Suppose that $w\in C^{\kappa }\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right),$$w^{\left(\kappa \right)}$ is of a fixed sign on $\left[{\upsilon }_0,\infty \right)$, $w^{\left(\kappa \right)}$ is not identically zero, and there exists a ${\upsilon }_1\ge {\upsilon }_0$, such that

$$w^{\left(\kappa -1\right)}\left(\upsilon \right)w^{\left(\kappa \right)}\left(\upsilon \right)\le 0$$

for all $\upsilon \ge {\upsilon }_1$. If we have ${lim}_{\upsilon \to \infty }w\left(\upsilon \right)\ne 0$, then there exists ${\upsilon }_{\theta }\ge {\upsilon }_1$, such that

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

for every $\theta \in \left(0,1\right)$ and $\upsilon \ge {\upsilon }_{\theta }$.

Lemma 2

(See [5]). If $w^{\left(i\right)}\left(\upsilon \right)>0$, $i=0,1,\dots ,\kappa$, and $w^{\left(\kappa +1\right)}\left(\upsilon \right)<0$, then

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

Lemma 3

(See [4]). Let β be a ratio of two odd numbers and $V>0$ and U be two constants. Then,

$$Ux-V{x}^{\left(\beta +1\right)/\beta }\le \frac{{\beta }^{\beta }}{{(\beta +1)}^{\beta +1}}\frac{U^{\beta +1}}{V^{\beta }}.$$

Lemma 4

(See [25]). If w is a positive solution of (1), then there exist three possible situations for $\upsilon \ge {\upsilon }_1$, where ${\upsilon }_1\ge {\upsilon }_0$ is sufficiently large:

$\left({\mathbf{S}}_1\right)$

$w\left(\upsilon \right)>0$, $w^{\prime }\left(\upsilon \right)>0$, $w^{\prime \prime }\left(\upsilon \right)>0$, $w^{\prime \prime \prime }\left(\upsilon \right)>0$, $w^{\left(4\right)}\left(\upsilon \right)<0$,

$\left({\mathbf{S}}_2\right)$

$w\left(\upsilon \right)>0$, $w^{\prime }\left(\upsilon \right)>0$, $w^{\prime \prime }\left(\upsilon \right)<0$, $w^{\prime \prime \prime }\left(\upsilon \right)>0$, $w^{\left(4\right)}\left(\upsilon \right)<0$,

$\left({\mathbf{S}}_3\right)$

$w\left(\upsilon \right)>0,w^{\prime \prime }\left(\upsilon \right)>0,w^{\prime \prime \prime }\left(\upsilon \right)<0$.

We are now in a position to formulate and prove our original results.

3. Main Results

In our first theorem, we employ an integral averaging technique to establish a Philos-type oscillation criterion.

Theorem 1

(Philos-type oscillation criterion for (1)). Under assumptions $A_1$–$A_5$, if there exist positive functions $\eta ,\vartheta \in C^1\left(\left[{\upsilon }_0,\infty \right),\mathbb{R}\right)$, such that

$$\underset{\upsilon \to \infty }{limsup}\frac{1}{H_1\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_1\left(\upsilon ,s\right)k\eta \left(s\right)q\left(s\right)-\pi \left(s,\upsilon \right)\right)\mathrm{d}s=\infty$$

(11)

for all $\theta \in \left(0,1\right)$, if

$$\underset{\upsilon \to \infty }{limsup}\frac{1}{H_2\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_2\left(\upsilon ,s\right)\vartheta \left(s\right)\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\upsilon ,s\right)}{4}\right)\mathrm{d}s=\infty$$

(12)

and, for every ${\theta }_1\in \left(0,1\right)$,

$$\underset{\upsilon \to \infty }{limsup}\frac{1}{H_3\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_3\left(\upsilon ,s\right)kq\left(s\right){\left(\frac{{\theta }_1{\delta }^2\left(s\right)}{2}\right)}^{\beta }{\zeta }^{\beta }\left(\delta \left(s\right)\right)-\tilde{\pi }\left(s,\upsilon \right)\right)\mathrm{d}s=\infty ,$$

then every solution of (1) is either oscillatory or satisfies $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$.

Proof. 

Assume the contrary, that w is a positive solution of (1). Then, we can suppose that $w\left(\upsilon \right)$ and $w\left(\delta \left(\upsilon \right)\right)$ are positive for all $\upsilon \ge {\upsilon }_1$ and are sufficiently large. From Lemma 4, we have three possible situations $\left({\mathbf{S}}_1\right)$, $\left({\mathbf{S}}_2\right)$ or $\left({\mathbf{S}}_3\right)$. Assume that $\left({\mathbf{S}}_1\right)$ holds. Using Lemma 1, we find that

$$w^{\prime }\left(\upsilon \right)\ge \frac{\theta }{2}{\upsilon }^2{w}^{\prime \prime \prime }\left(\upsilon \right)$$

(13)

for every $\theta \in (0,1)$ and for all large $\upsilon$. Define

$$\phi \left(\upsilon \right):=\eta \left(\upsilon \right)\left(\frac{a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }}{w^{\beta }\left(\upsilon \right)}\right).$$

(14)

We see that $\phi \left(\upsilon \right)>0$ for $\upsilon \ge {\upsilon }_1,$ where $\eta \in C^1\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right)$ and

$${\phi }^{\prime }\left(\upsilon \right)={\eta }^{\prime }\left(\upsilon \right)\frac{a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }}{w^{\beta }\left(\upsilon \right)}+\eta \left(\upsilon \right)\frac{{\left(a{\left(w^{\prime \prime \prime }\right)}^{\beta }\right)}^{\prime }\left(\upsilon \right)}{w^{\beta }\left(\upsilon \right)}-\beta \eta \left(\upsilon \right)\frac{w^{\beta -1}\left(\upsilon \right)w^{\prime }\left(\upsilon \right)a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }}{w^{2\beta }\left(\upsilon \right)}.$$

Using (13) and (14), we obtain that

$$\begin{array}{cc}\hfill {\phi }^{\prime }\left(\upsilon \right)& \le \frac{{\eta }^{\prime }\left(\upsilon \right)}{\eta \left(\upsilon \right)}\phi \left(\upsilon \right)+\eta \left(\upsilon \right)\frac{{\left(a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }}{w^{\beta }\left(\upsilon \right)}-\beta \eta \left(\upsilon \right)\frac{\theta }{2!}{\upsilon }^2\frac{a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta +1}}{w^{\beta +1}\left(\upsilon \right)}\hfill \\ \hfill & \le \frac{{\eta }^{\prime }\left(\upsilon \right)}{\eta \left(\upsilon \right)}\phi \left(\upsilon \right)+\eta \left(\upsilon \right)\frac{{\left(a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }}{w^{\beta }\left(\upsilon \right)}-\frac{\beta \theta {\upsilon }^2}{2{\left(\eta \left(\upsilon \right)a\left(\upsilon \right)\right)}^{\frac{1}{\beta }}}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}.\hfill \end{array}$$

(15)

From (1) and (15), it follows that

$${\phi }^{\prime }\left(\upsilon \right)\le \frac{{\eta }^{\prime }\left(\upsilon \right)}{\eta \left(\upsilon \right)}\phi \left(\upsilon \right)-k\eta \left(\upsilon \right)\frac{q\left(\upsilon \right)w^{\beta }\left(\delta \left(\upsilon \right)\right)}{w^{\beta }\left(\upsilon \right)}-\frac{\beta \theta {\upsilon }^2}{2{\left(\eta \left(\upsilon \right)a\left(\upsilon \right)\right)}^{\frac{1}{\beta }}}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}.$$

Note that $w^{\prime }\left(\upsilon \right)>0$ and $\delta \left(\upsilon \right)\ge \upsilon$. Thus, we find that

$${\phi }^{\prime }\left(\upsilon \right)\le \frac{{\eta }^{\prime }\left(\upsilon \right)}{\eta \left(\upsilon \right)}\phi \left(\upsilon \right)-k\eta \left(\upsilon \right)q\left(\upsilon \right)-\frac{\beta \theta {\upsilon }^2}{2{\left(\eta \left(\upsilon \right)a\left(\upsilon \right)\right)}^{\frac{1}{\beta }}}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}.$$

(16)

Multiplying (16) by $H_1\left(\upsilon ,s\right)$ and integrating the resulting inequality from ${\upsilon }_1$ to $\upsilon$, we find that

$$\begin{array}{ccc}\hfill {\int }_{{\upsilon }_1}^{\upsilon }{H}_1\left(\upsilon ,s\right)k\eta \left(s\right)q\left(s\right)\mathrm{d}s& \le & \phi \left({\upsilon }_1\right)H_1\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }\left(\frac{\partial }{\partial s}{H}_1\left(\upsilon ,s\right)+\frac{{\eta }^{\prime }\left(s\right)}{\eta \left(s\right)}{H}_1\left(\upsilon ,s\right)\right)\phi \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{\beta \theta s^2}{2!{\left(\eta \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}}{H}_1\left(\upsilon ,s\right){\phi }^{\frac{\beta +1}{\beta }}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

From (9), we obtain

$$\begin{array}{ccc}\hfill {\int }_{{\upsilon }_1}^{\upsilon }{H}_1\left(\upsilon ,s\right)k\eta \left(s\right)q\left(s\right)\mathrm{d}s& \le & \phi \left({\upsilon }_1\right)H_1\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }{h}_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)\phi \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{\beta \theta s^2}{2{\left(\eta \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}}{H}_1\left(\upsilon ,s\right){\phi }^{\frac{\beta +1}{\beta }}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(17)

Lemma 3 with $V=\beta \theta s^2/\left(2{\left(\eta \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}\right)H_1\left(\upsilon ,s\right),U=h_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)$, and $x=\phi \left(s\right)$, tell us that

$$h_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)\phi \left(s\right)-\frac{\beta \theta s^2}{2{\left(\eta \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}}{H}_1\left(\upsilon ,s\right){\phi }^{\frac{\beta +1}{\beta }}\left(s\right)\le \frac{h_1^{\beta +1}\left(\upsilon ,s\right)H_1^{\beta }\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{2^{\beta }\eta \left(s\right)a\left(s\right)}{{\left(\theta s^2\right)}^{\beta }},$$

which, with (17), gives

$$\frac{1}{H_1\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_1\left(\upsilon ,s\right)k\eta \left(s\right)q\left(s\right)-\pi \left(s,\upsilon \right)\right)\mathrm{d}s\le \phi \left({\upsilon }_1\right),$$

contradicting (11). Now, let us assume that $\left({\mathbf{S}}_2\right)$ holds. Define

$$\psi \left(\upsilon \right):=\vartheta \left(\upsilon \right)\frac{w^{\prime }\left(\upsilon \right)}{w\left(\upsilon \right)}.$$

We see that $\psi \left(\upsilon \right)>0$ for $\upsilon \ge {\upsilon }_1,$ where $\vartheta \in C^1\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right)$. Differentiating $\psi \left(\upsilon \right)$, we find

$${\psi }^{\prime }\left(\upsilon \right)=\frac{{\vartheta }^{\prime }\left(\upsilon \right)}{\vartheta \left(\upsilon \right)}\psi \left(\upsilon \right)+\vartheta \left(\upsilon \right)\frac{w^{\prime \prime }\left(\upsilon \right)}{w\left(\upsilon \right)}-\frac{1}{\vartheta \left(\upsilon \right)}\psi {\left(\upsilon \right)}^2.$$

(18)

Integrating (1) from $\upsilon$ to m and using the fact that $w^{\prime }\left(\upsilon \right)>0$, we find that

$$a\left(m\right){\left(w^{\prime \prime \prime }\left(m\right)\right)}^{\beta }-a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }=-{\int }_{\upsilon }^{m}q\left(s\right)g\left(w\left(\delta \left(s\right)\right)\right)ds.$$

By virtue of $w^{\prime }\left(\upsilon \right)>0$ and $\delta \left(\upsilon \right)\ge \upsilon$, we obtain

$$a\left(m\right){\left(w^{\prime \prime \prime }\left(m\right)\right)}^{\beta }-a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\le -k{w}^{\beta }\left(\upsilon \right){\int }_{\upsilon }^{u}q\left(s\right)ds.$$

Letting $m\to \infty$, we see that

$$a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\ge k{w}^{\beta }\left(\upsilon \right){\int }_{\upsilon }^{\infty }q\left(s\right)\mathrm{d}s$$

and so

$$w^{\prime \prime \prime }\left(\upsilon \right)\ge w\left(\upsilon \right){\left(\frac{k}{a\left(\upsilon \right)}{\int }_{\upsilon }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }.$$

Integrating again from $\upsilon$ to ∞, we obtain

$$w^{\prime \prime }\left(\upsilon \right)+w\left(\upsilon \right){\int }_{\upsilon }^{\infty }{\left(\frac{k}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma \le 0.$$

(19)

From (18) and (19), we obtain that

$${\psi }^{\prime }\left(\upsilon \right)\le \frac{{\vartheta }^{\prime }\left(\upsilon \right)}{\vartheta \left(\upsilon \right)}\psi \left(\upsilon \right)-\vartheta \left(\upsilon \right)\varpi \left(s\right)-\frac{1}{\vartheta \left(\upsilon \right)}\psi {\left(\upsilon \right)}^2.$$

(20)

Multiplying (20) by $H_2\left(\upsilon ,s\right)$ and integrating the resulting inequality from ${\upsilon }_1$ to $\upsilon$, it follows that

$$\begin{array}{cc}\hfill {\int }_{{\upsilon }_1}^{\upsilon }& H_2\left(\upsilon ,s\right)\vartheta \left(s\right)\varpi \left(s\right)\mathrm{d}s\le \psi \left({\upsilon }_1\right)H_2\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }\left(\frac{\partial }{\partial s}{H}_2\left(\upsilon ,s\right)+\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right)\right)\psi \left(s\right)\mathrm{d}s\hfill \\ \hfill & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{1}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right){\psi }^2\left(s\right)\mathrm{d}s.\hfill \end{array}$$

Thus, from (10), we obtain

$$\begin{array}{c}{\int }_{{\upsilon }_1}^{\upsilon }{H}_2\left(\upsilon ,s\right)\vartheta \left(s\right)\varpi \left(s\right)\mathrm{d}s\le \psi \left({\upsilon }_1\right)H_2\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }{h}_2\left(\upsilon ,s\right)\sqrt{H_2\left(\upsilon ,s\right)}\psi \left(s\right)\mathrm{d}s\hfill \\ \hfill -{\int }_{{\upsilon }_1}^{\upsilon }\frac{1}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right){\psi }^2\left(s\right)\mathrm{d}s\le \psi \left({\upsilon }_1\right)H_2\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }\frac{\vartheta \left(s\right)h_2^2\left(\upsilon ,s\right)}{4}\mathrm{d}s\end{array}$$

and so

$$\frac{1}{H_2\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_2\left(\upsilon ,s\right)\vartheta \left(s\right)\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\upsilon ,s\right)}{4}\right)\mathrm{d}s\le \psi \left({\upsilon }_1\right),$$

which contradicts (12). Finally, assume that $\left({\mathbf{S}}_3\right)$ holds and $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)\ne 0$. Recalling that $a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }$ is non-increasing, we obtain that

$$a^{1/\beta }\left(s\right)w^{\prime \prime \prime }\left(s\right)\le a^{1/\beta }\left(\upsilon \right)w^{\prime \prime \prime }\left(\upsilon \right),s\ge \upsilon \ge {\upsilon }_1.$$

Dividing the latter inequality by $a^{1/\beta }\left(s\right)$ and integrating the resulting inequality from $\upsilon$ to u, we find

$$w^{\prime \prime }\left(u\right)\le w^{\prime \prime }\left(\upsilon \right)+a^{1/\beta }\left(\upsilon \right)w^{\prime \prime \prime }\left(\upsilon \right){\int }_{\upsilon }^u{a}^{-1/\beta }\left(s\right)\mathrm{ds}.$$

Letting $u\to \infty$, we obtain

$$0\le w^{\prime \prime }\left(\upsilon \right)+a^{1/\beta }\left(\upsilon \right)w^{\prime \prime \prime }\left(\upsilon \right)\zeta \left(\upsilon \right).$$

Thus,

$$\frac{-a^{1/\beta }\left(\upsilon \right)w^{\prime \prime \prime }\left(\upsilon \right)\zeta \left(\upsilon \right)}{w^{\prime \prime }\left(\upsilon \right)}\le 1.$$

(21)

Furthermore, due to (21), we obtain that

$${\left(\frac{w^{\prime \prime }\left(\upsilon \right)}{\zeta \left(\upsilon \right)}\right)}^{\prime }\ge 0.$$

(22)

Now define

$$\varphi \left(\upsilon \right):=\frac{a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }}{{\left(w^{\prime \prime }\left(\upsilon \right)\right)}^{\beta }}.$$

(23)

We see that $\varphi \left(\upsilon \right)<0$ for $\upsilon \ge {\upsilon }_1$ and

$${\varphi }^{\prime }\left(\upsilon \right)=\frac{{\left(a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }}{{\left(w^{\prime \prime }\left(\upsilon \right)\right)}^{\beta }}-\frac{\beta a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta +1}}{{\left(w^{\prime \prime }\left(\upsilon \right)\right)}^{\beta +1}}.$$

It follows from (1) and (21) that

$${\varphi }^{\prime }\left(\upsilon \right)=\frac{-kq\left(\upsilon \right)w^{\beta }\left(\delta \left(\upsilon \right)\right)}{{\left(w^{\prime \prime }\left(\upsilon \right)\right)}^{\beta }}-\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}.$$

From Lemma 1, we find

$$w\left(\upsilon \right)\ge \frac{{\theta }_1}{2}{\upsilon }^2{w}^{\prime \prime }\left(\upsilon \right).$$

(24)

Thus, we have

$${\varphi }^{\prime }\left(\upsilon \right)=\frac{-kq\left(\upsilon \right)w^{\beta }\left(\delta \left(\upsilon \right)\right)}{{\left(w^{\prime \prime }\left(\delta \left(\upsilon \right)\right)\right)}^{\beta }}\frac{{\left(w^{\prime \prime }\left(\delta \left(\upsilon \right)\right)\right)}^{\beta }}{{\left(w^{\prime \prime }\left(\upsilon \right)\right)}^{\beta }}-\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}.$$

From (24), we obtain

$${\varphi }^{\prime }\left(\upsilon \right)\le -kq\left(\upsilon \right){\left(\frac{{\theta }_1{\delta }^2\left(\upsilon \right)}{2}\right)}^{\beta }{\left(\frac{\zeta \left(\delta \left(\upsilon \right)\right)}{\zeta \left(\upsilon \right)}\right)}^{\beta }-\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}.$$

(25)

Using (21) and (23), we see, due to (26), that

$$-\varphi \left(\upsilon \right){\zeta }^{\beta }\left(\upsilon \right)\le 1.$$

(26)

Multiplying (26) by ${\zeta }^{\beta }\left(\upsilon \right)$ and integrating the resulting inequality from ${\upsilon }_1$ to $\upsilon$, we obtain

$$\begin{array}{c}{\zeta }^{\beta }\left(\upsilon \right)\varphi \left(\upsilon \right)-{\zeta }^{\beta }\left({\upsilon }_1\right)\varphi \left({\upsilon }_1\right)+\beta {\int }_{{\upsilon }_1}^{\upsilon }{a}^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)\varphi \left(s\right)ds\hfill \\ \hfill \le -{\int }_{{\upsilon }_1}^{\upsilon }kq\left(s\right){\left(\frac{{\theta }_1{\delta }^2\left(s\right)}{2}\right)}^{\beta }{\zeta }^{\beta }\left(\delta \left(s\right)\right)ds-\beta {\int }_{{\upsilon }_1}^{\upsilon }\frac{{\varphi }^{\beta /\beta +1}\left(s\right)}{a^{1/\beta }\left(s\right)}{\zeta }^{\beta }\left(s\right)ds.\end{array}$$

(27)

Multiplying (27) by $H_3\left(\upsilon ,s\right)$, we find that

$$\begin{array}{ccc}\hfill {\int }_{{\upsilon }_1}^{\upsilon }{H}_3\left(\upsilon ,s\right)kq\left(s\right){\left(\frac{{\theta }_1{\delta }^2\left(s\right)}{2}\right)}^{\beta }{\zeta }^{\beta }\left(\delta \left(s\right)\right)\mathrm{d}s& \le & {\zeta }^{\beta }\left({\upsilon }_1\right)\varphi \left({\upsilon }_1\right)H_3\left(\upsilon ,{\upsilon }_1\right)-{\zeta }^{\beta }\left(\upsilon \right)\varphi \left(\upsilon \right)H_3\left(\upsilon ,{\upsilon }_1\right)\hfill \\ & & +{\int }_{{\upsilon }_1}^{\upsilon }\beta a^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)\varphi \left(s\right)H_3\left(\upsilon ,s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{\beta {\varphi }^{\beta /\beta +1}\left(s\right)}{a^{1/\beta }\left(s\right)}{\zeta }^{\beta }\left(s\right)H_3\left(\upsilon ,s\right)\mathrm{d}s.\hfill \end{array}$$

Using Lemma 3 with $V={\zeta }^{\beta }\left(s\right)H_3\left(\upsilon ,s\right)/a^{1/\beta }\left(s\right)$, $U=a^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)H_3\left(\upsilon ,s\right)$, and $x=\varphi \left(s\right)$, we obtain

$$\beta a^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)\varphi \left(s\right)H_3\left(\upsilon ,s\right)-\frac{\beta {\varphi }^{\beta /\beta +1}\left(s\right)}{a^{1/\beta }\left(s\right)}{\zeta }^{\beta }\left(s\right)H_3\left(\upsilon ,s\right)\le \frac{{\beta }^{\beta +1}{H}_3\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{1}{a^{1/\beta }\left(s\right)\zeta \left(s\right)}$$

and easily find, due to (26), that

$$\frac{1}{H_3\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_3\left(\upsilon ,s\right)kq\left(s\right){\left(\frac{{\theta }_1{\delta }^2\left(s\right)}{2}\right)}^{\beta }{\zeta }^{\beta }\left(\delta \left(s\right)\right)-\tilde{\pi }\left(s,\upsilon \right)\right)\mathrm{d}s\le {\zeta }^{\beta }\left({\upsilon }_1\right)\varphi \left({\upsilon }_1\right)+1,$$

which contradicts (11). This completes the proof. □

With the help of our Theorem 1, we now prove a generalized Hille–Nehari-type oscillation criterion (cf. Corollary 1).

Theorem 2

(Generalized Hille–Nehari-type oscillation criterion for (1)). Under assumptions $A_1$–$A_5$, if the differential equations

$${\left(\frac{2a^{\frac{1}{\beta }}\left(\upsilon \right)}{{\left(\theta {\upsilon }^2\right)}^{\beta }}{\left(w^{\prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+kq\left(\upsilon \right)w^{\beta }\left(\upsilon \right)=0,$$

(28)

for every $\theta \in (0,1)$,

$$w^{\prime \prime }\left(\upsilon \right)+w\left(\upsilon \right){\int }_{\upsilon }^{\infty }{\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma =0$$

(29)

and

$${\left(a\left(\upsilon \right){\left(w^{\prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+w^{\beta }\left(\upsilon \right)kq\left(\upsilon \right){\left(\frac{\zeta \left(\delta \left(\upsilon \right)\right)}{\zeta \left(\upsilon \right)}\right)}^{\beta }{\left(\frac{{\theta }_1}{2}{\delta }^2\left(\upsilon \right)\right)}^{\beta }=0,$$

(30)

for every ${\theta }_1\in (0,1)$, are oscillatory, then every solution of (1) is either oscillatory or satisfies $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$.

Proof. 

Assume the contrary, that w is a positive solution of (1). Then, we can suppose that $w\left(\upsilon \right)$ and $w\left(\delta \left(\upsilon \right)\right)$ are positive for all $\upsilon \ge {\upsilon }_1$ sufficiently large. From Lemma 4, we have three possible cases $\left({\mathbf{S}}_1\right)$, $\left({\mathbf{S}}_2\right)$, or $\left({\mathbf{S}}_3\right)$. Let situation $\left({\mathbf{S}}_1\right)$ hold. From Theorem 1, we obtain that (16) holds. If we set $\eta \left(\upsilon \right)=k=1$ in (16), then we find

$${\phi }^{\prime }\left(\upsilon \right)+\frac{\beta \theta {\upsilon }^2}{2a^{\frac{1}{\beta }}\left(\upsilon \right)}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}+q\left(\upsilon \right)\le 0.$$

Thus, we can see that Equation (28) is nonoscillatory, which is a contradiction. Let case $\left({\mathbf{S}}_2\right)$ holds. From Theorem 1, we obtain that (20) holds. If we now set $\vartheta \left(\upsilon \right)=k=1$ in (20), then we obtain

$${\psi }^{\prime }\left(\upsilon \right)+{\psi }^2\left(\upsilon \right)+\varpi \left(s\right)\varsigma \le 0.$$

Hence, we see that Equation (29) is nonoscillatory, which is a contradiction. Let case $\left({\mathbf{S}}_3\right)$ hold and $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)\ne 0$. From Theorem 1, we obtain that (25) holds. Thus, we see that

$${\varphi }^{\prime }\left(\upsilon \right)+\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}+kq\left(\upsilon \right){\left(\frac{{\theta }_1{\delta }^2\left(\upsilon \right)}{2}\right)}^{\beta }{\left(\frac{\zeta \left(\delta \left(\upsilon \right)\right)}{\zeta \left(\upsilon \right)}\right)}^{\beta }\le 0$$

and it follows that Equation (30) is nonoscillatory, which is a contradiction. Theorem 2 is proved. □

Let us now restrict ourselves to the case when $\beta =1$. Note that, if

$${\int }_{{\upsilon }_0}^{\infty }\frac{1}{a\left(\upsilon \right)}\mathrm{d}\upsilon <\infty$$

and

$$\underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^{2}q\left(s\right)\mathrm{d}s>\frac{1}{4},$$

then Equation (8) with $\beta =1$ is oscillatory [14]. For $\beta =1$ our Theorem 2 gives a Hille–Nehari-type oscillation criterion.

Corollary 1

(Hille–Nehari-type oscillation criterion). Let $\beta =1$. Under assumptions $A_1$–$A_5$ with $k=1$, if

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

and

$$\underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^2}{2a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^2}{2a\left(s\right)}\mathrm{d}s\right)}^{2}q\left(s\right)\mathrm{d}s>\frac{1}{4}$$

(31)

for every constant $\theta \in (0,1)$,

$$\underset{\upsilon \to \infty }{liminf}\upsilon {\int }_{{\upsilon }_0}^{\upsilon }{\int }_v^{\upsilon }\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\upsilon }q\left(s\right)\mathrm{d}s\right)\mathrm{d}\varsigma \mathrm{d}v>\frac{1}{4},$$

(32)

and

$$\underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^2\frac{{\theta }_1\zeta \left(\delta \left(s\right)\right){\delta }^2\left(s\right)q\left(s\right)}{2\zeta \left(s\right)}\mathrm{d}s>\frac{1}{4}$$

(33)

for every constant ${\theta }_1\in (0,1)$, then every solution of (1) is either oscillatory or satisfies $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$.

The next section shows that our results are new even in the very special situation covered by Corollary 1.

4. Illustrative Examples

We give two examples for which all results of [17,18] cannot be applied, since $\delta \left(\upsilon \right)=\upsilon +1>\upsilon$, while our Hille–Nehari-type oscillation criterion holds.

Example 1.

Let us consider the following equation:

$${\left({\mathrm{e}}^{\upsilon }{w}^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\prime }+\frac{1}{16}{\mathrm{e}}^{\upsilon +\frac{1}{2}}w\left(\upsilon +1\right)=0,\upsilon \ge 1,$$

(34)

where $\alpha >0$ is a constant. Note that $\beta =1,a\left(\upsilon \right)={\mathrm{e}}^{\upsilon },q\left(\upsilon \right)={\mathrm{e}}^{\upsilon +\frac{1}{2}}/16$ and $\delta \left(\upsilon \right)=\upsilon +1$. It is easy to see that all conditions of our Corollary 1 are satisfied. Hence, all solutions of (34) are either oscillatory or satisfy $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$.

Following is a second example, where previous results in the literature cannot be applied, while our Corollary 1 is conclusive.

Example 2.

Consider the following equation:

$${\left({\upsilon }^4{w}^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\prime }+\alpha w\left(\upsilon +1\right)=0,\upsilon \ge 1,$$

(35)

where $\alpha >0$ is a constant. Note that $\beta =1,a\left(\upsilon \right)={\upsilon }^4,q\left(\upsilon \right)=\alpha$, and $\delta \left(\upsilon \right)=\upsilon +1$. If we set $k=1$, then condition (31) becomes

$$\begin{array}{cc}\hfill \underset{\upsilon \to \infty }{liminf}& {\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^2}{2a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^2}{2a\left(s\right)}\mathrm{d}s\right)}^{2}q\left(s\right)\mathrm{d}s\hfill \\ \hfill & =\underset{\upsilon \to \infty }{liminf}\left(2\upsilon \right){\int }_{\upsilon }^{\infty }\frac{\alpha }{4s^2}ds=\underset{\upsilon \to \infty }{liminf}\left(2\upsilon \right)\left(\frac{\alpha }{4\upsilon }\right)\hfill \\ \hfill & =\frac{\alpha }{2}>\frac{1}{4},\hfill \end{array}$$

condition (32) becomes

$$\begin{array}{ccc}\hfill \underset{\upsilon \to \infty }{liminf}\upsilon {\int }_{{\upsilon }_0}^{\upsilon }{\int }_v^{\upsilon }\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\upsilon }q\left(s\right)\mathrm{d}s\right)\mathrm{d}\varsigma \mathrm{d}v& =& \underset{\upsilon \to \infty }{liminf}\upsilon \left(\frac{\alpha }{2\upsilon }\right)\hfill \\ & =& \frac{\alpha }{2}>\frac{1}{4},\hfill \end{array}$$

and (33) is satisfied. Therefore, from Corollary 1, any solution of Equation (35) is either oscillatory, if $\alpha >0.5$, or satisfies $\underset{\upsilon \to \infty }{lim}w\left(\upsilon \right)=0$.

5. Conclusions and Future Work

In this work, we obtained new Philos-type and Hille–Nehari-type oscillation criteria for equations of form (1). Our results are easy to generalize for the following equations:

$${\left(a\left(\upsilon \right){\left(w^{\prime \prime \prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+\sum _{i=1}^j{q}_i\left(\upsilon \right)w^{\alpha }\left({\delta }_i\left(\upsilon \right)\right)=0,j\ge 1,$$

(36)

where $\upsilon \ge {\upsilon }_0$, ${\delta }_i\left(\upsilon \right)\le \upsilon$, $\alpha \le \beta$, and $\alpha$ and $\beta$ are ratios of odd positive integers. However, it is not easy to find analogous results for Equation (36) in the case $\alpha >\beta$. We leave this as an interesting open question.