On the Oscillatory Behavior of Solutions of Second-Order Damped Differential Equations with Several Sub-Linear Neutral Terms

Abstract

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results.

1. Introduction

The aim of this paper is to establish new oscillation criteria for the second-order damped neutral differential equation of the type

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)+f\left(l,y\left(\xi \left(l\right)\right)\right)=0,l\ge l_0>0,$$

(1)

where $V\left(l\right)=y\left(l\right)+{\sum }_{i=1}^m{c}_i\left(l\right)y^{{\beta }_i}\left({\nu }_i\left(l\right)\right),m>0$ is an integer. We assume that:

• (A₁) $0<{\beta }_i\le 1$ for $i=1,2,....,m$ are the ratios of odd positive integers;
• (A₂) $a,p,c_i:\left[l_0,\infty \right)\to {\mathbb{R}}^{+}$ are continuous functions for $i=1,2,.....,m$;
• (A₃) ${\nu }_i\in C\left(\left[l_0,\infty \right),\mathbb{R}\right),\xi \in C^1\left(\left[l_0,\infty \right),\mathbb{R}\right)$ with ${\nu }_i\left(l\right)<l,\xi \left(l\right)\le l,{\xi }^{\prime }\left(l\right)>0$ and ${\nu }_i\left(l\right),\xi \left(l\right)\to \infty$ as $l\to \infty$ for $i=1,2,....,m$;
• (A₄) $f\left(l,y\right)\in C\left(\left[l_0,\infty \right)\times \mathbb{R},\mathbb{R}\right)$, and there exists $\gamma$ is a ratio of odd positive integers and a function $q\left(l\right)\in C\left(\left[l_0,\infty \right),\left(0,\infty \right)\right)$ such that $f\left(l,y\right)/y^{\gamma }\ge q\left(l\right)$, for all $y\ne 0$.

A solution of Equation (1) is defined by a nontrivial real-valued function y, which satisfies

$$y\left(l\right)+\sum _{i=1}^m{c}_i\left(l\right)y^{{\beta }_i}\left({\nu }_i\left(l\right)\right)\in C^1\left[l_y,\infty \right),a\left(l\right)V^{\prime }\left(l\right)\in C^1\left(\left[l_y,\infty \right),\mathbb{R}\right),$$

and satisfies Equation (1) on $\left[l_y,\infty \right),l_y\ge l_0$. In this article, we will be only concerned with solutions which satisfy

$$sup\left\{y\left(l\right):l\ge T\ge l_y\right\}>0.$$

A solution $y\left(l\right)$ of Equation (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called non-oscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

In dynamic models, delay and oscillation effects are often formulated by means of external sources and/or nonlinear diffusion, perturbing neutral evolution of related systems (see, e.g., [1,2,3]). During the past few decades, there has been increasing interests in discussing the oscillation property of solutions of several classes of damped second order differential equations, due to their numerous applications in physics, engineering, biology, economy, etc. (see [4,5,6,7,8,9,10,11]).

Recently, there has been considerable interest in studying the qualitative properties of solutions of neutral differential equations because they arise in several problems in electric networks containing lossless transmission lines in the study of vibrating masses attached to elastic bars or in variational problems with time delays. Moreover, second order damped differential equations are used in the study of NVH of vehicles. For further applications and technology, see [12].

It is well known that Equation (1) can be reduced via the Sturm–Liouville substitution

$$y\left(l\right)=V\left(l\right)exp\left(-{\int }_{l_0}^l{\displaystyle \frac{p\left(s\right)}{a\left(s\right)}}ds\right)$$

to the undamped form

$${\left(\pi \left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+\Omega \left(l,y\left(\xi \left(l\right)\right)\right)=0,$$

with $\pi \left(l\right)=a\left(l\right)exp\left({\int }_{l_0}^l{\displaystyle \frac{p\left(s\right)}{a\left(s\right)}}ds\right)$, and $\Omega \left(l,y\right)=exp\left({\int }_{l_0}^l{\displaystyle \frac{p\left(s\right)}{a\left(s\right)}}ds\right)f\left(l,y\right)$. However, an additional condition is needed for $p\left(l\right)$, such as $p\left(l\right)$, which must be differentiable, or at least $p\left(l\right)$, which is absolutely continuous, to guarantee that $p^{\prime }\left(l\right)$ is defined. Moreover, the damping term is crucial, since there are advantages in obtaining direct oscillation theorems for (1) besides the obvious practical advantage of eliminating the need for the integrating factor, and there is an incentive for developing methods which will lead to more general equations.

To the best of our knowledge, we note that many researchers were concerned with the special undamped form of Equation (1) (see [13,14,15,16,17,18,19,20,21]). Moreover, there are few authors who are dealing with differential equations with a sub-linear neutral term, (see [14,15,16,22,23]). So, this article is a contribution to fulfill a part of this gap in the literature. In particular, we mention here for instance some works which have been devoted to some special cases of (1), and motivated this work.

In [19], Grammatikopoulos et al. discussed the equation

$${\left(y\left(l\right)+u\left(l\right)y\left(l-\nu \right)\right)}^{″}+h\left(l\right)y\left(l-\nu \right)=0.$$

(2)

They deduced that if

$${\int }_{l_0}^{\infty }h\left(s\right)\left(1-u\left(s-\nu \right)\right)ds=\infty ,$$

then Equation (2) is oscillatory. Meanwhile, in [24], Grace et al. generalized the results of [19] to the equation

$${\left(b\left(l\right){\left(y\left(l\right)+u\left(l\right)y\left(l-\nu \right)\right)}^{\prime }\right)}^{\prime }+h\left(l\right)f\left(y\left(l-\nu \right)\right)=0,$$

(3)

with

$${\displaystyle \frac{f\left(y\right)}{y}}\ge L,L>0\mathrm{and}{\int }_{l_0}^{\infty }{\displaystyle \frac{ds}{b\left(s\right)}}=\infty .$$

They deduced that Equation (3) is oscillatory, provided that

$${\int }_{l_0}^{\infty }\left(\alpha \left(s\right)h\left(s\right)\left(1-u\left(s-\nu \right)\right)-{\displaystyle \frac{{\left({\alpha }^{\prime }\left(s\right)\right)}^{2}b\left(s-\nu \right)}{4L\alpha \left(s\right)}}\right)ds=\infty ,$$

for some continuously differentiable function $\alpha \left(l\right)$. Recently, in 2019, Baculikova [14] and Džurina et al. [16] discussed the D.E.,

$${\left(b\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+h\left(l\right)y^{\gamma }\left(\xi \left(l\right)\right)=0,l\ge l_0>0,$$

(4)

where $V\left(l\right)=y\left(l\right)+{\sum }_{i=1}^m{c}_i\left(l\right)y^{{\beta }_i}\left({\nu }_i\left(l\right)\right),m>0$ is an integer, $0<{\beta }_i\le 1$, for $i=1,2,...,m$, and $\gamma$ are the ratios of odd positive integers, provided that (A₂) and (A₃) hold. On the other hand, Equation (1) is more general than the D.E.,

$${\left(b\left(l\right)y^{\prime }\left(l\right)\right)}^{\prime }+g\left(l\right)y^{\prime }\left(l\right)+h\left(l\right)f\left(y\left(l\right)\right)=0,$$

(5)

which was studied by Çakmak [25] and Rogovchenko et al. [9,10,11], under the conditions $b\in C^1\left(\left[l_0,\infty \right),\mathbb{R}\right),g,h\in C\left(\mathbb{R},\mathbb{R}\right),yf\left(y\right)>0$, and $f^{\prime }\left(y\right)\ge L>0$. Equation (1) also generalizes the D.E.,

$${\left(b\left(l\right)y^{\prime }\left(l\right)\right)}^{\prime }+g\left(l\right)y^{\prime }\left(l\right)+h\left(l\right)f\left(y\left(\tau \left(l\right)\right)\right)=0,$$

which was studied by Fu et al. [7]. In this paper, we aim to improve and extend some of the results given in [14,16], by using some elementary inequalities and Riccati substitution. We claim that the results in this paper are of high generality, since it covers all possible cases of sub-linear, super-linear, and linear equations.

2. Preliminaries

We will need the following notation and lemmas:

$$E\left(l\right)=exp\left(-{\int }_{l_0}^l{\displaystyle \frac{p\left(s\right)}{a\left(s\right)}}ds\right),\Phi \left(l\right)={\int }_{l_1}^l{\displaystyle \frac{E\left(s\right)}{a\left(s\right)}}ds,l_1\ge l_0\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{large}\mathrm{enough},$$

(6)

and for any function $\rho :\left[l_0,\infty \right)\to {\mathbb{R}}^{+}$ which is positive, continuous, and decreasing to zero, we set

$$\Psi \left(l\right)=1-\sum _{i=1}^m{\beta }_i{c}_i\left(l\right)-{\displaystyle \frac{1}{\rho \left(l\right)}}\sum _{i=1}^m\left(1-{\beta }_i\right)c_i\left(l\right),$$

(7)

with $\Psi \left(l\right)>0$.

Lemma 1

([26]). If r is non-negative, then

$$r^{\beta }\le \beta r+\left(1-\beta \right)\mathrm{for}0<\beta \le 1.$$

(8)

Lemma 2.

Assume that

$${\int }_{l_0}^{\infty }{\displaystyle \frac{E\left(s\right)}{a\left(s\right)}}ds=\infty ,$$

(9)

and

$${\int }_{l_0}^{\infty }\left({\displaystyle \frac{E\left(u\right)}{a\left(u\right)}}{\int }_u^{\infty }{\displaystyle \frac{q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}}ds\right)du=\infty ,$$

(10)

hold where $E\left(l\right)$ is defined by (6), and $\Psi \left(l\right)$ is defined by (7). Suppose that Equation (1) has a positive solution $y\left(l\right)$ on $\left[l_0,\infty \right)$. Then, there may exist $T\in \left[l_0,\infty \right)$, large enough such that, for $l\ge T$, we have

(i) 

$V\left(l\right)>0,V^{\prime }\left(l\right)>0$, and ${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }<0$;

(ii) 

${\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}$ is decreasing;

(iii) 

$V\left(l\right)\to \infty$ for $l\to \infty$.

Proof. 

Since, $y\left(l\right)$ is a positive solution of Equation (1) on $\left[l_0,\infty \right)$, then by the assumption (A₃), there exists a $l_1\ge l_0$, such that $y\left({\nu }_i\left(l\right)\right)>0$ and $y\left(\xi \left(l\right)\right)>0$ on $\left[l_1,\infty \right)$. Now, since $V\left(l\right)=y\left(l\right)+{\sum }_{i=1}^m{c}_i\left(l\right)y^{{\beta }_i}\left({\nu }_i\left(l\right)\right)$, then $V\left(l\right)\ge y\left(l\right)>0$, for $l\ge l_1$, and by (1), we have

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)=-f\left(l,y\left(\xi \left(l\right)\right)\right)\le -q\left(l\right)y^{\gamma }\left(\xi \left(l\right)\right)<0.$$

(11)

Since, from (11), we have

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+{\displaystyle \frac{p\left(l\right)}{a\left(l\right)}}a\left(l\right)V^{\prime }\left(l\right)<0,$$

(12)

then

$${\left({\displaystyle \frac{a\left(l\right)}{E\left(l\right)}}{V}^{\prime }\left(l\right)\right)}^{\prime }=-{\displaystyle \frac{f\left(l,y\left(\xi \left(l\right)\right)\right)}{E\left(l\right)}}<0.$$

(13)

Thus, ${\displaystyle \frac{a\left(l\right)}{E\left(l\right)}}{V}^{\prime }\left(l\right)$ is decreasing.

Now, to prove that $V^{\prime }\left(l\right)>0$ on $\left[l_1,\infty \right)$, suppose that this is not true. Then, we may find a point $l_2>l_1$, such that $V^{\prime }\left(l_2\right)$ is negative. But since ${\displaystyle \frac{a\left(l\right)}{E\left(l\right)}}{V}^{\prime }\left(l\right)$ is decreasing, then

$${\displaystyle \frac{a\left(l\right)}{E\left(l\right)}}{V}^{\prime }\left(l\right)<{\displaystyle \frac{a\left(l_2\right)}{E\left(l_2\right)}}{V}^{\prime }\left(l_2\right)=c<0,\mathrm{for}l\ge l_2.$$

(14)

By integrating (14) from $l_2$ to l, we obtain

$$V\left(l\right)<V\left(l_2\right)+c{\int }_{l_2}^l{\displaystyle \frac{E\left(s\right)}{a\left(s\right)}}ds.$$

(15)

This, with (9), leads to $V\left(l\right)\to -\infty$ as $l\to \infty$, which is a contradiction with the fact that $V\left(t\right)>0$ eventually. This completes the proof of $V^{\prime }\left(l\right)>0$, and so, by (1), we have ${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }<0$. But since ${\displaystyle \frac{a\left(l\right)}{E\left(l\right)}}{V}^{\prime }\left(l\right)$ is decreasing, we have

$$V\left(l\right)>{\int }_{l_1}^l{\displaystyle \frac{a\left(s\right)}{E\left(s\right)}}{V}^{\prime }\left(s\right){\displaystyle \frac{E\left(s\right)}{a\left(s\right)}}ds>{\displaystyle \frac{a\left(l\right)}{E\left(l\right)}}{V}^{\prime }\left(l\right)\Phi \left(l\right).$$

(16)

Consequently, we obtain

$${\left({\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}\right)}^{\prime }<0,$$

i.e., ${\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}$ is decreasing for $l\ge l_1$.

Now, since $V\left(l\right)$ is increasing, then by applying the inequality (8) of Lemma 1, in view, of the definition of $V\left(l\right)$, we obtain

$$\begin{array}{ccc}y\left(l\right)\hfill & =\hfill & V\left(l\right)-{\displaystyle \sum _{i=1}^m}{c}_i\left(l\right)y^{{\beta }_i}\left({\nu }_i\left(l\right)\right)\ge V\left(l\right)-{\displaystyle \sum _{i=1}^m}{c}_i\left(l\right)V^{{\beta }_i}\left({\nu }_i\left(l\right)\right)\hfill \\ \hfill & \ge \hfill & V\left(l\right)-{\displaystyle \sum _{i=1}^m}{c}_i\left(l\right)\left({\beta }_{i}V\left({\nu }_i\left(l\right)\right)+\left(1-{\beta }_i\right)\right)\hfill \\ \hfill & \ge \hfill & \left(1-{\displaystyle \sum _{i=1}^m}{\beta }_i{c}_i\left(l\right)\right)V\left(l\right)-{\displaystyle \sum _{i=1}^m}\left(1-{\beta }_i\right)c_i\left(l\right)\hfill \\ \hfill & =\hfill & V\left(l\right)\left(1-{\displaystyle \sum _{i=1}^m}{\beta }_i{c}_i\left(l\right)-{\displaystyle \frac{1}{V\left(l\right)}}{\displaystyle \sum _{i=1}^m}\left(1-{\beta }_i\right)c_i\left(l\right)\right).\hfill \end{array}$$

(17)

But since $V\left(l\right)$, and $V^{\prime }\left(l\right)$ are both positive, and $\rho \left(l\right)$ is positive and decreasing to zero, then there may exist $l_2\ge l_1$, such that

$$V\left(l\right)\ge \rho \left(l\right)\mathrm{for}l\ge l_2.$$

(18)

This, with (17), leads to

$$y\left(l\right)\ge V\left(l\right)\left(1-\sum _{i=1}^m{\beta }_i{c}_i\left(l\right)-{\displaystyle \frac{1}{\rho \left(l\right)}}\sum _{i=1}^m\left(1-{\beta }_i\right)c_i\left(l\right)\right)=\Psi \left(l\right)V\left(l\right).$$

(19)

Moreover, substituting from (19) into (11), it follows that

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)+q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)V^{\gamma }\left(\xi \left(l\right)\right)\le 0,l\ge l_2,$$

(20)

or

$${\left({\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{E\left(l\right)}}\right)}^{\prime }+{\displaystyle \frac{q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)V^{\gamma }\left(\xi \left(l\right)\right)}{E\left(l\right)}}\le 0.$$

(21)

By integrating (21) from l to ∞, we obtain

$${\displaystyle \frac{a\left(\infty \right)V^{\prime }\left(\infty \right)}{E\left(\infty \right)}}-{\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{E\left(l\right)}}\le -{\int }_l^{\infty }{\displaystyle \frac{q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)V^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}}ds,$$

then

$${\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{E\left(l\right)}}\ge {\int }_l^{\infty }{\displaystyle \frac{q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)V^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}}ds,$$

and since $\xi$ and V are increasing, then we have

$$V^{\prime }\left(l\right)\ge V^{\gamma }\left(\xi \left(l\right)\right){\displaystyle \frac{E\left(l\right)}{a\left(l\right)}}{\int }_l^{\infty }{\displaystyle \frac{q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}}ds.$$

By integrating again from $l_1$ to l, we have

$$V\left(l\right)\ge V\left(l_1\right)+{\int }_{l_1}^l{V}^{\gamma }\left(\xi \left(u\right)\right){\displaystyle \frac{E\left(u\right)}{a\left(u\right)}}{\int }_u^{\infty }{\displaystyle \frac{q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}}dsdu.$$

This, with (10), leads to ${\displaystyle \underset{l\to \infty }{lim}}V\left(l\right)=\infty$, and completes the proof. □

Remark 1.

In the special case $p\left(l\right)=0$, our condition (10) of Lemma 2 would take the form

$${\int }_{l_0}^{\infty }\left({\displaystyle \frac{1}{a\left(u\right)}}{\int }_u^{\infty }q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)ds\right)du=\infty ,$$

which shows that our method and result are more general than the criterion (2.2) of Lemma 2.2 in [14].

3. Main Results

We first consider the case when (1) is super-linear.

Theorem 1.

Let (A₁)–(A₄) be satisfied. Assume that the assumptions of Lemma 2 are satisfied with $\gamma >1$. Suppose further that

$$\underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-p\left(s\right)\mu \left(s\right)\right)}^2}{4\mu \left(s\right){\xi }^{\prime }\left(s\right)a^2\left(s\right)}}\right]ds=\infty ,$$

(22)

for some positive function $\mu \left(l\right)\in C^1\left(\left[l_0,\infty \right),\mathbb{R}\right)$, and $\Psi \left(l\right)$ which defined by (7). Then, Equation (1) is oscillatory.

Proof. 

Suppose the contrary, that $y\left(l\right)>0$. Thus, there exists $l\ge l_1$, such that $y\left({\nu }_i\left(l\right)\right)>0,y\left(\xi \left(l\right)\right)>0$ for $i=1,2,...,m$. It is easy to see that $V\left(l\right)>0$ for $l\ge l_1$, which means that Lemma 2 (i) holds. Now, from the proof of Lemma 2, we obtain (20), from which we obtain, for $l\ge l_2\ge l_1$,

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)\le -q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)V^{\gamma }\left(\xi \left(l\right)\right).$$

(23)

Thus, by (iii) of Lemma 2, for $\gamma >1$, the inequality

$$V^{\gamma }\left(\xi \left(l\right)\right)>V\left(\xi \left(l\right)\right)$$

is satisfied. This, with (23), leads to

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)\le -q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)V\left(\xi \left(l\right)\right),l\ge l_2.$$

(24)

Define

$$w\left(l\right)=\mu \left(l\right){\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{V\left(\xi \left(l\right)\right)}},l\ge l_2.$$

(25)

Thus, clearly, $w\left(l\right)>0$ for $l\ge l_2$, and

$$w^{\prime }\left(l\right)={\mu }^{\prime }\left(l\right){\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{V\left(\xi \left(l\right)\right)}}+\mu \left(l\right){\displaystyle \frac{{\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }}{V\left(\xi \left(l\right)\right)}}-{\displaystyle \frac{\mu \left(l\right)a\left(l\right)V^{\prime }\left(l\right)}{V^2\left(\xi \left(l\right)\right)}}{V}^{\prime }\left(\xi \left(l\right)\right){\xi }^{\prime }\left(l\right).$$

(26)

But since $a\left(l\right)V^{\prime }\left(l\right)$ is positive and decreasing, we obtain

$$a\left(l\right)V^{\prime }\left(l\right)\le a\left(\xi \left(l\right)\right)V^{\prime }\left(\xi \left(l\right)\right).$$

(27)

Now, in view of (24) and (27), Equation (26) leads to

$$\begin{array}{ccc}\hfill w^{\prime }\left(l\right)& \le & {\mu }^{\prime }\left(l\right){\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{V\left(\xi \left(l\right)\right)}}-\mu \left(l\right){\displaystyle \frac{p\left(l\right)V^{\prime }\left(l\right)}{V\left(\xi \left(l\right)\right)}}\hfill \\ & & -\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)-w\left(l\right){\displaystyle \frac{V^{\prime }\left(\xi \left(l\right)\right)}{V\left(\xi \left(l\right)\right)}}{\xi }^{\prime }\left(l\right)\hfill \\ & \le & {\displaystyle \frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}}w\left(l\right)-{\displaystyle \frac{p\left(l\right)}{a\left(l\right)}}w\left(l\right)-\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)\hfill \\ & & -w\left(l\right){\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{a\left(\xi \left(l\right)\right)V\left(\xi \left(l\right)\right)}}{\xi }^{\prime }\left(l\right)\hfill \\ & \le & {\displaystyle \frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}}w\left(l\right)-{\displaystyle \frac{p\left(l\right)}{a\left(l\right)}}w\left(l\right)-{\displaystyle \frac{{\xi }^{\prime }\left(l\right)}{a\left(\xi \left(l\right)\right)\mu \left(l\right)}}{w}^2\left(l\right)-\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right).\hfill \end{array}$$

By completing the squares, we obtain

$$\begin{array}{ccc}\hfill w^{\prime }\left(l\right)& \le & -\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)+{\displaystyle \frac{a\left(\xi \left(l\right)\right){\left({\mu }^{\prime }\left(l\right)a\left(l\right)-p\left(l\right)\mu \left(l\right)\right)}^2}{4\mu \left(l\right){\xi }^{\prime }\left(l\right)a^2\left(l\right)}}\hfill \\ & & -{\left(\sqrt{{\displaystyle \frac{{\xi }^{\prime }\left(l\right)}{a\left(\xi \left(l\right)\right)\mu \left(l\right)}}}w\left(l\right)-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a\left(\xi \left(l\right)\right)\mu \left(l\right)}{{\xi }^{\prime }\left(l\right)}}}{\displaystyle \frac{\left({\mu }^{\prime }\left(l\right)a\left(l\right)-p\left(l\right)\mu \left(l\right)\right)}{\mu \left(l\right)a\left(l\right)}}\right)}^2\hfill \\ & \le & -\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)+{\displaystyle \frac{a\left(\xi \left(l\right)\right){\left({\mu }^{\prime }\left(l\right)a\left(l\right)-p\left(l\right)\mu \left(l\right)\right)}^2}{4\mu \left(l\right){\xi }^{\prime }\left(l\right)a^2\left(l\right)}}.\hfill \end{array}$$

By integrating from $l_2$ to l, and taking $limsup$ as $l\to \infty$, we obtain

$$\underset{l\to \infty }{limsup}{\int }_{l_2}^l\left[\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right)-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-p\left(s\right)\mu \left(s\right)\right)}^2}{4\mu \left(s\right){\xi }^{\prime }\left(s\right)a^2\left(s\right)}}\right]ds\le w\left(l_2\right).$$

Thus, we arrive at a contradiction with (22), and so the proof is completed. □

Using a generalized Riccati transformation and Kamenev techniques, we give the following new result.

Theorem 2.

Assume that (A₁)–(A₄) hold. Suppose further that the assumptions of Lemma 2 are satisfied with $\gamma >1$. If there exist $\mu \left(l\right)\in C^1\left(\left[l_0,\infty \right),\left(0,\infty \right)\right),b\left(l\right)\in C^1\left(\left[l_0,\infty \right),\left[0,\infty \right)\right)$ such that

$$\underset{l\to \infty }{limsup}{\displaystyle \frac{1}{l^n}}{\int }_{l_0}^l{\left(l-s\right)}^n\left[L\left(s\right)-{\displaystyle \frac{\mu \left(s\right)a\left(\xi \left(s\right)\right){\left[\frac{{\mu }^{\prime }\left(s\right)}{\mu \left(s\right)}-\frac{p\left(s\right)}{a\left(s\right)}+\frac{2{\xi }^{\prime }\left(s\right)a\left(s\right)}{a\left(\xi \left(s\right)\right)}\right]}^2}{4{\xi }^{\prime }\left(s\right)}}\right]ds=\infty ,$$

(28)

where

$$L\left(l\right)=\mu \left(l\right)\left[q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)-b\left(l\right)p\left(l\right)-{\left(a\left(l\right)b\left(l\right)\right)}^{\prime }\right],$$

(29)

and $\Psi \left(l\right)$ is defined by (7), then Equation (1) is oscillatory.

Proof. 

Suppose the contrary, that $y\left(l\right)>0$. Thus, there exists $l\ge l_1$ such that $y\left({\nu }_i\left(l\right)\right)>0,y\left(\xi \left(l\right)\right)>0$ for $i=1,2,...,m$. As in the proof of Theorem 1, we arrive at (24). Now, consider the generalized Riccati transformation,

$$\theta \left(l\right)=\mu \left(l\right)a\left(l\right)\left[{\displaystyle \frac{V^{\prime }\left(l\right)}{V\left(\xi \left(l\right)\right)}}+b\left(l\right)\right].$$

Then, clearly, $\theta \left(l\right)>0$, and

$$\begin{array}{ccc}\hfill {\theta }^{\prime }\left(l\right)& =& {\displaystyle \frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}}\theta \left(l\right)+\mu \left(l\right){\displaystyle \frac{{\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }}{V\left(\xi \left(l\right)\right)}}\hfill \\ & & -{\displaystyle \frac{\mu \left(l\right){\xi }^{\prime }\left(l\right)a\left(l\right)V^{\prime }\left(l\right)V^{\prime }\left(\xi \left(l\right)\right)}{V^2\left(\xi \left(l\right)\right)}}+\mu \left(l\right){\left(a\left(l\right)b\left(l\right)\right)}^{\prime }.\hfill \end{array}$$

Going through as in Theorem 1, we have

$$\begin{array}{ccc}\hfill {\theta }^{\prime }\left(l\right)& \le & \left[{\displaystyle \frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}}-{\displaystyle \frac{p\left(l\right)}{a\left(l\right)}}+{\displaystyle \frac{2{\xi }^{\prime }\left(l\right)a\left(l\right)b\left(l\right)}{a\left(\xi \left(l\right)\right)}}\right]\theta \left(l\right)-{\displaystyle \frac{{\xi }^{\prime }\left(l\right){\theta }^2\left(l\right)}{\mu \left(l\right)a\left(\xi \left(l\right)\right)}}\hfill \\ & & +\mu \left(l\right)p\left(l\right)b\left(l\right)-\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)+\mu \left(l\right){\left(a\left(l\right)b\left(l\right)\right)}^{\prime }.\hfill \end{array}$$

Now, by completing the squares, we obtain

$$\begin{array}{ccc}{\theta }^{\prime }\left(l\right)\hfill & \le \hfill & -L\left(l\right)+{\displaystyle \frac{{\left[\frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}-\frac{p\left(l\right)}{a\left(l\right)}+\frac{2{\xi }^{\prime }\left(l\right)a\left(l\right)b\left(l\right)}{a\left(\xi \left(l\right)\right)}\right]}^2}{4\frac{{\xi }^{\prime }\left(l\right)}{\mu \left(l\right)a\left(\xi \left(l\right)\right)}}}\hfill \\ \hfill & \hfill & -{\left(\sqrt{{\displaystyle \frac{{\xi }^{\prime }\left(l\right)}{\mu \left(l\right)a\left(\xi \left(l\right)\right)}}}\theta \left(l\right)-{\displaystyle \frac{\frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}-\frac{p\left(l\right)}{a\left(l\right)}+\frac{2{\xi }^{\prime }\left(l\right)a\left(l\right)b\left(l\right)}{a\left(\xi \left(l\right)\right)}}{2\sqrt{\frac{{\xi }^{\prime }\left(l\right)}{\mu \left(l\right)a\left(\xi \left(l\right)\right)}}}}\right)}^2\hfill \\ \hfill & \le \hfill & -L\left(l\right)+{\displaystyle \frac{{\left[\frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}-\frac{p\left(l\right)}{a\left(l\right)}+\frac{2{\xi }^{\prime }\left(l\right)a\left(l\right)b\left(l\right)}{a\left(\xi \left(l\right)\right)}\right]}^2}{4\frac{{\xi }^{\prime }\left(l\right)}{\mu \left(l\right)a\left(\xi \left(l\right)\right)}}},\hfill \end{array}$$

(30)

where $L\left(l\right)$ is defined in (29). Multiplying (30) by ${\left(l-s\right)}^n$, and integrating the resulting inequality from $l_0$ to l, we have

$$\begin{array}{ccc}& & -{\int }_{l_0}^l{\left(l-s\right)}^n{\theta }^{\prime }\left(s\right)ds\hfill \\ & \ge & {\int }_{l_0}^l{\left(l-s\right)}^n\left[L\left(s\right)-{\displaystyle \frac{\mu \left(s\right)a\left(\xi \left(s\right)\right){\left[\frac{{\mu }^{\prime }\left(s\right)}{\mu \left(s\right)}-\frac{p\left(s\right)}{a\left(s\right)}+\frac{2{\xi }^{\prime }\left(s\right)a\left(s\right)b\left(s\right)}{a\left(\xi \left(s\right)\right)}\right]}^2}{4{\xi }^{\prime }\left(s\right)}}\right]ds.\hfill \end{array}$$

(31)

However, since

$${\int }_{l_0}^l{\left(l-s\right)}^n{\theta }^{\prime }\left(s\right)ds=n{\int }_{l_0}^l{\left(l-s\right)}^{n-1}\theta \left(s\right)ds-{\left(l-l_0\right)}^n\theta \left(l_0\right),$$

then from (31), we obtain

$$\begin{array}{ccc}& & {\left(l-l_0\right)}^n\theta \left(l_0\right)-n{\int }_{l_0}^l{\left(l-s\right)}^{n-1}\theta \left(s\right)ds\hfill \\ & \ge & {\int }_{l_0}^l{\left(l-s\right)}^n\left[L\left(s\right)-{\displaystyle \frac{\mu \left(s\right)a\left(\xi \left(s\right)\right){\left[\frac{{\mu }^{\prime }\left(s\right)}{\mu \left(s\right)}-\frac{p\left(s\right)}{a\left(s\right)}+\frac{2{\xi }^{\prime }\left(s\right)a\left(s\right)b\left(s\right)}{a\left(\xi \left(s\right)\right)}\right]}^2}{4{\xi }^{\prime }\left(s\right)}}\right]ds.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{l^n}}{\int }_{l_0}^l{\left(l-s\right)}^n\left[L\left(s\right)-{\displaystyle \frac{\mu \left(s\right)a\left(\xi \left(s\right)\right){\left[\frac{{\mu }^{\prime }\left(s\right)}{\mu \left(s\right)}-\frac{p\left(s\right)}{a\left(s\right)}+\frac{2{\xi }^{\prime }\left(s\right)a\left(s\right)b\left(s\right)}{a\left(\xi \left(s\right)\right)}\right]}^2}{4{\xi }^{\prime }\left(s\right)}}\right]ds\hfill \\ & \le & {\left({\displaystyle \frac{l-l_0}{l}}\right)}^n\theta \left(l_0\right),\hfill \end{array}$$

and so

$$\underset{l\to \infty }{limsup}{\displaystyle \frac{1}{l^n}}{\int }_{l_0}^l{\left(l-s\right)}^n\left[L\left(s\right)-{\displaystyle \frac{\mu \left(s\right)a\left(\xi \left(s\right)\right){\left[\frac{{\mu }^{\prime }\left(s\right)}{\mu \left(s\right)}-\frac{p\left(s\right)}{a\left(s\right)}+\frac{2{\xi }^{\prime }\left(s\right)a\left(s\right)b\left(s\right)}{a\left(\xi \left(s\right)\right)}\right]}^2}{4{\xi }^{\prime }\left(s\right)}}\right]ds\to \theta \left(l_0\right).$$

This is a contradiction with (28), and so the proof is completed. □

Remark 2.

In the special case (4), our methods and criteria (10) and (22) of Theorem 1, include the conditions (7) and (8) of Theorem 2.1 in [16]. The same can be said for our criteria (10) and (28) of Theorem 2.

Theorem 3.

Assume that (A₁)–(A₄), (9), and (10) hold. Suppose further that

$$\underset{l\to \infty }{lim}{c}_i\left(l\right)=0,i=1,2,...,m.$$

(32)

If, for $\gamma >1$, we have

$$\underset{l\to \infty }{limsup}\left\{\begin{array}{c}{\Phi }^{-\gamma }\left(\xi \left(l\right)\right){\int }_{l_1}^{\xi \left(l\right)}\frac{\Phi \left(s\right)q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}{ds+\Phi }^{1-\gamma }\left(\xi \left(l\right)\right){\int }_{\xi \left(l\right)}^l\frac{q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}ds\\ +\Phi \left(\xi \left(l\right)\right){\int }_l^{\infty }\frac{q\left(s\right)}{E\left(s\right)}ds\end{array}\right\}>0,$$

(33)

where $E\left(l\right)$ and $\Phi \left(l\right)$ are defined by (6), then Equation (1) is oscillatory.

Proof. 

For the sake of contradiction, assume that (1) has an eventually positive solution $y\left(l\right)$. Moreover, in view of the properties of $V\left(l\right)$, (A₂), and (32), one can see that for any $\epsilon \in \left(0,1\right)$, we have

$$\sum _{i=1}^m{\beta }_i{c}_i\left(l\right)+{\displaystyle \frac{1}{V\left(l\right)}}\sum _{i=1}^m\left(1-{\beta }_i\right)c_i\left(l\right)<\epsilon .$$

Thus in view of (17), it follows that

$$y\left(l\right)\ge \lambda V\left(l\right)$$

(34)

where $\lambda =\left(1-\epsilon \right)\in \left(0,1\right)$. Substituting from (34) into (11), we have

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)+q\left(l\right)V^{\gamma }\left(\xi \left(l\right)\right){\lambda }^{\gamma }\le 0,$$

or

$${\left({\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{E\left(l\right)}}\right)}^{\prime }+{\displaystyle \frac{q\left(l\right)}{E\left(l\right)}}{V}^{\gamma }\left(\xi \left(l\right)\right){\lambda }^{\gamma }\le 0.$$

(35)

By integrating (35) from l to ∞, we obtain

$${\displaystyle \frac{a\left(\infty \right)V^{\prime }\left(\infty \right)}{E\left(\infty \right)}}-{\displaystyle \frac{a\left(l\right)V^{\prime }\left(l\right)}{E\left(l\right)}}\le -{\int }_l^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds,$$

then

$$V^{\prime }\left(l\right)\ge {\displaystyle \frac{E\left(l\right)}{a\left(l\right)}}{\int }_l^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds,$$

again integrating from $l_1$ to l, we obtain

$$\begin{array}{ccc}\hfill V\left(l\right)& \ge & V\left(l\right)-V\left(l_1\right)\hfill \\ & \ge & {\int }_{l_1}^l{\displaystyle \frac{E\left(u\right)}{a\left(u\right)}}{\int }_u^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}dsdu\hfill \\ & =& {\int }_{l_1}^l{\displaystyle \frac{E\left(u\right)}{a\left(u\right)}}{\int }_u^l{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}dsdu\hfill \\ & & +{\int }_{l_1}^l{\displaystyle \frac{E\left(u\right)}{a\left(u\right)}}{\int }_l^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}dsdu\hfill \\ & =& {\int }_{l_1}^l{\displaystyle \frac{\Phi \left(s\right)q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds+\Phi \left(l\right){\int }_l^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill V\left(\xi \left(l\right)\right)& \ge & {\int }_{l_1}^{\xi \left(l\right)}{\displaystyle \frac{\Phi \left(s\right)q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds+\Phi \left(\xi \left(l\right)\right){\int }_{\xi \left(l\right)}^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ & =& {\int }_{l_1}^{\xi \left(l\right)}{\displaystyle \frac{\Phi \left(s\right)q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds+\Phi \left(\xi \left(l\right)\right){\int }_{\xi \left(l\right)}^l{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ & & +\Phi \left(\xi \left(l\right)\right){\int }_l^{\infty }{\displaystyle \frac{q\left(s\right)V^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds.\hfill \end{array}$$

Since $V\left(l\right)$ is increasing while ${\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}$ is decreasing, we have

$$\begin{array}{ccc}V\left(\xi \left(l\right)\right)\hfill & \ge \hfill & {\displaystyle \frac{V^{\gamma }\left(\xi \left(l\right)\right)}{{\Phi }^{\gamma }\left(\xi \left(l\right)\right)}}{\int }_{l_1}^{\xi \left(l\right)}{\displaystyle \frac{\Phi \left(s\right)q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ \hfill & \hfill & +{\Phi }^{1-\gamma }\left(\xi \left(l\right)\right)V^{\gamma }\left(\xi \left(l\right)\right){\int }_{\xi \left(l\right)}^l{\displaystyle \frac{q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ \hfill & \hfill & +\Phi \left(\xi \left(l\right)\right)V^{\gamma }\left(\xi \left(l\right)\right){\int }_l^{\infty }{\displaystyle \frac{q\left(s\right){\lambda }^{\gamma }}{E\left(s\right)}}ds.\hfill \end{array}$$

(36)

Thus,

$$\begin{array}{ccc}\hfill V^{1-\gamma }\left(\xi \left(l\right)\right)& \ge & {\Phi }^{-\gamma }\left(\xi \left(l\right)\right){\int }_{l_1}^{\xi \left(l\right)}{\displaystyle \frac{\Phi \left(s\right)q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ & & +{\Phi }^{1-\gamma }\left(\xi \left(l\right)\right){\int }_{\xi \left(l\right)}^l{\displaystyle \frac{q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ & & +\Phi \left(\xi \left(l\right)\right){\int }_l^{\infty }{\displaystyle \frac{q\left(s\right){\lambda }^{\gamma }}{E\left(s\right)}}ds.\hfill \end{array}$$

Since ${\displaystyle \underset{l\to \infty }{lim}}V\left(l\right)=\infty$, then taking $limsup$ on both sides of the above inequality as $l\to \infty$, we arrive at a contradiction with (33) and, consequently, the proof is completed. □

Remark 3.

Although the technique of the above theorem depends on Theorem 2.3 of [14], our criterion (33) includes the criterion of Theorem 2.3 of [14] in the special case (4).

Now, we consider the case when (1) is sub-linear.

Theorem 4.

Assume that (A₁)–(A₄), (9) hold. If for $0<\gamma <1$, there exists a positive function $\mu \left(l\right)\in C^1\left(\left[l_0,\infty \right),\mathbb{R}\right)$, such that

$$\underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[{\displaystyle \frac{\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right){\Phi }^{\gamma -1}\left(\xi \left(s\right)\right)}{K^{1-\gamma }}}-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-\mu \left(s\right)p\left(s\right)\right)}^2}{4{\xi }^{\prime }\left(s\right)\mu \left(s\right)a^2\left(s\right)}}\right]ds=\infty ,$$

(37)

where K is any positive constant and $\Phi \left(l\right),\Psi \left(l\right)$ are defined by (6), and (7), respectively, then every solution of Equation (1) is oscillatory.

Proof. 

Suppose the contrary, that $y\left(l\right)>0$. Thus, there exists $l\ge l_1$, such that $y\left({\nu }_i\left(l\right)\right)>0,y\left(\xi \left(l\right)\right)>0$ for $i=1,2,...,m$. Since, from (20), we have

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)\le -q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right)V^{\gamma }\left(\xi \left(l\right)\right),$$

the above inequality can be written as

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)+q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right){\Phi }^{\gamma -1}\left(\xi \left(l\right)\right){\displaystyle \frac{V^{\gamma -1}\left(\xi \left(l\right)\right)}{{\Phi }^{\gamma -1}\left(\xi \left(l\right)\right)}}V\left(\xi \left(l\right)\right)\le 0,l\ge l_2.$$

(38)

But since ${\left({\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}\right)}^{\prime }<0$, then there exists a constant $K>0$, such that

$${\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}\le K\mathrm{for}l\ge l_2.$$

(39)

This, with (38), in view of the fact that $\gamma <1$, leads to

$${\left(a\left(l\right)V^{\prime }\left(l\right)\right)}^{\prime }+p\left(l\right)V^{\prime }\left(l\right)+{\displaystyle \frac{q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right){\Phi }^{\gamma -1}\left(\xi \left(l\right)\right)}{K^{1-\gamma }}}V\left(\xi \left(l\right)\right)\le 0.$$

(40)

Going through as in Theorem 1 with the same $w\left(l\right)$, we arrive at

$$w^{\prime }\left(l\right)\le {\displaystyle \frac{{\mu }^{\prime }\left(l\right)}{\mu \left(l\right)}}w\left(l\right)-{\displaystyle \frac{p\left(l\right)}{a\left(l\right)}}w\left(l\right)-{\displaystyle \frac{{\xi }^{\prime }\left(l\right)}{a\left(\xi \left(l\right)\right)\mu \left(l\right)}}{w}^2\left(l\right)-{\displaystyle \frac{\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right){\Phi }^{\gamma -1}\left(\xi \left(l\right)\right)}{K^{1-\gamma }}}.$$

Then, by completing squares, we obtain

$$w^{\prime }\left(l\right)\le -{\displaystyle \frac{\mu \left(l\right)q\left(l\right){\Psi }^{\gamma }\left(\xi \left(l\right)\right){\Phi }^{\gamma -1}\left(\xi \left(l\right)\right)}{K^{1-\gamma }}}+{\displaystyle \frac{a\left(\xi \left(l\right)\right){\left({\mu }^{\prime }\left(l\right)a\left(l\right)-\mu \left(l\right)p\left(l\right)\right)}^2}{4{\xi }^{\prime }\left(l\right)\mu \left(l\right)a^2\left(l\right)}}.$$

Integrating from $l_2$ to l, we obtain

$${\int }_{l_2}^l\left[{\displaystyle \frac{\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right){\Phi }^{\gamma -1}\left(\xi \left(s\right)\right)}{K^{1-\gamma }}}-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-\mu \left(s\right)p\left(s\right)\right)}^2}{4{\xi }^{\prime }\left(s\right)\mu \left(s\right)a^2\left(s\right)}}\right]ds\le w\left(l_2\right).$$

Taking $limsup$ as l tends to ∞, we obtain a contradiction with (37), and so the proof is completed. □

Remark 4.

In the special case (4), the method and Criterion (37) of Theorem 4 are more general than those in (17) of Theorem 2.4 of [16].

Remark 5.

If we consider (21) from the proof of Lemma 2, then the criterion of Theorem 4 becomes

$$\underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[{\displaystyle \frac{\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right){\Phi }^{\gamma -1}\left(\xi \left(s\right)\right)}{K^{1-\gamma }}}-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left[{\mu }^{\prime }\left(s\right)E\left(s\right)+\mu \left(s\right)E^{\prime }\left(s\right)\right]}^2}{4{\xi }^{\prime }\left(s\right)\mu \left(s\right)E^2\left(s\right)}}\right]ds=\infty$$

for every constant $K>0$, where $E\left(l\right)$ and $\Phi \left(l\right)$ are as defined in (6) and $\Psi \left(l\right)$ is as defined in (7).

Theorem 5.

Assume that (A₁)–(A₄), (9), and (32) hold. Suppose that, for $0<\gamma <1$,

$${\int }_{l_1}^{\infty }{\displaystyle \frac{q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}}ds=\infty ,l_1\ge l_0,$$

(41)

where $E\left(l\right)$, and $\Phi \left(l\right)$ are as defined by (6). If

$$\underset{l\to \infty }{limsup}\left\{\frac{1}{\Phi \left(\xi \left(l\right)\right)}{\int }_{l_1}^{\xi \left(l\right)}\frac{\Phi \left(s\right)q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}ds+{\int }_{\xi \left(l\right)}^l\frac{q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right)}{E\left(s\right)}ds+{\Phi }^{\gamma }\left(\xi \left(l\right)\right){\int }_l^{\infty }\frac{q\left(s\right)}{E\left(s\right)}ds\right\}>0,$$

(42)

then Equation (1) is oscillatory.

Proof. 

Suppose, for the sake of contradiction, that there exists an eventually positive solution $y\left(l\right)$ of (1). Then, $V\left(l\right)>0$ for $l\ge l_1\ge l_0$. Moreover, $V\left(l\right)$ is an increasing function, and ${\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}$ is decreasing. Now, we claim that (41) implies

$$\underset{l\to \infty }{lim}{\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}=0.$$

(43)

If this is false, then there exists $\delta >0$, such that

$$\underset{l\to \infty }{lim}{\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}=\delta >0.$$

Then, for large $l,{\displaystyle \frac{V\left(l\right)}{\Phi \left(l\right)}}>\delta$, from which we have

$$V^{\gamma }\left(\xi \left(l\right)\right)\ge {\delta }^{\gamma }{\Phi }^{\gamma }\left(\xi \left(l\right)\right).$$

Now, integrating (35) from $l_1$ to ∞, we obtain

$${\displaystyle \frac{a\left(l_1\right)V^{\prime }\left(l_1\right)}{E\left(l_1\right)}}\ge {\lambda }^{\gamma }{\delta }^{\gamma }{\int }_{l_1}^{\infty }{\displaystyle \frac{q\left(s\right)}{E\left(s\right)}}{\Phi }^{\gamma }\left(\xi \left(s\right)\right)ds.$$

This contradicts (41), which means that (43) holds. Now, putting

$$\phi \left(l\right)={\displaystyle \frac{V\left(\xi \left(l\right)\right)}{\Phi \left(\xi \left(l\right)\right)}}$$

into (36), we obtain

$$\begin{array}{ccc}\hfill {\phi }^{1-\gamma }\left(l\right)& \ge & {\displaystyle \frac{1}{\Phi \left(\xi \left(l\right)\right)}}{\int }_{l_1}^{\xi \left(l\right)}{\displaystyle \frac{\Phi \left(s\right)q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds+{\int }_{\xi \left(l\right)}^l{\displaystyle \frac{q\left(s\right){\Phi }^{\gamma }\left(\xi \left(s\right)\right){\lambda }^{\gamma }}{E\left(s\right)}}ds\hfill \\ & & +{\Phi }^{\gamma }\left(\xi \left(l\right)\right){\int }_l^{\infty }{\displaystyle \frac{q\left(s\right){\lambda }^{\gamma }}{E\left(s\right)}}ds.\hfill \end{array}$$

Taking $limsup$ on both sides as $l\to \infty$, we obtain a contradiction. This completes the proof. □

Remark 6.

The conditions (41) and (42) include the criteria (2.10) and (2.11) of Theorem 2.4 in [14].

Now, we consider the linear case of (1).

Theorem 6.

Let the conditions (A₁)–(A₄) be satisfied with $\gamma =1$. Suppose further that (9) and (10) hold. If there exists a positive function $\mu \left(l\right)\in C^1\left(\left[l_0,\infty \right),\mathbb{R}\right)$, such that

$$\underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[\mu \left(s\right)q\left(s\right)\Psi \left(\xi \left(s\right)\right)-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-p\left(s\right)\mu \left(s\right)\right)}^2}{4\mu \left(s\right){\xi }^{\prime }\left(s\right)a^2\left(s\right)}}\right]ds=\infty ,$$

(44)

where $\Psi \left(l\right)$ is defined in (7), then every solution of Equation (1) is oscillatory.

Proof. 

For the sake of contradiction, assume that Equation (1) has an eventually positive solution $y\left(l\right)$. Proceeding as in the proof of Theorem 1 with $f\left(l,y\left(\xi \left(l\right)\right)\right)\ge q\left(l\right)y\left(\xi \left(l\right)\right)$, we obtain a contradiction with (44). So, the proof is completed. □

Theorem 7.

Let $\gamma =1$, (9), (10), and (32) hold, and assume that

$$\underset{l\to \infty }{limsup}\left\{\frac{1}{\Phi \left(\xi \left(l\right)\right)}{\int }_{l_1}^{\xi \left(l\right)}\frac{\Phi \left(s\right)q\left(s\right)\Phi \left(\xi \left(s\right)\right)}{E\left(s\right)}ds+{\int }_{\xi \left(l\right)}^l\frac{q\left(s\right)\Phi \left(\xi \left(s\right)\right)}{E\left(s\right)}ds+\Phi \left(\xi \left(l\right)\right){\int }_l^{\infty }\frac{q\left(s\right)}{E\left(s\right)}ds\right\}>1,$$

(45)

where $E\left(l\right)$ and $\Phi \left(l\right)$ are defined by (6). Then, Equation (1) is oscillatory.

Proof. 

For the sake of contradiction, assume that Equation (1) has an eventually positive solution $y\left(l\right)$, when $f\left(l,y\left(\xi \left(l\right)\right)\right)\ge q\left(l\right)y\left(\xi \left(l\right)\right)$. Going through as in the proof of Theorem 3, we are led to (36) with $\gamma =1$. Consequently,

$$\begin{array}{ccc}\hfill 1& \ge & {\displaystyle \frac{1}{\Phi \left(\xi \left(l\right)\right)}}{\int }_{l_1}^{\xi \left(l\right)}{\displaystyle \frac{\Phi \left(s\right)q\left(s\right)\Phi \left(\xi \left(s\right)\right)\lambda }{E\left(s\right)}}ds+{\int }_{\xi \left(l\right)}^l{\displaystyle \frac{q\left(s\right)\Phi \left(\xi \left(s\right)\right)\lambda }{E\left(s\right)}}ds\hfill \\ & & +\Phi \left(\xi \left(l\right)\right){\int }_l^{\infty }{\displaystyle \frac{q\left(s\right)\lambda }{E\left(s\right)}}ds.\hfill \end{array}$$

Taking $limsup$ as $l\to \infty$ on both sides of the above inequality, we obtain a contradiction with (45), and so the proof is completed. □

Remark 7.

For those who are concerned with the linear case $f\left(l,y\left(\xi \left(l\right)\right)\right)=q\left(l\right)y\left(\xi \left(l\right)\right)$, it is clear that our methods and results include the criterion (16) of Theorem 2.3 of [16] and those of Theorem 2.5 of [14].

4. Examples

To show the effectiveness and efficiency of our new oscillation criteria, which enhance numerous earlier efforts, we give the following four examples to support the obtained results.

Example 1.

Consider the equation

$${\left(\frac{1}{l}{\left(y\left(l\right)+\frac{1}{l}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^2}{y}^{{\displaystyle \frac{1}{5}}}\left(\frac{l}{5}\right)\right)}^{\prime }\right)}^{\prime }+\frac{1}{l^2}{\left(y\left(l\right)+\frac{1}{l}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^2}{y}^{{\displaystyle \frac{1}{5}}}\left(\frac{l}{5}\right)\right)}^{\prime }+\frac{\sigma }{l^3}{y}^3\left(\frac{l}{3}\right)=0,l\ge 4.$$

(46)

Here, $a\left(l\right)={\displaystyle \frac{1}{l}},p\left(l\right)={\displaystyle \frac{1}{l^2}},c_1\left(l\right)={\displaystyle \frac{1}{l}},c_2\left(l\right)={\displaystyle \frac{1}{l^2}},{\nu }_1\left(l\right)={\displaystyle \frac{l}{3}},{\nu }_2\left(l\right)={\displaystyle \frac{l}{5}},{\beta }_1={\displaystyle \frac{1}{3}},{\beta }_2={\displaystyle \frac{1}{5}}$, and $f\left(l,y\left(\xi \left(l\right)\right)\right)={\displaystyle \frac{\sigma }{l^3}}{y}^3\left({\displaystyle \frac{l}{3}}\right)$, i.e., $q\left(l\right)={\displaystyle \frac{\sigma }{l^3}},\xi \left(l\right)={\displaystyle \frac{l}{3}},\gamma =3$. It is easy to see that $E\left(l\right)={\displaystyle \frac{4}{l}}$. Choosing $\rho \left(l\right)={\displaystyle \frac{1}{l}}$, then $\rho \left(l\right)\to 0$ as $l\to \infty$ and

$$\begin{array}{ccc}\hfill \Psi \left(l\right)& =& \left(1-{\displaystyle \frac{1}{3l}}-{\displaystyle \frac{1}{5l^2}}-l\left({\displaystyle \frac{2}{3l}}+{\displaystyle \frac{4}{5l^2}}\right)\right)\hfill \\ & =& \left({\displaystyle \frac{1}{3}}-{\displaystyle \frac{17}{15l}}-{\displaystyle \frac{1}{5l^2}}\right)={\displaystyle \frac{5l^2-17l-3}{15l^2}}>0\mathrm{for}l\ge 4.\hfill \end{array}$$

Thus, the conditions (9) and (10) are satisfied. Taking $\mu \left(l\right)=l^2$, we see that

$$\underset{l\to \infty }{limsup}{\int }_4^l[{\displaystyle \frac{\sigma }{s}}{\left({\displaystyle \frac{1}{3}}-{\displaystyle \frac{9}{5s^2}}-{\displaystyle \frac{17}{5s}}\right)}^3-{\displaystyle \frac{9}{4s}}]ds=\infty$$

for $\sigma \ge {\displaystyle \frac{243}{4}}$. So, by Theorem 1, every solution of Equation (46) is oscillatory.

Example 2.

Consider the equation

$${\left(l^{{\displaystyle \frac{1}{4}}}{\left(y\left(l\right)+\frac{1}{l}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^2}{y}^{{\displaystyle \frac{5}{7}}}\left(\frac{l}{5}\right)\right)}^{\prime }\right)}^{\prime }+\frac{1}{4}{l}^{-{\displaystyle \frac{3}{4}}}{\left(y\left(l\right)+\frac{1}{l}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^2}{y}^{{\displaystyle \frac{5}{7}}}\left(\frac{l}{5}\right)\right)}^{\prime }+l^{{\displaystyle \frac{2}{15}}}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{2}\right)=0,l\ge 8.$$

(47)

Here, $a\left(l\right)=l^{{\displaystyle \frac{1}{4}}},p\left(l\right)={\displaystyle \frac{1}{4}}{l}^{-{\displaystyle \frac{3}{4}}},c_1\left(l\right)={\displaystyle \frac{1}{l}},c_2\left(l\right)={\displaystyle \frac{1}{l^2}},{\beta }_1={\displaystyle \frac{1}{3}},{\beta }_2={\displaystyle \frac{5}{7}},{\nu }_1\left(l\right)={\displaystyle \frac{l}{3}},{\nu }_2\left(l\right)={\displaystyle \frac{l}{5}}$, and $f\left(l,y\left(\xi \left(l\right)\right)\right)=l^{{\displaystyle \frac{2}{15}}}{y}^{{\displaystyle \frac{1}{3}}}\left({\displaystyle \frac{l}{2}}\right)$, i.e., $q\left(l\right)=l^{{\displaystyle \frac{2}{15}}},\gamma ={\displaystyle \frac{1}{3}},\xi \left(l\right)={\displaystyle \frac{l}{2}}$. Choosing $\rho \left(l\right)={\displaystyle \frac{1}{l}}$, then $\rho \left(l\right)\to 0$ as $l\to \infty$, and

$$\begin{array}{ccc}\hfill \Psi \left(l\right)& =& 1-{\displaystyle \frac{1}{3l}}-{\displaystyle \frac{5}{7l^2}}-l\left[{\displaystyle \frac{2}{3l}}+{\displaystyle \frac{2}{7l^2}}\right]\hfill \\ & =& {\displaystyle \frac{1}{3}}-{\displaystyle \frac{13}{21l}}-{\displaystyle \frac{5}{7l^2}}={\displaystyle \frac{7l^2-13l-15}{21l^2}},\hfill \end{array}$$

$$\Psi \left(\xi \left(l\right)\right)={\displaystyle \frac{7l^2-26l-60}{21l^2}}>0\mathrm{for}l\ge 8.$$

It is easy to see that $E\left(l\right)={\displaystyle \frac{8^{\frac{1}{4}}}{l^{\frac{1}{4}}}}$,

$$\begin{array}{ccc}\hfill \Phi \left(l\right)& =& {\int }_{l_1}^l{\displaystyle \frac{E\left(s\right)}{a\left(s\right)}}ds={\int }_8^l{\displaystyle \frac{8^{\frac{1}{4}}}{s^{\frac{1}{2}}}}ds=8^{{\displaystyle \frac{1}{4}}}\left[2\sqrt{l}-4\sqrt{2}\right]\hfill \\ & =& 8^{{\displaystyle \frac{1}{4}}}\sqrt{2}\left[\sqrt{2l}-4\right],\hfill \\ \hfill \Phi \left(\xi \left(l\right)\right)& =& 2^{{\displaystyle \frac{5}{4}}}\left[\sqrt{l}-4\right]\hfill \end{array}$$

and so Condition (9) is satisfied.

Letting $\mu \left(l\right)=l^{{\displaystyle \frac{1}{5}}}$, then

$$\begin{array}{ccc}& & \underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[{\displaystyle \frac{\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right){\Phi }^{\gamma -1}\left(\xi \left(s\right)\right)}{K^{1-\gamma }}}-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left[{\mu }^{\prime }\left(s\right)a\left(s\right)-\mu \left(s\right)p\left(s\right)\right]}^2}{4{\xi }^{\prime }\left(s\right)\mu \left(s\right)a^2\left(s\right)}}\right]ds\hfill \\ & =& \underset{l\to \infty }{limsup}{\int }_8^l[K^{-{\displaystyle \frac{2}{3}}}{s}^{{\displaystyle \frac{1}{5}}}{s}^{{\displaystyle \frac{2}{15}}}{\left(2^{{\displaystyle \frac{5}{4}}}\right)}^{-{\displaystyle \frac{2}{3}}}{\left({\displaystyle \frac{7s^2-26s-60}{21s^2}}\right)}^{{\displaystyle \frac{1}{3}}}{\left(s^{{\displaystyle \frac{1}{2}}}-4\right)}^{-{\displaystyle \frac{2}{3}}}\hfill \\ & & -{\displaystyle \frac{\frac{1}{2^{\frac{1}{4}}}{s}^{\frac{1}{4}}{\left[\frac{1}{5}{s}^{-\frac{4}{5}}{s}^{\frac{1}{4}}-\frac{1}{4}{s}^{-\frac{3}{4}}{s}^{\frac{1}{5}}\right]}^2}{2s^{\frac{1}{5}}{s}^{\frac{1}{2}}}}]ds\hfill \\ & =& \underset{l\to \infty }{limsup}{\int }_8^l\left[K^{-{\displaystyle \frac{2}{3}}}{2}^{-{\displaystyle \frac{5}{6}}}{s}^{{\displaystyle \frac{1}{3}}}{\left({\displaystyle \frac{7s^2-26s-60}{21s^2}}\right)}^{{\displaystyle \frac{1}{3}}}{\left(s^{{\displaystyle \frac{1}{2}}}-4\right)}^{-{\displaystyle \frac{2}{3}}}-{\displaystyle \frac{{\left(\frac{1}{20}\right)}^2}{2^{\frac{5}{4}}{s}^{\frac{62}{40}}}}\right]ds=\infty .\hfill \end{array}$$

Then, by Theorem 4, every solution of Equation (47) is oscillatory.

Example 3.

Consider the equation

$${\left(\frac{1}{l}{(y\left(l\right)+\frac{1}{l^3}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^6}{y}^{{\displaystyle \frac{1}{5}}}\left(\frac{l}{5}\right))}^{\prime }\right)}^{\prime }+\frac{1}{l^2}{\left(y\left(l\right)+\frac{1}{l^3}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^6}{y}^{{\displaystyle \frac{1}{5}}}\left(\frac{l}{5}\right)\right)}^{\prime }+\frac{1}{l^{\frac{4}{3}}}{y}^{{\displaystyle \frac{1}{3}}}\left(l\right)=0,l\ge 2.$$

(48)

Here, $a\left(l\right)={\displaystyle \frac{1}{l}},p\left(l\right)={\displaystyle \frac{1}{l^2}},c_1\left(l\right)={\displaystyle \frac{1}{l^3}},c_2\left(l\right)={\displaystyle \frac{1}{l^6}},{\beta }_1={\displaystyle \frac{1}{3}},{\beta }_2={\displaystyle \frac{1}{5}},{\nu }_1\left(l\right)={\displaystyle \frac{l}{3}},{\nu }_2\left(l\right)={\displaystyle \frac{l}{5}}$, and $f\left(l,y\left(\xi \left(l\right)\right)\right)={\displaystyle \frac{1}{l^{\frac{4}{3}}}}{y}^{{\displaystyle \frac{1}{3}}}\left(l\right)$, i.e., $q\left(l\right)={\displaystyle \frac{1}{l^{\frac{4}{3}}}},\gamma ={\displaystyle \frac{1}{3}},\xi \left(l\right)=l$.

Choosing $\rho \left(l\right)={\displaystyle \frac{1}{l^3}}$, then $\rho \left(l\right)\to 0$ as $l\to \infty$ and

$$\begin{array}{ccc}\hfill \Psi \left(l\right)& =& 1-{\displaystyle \frac{1}{3l^3}}-{\displaystyle \frac{1}{5l^6}}-l^3\left[{\displaystyle \frac{2}{3l^3}}+{\displaystyle \frac{4}{5l^6}}\right]\hfill \\ & =& {\displaystyle \frac{1}{3}}-{\displaystyle \frac{17}{15l^3}}-{\displaystyle \frac{1}{5l^6}}={\displaystyle \frac{5l^6-17l^3-3}{15l^6}}>0\mathrm{for}l\ge 2.\hfill \end{array}$$

It is easy to see that $E\left(l\right)={\displaystyle \frac{2}{l}}$,

$$\Phi \left(l\right)={\int }_{l_1}^l{\displaystyle \frac{E\left(s\right)}{a\left(s\right)}}ds={\int }_2^{l}2ds={\left[2s\right]}_2^l=2\left(l-2\right)$$

and so the condition (9) is satisfied. Letting $\mu \left(l\right)=l$, then

$$\begin{array}{ccc}& & \underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[{\displaystyle \frac{\mu \left(s\right)q\left(s\right){\Psi }^{\gamma }\left(\xi \left(s\right)\right){\Phi }^{\gamma -1}\left(\xi \left(s\right)\right)}{K^{1-\gamma }}}-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-\mu \left(s\right)p\left(s\right)\right)}^2}{4{\xi }^{\prime }\left(s\right)\mu \left(s\right)a^2\left(s\right)}}\right]\hfill \\ & =& \underset{l\to \infty }{limsup}{\int }_2^l\left[{\displaystyle \frac{s}{2^{\frac{2}{3}}{K}^{\frac{2}{3}}{s}^{\frac{4}{3}}}}{\left({\displaystyle \frac{1}{3}}-{\displaystyle \frac{17}{15s^3}}-{\displaystyle \frac{1}{5s^6}}\right)}^{{\displaystyle \frac{1}{3}}}{\left(s-2\right)}^{-{\displaystyle \frac{2}{3}}}-{\displaystyle \frac{\frac{1}{s}\left(\frac{1}{s}-\frac{s}{s^2}\right)}{4s\frac{1}{s^2}}}\right]ds\hfill \\ & =& \underset{l\to \infty }{limsup}{\int }_2^l{\displaystyle \frac{s}{2^{\frac{2}{3}}{K}^{\frac{2}{3}}{s}^2{s}^{-\frac{2}{3}}}}{\left({\displaystyle \frac{1}{3}}-{\displaystyle \frac{17}{15s^3}}-{\displaystyle \frac{1}{5s^6}}\right)}^{{\displaystyle \frac{1}{3}}}{\left(s-2\right)}^{-{\displaystyle \frac{2}{3}}}ds\hfill \\ & =& \underset{l\to \infty }{limsup}{\int }_2^l{\displaystyle \frac{1}{{\left(2K\right)}^{\frac{2}{3}}s}}{\left({\displaystyle \frac{1}{3}}-{\displaystyle \frac{17}{15s^3}}-{\displaystyle \frac{1}{5s^6}}\right)}^{{\displaystyle \frac{1}{3}}}{\left(1-{\displaystyle \frac{2}{s}}\right)}^{-{\displaystyle \frac{2}{3}}}ds=\infty .\hfill \end{array}$$

Then, by Theorem 4, every solution of Equation (48) is oscillatory.

Example 4.

Consider the equation

$${\left(\frac{1}{l}{\left(y\left(l\right)+\frac{1}{l^4}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^6}{y}^{{\displaystyle \frac{3}{5}}}\left(\frac{l}{5}\right)\right)}^{\prime }\right)}^{\prime }+\frac{1}{l^2}{\left(y\left(l\right)+\frac{1}{l^4}{y}^{{\displaystyle \frac{1}{3}}}\left(\frac{l}{3}\right)+\frac{1}{l^6}{y}^{{\displaystyle \frac{3}{5}}}\left(\frac{l}{5}\right)\right)}^{\prime }+\frac{1}{l^3}y\left(l\right)=0,l\ge 2.$$

(49)

Here, $a\left(l\right)={\displaystyle \frac{1}{l}},p\left(l\right)={\displaystyle \frac{1}{l^2}},c_1\left(l\right)={\displaystyle \frac{1}{l^4}},c_2\left(l\right)={\displaystyle \frac{1}{l^6}},{\nu }_1\left(l\right)={\displaystyle \frac{l}{3}},{\nu }_2\left(l\right)={\displaystyle \frac{l}{5}},{\beta }_1={\displaystyle \frac{1}{3}},{\beta }_2={\displaystyle \frac{3}{5}}$, and $f\left(l,y\left(\xi \left(l\right)\right)\right)={\displaystyle \frac{1}{l^3}}y\left(l\right)$, i.e., $q\left(l\right)={\displaystyle \frac{1}{l^3}},\xi \left(l\right)=l,\gamma =1$. It is easy to see that $E\left(l\right)={\displaystyle \frac{2}{l}}$. Letting $\rho \left(l\right)={\displaystyle \frac{1}{l^2}}$ then $\rho \left(l\right)\to 0$ as $l\to \infty$ and

$$\begin{array}{ccc}\hfill \Psi \left(l\right)& =& 1-{\displaystyle \frac{1}{3l^4}}-{\displaystyle \frac{3}{5l^6}}-l^2\left[{\displaystyle \frac{2}{3l^4}}+{\displaystyle \frac{2}{5l^6}}\right]\hfill \\ & =& 1-{\displaystyle \frac{2}{3l^2}}-{\displaystyle \frac{11}{15l^4}}-{\displaystyle \frac{3}{5l^6}}\hfill \\ & =& {\displaystyle \frac{15l^6-11l^2-10l^4-9}{15l^6}}>0\mathrm{for}l\ge 2.\hfill \end{array}$$

The conditions (9) and (10) are satisfied. Taking $\mu \left(l\right)=l^2$, we see that

$$\begin{array}{ccc}& & \underset{l\to \infty }{limsup}{\int }_{l_0}^l\left[\mu \left(s\right)q\left(s\right)\Psi \left(\xi \left(s\right)\right)-{\displaystyle \frac{a\left(\xi \left(s\right)\right){\left({\mu }^{\prime }\left(s\right)a\left(s\right)-p\left(s\right)\mu \left(s\right)\right)}^2}{4\mu \left(s\right){\xi }^{\prime }\left(s\right)a^2\left(s\right)}}\right]ds\hfill \\ & =& \underset{l\to \infty }{limsup}{\int }_2^l\left[{\displaystyle \frac{1}{s}}\left[1-{\displaystyle \frac{2}{3s^2}}-{\displaystyle \frac{11}{15s^4}}-{\displaystyle \frac{3}{5s^6}}\right]-{\displaystyle \frac{1}{4s}}\right]ds=\infty ,\hfill \end{array}$$

then, by Theorem 6, every solution of Equation (49) is oscillatory.

Remark 8.

We may note that the results of [16] do not work for (46), (47), (48), and (49). So, our criteria (22) of Theorem 1, (37) of Theorem 4 and (44) of Theorem 6 are new.