Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument

Abstract

The chapter is devoted to study the oscillation of all solutions to second‐order nonlinear neutral damped differential equations with delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and integral averaging techniques.

1. Introduction

In the chapter, we are mainly concerned with the oscillatory behavior of solutions to second‐order nonlinear neutral damped differential equations with delay argument of the form

${\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+p(t){\left(z^{\prime }(t)\right)}^{\alpha }+q(t)f\left(x(\sigma (t))\right)=0,t\ge t_0,$ E1

where $\alpha \ge 1$ is a quotient of positive odd integers and

$z(t)=x(t)+a(t)x(\tau (t)).$ E2

Throughout, we suppose that the following hypotheses hold:

1. $r,p,q\in C(ℐ,{ℝ}^{+})$ , where $ℐ=[t_0,\infty )$ and ${ℝ}^{+}=(\mathrm{0,}\infty )$ ;

2. $a\in C(ℐ,ℝ)$ , $0\le a(t)\le 1;$

3. $\tau \in C(ℐ,ℝ)$ , $\tau (t)\le t$ , $\tau (t)\to \infty$ as $t\to \infty$ ;

4. $\sigma \in C^1(ℐ,ℝ)$ , $\sigma (t)\le t$ , ${\sigma }^{\prime }(t)\ge 0$ , $\sigma (t)\to \infty$ as $t\to \infty$ ;

5. $f\in C(ℝ,ℝ)$ , such that $xf(x)>0$ and $f(x)/x^{\beta }\ge k>0$ for $x\cancel{=}0$ , where $k$ is a constant and $\beta$ is the ratio of odd positive integers.

By a solution of Eq. (1), we mean a nontrivial real‐valued function $x(t)$ , which has the property $z(t)\in C^1([T_x,\infty ))$ , $r(t){\left(z^{\prime }(t)\right)}^{\alpha }\in C^1([T_x,\infty ))$ , $T_x\ge t_0$ , and satisfies Eq. (1) on $[T_x,\infty )$ . In the sequel, we will restrict our attention to those solutions $x(t)$ of Eq. (1) that satisfy the condition

$\sup \{\left|x(t)\right|:T\le t<\infty \}>0\text{for}T\ge T_x.$ E3

We make the standing hypothesis that Eq. (1) admits such a solution. As is customary, a solution of Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative on $[T_x,\infty )$ and otherwise, it is termed nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

Remark 1. All the functional inequalities considered in the sequel are assumed to hold eventually, that is, they are satisfied for all $t$ large enough.

Oscillation theory was created in 1836 with a paper of Jacques Charles François Sturm published in Journal des Mathematiqués Pures et Appliqueés. His long and detailed memoir [1] was one of the first contributions in Liouville's newly founded journal and initiated a whole new research into the qualitative analysis of differential equations. Heretofore, the theory of differential equations was primarily about finding solutions of a given equation and so was very limited. Contrarily, the main idea of Sturm was to obtain geometric properties of solutions (such as sign changes, zeros, boundaries, and oscillation) directly from the differential equation, without benefit of solutions themselves.

Henceforth, the oscillation theory for ordinary differential equations has undergone a significant development. Nowadays, it is considered as coherent, self‐contained domain in the qualitative theory of differential equations that is turning mainly toward the study of solution properties of functional differential equations (FDEs).

The problem of obtaining sufficient conditions for asymptotic and oscillatory properties of different classes of FDEs has experienced long‐term interest of many researchers. This is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of the countless processes in all areas of science. For a summary of the most significant efforts and recent findings in the oscillation theory of FDEs and vast bibliography therein, we refer the reader to the excellent monographs [2–6].

In a neutral delay differential equation the highest‐order derivative of the unknown function appears both with and without delay. The study of qualitative properties of solutions of such equations has, besides its theoretical interest, significant practical importance. This is due to the fact that neutral differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high‐speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays. We refer the reader to the monograph [7] for further applications in science and technology.

So far, most of the results obtained in the literature has centered around the special undamped form of Eq. (1), i.e., when $p(t)=0$ (for example, see Refs. [8–18]). For instance, in one of the pioneering works on the subject, Grammatikopoulos et al. [8] studied the second‐order neutral differential equation with constant delay of the form

$(x(t)+a(t)x(t-\tau {)}^{″}+q(t)x(t-\tau )=0$ E4

and proved that Eq. (4) is oscillatory if

${\displaystyle {\int }_{t_0}^{\infty }}q(s)\left(1-a(s-\tau )\right)\text{d}s=\infty .$ E5

Later on, Grace and Lalli [9] extended the results from [8] to the more general equation

${\left(r(t)(x(t)+a(t)x(t-\tau {)}^{\prime }\right)}^{\prime }+q(t)f\left(x(t-\tau )\right)=0,$ E6

with

$\frac{f(x)}{x}\ge k,k>0\text{and}{\displaystyle {\int }_{t_0}^{\infty }}\frac{\text{d}s}{r(s)}=\infty$ E7

and showed that Eq. (6) is oscillatory if there exists a continuously differentiable function $\rho (t)$ such that

${\displaystyle {\int }_{t_0}^{\infty }}\left(\rho (s)q(s)(1-a(s-\tau ))-\frac{{\left({\rho }^{\prime }(s)\right)}^{2}r(s-\tau )}{4k\rho (s)}\right)\text{d}s=\infty .$ E8

In Ref. [10], Dong has involved to study the oscillation problem for a half‐linear case of Eq. (1) and by defining a sequence of continuous functions has obtained various kinds of better results. Afterward, his approach has been further developed by several authors, see, e.g., [11–14]. However, it appears that very little is known regarding the oscillation of Eq. (1) with $p(t)\ne 0$ and $\alpha \ne \beta$ . Motivated by the results of Ref. [10], this chapter presents some new oscillation criteria, which are applicable on Eq. (1).

On the other hand, Eq. (1) can be considered as a natural generalization of the second‐order delay differential equation of the form

$\left(r(t){\left(x^{\prime }(t)\right)}^{\alpha }\right)+p(t){\left(x^{\prime }(t)\right)}^{\alpha }+q(t)f\left(x(\sigma (t))\right)=0.$ E9

Very recently, the authors of [19] studied the oscillation problem of Eq. (9) with $p(t)=0$ and $\alpha =\beta$ . Their ideas, which are based on careful investigation of classical techniques covering Riccati transformations and integral averages, will be extended to the more general equation (1).

2. Main results

For the simplicity and without further mention, we use the following notations:

$A(t)=\exp \left(-{\displaystyle {\int }_{t_0}^t}\frac{p(s)}{r(s)}\text{d}s\right),Q(t)=kq(t){\left(1-a(\sigma (t))\right)}^{\beta },$ E10

$R(t)={\displaystyle {\int }_t^{\infty }}{\left(\frac{A(s)}{r(s)}\right)}^{\frac{1}{\alpha }}\text{d}s,\tilde{Q}(t)=q(t){\left(1-a(\sigma (t))\frac{R(\tau (\sigma (t)))}{R(\sigma (t))}\right)}^{\beta },$ E11

$P(t)=\frac{{\varphi }^{\prime }(t)}{\varphi (t)}-\frac{p(t)}{r(t)},\tilde{q}(t)=Q(t)+\frac{p(t)A(t)}{r(t)}{\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s,$ E12

where $\varphi (t)\in C^1(ℐ,ℝ)$ is a given function and will be specified later.

The organization of this chapter is as follows. Before stating our main results, we present two lemmas that ensure that any solution $x(t)$ of Eq. (1) satisfies the condition

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}$ E13

for $t$ sufficiently large. Next, we get our main oscillation results for Eq. (1) by employing the generalized Riccati transformations and integral averaging techniques. We base our arguments on the assumption that the function $P(t)$ is positive or negative.

Lemma 1. Assume that

${\displaystyle {\int }_{t_0}^{\infty }}{\left(\frac{A(s)}{r(s)}\right)}^{\frac{1}{\alpha }}\text{d}s=\infty$ E14

holds and Eq. (1) has a positive solution $x(t)$ on $ℐ$ . Then there exists a $T\in ℐ$ , sufficiently large, such that

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}$ E15

on $[T,\infty )$ .

Proof. Since, $x(t)$ is a positive solution of Eq. (1) on $ℐ$ , then, by the assumptions $\text{(iii)}$ and $\text{(iv)}$ , there exists a $t_1\in ℐ$ such that $x(\tau (t))>0$ and $x(\sigma (t))>0$ on $[t_1,\infty )$ . Define the function $z(t)$ as in Eq. (2). Then it is easy to see that $z(t)\ge x(t)>0$ , for $t\ge t_1$ , and at the same time, from Eq. (1), we get

$\begin{array}{ll}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+p(t){\left(z^{\prime }(t)\right)}^{\alpha }\hfill & =-q(t)f\left(x(\sigma (t))\right)<0.\hfill \end{array}$ E16

We assert that $\frac{r(t)}{A(t)}{\left(z^{\prime }(t)\right)}^{\alpha }$ is decreasing. Clearly, by writing the left‐hand side of Eq. (16) in the form

${\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+\frac{p(t)}{r(t)}r(t){\left(z^{\prime }(t)\right)}^{\alpha }<\mathrm{0,}$ E17

we get

${\left(\frac{r(t)}{A(t)}{\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }=-\frac{q(t)}{A(t)}f(x(\sigma (t)))<0$ E18

and so the assertion is proved.

Now, we claim that $z^{\prime }(t)>0$ on $[t_1,\infty )$ . If not, then there exists $t_2\in [t_1,\infty )$ such that $z^{\prime }(t_2)<0$ . Using the fact that $\frac{r(t)}{A(t)}{\left(z^{\prime }(t)\right)}^{\alpha }$ is decreasing, we obtain, for $t\ge t_2$ ,

$\frac{r(t)}{A(t)}{\left(z^{\prime }(t)\right)}^{\alpha }<c:=\frac{r(t_2)}{A(t_2)}{\left(z^{\prime }(t_2)\right)}^{\alpha }<0.$ E19

Integrating the above inequality from $t_2$ to $t$ , we find that

$z(t)<z(t_2)+c^{\frac{1}{\alpha }}{\displaystyle {\int }_{t_2}^t}{\left(\frac{A(s)}{r(s)}\right)}^{\frac{1}{\alpha }}\text{d}s$ E20

for $t\ge t_2.$ By condition (14), $z(t)$ approaches to $-\infty$ as $t\to \infty$ , which contradicts the fact that $z(t)$ is eventually positive. Therefore, $z^{\prime }(t)>0$ and from Eq. (1), we have that ${\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<0$ . The proof is complete.

Lemma 2. Assume that

${\displaystyle {\int }_{t_0}^{\infty }}{\left(\frac{A(u)}{r(u)}{\displaystyle {\int }_{t_0}^u}\frac{\tilde{Q}(s)R^{\beta }(\sigma (s))}{A(s)}\text{d}s\right)}^{\frac{1}{\alpha }}\text{d}u=\infty ,$ E21

holds and Eq. (1) has a positive solution $x(t)$ on $ℐ$ . Then there exists $T\in ℐ$ , sufficiently large, such that

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}$ E22

on $[T,\infty )$ .

Proof. Similarly to the proof of Lemma 1, we assume that there exists $t_2\in ℐ$ such that $z^{\prime }(t)<0$ on $[t_2,\infty )$ . Taking Eq. (18) into account, we have

$z^{\prime }(s)\le {\left(\frac{r(t)}{A(t)}\frac{A(s)}{r(s)}\right)}^{\frac{1}{\alpha }}{z}^{\prime }(t),$ E23

for $s\ge t\ge t_2$ . Integrating the above inequality from $t$ to $t^{\prime }$ , $t^{\prime }\ge t\ge t_2$ , we get

$z(t^{\prime })\le z(t)+{\left(\frac{r(t)}{A(t)}\right)}^{\frac{1}{\alpha }}{z}^{\prime }(t){\displaystyle {\int }_t^{t^{\prime }}}{\left(\frac{r(s)}{A(s)}\right)}^{-\frac{1}{\alpha }}\text{d}s.$ E24

Letting $t^{\prime }\to \infty$ , we have

$z(t)\ge -R(t){\left(\frac{r(t)}{A(t)}\right)}^{\frac{1}{\alpha }}{z}^{\prime }(t),$ E25

which yields

$\left(\frac{z(t)}{R(t)}\right)\ge 0$ E26

and hence we see that $\frac{z(t)}{R(t)}$ is nondecreasing. By Eq. (2) and $\text{(iii)}$ , we have

$\begin{array}{ll}x(t)\hfill & =z(t)-a(t)x(\tau (t))\hfill \\ \hfill & \ge z(t)-a(t)z(\tau (t))\hfill \\ \hfill & \ge \left(1-a(t)\frac{R(\tau (t))}{R(t)}\right)z(t),\hfill \end{array}$ E27

which together with Eq. (1) and the assumption $(\text{v})$ yields

$\begin{array}{ll}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+p(t){\left(z^{\prime }(t)\right)}^{\alpha }\hfill & \le -kq(t){\left(1-a(\sigma (t))\frac{R(\tau (\sigma (t)))}{R(\sigma (t))}\right)}^{\beta }{z}^{\beta }(\sigma (t))\hfill \\ \hfill & =-k\tilde{Q}\left(t\right)z^{\beta }(\sigma (t)).\hfill \end{array}$ E28

On the other hand, from Eq. (23), we have

$\frac{r(t){\left(z^{\prime }(t)\right)}^{\alpha }}{A(t)}\le \frac{r(t_2){\left(z^{\prime }(t_2)\right)}^{\alpha }}{A(t_2)},$ E29

that is,

$\frac{r(t)}{A(t)}{\left(-z^{\prime }(t)\right)}^{\alpha }\ge \frac{r(t_2)}{A(t_2)}{\left(-z^{\prime }(t_2)\right)}^{\alpha }:={\gamma }^{\alpha }$ E30

for some positive constant $\gamma$ . Setting Eq. (30) into Eq. (25), we obtain

$z(t)\ge \gamma R(t)$ E31

and so, Eq. (28) becomes

$\begin{array}{cc}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+p(t){\left(z^{\prime }(t)\right)}^{\alpha }& \le -\tilde{\gamma }\tilde{Q}(t)R^{\beta }(\sigma (t)),\hfill \end{array}$ E32

where $\tilde{\gamma }:=k{\gamma }^{\beta }$ . Now, if we define the function

$U(t)=r(t){\left(-z^{\prime }(t)\right)}^{\alpha }>\mathrm{0,}$ E33

then

$U^{\prime }(t)+\frac{p(t)}{r(t)}U(t)\ge \tilde{\gamma }\tilde{Q}(t)R^{\beta }(\sigma (t)),$ E34

or, equally

${\left(\frac{U(t)}{A(t)}\right)}^{\prime }\ge \tilde{\gamma }\frac{\tilde{Q}(t)R^{\beta }(\sigma (t))}{A(t)}.$ E35

Integrating the above inequality from $t_2$ to $t$ , we get

$U(t)\ge \tilde{\gamma }A(t){\displaystyle {\int }_{t_2}^t}\frac{\tilde{Q}(s)R^{\beta }(\sigma (s))}{A(s)}\text{d}s$ E36

or

$r(t){\left(-z^{\prime }(t)\right)}^{\alpha }\ge \tilde{\gamma }A(t){\displaystyle {\int }_{t_2}^t}\frac{\tilde{Q}(s)R^{\beta }(\sigma (s))}{A(s)}\text{d}s.$ E37

It follows from this last inequality that

$0<z(t)\le z(t_2)-\tilde{\gamma }{\displaystyle {\int }_{t_2}^t}{\left(\frac{A(u)}{r(u)}{\displaystyle {\int }_{t_2}^u}\frac{\tilde{Q}(s)R^{\beta }(\sigma (s))}{A(s)}\text{d}s\right)}^{\frac{1}{\alpha }}\text{d}u$ E38

for $t\ge t_2.$ As $t\to \infty$ , then by condition Eq. (21), $z(t)$ approaches to $-\infty$ , which contradicts the fact that $z(t)$ is eventually positive. Therefore, $z^{\prime }(t)>0$ and from Eq. (1), we have ${\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<0$ . The proof is complete.

Lemma 3. Assume that

${\displaystyle {\int }_{t_0}^{\infty }}{\left(\frac{A(u)}{r(u)}{\displaystyle {\int }_{t_0}^u}\frac{\tilde{Q}(s)}{A(s)}\text{d}s\right)}^{\frac{1}{\alpha }}\text{d}u=\infty ,$ E39

holds and Eq. (1) has a positive solution $x(t)$ on $ℐ$ . Then there exists $T\in ℐ$ , sufficiently large, such that either

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}$ E40

on $[T,\infty )$ or $\underset{t\to \infty }{{\displaystyle \lim }}x(t)=0$ .

Proof. As in the proof of Lemma 1, we assume that there exists $t_2\in ℐ$ such that $z^{\prime }(t)<0$ on $[t_2,\infty )$ . So, $z(t)$ is decreasing and

$\underset{t\to \infty }{{\displaystyle \lim }}z(t)=:b\ge 0$ E41

exists. Therefore, there exists $t_3\in [t_2,\infty )$ such that

$z(\sigma (t))>z(t)\ge b>0.$ E42

As in the proof of Lemma 2, we obtain Eq. (27), i.e.,

$\begin{array}{ll}x(\sigma (t))\hfill & \ge \left(1-a(\sigma (t))\frac{R(\tau (\sigma (t)))}{R(\sigma (t))}\right)z(\sigma (t))\hfill \\ \hfill & \ge b\left(1-a(\sigma (t))\frac{R(\tau (\sigma (t)))}{R(\sigma (t))}\right),\text{for}t\ge t_3.\hfill \end{array}$ E43

Thus,

$\begin{array}{lll}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+p(t){\left(z^{\prime }(t)\right)}^{\alpha }\hfill & \le -\tilde{b}q(t){\left(1-a(\sigma (t))\frac{R(\tau (\sigma (t)))}{R(\sigma (t))}\right)}^{\beta }\hfill & =-\tilde{b}\tilde{Q}(t),\hfill \end{array}$ E44

where $\tilde{b}:=k{b}^{\beta }$ .

Define the function $U(t)$ as in Eq. (103). Then Eq. (44) becomes

${\left(\frac{U(t)}{A(t)}\right)}^{\prime }\ge \tilde{b}\frac{\tilde{Q}(t)}{A(t)}.$ E45

Integrating the above inequality twice from $t_3$ to $t$ , one gets

$0<z(t)\le z(t_3)-\tilde{b}{\displaystyle {\int }_{t_3}^t}{\left(\frac{A(u)}{r(u)}{\displaystyle {\int }_{t_3}^u}\frac{\tilde{Q}(s)}{A(s)}\text{d}s\right)}^{\frac{1}{\alpha }}\text{d}u,$ E46

for $t\ge t_3.$ As $t\to \infty$ , then by condition (39), $z(t)$ approaches to $-\infty$ , which contradicts the fact that $z(t)$ is eventually positive. Thus, $b=0$ and from $0\le x(t)\le z(t),$ we see that $\underset{t\to \infty }{{\displaystyle \lim }}x(t)=0$ . The proof is complete.

Using results of Lemmas 1 and 2, we can obtain the following oscillation criteria for Eq. (1).

Theorem 1. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a positive continuously differentiable function $\varphi (t)$ such that, for all sufficiently large, $T$ , $T_1\ge T$ ,

$P(t)\ge 0$ E47

on $[T,\infty )$ and

$\underset{t\to \infty }{{\displaystyle \lim sup}}\{\varphi (t)A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s+{\displaystyle {\int }_{T_1}^t}\left[\varphi (s)Q(s)-\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{\varphi (s)r(\sigma (s)){\left(P(s)\right)}^{\alpha +1}}{{\left(\beta {\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\right]\text{d}s\}=\infty ,$ E48

where

$\psi (t)=\{\begin{array}{ll}{c}_1,\hfill & c_1\text{is}\text{some}\text{positive}\text{constant}\text{if}\beta >\alpha \hfill \\ \mathrm{1,}\hfill & \text{if}\beta =\alpha \hfill \\ c_2{\left({\displaystyle {\int }_T^t}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\frac{\beta -\alpha }{\alpha }},\hfill & c_2\text{is}\text{some}\text{positive}\text{constant}\text{if}\beta <\alpha .\hfill \end{array}$ E49

Then, Eq. (1) is oscillatory.

Proof. Suppose to the contrary that $x(t)$ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists $T\in ℐ$ large enough, so that $x(t)$ satisfies the conclusions of Lemma 1 or 2 on $[T,\infty )$ with

$x(t)>\mathrm{0,}x(\tau (t))>\mathrm{0,}x(\sigma (t))>0$ E50

on $[T,\infty )$ . In particular, we have

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}\text{for}t\ge T.$ E51

By Eq. (2) and the assumption $\text{(iii)}$ , we get

$\begin{array}{ll}x(t)\hfill & =z(t)-a(t)x(\tau (t))\hfill \\ \hfill & \ge z(t)-a(t)z(\tau (t))\hfill \\ \hfill & \ge (1-a(t))z(t),\hfill \end{array}$ E52

which together with Eq. (1) implies

$\begin{array}{ll}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }+\frac{p(t)}{r(t)}{\left(z^{\prime }(t)\right)}^{\alpha }\hfill & \le -kq(t){\left(1-a(\sigma (t))\right)}^{\beta }{z}^{\beta }(\sigma (t))\hfill \\ \hfill & =-Q(t)z^{\beta }(\sigma (t)).\hfill \end{array}$ E53

We consider the generalized Riccati substitution

$w(t)=\varphi (t)\frac{r(t){\left(z^{\prime }(t)\right)}^{\alpha }}{z^{\beta }(\sigma (t))}>\mathrm{0,}\text{for}t\ge T.$ E54

As in the proof of Lemma 1, we get Eq. (18), which in view of the assumption $(\text{v})$ yields

${\left(\frac{r(t)}{A(t)}{\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }\le -\frac{Q(t)}{A(t)}{z}^{\beta }(\sigma (t)).$ E55

Integrating Eq. (55) from $t$ to $\infty$ and using the fact that $z(t)$ is increasing, we have

$\begin{array}{ll}\frac{r(t)}{A(t)}{\left(z^{\prime }(t)\right)}^{\alpha }\hfill & \ge {\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}{z}^{\beta }(\sigma (s))\text{d}s\hfill \\ \hfill & \ge z^{\beta }(\sigma (t)){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s.\hfill \end{array}$ E56

So it follows from Eq. (56) and the definition (54) of $w(t)$ that

$w(t)=\varphi (t)\frac{r(t){\left(z^{\prime }(t)\right)}^{\alpha }}{z^{\beta }(\sigma (t))}\ge \varphi (t)A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s.$ E57

By Eq. (53) we can easily prove that

$\begin{array}{ll}{w}^{\prime }(t)\hfill & ={\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }\frac{\varphi (t)}{z^{\beta }(\sigma (t))}+{\left(\frac{\varphi (t)}{z^{\beta }(\sigma (t))}\right)}^{\prime }r(t){\left(z^{\prime }(t)\right)}^{\alpha }\hfill \\ \hfill & \le -\frac{\varphi (t)}{z^{\beta }(\sigma (t))}\left(p(t){\left(z^{\prime }(t)\right)}^{\beta }+Q(t)z^{\beta }(\sigma (t))\right)\hfill \\ \hfill & +r(t){\left(z^{\prime }(t)\right)}^{\alpha }\left(\frac{{\varphi }^{\prime }(t)}{z^{\beta }(\sigma (t))}-\frac{\varphi (t)\left(z^{\beta }(\sigma (t))\right)}{z^{\beta +1}(\sigma (t))}\right)\hfill \\ \hfill & \le -\varphi (t)Q(t)+w(t)\left(\frac{{\varphi }^{\prime }(t)}{\varphi (t)}-\frac{p(t)}{r(t)}\right)\hfill \\ \hfill & -\beta \varphi (t)\frac{r(t)(z^{\prime }{(t)}^{\beta }{z}^{\prime }(\sigma (t){\sigma }^{\prime }(t)}{z^{\beta +1}(\sigma (t))}.\hfill \end{array}$ E58

On the other hand, since $r(t){\left(z^{\prime }(t)\right)}^{\alpha }$ is decreasing, we have

$\frac{z^{\prime }(\sigma (t))}{z^{\prime }(t)}\ge {\left(\frac{r(t)}{r(\sigma (t))}\right)}^{\frac{1}{\alpha }}$ E59

and thus Eq. (58) becomes

$\begin{array}{ll}{w}^{\prime }(t)\le \hfill & -\varphi (t)Q(t)+P(t)w(t)\hfill \\ \hfill & -\frac{\beta \varphi (t){\sigma }^{\prime }(t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{\left(\frac{w(t)}{\varphi (t)}\right)}^{\frac{\alpha +1}{\alpha }}{z}^{\frac{\beta -\alpha }{\alpha }}(\sigma (t)).\hfill \end{array}$ E60

Now, we consider the following three cases:

Case I: $\beta >\alpha$ .

In this case, since $z^{\prime }(t)>0$ for $t\ge T$ , then there exists $T_1\ge T$ such that $z(\sigma (t))\ge c$ for $t\ge T_1$ . This implies that

$z^{\frac{\beta -\alpha }{\alpha }}(\sigma (t))\ge c^{\frac{\beta -\alpha }{\alpha }}:=c_1$ E61

Case II: $\beta =\alpha$ .

In this case, we see that $z^{\frac{\beta -\alpha }{\alpha }}(\sigma (t))=1.$

Case III: $\beta <\alpha$ .

Since $r(t){\left(z^{\prime }(t)\right)}^{\alpha }$ is decreasing, there exists a constant $d$ such that

$r(t){\left(z^{\prime }(t)\right)}^{\alpha }\le d$ E62

for $t\ge T$ . Integrating the above inequality from $T$ to $t$ , we have

$z(t)\le z(T)+{\displaystyle {\int }_T^t}{\left(\frac{d}{r(s)}\right)}^{\frac{1}{\alpha }}\text{d}s.$ E63

Hence, there exists $T_1\ge T$ and a constant $d_1$ depending on $d$ such that

$z(t)\le d_1{\displaystyle {\int }_T^t}{r}^{-\frac{1}{\alpha }}(s)\text{d}s,\text{for}t\ge T_1$ E64

and thus

$z^{\frac{\beta -\alpha }{\alpha }}(\sigma (t))\ge d_1^{\frac{\beta -\alpha }{\alpha }}{\left({\displaystyle {\int }_T^t}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\frac{\beta -\alpha }{\alpha }}=d_2{\left({\displaystyle {\int }_T^t}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\frac{\beta -\alpha }{\alpha }}$ E65

for some positive constant $d_2$ .

Using these three cases and the definition of $\psi (t)$ , we get

$\begin{array}{l}{w}^{\prime }(t)\le -\varphi (t)Q(t)+P(t)w(t)-\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{{\left(\varphi (t)r(\sigma (t))\right)}^{\frac{1}{\alpha }}}{w}^{\frac{1+\alpha }{\alpha }}(t)\hfill \end{array}$ E66

for $t\ge T_1\ge T$ . Setting

$A:=P(t),$ E67

$B:=\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{{\left(\varphi (t)r(\sigma (t))\right)}^{\frac{1}{\alpha }}},$ E68

and using the inequality

$Au-B{u}^{\frac{1+\alpha }{\alpha }}\le \frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{A^{\alpha +1}}{B^{\alpha }},$ E69

we obtain

$w^{\prime }(t)\le -\varphi (t)Q(t)+\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{\varphi (t)r(\sigma (t)){\left(P(t)\right)}^{\alpha +1}}{{\left(\beta {\sigma }^{\prime }(t)\psi (t)\right)}^{\alpha }}.$ E70

Integrating the above inequality from $T_1$ to $t$ , we have

$w(t)\le w(T_1)-{\displaystyle {\int }_{T_1}^t}\left(\varphi (s)Q(s)-\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{\varphi (s)r(\sigma (s)){\left(P(s)\right)}^{\alpha +1}}{{\left(\beta {\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\right)\text{d}s.$ E71

Taking Eq. (57) into account, we get

$\begin{array}{ll}w(T_1)\hfill & \ge \varphi (t)A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s\hfill \\ \hfill & +{\displaystyle {\int }_{T_1}^t}\left(\varphi (s)Q(s)-\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{\varphi (s)r(\sigma (s)){\left(P(s)\right)}^{\alpha +1}}{{\left(\beta {\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\right)\text{d}s.\hfill \end{array}$ E72

Taking the lim sup on both sides of the above inequality as $t\to \infty$ , we obtain a contradiction to the condition (48). This completes the proof.

Remark 2. Note that the presence of the term $\varphi (t)A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s$ in Eq. (57) improves a number of related results in, e.g., [9, 13–18, 20].

Setting $\varphi (t)=t$ in Eq. (57), then the following corollary becomes immediate.

Corollary 1. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. Assume that, for all sufficiently large, $T$ , $T_1\ge T$ ,

$tp(t)\le r(t)$ E73

on $[T,\infty )$ and

$\begin{array}{ll}\hfill & \underset{t\to \infty }{{\displaystyle \lim sup}}\{tA(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s\hfill \\ \hfill & +{\displaystyle {\int }_{T_1}^t}\left[sQ(s)-\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{sr(\sigma (s)){\left(\frac{1}{s}-\frac{p(s)}{r(s)}\right)}^{\alpha +1}}{{\left(\beta {\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\right]\text{d}s\}=\infty ,\hfill \end{array}$ E74

where $\psi (t)$ is as in Theorem $1$ . Then Eq. (1) is oscillatory.

Corollary 2. Assume that the conditions (39) and (74) hold. Then Eq. (1) is oscillatory or $\underset{t\to \infty }{{\displaystyle \lim }}x(t)=0$ .

Next, we present some complementary oscillation results for Eq. (1) by using an integral averaging technique due to Philos. We need the class of functions ${F}$ . Let

${\mathcal{D}}_0=\left\{(t,s):t>s\ge t_0\right\}\text{and}\mathcal{D}=\left\{(t,s):t>s\ge t_0\right\}$ E75

The function $H(t,s)\in C(\mathcal{D},ℝ)$ is said to belong to a class ${F}$ if

(a) $H(t,t)=0$ for $t\ge T$ , $H(t,s)>0$ for $(t,s)\in {\mathcal{D}}_0$

(b) $H(t,s)$ has a continuous and nonpositive partial derivative on ${\mathcal{D}}_0$ with respect to the second variable such that

$\frac{\partial }{\partial s}\left(H(t,s)\varphi (s)\right)-H(t,s)\frac{\varphi (s)p(s)}{r(s)}=-h(t,s){\left(H(t,s)\varphi (s)\right)}^{\frac{\alpha }{\alpha +1}}$ E76

for all $(t,s)\in {\mathcal{D}}_0$ .

Theorem 2. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. Furthermore, assume that there exist functions $H(t,s)$ , $h(t,s)\in {F}$ such that, for all sufficiently large, $T$ , for $T_1\ge T$ ,

$\begin{array}{ll}\underset{t\to \infty }{{\displaystyle \lim sup}}\frac{1}{H(t,T_1)}{\displaystyle {\int }_{T_1}^t}\hfill & (H(t,s)\left(\varphi (s)Q(s)+\rho (s)\varphi (s)p(s)\right)\hfill \\ \hfill & -\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{h^{\alpha +1}(t,s)r(\sigma (s))}{{\beta }^{\alpha }{\left({\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }})\text{d}s=\infty \hfill \end{array}$ E77

where $\varphi (t)$ and $\rho (t)$ are continuously differentiable functions and $\psi (t)$ is as in Theorem 1. Then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that $x(t)$ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists $T\in ℐ$ large enough, so that $x(t)$ satisfies the conclusions of Lemma 1 or 2 on $[T,\infty )$ with

$x(t)>\mathrm{0,}x(\tau (t))>\mathrm{0,}x(\sigma (t))>0$ E78

on $[T,\infty )$ . In particular, we have

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}\text{for}t\ge T.$ E79

Define the function $w(t)$ as

$w(t)=\varphi (t)r(t)\left(\frac{{\left(z^{\prime }(t)\right)}^{\alpha }}{z^{\beta }(\sigma (t))}+\rho (t)\right)\ge \varphi (t)r(t)\rho (t),$ E80

where $\rho (t)\in C^1(ℐ,ℝ)$ . Similarly to the proof of Theorem 1, we obtain the inequality

$\begin{array}{ll}{w}^{\prime }(t)\le \hfill & -\varphi (t)Q(t)+\varphi (t){\left(r(t)\rho (t)\right)}^{\prime }+\left(\frac{{\varphi }^{\prime }(t)}{\varphi (t)}-\frac{p(t)}{r(t)}\right)w(t)\hfill \\ \hfill & -\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{{\left(\varphi (t)r(\sigma (t))\right)}^{\frac{1}{\alpha }}}{\left(w(t)-\varphi (t)r(t)\rho (t)\right)}^{\frac{1+\alpha }{\alpha }}.\hfill \end{array}$ E81

Multiplying Eq. (81) by $H(t,s)$ , integrating with respect to $s$ from $T_1$ to $t$ for $t\ge T_1\ge T$ , and using $(a)$ and $(b)$ , we find that

$\begin{array}{l}{\displaystyle {\int }_{T_1}^t}H(t,s)\varphi (s){\left(Q(s)-\left(r(s)\rho (s)\right)\right)}^{\prime }\text{d}s\\ \le -{\displaystyle {\int }_T^t}H(t,s)w^{\prime }(s)\text{d}s+{\displaystyle {\int }_{T_1}^t}H(t,s)\left(\frac{{\varphi }^{\prime }(s)}{\varphi (s)}-\frac{p(s)}{r(s)}\right)w(s)\text{d}s\\ -{\displaystyle {\int }_{T_1}^t}\frac{\beta H(t,s){\sigma }^{\prime }(s)\psi (s)}{{\left(\varphi (s)r(\sigma (s))\right)}^{\frac{1}{\alpha }}}{\left(w(s)-\varphi (s)r(s)\rho (s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s\\ =H(t,s)w(s{)}_{|}^{T_1}+{\displaystyle {\int }_{T_1}^t}\left(\frac{\partial }{\partial s}H(t,s)+H(t,s)\left(\frac{{\varphi }^{\prime }(s)}{\varphi (s)}-\frac{p(s)}{r(s)}\right)\right)w(s)\text{d}s\\ -{\displaystyle {\int }_{T_1}^t}\frac{\beta H(t,s){\sigma }^{\prime }(s)\psi (s)}{{\left(\varphi (s)r(\sigma (s))\right)}^{\frac{1}{\alpha }}}{\left(w(s)-\varphi (s)r(s)\rho (s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s\\ =H(t,T_1)w(T_1)+{\displaystyle {\int }_{T_1}^t}-\frac{h(t,s)}{\varphi (s)}{\left(H(t,s)\varphi (s)\right)}^{\frac{\alpha }{\alpha +1}}w(s)\text{d}s\\ -{\displaystyle {\int }_{T_1}^t}\frac{\beta H(t,s){\sigma }^{\prime }(s)\psi (s)}{{\left(\varphi (s)r(\sigma (s))\right)}^{\frac{1}{\alpha }}}{\left(w(s)-\varphi (s)r(s)\rho (s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s\end{array}$ E82

Setting

$A:=-\frac{h(t,s)}{\varphi (s)}{\left[H(t,s)\varphi (s)\right]}^{\frac{\alpha }{\alpha +1}},B:=\frac{\beta H(t,s){\sigma }^{\prime }(s)\psi (s)}{{\left(\varphi (s)r(\sigma (s))\right)}^{\frac{1}{\alpha }}}$ E83

and

$C:=\varphi (s)r(s)\rho (s)$ E84

and using the inequality

$Au-B{(u-C)}^{\frac{1+\alpha }{\alpha }}\le AC+\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{A^{\alpha +1}}{B^{\alpha }},$ E85

we obtain

$\begin{array}{l}{\displaystyle {\int }_{T_1}^t}H(t,s)\varphi (s)\left(Q(s)-{\left(r(s)\rho (s)\right)}^{\prime }\right)\text{d}s\\ \le H(t,T_1)w(T_1)+{\displaystyle {\int }_{T_1}^t}-h(t,s)r(s)\rho (s){\left[H(t,s)\varphi (s)\right]}^{\frac{\alpha }{\alpha +1}}\text{d}s\\ +{\displaystyle {\int }_{T_1}^t}\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{h^{\alpha +1}(t,s)r(\sigma (s))}{{\beta }^{\alpha }{\left({\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\text{d}s\end{array}$ E86

Thus,

$\begin{array}{ll}H(t,T_1)w(T_1)\hfill & \ge {\displaystyle {\int }_{T_1}^t}H(t,s)\varphi (s)\left(Q(s)-{\left(r(s)\rho (s)\right)}^{\prime }\right)\text{d}s\hfill \\ \hfill & +{\displaystyle {\int }_{T_1}^t}h(t,s)r(s)\rho (s){\left[H(t,s)\varphi (s)\right]}^{\frac{\alpha }{\alpha +1}}\text{d}s\hfill \\ \hfill & -{\displaystyle {\int }_{T_1}^t}\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{h^{\alpha +1}(t,s)r(\sigma (s))}{{\beta }^{\alpha }{\left({\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\text{d}s.\hfill \end{array}$ E87

That is,

$\begin{array}{l}H(t,T_1)w(T_1)\\ \ge {\displaystyle {\int }_{T_1}^t}H(t,s)\varphi (s)\left(Q(s)-{\left(r(s)\rho (s)\right)}^{\prime }\right)\text{d}s\\ +{\displaystyle {\int }_{T_1}^t}-r(s)\rho (s)\left(\frac{\partial }{\partial s}\left(H(t,s)\varphi (s)\right)-H(t,s)\frac{\varphi (s)p(s)}{r(s)}\right)\text{d}s\\ -{\displaystyle {\int }_{T_1}^t}\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{h^{\alpha +1}(t,s)r(\sigma (s))}{{\beta }^{\alpha }{\left({\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\text{d}s\\ ={\displaystyle {\int }_{T_1}^t}H(t,s)\left(\varphi (s)Q(s)+\rho (s)\varphi (s)p(s)\right)\text{d}s\\ -H(t,s)\varphi (s)r(s)\rho (s{)}_{|}^{T_1}-{\displaystyle {\int }_{T_1}^t}\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{h^{\alpha +1}(t,s)r(\sigma (s))}{{\beta }^{\alpha }{\left({\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\text{d}s\end{array}$ E88

It follows that

$\begin{array}{ll}\hfill & {\displaystyle {\int }_{T_1}^t}H(t,s)\left(\varphi (s)Q(s)+\rho (s)\varphi (s)p(s)\right)\text{d}s\hfill \\ -\hfill & {\displaystyle {\int }_{T_1}^t}\frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{h^{\alpha +1}(t,s)r(\sigma (s))}{{\beta }^{\alpha }{\left({\sigma }^{\prime }(s)\psi (s)\right)}^{\alpha }}\text{d}s\hfill \\ \hfill & \le H(t,T_1)\left(w(T_1)-\varphi (T_1)r(T_1)\rho (T_1)\right),\hfill \end{array}$ E89

which is a contradiction to Eq. (77). The proof is complete.

Remark 3. Authors in [15, 20] studied a partial case of Eq. (1) by employing the generalized Riccati substitution (80). Note that the function $\rho (t)$ used in the generalized Riccati substitution (80) finally becomes unimportant. Thus, we can put $\rho (t)=0$ and obtain similar results to those from [15, 20].

In the next part, we provide several oscillation results for Eq. (1) under the assumption that the function $P(t)$ is nonpositive. These results generalize those from [10] for Eq. (1) in such sense that $\alpha \ne \beta$ and $p(t)\ne 0$ .

Theorem 3. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a continuously differentiable function $\varphi (t)$ such that, for all sufficiently large, $T$ , $T_1\ge T$ ,

$P(t)\le 0$ E90

on $[T,\infty )$ and

$\begin{array}{l}\underset{t\to \infty }{{\displaystyle \text{lim}\text{sup}}}[\varphi (t)A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s\\ +{\displaystyle {\int }_{T_1}^t}\varphi (s)\left(Q(s)-A(s)P(s){\displaystyle {\int }_s^{\infty }}\frac{Q(u)}{A(u)}\text{d}u\right)\text{d}s]=\infty .\end{array}$ E91

Then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that $x(t)$ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists $T\in ℐ$ large enough, so that $x(t)$ satisfies the conclusions of Lemma 1 or 2 on $[T,\infty )$ with

$x(t)>\mathrm{0,}x(\tau (t))>\mathrm{0,}x(\sigma (t))>0$ E92

on $[T,\infty )$ . In particular, we have

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}\text{for}t\ge T.$ E93

Proceeding as in the proof of Theorem 1, we obtain the inequality (66), i.e.,

$\begin{array}{ll}{w}^{\prime }(t)\le \hfill & -\varphi (t)Q(t)+P(t)w(t)-\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{{\left(\varphi (t)r(\sigma (t))\right)}^{\frac{1}{\alpha }}}{w}^{\frac{1+\alpha }{\alpha }}(t)\hfill \end{array}$ E94

for $t\ge T_1\ge T$ . Using Eq. (90), and setting Eq. (57) in Eq. (94), we get

$\begin{array}{ll}{w}^{\prime }(t)\hfill & \le -\varphi (t)Q(t)+\varphi (t)A(t)P(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s\hfill \\ \hfill & -\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{{\left(\varphi (t)r(\sigma (t))\right)}^{\frac{1}{\alpha }}}{w}^{\frac{1+\alpha }{\alpha }}(t)\hfill \\ \hfill & \le -\varphi (t)Q(t)+\varphi (t)A(t)P(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s,\hfill \end{array}$ E95

that is,

$w^{\prime }(t)+\varphi (t)Q(t)-\varphi (t)A(t)P(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s\le 0.$ E96

Integrating the above inequality from $T_1$ to $t$ , we have

$\begin{array}{ll}w(T_1)\hfill & \ge w(t)+{\displaystyle {\int }_{T_1}^t}\left(\varphi (s)Q(s)-\varphi (s)A(s)P(s){\displaystyle {\int }_s^{\infty }}\frac{Q(u)}{A(u)}\text{d}u\right)\text{d}s\hfill \\ \hfill & \ge \varphi (t)A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s+{\displaystyle {\int }_{T_1}^t}\left(\varphi (s)Q(s)-\varphi (s)A(s)P(s){\displaystyle {\int }_s^{\infty }}\frac{Q(u)}{A(u)}\text{d}u\right)\text{d}s\hfill \end{array}$ E97

Taking the lim sup on both sides of the above inequality as $t\to \infty$ , we obtain a contradiction to condition Eq. (91). This completes the proof.

Setting $\varphi (t)=1$ , we have the following consequence.

Corollary 3. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. Assume that

$\underset{t\to \infty }{{\displaystyle \lim \sup }}\left[A(t){\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s+{\displaystyle {\int }_{T_1}^t}\tilde{q}(s)\text{d}s\right]=\infty ,$ E98

for all sufficiently large $T$ , for $T_1\ge T$ . Then Eq. (1) is oscillatory.

Define a sequence of functions ${\{y_n(t)\}}_{n=0}^{\infty }$ as

$y_0(t)={\displaystyle {\int }_t^{\infty }}\tilde{q}(s)\text{d}s,t\ge T$ E99

$y_n(t)={\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{\left(y_{n-1}(s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s+y_0(t),t\ge T,n=\mathrm{1,2,3,}\dots ,$ E100

for $T\ge t_0$ sufficiently large.

By induction, we can see that $y_n\le y_{n+1}$ , $n=\mathrm{1,2,3,}\dots$ .

Lemma 4. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. Assume that $x(t)$ is a positive solution of Eq. (1) on $ℐ$ . Then there exists $T\in ℐ$ , sufficiently large, such that

$w(t)\ge y_n(t),$ E101

where $w(t)$ and $y_n(t)$ are defined as Eqs. (54) and (100), respectively. Furthermore, there exists a positive function $y(t)$ on $[T_1,\infty )$ , $T_1\ge T$ , such that

$\underset{n\to \infty }{{\displaystyle \lim }}{y}_n(t)=y(t)$ E102

and

$y(t)={\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{\left(y(s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s+y_0(t).$ E103

Proof. Similarly to the proof of Theorem 3, we obtain Eq. (95). Setting $\varphi (t)=1$ in Eq. (95), we get

$w^{\prime }(t)+Q(t)+\frac{p(t)A(t)}{r(t)}{\displaystyle {\int }_t^{\infty }}\frac{Q(s)}{A(s)}\text{d}s+\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{w}^{\frac{1+\alpha }{\alpha }}(t)\le 0$ E104

for $t\ge T_1\ge T$ . Integrating Eq. (104) from $t$ to $t^{\prime }$ , we get

$w(t^{\prime })-w(t)+{\displaystyle {\int }_t^{t^{\prime }}}\tilde{q}(s)\text{d}s+{\displaystyle {\int }_t^{t^{\prime }}}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{w}^{\frac{1+\alpha }{\alpha }}(s)\text{d}s\le 0$ E105

or

$w(t^{\prime })-w(t)+{\displaystyle {\int }_t^{t^{\prime }}}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{w}^{\frac{1+\alpha }{\alpha }}(s)\text{d}s\le 0.$ E106

We assert that

${\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{w}^{\frac{1+\alpha }{\alpha }}(s)\text{d}s<\infty .$ E107

If not, then

$w(t^{\prime })\le w(t)-{\displaystyle {\int }_t^{t^{\prime }}}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{w}^{\frac{1+\alpha }{\alpha }}(s)\text{d}s\to -\infty$ E108

as $t^{\prime }\to \infty$ , which contradicts to the positivity of $w(t)$ and thus the assertion is proved. By Eq. (104), we see that $w(t)$ is decreasing that means

$\underset{t\to \infty }{{\displaystyle \lim }}w(t)=k,k\ge 0.$ E109

By virtue of Eq. (107), we have $k=0$ . Thus, letting $t^{\prime }\to \infty$ in Eq. (105), we get

$\begin{array}{ll}w(t)\hfill & \ge {\displaystyle {\int }_t^{\infty }}\tilde{q}(s)\text{d}s+{\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{w}^{\frac{1+\alpha }{\alpha }}(s)\text{d}s\hfill \\ \hfill & =y_0(t)+{\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{w}^{\frac{1+\alpha }{\alpha }}(s)\text{d}s,\hfill \end{array}$ E110

that is,

$w(t)\ge {\displaystyle {\int }_t^{\infty }}\tilde{q}(s)\text{d}s=y_0(t).$ E111

Moreover, by induction, we have that

$w(t)\ge y_n(t),\text{for}t\ge T_1,n=\mathrm{1,2,3,}\dots .$ E112

Thus, since the sequence ${\{y_n(t)\}}_{n=0}^{\infty }$ is monotone increasing and bounded above, it converges to $y(t)$ . Letting $n\to \infty$ and using Lebesgue monotone convergence theorem in Eq. (100), we get Eq. (103). The proof is complete.

Theorem 4. Let conditions $\text{(i)}$ – $\text{(v)}$ and one of the conditions (14) or (21) hold. If

$\underset{t\to \infty }{{\displaystyle \lim inf}}\left(\frac{1}{y_0(t)}{\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{\left(y_0(s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s\right)>\frac{\alpha }{{(\alpha +1)}^{\frac{1+\alpha }{\alpha }}},$ E113

where $\psi (t)$ is as in Theorem 1, then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that $x(t)$ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists $T\in ℐ$ large enough, so that $x(t)$ satisfies the conclusions of Lemma 1 or 2 on $[T,\infty )$ with

$x(t)>\mathrm{0,}x(\tau (t))>\mathrm{0,}x(\sigma (t))>0$ E114

on $[T,\infty )$ . In particular, we have

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}\text{for}t\ge T.$ E115

By Eq. (113), there exists a constant $\gamma >\frac{\alpha }{{(\alpha +1)}^{\frac{1+\alpha }{\alpha }}}$ such that

$\underset{t\to \infty }{{\displaystyle \lim inf}}\frac{1}{y_0(t)}{\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{\left(y_0(s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s>\gamma .$ E116

Proceeding as in the proof of Lemma 4, we obtain Eq. (110) and from that, we have

$\frac{w(t)}{y_0(t)}\ge 1+\frac{1}{y_0(t)}{\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{\left(y_0(s)\right)}^{\frac{1+\alpha }{\alpha }}{\left(\frac{w(s)}{y_0(s)}\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s$ E117

Let

$\lambda =\underset{t\ge t_1}{{\displaystyle \mathrm{in}f}}\frac{w(t)}{y_0(t)}.$ E118

Then it is easy to see that $\lambda \ge 1$ and

$\lambda \ge 1+{\lambda }^{\frac{1+\alpha }{\alpha }}\gamma ,$ E119

which contradicts the admissible value of $\lambda$ and $\gamma$ , and thus completes the proof.

Theorem 5. Let conditions $\text{(i)}$ – $\text{(v)}$ , one of the conditions (14) or (21) hold, and $y_n(t)$ be defined as in Eq. (100). If there exists some $y_n(t)$ such that, for $T$ sufficiently large,

$\underset{t\to \infty }{{\displaystyle \lim sup}}{y}_n(t){\left({\displaystyle {\int }_T^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\alpha }>\frac{1}{\psi (t)},$ E120

where $\psi (t)$ is as in Theorem 1, then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that $x(t)$ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists $T\in ℐ$ large enough, so that $x(t)$ satisfies the conclusions of Lemma 1 or 2 on $[T,\infty )$ with

$x(t)>\mathrm{0,}x(\tau (t))>\mathrm{0,}x(\sigma (t))>0$ E121

on $[T,\infty )$ . In particular, we have

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}\text{for}t\ge T.$ E122

Proceeding as in the proof of Theorem 3 and using defining $w(t)$ as in Eq. (54), for $T_1\ge T$ , we get

$\begin{array}{ll}\frac{1}{w(t)}\hfill & =\frac{z^{\beta }(\sigma (t))}{r(t){\left(z^{\prime }(t)\right)}^{\alpha }}\hfill \\ \hfill & \ge \frac{\psi (t)}{r(t)}{\left(\frac{z(\sigma (t))}{z^{\prime }(t)}\right)}^{\alpha }\hfill \\ \hfill & =\frac{\psi (t)}{r(t)}{\left(\frac{z(T_1)+{\displaystyle {\int }_{T_1}^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)r^{\frac{1}{\alpha }}(s)z^{\prime }(s)\text{d}s}{z^{\prime }(t)}\right)}^{\alpha }\hfill \\ \hfill & \ge \psi (t){\left({\displaystyle {\int }_{T_1}^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\alpha }\hfill \end{array}$ E123

Thus,

$w(t){\left({\displaystyle {\int }_T^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\alpha }\le \frac{1}{\psi (t)}{\left(\frac{{\displaystyle {\int }_T^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)\text{d}s}{{\displaystyle {\int }_{T_1}^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)\text{d}s}\right)}^{\alpha }$ E124

And therefore,

$\underset{t\to \infty }{{\displaystyle \lim sup}}w(t){\left({\displaystyle {\int }_T^{\sigma (t)}}{r}^{-\frac{1}{\alpha }}(s)\text{d}s\right)}^{\alpha }\le \frac{1}{\psi (t)},$ E125

which contradicts Eq. (120). The proof is complete.

Theorem 6. Let conditions $\text{(i)}$ – $\text{(v)}$ , one of the conditions (14) or (21) hold, and $y_n(t)$ be defined as in Eq. (100). If there exists some $y_n(t)$ such that

${\displaystyle {\int }_{T_1}^{\infty }}\tilde{q}(t)\exp \left({\displaystyle {\int }_{T_1}^t}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{y}_n^{\frac{1}{\alpha }}(s)\text{d}s\right)\text{d}t=\infty$ E126

or

${\displaystyle {\int }_{T_1}^{\infty }}\frac{\beta {\sigma }^{\prime }(t)\psi (t)y_n^{\frac{1}{\alpha }}(t)y_0(t)}{r^{\frac{1}{\alpha }}(\sigma (t))}\exp \left({\displaystyle {\int }_{T_1}^t}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{y}_n^{\frac{1}{\alpha }}(s)\text{d}s\right)\text{d}t=\infty ,$ E127

for $T$ sufficiently large and $T_1\ge T$ , where $\psi (t)$ is as in Theorem 1, then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that $x(t)$ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists $T\in ℐ$ large enough, so that $x(t)$ satisfies the conclusions of Lemma 1 or 2 on $[T,\infty )$ with

$x(t)>\mathrm{0,}x(\tau (t))>\mathrm{0,}x(\sigma (t))>0$ E128

on $[T,\infty )$ . In particular, we have

$z(t)>\mathrm{0,}{z}^{\prime }(t)>\mathrm{0,}{\left(r(t){\left(z^{\prime }(t)\right)}^{\alpha }\right)}^{\prime }<\mathrm{0,}\text{for}t\ge T.$ E129

From Eq. (103), we have

$y^{\prime }(t)=-\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{\left(y(t)\right)}^{\frac{1+\alpha }{\alpha }}-\tilde{q}(t),$ E130

for all $t\ge T_1\ge T$ . Since $y(t)\ge y_n(t)$ , Eq. (130) yields

$y^{\prime }(t)\le -\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{y}_n^{\frac{1}{\alpha }}(t)y(t)-\tilde{q}(t).$ E131

Multiplying the above inequality by the integration factor

$\exp \left({\displaystyle {\int }_{T_1}^t}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{y}_n^{\frac{1}{\alpha }}(s)\text{d}s\right),$ E132

one gets

$\begin{array}{ll}y(t)\le \hfill & \exp \left(-{\displaystyle {\int }_{T_1}^t}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{y}_n^{\frac{1}{\alpha }}(s)\text{d}s\right)\hfill \\ \hfill & \times \left(y(t_1)-{\displaystyle {\int }_{T_1}^t}\tilde{q}(s)\exp \left({\displaystyle {\int }_{T_1}^s}\frac{\beta {\sigma }^{\prime }(u)\psi (u)}{r^{\frac{1}{\alpha }}(\sigma (u))}{y}_n^{\frac{1}{\alpha }}(u)\text{d}u\right)\text{d}s\right),\hfill \end{array}$ E133

from which we have that

${\displaystyle {\int }_{T_1}^t}\tilde{q}(s)\exp \left({\displaystyle {\int }_{T_1}^s}\frac{\beta {\sigma }^{\prime }(u)\psi (u)}{r^{\frac{1}{\alpha }}(\sigma (u))}{y}_n^{\frac{1}{\alpha }}(u)\text{d}u\right)\text{d}s\le y(T_1)<\infty .$ E134

This is a contradiction with Eq. (126).

Now denote

$u(t)={\displaystyle {\int }_t^{\infty }}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{\left(y(s)\right)}^{\frac{1+\alpha }{\alpha }}\text{d}s$ E135

Taking the derivative of $u(t)$ , one gets

$\begin{array}{ll}{u}^{\prime }(t)\hfill & =-\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{\left(y(t)\right)}^{\frac{1+\alpha }{\alpha }}\hfill \\ \hfill & \le -\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{y}_n^{\frac{1}{\alpha }}(t)y(t)\hfill \\ \hfill & =\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{y}_n^{\frac{1}{\alpha }}(t)\left(u(t)+y_0(t)\right)\hfill \end{array}$ E136

Proceeding in a similar manner to that above, we conclude that

${\displaystyle {\int }_{T_1}^{\infty }}\frac{\beta {\sigma }^{\prime }(t)\psi (t)}{r^{\frac{1}{\alpha }}(\sigma (t))}{y}_n^{\frac{1}{\alpha }}(t)y_0(t)\exp \left({\displaystyle {\int }_{T_1}^t}\frac{\beta {\sigma }^{\prime }(s)\psi (s)}{r^{\frac{1}{\alpha }}(\sigma (s))}{y}_n^{\frac{1}{\alpha }}(s)\text{d}s\right)\text{d}t<\infty ,$ E137

which contradicts to Eq. (127). The proof is complete.