On the oscillation of second-order half-linear functional differential equations with mixed neutral term

Abstract

In this article, the authors establish new sufficient conditions for the oscillation of solutions to a class of second-order half-linear functional differential equations with mixed neutral term. The results obtained improve and complement some known results in the relevant literature. Examples illustrating the results are included.

Keywords:

• Oscillation
• second-order
• neutral differential equations

1991 Mathematics Subject Classifications:

• 34K11
• 34K40
• 39A21

1. Introduction

This paper deals with the oscillatory behaviour of solutions to a class of second order half-linear functional differential equations with mixed neutral term of the form (1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) where $z\left(t\right)=x\left(t\right)+p_1\left(t\right)x\left(g_1\right(t\left)\right)+p_2\left(t\right)x\left(g_2\right(t\left)\right)$, α is a quotient of odd positive integers, and the following conditions are always assumed to hold:

(C1)	$r,q:[t_0,\mathrm{\infty })\to (0,\mathrm{\infty })$ are real valued continuous functions with ${\int }_{t_0}^{\mathrm{\infty }}{r}^{-1/\alpha }\left(s\right)\mathrm{d}s=\mathrm{\infty };$
(C2)	$g_1,g_2,h:[t_0,\mathrm{\infty })\to \mathbb{R}$ are real valued continuous functions such that $g_1\left(t\right)<t$, $g_2\left(t\right)>t$, $g_1$ and $g_2$ are strictly increasing, and $\underset{t\to \mathrm{\infty }}{lim}{g}_1\left(t\right)=\underset{t\to \mathrm{\infty }}{lim}{g}_2\left(t\right)=\underset{t\to \mathrm{\infty }}{lim}h\left(t\right)=\mathrm{\infty }$;
(C3ξ)	$p_1,p_2:[t_0,\mathrm{\infty })\to \mathbb{R}$ are real valued continuous functions with $p_1\left(t\right)\ge 0$, $p_2\left(t\right)\ge 1$, and $p_2\left(t\right)\not\equiv 1$ eventually; or
(C3ϕ)	$p_1,p_2:[t_0,\mathrm{\infty })\to \mathbb{R}$ are real valued continuous functions with $p_2\left(t\right)\ge 0$, $p_1\left(t\right)\ge 1$, and $p_1\left(t\right)\not\equiv 1$ eventually.

By a solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) we mean a function $x:[t_x,\mathrm{\infty })\to \mathbb{R}$, $t_x\ge t_0$, such that $z\in C^1\left(\right[t_x,\mathrm{\infty }),\mathbb{R})$, $r(z^{\prime }{)}^{\alpha }\in C^1([t_x,\mathrm{\infty }),\mathbb{R})$, and which satisfies (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) on $[t_x,\mathrm{\infty })$. We consider only those solutions $x\left(t\right)$ of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) that satisfy $sup\left\{\right|x\left(t\right)|:t\ge T\}>0$ for all $T\ge t_x$; moreover, we tacitly assume that (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) possesses such solutions. Such a solution $x\left(t\right)$ of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is said to be oscillatory if it has arbitrarily large zeros on $[t_x,\mathrm{\infty })$; otherwise, it is called nonoscillatory. Equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is said to be oscillatory if all its solutions are oscillatory.

Oscillation and asymptotic behaviour of solutions to various classes of delay and advanced neutral differential and dynamic equations have been widely discussed in the literature; see, for example, [1–19], and the references contained therein.

However, oscillation results for mixed neutral differential and dynamic equations are relatively scarce in the literature; some results can be found, for example, in [20–32], and the references cited therein. We would like to point out that the results obtained in [20–32] require both of $p_1$ and $p_2$ to be constants or bounded functions, and hence, the results established in these papers cannot be applied to the cases where $\underset{t\to \mathrm{\infty }}{lim}{p}_1\left(t\right)=\mathrm{\infty }$ and /or $\underset{t\to \mathrm{\infty }}{lim}{p}_2\left(t\right)=\mathrm{\infty }$. In view of the observations above, we wish to develop new sufficient conditions which can be applied to the cases where $\underset{t\to \mathrm{\infty }}{lim}{p}_1\left(t\right)=\mathrm{\infty }$ and /or $\underset{t\to \mathrm{\infty }}{lim}{p}_2\left(t\right)=\mathrm{\infty }$. In this connection, the results obtained in the present paper are new, improve and complement some existing results in the relevant literature. Furthermore, the results in this paper can easily be extended to more general second-order mixed neutral differential and dynamic equations to obtain more general oscillation results. It is therefore hoped that the present paper will contribute significantly to the study of oscillation of solutions of second-order mixed neutral differential equations.

2. Some preliminary lemmas

In this section, we present some lemmas that will play an important role in establishing our main results. For notational purposes, we let, for any continuous function d, $d_{+}\left(t\right):=max\left\{0,d\left(t\right)\right\},R\left(t,t_1\right):={\int }_{t_1}^t{r}^{-1/\alpha }\left(s\right)\mathrm{d}s,$ and throughout this paper, we define $\begin{array}{rl}\xi \left(t\right)& :=\frac{1}{p_2\left(g_2^{-1}\right(t\left)\right)}\\ & \left[1-\frac{1}{p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}-\frac{p_1\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right],\\ \varphi \left(t\right)& :=\frac{1}{p_1\left(g_1^{-1}\right(t\left)\right)}\\ & \left[1-\frac{1}{p_1\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}\frac{m\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\right.\\ & \left.-\frac{p_2\left(g_1^{-1}\right(t\left)\right)}{p_1\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}\frac{m\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\right]\end{array}$ for all sufficiently large t, where $g_1^{-1}$ and $g_2^{-1}$ denote the inverse functions of $g_1$ and $g_2$, respectively, and m is a function to be specified later.

Lemma 2.1

[33]

If D and E are nonnegative and $\lambda >1,$ then $\lambda D{E}^{\lambda -1}-D^{\lambda }\le \left(\lambda -1\right)E^{\lambda },$ where equality holds if and only if D=E.

Lemma 2.2

Assume that conditions (C1)–$\left({\mathrm{C}3}_{\xi }\right)$ (or (C1), (C2), and $\left({\mathrm{C}3}_{\varphi }\right)$) hold, and let $x\left(t\right)$ be an eventually positive solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Then there exists $t_1\ge t_0$ such that, for $t\ge t_1,$ (2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2)

Proof.

The proof is standard and so we omit the details of its proof.

Lemma 2.3

In addition to conditions $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ (or $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$), assume that $x\left(t\right)$ is an eventually positive solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) such that (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) holds. If there exist a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ and a $t_2\in [t_1,\mathrm{\infty })$ such that (3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) then (4) ${\left(z\left(t\right)/m\left(t\right)\right)}^{\prime }\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(4)

Proof.

Since $r\left(t\right)\left(z^{\prime }\right(t){)}^{\alpha }$ is decreasing on $[t_1,\mathrm{\infty })$, we obtain (5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) Thus, in view of (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) and (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ), we see that (6) $\begin{array}{rl}{\left(\frac{z\left(t\right)}{m\left(t\right)}\right)}^{\prime }& =\frac{z^{\prime }\left(t\right)m\left(t\right)-z\left(t\right)m^{\prime }\left(t\right)}{m^2\left(t\right)}\\ & \le \frac{z\left(t\right)}{m^2\left(t\right)}\left(\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\right)\le 0,\end{array}$(6) for $t\ge t_2$ and for some $t_2\in [t_1,\mathrm{\infty })$. This completes the proof of the lemma.

Lemma 2.4

Let conditions $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold and $\xi \left(t\right)>0$. If $x\left(t\right)$ is an eventually positive solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) such that (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) holds, then $z\left(t\right)$ satisfies the inequality (7) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)\le 0$(7) for sufficiently large t.

Proof.

Let $x\left(t\right)$ be an eventually positive solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\xi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$ and for some $t_1\in [t_0,\mathrm{\infty })$. From the definition of $z\left(t\right)$, (see also (8.6) in [1]), we obtain (8) $\begin{array}{rl}x\left(t\right)& =\frac{1}{p_2\left(g_2^{-1}\right(t\left)\right)}\left(z\left(g_2^{-1}\right(t\left)\right)-x\left(g_2^{-1}\right(t\left)\right)\right.\\ & \left.-p_1\left(g_2^{-1}\right(t\left)\right)x\left(g_1\right(g_2^{-1}\left(t\right)\left)\right)\right)\\ & =\frac{z\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)}\\ & -\frac{\begin{array}{c}z\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)-x\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)\\ -p_1\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)x\left(g_1\right(g_2^{-1}\left(g_2^{-1}\right(t\left)\right)\left)\right)\end{array}}{p_2\left(g_2^{-1}\right(t\left)\right)p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}\\ & -\frac{p_1\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)}\left[\frac{\begin{array}{c}z\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)\\ -x\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)\end{array}}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right.\\ & -\left.\frac{p_1\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)x\left(g_1\right(g_2^{-1}\left(g_1\right(g_2^{-1}\left(t\right)\left)\right)\left)\right)}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right]\\ & \ge \frac{z\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)}-\frac{z\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}\\ & -\frac{p_1\left(g_2^{-1}\right(t\left)\right)z\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}.\end{array}$(8) Using the fact that the functions z, $g_1$ and $g_2$ are strictly increasing, and noting that $g_1\left(t\right)<t<g_2\left(t\right)$, we get (9) $z\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)<z\left(g_2^{-1}\right(t\left)\right)$(9) and (10) $z\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)<z\left(g_2^{-1}\right(t\left)\right).$(10) Using (9(9) $z\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)<z\left(g_2^{-1}\right(t\left)\right)$(9) ) and (10(10) $z\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)<z\left(g_2^{-1}\right(t\left)\right).$(10) ) in (8(8) $\begin{array}{rl}x\left(t\right)& =\frac{1}{p_2\left(g_2^{-1}\right(t\left)\right)}\left(z\left(g_2^{-1}\right(t\left)\right)-x\left(g_2^{-1}\right(t\left)\right)\right.\\ & \left.-p_1\left(g_2^{-1}\right(t\left)\right)x\left(g_1\right(g_2^{-1}\left(t\right)\left)\right)\right)\\ & =\frac{z\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)}\\ & -\frac{\begin{array}{c}z\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)-x\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)\\ -p_1\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)x\left(g_1\right(g_2^{-1}\left(g_2^{-1}\right(t\left)\right)\left)\right)\end{array}}{p_2\left(g_2^{-1}\right(t\left)\right)p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}\\ & -\frac{p_1\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)}\left[\frac{\begin{array}{c}z\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)\\ -x\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)\end{array}}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right.\\ & -\left.\frac{p_1\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)x\left(g_1\right(g_2^{-1}\left(g_1\right(g_2^{-1}\left(t\right)\left)\right)\left)\right)}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right]\\ & \ge \frac{z\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)}-\frac{z\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}\\ & -\frac{p_1\left(g_2^{-1}\right(t\left)\right)z\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}{p_2\left(g_2^{-1}\right(t\left)\right)p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}.\end{array}$(8) ) gives (11) $\begin{array}{rl}x\left(t\right)& \ge \frac{1}{p_2\left(g_2^{-1}\right(t\left)\right)}\left[1-\frac{1}{p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}\right.\\ & \left.-\frac{p_1\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right]z\left(g_2^{-1}\right(t\left)\right)\end{array}$(11) for $t\ge t_1$. Since $\underset{t\to \mathrm{\infty }}{lim}h\left(t\right)=\mathrm{\infty }$, we can choose $t_2\ge t_1$ such that $h\left(t\right)\ge t_1$ for all $t\ge t_2$. Thus, from (11(11) $\begin{array}{rl}x\left(t\right)& \ge \frac{1}{p_2\left(g_2^{-1}\right(t\left)\right)}\left[1-\frac{1}{p_2\left(g_2^{-1}\right(g_2^{-1}\left(t\right)\left)\right)}\right.\\ & \left.-\frac{p_1\left(g_2^{-1}\right(t\left)\right)}{p_2\left(g_2^{-1}\right(g_1\left(g_2^{-1}\right(t\left)\right)\left)\right)}\right]z\left(g_2^{-1}\right(t\left)\right)\end{array}$(11) ) we have (12) $x\left(h\left(t\right)\right)\ge \xi \left(h\left(t\right)\right)z\left(g_2^{-1}\left(h\left(t\right)\right)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(12) Substituting (12(12) $x\left(h\left(t\right)\right)\ge \xi \left(h\left(t\right)\right)z\left(g_2^{-1}\left(h\left(t\right)\right)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(12) ) into (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) gives (7(7) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)\le 0$(7) ) and completes the proof.

Lemma 2.5

Let conditions $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$ hold. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If $x\left(t\right)$ is an eventually positive solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) such that (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) holds, then $z\left(t\right)$ satisfies the inequality (13) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)\le 0$(13) for sufficiently large t.

Proof.

Let $x\left(t\right)$ be an eventually positive solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\varphi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$ and for some $t_1\in [t_0,\mathrm{\infty })$. Following a similar argument as in the proof of Lemma 2.4, we obtain (14) $\begin{array}{rl}x\left(t\right)& \ge \frac{z\left(g_1^{-1}\right(t\left)\right)}{p_1\left(g_1^{-1}\right(t\left)\right)}-\frac{z\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}{p_1\left(g_1^{-1}\right(t\left)\right)p_1\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}\\ & -\frac{p_2\left(g_1^{-1}\right(t\left)\right)z\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}{p_1\left(g_1^{-1}\right(t\left)\right)p_1\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}.\end{array}$(14) Using the fact that $g_1$ and $g_2$ are strictly increasing, and noting that $g_1\left(t\right)<t<g_2\left(t\right)$, we get (15) $g_1^{-1}\left(g_1^{-1}\right(t\left)\right)>g_1^{-1}\left(t\right)$(15) and (16) $g_1^{-1}\left(g_2\right(g_1^{-1}\left(t\right)\left)\right)>g_1^{-1}\left(t\right).$(16) Since (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds, we again have (4(4) ${\left(z\left(t\right)/m\left(t\right)\right)}^{\prime }\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(4) ) holds, i.e. z/m is nonincreasing on $[t_2,\mathrm{\infty })\subseteq [t_1,\mathrm{\infty })$. Thus, we deduce from (15(15) $g_1^{-1}\left(g_1^{-1}\right(t\left)\right)>g_1^{-1}\left(t\right)$(15) ) and (16(16) $g_1^{-1}\left(g_2\right(g_1^{-1}\left(t\right)\left)\right)>g_1^{-1}\left(t\right).$(16) ) that (17) $\begin{array}{rl}& \frac{m\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)z\left(g_1^{-1}\right(t\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\\ & \ge z\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,\end{array}$(17) and (18) $\begin{array}{rl}& \frac{m\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)z\left(g_1^{-1}\right(t\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\\ & \ge z\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,\end{array}$(18) respectively.

Using (17(17) $\begin{array}{rl}& \frac{m\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)z\left(g_1^{-1}\right(t\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\\ & \ge z\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,\end{array}$(17) ) and (18(18) $\begin{array}{rl}& \frac{m\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)z\left(g_1^{-1}\right(t\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\\ & \ge z\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,\end{array}$(18) ) in (14(14) $\begin{array}{rl}x\left(t\right)& \ge \frac{z\left(g_1^{-1}\right(t\left)\right)}{p_1\left(g_1^{-1}\right(t\left)\right)}-\frac{z\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}{p_1\left(g_1^{-1}\right(t\left)\right)p_1\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}\\ & -\frac{p_2\left(g_1^{-1}\right(t\left)\right)z\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}{p_1\left(g_1^{-1}\right(t\left)\right)p_1\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}.\end{array}$(14) ) yields (19) $\begin{array}{rl}x\left(t\right)& \ge \frac{1}{p_1\left(g_1^{-1}\right(t\left)\right)}\left[1-\frac{1}{p_1\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}\right.\\ & \frac{m\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}-\frac{p_2\left(g_1^{-1}\right(t\left)\right)}{p_1\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}\\ & \left.\frac{m\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\right]z\left(g_1^{-1}\right(t\left)\right)\end{array}$(19) for $t\ge t_2$. Since $\underset{t\to \mathrm{\infty }}{lim}h\left(t\right)=\mathrm{\infty }$, we can choose $t_3\ge t_2$ such that $h\left(t\right)\ge t_2$ for all $t\ge t_3$. Thus, from (19(19) $\begin{array}{rl}x\left(t\right)& \ge \frac{1}{p_1\left(g_1^{-1}\right(t\left)\right)}\left[1-\frac{1}{p_1\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}\right.\\ & \frac{m\left(g_1^{-1}\right(g_1^{-1}\left(t\right)\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}-\frac{p_2\left(g_1^{-1}\right(t\left)\right)}{p_1\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}\\ & \left.\frac{m\left(g_1^{-1}\right(g_2\left(g_1^{-1}\right(t\left)\right)\left)\right)}{m\left(g_1^{-1}\right(t\left)\right)}\right]z\left(g_1^{-1}\right(t\left)\right)\end{array}$(19) ) we have (20) $x\left(h\left(t\right)\right)\ge \varphi \left(h\left(t\right)\right)z\left(g_1^{-1}\left(h\left(t\right)\right)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.$(20) Substituting (20(20) $x\left(h\left(t\right)\right)\ge \varphi \left(h\left(t\right)\right)z\left(g_1^{-1}\left(h\left(t\right)\right)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.$(20) ) into (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) gives (13(13) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)\le 0$(13) ) and completes the proof.

3. Main results

In this section, we present some sufficient conditions for the oscillation of all solutions of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). We begin with the following result.

Theorem 3.1

Assume that $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $\xi \left(t\right)>0$ and $g_2\left(t\right)\ge h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in (t_1,\mathrm{\infty })$, (21) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s=\mathrm{\infty },$(21) where ${\chi }_1\left(t\right)=\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)},$ then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Without loss of generality, we may assume that there exists $t_1\in [t_0,\mathrm{\infty })$ such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\xi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$. (The proof if $x\left(t\right)$ is eventually negative is similar, so we omit the details of that case here as well as in the remaining proofs in this paper). Proceeding as in the proofs of Lemmas 2.3 and 2.4, we see that (4(4) ${\left(z\left(t\right)/m\left(t\right)\right)}^{\prime }\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(4) ), (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ) and (7(7) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)\le 0$(7) ) hold on $[t_2,\mathrm{\infty })\subseteq [t_1,\mathrm{\infty })$. Define the Riccati substitution (22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) Clearly $\omega \left(t\right)>0$ for $t\ge t_1$, and from (7(7) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)\le 0$(7) ) we obtain (23) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(23) From $g_2\left(t\right)\ge h\left(t\right)$ and the fact that $g_2$ is strictly increasing, we see that $t\ge g_2^{-1}\left(h\right(t\left)\right)$. Hence, by (4(4) ${\left(z\left(t\right)/m\left(t\right)\right)}^{\prime }\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(4) ) we get (24) $\begin{array}{rl}& \frac{z\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\\ & \ge \frac{m\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(24) Substituting (24(24) $\begin{array}{rl}& \frac{z\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\\ & \ge \frac{m\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(24) ) into (23(23) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(23) ) gives (25) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(25) In view of (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) and (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ), it is easy to see that (26) $\frac{m^{\prime }\left(t\right)}{m\left(t\right)}\ge \frac{1}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}\ge \frac{z^{\prime }\left(t\right)}{z\left(t\right)}.$(26) From (26(26) $\frac{m^{\prime }\left(t\right)}{m\left(t\right)}\ge \frac{1}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}\ge \frac{z^{\prime }\left(t\right)}{z\left(t\right)}.$(26) ), $z\left(t\right)>0$, and $z^{\prime }\left(t\right)>0$, (25(25) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(25) ) yields (27) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)r\left(t\right){\left(m^{\prime }\left(t\right)\right)}^{\alpha }}{m^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\end{array}$(27) for $t\ge t_2$. An integration of (27(27) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)r\left(t\right){\left(m^{\prime }\left(t\right)\right)}^{\alpha }}{m^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\end{array}$(27) ) from $t_2$ to t gives ${\int }_{t_2}^t\left({\chi }_1\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s\le \omega \left(t_2\right),$ which contradicts condition (21(21) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s=\mathrm{\infty },$(21) ). This completes the proof.

Theorem 3.2

Assume that $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $\xi \left(t\right)>0$ and $g_2\left(t\right)\ge h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in (t_1,\mathrm{\infty }),$ (28) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(s\right)}\right)\\ & \mathrm{d}s=\mathrm{\infty },\end{array}$(28) where ${\chi }_1\left(t\right)$ is as in Theorem 3.1, then every solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) oscillates.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Without loss of generality, we may assume that there exists $t_1\in [t_0,\mathrm{\infty })$ such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\xi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$. Proceeding as in the proof of Theorem 3.1, we again arrive at (25(25) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(25) ) for $t\ge t_2$. From (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ) and the definition of ${\chi }_1\left(t\right)$, inequality (25(25) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(25) ) can be written as (29) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(29) Applying Lemma 2.1 with $\lambda :=(\alpha +1)/\alpha$, $\begin{array}{rl}D& =\frac{{\alpha }^{1/\lambda }}{{\left[{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }\right]}^{1/\lambda }}\omega \left(t\right),\mathrm{a}\mathrm{n}\mathrm{d}\\ E& ={\left[\frac{\alpha }{\alpha +1}\frac{{\left[{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }\right]}^{1/\lambda }}{{\alpha }^{1/\lambda }}\frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\right]}^{\alpha },\end{array}$ we obtain (30) $\begin{array}{rl}& \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\lambda }\left(t\right)\\ & \le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\eta }_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(t\right)}.\end{array}$(30) Substituting (30(30) $\begin{array}{rl}& \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\lambda }\left(t\right)\\ & \le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\eta }_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(t\right)}.\end{array}$(30) ) into (29(29) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(29) ) gives ${\omega }^{\prime }\left(t\right)\le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\eta }_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(t\right)}-{\chi }_1\left(t\right).$ Integrating the last inequality from $t_2$ to t yields $\begin{array}{rl}& {\int }_{t_2}^t\left({\chi }_1\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(s\right)}\right)\\ & ds\le \omega \left(t_2\right),\end{array}$ which is a contradiction to our assumption (28(28) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(s\right)}\right)\\ & \mathrm{d}s=\mathrm{\infty },\end{array}$(28) ). This proves the theorem.

Theorem 3.3

Let $\alpha \ge 1$. Assume that $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $\xi \left(t\right)>0$ and $g_2\left(t\right)\ge h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in (t_1,\mathrm{\infty }),$ (31) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{r^{1/\alpha }\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^2}{4\alpha \eta \left(s\right){\left(R\left(s,t_1\right)\right)}^{\alpha -1}}\right)\mathrm{d}s=\mathrm{\infty },$(31) where ${\chi }_1\left(t\right)$ is as in Theorem 3.1, then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) oscillates.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Without loss of generality, we may assume that there exists $t_1\in [t_0,\mathrm{\infty })$ such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\xi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$. Proceeding as in the proof of Theorem 3.2, we again arrive at (29(29) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(29) ) which can be written as (32) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\frac{\alpha {\omega }^{1/\alpha -1}\left(t\right)}{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^2\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(32) From (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ) and (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ), we obtain (33) $\begin{array}{rl}{\left(\omega \left(t\right)\right)}^{1/\alpha -1}& ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z^{\prime }\left(t\right)\right)}^{1-\alpha }}{{\left(z\left(t\right)\right)}^{1-\alpha }}\\ & ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z\left(t\right)\right)}^{\alpha -1}}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha -1}}\\ & \ge {\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(r^{1/\alpha }\left(t\right)R\left(t,t_1\right)\right)}^{\alpha -1}\\ & ={\left(\eta \left(t\right)\right)}^{1/\alpha -1}{\left(R\left(t,t_1\right)\right)}^{\alpha -1}.\end{array}$(33) Using (33(33) $\begin{array}{rl}{\left(\omega \left(t\right)\right)}^{1/\alpha -1}& ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z^{\prime }\left(t\right)\right)}^{1-\alpha }}{{\left(z\left(t\right)\right)}^{1-\alpha }}\\ & ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z\left(t\right)\right)}^{\alpha -1}}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha -1}}\\ & \ge {\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(r^{1/\alpha }\left(t\right)R\left(t,t_1\right)\right)}^{\alpha -1}\\ & ={\left(\eta \left(t\right)\right)}^{1/\alpha -1}{\left(R\left(t,t_1\right)\right)}^{\alpha -1}.\end{array}$(33) ) in (32(32) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\frac{\alpha {\omega }^{1/\alpha -1}\left(t\right)}{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^2\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(32) ), we conclude that (34) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\alpha \frac{{\left(R\left(t,t_1\right)\right)}^{\alpha -1}}{\eta \left(t\right)r^{1/\alpha }\left(t\right)}{\omega }^2\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(34) Completing the square with respect to ω, it follows from (34(34) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-{\chi }_1\left(t\right)\\ & -\alpha \frac{{\left(R\left(t,t_1\right)\right)}^{\alpha -1}}{\eta \left(t\right)r^{1/\alpha }\left(t\right)}{\omega }^2\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(34) ) that (35) ${\omega }^{\prime }\left(t\right)\le -{\chi }_1\left(t\right)+\frac{r^{1/\alpha }\left(t\right)}{4\alpha {\left(R\left(t,t_1\right)\right)}^{\alpha -1}}\frac{{\left({\eta }_{+}^{\prime }\left(t\right)\right)}^2}{\eta \left(t\right)}.$(35) Integrating (35(35) ${\omega }^{\prime }\left(t\right)\le -{\chi }_1\left(t\right)+\frac{r^{1/\alpha }\left(t\right)}{4\alpha {\left(R\left(t,t_1\right)\right)}^{\alpha -1}}\frac{{\left({\eta }_{+}^{\prime }\left(t\right)\right)}^2}{\eta \left(t\right)}.$(35) ) from $t_2$ to t gives ${\int }_{t_2}^t\left({\chi }_1\left(s\right)-\frac{r^{1/\alpha }\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^2}{4\alpha \eta \left(s\right){\left(R\left(s,t_1\right)\right)}^{\alpha -1}}\right)\mathrm{d}s\le \omega \left(t_2\right),$ which contradicts condition (31(31) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{r^{1/\alpha }\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^2}{4\alpha \eta \left(s\right){\left(R\left(s,t_1\right)\right)}^{\alpha -1}}\right)\mathrm{d}s=\mathrm{\infty },$(31) ). This completes the proof of the theorem.

Theorem 3.4

Assume that $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $\xi \left(t\right)>0$ and $g_2\left(t\right)\le h\left(t\right)$. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in (t_1,\mathrm{\infty }),$ (36) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_2\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)}{R^{\alpha }\left(s,t_1\right)}\right)\mathrm{d}s=\mathrm{\infty },$(36) where ${\chi }_2\left(t\right)=\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right),$ then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Without loss of generality, we may assume that there exists $t_1\in [t_0,\mathrm{\infty })$ such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\xi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$. Proceeding exactly as in the proof of Theorem 3.1, we again arrive at (23(23) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(23) ) for $t\ge t_2$. From $g_2\left(t\right)\le h\left(t\right)$ and the fact that $g_2$ is strictly increasing, we see that (37) $t\le g_2^{-1}\left(h\left(t\right)\right),$(37) and so (38) $\frac{z\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\ge 1.$(38) Using (38(38) $\frac{z\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\ge 1.$(38) ) in (23(23) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\xi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_2^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(23) ) yields (39) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}-{\chi }_2\left(t\right)\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(39) Taking into account that (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ) holds, and using the fact that $z^{\prime }\left(t\right)>0$, inequality (39(39) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}-{\chi }_2\left(t\right)\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(39) ) takes the form ${\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{R^{\alpha }\left(t,t_1\right)}-{\chi }_2\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$ The remainder of the proof is similar to that of Theorem 3.1, and so the details are omitted.

Theorem 3.5

Assume that $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $\xi \left(t\right)>0$ and $g_2\left(t\right)\le h\left(t\right)$. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in (t_1,\mathrm{\infty }),$ (40) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_2\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(s\right)}\right)\\ & \mathrm{d}s=\mathrm{\infty },\end{array}$(40) where ${\chi }_2\left(t\right)$ is as in Theorem 3.4, then equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

The proof follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ), (30(30) $\begin{array}{rl}& \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\lambda }\left(t\right)\\ & \le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\eta }_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(t\right)}.\end{array}$(30) ), (39(39) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}-{\chi }_2\left(t\right)\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(39) ), and Theorem 3.2.

Theorem 3.6

Let $\alpha \ge 1$. Assume that $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $\xi \left(t\right)>0$ and $g_2\left(t\right)\le h\left(t\right)$. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in (t_1,\mathrm{\infty }),$ (41) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_2\left(s\right)-\frac{r^{1/\alpha }\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^2}{4\alpha \eta \left(s\right){\left(R\left(s,t_1\right)\right)}^{\alpha -1}}\right)\mathrm{d}s=\mathrm{\infty },$(41) where ${\chi }_2\left(t\right)$ is as in Theorem 3.4, then equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

The proof follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ), (33(33) $\begin{array}{rl}{\left(\omega \left(t\right)\right)}^{1/\alpha -1}& ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z^{\prime }\left(t\right)\right)}^{1-\alpha }}{{\left(z\left(t\right)\right)}^{1-\alpha }}\\ & ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z\left(t\right)\right)}^{\alpha -1}}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha -1}}\\ & \ge {\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(r^{1/\alpha }\left(t\right)R\left(t,t_1\right)\right)}^{\alpha -1}\\ & ={\left(\eta \left(t\right)\right)}^{1/\alpha -1}{\left(R\left(t,t_1\right)\right)}^{\alpha -1}.\end{array}$(33) ), (39(39) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}-{\chi }_2\left(t\right)\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.\end{array}$(39) ), and Theorem 3.3.

Theorem 3.7

Assume that conditions $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$ hold and $g_1\left(t\right)\ge h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in [t_2,\mathrm{\infty })\subseteq (t_1,\mathrm{\infty }),$ (42) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_3\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s=\mathrm{\infty },$(42) where ${\chi }_3\left(t\right)=\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)},$ then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Without loss of generality, we may assume that there exists $t_1\in [t_0,\mathrm{\infty })$ such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\varphi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$. Proceeding as in the proofs of Lemmas 2.3 and 2.5, we see that (4(4) ${\left(z\left(t\right)/m\left(t\right)\right)}^{\prime }\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(4) ), (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ) and (13(13) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)\le 0$(13) ) hold for $t\ge t_3$ and for some $t_3\in [t_2,\mathrm{\infty })\subseteq [t_1,\mathrm{\infty })$. Define again Riccati substitution w by (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ). Then, it follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ) and (13(13) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)\le 0$(13) ) that (43) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(43) Since $g_1\left(t\right)\ge h\left(t\right)$ and $g_1$ is strictly increasing, we see that $t\ge g_1^{-1}\left(h\right(t\left)\right)$. Hence, from (4(4) ${\left(z\left(t\right)/m\left(t\right)\right)}^{\prime }\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(4) ) we get (44) $\frac{z\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\ge \frac{m\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{m\left(t\right)}.$(44) Substituting (44(44) $\frac{z\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\ge \frac{m\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{m\left(t\right)}.$(44) ) into (43(43) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(43) ) gives (45) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(45) The remainder of the proof is similar to that of Theorem 3.1, and so the details are omitted.

Theorem 3.8

Assume that conditions $\left(\mathrm{C}1\right),$ $\left(C2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$ hold and $g_1\left(t\right)\ge h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in [t_2,\mathrm{\infty })\subseteq (t_1,\mathrm{\infty }),$ (46) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_3\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(s\right)}\right)\\ & \mathrm{d}s=\mathrm{\infty },\end{array}$(46) where ${\chi }_3\left(t\right)$ is as in Theorem 3.7, then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) oscillates.

Proof.

The proof follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ), (30(30) $\begin{array}{rl}& \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\lambda }\left(t\right)\\ & \le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\eta }_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(t\right)}.\end{array}$(30) ), (45(45) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(45) ), and Theorem 3.2.

Theorem 3.9

Let $\alpha \ge 1$. Assume that conditions $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$ hold and $g_1\left(t\right)\ge h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in [t_2,\mathrm{\infty })\subseteq (t_1,\mathrm{\infty }),$ (47) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_3\left(s\right)-\frac{r^{1/\alpha }\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^2}{4\alpha \eta \left(s\right){\left(R\left(s,t_1\right)\right)}^{\alpha -1}}\right)\mathrm{d}s=\mathrm{\infty },$(47) where ${\chi }_3\left(t\right)$ is as in Theorem 3.7, then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) oscillates.

Proof.

The proof follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ), (33(33) $\begin{array}{rl}{\left(\omega \left(t\right)\right)}^{1/\alpha -1}& ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z^{\prime }\left(t\right)\right)}^{1-\alpha }}{{\left(z\left(t\right)\right)}^{1-\alpha }}\\ & ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z\left(t\right)\right)}^{\alpha -1}}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha -1}}\\ & \ge {\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(r^{1/\alpha }\left(t\right)R\left(t,t_1\right)\right)}^{\alpha -1}\\ & ={\left(\eta \left(t\right)\right)}^{1/\alpha -1}{\left(R\left(t,t_1\right)\right)}^{\alpha -1}.\end{array}$(33) ), (45(45) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{m^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{m^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(45) ), and Theorem 3.3.

Theorem 3.10

Assume that conditions $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$ hold and $g_1\left(t\right)\le h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in [t_2,\mathrm{\infty })\subseteq (t_1,\mathrm{\infty }),$ (48) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_4\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)}{R^{\alpha }\left(s,t_1\right)}\right)\mathrm{d}s=\mathrm{\infty },$(48) where ${\chi }_4\left(t\right)=\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right),$ then every solution of equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ). Without loss of generality, we may assume that there exists $t_1\in [t_0,\mathrm{\infty })$ such that $x\left(t\right)>0$, $x\left(g_1\right(t\left)\right)>0$, $x\left(g_2\right(t\left)\right)>0$, $x\left(h\right(t\left)\right)>0$, $\varphi \left(t\right)>0$ and $z\left(t\right)$ satisfies (2(2) ${\left(r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }<0,z^{\prime }\left(t\right)>0,\mathrm{a}\mathrm{n}\mathrm{d}z\left(t\right)>0.$(2) ) for $t\ge t_1$. Proceeding as in the proof of Theorem 3.7, we again arrive at (43(43) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(43) ). From $g_1\left(t\right)\le h\left(t\right)$ and the fact that $g_1$ is strictly increasing, we see that (49) $t\le g_1^{-1}\left(h\left(t\right)\right),$(49) and so, from the fact that $z^{\mathrm{\prime }}\left(t\right)>0$, we have (50) $\frac{z\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\ge 1.$(50) Using (50(50) $\frac{z\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z\left(t\right)}\ge 1.$(50) ) in (43(43) $\begin{array}{rl}{\omega }^{\prime }\left(t\right)& \le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)\frac{z^{\alpha }\left(g_1^{-1}\left(h\left(t\right)\right)\right)}{z^{\alpha }\left(t\right)}\\ & -\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(43) ) gives (51) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)-\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}.\end{array}$(51) Taking into account that (5(5) $\begin{array}{rl}z\left(t\right)& =z\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}\mathrm{d}s\\ & \ge r^{1/\alpha }\left(t\right)R\left(t,t_1\right)z^{\prime }\left(t\right).\end{array}$(5) ) holds, and from the fact that $z^{\prime }\left(t\right)>0$, inequality (51(51) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)-\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}.\end{array}$(51) ) takes the form ${\omega }^{\prime }\left(t\right)\le \frac{{\eta }_{+}^{\prime }\left(t\right)}{R^{\alpha }\left(t,t_1\right)}-\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right).$ The remainder of the proof is similar to that of Theorem 3.1, and so the details are omitted.

Theorem 3.11

Assume that conditions $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left({\mathrm{C}3}_{\varphi }\right)$ hold and $g_1\left(t\right)\le h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in [t_2,\mathrm{\infty })\subseteq (t_1,\mathrm{\infty }),$ (52) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_4\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(s\right)}\right)\\ & \mathrm{d}s=\mathrm{\infty },\end{array}$(52) where ${\chi }_4\left(t\right)$ is as in Theorem 3.10, then equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

The proof follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ), (30(30) $\begin{array}{rl}& \frac{{\eta }_{+}^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)-\frac{\alpha }{{\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha }}{\omega }^{\lambda }\left(t\right)\\ & \le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\eta }_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\eta }^{\alpha }\left(t\right)}.\end{array}$(30) ), (51(51) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)-\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}.\end{array}$(51) ), and Theorem 3.2.

Theorem 3.12

Let $\alpha \ge 1$. Assume that conditions $\left(\mathrm{C}1\right),$ $\left(\mathrm{C}2\right),$ and $\left(C{3}_{\varphi }\right)$ hold and $g_1\left(t\right)\le h\left(t\right)$. Assume further that there exists a positive function $m\in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. If there exists a positive function $\eta \in C^1\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ such that, for all sufficiently large $t_1\in [t_0,\mathrm{\infty })$ and for some $T\in [t_2,\mathrm{\infty })\subseteq (t_1,\mathrm{\infty }),$ (53) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_4\left(s\right)-\frac{r^{1/\alpha }\left(s\right){\left({\eta }_{+}^{\prime }\left(s\right)\right)}^2}{4\alpha \eta \left(s\right){\left(R\left(s,t_1\right)\right)}^{\alpha -1}}\right)\mathrm{d}s=\mathrm{\infty },$(53) where ${\chi }_4\left(t\right)$ is as in Theorem 3.10, then equation (1(1) $(r\left(t\right)(z^{\prime }\left(t\right){)}^{\alpha }{)}^{\prime }+q\left(t\right)x^{\alpha }(h\left(t\right))=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

The proof follows from (22(22) $\omega \left(t\right)=\eta \left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1.$(22) ), (33(33) $\begin{array}{rl}{\left(\omega \left(t\right)\right)}^{1/\alpha -1}& ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z^{\prime }\left(t\right)\right)}^{1-\alpha }}{{\left(z\left(t\right)\right)}^{1-\alpha }}\\ & ={\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}\frac{{\left(z\left(t\right)\right)}^{\alpha -1}}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha -1}}\\ & \ge {\left(\eta \left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(r^{1/\alpha }\left(t\right)R\left(t,t_1\right)\right)}^{\alpha -1}\\ & ={\left(\eta \left(t\right)\right)}^{1/\alpha -1}{\left(R\left(t,t_1\right)\right)}^{\alpha -1}.\end{array}$(33) ), (51(51) $\begin{array}{rl}& {\omega }^{\prime }\left(t\right)\le {\eta }_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{\alpha }}{z^{\alpha }\left(t\right)}\\ & -\eta \left(t\right)q\left(t\right){\varphi }^{\alpha }\left(h\left(t\right)\right)-\alpha \eta \left(t\right)r\left(t\right){\left(\frac{z^{\prime }\left(t\right)}{z\left(t\right)}\right)}^{\alpha +1}.\end{array}$(51) ), and Theorem 3.3.

We conclude this paper with the following examples to illustrate the above results. First example is concerned with the case where $p_2\left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$, second example is concerned with the case where $p_1$ and $p_2$ are constants or bounded functions, third example is concerned with the case where $p_1\left(t\right)\to \mathrm{\infty }$ and $p_2\left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$, and fourth example is concerned with the case where $p_1\left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$.

Example 3.13

Consider the neutral differential equation (54) $\begin{array}{rl}& {\left({\left({\left(x\left(t\right)+x\left(\frac{t}{2}\right)+tx\left(2t\right)\right)}^{\prime }\right)}^3\right)}^{\prime }\\ & +\left(t^4+1\right)x^3\left(2t-1\right)=0,t\ge 13.\end{array}$(54) Here we have $\alpha =3$, $r\left(t\right)=p_1\left(t\right)=1$, $p_2\left(t\right)=t$, $q\left(t\right)=t^4+1$, $g_1\left(t\right)=t/2$, $g_2\left(t\right)=2t$ and $h\left(t\right)=2t-1$. It is clear that conditions $\left(C1\right)-\left(C{3}_{\xi }\right)$ hold, $g_2\left(t\right)\ge h\left(t\right)$, and (55) $\xi \left(t\right)=\frac{1}{t/2}\left[1-\frac{1}{t/4}-\frac{1}{t/8}\right]=\frac{2t-24}{t^2}>0.$(55) On the other hand, if we choose $m\left(t\right)=R(t,t_1)$, we see that $m\left(t\right)={\int }_{t_1}^t\frac{1}{r^{1/\alpha }\left(s\right)}\mathrm{d}s={\int }_{13}^t\mathrm{d}s=t-13,$ and so (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds. With $\eta \left(t\right)=t$, condition (21(21) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s=\mathrm{\infty },$(21) ) with T>13 becomes $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_1\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s\\ & =\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left[s(s^4+1){\left(\frac{4s-26}{(2s-1{)}^2}\right)}^3{\left(\frac{s-27/2}{s-13}\right)}^3\right.\\ & \left.-\frac{1}{(s-13{)}^3}\right]\mathrm{d}s=\mathrm{\infty },\end{array}$ due to $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^{t}s(s^4+1){\left(\frac{4s-26}{(2s-1{)}^2}\right)}^3\\ & {\left(\frac{s-27/2}{s-13}\right)}^3\mathrm{d}s=\mathrm{\infty }\end{array}$ and $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\frac{1}{(s-13{)}^3}\mathrm{d}s<\mathrm{\infty }.$ Hence, by Theorem 3.1, every solution of (54(54) $\begin{array}{rl}& {\left({\left({\left(x\left(t\right)+x\left(\frac{t}{2}\right)+tx\left(2t\right)\right)}^{\prime }\right)}^3\right)}^{\prime }\\ & +\left(t^4+1\right)x^3\left(2t-1\right)=0,t\ge 13.\end{array}$(54) ) is oscillatory.

Example 3.14

Consider the neutral differential equation (56) $\begin{array}{rl}& {\left(\frac{1}{t^{1/3}}{\left({\left(x\left(t\right)+2x\left(\frac{t}{3}\right)+8x\left(4t\right)\right)}^{\prime }\right)}^{1/3}\right)}^{\prime }\\ & +\left(t^2+t\right)x^{1/3}\left(5t\right)=0,t\ge 2.\end{array}$(56) Here we have $\alpha =1/3$, $r\left(t\right)=1/t^{1/3}$, $p_1\left(t\right)=2$, $p_2\left(t\right)=8$, $q\left(t\right)=t^2+t$, $g_1\left(t\right)=t/3$, $g_2\left(t\right)=4t$ and $h\left(t\right)=5t$. It is clear that conditions $\left(\mathrm{C}1\right)$–$\left({\mathrm{C}3}_{\xi }\right)$ hold, $g_2\left(t\right)\le h\left(t\right)$, and (57) $\xi \left(t\right)=\frac{1}{8}\left[1-\frac{1}{8}-\frac{2}{8}\right]=\frac{5}{64}>0.$(57) On the other hand, if we choose $m\left(t\right)=R(t,t_1)$, we see that $m\left(t\right)={\int }_{t_1}^t\frac{1}{r^{1/\alpha }\left(s\right)}\mathrm{d}s={\int }_2^{t}s\mathrm{d}s=\frac{t^2-4}{2},$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds. With $\eta \left(t\right)=c>0$ is a constant, condition (36(36) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_2\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)}{R^{\alpha }\left(s,t_1\right)}\right)\mathrm{d}s=\mathrm{\infty },$(36) ) with T>2 becomes $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_2\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)}{R^{\alpha }\left(s,t_1\right)}\right)\mathrm{d}s\\ & =\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^{t}c\left(s^2+s\right){\left(\frac{5}{64}\right)}^{1/3}\mathrm{d}s=\mathrm{\infty },\end{array}$ i.e. condition (36(36) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_2\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)}{R^{\alpha }\left(s,t_1\right)}\right)\mathrm{d}s=\mathrm{\infty },$(36) ) holds. Hence, by Theorem 3.4, every solution of (56(56) $\begin{array}{rl}& {\left(\frac{1}{t^{1/3}}{\left({\left(x\left(t\right)+2x\left(\frac{t}{3}\right)+8x\left(4t\right)\right)}^{\prime }\right)}^{1/3}\right)}^{\prime }\\ & +\left(t^2+t\right)x^{1/3}\left(5t\right)=0,t\ge 2.\end{array}$(56) ) is oscillatory.

Example 3.15

Consider the neutral differential equation (58) $\begin{array}{rl}& {\left({\left({\left(x\left(t\right)+3tx\left(\frac{t}{3}\right)+tx\left(2t\right)\right)}^{\prime }\right)}^3\right)}^{\prime }\\ & +\left(t^5+t\right)x^3\left(\frac{t}{4}\right)=0,t\ge 2.\end{array}$(58) Here we have $\alpha =3$, $r\left(t\right)=1$, $p_1\left(t\right)=3t$, $p_2\left(t\right)=t$, $q\left(t\right)=t^5+t$, $g_1\left(t\right)=t/3$, $g_2\left(t\right)=2t$ and $h\left(t\right)=t/4$. It is clear that conditions $\left(C1\right)-\left(C{3}_{\varphi }\right)$ hold and $g_1\left(t\right)\ge h\left(t\right)$. If we choose $m\left(t\right)=R(t,t_1)$, we see that $m\left(t\right)=t-2$, (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) holds, and (59) $\begin{array}{rl}\varphi \left(t\right)& =\frac{1}{9t}\left[1-\frac{1}{27t}\frac{9t-2}{3t-2}-\frac{3t}{54t}\frac{18t-2}{3t-2}\right]\\ & =\frac{108t^2-120t+4}{1458t^3-972t^2}>0.\end{array}$(59) With $\eta \left(t\right)=t^2$, we see that condition (42(42) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_3\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)r\left(s\right){\left(m^{\prime }\left(s\right)\right)}^{\alpha }}{m^{\alpha }\left(s\right)}\right)\mathrm{d}s=\mathrm{\infty },$(42) ) holds for T>2. Hence, by Theorem 3.7, every solution of (58(58) $\begin{array}{rl}& {\left({\left({\left(x\left(t\right)+3tx\left(\frac{t}{3}\right)+tx\left(2t\right)\right)}^{\prime }\right)}^3\right)}^{\prime }\\ & +\left(t^5+t\right)x^3\left(\frac{t}{4}\right)=0,t\ge 2.\end{array}$(58) ) is oscillatory.

Example 3.16

Consider the neutral differential equation (60) $\begin{array}{rl}& {\left(x\left(t\right)+e^{t}x(t-2\pi )+x(t+\pi )\right)}^{″}\\ & +2e^{t}x\left(t-\frac{\pi }{2}\right)=0,t\ge 5.\end{array}$(60) Here we have $\alpha =1$, $r\left(t\right)=p_2\left(t\right)=1$, $p_1\left(t\right)=e^t$, $q\left(t\right)=2e^t$, $g_1\left(t\right)=t-2\pi$, $g_2\left(t\right)=t+\pi$ and $h\left(t\right)=t-\pi /2$. It is clear that conditions $\left(C1\right)$, $\left(C2\right)$, and $\left(C{3}_{\varphi }\right)$ hold, and $g_1\left(t\right)\le h\left(t\right)$. On the other hand, if we choose $m\left(t\right)=R(t,t_1)$, we see that $m\left(t\right)=t-5$, and so, $\varphi \left(t\right)>0$ and (3(3) $\frac{m\left(t\right)}{r^{1/\alpha }\left(t\right)R\left(t,t_1\right)}-m^{\prime }\left(t\right)\le 0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$(3) ) hold. With $\eta \left(t\right)=c$, it is easy to see that condition (48(48) $\underset{t\to \mathrm{\infty }}{limsup}{\int }_T^t\left({\chi }_4\left(s\right)-\frac{{\eta }_{+}^{\prime }\left(s\right)}{R^{\alpha }\left(s,t_1\right)}\right)\mathrm{d}s=\mathrm{\infty },$(48) ) holds. Hence, by Theorem 3.10, every solution of (60(60) $\begin{array}{rl}& {\left(x\left(t\right)+e^{t}x(t-2\pi )+x(t+\pi )\right)}^{″}\\ & +2e^{t}x\left(t-\frac{\pi }{2}\right)=0,t\ge 5.\end{array}$(60) ) is oscillatory. In fact, $x\left(t\right)=\sin t$ is such a solution.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ercan Tunç  http://orcid.org/0000-0001-8860-608X

Orhan Özdemir  http://orcid.org/0000-0003-1294-5346