Third-Order Neutral Delay Differential Equations with Mixed Nonlinearities: Almost Oscillation via Linearization Method and Arithmetic-Geometric Inequality

1. Introduction

This paper deals with third-order nonlinear neutral delay differential equations with mixed nonlinearities of the form

$${\left(a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)=0,t\ge t_0\ge 0,\left(E\right)$$

where $z\left(t\right)=x\left(t\right)+bx(t-\sigma ).$ In the sequel, the following conditions are assumed without further mention:

(H₁) 

$\alpha ,{\alpha }_1,{\alpha }_2,...,{\alpha }_n$ are ratios of odd positive integers such that ${\alpha }_1>{\alpha }_2>...>{\alpha }_m>\alpha >{\alpha }_{m+1}>...>{\alpha }_n,$ with $\alpha \ge 1,$$b\in [0,\infty ),$$b\ne 1$ and $\sigma \in [0,\infty )$ are constants;

(H₂) 

$a,p_i\in C([t_0,\infty ),(0,\infty ))$ for $i=1,2,3,...,n$ and

$$A(t,t_0)={\int }_{t_0}^t{a}^{-1/\alpha }\left(t\right)dt\mathrm{with}A(t,t_0)\to \infty \mathrm{as}t\to \infty ;$$

(H₃) 

${\tau }_i\in C^1([t_0,\infty ),\mathbb{R})$ with ${\tau }_i\left(t\right)<t$ and ${lim}_{t\to \infty }{\tau }_i\left(t\right)=\infty$ for $i=1,2,3,...,n.$

Let $\tau \left(t\right)=min\{{\tau }_1\left(t\right),{\tau }_2\left(t\right),...,{\tau }_n\left(t\right)\}.$ By a solution of $\left(E\right),$ we mean a function $x\in C([t_x,\infty ),\mathbb{R}),$$t_x=min\{t-\sigma ,\tau \left(t\right)\}$ such that $a{\left(z^{″}\right)}^{\alpha }\in C^{\prime }([t_x,\infty ),\mathbb{R})$ and x satisfies $\left(E\right)$ on $[t_x,\infty ).$ We consider only solutions of $\left(E\right)$ which satisfy $sup\left\{\right|x\left(t\right)|:t\ge T\}>0$ for all $T\in [t_x,\infty )$ and tacitly assume that $\left(E\right)$ possesses such solutions. If such a solution contains an unbounded number of zeros, it is said to be $oscillatory;$ otherwise it is called $nonoscillatory.$ The equation is said to be $almost$$oscillatory$ if its solutions are either oscillatory or tend to zero monotonically.

The theory and applications of neutral type differential equations has drawn great interest over the past four decades since such equations are used to describe a variety of real world problems in physics, engineering, mathematical biology and so on, see, for example [1,2,3]. For recent applications and general theory of these equations, the reader is referred to the monographs [4,5].

The oscillatory character of third-order delay differential equations are peculiar in the sense that they may have both oscillatory and nonoscillatory solutions, or they have only oscillatory solutions. For example, in [6], all the solutions of the third-order delay differential equation

$$x^{‴}\left(t\right)+x(t-\pi )=0,$$

are oscillatory if $\pi e>3.$ However in [3], the third-order delay differential equation

$$x^{‴}\left(t\right)+2x^{\prime }\left(t\right)-x(t-{\displaystyle \frac{3\pi }{2}})=0,$$

has the oscillatory solution $x_1\left(t\right)=sint$ and a nonoscillatory solution $x_2\left(t\right)=exp\left(\beta t\right),$ where $\beta >0$ such that

$${\beta }^3+2\beta -exp\left(-{\displaystyle \frac{3\pi }{2}}\beta \right)=0.$$

Because of the above mentioned behavior of solutions of third-order differential equations, there has been great interest in establishing sufficient conditions for the oscillation or nonoscillation of solutions of different classes of differential equations of third-order, see, for example [3,4,7,8,9,10,11,12,13,14,15,16,17,18,19,20] and the references are contained therein.

Recently in [21], the authors studied the oscillatory behavior of $\left(E\right)$ for the case $n=1$ and ${\alpha }_1=\alpha ,$ and in [22], the authors studied the following equation

$${\left(b_2\left(t\right){\left({\left(b_1\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{{\gamma }_1}\right)}^{\prime }\right)}^{{\gamma }_2}\right)}^{\prime }+\sum _{i=1}^m{q}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)=0,\left(E_1\right)$$

where $z\left(t\right)=x\left(t\right)+bx(t-{\tau }_0),$ and obtained some sufficient conditions which state that every solution of $\left(E_1\right)$ is either oscillatory or tends to zero eventually(almost oscillatory) under the assumption

$${\int }_{t_0}^{\infty }{b}_1^{-1/{\gamma }_1}\left(t\right)dt={\int }_{t_0}^{\infty }{b}_2^{-1/{\gamma }_2}\left(t\right)dt=\infty .$$

Since the positive solution of $\left(E_1\right)$ satisfies the condition

$$z^{\prime }\left(t\right)>0,{\left(b_1\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{{\gamma }_1}\right)}^{\prime }>0,{\left(b_2\left(t\right){\left({\left(b_1\left(t\right){\left(z^{\prime }\left(t\right)\right)}^{{\gamma }_1}\right)}^{\prime }\right)}^{{\gamma }_2}\right)}^{\prime }\le 0$$

and using this the authors infer that $z^{″}\left(t\right)>0$ for $t\ge t_0.$ This is not true in general, for example, if $b_1\left(t\right)=t$ and ${\gamma }_1=1$ then we have $z^{\prime }\left(t\right)+t{z}^{″}\left(t\right)>0,$ and this may not imply that $z^{″}\left(t\right)>0$ for $t\ge t_0.$ However, this is used in [22] to obtain the main results and hence the results in [22] may not be correct unless they have to assume that $b_1\left(t\right)$ is either constant or monotonically decreasing. Note that the authors used the function $b_1\left(t\right)={\displaystyle \frac{1}{t}}$ in their examples which is clearly monotonically decreasing.

Motivated by the above observations and inspired by recent works [21,22], in this study we consider equation $\left(E\right)$ which is same as $\left(E_1\right)$ if ${\gamma }_1=1$ and $b_1\left(t\right)\equiv 1$ and then using linearization method and arithmetic-geometric inequality, we obtain some new criteria for the oscillation and asymptotic behavior of solutions of $\left(E\right).$ This modified and corrected the results in [22]. Examples are provided to illustrate the importance and novelty of the main results.

2. Main Results

We begin with the following preliminary results, which will be used in the proof of the main results.

Lemma 2.1.

Assume that

$${\alpha }_i>\alpha ,i=1,2,3,...,mand{\alpha }_i<\alpha ,i=m+1,m+2,...,n.$$

(2.1)

Then an n-tuple $({\eta }_1,{\eta }_2,...,{\eta }_n)$ exists with ${\eta }_i>0$ satisfying the conditions

$$\sum _{i=1}^n{\alpha }_i{\eta }_i=\alpha and\sum _{i=1}^n{\eta }_i=1.$$

(2.2)

Proof. 

From (2.2), we see that

$$\sum _{i=1}^n{\beta }_i{\eta }_i=1and\sum _{i=1}^n{\eta }_i=1$$

where ${\beta }_i={\displaystyle \frac{{\alpha }_i}{\alpha }}.$ The rest of the proof is similar to Lemma 1 of [23] and hence the details are omitted. □

Lemma 2.2

([5], Lemma 1.5.1). Let $h,g:[t_0,\infty )\to \mathbb{R}$ such that $h\left(t\right)=g\left(t\right)+bg(t-c),$$t\ge t_0+max\{0,c\},$ where $p\ne 1$ and c are constants. Assume that there exists a constant $l\in \mathbb{R}$ such that ${lim}_{t\to \infty }h\left(t\right)=l.$

(i)

If ${lim}_{t\to \infty }infg\left(t\right)=g_{*}\in \mathbb{R},$ then $l=(1+b)g_{*};$

(ii)

If ${lim}_{t\to \infty }supg\left(t\right)=g^{*}\in \mathbb{R},$ then $l=(1+b)g^{*}.$

Lemma 2.3

([21], Lemma 1). Let $x\left(t\right)$ be an eventually positive solution of equation $\left(E\right).$ Then there exists a sufficiently large $t_1\ge t_0$ such that, for all $t\ge t_1$ either

(I)

$z\left(t\right)>0,$$z^{\prime }\left(t\right)>0,$$z^{″}\left(t\right)>0,$${\left(a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le 0,$

(II)

$z\left(t\right)>0,$$z^{\prime }\left(t\right)<0,$$z^{″}\left(t\right)>0,$${\left(a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le 0.$

Lemma 2.4.

Let $x\left(t\right)$ be an eventually positive solution of equation $\left(E\right)$ and assume that Case (II) of Lemma 2.3 holds. If

$${\int }_{t_0}^{\infty }\xi {\left({\displaystyle \frac{1}{a\left(\xi \right)}}{\int }_{\xi }^{\infty }Q\left(s\right)ds\right)}^{1/\alpha }d\xi =\infty ,$$

(2.3)

where

$$Q\left(t\right)=\prod _{i=1}^n{\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right)}^{{\eta }_i},$$

(2.4)

with ${\eta }_i$ defined as in Lemma 2.1, then

$$\underset{t\to \infty }{lim}x\left(t\right)=0.$$

(2.5)

Proof. 

Since $z\left(t\right)>0$ and $z^{\prime }\left(t\right)<0,$ there exists a constant $M>0$ such that ${lim}_{t\to \infty }z\left(t\right)=M\ge 0.$ We claim that $M=0.$ If not, then by using Lemma 2.2, we see that ${lim}_{t\to \infty }x\left(t\right)={\displaystyle \frac{M}{1+b}}>0.$ Then there exists $t_1\ge t_0$ such that for all $t\ge t_1,$ we have

$$x\left({\tau }_i\left(t\right)\right)\ge {\displaystyle \frac{M}{2(1+b)}},i=1,2,3,...,n.$$

Using the last inequality, we see that

$$\begin{array}{ccc}\hfill \sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)& \ge & \sum _{i=1}^n{p}_i\left(t\right){\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{{\alpha }_i}\hfill \\ & =& {\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{\alpha }\sum _{i=1}^n{p}_i\left(t\right){\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{{\alpha }_i-\alpha }.\hfill \end{array}$$

(2.6)

By Lemma 2.2, there exists ${\eta }_1,{\eta }_2,...,{\eta }_n$ with

$$\sum _{i=1}^n{\alpha }_i{\eta }_i-\alpha \sum _{i=1}^n{\eta }_i=0.$$

The arithmetic-geometric mean inequality (see [24]) leads to

$$\sum _{i=1}^n{\eta }_i{u}_i\ge \prod _{i=1}^n{u}_i^{{\eta }_i},\mathrm{for}\mathrm{any}{u}_i\ge 0,i=1,2,3,...,n.$$

In view of the above inequality, we obtain

$$\begin{array}{ccc}\hfill \sum _{i=1}^n{p}_i\left(t\right){\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{{\alpha }_i-\alpha }& =& \sum _{i=1}^n{\eta }_i\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right){\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{{\alpha }_i-\alpha }\hfill \\ & \ge & \prod _{i=1}^n{\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right)}^{{\eta }_i}{\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{{\eta }_i({\alpha }_i-\alpha )}\hfill \\ & =& \prod _{i=1}^n{\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right)}^{{\eta }_i}=Q\left(t\right).\hfill \end{array}$$

This together with (2.6) yields that

$$\begin{array}{c}\hfill \sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)\ge Q\left(t\right){\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{\alpha }.\end{array}$$

(2.7)

Combining $\left(E\right)$ and (2.7), we take

$${\left(a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+{\left({\displaystyle \frac{M}{2(1+b)}}\right)}^{\alpha }Q\left(t\right)\le 0.$$

Further note that there exist constants $M_1$ and $M_2$ such that ${lim}_{t\to \infty }a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }=M_1\ge 0$ and ${lim}_{t\to \infty }{z}^{\prime }\left(t\right)=M_2\le 0.$

Now a method similar to that in Theorem 15 of [16] leads to the conclusion that ${lim}_{t\to \infty }x\left(t\right)=0.$ This completes the proof. □

Lemma 2.5.

Let $x\left(t\right)$ be a positive solution of $\left(E\right)$ with corresponding function $z\left(t\right)\in$ class (I) for all $t\ge t_1.$ Then

(i)

$z^{\prime }\left(t\right)\ge \left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)A(t,t_1),$

(ii)

${\displaystyle \frac{z^{\prime }\left(t\right)}{A(t,t_1)}}$ is decreasing,

(iii)

$z\left(t\right)\ge \left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)A_1(t,t_1),$

(iv)

$z\left(t\right)\ge A_1(t,t_1){\displaystyle \frac{z^{\prime }\left(t\right)}{A(t,t_1)}},$

(ii)

${\displaystyle \frac{z\left(t\right)}{A_1(t,t_1)}}$ is decreasing,

where $A_1(t,t_1)={\int }_{t_1}^{t}A(s,t_1)ds.$

Proof. 

Since $z\left(t\right)\in$ class (I), we see that $a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }>0$ and decreasing for all $t\ge t_1.$ Then

$$\begin{array}{ccc}\hfill z^{\prime }\left(t\right)\ge z^{\prime }\left(t\right)-z^{\prime }\left(t_1\right)& =& {\int }_{t_1}^t{\displaystyle \frac{{\left(a\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{a^{1/\alpha }\left(s\right)}}ds\hfill \\ & \ge & A(t,t_1)a^{1/\alpha }\left(t\right)z^{″}\left(t\right),\hfill \end{array}$$

(2.8)

which proves (i).

Moreover

$${\left({\displaystyle \frac{z^{\prime }\left(t\right)}{A(t,t_1)}}\right)}^{\prime }={\displaystyle \frac{A(t,t_1)a^{1/\alpha }\left(t\right)z^{″}\left(t\right)-z^{\prime }\left(t\right)}{a^{1/\alpha }\left(t\right)A^2(t,t_1)}}\le 0,$$

which implies that ${\displaystyle \frac{z^{\prime }\left(t\right)}{A(t,t_1)}}$ is decreasing.

Integrating (2.8) from $t_1$ to t yields

$$z\left(t\right)\ge A_1(t,t_1)a^{1/\alpha }\left(t\right)z^{″}\left(t\right),$$

which proves (iii).

Since

$$z\left(t\right)-z\left(t_1\right)={\int }_{t_1}^{t}A(s,t_1){\displaystyle \frac{z^{\prime }\left(s\right)}{A(s,t_1)}}ds,$$

or

$$z\left(t\right)\ge A_1(t,t_1){\displaystyle \frac{z^{\prime }\left(t\right)}{A(t,t_1)}},$$

where we have used (ii). This proves (iv).

Finally,

$${\left({\displaystyle \frac{z\left(t\right)}{A_1(t,t_1)}}\right)}^{\prime }={\displaystyle \frac{A_1(t,t_1)z^{\prime }\left(t\right)-A(t,t_1)z\left(t\right)}{A_1^2(t,t_1)}}\le 0$$

by (iv). Hence ${\displaystyle \frac{z\left(t\right)}{A_1(t,t_1)}}$ is decreasing. This completes the proof. □

Next, we state and prove the main theorems.

Theorem 2.6.

Let condition (2.3) holds. If the first-order delay differential equation

$$w^{\prime }\left(t\right)+{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }{Q}_1\left(t\right)w\left(\tau \left(t\right)\right)=0,$$

(2.9)

where

$$Q_1\left(t\right)=Q\left(t\right)A_1^{\alpha }(\tau \left(t\right),t_2),$$

with $Q\left(t\right)$ defined as in (2.4), is oscillatory for all large $t_1\ge t_0$ and for some $t_2\ge t_1$, then the equation $\left(E\right)$ is almost oscillatory.

Proof. 

Let $x\left(t\right)$ be a nonoscillatory solution of $\left(E\right).$ Then with no loss of generality, assume $x\left(t\right)>0,$$x(t-\sigma )>0$ and $x\left({\tau }_i\left(t\right)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0.$ Then from Lemma 2.3 that the corresponding function $z\left(t\right)>0$ for all $t\ge t_1$ and satisfies either $\left(I\right)$ or $\left(II\right)$. If $z\left(t\right)$ satisfies case $\left(II\right)$, then from Lemma 2.4 that (2.5) holds, and we need to consider the other case (I).

From $\left(E\right)$, we see that

$$\begin{array}{ccc}\hfill {\left(a\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }& =& {\left({\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & =& \alpha {\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\alpha -1}{\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }\hfill \\ & =& -\sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)\hfill \end{array}$$

and so

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha }}{\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{1-\alpha }\sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)=0.$$

(2.10)

Since $z^{\prime }\left(t\right)>0$ and $z^{″}\left(t\right)>0,$ there exists a constant $d_0$(it is also possible that $d_0=\infty$) such that ${lim}_{t\to \infty }{z}^{\prime }\left(t\right)=d_0>0.$ Consequently, by Lemma 2.2, ${lim}_{t\to \infty }{x}^{\prime }\left(t\right)={\displaystyle \frac{d_0}{1+b}}>0,$ and we conclude that

$$x^{\prime }\left(t\right)>0.$$

(2.11)

Using (2.11), we see that $z\left(t\right)=x\left(t\right)+bx(t-\sigma )\le (1+b)x\left(t\right)$, that is,

$$x\left(t\right)\ge \left({\displaystyle \frac{1}{1+b}}\right)z\left(t\right).$$

(2.12)

In addition, we have

$$\begin{array}{ccc}\hfill \sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)& \ge & \sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left(\tau \left(t\right)\right)\hfill \\ & =& x^{\alpha }\left(\tau \left(t\right)\right)\sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i-\alpha }\left(\tau \left(t\right)\right).\hfill \end{array}$$

(2.13)

In view of Lemma 2.1, there exists ${\eta }_1,{\eta }_2,...,{\eta }_n$ with

$$\sum _{i=1}^n{\alpha }_i{\eta }_i-\alpha \sum _{i=1}^n{\eta }_1=0.$$

The arithmetic - geometric mean inequality (see [24]) gives

$$\sum _{i=1}^n{\eta }_i{u}_i\ge \prod _{i=1}^n{u}_i^{{\eta }_i}\mathrm{for}\mathrm{any}{u}_i\ge 0,i=1,2,...,n.$$

Therefore, we have

$$\begin{array}{ccc}\hfill \sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i-\alpha }\left(\tau \left(t\right)\right)& =& \sum _{i=1}^n{\eta }_i\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right)x^{{\alpha }_i-\alpha }\left(\tau \left(t\right)\right)\hfill \\ & \ge & \prod _{i=1}^n{\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right)}^{{\eta }_i}{\left(x\left(\tau \right(t\left)\right)\right)}^{{\eta }_i({\alpha }_i-\alpha )}\hfill \\ & =& \prod _{i=1}^n{\left({\displaystyle \frac{p_i\left(t\right)}{{\eta }_i}}\right)}^{{\eta }_i}=Q\left(t\right).\hfill \end{array}$$

This together with (2.13) yields that

$$\sum _{i=1}^n{p}_i\left(t\right)x^{{\alpha }_i}\left({\tau }_i\left(t\right)\right)\ge Q\left(t\right)x^{\alpha }\left(\tau \left(t\right)\right).$$

(2.14)

Using (2.12), (2.14) in (2.10), we obtain

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha }}{\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{1-\alpha }Q\left(t\right){\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }{z}^{\alpha }\left(\tau \left(t\right)\right)\le 0.$$

(2.15)

From Lemma 2.5(iii), we see that

$$z\left(\tau \left(t\right)\right)\ge A_1(\tau \left(t\right),t_1)\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)$$

(2.16)

for $t\ge t_1.$ Since $a^{1/\alpha }\left(t\right)z^{″}\left(t\right)$ is nonincreasing and $\alpha \ge 1,$ we have

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{1-\alpha }\ge {\left(a^{1/\alpha }\left(\tau \left(t\right)\right)z^{″}\left(\tau \left(t\right)\right)\right)}^{1-\alpha }.$$

(2.17)

Using (2.17) in (2.15) yields

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }{\left(a^{1/\alpha }\left(\tau \left(t\right)\right)z^{″}\left(t\right)\right)}^{1-\alpha }Q\left(t\right)z^{\alpha }\left(\tau \left(t\right)\right)\le 0.$$

(2.18)

From (2.16) and (2.18), we observe that

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }Q\left(t\right)A_1^{\alpha }(\tau \left(t\right),t_1)\left(a^{1/\alpha }\left(\tau \left(t\right)\right)z^{″}\left(t\right)\right)\le 0.$$

(2.19)

Let $w\left(t\right)=a^{1/\alpha }\left(t\right)z^{″}\left(t\right)$ in (2.19), we see that w is a positive solution of the first-order linear delay differential inequality

$$w^{\prime }\left(t\right)+{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }{Q}_1\left(t\right)w\left(\tau \left(t\right)\right)\le 0.$$

The function w is clearly strictly decreasing for all $t\ge t_2$ and so by Theorem 1 of [25], there exists a positive solution of the equation (2.9), which contradicts the fact that the equation (2.9) is oscillatory. The proof of the theorem is complete. □

The next result immediately follows from Theorem 2.6 and [Theorem 2.11, [14].

Corollary 2.7.

Let condition (2.3) holds. If

$$\underset{t\to \infty }{lim}inf{\int }_{\tau \left(s\right)}^t{Q}_1\left(s\right)ds\ge {\displaystyle \frac{\alpha {(1+b)}^{\alpha }}{e}}$$

(2.20)

where $Q_1\left(t\right)$ is defined as in Theorem 2.6, then the equation $\left(E\right)$ is almost oscillatory.

In our next theorem we use Riccati transformation and integral averaging technique to obtain oscillation results.

Theorem 2.8.

Let condition (2.3) holds and $\tau \left(t\right)\in C^{\prime }\left([t_0,\infty )\right)$ with ${\tau }^{\prime }\left(t\right)>0.$ Assume that there exists a function $\rho \in C^{\prime }([t_0,\infty ),(0,\infty )),$ for sufficiently large $t_1\ge t_0,$ there is a $t_2\ge t_1$ such that

$$\underset{t\to \infty }{lim}sup{\int }_{t_2}^t\left[\rho \left(s\right)Q_2\left(s\right)-{\displaystyle \frac{\alpha {(1+b)}^{\alpha }{\left({\rho }^{\prime }\left(s\right)\right)}^2}{4\rho \left(s\right)A(\tau \left(s\right),t_1){\tau }^{\prime }\left(s\right)}}\right]ds=\infty ,$$

(2.21)

where $Q_2\left(t\right)=Q\left(t\right){(A_1\left(\tau \left(t\right)\right),t_1)}^{\alpha -1}$ with $Q\left(t\right)$ defined as in (2.4) and ${\left({\rho }^{\prime }\left(t\right)\right)}_{+}=max\{0,{\rho }^{\prime }\left(t\right)\}.$ Then the equation $\left(E\right)$ is almost oscillatory.

Proof. 

Let $x\left(t\right)$ be a nonoscillatory solution of $\left(E\right).$ Then with no loss of generality, assume $x\left(t\right)>0,$$x(t-\sigma )>0$ and $x\left({\tau }_i\left(t\right)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0.$ Then from Lemma 2.3, we see that the corresponding function $z\left(t\right)>0$ and satisfies either case $\left(I\right)$ or case $\left(II\right)$ for all $t\ge t_1.$ If $z\left(t\right)$ satisfies case $\left(II\right)$ then from Lemma 2.4 that (2.5) holds, and we need to consider the other case (I). It follows from (2.16), and the fact that $a^{1/\alpha }\left(t\right)z^{″}\left(t\right)$ is nonincreasing that

$$\begin{array}{ccc}\hfill a^{1/\alpha }\left(t\right)z^{″}\left(t\right)& \le & a^{1/\alpha }\left(\tau \left(t\right)\right)z^{″}\left(t\right)\le A_1{(\tau \left(t\right),t_1)}^{-1}z\left(\tau \left(t\right)\right)\hfill \end{array}$$

and so

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{1-\alpha }\ge (A_1{(\tau \left(t\right),t_1)}^{\alpha -1}{\left(z\left(\tau \left(t\right)\right)\right)}^{1-\alpha }.$$

Using this inequality in (2.18) yields

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }{Q}_2\left(t\right)z\left(\tau \left(t\right)\right)\le 0,t\ge t_1.$$

(2.22)

Define

$$w\left(t\right)={\displaystyle \frac{\rho \left(t\right)a^{1/\alpha }\left(t\right)z^{″}\left(t\right)}{z\left(\tau \right(t\left)\right)}}.$$

Then $w\left(t\right)>0$ and using (2.22), we obtain

$$w^{\prime }\left(t\right)\le {\displaystyle \frac{{\rho }^{\prime }\left(t\right)}{\rho \left(t\right)}}w\left(t\right)-{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }\rho \left(t\right)Q_2\left(t\right)-{\displaystyle \frac{w\left(t\right)z^{\prime }\left(\tau \left(t\right)\right){\tau }^{\prime }\left(t\right)}{z\left(\tau \right(t\left)\right)}}.$$

(2.23)

From Lemma 2.5(i), we see that

$$z^{\prime }\left(\tau \left(t\right)\right)\ge \left(a^{1/\alpha }\left(\tau \left(t\right)\right)z^{″}\left(\tau \left(t\right)\right)\right)A(\tau \left(t\right),t_1)\ge \left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)A(\tau \left(t\right),t_1).$$

Combining the last inequality with (2.23), we obtain

$$w^{\prime }\left(t\right)\le {\displaystyle \frac{-1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }\rho \left(t\right)Q_2\left(t\right)+{\displaystyle \frac{{\rho }^{\prime }\left(t\right)}{\rho \left(t\right)}}w\left(t\right)-{\displaystyle \frac{A(\tau \left(t\right),t_1){\tau }^{\prime }\left(t\right)}{\rho \left(t\right)}}{w}^2\left(t\right).$$

(2.24)

Using the last inequality $Bu-A{u}^2\le {\displaystyle \frac{1}{4}}{\displaystyle \frac{B^2}{A}},$$A>0$ in (2.24), we have

$$w^{\prime }\left(t\right)\le {\displaystyle \frac{-1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }\rho \left(t\right)Q_2\left(t\right)+{\displaystyle \frac{{\left({\rho }^{\prime }\left(t\right)\right)}^2}{4\rho \left(t\right)A(\tau \left(t\right),t_1){\tau }^{\prime }\left(t\right)}},t\ge t_2\ge t_1.$$

Integrating from $t_2$ to $t,$ we get

$${\int }_{t_2}^t\left[{\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{1}{1+b}}\right)}^{\alpha }\rho \left(s\right)Q_2\left(s\right)-{\displaystyle \frac{{\left({\rho }^{\prime }\left(s\right)\right)}^2}{4\rho \left(s\right)A(\tau \left(s\right),t_1){\tau }^{\prime }\left(s\right)}}\right]ds\le w\left(t_2\right)$$

which contradicts (2.21). The proof of the theorem is complete. □

Theorem 2.9.

Let condition (2.3) holds and

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}sup\left\{{\displaystyle \frac{1}{A(\tau \left(t\right),t_{*})}}{\int }_{t_{*}}^{\tau \left(t\right)}{Q}_2\left(s\right)A(s,t_{*})A_1(\tau \left(s\right),t_{*})ds+{\int }_{\tau \left(t\right)}^t{Q}_2\left(s\right)A_1(\tau \left(s\right),t_{*})ds\right.\\ \hfill \left.+A_1(\tau \left(s\right),t_{*}){\int }_t^{\infty }{Q}_2\left(s\right)ds\right\}>\alpha {(1+b)}^{\alpha },\end{array}$$

(2.25)

where $Q_2\left(t\right)$ is as defined in Theorem 2.8. Then the equation $\left(E\right)$ is almost oscillatory.

Proof. 

Let $x\left(t\right)$ be a positive solution of $\left(E\right)$ with $x(t-\sigma )>0$ and $x\left({\tau }_i\left(t\right)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0.$ Then the corresponding function $z\left(t\right)>0$ and satisfies case $\left(I\right)$ or case $\left(II\right)$ of Lemma 2.3 for all $t\ge t_1.$ If $z\left(t\right)$ satisfies case $\left(II\right)$ then from Lemma 2.4 that (2.5) holds, and therefore we need to consider the other case (I). Proceeding as in the proof of Theorem 2.8, we arrive at (2.22).

Integrating (2.22) from t to ∞ yields

$$\begin{array}{c}\hfill z^{″}\left(t\right)\ge {\displaystyle \frac{1}{\alpha {(1+b)}^{\alpha }}}{\displaystyle \frac{1}{a^{1/\alpha }\left(t\right)}}{\int }_t^{\infty }{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds.\end{array}$$

Integrating again from $t_1$ to t, we get

$$\begin{array}{ccc}\hfill \alpha {(1+b)}^{\alpha }{z}^{\prime }\left(t\right)& \ge & {\int }_{t_1}^t{\displaystyle \frac{1}{a^{1/\alpha }\left(u\right)}}{\int }_u^{\infty }{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds\hfill \\ & =& {\int }_{t_1}^t{\displaystyle \frac{1}{a^{1/\alpha }\left(u\right)}}{\int }_u^t{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds+{\int }_{t_1}^t{\displaystyle \frac{1}{a^{1/\alpha }\left(u\right)}}{\int }_t^{\infty }{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds\hfill \\ & =& {\int }_{t_1}^{t}A(u,t_1)Q_2\left(s\right)z\left(\tau \left(s\right)\right)ds+A(t,t_1){\int }_t^{\infty }{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds.\hfill \end{array}$$

Employing Lemma 2.5(iv), we have

$$\begin{array}{ccc}\hfill \alpha {(1+b)}^{\alpha }{\displaystyle \frac{z\left(t\right)A(t,t_1)}{A_1(t,t_1)}}& \ge & {\int }_{t_1}^{t}A(s,t_1)Q_2\left(s\right)z\left(\tau \left(s\right)\right)ds+A(t,t_1){\int }_t^{\infty }{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds,\hfill \end{array}$$

or

$$\begin{array}{ccc}\hfill \alpha {(1+b)}^{\alpha }{\displaystyle \frac{z\left(\tau \left(t\right)\right)A(\tau \left(t\right),t_1)}{A_1(\tau \left(t\right),t_1)}}& \ge & {\int }_{t_1}^{\tau \left(t\right)}A(s,t_1)Q_2\left(s\right)z\left(\tau \left(s\right)\right)ds+A(\tau \left(t\right),t_1){\int }_{\tau \left(t\right)}^t{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds\hfill \\ & & +A(\tau \left(t\right),t_1){\int }_t^{\infty }{Q}_2\left(s\right)z\left(\tau \left(s\right)\right)ds.\hfill \end{array}$$

Taking into account that $z\left(t\right)$ is increasing and ${\displaystyle \frac{z\left(t\right)}{A_1(t,t_1)}}$ is decreasing, one can verify that

$$\begin{array}{ccc}\hfill \alpha {(1+b)}^{\alpha }{\displaystyle \frac{z\left(\tau \left(t\right)\right)A(\tau \left(t\right),t_1)}{A_1(\tau \left(t\right),t_1)}}& \ge & {\displaystyle \frac{z\left(\tau \right(t\left)\right)}{A_1(\tau \left(t\right),t_1)}}{\int }_{t_1}^{\tau \left(t\right)}A(s,t_1)A_1(\tau \left(s\right),t_1)Q_2\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{A(\tau \left(t\right),t_1)z\left(\tau \left(t\right)\right)}{A_1(\tau \left(t\right),t_1)}}{\int }_{\tau \left(t\right)}^t{Q}_2\left(s\right)A_1(\tau \left(s\right),t_1)ds\hfill \\ & & +A(\tau \left(t\right),t_1)z\left(\tau \left(t\right)\right){\int }_t^{\infty }{Q}_2\left(s\right)ds,\hfill \end{array}$$

which yields

$$\begin{array}{ccc}\hfill \alpha {(1+b)}^{\alpha }& \ge & {\displaystyle \frac{1}{A(\tau \left(t\right),t_1)}}{\int }_{t_1}^{\tau \left(t\right)}A(s,t_1)A_1(\tau \left(s\right),t_1)Q_2\left(s\right)ds+{\int }_{\tau \left(t\right)}^t{Q}_2\left(s\right)A_1(\tau \left(s\right),t_1)ds\hfill \\ & & +A_1(\tau \left(t\right),t_1){\int }_t^{\infty }{Q}_2\left(s\right)ds.\hfill \end{array}$$

Taking $limsup$ as $t\to \infty$ on both sides of the last inequality, we are led to a contradiction with (2.25). The proof of the theorem is complete. □

Theorem 2.10.

Let condition (2.3) holds, and

$$\underset{t\to \infty }{lim}infA(t,t_1){\int }_t^{\infty }{\displaystyle \frac{Q\left(s\right){\left(A_1(\tau \left(s\right),t_1)\right)}^{\alpha }}{A(s,t_1)}}ds>{\displaystyle \frac{\alpha {(1+b)}^{\alpha }}{4}},$$

(2.26)

where $Q\left(t\right)$ defined as in (2.4). Then the equation $\left(E\right)$ is almost oscillatory.

Proof. 

Let $x\left(t\right)$ be a positive solution of $\left(E\right)$ with $x(t-\sigma )>0$ and $x\left({\tau }_i\left(t\right)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0.$ Then the corresponding function $z\left(t\right)>0$ and satisfies case $\left(I\right)$ or case $\left(II\right)$ of Lemma 2.3 for all $t\ge t_1.$ If $z\left(t\right)$ satisfies case $\left(II\right)$ then by Lemma 2.4 that (2.5) holds and therefore we need to consider the other case(I). Proceeding as in the proof of Theorem 2.8, we arrive at (2.22).

From Lemma 2.5$\left(ii\right)$ and $\left(iv\right),$ we have

$$\begin{array}{ccc}\hfill z\left(\tau \right(t\left)\right)& \ge & A_1(\tau \left(t\right),t_1){\displaystyle \frac{z^{\prime }\left(t\right)}{A(t,t_1)}}\hfill \end{array}$$

and using this inequality in (2.22), we obtain

$${\left(a^{1/\alpha }\left(t\right)z^{″}\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha {(1+b)}^{\alpha }}}{\displaystyle \frac{Q\left(t\right)A^{\alpha }(\tau \left(t\right),t)}{A(t,t_1)}}{z}^{\prime }\left(t\right)\le 0.$$

Let $u\left(t\right)=z^{\prime }\left(t\right).$ Then, we see that $u\left(t\right)$ is a positive solution of the inequality

$${\left(a^{1/\alpha }\left(t\right)u^{\prime }\left(t\right)\right)}^{\prime }+{\displaystyle \frac{1}{\alpha {(1+b)}^{\alpha }}}{\displaystyle \frac{Q\left(t\right)A_1^{\alpha }(\tau \left(t\right),t)}{A(t,t_1)}}u\left(t\right)\le 0.$$

Define

$$w\left(t\right)={\displaystyle \frac{a^{1/\alpha }\left(t\right)u^{\prime }\left(t\right)}{u\left(t\right)}},t\ge t_1.$$

Then $w\left(t\right)>0$ and satisfies

$$w\left(t\right)\ge {\displaystyle \frac{1}{\alpha {(1+b)}^{\alpha }}}{\int }_t^{\infty }{\displaystyle \frac{Q\left(s\right)A_1^{\alpha }(\tau \left(s\right),t_1)}{A(s,t_1)}}ds+{\int }_t^{\infty }{\displaystyle \frac{w^2\left(s\right)}{a^{1/\alpha }\left(s\right)}}ds.$$

(2.27)

Multiply (2.27) by $A(t,t_1)$ and letting $M={inf}_{t\ge t_1}R\left(t\right)w\left(t\right),$ we see that

$$\begin{array}{ccc}\hfill M& >& {\displaystyle \frac{1}{4}}+M^{2}R\left(t\right){\int }_t^{\infty }{\displaystyle \frac{1}{a^{1/\alpha }\left(s\right)R^2\left(s\right)}}ds\hfill \\ & =& {\displaystyle \frac{1}{4}}+M^{2}R\left(t\right){\displaystyle \frac{1}{R\left(t\right)}}ds={\displaystyle \frac{1}{4}}+M^2,\hfill \end{array}$$

which contradicts the admissible value of $M.$ The proof of the theorem is complete. □

3. Examples

In this section, we present some examples to illustrate the main results.

Example 3.1.

Consider the third-order nonlinear neutral differential equation of the form

$${\left({\displaystyle \frac{1}{t}}{\left(x\left(t\right)+2x(t-1)\right)}^{″}\right)}^{\prime }+{\displaystyle \frac{d_1}{t^4}}{x}^3\left({\displaystyle \frac{t}{2}}\right)+{\displaystyle \frac{d_2}{t^4}}{x}^{1/3}\left({\displaystyle \frac{t}{3}}\right)=0,t\ge 1,$$

(3.1)

where $d_1>0$ and $d_2>0$ are constants.

Here $a\left(t\right)={\displaystyle \frac{1}{t}},$$b=2,$$\sigma =1,$$p_1\left(t\right)={\displaystyle \frac{d_1}{t^4}},$$p_2\left(t\right)={\displaystyle \frac{d_2}{t^4}},$$\alpha =1,$${\alpha }_1=3,$${\alpha }_2={\displaystyle \frac{1}{3}},$${\tau }_1\left(t\right)={\displaystyle \frac{t}{2}},$${\tau }_2\left(t\right)={\displaystyle \frac{t}{3}}.$ A simple computation shows that $\tau \left(t\right)={\displaystyle \frac{t}{3}},$${\eta }_1={\displaystyle \frac{1}{4}},$${\eta }_2={\displaystyle \frac{3}{4}},$

$$Q\left(t\right)={\left(4d_1\right)}^{1/4}{\left({\displaystyle \frac{4}{3}}{d}_2\right)}^{3/4}{\displaystyle \frac{1}{t^4}},A(t,1)\approx {\displaystyle \frac{t^2}{2}}\mathrm{and}{A}_1(t,1)\approx {\displaystyle \frac{t^3}{6}}.$$

The condition (2.3) becomes

$${\int }_1^{\infty }\xi \left(\xi {\int }_{\xi }^{\infty }{\left(4d_1\right)}^{{\displaystyle \frac{1}{4}}}{\left({\displaystyle \frac{4}{3}}{d}_2\right)}^{{\displaystyle \frac{3}{4}}}{\displaystyle \frac{1}{s^4}}ds\right)d\xi ={\left(4d_1\right)}^{{\displaystyle \frac{1}{4}}}{\left({\displaystyle \frac{4}{3}}{d}_2\right)}^{{\displaystyle \frac{3}{4}}}{\displaystyle \frac{1}{3}}{\int }_1^{\infty }{\displaystyle \frac{1}{\xi }}d\xi =\infty ,$$

that is, condition (2.3) is satisfied. The condition (2.20) becomes

$$\underset{t\to \infty }{lim}inf{\int }_{t/3}^t{\left(4d_1\right)}^{{\displaystyle \frac{1}{4}}}{\left({\displaystyle \frac{4}{3}}{d}_2\right)}^{{\displaystyle \frac{3}{4}}}{\displaystyle \frac{1}{162}}{\displaystyle \frac{1}{s}}ds={\left(4d_1\right)}^{{\displaystyle \frac{1}{4}}}{\left({\displaystyle \frac{4}{3}}{d}_2\right)}^{{\displaystyle \frac{3}{4}}}{\displaystyle \frac{1}{162}}ln3>{\displaystyle \frac{3}{e}},$$

that is, condition (2.20) is satisfied if $d_1^{{\displaystyle \frac{1}{4}}}{d}_2^{{\displaystyle \frac{3}{4}}}>{\displaystyle \frac{243{\left(3\right)}^{3/4}}{2eln3}}.$ Thus by Corollary 2.7, the equation (3.1) is almost oscillatory if $d_1^{{\displaystyle \frac{1}{4}}}{d}_2^{{\displaystyle \frac{3}{4}}}>92.7423.$

Example 3.2.

Consider the third-order nonlinear neutral delay differential equation

$${\left({\left({\left(x\left(t\right)+2x(t-1)\right)}^{″}\right)}^{{\displaystyle \frac{5}{3}}}\right)}^{\prime }+{\displaystyle \frac{d_1}{t^2}}{x}^3\left({\displaystyle \frac{t}{2}}\right)+{\displaystyle \frac{d_2}{t^2}}x\left({\displaystyle \frac{t}{3}}\right)=0,t\ge 1,$$

(3.2)

where $d_1>0$ and $d_2>0$ are constants.

Here $a\left(t\right)=1,$$b=2,$$\sigma =1,$$p_1\left(t\right)={\displaystyle \frac{d_1}{t^2}},$$p_2\left(t\right)={\displaystyle \frac{d_2}{t^2}},$$\alpha ={\displaystyle \frac{5}{3}},$${\alpha }_1=3,$${\alpha }_2=1,$${\tau }_1\left(t\right)={\displaystyle \frac{t}{2}},$${\tau }_2\left(t\right)={\displaystyle \frac{t}{3}}.$ By a simple calculation, we see that $\tau \left(t\right)={\displaystyle \frac{t}{3}},$${\eta }_1={\displaystyle \frac{1}{3}},$${\eta }_2={\displaystyle \frac{2}{3}},$

$$Q\left(t\right)=3d_1^{1/3}{\left({\displaystyle \frac{d_2}{2}}\right)}^{2/3}{\displaystyle \frac{1}{t^2}},A(t,1)\approx t\mathrm{and}{A}_1(t,1)\approx {\displaystyle \frac{t^2}{2}}.$$

The condition (2.3) becomes

$${\int }_1^{\infty }\xi {\left({\int }_{\xi }^{\infty }3d_1^{1/3}{\left({\displaystyle \frac{d_2}{2}}\right)}^{2/3}{\displaystyle \frac{1}{s^2}}ds\right)}^{{\displaystyle \frac{3}{5}}}d\xi ={\left(3d_1^{1/3}{\left({\displaystyle \frac{d_2}{2}}\right)}^{2/3}\right)}^{{\displaystyle \frac{3}{5}}}{\int }_1^{\infty }{\xi }^{2/5}d\xi =\infty ,$$

that is, condition (2.3) holds. Since $Q_2\left(t\right)={\displaystyle \frac{3d_1^{1/3}{\left(\frac{d_2}{2}\right)}^{2/3}}{{\left(18\right)}^{\frac{2}{3}}}}{\displaystyle \frac{1}{t^{2/3}}},$ then by choosing $\rho \left(t\right)=1,$ the condition (2.21) is clearly satisfied for all $d_1>0$ and $d_2>0.$ Therefore by Theorem 2.8, the equation (3.2) is almost oscillatory if $d_1>0$ and $d_2>0.$

Example 3.3.

Consider the third-order nonlinear neutral delay differential equation

$${\left(t^{{\displaystyle \frac{1}{2}}}{\left(x\left(t\right)+2x(t-1)\right)}^{″}\right)}^{\prime }+{\displaystyle \frac{d_1}{t^4}}{x}^{{\displaystyle \frac{5}{3}}}\left({\displaystyle \frac{t}{2}}\right)+{\displaystyle \frac{d_2}{t}}{x}^{{\displaystyle \frac{1}{3}}}\left({\displaystyle \frac{t}{4}}\right)=0,t\ge 1,$$

(3.3)

where $d_1>0$ and $d_2>0$ are constants.

Here $a\left(t\right)=t^{{\displaystyle \frac{1}{2}}},$$b=2,$$\sigma =1,$$p_1\left(t\right)={\displaystyle \frac{d_1}{t^4}},$$p_2\left(t\right)={\displaystyle \frac{d_2}{t}},$$\alpha =1,$${\alpha }_1={\displaystyle \frac{5}{3}},$${\alpha }_2={\displaystyle \frac{1}{3}},$${\tau }_1\left(t\right)={\displaystyle \frac{t}{2}},$${\tau }_2\left(t\right)={\displaystyle \frac{t}{4}}.$ By a simple calculation, we see that $\tau \left(t\right)={\displaystyle \frac{t}{4}},$${\eta }_1={\eta }_2={\displaystyle \frac{1}{2}},$

$$Q\left(t\right)={\displaystyle \frac{2\sqrt{d_1{d}_2}}{t^{\frac{5}{2}}}},A(t,1)\approx 2t^{{\displaystyle \frac{1}{2}}}\mathrm{and}{A}_1(t,1)\approx {\displaystyle \frac{4}{3}}{t}^{{\displaystyle \frac{3}{2}}}.$$

The condition (2.3) becomes

$$2\sqrt{d_1{d}_2}{\int }_1^{\infty }\xi \left({\displaystyle \frac{1}{\sqrt{\xi }}}{\int }_{\xi }^{\infty }{\displaystyle \frac{1}{s^{\frac{5}{2}}}}ds\right)d\xi ={\displaystyle \frac{4}{3}}\sqrt{d_1{d}_2}{\int }_1^{\infty }{\displaystyle \frac{d\xi }{\xi }}=\infty ,$$

that is, condition (2.3) holds. The condition (2.26) becomes

$$\underset{t\to \infty }{lim}inf{\displaystyle \frac{\sqrt{t}}{3}}{\int }_t^{\infty }{\displaystyle \frac{\sqrt{d_1{d}_2}}{s^{\frac{3}{2}}}}ds=\sqrt{d_1{d}_2}>{\displaystyle \frac{9}{8}},$$

that is, condition (2.26) holds if $\sqrt{d_1{d}_2}>1.125.$ Also the condition (2.25) holds if $\sqrt{d_1{d}_2}>0.81691.$ Hence, by Theorem 2.10, the equation (3.3) is almost oscillatory if $\sqrt{d_1{d}_2}>1.125$ and the same conclusion holds by Theorem 2.9 if $\sqrt{d_1{d}_2}>0.81691).$ Therefore, Theorem 2.9 is better than Theorem 2.10.

Note that using Corollary 1 of [22], we see that (3.3) is almost oscillatory if $\sqrt{d_1{d}_2}>2.3883203.$ So our Theorems 2.9 and 2.10 significantly improve Corollary 1 of [22].

4. Conclusion

In this paper, we have obtained some new oscillation criteria by using arithmetic-geometric mean inequality along with linearization technique and then applying comparison method and integral averaging technique. The obtained results improve that of in [22] and this is illustrated via an example.The results already reported in the literature [3,9,10,11,12,13,15,16,17,18,19,21] cannot be applied to equations (3.1) to (3.3) since the number of nonlinear terms are more than one.