Oscillation and Asymptotic Properties of Differential Equations of Third-Order

Abstract

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator ${\left(a\left(\iota \right){\left[\right(b\left(\iota \right){\left[x\left(\iota \right)+p\left(\iota \right)x(\iota -\tau )\right]}^{\prime })}^{\prime }\right]}^{\beta }{)}^{\prime }$$+{\int }_c^{d}q(\iota ,\mu )x^{\beta }\left(\sigma (\iota ,\mu )\right)d\mu =0,$ where $\iota \ge {\iota }_0$ and $w\left(\iota \right):=x\left(\iota \right)+p\left(\iota \right)x(\iota -\tau )$. New oscillation results are established by using the generalized Riccati technique under the assumption of ${\int }_{{\iota }_0}^{\iota }{a}^{-1/\beta }\left(s\right)ds$<${\int }_{{\iota }_0}^{\iota }\frac{1}{b\left(s\right)}ds=\infty \mathrm{as}\iota \to \infty$. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

1. Introduction

The objective of this paper is to provide oscillation theorems for the third order equation as follows:

$$\begin{array}{c}\hfill {\left(a\left(\iota \right){\left[{\left(b\left(\iota \right){\left[x\left(\iota \right)+p\left(\iota \right)x(\iota -\tau )\right]}^{\prime }\right)}^{\prime }\right]}^{\beta }\right)}^{\prime }+{\int }_c^{d}q(\iota ,\mu )x^{\beta }\left(\sigma (\iota ,\mu )\right)d\mu =0,\end{array}$$

(1)

where $a\left(\iota \right),b\left(\iota \right),p\left(\iota \right),q\left(\iota \right)\in C\left([{\iota }_0,+\infty )\right)$, $a\left(\iota \right),b\left(\iota \right)>0$, $a^{\prime }\left(\iota \right)\ge 0$, $q\left(\iota \right)\ge 0$, $\beta \ge 1$ and $0\le p\left(\iota \right)\le p_0\le 1$. The main results are obtained under the following assumptions:

(A₁)

$q(\iota ,\mu )\in C([{\iota }_0,+\infty )\times [c,d],[0,+\infty ))$ and $q(\iota ,\mu )$ does not vanish identically for any half line $[{\iota }_{*},+\infty )\times [c,d]$, ${\iota }_{*}\ge \iota$;

(A₂)

$\sigma (\iota ,\mu )\in C([{\iota }_0,+\infty )\times [c,d],[0,+\infty ))$, $\sigma (\iota ,\mu )+\tau \le \iota$, $\sigma (\iota ,\mu )$ is nondecreasing with respect to $\iota$ and $\mu$ respectively, ${\displaystyle \underset{\iota \to +\infty }{lim\; inf}}$$\sigma (\iota ,\mu )=\infty$ and ${\displaystyle \underset{\iota \to +\infty }{lim\; inf}}$$\tau \left(\iota \right)=\infty$.

We set

$$\begin{array}{c}\hfill w\left(\iota \right):=x\left(\iota \right)+p\left(\iota \right)x(\iota -\tau )\end{array}$$

and

$$\begin{array}{c}\hfill A(\iota ,{\iota }_0)={\int }_{{\iota }_0}^{\iota }{a}^{-1/\beta }\left(s\right)ds,B(\iota ,{\iota }_0)={\int }_{{\iota }_0}^{\iota }\frac{1}{b\left(s\right)}ds.\end{array}$$

We intend that for a solution of (1), we mean a function $x\left(\iota \right)\in$$C\left([T_x,\infty )\right)$, $T_x$ ≥ ${\iota }_0$, which has the property $w^{\prime }\in C^1\left([T_x,\infty )\right)$, $b{w}^{\prime }$ ∈ $C^1\left([T_x,\infty )\right)$, $a{\left({\left(b{w}^{\prime }\right)}^{\prime }\right)}^{\beta }$ ∈ $C^1\left([T_x,\infty )\right)$ and satisfies (1) on $[T_x,\infty )$. We only consider those solutions x of (1) which satisfy $sup\left\{\right|x\left(\iota \right)|:\iota \ge T\}>0$ for all $T\ge T_x$. We start with the assumption that Equation (1) does possess a proper solution. A proper solution of Equation (1) is called oscillatory if it has a sequence of large zeros lending to ∞; otherwise, we call it non-oscillatory.

Neutral/delay differential equations of the third order are used in a variety of problems in economics, biology, and physics, including lossless transmission lines, vibrating masses attached to an elastic bar, and as the Euler equation in some variational problems; see Hale [1]. As a result, there is an ongoing interest in obtaining several sufficient conditions for the oscillation or non-oscillation of the solutions of different kinds of differential equations; see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] as examples of instant results on this topic.

However, to the best of our knowledge, only a few papers have studied the oscillation of nonlinear neutral delay differential equations of third order with distributed deviating arguments; see, for example, [2,3,4,5]. Recently, Haifei Xiang [6] and Haixia Wang et. al [7] studied the oscillatory behavior of Equation (1) under the following assumption:

$$\begin{array}{c}\hfill A(\iota ,{\iota }_0)=\infty ,B(\iota ,{\iota }_0)=\infty \mathrm{as}\iota \to \infty .\end{array}$$

Motivated by this above observation, in this paper, we extend the results under the following assumption:

$$\begin{array}{c}\hfill A(\iota ,{\iota }_0)<B(\iota ,{\iota }_0)=\infty \mathrm{as}\iota \to \infty .\end{array}$$

(2)

Motivated by these reasons mentioned above, in this paper, we extend the results using generalized Riccati transformation and the integral averaging technique. We establish criteria for Equation (1) to be oscillatory or converge to zero asymptotically with the assumption of (2). As is customary, all observed functional inequalities are assumed to support eventually; that is, they are satisfied for all $\iota$ that are large enough.

2. Main Results

For our further reference, let us denote the following:

$$\begin{array}{c}\hfill E_{0}w=w,E_{1}w=b{\left(E_{0}w\right)}^{\prime },E_{2}w=a{\left({\left(E_{1}w\right)}^{\prime }\right)}^{\beta },E_{3}w={\left(E_{2}w\right)}^{\prime },\end{array}$$

(3)

and

$$\begin{array}{ccc}& & C(\iota ,{\iota }_0)={\int }_{{\iota }_2}^{\iota }\frac{A(s,{\iota }_1)}{b\left(s\right)}ds,D\left(\iota \right):={\int }_{\iota }^{\infty }\frac{1}{a^{1/\beta }\left(s\right)}ds,{\sigma }_1\left(\iota \right)=\sigma (\iota ,c),\hfill \\ & & q_1\left(\iota \right)={(1-p_0)}^{\beta }{\int }_c^{d}q(\iota ,\mu )d\mu ,{\phi }_{+}^{\prime }\left(\iota \right)=max\{0,{\phi }^{\prime }\left(\iota \right)\}.\hfill \end{array}$$

Theorem 1.

Assume $\left(A_1\right)-\left(A_2\right)$ and (2) hold. If there exists a $\phi \in C^1([{\iota }_0,\infty ),\mathbb{R})$, such that ${\iota }_1\ge {\iota }_0$ for some ${\iota }_m>{\iota }_1$, we have the following:

$$\begin{array}{c}\hfill \underset{\iota \to \infty }{lim\; sup}{\int }_{{\iota }_3}^{\iota }\left(\phi \left(s\right)q_1\left(s\right)\frac{C^{\beta }({\sigma }_1\left(s\right),{\iota }_2)}{A^{\beta }({\sigma }_1\left(s\right),{\iota }_1)}-\frac{1}{{(1+\beta )}^{1+\beta }}\frac{a\left(s\right){\left({\phi }_{+}^{\prime }\left(s\right)\right)}^{1+\beta }}{{\phi }^{\beta }\left(s\right)}\right)ds=\infty ,\end{array}$$

(4)

$$\begin{array}{c}\hfill {\int }_{{\iota }_4}^{\infty }{b}^{-1}\left(v\right){\int }_v^{\infty }{\left(a^{-1}\left(u\right){\int }_u^{\infty }{\int }_c^{d}q(s,\mu )d\mu ds\right)}^{1/\beta }dudv=\infty ,\end{array}$$

(5)

and

$$\begin{array}{c}\hfill \underset{\iota \to \infty }{lim\; sup}{\int }_{{\iota }_6}^{\iota }\left(D^{\beta }\left(s\right)q_1\left(s\right)B^{\beta }({\sigma }_1\left(s\right),{\iota }_5)-{\left(\frac{\beta }{\beta +1}\right)}^{\beta +1}\frac{1}{D^{\beta }\left(s\right)a^{1/\beta }\left(s\right)}\right)ds=\infty ,\end{array}$$

(6)

Then, every solution $x\left(\iota \right)$ of (1) is either oscillatory or tends to 0.

Proof. 

Suppose that (1) has a non-oscillatory solution x. Now, we may take $x\left(\iota \right)>0$, $x(\iota -\tau )>0$ and $x\left(\sigma \right(\iota ,\mu \left)\right)>0$ for $\iota \ge {\iota }_1$ some ${\iota }_1\ge {\iota }_0$ and $\mu \in [c,d]$. By condition (2), there exist three possible cases:

(I)

$w\left(\iota \right)>0$, $w^{\prime }\left(\iota \right)>0$, ${\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }>0$, ${\left(a\left(\iota \right){\left[{\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }\right]}^{\beta }\right)}^{\prime }<0$,

(II)

$w\left(\iota \right)>0$, $w^{\prime }\left(\iota \right)<0$, ${\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }>0$, ${\left(a\left(\iota \right){\left[{\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }\right]}^{\beta }\right)}^{\prime }<0$, or

(III)

$w\left(\iota \right)>0$, $w^{\prime }\left(\iota \right)>0$, ${\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }<0$, ${\left(a\left(\iota \right){\left[{\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }\right]}^{\beta }\right)}^{\prime }<0$, for $\iota \ge {\iota }_1$, ${\iota }_1$ is large enough.

Assume first the case (I) holds for $\iota \ge {\iota }_2$. From the definition of $w\left(\iota \right)$, $w\left(\iota \right)\ge x\left(\iota \right)$ for $\iota \ge {\iota }_2$ and

$$w\left(\sigma (\iota ,\mu )\right)\ge w(\sigma (\iota ,\mu )-\tau )\ge x(\sigma (\iota ,\mu )-\tau ),\iota \ge {\iota }_3\ge {\iota }_2,$$

(7)

and

$$x\left(\iota \right)=w\left(\iota \right)-p\left(\iota \right)x(\iota -\tau )\ge w\left(\iota \right)-p\left(\iota \right)w(\iota -\tau )\ge (1-p_0)w\left(\iota \right).$$

(8)

Thus from (1) and (8), we have the following:

$$\begin{array}{ccc}\hfill E_{3}w\left(\iota \right)& =& -{\int }_c^{d}q(\iota ,\mu )x^{\beta }\left(\sigma (\iota ,\mu )\right)d\mu \hfill \\ & \le & -{(1-p_0)}^{\beta }{\int }_c^{d}q(\iota ,\mu )w^{\beta }\left(\sigma (\iota ,\mu )\right)d\mu \hfill \\ & \le & -{(1-p_0)}^{\beta }{w}^{\beta }\left(\sigma (\iota ,c)\right){\int }_c^{d}q(\iota ,\mu )d\mu \hfill \\ & =& -q_1\left(\iota \right)w^{\beta }\left({\sigma }_1\left(\iota \right)\right).\hfill \end{array}$$

(9)

Using the fact that $w^{\prime }\left(\iota \right)>0$, we have the following:

$$E_{1}w\left(\iota \right)\ge {\int }_{{\iota }_1}^{\iota }\frac{a^{1/\beta }\left(s\right){\left(E_{1}w\left(s\right)\right)}^{\prime }}{a^{1/\beta }\left(s\right)}ds\ge a^{1/\beta }\left(\iota \right){\left(E_{1}w\left(\iota \right)\right)}^{\prime }A(\iota ,{\iota }_1).$$

Thus, we have the following:

$$\begin{array}{ccc}\hfill w\left(\iota \right)& =& w\left({\iota }_2\right)+{\int }_{{\iota }_2}^{\iota }\frac{E_{1}w\left(s\right)}{A(s,{\iota }_1)}\frac{A(s,{\iota }_1)}{b\left(s\right)}ds\ge \frac{E_{1}w\left(\iota \right)}{A(\iota ,{\iota }_1)}{\int }_{{\iota }_2}^{\iota }\frac{A(s,{\iota }_1)}{b\left(s\right)}ds.\hfill \end{array}$$

Then, we have the following:

$$\begin{array}{c}\hfill \frac{w\left({\sigma }_1\left(\iota \right)\right)}{E_{1}w\left({\sigma }_1\left(\iota \right)\right)}\ge \frac{C({\sigma }_1\left(\iota \right),{\iota }_2)}{A({\sigma }_1\left(\iota \right),{\iota }_1)},\end{array}$$

(10)

and

$$\begin{array}{c}\hfill \frac{E_{1}w\left({\sigma }_1\left(\iota \right)\right)}{E_{1}w\left(\iota \right)}\ge \frac{A({\sigma }_1\left(\iota \right),{\iota }_1)}{A(\iota ,{\iota }_1)}\end{array}$$

(11)

We define a function as follows:

$$\begin{array}{c}\hfill \psi \left(\iota \right):=\phi \left(\iota \right)\frac{E_{2}w\left(\iota \right)}{E_1^{\beta }w\left(\iota \right)},\end{array}$$

(12)

and note that $\psi \left(\iota \right)>0$ for $\iota \ge {\iota }_1$. Differentiating (12), we obtain the following:

$$\begin{array}{c}\hfill {\psi }^{\prime }\left(\iota \right)=\frac{{\phi }^{\prime }\left(\iota \right)}{\phi \left(\iota \right)}\psi \left(\iota \right)+\phi \left(\iota \right)\frac{E_{3}w\left(\iota \right)}{E_1^{\beta }w\left(\iota \right)}-\beta \phi \left(\iota \right)a\left(\iota \right){\left[\frac{{\left(E_{1}w\left(\iota \right)\right)}^{\prime }}{E_{1}w\left(\iota \right)}\right]}^{\beta +1}.\end{array}$$

(13)

It follows from (9), (12), and (13) that the following holds:

$$\begin{array}{ccc}\hfill {\psi }^{\prime }\left(\iota \right)& \le & \frac{{\phi }^{\prime }\left(\iota \right)}{\phi \left(\iota \right)}\psi \left(\iota \right)-\phi \left(\iota \right)q_1\left(\iota \right){\left(\frac{w\left(\sigma \right(\iota \left)\right)}{E_{1}w\left(\iota \right)}\right)}^{\beta }-\beta \frac{{\psi }^{\frac{(\beta +1)}{\beta }}\left(\iota \right)}{{\left[\phi \left(\iota \right)a\left(\iota \right)\right]}^{1/\beta }}\hfill \\ & =& \frac{{\phi }^{\prime }\left(\iota \right)}{\phi \left(\iota \right)}\psi \left(\iota \right)-\phi \left(\iota \right)q_1\left(\iota \right){\left(\frac{w\left({\sigma }_1\left(\iota \right)\right)}{E_{1}w\left({\sigma }_1\left(\iota \right)\right))}\frac{E_{1}w\left({\sigma }_1\left(\iota \right)\right))}{E_{1}w\left(\iota \right))}\right)}^{\beta }-\beta \frac{{\psi }^{\frac{(\beta +1)}{\beta }}\left(\iota \right)}{{\left[\phi \left(\iota \right)a\left(\iota \right)\right]}^{1/\beta }}.\hfill \end{array}$$

(14)

Now, (10) and (14) implies the following:

$$\begin{array}{ccc}\hfill {\psi }^{\prime }\left(\iota \right)& \le & \frac{{\phi }_{+}^{\prime }\left(\iota \right)}{\phi \left(\iota \right)}\psi \left(\iota \right)-\beta \frac{{\psi }^{\frac{(\beta +1)}{\beta }}\left(\iota \right)}{{\left[\phi \left(\iota \right)a\left(\iota \right)\right]}^{1/\beta }}-\phi \left(\iota \right)q_1\left(\iota \right)\frac{C^{\beta }({\sigma }_1\left(\iota \right),{\iota }_2)}{A^{\beta }({\sigma }_1\left(\iota \right),{\iota }_1)}.\hfill \end{array}$$

(15)

Then, using (15) and inequality, we have the following:

$$\begin{array}{c}\hfill Bu-A{u}^{(m+1)/m}\le \frac{m^m}{{(m+1)}^{m+1}}\frac{B^{m+1}}{A^m}.\end{array}$$

(16)

We find the following:

$$\begin{array}{c}\hfill {\psi }^{\prime }\left(\iota \right)\le -\phi \left(\iota \right)q_1\left(\iota \right)\frac{C^{\beta }({\sigma }_1\left(\iota \right),{\iota }_2)}{A^{\beta }({\sigma }_1\left(\iota \right),{\iota }_1)}+\frac{1}{{(1+\beta )}^{1+\beta }}\frac{a\left(\iota \right){\left({\phi }_{+}^{\prime }\left(\iota \right)\right)}^{1+\beta }}{{\phi }^{\beta }\left(\iota \right)}.\end{array}$$

Integrating the last inequality from ${\iota }_3$$(>{\iota }_2)$ to $\iota$ gives

$$\begin{array}{c}\hfill \underset{\iota \to \infty }{lim\; sup}{\int }_{{\iota }_3}^{\iota }\left(\phi \left(s\right)q_1\left(s\right)\frac{C^{\beta }({\sigma }_1\left(s\right),{\iota }_2)}{A^{\beta }({\sigma }_1\left(s\right),{\iota }_1)}-\frac{1}{{(1+\beta )}^{1+\beta }}\frac{a\left(s\right){\left({\phi }_{+}^{\prime }\left(s\right)\right)}^{1+\beta }}{{\phi }^{\beta }\left(s\right)}\right)ds\le \psi \left({\iota }_3\right),\end{array}$$

(17)

which contradicts (4).

Next, if (II) holds. Since $w\left(\iota \right)>0$ and $w^{\prime }\left(\iota \right)<0$, we have $w\left(\iota \right)\to l\ge 0$. If $L>0$, then for $ϵ=\frac{L(1-p_0)}{2p_0}>0$, there exists ${\iota }_4\ge {\iota }_1$ such that $L<w\left(\iota \right)<L+ϵ$ for $\iota \ge {\iota }_4$. Then, for $\iota \ge {\iota }_4$, we have the following:

$$\begin{array}{c}\hfill x\left(\iota \right)=w\left(\iota \right)-p\left(\iota \right)x(\iota -\tau )>L-p_{0}w\left(\iota \right)>L-p_0(L+ϵ)=L_1\end{array}$$

Using the above inequality, which we obtained from (9), we have the following:

$$\begin{array}{c}\hfill E_{3}w\left(\iota \right)>-L_1^{\beta }{\int }_c^{d}q(\iota ,\mu )d\mu \end{array}$$

Integrating from $\iota (\ge {\iota }_4)$ to ∞ and using the fact that $a\left(\iota \right){\left[{\left(b\left(\iota \right)w^{\prime }\left(\iota \right)\right)}^{\prime }\right]}^{\beta }$ is positive and decreasing, we obtain the following:

$$\begin{array}{c}\hfill {\left(E_{1}w\left(\iota \right)\right)}^{\prime }\ge L_1{\left(\frac{1}{a\left(\iota \right)}{\int }_{\iota }^{\infty }{\int }_c^{d}q(s,\mu )d\mu ds\right)}^{1/\beta }.\end{array}$$

Again integrating the following,

$$\begin{array}{c}\hfill E_{1}w\left(\iota \right)\ge -L_1{\int }_{\iota }^{\infty }{\left(\frac{1}{a\left(u\right)}{\int }_{\iota }^{\infty }{\int }_c^{d}q(s,\mu )d\mu ds\right)}^{1/\beta }du,\end{array}$$

and again with the integration from ${\iota }_4$ to ∞, we obtain the following:

$$\begin{array}{c}\hfill w\left({\iota }_4\right)\ge L_1{\int }_{{\iota }_4}^{\infty }\frac{1}{b\left(v\right)}{\int }_{\iota }^{\infty }{\left(\frac{1}{a\left(u\right)}{\int }_{\iota }^{\infty }{\int }_c^{d}q(s,\mu )d\mu ds\right)}^{1/\beta }dudv,\end{array}$$

which contradicts (5) and shows that $L=0$, i.e., $w\left(\iota \right)\to 0$. Since $0<x\left(\iota \right)<w\left(\iota \right)$, we have $x\left(\iota \right)\to 0$ as $\iota \to \infty$.

Finally, assume that case (III) holds, $E_{3}w\left(\iota \right)\le 0$, and is non-increasing. Thus, we obtain the following:

$$\begin{array}{c}\hfill E_{2}w\left(s\right)\le E_{2}w\left(\iota \right),s\ge \iota \ge {\iota }_5.\end{array}$$

(18)

for some ${\iota }_5\ge {\iota }_0$. Dividing (18) by $a\left(s\right)$ and integrating from $\iota$ to l, we obtain the following:

$$\begin{array}{c}\hfill E_{1}w\left(l\right)\le E_{1}w\left(\iota \right)+a^{1/\beta }\left(\iota \right){\left(E_{1}w\left(\iota \right)\right)}^{\prime }A(l,\iota ).\end{array}$$

Letting $l\to \infty$, we have the following:

$$\begin{array}{c}\hfill -\frac{a^{1/\beta }\left(\iota \right){\left(E_{1}w\left(\iota \right)\right)}^{\prime }}{E_{1}w\left(\iota \right)}D\left(\iota \right)\le 1.\end{array}$$

(19)

Define function $\varphi$ by the following:

$$\begin{array}{c}\hfill \varphi \left(\iota \right):=\frac{E_{2}w\left(\iota \right)}{E_1^{\beta }w\left(\iota \right)},\iota \ge {\iota }_5.\end{array}$$

(20)

Then $\varphi \left(\iota \right)<0$ for $\iota \ge {\iota }_5$. Hence, from (19) and (20), we obtain the following:

$$\begin{array}{c}\hfill -D^{\beta }\left(\iota \right)\varphi \left(\iota \right)\le 1.\end{array}$$

(21)

Differentiating (20) gives the following:

$$\begin{array}{c}\hfill {\varphi }^{\prime }\left(\iota \right)=\frac{E_{3}w\left(\iota \right)}{E_1^{\beta }w\left(\iota \right)}-\beta a\left(\iota \right){\left[\frac{{\left(E_{1}w\left(\iota \right)\right)}^{\prime }}{E_{1}w\left(\iota \right)}\right]}^{\beta +1}.\end{array}$$

Now $w^{\prime }\left(\iota \right)>0$, so from (9) and (20), we have the following:

$$\begin{array}{c}\hfill {\varphi }^{\prime }\left(\iota \right)\le -q_1\left(\iota \right){\left[\frac{w\left({\sigma }_1\left(\iota \right)\right)}{E_{1}w\left(\iota \right)}\right]}^{\beta }-\beta \frac{{\varphi }^{1+\frac{1}{\beta }}\left(\iota \right)}{a^{1/\beta }\left(\iota \right)}.\end{array}$$

(22)

In case (III), we see that the following holds:

$$\begin{array}{c}\hfill w\left(\iota \right)\ge b\left(\iota \right)w^{\prime }\left(\iota \right){\int }_{{\iota }_5}^{\iota }\frac{ds}{b\left(s\right)}=E_{1}w\left(\iota \right)B(\iota ,{\iota }_5).\end{array}$$

(23)

Hence

$${\left[\frac{w\left(\iota \right)}{B(\iota ,{\iota }_5)}\right]}^{\prime }\le 0,$$

which implies the following:

$$\begin{array}{c}\hfill \frac{w\left({\sigma }_1\left(\iota \right)\right)}{w\left(\iota \right)}\ge \frac{B({\sigma }_1\left(\iota \right),{\iota }_5)}{B(\iota ,{\iota }_5)}.\end{array}$$

(24)

Using (23) and (24) in (22), we obtain the following:

$$\begin{array}{c}\hfill {\varphi }^{\prime }\left(\iota \right)\ge -q_1\left(\iota \right)B^{\beta }({\sigma }_1\left(\iota \right),{\iota }_5)-\beta \frac{{\varphi }^{1+\frac{1}{\beta }}\left(\iota \right)}{a^{1/\beta }\left(\iota \right)}.\end{array}$$

(25)

Hence from (25), we have the following:

$$\begin{array}{ccc}& & \varphi \left(\iota \right)D^{\beta }\left(\iota \right)-\varphi \left({\iota }_6\right)D^{\beta }\left({\iota }_6\right)\le -{\int }_{{\iota }_6}^{\iota }{q}_1\left(s\right)D^{\beta }\left(s\right)B^{\beta }({\sigma }_1\left(s\right),{\iota }_5)ds\hfill \\ & & -{\int }_{{\iota }_6}^{\iota }\beta \frac{D^{\beta -1}\left(s\right)\varphi \left(s\right)}{a^{1/\beta }\left(s\right)}ds-{\int }_{{\iota }_6}^{\iota }\beta \frac{D^{\beta }\left(s\right){\varphi }^{{}^{1+\frac{1}{\beta }}}\left(s\right)}{a^{1/\beta }\left(s\right)}ds,\hfill \end{array}$$

or

$$\begin{array}{ccc}\hfill & & \varphi \left(\iota \right)D^{\beta }\left(\iota \right)-\varphi \left({\iota }_6\right)D^{\beta }\left({\iota }_6\right)\le -{\int }_{{\iota }_6}^{\iota }{q}_1\left(s\right)D^{\beta }\left(s\right)B^{\beta }({\sigma }_1\left(s\right),{\iota }_5)ds\hfill \\ & & -{\int }_{{\iota }_6}^{\iota }\beta \left[\frac{D^{\beta -1}\left(s\right)\varphi \left(s\right)}{a^{1/\beta }\left(s\right)}+\frac{D^{\beta }\left(s\right){\varphi }^{{}^{1+\frac{1}{\beta }}}\left(s\right)}{a^{1/\beta }\left(s\right)}\right]ds.\hfill \end{array}$$

(26)

Set $\varphi :=-u\left(s\right)$ and using inequality

$$\begin{array}{c}\hfill A{u}^{(\beta +1)/\beta }-Bu\le -\frac{{\beta }^{\beta }}{{(\beta +1)}^{\beta +1}}\frac{B^{\beta +1}}{A^{\beta }},A>0,\end{array}$$

we obtain the following:

$$\begin{array}{c}\hfill {\int }_{{\iota }_6}^{\iota }\left[q_1\left(s\right)D^{\beta }\left(s\right)B^{\beta }({\sigma }_1\left(s\right),{\iota }_5)-{\left(\frac{\beta }{\beta +1}\right)}^{\beta +1}\frac{1}{D^{\beta }\left(s\right)a^{1/\beta }\left(s\right)}\right]ds\le -\varphi \left(\iota \right)D^{\beta }\left(\iota \right)+\varphi \left({\iota }_6\right)D^{\beta }\left({\iota }_6\right)\end{array}$$

(27)

Using (21) in (26) and then taking $\iota \to \infty$, we obtain the following:

$$\begin{array}{c}\hfill {\int }_{{\iota }_6}^{\infty }\left[q_1\left(s\right)D^{\beta }\left(s\right)B^{\beta }({\sigma }_1\left(s\right),{\iota }_5)-{\left(\frac{\beta }{\beta +1}\right)}^{\beta +1}\frac{1}{D^{\beta }\left(s\right)a^{1/\beta }\left(s\right)}\right]ds\le 1+\varphi \left({\iota }_6\right)D^{\beta }\left({\iota }_6\right)\end{array}$$

which contradicts (6). This completes the proof. □

We will present an example to illustrate the main results.

Example 1.

Consider the following 3rd-order equation:

$${\left({\iota }^2{\left[x\left(\iota \right)+p\left(\iota \right)x(\iota -\pi )\right]}^{\prime \prime }\right)}^{\prime }+{\int }_{\pi }^{3\pi /2}x(\iota -\mu )d\mu =0.$$

(28)

where $a\left(\iota \right)={\iota }^2$, $b\left(\iota \right)=1$, $\tau \left(\iota \right)=\iota -\pi$, $p\left(\iota \right)=1$, $\sigma (\iota ,\mu )=\iota -\mu$, $\beta =1$, $a=\pi$, $b=3\pi /2$. Moreover $0<p\left(\iota \right)\le p_0$ and $\phi \left(\iota \right)=1$. Then, we obtain the following: $q_1\left(\iota \right)=(1-p_0)\pi /2$, $D\left(\iota \right)=1/\iota$, ${\int }_{{\iota }_5}^{\iota -\mu }\frac{ds}{b\left(s\right)}=\iota -\mu -{\iota }_5$. The condition (4) becomes the following:

$$\begin{array}{c}\hfill {\int }_{{\iota }_3}^{\infty }\mathsf{\Phi }\left(s\right)ds=\frac{\pi (1-p_0)}{2}{\int }_{{\iota }_3}^{\infty }\frac{\frac{{\iota }_1}{{\iota }_2}{(\iota -\mu )}^2-(\iota -\mu )({\iota }_{1}log(\iota -\mu )+{\iota }_{1}log{\iota }_2-{\iota }_2)}{(\iota -{\iota }_1-\mu )}=\infty ,\end{array}$$

and

$$\begin{array}{ccc}& & {\int }_{{\iota }_6}^{\infty }\left(D^{\beta }\left(s\right)q_1\left(s\right){\int }_{{\iota }_5}^{{\sigma }_1\left(\iota \right)}\frac{dv}{b\left(v\right)}-\frac{1}{D\left(s\right)a^{1/\beta }\left(s\right)}\right)ds\hfill \\ & & =\frac{\pi (1-p_0)}{2}{\int }_{{\iota }_6}^{\infty }\left(1-\frac{(\mu -{\iota }_5-1)}{s}\right)ds=\infty ,\hfill \end{array}$$

so condition (6) also holds. Hence, by Theorem (1), it holds that every solution x of (28) is almost oscillatory.

3. A Concluding Remark

We established new oscillation theorems for (1) in this paper. The main outcomes are proved via the means of the integral averaging condition, and the generalized Riccati technique under the assumptions of ${\int }_{{\iota }_0}^{\iota }{a}^{-1/\beta }\left(s\right)ds$ < ${\int }_{{\iota }_0}^{\iota }\frac{1}{b\left(s\right)}ds=\infty \mathrm{as}\iota \to \infty$. Examples are given to prove the significance of the new results. The main results in this paper are presented in an essentially new form and of a high degree of generality. For future consideration, it will be of great importance to study the oscillation of (1) when $-\infty <p\left(\iota \right)\le -1$ and $\left|p\right(\iota \left)\right|<\infty$.