New Theorems for Oscillations to Differential Equations with Mixed Delays

Abstract

The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new oscillatory properties which describe both the necessary and sufficient conditions for a class of nonlinear second-order differential equations with neutral term and mixed delays of the form ${\left(p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{\prime }+r\left(\iota \right)u^{\beta }\left(\nu \left(\iota \right)\right)=0,\iota \ge {\iota }_0$ where $w\left(\iota \right)=u\left(\iota \right)+q\left(\iota \right)u\left(\zeta \right(\iota \left)\right).$ Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

1. Introduction

In this paper we present our work in the study of certain oscillation properties of second-order differential equations containing mixed delays.

Nowadays, the analysis of qualitative properties of ordinary differential equations is attracting considerable attention from the scientific community due to numerous applications in several contexts as Biology, Physics, Chemistry, and Dynamical Systems. For some details related the recent studies on oscillation and non-oscillation properties, exponential stability, instability, existence of unbounded solutions of the equations under consideration, we refer the reader to the books [1,2]. It is worth pointing out that both oscillation and stability criteria are currently used in the studies of nonlinear mathematical models with delay for single species and several species with interactions, in logistic models, $\alpha$-delay models, mathematical models with varying capacity, mathematical models for food-limited population dynamics with periodic coefficients, diffusive logistic models (for instance, diffusive Malthus-type models with several delays, autonomous diffusive delayed logistic models with Neumann boundary conditions, periodic diffusive logistic Volterra-type models with delays, and so on). In the last few years, the research activity concerning the oscillation of solutions of neutral differential equations has been received considerable attention. Moreover, neutral equations contribute to many applications in economics, physics, medicine, engineering and biology, see [3,4,5,6,7,8]. The literature is full of very interesting results linked with the oscillation properties for second-order differential equations. Now we recall some studies that have a strong connection with the content of this paper. In [9], the authors obtained some oscillation criteria of the following second-order neutral differential equations

$${\left(p\left(\iota \right){[u\left(\iota \right)+q\left(\iota \right)u\left(\zeta \left(\iota \right)\right)]}^{\prime }\right)}^{\prime }+r\left(\iota \right)u\left(\sigma \left(\iota \right)\right)+v\left(\iota \right)u\left(\eta \left(\iota \right)\right)=0$$

considering the cases in which the arguments are delayed, advanced or mixed. In [10], the authors had investigated some oscillation properties of the solutions of the following equation

$${\left(p\left(\iota \right)z^{\prime }\left(\iota \right)\right)}^{\prime }+r\left(\iota \right)u\left(\sigma \left(\iota \right)\right)=0,\iota \ge {\iota }_0\ge 0,$$

where $z\left(\iota \right)=u\left(\iota \right)+a\left(\iota \right)u(\iota -\tau )+b\left(\iota \right)u(\iota +\delta )$. It is interesting to notice that, in the aforementioned works, the authors obtained only sufficient conditions that ensure the oscillation of the solutions of the considered equations. A problem worthy of investigations is the study of necessary and sufficient conditions for oscillation, and some satisfactory answers were given in [11,12,13,14,15,16,17,18]. Finally, the interested readers are referred to the following papers and to the references therein for some recent results on the oscillation theory for ordinary differential equations of several orders [19,20,21,22,23,24,25,26,27].

In this work, we obtained the necessary and sufficient conditions for the oscillation of solutions to second-order non-linear differential equations in the form

$$\begin{array}{c}{\left(p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{\prime }+r\left(\iota \right)u^{\beta }\left(\nu \left(\iota \right)\right)=0,\iota \ge {\iota }_0,\end{array}$$

(1)

where

$$w\left(\iota \right)=u\left(\iota \right)+q\left(\iota \right)u\left(\zeta \right(\iota \left)\right).$$

The functions $r,p,q,\nu ,\zeta$ are continuous and satisfy the conditions stated below;

(a)

$\nu \in C\left(\right[0,\infty ),\mathbb{R})$, $\zeta \in C^2([0,\infty ),\mathbb{R})$, $\nu \left(\iota \right)<\iota$, $\zeta \left(\iota \right)<\iota$, ${lim}_{\iota \to \infty }\nu \left(\iota \right)=\infty$, ${lim}_{\iota \to \infty }\zeta \left(\iota \right)=\infty$.

(b)

$\nu \in C\left(\right[0,\infty ),\mathbb{R})$, $\zeta \in C^2([0,\infty ),\mathbb{R})$, $\nu \left(\iota \right)>\iota$, $\zeta \left(\iota \right)<\iota$, ${lim}_{\iota \to \infty }\nu \left(\iota \right)=\infty$, ${lim}_{\iota \to \infty }\zeta \left(\iota \right)=\infty$.

(c)

$p\in C^1([0,\infty ),\mathbb{R})$, $r,\tilde{r}\in C([0,\infty ),\mathbb{R})$; $0<p\left(\iota \right)$, $0\le r\left(\iota \right)$, $0\le \tilde{r}\left(\iota \right)$, for all $\iota \ge 0$.

(d)

$q\in C^2([0,\infty ),{\mathbb{R}}_{+})$ with $0\le q\left(\iota \right)\le a<1$.

(e)

${lim}_{\iota \to \infty }P\left(\iota \right)=\infty$ where $P\left(\iota \right)={\int }_0^{\iota }{p}^{-1/\alpha }\left(s\right)\mathrm{d}s$.

(f)

$\alpha$ and $\beta$ are the quotient of two positive odd integers.

2. Preliminary Results

To make our notations simpler, we set

$$\begin{array}{c}\hfill R_1\left(\iota \right)=r\left(\iota \right){\left((1-a)w\left(\nu \left(\iota \right)\right)\right)}^{\beta }.\end{array}$$

Lemma 1.

Suppose(a)–(f) holds for $\iota \ge {\iota }_0$, and if u is an eventually positive solution of (1). Then w satisfies

$$\begin{array}{c}0<w\left(\iota \right),w^{\prime }\left(\iota \right)>0,and{\left(p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{\prime }\le 0for\iota \ge {\iota }_1.\end{array}$$

(2)

Proof. 

Let u be an eventually positive solution. Then $w\left(\iota \right)>0$ and there exists ${\iota }_0\ge 0$ such that $u\left(\iota \right)>0$, $u\left(\nu \right(\iota \left)\right)>0$, $u\left(\zeta \right(\iota \left)\right)>0$ for all $\iota \ge {\iota }_0$. Then (1) gives that

$$\begin{array}{c}{\left(p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{\prime }=-r\left(\iota \right)u^{\beta }\left(\nu \left(\iota \right)\right)\le 0\end{array}$$

(3)

which shows that $p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }$ is non-increasing for $\iota \ge {\iota }_0$. Next we claim that for $w>0$, $p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }$ is positive for $\iota \ge {\iota }_1>{\iota }_0$. If not, let $p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\le 0$ for $\iota \ge {\iota }_1$, we can choose $c>0$ such that

$$p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\le -c,$$

that is,

$$w^{\prime }\left(\iota \right)\le {(-c)}^{1/\alpha }{p}^{-1/\alpha }\left(\iota \right).$$

Integrating both sides from ${\iota }_1$ to $\iota$ we get

$$w\left(\iota \right)-w\left({\iota }_1\right)\le {(-c)}^{1/\alpha }\left(P\left(\iota \right)-P\left({\iota }_1\right)\right).$$

Taking limit both sides as $\iota \to \infty$, we have ${lim}_{\iota \to \infty }w\left(\iota \right)\le -\infty$ which leads to a contradiction to $w\left(\iota \right)>0$. Hence, $p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }>0$ for $\iota \ge {\iota }_1$ i.e., $w^{\prime }\left(\iota \right)>0$ for $\iota \ge {\iota }_1$. Hence proved. □

Lemma 2.

Suppose(a)–(f) hold for $\iota \ge {\iota }_0$, and if u is an eventually positive solution of (1). Then w satisfies

$$\begin{array}{c}\hfill u\left(\iota \right)\ge (1-a)w\left(\iota \right)for\iota \ge {\iota }_1.\end{array}$$

(4)

Proof. 

Assume that u be an eventually positive solution of (1). Then $w\left(\iota \right)>0$ and there exists $\iota \ge {\iota }_1>{\iota }_0$ such that

$$\begin{array}{cc}\hfill u\left(\iota \right)& =w\left(\iota \right)-q\left(\iota \right)u\left(\zeta \right(\iota \left)\right)\hfill \\ \hfill & \ge w\left(\iota \right)-q\left(\iota \right)w\left(\zeta \right(\iota \left)\right)\hfill \\ \hfill & \ge w\left(\iota \right)-q\left(\iota \right)w\left(\iota \right)\hfill \\ \hfill & =\left(1-q\left(\iota \right)\right)w\left(\iota \right)\hfill \\ \hfill & \ge (1-a)w\left(\iota \right).\hfill \end{array}$$

Hence w satisfies (4) for $\iota \ge {\iota }_1$. □

Remark 1.

The above two lemmas hold for any α and β (i.e., $\alpha \ge \beta$ or $\alpha \le \beta$).

3. Main Results

Theorem 1.

Let(b)–(f) hold for $\iota \ge {\iota }_0$ and $\beta >\alpha$. Then every solution of (1) is oscillatory if and only if

$${\int }_0^{\infty }{p}^{-1/\alpha }\left(s\right){\left[{\int }_s^{\infty }r\left(\psi \right)\mathrm{d}\psi \right]}^{1/\alpha }\mathrm{d}s=\infty .$$

(5)

Proof. 

Let u is an eventually positive solution of (1). Then $w\left(\iota \right)>0$ and there exists ${\iota }_0\ge 0$ such that $u\left(\iota \right)>0$, $u\left(\nu \right(\iota \left)\right)>0$, $u\left(\zeta \right(\iota \left)\right)>0$ for all $\iota \ge {\iota }_0$. Thus, Lemmas 1 and 2 holds for $\iota \ge {\iota }_1$. By Lemma 1, there exists ${\iota }_2>{\iota }_1$ such that $w^{\prime }\left(\iota \right)>0$ for all $\iota \ge {\iota }_2$. Then there exists ${\iota }_3>{\iota }_2$ and $c>0$ such that $w\left(\iota \right)\ge c$ for all $\iota \ge {\iota }_3$. Next using Lemma 2, we wet $u\left(\iota \right)\ge (1-a)w\left(\iota \right)$ for all $\iota \ge {\iota }_3$ and (1) become

$$\begin{array}{c}{\left(p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{\prime }+R_1\left(\iota \right)\le 0.\end{array}$$

(6)

Integrating (6) from $\iota$ to ∞ we get

$$\begin{array}{cc}\hfill & {\left[p\left(s\right){\left(w^{\prime }\left(s\right)\right)}^{\alpha }\right]}_{\iota }^{\infty }+{\int }_{\iota }^{\infty }{R}_1\left(s\right)\mathrm{d}s\le 0.\hfill \end{array}$$

Since $p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }$ is positive and non-decreasing, ${lim}_{\iota \to \infty }p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }$ finitely exists and is positive.

$$\begin{array}{cc}\hfill p\left(\iota \right){\left(w^{\prime }\left(\iota \right)\right)}^{\alpha }& \ge {\int }_{\iota }^{\infty }{R}_1\left(s\right)\mathrm{d}s,\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill w^{\prime }\left(\iota \right)& \ge p^{-1/\alpha }\left(\iota \right){\left[{\int }_{\iota }^{\infty }{R}_1\left(s\right)\mathrm{d}s\right]}^{1/\alpha }\hfill \\ \hfill & ={(1-a)}^{\beta /\alpha }{p}^{-1/\alpha }\left(\iota \right){\left[{\int }_{\iota }^{\infty }r\left(s\right)w^{\beta }\left(\nu \left(s\right)\right)\mathrm{d}s\right]}^{1/\alpha }.\hfill \end{array}$$

(7)

Using the assumption (b) and $w\left(\iota \right)$ is non-decreasing,

$$\begin{array}{cc}\hfill w^{\prime }\left(\iota \right)& \ge {(1-a)}^{\beta /\alpha }{p}^{-1/\alpha }\left(\iota \right){\left[{\int }_{\iota }^{\infty }r\left(s\right)\mathrm{d}s\right]}^{1/\alpha }{w}^{\beta /\alpha }\left(\iota \right),\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill \frac{w^{\prime }\left(\iota \right)}{w^{\beta /\alpha }\left(\iota \right)}& \ge {(1-a)}^{\beta /\alpha }{p}^{-1/\alpha }\left(\iota \right){\left[{\int }_{\iota }^{\infty }r\left(s\right)\mathrm{d}s\right]}^{1/\alpha }.\hfill \end{array}$$

Taking integration both sides from ${\iota }_3$ to ∞ we have,

$$\begin{array}{c}\hfill {(1-a)}^{\beta /\alpha }{\int }_{{\iota }_3}^{\infty }{p}^{-1/\alpha }\left(s\right){\left[{\int }_s^{\infty }r\left(\psi \right)\mathrm{d}\psi \right]}^{1/\alpha }\mathrm{d}s\le {\int }_{{\iota }_3}^{\infty }\frac{w^{\prime }\left(s\right)}{w^{\beta /\alpha }\left(s\right)}\mathrm{d}s<\infty \end{array}$$

due to $\beta >\alpha$, which is a contradiction to (5) and hence the sufficient part of the theorem is proved.

Next by applying contrapositive argument we proved the necessary part. If (5) does not hold, then for every $\epsilon >0$ there exists $\iota \ge {\iota }_0$ for which

$$\begin{array}{c}\hfill {\int }_{\iota }^{\infty }{p}^{-1/\alpha }\left(s\right){\left[{\int }_s^{\infty }r\left(\psi \right)\mathrm{d}\psi \right]}^{1/\alpha }\mathrm{d}s<\epsilon \mathrm{for}\iota \ge T,\end{array}$$

where $2\epsilon ={\left[\frac{1}{1-a}\right]}^{-\beta /\alpha }>0$. Let us define a set

$$V=\left\{u\in C\left([0,\infty )\right):\frac{1}{2}\le u\left(\iota \right)\le \frac{1}{1-a}\mathrm{for}\mathrm{all}\iota \ge T\right\}$$

and $\Phi :V\to V$ as

$$(\Phi u)\left(\iota \right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}\iota \le T,\hfill \\ \frac{1+a}{2(1-a)}-q\left(\iota \right)u\left(\zeta \left(\iota \right)\right)\hfill \\ +{\int }_{\iota }^{\iota }{p}^{-1/\alpha }\left(s\right){\left[{\int }_s^{\infty }r\left(\psi \right)u^{\beta }\left(\nu \left(\psi \right)\right)\mathrm{d}\psi \right]}^{1/\alpha }\mathrm{d}s\hfill & \mathrm{if}\iota >T.\hfill \end{array}\right.$$

Next we prove $(\Phi u)\left(\iota \right)\in V$. For $u\left(\iota \right)\in V$,

$$\begin{array}{cc}\hfill (\Phi u)\left(\iota \right)& \le \frac{1+a}{2(1-a)}+{\int }_T^{\iota }{p}^{-1/\alpha }\left(s\right){\left[{\int }_s^{\infty }r\left(\psi \right){\left(\frac{1}{1-a}\right)}^{\beta }\mathrm{d}\psi \right]}^{1/\alpha }\mathrm{d}s\hfill \\ \hfill & \le \frac{1+a}{2(1-a)}+{\left(\frac{1}{1-a}\right)}^{\beta /\alpha }\times \epsilon \hfill \\ \hfill & =\frac{1+a}{2(1-a)}+\frac{1}{2}=\frac{1}{1-a}\hfill \end{array}$$

and further, for $u\left(\iota \right)\in V$

$$\begin{array}{c}\hfill (\Phi u)\left(\iota \right)\ge \frac{1+a}{2(1-a)}-q\left(\iota \right)\times \frac{1}{1-a}+0\ge \frac{1+a}{2(1-a)}-\frac{a}{1-a}=\frac{1}{2}.\end{array}$$

Hence, $\Phi$ maps from V to V.

Now we are going to find a fixed point for $\Phi$ in V which will eventually give a positive solution of (1).

First, we define a sequence of functions in V by

$$\begin{array}{c}{u}_0\left(\iota \right)=0\mathrm{for}\iota \ge {\iota }_0,\\ u_1\left(\iota \right)=(\Phi u_0)\left(\iota \right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}\iota <T\hfill \\ \frac{1}{2}\hfill & \mathrm{if}\iota \ge T\hfill \end{array}\right.,\\ u_{n+1}\left(\iota \right)=(\Phi u_n)\left(\iota \right)\mathrm{for}n\ge 1,\iota \ge T.\end{array}$$

Here, we see $u_1\left(\iota \right)\ge u_0\left(\iota \right)$ for each fixed $\iota$ and $\frac{1}{2}\le u_{n-1}\left(\iota \right)\le u_n\left(\iota \right)\le \frac{1}{1-a},\iota \ge T$ for all $n\ge 1$. Thus, $u_n$ converges point-wise to a function u. By Lebesgue’s Dominated Convergence Theorem u is a fixed point of $\Phi$ in V, which shows that there has a non-oscillatory solution. This completes the proof of the theorem. □

Theorem 2.

Let (a), (c)–(f) hold for $\iota \ge {\iota }_0$ and $\beta <\alpha$. Then every solution of (1) is oscillatory if and only if

$$\begin{array}{cc}\hfill {\int }_0^{\infty }& r\left(\psi \right){\left[(1-a)P\left(\nu \left(\psi \right)\right)\right]}^{\beta }\mathrm{d}\psi =\infty .\hfill \end{array}$$

(8)

Proof. 

Let $u\left(\iota \right)$ be an eventually positive solution of (1). Then proceeding as in Theorem 1, we have ${\iota }_2>{\iota }_1>{\iota }_0$ such that (7) holds for all $\iota \ge {\iota }_2$. Using (e), there exists ${\iota }_3>{\iota }_2$ for which $P\left(\iota \right)-P\left({\iota }_3\right)\ge \frac{1}{2}P\left(\iota \right)$ for $\iota \ge {\iota }_3$. Integrating (7) from ${\iota }_3$ to $\iota$, we have

$$\begin{array}{cc}\hfill w\left(\iota \right)-w\left({\iota }_3\right)& \ge {\int }_{{\iota }_3}^{\iota }{p}^{-1/\alpha }\left(s\right){\left[{\int }_s^{\infty }{R}_1\left(\kappa \right)\mathrm{d}\kappa \right]}^{1/\alpha }\mathrm{d}s\hfill \\ \hfill & \ge {\int }_{{\iota }_3}^{\iota }{p}^{-1/\alpha }\left(s\right){\left[{\int }_{\iota }^{\infty }{R}_1\left(\kappa \right)\mathrm{d}\kappa \right]}^{1/\alpha }\mathrm{d}s,\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill w\left(\iota \right)& \ge (P\left(\iota \right)-P\left({\iota }_3\right)){\left[{\int }_{\iota }^{\infty }{R}_1\left(\kappa \right)\mathrm{d}\kappa \right]}^{1/\alpha }\hfill \\ \hfill & \ge \frac{1}{2}P\left(\iota \right){\left[{\int }_{\iota }^{\infty }{R}_1\left(\kappa \right)\mathrm{d}\kappa \right]}^{1/\alpha }.\hfill \end{array}$$

(9)

Hence,

$$\begin{array}{c}\hfill w\left(\iota \right)\ge \frac{1}{2}P\left(\iota \right)U^{1/\alpha }\left(\iota \right)\mathrm{for}\iota \ge {\iota }_3\end{array}$$

where

$$\begin{array}{c}\hfill U\left(\iota \right)={\int }_{\iota }^{\infty }r\left(\kappa \right){\left((1-a)w\left(\nu \left(\kappa \right)\right)\right)}^{\beta }\mathrm{d}\kappa .\end{array}$$

Now,

$$\begin{array}{cc}\hfill U^{\prime }\left(\iota \right)& =-r\left(\iota \right){\left((1-a)w\left(\nu \left(\iota \right)\right)\right)}^{\beta }\hfill \\ \hfill & \le -\frac{1}{2^{\beta }}r\left(\iota \right){\left[(1-a)P\left(\nu \left(\iota \right)\right)\right]}^{\beta }{U}^{\beta /\alpha }\left(\nu \left(\iota \right)\right)\le 0\hfill \end{array}$$

(10)

which shows that $U\left(\iota \right)$ is non-increasing on $[{\iota }_4,\infty )$ and ${lim}_{\iota \to \infty }U\left(\iota \right)$ exists. Using (10) and (a), we find

$$\begin{array}{cc}\hfill {\left[U^{1-\beta /\alpha }\left(\iota \right)\right]}^{\prime }& =(1-\beta /\alpha )U^{-\beta /\alpha }\left(\iota \right)U^{\prime }\left(\iota \right)\hfill \\ \hfill & \le -\frac{1-\beta /\alpha }{2^{\beta }}r\left(\iota \right){\left[(1-a)P\left(\nu \left(\iota \right)\right)\right]}^{\beta }{U}^{\beta /\alpha }\left(\nu \left(\iota \right)\right)U^{-\beta /\alpha }\left(\iota \right)\hfill \\ \hfill & \le -\frac{1-\beta /\alpha }{2^{\beta }}r\left(\iota \right){\left[(1-a)P\left(\nu \left(\iota \right)\right)\right]}^{\beta }.\hfill \end{array}$$

(11)

Integrating (11) from ${\iota }_3$ to $\iota$ we have,

$$\begin{array}{c}\hfill {\left[U^{1-\beta /\alpha }\left(s\right)\right]}_{{\iota }_4}^{\iota }\le -\frac{1-\beta /\alpha }{2^{\beta }}{\int }_{{\iota }_3}^{\iota }r\left(s\right){\left[(1-a)P\left(\nu \left(s\right)\right)\right]}^{\beta }\mathrm{d}s,\end{array}$$

that is,

$$\begin{array}{c}\hfill \frac{1-\beta /\alpha }{2^{\beta }}\left[{\int }_0^{\infty }r\left(s\right){\left[(1-a)P\left(\nu \left(s\right)\right)\right]}^{\beta }{\mathrm{d}s]}^{\beta }\right]\le -{\left[U^{1-\beta /\alpha }\left(s\right)\right]}_{{\iota }_3}^{\iota }<U^{1-\beta /\alpha }\left({\iota }_3\right)<\infty \end{array}$$

which contradicts (8). This completes the proof of the theorem. □

Example 1.

Consider the neutral differential equations

$$\begin{array}{c}{\left({\left({\left(u\left(\iota \right)+e^{-\iota }u\left(\zeta \left(\iota \right)\right)\right)}^{\prime }\right)}^{1/3}\right)}^{\prime }+\iota {\left(u(\iota +2)\right)}^{5/3}=0.\end{array}$$

(12)

Here $\alpha =1/3$, $p\left(\iota \right)=1$, $0<q\left(\iota \right)=e^{-\iota }<1$, $\nu \left(\iota \right)=\iota +2$. For $\beta =5/3$, we have $\beta =5/3>\alpha =1/3$. To check (5) we have

$$\begin{array}{c}\hfill {\int }_{{\iota }_0}^{\infty }{\left[\frac{1}{p\left(s\right)}\left[{\int }_s^{\infty }r\left(\psi \right)d\psi \right]\right]}^{1/\alpha }ds={\int }_2^{\infty }{\left[{\int }_s^{\infty }\psi d\psi \right]}^{3}ds=\infty .\end{array}$$

So, all the conditions of of Theorem 1 hold. Thus, each solution of (12) is oscillatory.

Example 2.

Consider the neutral differential equations

$$\begin{array}{c}{\left(e^{-\iota }{\left({\left(u\left(\iota \right)+e^{-\iota }u\left(\zeta \left(\iota \right)\right)\right)}^{\prime }\right)}^{11/3}\right)}^{\prime }+\frac{1}{\iota +1}{\left(u(\iota -2)\right)}^{7/3}=0.\end{array}$$

(13)

Here $\alpha =11/3$, $p\left(\iota \right)=e^{-\iota }$, $0<q\left(\iota \right)=e^{-\iota }<1$, $\nu \left(\iota \right)=\iota -2$, $P\left(\iota \right)={\int }_0^{\iota }{e}^{3s/11}ds=\frac{11}{3}(e^{3\iota /11}-1)$. For $\beta =7/3$, we get $\beta =7/3<\alpha =11/3$. To check (8) we have

$$\begin{array}{cc}\hfill & \frac{1}{{\left(2\right)}^{\beta }}\left[{\int }_0^{\infty }r\left(\psi \right){\left[(1-a)P\left(\nu \left(\psi \right)\right)\right]}^{\beta }d\psi \right]\hfill \\ \hfill & =\frac{1}{{\left(2\right)}^{7/3}}{\int }_0^{\infty }\frac{1}{\psi +1}{\left[(1-a)\frac{11}{3}\left(e^{3(\psi -2)/11}-1\right)\right]}^{7/3}d\psi =\infty .\hfill \end{array}$$

So, all the conditions of Theorem 2 hold, and therefore, each solution of (13) is oscillatory.

4. Conclusions

In this work, we studied second order highly nonlinear neutral differential equations and established necessary and sufficient conditions for the oscillation of (1) when the neutral coefficient lies in $[0,1)$. We already studied this for the case when $-1\le q\left(\iota \right)\le 0$. The obtained method is applicable for any type of second-order delay differential equation. In this direction, we have an open problem, namely: "Can we find the necessary and sufficient conditions for the oscillation of the solutions to the equations (1) for the range $-\infty <q\left(\iota \right)<-1$ or $1<q\left(\iota \right)<\infty$?".