New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms

Abstract

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new classes of equations that were not covered by the previous criteria. According to symmetry between the negative and positive solutions of the studied equation, we present some new criteria that ensure the oscillation of all solutions by using the Riccati transformation and comparison method. We provide an example to illustrate our main results.

1. Introduction

In this paper, we study the oscillatory behavior of solutions of the following second-order damped differential Equations (DEs) with a neutral term:

$${\left(\iota y^{\prime }\right)}^{\prime }\left(\top \right)+d\left(\top \right)y^{\prime }\left(\top \right)+{\displaystyle Ϝ}\left(\top \right){\varkappa }^{\lambda }\left(\left(\tau \left(\top \right)\right)\right)=0,$$

(1)

$\top \ge {\top }_0>0,$ where

$$y\left(\top \right)=\varkappa \left(\top \right)+\nu \left(\top \right){\varkappa }^{\gamma }\left(\zeta \left(\top \right)\right).$$

Throughout this paper, we assume that:

(Y₁)

$\gamma$ and $\lambda$ are quotients of odd positive integers and $\gamma \in [1,\infty );$

(Y₂)

$\nu ,$ ${\displaystyle Ϝ}\in C\left([{\top }_0,\infty ),\left(0,\infty \right)\right)$, $\nu \left(\top \right)<1$ and ${\displaystyle Ϝ}\left(\top \right)$ is not identically zero for large ⊤;

(Y₃)

$\iota \in C\left([{\top }_0,\infty ),\left(0,\infty \right)\right)$, $d\in C\left([{\top }_0,\infty ),\mathbb{R}\right)$ and

$$A\left(\top \right):={\int }_{{\top }_0}^{\top }\frac{1}{\iota \left(u\right)}exp\left(-{\int }_{{\top }_0}^u\left|\frac{d\left(s\right)}{\iota \left(s\right)}\right|\mathrm{d}s\right)\mathrm{d}u<\infty ;$$

(2)

(Y₄)

$\zeta ,$ $\tau \in C\left([{\top }_0,\infty ),\mathbb{R}\right)$ satisfy $\zeta \left(\top \right)\ge \top$, $\tau \left(\top \right)\le \top ,$ and ${lim}_{\top \to \infty }\tau \left(\top \right)=\infty .$

A solution (1) refers to a function $\varkappa \in C\left([{\top }_{\varkappa },\infty ),\mathbb{R}\right)$ for some ${\top }_{\varkappa }\ge {\top }_0$ such that $\iota y^{\prime }\in C^1\left([{\top }_{\varkappa },\infty ),\mathbb{R}\right),$ and the solution $\varkappa$ satisfies (1) on $[{\top }_{\varkappa },\infty ).$ We focus only on those solutions of (1) that exist on some half-line $[{\top }_{\varkappa },\infty )$ and satisfy the condition

$$sup\left[\varkappa \left(\top \right):\overline{T}\le \top <\infty \right]>0,$$

for $\overline{T}\le {\top }_{\varkappa }.$ A solution $\varkappa \left(\top \right)$ is said to be oscillatory if it has arbitrarily large zeros on $[{\top }_{\varkappa },\infty );$ otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

The study of DEs is important in mathematics, as it is a tool for describing different life patterns. Therefore, it is indispensable for linking mathematics with applied sciences, engineering, biology, and other sciences.

Qualitative theory is concerned with analyzing and understanding these models by studying the qualitative characteristics (such as oscillation, periodicity, stability, etc.) of different solutions of DEs (see [1,2,3]). One of the most important branches of this theory is oscillation theory, which is concerned with studying the oscillatory behavior of solutions of DEs by providing conditions that guarantee the oscillation or non-oscillation of these solutions.

The science of calculus remained elusive to scientists until its features were identified by notable figures such as Leibniz, Liouville, Riemann, etc. [4,5]. Initially regarded as a mathematical field of research with practical applications, calculus has emerged as a separate discipline through persistent efforts. It has gained significant importance in various domains, including viscoelasticity, electromagnetics, electrochemistry, fluid mechanics, optics, and biological assembly models. The applications of partial calculus extend to the simulation of complex technical and physical processes, which are described and defined as fractional DEs.

Over the past few decades, numerous results have emerged concerning the study of the oscillatory behavior of solutions for various classes of second-order neutral DEs without damping term. For the latest contributions, refer to [6,7,8,9,10,11,12,13,14] and the references therein, although we find limited results regarding the oscillation of solutions for second-order neutral DEs with damping.

Damped DEs find numerous important applications in various fields of science and engineering. This type of equation is involved in the study of dynamical systems that exhibit oscillatory behavior and are influenced by damping effects [4,15]. For instance, in the field of civil engineering, these equations help understand the vibrations of buildings, bridges, and other structures subjected to external forces or disturbances. By incorporating damping terms into the equations, engineers can assess the energy dissipation and stability of these systems.

In contrast to all previous studies (to the best knowledge of the authors), we do not impose any additional restrictions on the damping term. Even when $d\in C\left([{\top }_0,\infty ),\left(0,\infty \right)\right)$, the results of this paper complement, improve, and generalize the previous findings.

In the following, we briefly comment on the relevant findings that motivated our study. Baculikova and Dzurina [7] investigated a damped DE

$${\left(\iota \left(\top \right){\left(\varkappa \left(\top \right)+\nu \left(\top \right)\varkappa \left(\zeta \left(\top \right)\right)\right)}^{\prime }\right)}^{\prime }+d\left(\top \right)y^{\prime }\left(\top \right)+{\displaystyle Ϝ}\left(\top \right)\varkappa \left(\left(\tau \left(\top \right)\right)\right)=0$$

(3)

under the conditions

$$d\in C\left([{\top }_0,\infty ),\mathbb{R}\right),$$

(4)

$$\zeta \circ \tau =\tau \circ \zeta$$

(5)

and

$$A\left(\top \right):={\int }_{{\top }_0}^{\top }\frac{1}{\iota \left(u\right)}\mathrm{d}u=\infty .$$

On the other hand, in [16], sufficient conditions were presented to guarantee that the solutions of the following damped DE:

$${\left(\iota \left(\top \right){\left({\left(\varkappa \left(\top \right)+\nu \left(\top \right)\varkappa \left(\zeta \left(\top \right)\right)\right)}^{\prime }\right)}^{\lambda }\right)}^{\prime }+d\left(\top \right){\left(y^{\prime }\left(\top \right)\right)}^{\lambda }+{\int }_a^b{\displaystyle Ϝ}\left(\top ,u\right){\varkappa }^{\lambda }\left(\left(\tau \left(\top ,u\right)\right)\right)\mathrm{d}u=0$$

either oscillate or approach zero under conditions (4),

$${\int }_{{\top }_0}^{\top }\frac{1}{\iota \left(u\right)}exp\left({\int }_{{\top }_0}^u\frac{-d\left(s\right)}{\iota \left(s\right)}\mathrm{d}s\right)\mathrm{d}u<\infty$$

and

$$\nu \left(\top \right)\in [0,1).$$

(6)

Specifically, by using the Riccati type transformation and an integral criterion, Tunc, in [17], obtained some oscillation results where (4) holds, $\nu \left(\top \right)\in [1,\infty ),$ $\iota \left(\top \right)=1$ and

$$A\left(\top \right):={\int }_{{\top }_0}^{\top }exp\left(-{\int }_{{\top }_0}^{u}d\left(s\right)\mathrm{d}s\right)\mathrm{d}\top =\infty .$$

Moreover, the authors in [15,16,18,19,20,21,22,23] studied Equation (1) and special forms thereof and concluded that some oscillatory results occur in the presence of the condition $\gamma \in [0,1)$. Several significant results have been presented regarding the oscillatory behavior of second-order neutral differential equation solutions with a damping term. However, these results have not been studied when the function $d\left(t\right)$ allows them to be negative and satisfy (6) and

$$\gamma >1.$$

(7)

As a result, in this paper, we extend the results presented in the reference [24] to include a new class of DEs with variable damping term in which the function allows $d\left(t\right)$ to be negative and fulfill the conditions (6) and (7) at the same time. Thus, we believe that the results presented in this paper are a good contribution to the least developed oscillation theory of second-order superlinear neutral DEs with a variable damping term, which can be more generally be made into second-order DEs with a superlinear neutral term and thus obtain more general oscillation conditions.

Definition 1.

We say that the property (R) is satisfied if

$$\underset{\top \to \infty }{\lim }\nu \left(\top \right)\frac{A^{\gamma }\left(\zeta \left(\top \right)\right)}{A\left(\top \right)}=0$$

2. Preliminaries

Lemma 1.

Assume that $\varkappa >0$ is a solution of (1). Then

$${\left(N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{\prime }=-N\left(\top \right){\displaystyle Ϝ}\left(\top \right){\varkappa }^{\lambda }\left(\tau \left(\top \right)\right),$$

(8)

where $N\left(\top \right)=exp\left({\int }_{{\top }_0}^{\top }\left|\frac{d\left(s\right)}{\iota \left(s\right)}\right|\mathrm{d}s\right).$ Moreover, one of the following cases:

$$\left(Y_1\right)y\left(\top \right)>0\mathrm{and}{y}^{\prime }\left(\top \right)>0$$

or

$$\left(Y_2\right)y\left(\top \right)>0\mathrm{and}{y}^{\prime }\left(\top \right)<0$$

is satisfied.

Proof. 

Let $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive for all $\top \ge {\top }_1\ge {\top }_0$. We see that$y\left(\top \right)$ is positive. From (1), we find that

$${\left(N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{\prime }+N\left(\top \right){\displaystyle Ϝ}\left(\top \right){\varkappa }^{\lambda }\left(\tau \left(\top \right)\right)=0.$$

So, we note that $N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)$ eventually has one sign. Thus, for all sufficiently large $\top ,$ we note that $y^{\prime }(\top )$ has a fixed sign. This ends the proof. □

Lemma 2.

Assume that $\varkappa >0$ is a solution of (1) such that $\left(Y_1\right)$ holds. Then,

$$\frac{y\left(\top \right)}{y^{\prime }\left(\top \right)}>A\left(\top \right)N\left(\top \right)\iota \left(\top \right),\top \ge {\top }_1$$

(9)

and

$${\left(\frac{y}{A}\right)}^{\prime }\left(\top \right)<0.$$

Proof. 

Let $\left(Y_1\right)$ hold. Because ${\left(\iota y^{\prime }\right)}^{\prime }\left(\top \right)\le 0$ and from (1), we see that

$$y\left(\top \right)-y\left({\top }_1\right)={\int }_{{\top }_1}^{\top }\frac{N\left(s\right)\iota \left(s\right)y^{\prime }\left(s\right)}{N\left(s\right)\iota \left(s\right)}\mathrm{d}s>A\left(\top \right)N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right).$$

By (9), we have

$$\begin{array}{ccc}\hfill {\left(\frac{y}{A}\right)}^{\prime }\left(\top \right)& =& \left(y^{\prime }\left(\top \right)A\left(\top \right)-y\left(\top \right)A^{\prime }\left(\top \right)\right)\frac{1}{A^2\left(\top \right)}\hfill \\ & =& \left(y^{\prime }\left(\top \right)A\left(\top \right)\iota \left(\top \right)N\left(\top \right)-y\left(\top \right)\right)\frac{1}{\iota \left(\top \right)N\left(\top \right)A^2\left(\top \right)}<0.\hfill \end{array}$$

Thus, it is proven that $y/A$ is decreasing on $[{\top }_1,\infty )$. This ends the proof. □

Lemma 3.

Assume that $\varkappa >0$ is a solution of (1), and property (R) is satisfied. If

$${\int }_{{\top }_0}^{\infty }N\left(\top \right){\displaystyle Ϝ}\left(\top \right)\mathrm{d}\top =\infty ,$$

(10)

then $\left(Y_2\right)$ holds.

Proof. 

Let $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive for all $\top \ge {\top }_1\ge {\top }_0$. By Lemma 1, y satisfies either $\left(Y_1\right)$ or $\left(Y_2\right)$. Let $\left(Y_1\right)$ hold. By Lemma 2, ${\left(y\left(\top \right)/A\left(\top \right)\right)}^{\prime }<0$ for $\top \ge {\top }_2\ge {\top }_1$. Because $y/A$ is decreasing and y is increasing, we obtain

$$y\left(\top \right)\ge M_1\mathrm{and}y\left(\top \right)\le M_{2}A\left(\top \right),\mathrm{for}\top >{\top }_3,$$

where $M_1,$ $M_2>0$ and ${\top }_3>{\top }_2.$ Let $\epsilon \in (0,1)$. From property (R), we note that there exists a ${\top }_4\ge {\top }_3$ such that

$$\nu \left(\top \right)A^{\gamma }\left(\zeta \left(\top \right)\right)\le A\left(\top \right)M_2^{1-\gamma }\left(1-\epsilon \right)\mathrm{for}\top \ge {\top }_4.$$

and

$$\begin{array}{ccc}\hfill \varkappa \left(\top \right)& =& y\left(\top \right)-\nu \left(\top \right){\varkappa }^{\gamma }\left(\zeta \left(\top \right)\right)\ge y\left(\top \right)-\nu \left(\top \right)y^{\gamma }\left(\zeta \left(\top \right)\right)\hfill \\ & =& y\left(\top \right)-\nu \left(\top \right){\left(\frac{y\left(\zeta \left(\top \right)\right)}{A\left(\zeta \left(\top \right)\right)}\right)}^{\gamma }{A}^{\gamma }\left(\zeta \left(\top \right)\right)\hfill \\ & \ge & y\left(\top \right)-\nu \left(\top \right){\left(\frac{y\left(\top \right)}{A\left(\top \right)}\right)}^{\gamma }{A}^{\gamma }\left(\zeta \left(\top \right)\right)\hfill \\ & =& y\left(\top \right)\left(1-\left(\nu \left(\top \right)\frac{A^{\gamma }\left(\zeta \left(\top \right)\right)}{A\left(\top \right)}\right){\left(\frac{y\left(\top \right)}{A\left(\top \right)}\right)}^{\gamma -1}\right)\hfill \\ & \ge & M_1\left(1-M_2^{1-\gamma }\left(1-\epsilon \right)M_2^{\gamma -1}\right)=M_1\epsilon =:M,\mathrm{for}\top \ge {\top }_4.\hfill \end{array}$$

Thus, $M>0.$ In (8), we have

$${\left(N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{\prime }+N\left(\top \right){\displaystyle Ϝ}\left(\top \right)M^{\lambda }\le 0.$$

Integrating from ${\top }_4$ to ⊤, (10) implies

$$\begin{array}{ccc}\hfill 0& <& N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\hfill \\ & \le & N\left({\top }_4\right)\iota \left({\top }_4\right)y^{\prime }\left({\top }_4\right)-M^{\lambda }{\int }_{{\top }_4}^{\top }N\left(\top \right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\to -\infty .\hfill \end{array}$$

Thus, case $\left(Y_1\right)$ is impossible. The proof is complete. □

Lemma 4.

Assume that $\varkappa >0$ is a solution of (1), $\left(Y_2\right)$ holds, and property (R) is satisfied. If there exists a function $\rho \in C^1\left([{\top }_0,\infty ),{\mathbb{R}}^{+}\right),$ ${\rho }^{\prime }\left(\top \right)>0$ such that

$${\int }_{{\top }_0}^{\infty }\frac{1}{\iota \left(\top \right)N\left(\top \right)\rho \left(\top \right)}\left({\int }_{{\top }_0}^{\top }\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\right)\mathrm{d}\top =\infty ,$$

(11)

then

$$\underset{\top \to \infty }{lim}\varkappa \left(\top \right)=\underset{\top \to \infty }{lim}y\left(\top \right)=0.$$

Proof. 

Let $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive for all $\top \ge {\top }_1\ge {\top }_0$. Because y is decreasing, we note that

$$y(\top )\le M_3,$$

where $M_3>0$ is constant and ${\top }_2\ge {\top }_1$. In view of property (R) and the fact that $A(\top )$ is bounded and increasing, we see that

$$\underset{\top \to \infty }{lim}\nu \left(\top \right)=0,$$

there exists ${\top }_3\ge {\top }_2$ such that

$$\nu \left(\top \right)\frac{1}{\left(1-{\epsilon }_2\right)}\le M_3^{1-\gamma },{\epsilon }_2\in (0,1),$$

for $\top \ge {\top }_3$. Additionally,

$$\begin{array}{ccc}\hfill \varkappa \left(\top \right)& =& y\left(\top \right)-\nu \left(\top \right){\varkappa }^{\gamma }\left(\zeta \left(\top \right)\right)\ge y\left(\top \right)-\nu \left(\top \right)y^{\gamma }\left(\zeta \left(\top \right)\right)\hfill \\ & \ge & y\left(\top \right)-\nu \left(\top \right)y^{\gamma }\left(\top \right)\hfill \\ & =& y\left(\top \right)\left(1-\nu \left(\top \right)y^{\gamma -1}\left(\top \right)\right)\hfill \\ & \ge & y\left(\top \right)\left(1-M_3^{1-\gamma }\left(1-{\epsilon }_2\right)M_3^{\gamma -1}\left(\top \right)\right)\hfill \\ & =& {\epsilon }_{2}y\left(\top \right).\hfill \end{array}$$

In (8), we find that

$${\left(N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{\prime }+\epsilon N\left(\top \right){\displaystyle Ϝ}\left(\top \right)y^{\lambda }\left(\tau \left(\top \right)\right)\le 0,$$

(12)

for $\epsilon :={\epsilon }_2^{\lambda }$ and $\top \ge {\top }_3$. Because $y^{\prime }<0$, we obtain

$$\underset{\top \to \infty }{lim}y\left(\top \right)=:\varrho \ge 0.$$

Let $\varrho >0$. There exists ${\top }_4\ge {\top }_3$ such that $y\left(\tau \right(\top \left)\right)\ge \varrho ,$ for $\top \ge {\top }_4$. Thus,

$${\left(N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{\prime }+{\varrho }_{1}N\left(\top \right){\displaystyle Ϝ}\left(\top \right)\le 0,{\varrho }_1:=\epsilon {\varrho }^{\lambda }>0,$$

(13)

for $\top \ge {\top }_4.$ Define the function w as the following:

$$w\left(\top \right):=\rho \left(\top \right)N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right).$$

From (13), we have

$$\begin{array}{ccc}\hfill w^{\prime }\left(\top \right)& =& \rho \left(\top \right){\left(N\iota y^{\prime }\right)}^{\prime }\left(\top \right)+{\rho }^{\prime }\left(\top \right)N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\hfill \\ & \le & -{\varrho }_1\rho \left(\top \right)N\left(\top \right){\displaystyle Ϝ}\left(\top \right)+{\rho }^{\prime }\left(\top \right)N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\hfill \\ & \le & -{\varrho }_1\rho \left(\top \right)N\left(\top \right){\displaystyle Ϝ}\left(\top \right),\top \ge {\top }_4.\hfill \end{array}$$

Integrating from ${\top }_4$ to $\top ,$ we obtain

$$\begin{array}{ccc}\hfill w(\top )& \le & w\left({\top }_4\right)-{\varrho }_1{\int }_{{\top }_4}^{\top }\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\hfill \\ & \le & -{\varrho }_1{\int }_{{\top }_4}^{\top }\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

That is,

$$y^{\prime }\left(\top \right)\le -\frac{{\varrho }_1}{\rho \left(\top \right)N\left(\top \right)\iota \left(\top \right)}{\int }_{{\top }_4}^{\top }\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s.$$

Again, integrating from ${\top }_4$ to ⊤, we obtain

$$y(\top )\le y\left({\top }_4\right)-{\varrho }_1{\int }_{{\top }_4}^{\top }\frac{1}{\iota \left(u\right)N\left(u\right)\rho \left(u\right)}\left({\int }_{{\top }_4}^s\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\right)\mathrm{d}u\to -\infty .$$

We see that, $\varrho =0$. On the other hand, from the definition of y, we note that ${lim}_{\top \to \infty }\varkappa (\top )=0$. This ends the proof. □

3. Oscillation Results

In this section, we introduce some theorems that include new criteria for ensuring the oscillation of solutions of (1).

Theorem 1.

Let property (R) be satisfied. If there exists a function $\rho \in C^1\left([{\top }_0,\infty ),{\mathbb{R}}^{+}\right),$ ${\rho }^{\prime }>0,$ and (11) holds, then every solution of (1) is either ${lim}_{\top \to \infty }\varkappa (\top )=0$ or oscillatory.

Proof. 

Let $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive on $[{\top }_1,\infty )$ for ${\top }_1\ge {\top }_0$. Because $\rho$ is an increasing function and

$${\int }_{{\top }_0}^{\infty }\frac{1}{\iota \left(\top \right)N\left(\top \right)\rho \left(\top \right)}\left({\int }_{{\top }_0}^{\top }\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\right)\mathrm{d}\top =\infty$$

this leads us to note that

$${\int }_{{\top }_0}^{\top }\rho \left(s\right)N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s=\infty .$$

By using, (10) in Lemma 3 is satisfied. □

Theorem 2.

Assume that property (R) is satisfied and (10) holds. If

$${\int }_{{\top }_0}^{\infty }\frac{\left({\int }_{{\top }_0}^{\top }{\displaystyle Ϝ}\left(s\right)N\left(s\right)A^{\lambda }\left(\tau \left(s\right)\right)\mathrm{d}s\right)}{\iota \left(\top \right)N\left(\top \right)}\mathrm{d}\top =\infty ,$$

(14)

then (1) is oscillatory.

Proof. 

Let $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive on $[{\top }_1,\infty )$ for ${\top }_1\ge {\top }_0$. By (10) and Lemma 3, y satisfies $\left(Y_2\right)$ for all $\top \ge {\top }_2\ge {\top }_1$. In the same manner as the proof of Lemma 4, we obtain (12). Because ${\left(N\iota y^{\prime }\right)}^{\prime }\le 0$, we have

$$\begin{array}{ccc}\hfill y\left(\top \right)& \ge & -{\int }_{\top }^{\infty }\frac{1}{\iota \left(s\right)N\left(s\right)}N\left(s\right)\iota \left(s\right)y^{\prime }\left(s\right)\mathrm{d}s\hfill \\ & \ge & -A\left(\top \right)N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right),\hfill \end{array}$$

therefore

$$\left(\frac{y}{A}\right)\mathrm{is}\mathrm{nondecreasing}.$$

So,

$$\frac{y\left(\top \right)}{A\left(\top \right)}\ge \overline{M}\mathrm{for}\top \ge T_a,$$

where $\overline{M}>0$ is a constant and $\top \ge {\top }_3.$ In (12), we see that

$${\left(N\iota y^{\prime }\right)}^{\prime }\left(\top \right)+M_{4}N\left(\top \right){\displaystyle Ϝ}\left(\top \right)A^{\lambda }\left(\tau \left(\top \right)\right)\le 0,M_4:=\epsilon {\overline{M}}^{\lambda }\mathrm{for}\top \ge T_a.$$

(15)

By integrating from $T_a$ to ⊤ implies

$$\begin{array}{ccc}\hfill N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)& \le & N\left(T_a\right)\iota \left(T_a\right)y^{\prime }\left(T_a\right)-M_4{\int }_{T_a}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)A^{\lambda }\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ & \le & -M_4{\int }_{T_a}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)A^{\lambda }\left(\tau \left(s\right)\right)\mathrm{d}s.\hfill \end{array}$$

Integrating from $T_a$ to ⊤ and taking (1) into account, we obtain

$$0<y\left(\top \right)\le y\left(T_a\right)-M_4{\int }_{T_a}^{\top }\frac{\left({\int }_{T_a}^{\top }N\left(u\right){\displaystyle Ϝ}\left(u\right)A^{\lambda }\left(\tau \left(u\right)\right)\mathrm{d}u\right)}{\iota \left(s\right)N\left(s\right)}\mathrm{d}s\to -\infty .$$

The proof is complete. □

Theorem 3.

Let property (R) be satisfied and (10) hold. If

$$\underset{\top \to \infty }{lim\; sup}\left(A\left(\top \right){\int }_{{\top }_0}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\right)>1$$

(16)

and $\lambda =1$, then (1) is oscillatory.

Proof. 

Assume that $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive on $[{\top }_1,\infty )$ for some ${\top }_1\ge {\top }_0$. By (2) and Lemma 3, we find that y satisfies $\left(Y_2\right)$ for all $\top \ge {\top }_2\ge {\top }_1$. Furthermore, by Lemma 4, y satisfies (12). Integrating (12), we have

$$\begin{array}{ccc}\hfill -N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)& \ge & \epsilon {\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)y^{\lambda }\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ & \ge & \epsilon y^{\lambda }\left(\tau \left(\top \right)\right){\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(17)

Using (12) in (17), we have

$$\begin{array}{ccc}\hfill -N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)& \ge & \epsilon y^{\lambda }\left(\top \right){\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\hfill \\ & \ge & \epsilon A^{\lambda }\left(\top \right){\left(-N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{\lambda }{\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s,\hfill \end{array}$$

that is,

$${\left(-N\left(\top \right)\iota \left(\top \right)y^{\prime }\left(\top \right)\right)}^{1-\lambda }\ge \epsilon A^{\lambda }\left(\top \right){\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s,\mathrm{for}\mathrm{any}\epsilon \in (0,1)\mathrm{and}\top \ge {\top }_3.$$

(18)

If $\lambda =1$, then (18) becomes

$$1\ge \epsilon A\left(\top \right){\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s,\mathrm{for}\mathrm{any}\epsilon \in (0,1).$$

The proof is complete. □

Theorem 4.

Let property (R) be satisfied, ${\tau }^{\prime }\left(\top \right)\ge 0$, and (2) hold. Then, (1) is oscillatory if one of the following statements holds:

$$\left(\mathbf{i}\right)\underset{\top \to \infty }{lim\; inf}{\int }_{\tau \left(\top \right)}^{\top }\frac{1}{\iota \left(s\right)N\left(s\right)}\left({\int }_{{\top }_0}^{s}N\left(u\right){\displaystyle Ϝ}\left(u\right)\mathrm{d}u\right)\mathrm{d}s>\frac{1}{e}\mathit{when}\lambda =1$$

(19)

or

$$\left(\mathbf{ii}\right){\int }_{{\top }_0}^{\infty }\frac{1}{\iota \left(\top \right)N\left(\top \right)}\left({\int }_{{\top }_0}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s\right)\mathrm{d}\top =\infty \mathit{when}\lambda <1.$$

(20)

Proof. 

Assume that $\varkappa \left(\top \right)>0,$ that is, $\varkappa \left(\zeta \left(\top \right)\right)$ and $\varkappa \left(\tau \left(\top \right)\right)$ are positive on $[{\top }_1,\infty )$ for some ${\top }_1\ge {\top }_0$. From (10), Lemma 3 implies that y satisfies $\left(Y_2\right)$ for all $\top \ge {\top }_2\ge {\top }_1$. From the proof of Theorem 3, we obtain (17). Thus, $y>0$ is a solution of the following inequality

$$y^{\prime }\left(\top \right)+\epsilon \frac{{\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s}{\iota \left(\top \right)N\left(\top \right)}{y}^{\lambda }\left(\tau \left(\top \right)\right)\le 0,\mathrm{for}\mathrm{all}\epsilon \in (0,1),$$

for $\top \ge {\top }_3$. By Theorem 5.1.1 in [25], the associated DDE

$$y^{\prime }\left(\top \right)+\epsilon \frac{{\int }_{{\top }_3}^{\top }N\left(s\right){\displaystyle Ϝ}\left(s\right)\mathrm{d}s}{\iota \left(\top \right)N\left(\top \right)}{y}^{\lambda }\left(\tau \left(\top \right)\right)=0$$

(21)

also has a positive solution. Hence, Theorem 2.1.1 in [26] implies that if $\lambda =1,$ then (19) ensures oscillation of (21) and if $\lambda <1,$ then (20) ensures oscillation of (21). Thus, (1) cannot have an eventually positive solution; this contradiction completes the proof. □

Example 1.

Let us consider following the damped DE with a superlinear neutral term:

$${\left({\top }^2{\left(\varkappa \left(\top \right)+{\top }^{-2}{\varkappa }^3\left(2\top \right)\right)}^{\prime }\right)}^{\prime }+\top sin\left(\top \right)y^{\prime }\left(\top \right)+{\top }^3{\varkappa }^3\left(\frac{\top }{3}\right)=0,\top \in [1,\infty ).$$

(22)

Note that $\iota \left(\top \right)={\top }^2,$ $\nu \left(\top \right)=\frac{1}{{\top }^2},$ ${\displaystyle Ϝ}\left(\top \right)={\top }^3,$ $\zeta \left(\top \right)=2\top ,$ $d\left(\top \right)=\top sin\left(\top \right),$ $\tau \left(\top \right)=\frac{1}{3}\top$. We see that

$${\int }_{{\top }_0}^{\top }\left|\frac{sin\left(s\right)}{s}\right|\mathrm{d}s=\Psi >0.$$

Therefore,

$$N\left(\top \right)=W>0.$$

From (10), we get

$$W{\int }_{{\top }_0}^{\infty }{\top }^{3}d\top =\infty$$

and

$$\underset{\top \to \infty }{lim}\frac{A^{\gamma }\left(\zeta \left(\top \right)\right)}{A\left(\top \right)}\nu \left(\top \right)=0,$$

hence, property (R) is satisfied. Furthermore,

$$\begin{array}{ccc}\hfill {\int }_{{\top }_0}^{\infty }\frac{\left({\int }_{{\top }_0}^{\top }{\displaystyle Ϝ}\left(s\right)N\left(s\right)A^{\lambda }\left(\tau \left(s\right)\right)\mathrm{d}s\right)}{\iota \left(\top \right)N\left(\top \right)}\mathrm{d}\top & =& \\ \hfill 3{\int }_{{\top }_0}^{\infty }\frac{1}{{\top }^2}\left({\int }_{{\top }_0}^{\top }s\mathrm{d}s\right)\mathrm{d}\top & =& \infty .\hfill \end{array}$$

Thus, (14) holds. By Theorem 2, DE (22) is oscillatory.

Remark 1.

All results presented in references [15,16,18,19,20,21,22,23,24] are not applicable to example (22); this is because the function is oscillatory, whereas all the results in the references are applied by assuming $d\left(\top \right)$ is a positive function.

4. Conclusions

In this paper, we studied the oscillatory behavior of the solutions of the Equation (1), and we obtained some new criteria that guarantee that all the solutions of the studied equation oscillate. We find that the results previously obtained in [16,21,27,28,29] include the conditions $\nu \left(\top \right)\to 0$ as $\top \to \infty$ and $\gamma \in [1,\infty ).$ Furthermore, the results mentioned in [7,20,23,30] require the condition $\gamma =1$, and this means that all these results are not applicable to our examples, so the results presented in this paper are new and complementary to the results found in previous literature. It will be interesting to discuss the results for Equation (1) in a more general form, such as:

$${\left(\iota \left(\top \right){\left(\varkappa \left(\top \right)+\nu \left(\top \right){\varkappa }^{\gamma }\left(\zeta \left(\top \right)\right)\right)}^{\prime }\right)}^{\prime }+d\left(\top \right)y^{\prime }\left(\top \right)+\sum _{n=1}^m{{\displaystyle Ϝ}}_n\left(\top \right){\varkappa }^{\lambda }\left({\tau }_n\left(\top \right)\right)=0.$$