Second-Order Neutral Differential Equations with a Sublinear Neutral Term: Examining the Oscillatory Behavior

Abstract

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings.

1. Introduction

Differential equations (DEs) are crucial in numerous scientific and engineering fields because they effectively represent complex systems and natural phenomena. They are essential in describing how physical quantities change over time or space, making them fundamental in fields such as physics, chemistry, biology, economics, and engineering. For instance, in physics, DEs are used to model the motion of particles, the propagation of heat, and the behavior of electrical circuits. In biology, they help in understanding population dynamics and the spread of diseases. In economics, DEs are employed to model market dynamics and economic growth. Their versatility and wide-ranging applications underscore their importance in both theoretical research and practical problem-solving, driving advancements in technology and contributing to our understanding of the world around us (see [1,2,3,4,5,6,7]).

Neutral differential equations (NDEs) are of significant importance in various scientific and engineering disciplines due to their ability to model systems where delays play a crucial role. Unlike traditional differential equations, neutral equations allow for the inclusion of delays that depend not only on the current state but also on past states of the system. This feature makes them particularly suitable for describing phenomena such as biological processes with memory effects, chemical reactions with transport delays, and mechanical systems with feedback mechanisms. The utility of NDEs extends to diverse fields, including control theory, population dynamics, neuroscience, and, more recently, the study of complex networks and cyber-physical systems. Their ability to capture intricate temporal dynamics makes them a powerful tool for theoretical analysis and practical applications alike (see [8,9,10,11,12,13,14,15,16]).

Emden–Fowler delay differential equations (DDEs) are a class of DEs characterized by their delay terms and specific functional form, originally introduced by Robert Emden and later expanded upon by Arthur Fowler. They are named after these mathematicians due to their significant contributions to celestial mechanics and astrophysics, where these equations find critical applications in modeling physical phenomena such as stellar structure and planetary atmospheres. Their importance lies in their ability to describe systems with time delays, allowing for more accurate and realistic modeling of dynamical systems where past states influence present behavior. These equations are pivotal in various fields of science and engineering, facilitating the study of oscillatory and asymptotic behaviors in complex systems with delayed feedback (see [17,18,19,20,21]).

Various oscillation criteria for NDEs impose specific constraints on their coefficients.

Agarwal et al. [22], Kusano et al. [23], Sun and Meng [24], and Džurina and Stavroulakis [25] noted analogous properties between the nonlinear NDE

$${\left(\mathfrak{a}\left(s\right){\left|y^{\prime }\left(s\right)\right|}^{\gamma -1}{y}^{\prime }\left(s\right)\right)}^{\prime }+\mathfrak{q}\left(s\right){\left|y\left(\mu \left(s\right)\right)\right|}^{\gamma -1}y\left(\mu \left(s\right)\right)=0,$$

and its corresponding linear form

$${\left(\mathfrak{a}\left(s\right)y^{\prime }\left(s\right)\right)}^{\prime }+\mathfrak{q}\left(s\right)y\left(s\right)=0.$$

Zhang et al. [26] and Han et al. [27,28] studied the oscillation of second-order linear NDEs of the form

$${\left(\mathfrak{a}\left(s\right){\left(y\left(s\right)+\mathrm{u}\left(s\right)y\left(\eta \left(s\right)\right)\right)}^{\prime }\right)}^{\prime }+\mathfrak{q}\left(s\right)y\left(\mu \left(s\right)\right)=0,$$

introducing new criteria under the condition $0\le \mathrm{u}\left(s\right)\le {\mathrm{u}}_0<\infty .$

Zhang et al. [29] investigated a specific class of second-order NDEs given by

$${\left(\mathfrak{a}\left(s\right){\left|h^{\prime }\left(s\right)\right|}^{\gamma -1}{h}^{\prime }\left(s\right)\right)}^{\prime }+\mathfrak{q}\left(s\right){\left|y\left(\mu \left(s\right)\right)\right|}^{\gamma -1}y\left(\mu \left(s\right)\right)=0,$$

where $h\left(s\right)=y\left(s\right)+{\sum }_{i=1}^m{\mathrm{u}}_i\left(s\right)y\left({\eta }_i\left(s\right)\right),$ streamlining the analysis of these equations.

Moaaz et al. [12] examined the oscillatory characteristics of NDEs of the form

$${\left(\mathfrak{a}\left(s\right){\left({\left(y\left(s\right)+\mathrm{u}\left(s\right)y\left(\eta \left(s\right)\right)\right)}^{\prime }\right)}^{\gamma }\right)}^{\prime }+\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right)y^{\gamma }\left({\mu }_i\left(s\right)\right)=0.$$

They introduced new monotonic properties for the solutions, characterized by an iterative nature, and derived oscillation conditions ensuring that all solutions oscillate.

Sun [30] established new oscillation criteria for second-order nonlinear NDEs of the form

$${\left(\mathfrak{a}\left(s\right){\left|h^{\prime }\left(s\right)\right|}^{\gamma -1}{h}^{\prime }\left(s\right)\right)}^{\prime }+\mathfrak{q}\left(s\right)f\left(s,y\left(\mu \left(s\right)\right)\right)=0,$$

where $h\left(s\right):=y\left(s\right)+\mathrm{u}\left(s\right)y\left(\eta \left(s\right)\right).$ They utilized a novel inequality to derive these oscillation criteria.

Agarwal et al. [31] established criteria for the oscillatory behavior of second-order DEs featuring a neutral term:

$${\left(\mathfrak{a}\left(s\right){\left(y\left(s\right)+\mathrm{u}\left(s\right)y^{\alpha }\left(\eta \left(s\right)\right)\right)}^{\prime }\right)}^{\prime }+\mathfrak{q}\left(s\right)y\left(\mu \left(s\right)\right)=0$$

under the conditions

$${\int }_{s_0}^{\infty }{\displaystyle \frac{1}{\mathfrak{a}\left(\mathsf{\ell }\right)}}\mathrm{d}\mathsf{\ell }=\infty ,$$

and

$${\int }_{s_0}^{\infty }{\displaystyle \frac{1}{\mathfrak{a}\left(\mathsf{\ell }\right)}}\mathrm{d}\mathsf{\ell }<\infty .$$

Tamilvanan et al. [32] explored the oscillatory characteristics of second-order Emden–Fowler DEs incorporating sublinear neutral terms, represented by

$${\left(\mathfrak{a}\left(s\right){\left(y\left(s\right)+\mathrm{u}\left(s\right)y^{\alpha }\left(\eta \left(s\right)\right)\right)}^{\prime }\right)}^{\prime }+\mathfrak{q}\left(s\right)y^{\beta }\left(\mu \left(s\right)\right)=0.$$

Wu et al. [33] focused on the oscillatory properties of second-order Emden–Fowler DDEs with a sublinear neutral term given by

$${\left(\mathfrak{a}\left(s\right){\left({\left(y\left(s\right)+\mathrm{u}\left(s\right)y^{\alpha }\left(\eta \left(s\right)\right)\right)}^{\prime }\right)}^{\gamma }\right)}^{\prime }+\mathfrak{q}\left(s\right)y^{\beta }\left(\mu \left(s\right)\right)=0.$$

They employed the Riccati method to derive several oscillation criteria for this equation.

In this paper, we investigate the second-order half-linear Emden–Fowler DDEs featuring a sublinear neutral term, given by

$${\left(\mathfrak{a}\left(s\right){\left({\left(y\left(s\right)+\mathrm{u}\left(s\right)y^{\alpha }\left(\eta \left(s\right)\right)\right)}^{\prime }\right)}^{\gamma }\right)}^{\prime }+\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right)y^{\beta }\left({\mu }_i\left(s\right)\right)=0,$$

(1)

where $s\ge s_0.$ Throughout this study, we make the following assumptions:

(H₁)

$0<\alpha \le 1,$ and $\beta$ and $\gamma$ are ratios of two positive odd integers;

(H₂)

$\eta ,$ ${\mu }_i\in C^1([s_0,\infty ),\mathbb{R}),$ $\eta \left(s\right)\le s,$ ${\mu }_i\left(s\right)\le s,$ ${\mu }_i^{\prime }\left(s\right)>0,$ and${lim}_{s\to \infty }\eta \left(s\right)={lim}_{s\to \infty }{\mu }_i\left(s\right)$ $=\infty ,$ $i=1,2,\cdots ,n;$

(H₃)

$\mathfrak{a}\in C^1([s_0,\infty ),\left(0,\infty \right)),$ ${\mathfrak{a}}^{\prime }\left(s\right)\ge 0,$ $\mathrm{u},$ $\mathfrak{q}\in C([s_0,\infty ),\left(0,\infty \right)),$ ${lim}_{s\to \infty }$u$\left(s\right)=0$, and $\mathfrak{q}\left(s\right)$ is not eventually zero on $[s^{*},\infty )$ for $s^{*}\ge s_0.$

A function $y\in C^1([S_y,\infty ),\mathbb{R})$, $S_y⩾s_0,$ is said to be a solution of (1), which has the property $\mathfrak{a}{\left(h^{\prime }\right)}^{\gamma }\in C^1[S_y,\infty )$ and satisfies Equation (1) on $[S_y,\infty )$. We consider only that those solutions y of (1) which exist on some half-line $[S_y,\infty )$ and satisfy the condition

$$sup\left\{\right|y\left(s\right)|:s⩾S\}>0,\mathrm{for}\mathrm{all}S\ge S_y.$$

A solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.

In this paper, we derive new oscillation criteria for Equation (1) under specific conditions using generalized Riccati inequalities. Our approach is aligned with the methodology employed in [33]. Our results are also an extension of the results in [33], which consider the special case of (1) when $n=1$.

Therefore, our aim in this paper is to provide oscillation criteria that cover a larger area when applied by taking into account the existence of multiple delay arguments and studying Equation (1), which is a generalization of many different types of differential equations, such as linear, half-linear, nonlinear, and quasilinear differential equations. This, in turn, motivates us to generalize many oscillation criteria in previous works, such as [30,31,33,34]. Also, our study is distinguished by its coverage of canonical and noncanonical cases, unlike its predecessors, which are limited to studying only one of them.

Here, we focus on the qualitative oscillatory behavior of solutions to functional differential equations, which is generally independent of specific initial conditions. Including initial functions would limit the general applicability of our results, as oscillation criteria are derived from the equation’s structural properties (Győri and Ladas [5]). Oscillatory behavior is an asymptotic property and is thus determined by the coefficient of the equation rather than initial data (Zhang and Erbe [35] and Kiguradze and Chanturia [36]). This approach is consistent with established practices in oscillation theory, aiming to provide a broad analysis of solution behavior (Agarwal et al. [21]). We have chosen to omit initial functions to maintain focus on the general oscillatory properties and avoid unnecessary complexity.

2. Auxiliary Results

This section introduces several key lemmas related to the monotonic behavior of nonoscillatory solutions for the equations under investigation. These results are useful in deriving the main findings. For analytical convenience, we define

$$h\left(s\right):=y\left(s\right)+\mathrm{u}\left(s\right)y^{\alpha }\left(\eta \left(s\right)\right).$$

(2)

$$\mu \left(s\right):=min\left\{{\mu }_i\left(s\right):i=1,2,...,n\right\},$$

$$\widehat{\mu }\left(s\right):=max\left\{{\mu }_i\left(s\right):i=1,2,...,n\right\},$$

$$\pi \left(s\right):={\int }_s^{\infty }{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\alpha }\left(\mathsf{\ell }\right)}}\mathrm{d}\mathsf{\ell },$$

$$\tilde{\mathrm{k}}\left(s\right):=\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-{\displaystyle \frac{\mathrm{u}\left({\mu }_i\left(s\right)\right)}{c_1^{1-\alpha }}}\right)}^{\beta },$$

and

$$\widehat{\mathrm{k}}\left(s\right):=\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-\mathrm{u}\left({\mu }_i\left(s\right)\right){\displaystyle \frac{{\pi }^{\alpha }\left(\eta \left({\mu }_i\left(s\right)\right)\right)}{c_0^{1-\alpha }\pi \left({\mu }_i\left(s\right)\right)}}\right)}^{\beta }.$$

Lemma 1 

([37]). Let γ be a ratio of two odd positive integers, and let $\mathfrak{a}>0$ and B be constants. Then,

$$Bu-\mathfrak{a}{u}^{\left(\gamma +1\right)/\gamma }\le {\displaystyle \frac{{\gamma }^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{B^{\gamma +1}}{{\mathfrak{a}}^{\gamma }}},\mathfrak{a}>0.$$

(3)

Lemma 2 

([36]). Let $y\in {\mathbf{C}}^n\left([s_0,\infty ),\left(0,\infty \right)\right)$, $y^{\left(i\right)}\left(s\right)>0$ for $i=1,2,\cdots ,n$, and $y^{\left(n+1\right)}\left(s\right)\le 0$, eventually. Then, eventually, $y\left(s\right)/y^{\prime }\left(s\right)\ge ϵs/n$ for every $ϵ\in \left(0,1\right)$.

Lemma 3. 

Assume that $y\left(s\right)$ is an eventually positive solution of (1). Then, $y\left(s\right)$ eventually satisfies the following cases:

$$\begin{array}{ccc}\hfill {\mathrm{C}}_1& :& h>0,h^{\prime }>0,{\left(\mathfrak{a}{\left(h^{\prime }\right)}^{\gamma }\right)}^{\prime }<0,\hfill \\ \hfill {\mathrm{C}}_2& :& h>0,h^{\prime }<0,h^{″}<0,\hfill \end{array}$$

for $s⩾s_1⩾s_0.$

Proof. 

Let us assume that $y\left(s\right)$ is an eventually positive solution of (1). Given condition $\left({\mathrm{H}}_2\right)$, there exists an $s_1\ge s_0$ such that $y\left({\mu }_i\left(s\right)\right)>0$ and $y\left(\eta \left(s\right)\right)>0$ for all $s>s_1$. Consequently, $h\left(s\right)>0$ for all $s>s_1$. From (1), we can write

$${\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }=-\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right)y^{\beta }\left({\mu }_i\left(s\right)\right)<0.$$

This implies that $\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }$ is a decreasing function. Therefore, $\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }$ must eventually be either positive or negative. Hence, the proof is now complete. □

Lemma 4. 

If $h^{\prime }\left(s\right)>0$, then

$$y\left(s\right)\ge \left(1-{\displaystyle \frac{\mathrm{u}\left(s\right)}{c_1^{1-\alpha }}}\right)h\left(s\right),$$

(4)

and

$${\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le -\tilde{\mathrm{k}}\left(s\right)h^{\beta }\left(\mu \left(s\right)\right).$$

(5)

Proof. 

Notice that $h^{\prime }\left(s\right)>0$. Hence, there is a constant $c_1>0$ such that $h\left(s\right)\ge c_1$ for all sufficiently large s. By (2), we obtain

$$\begin{array}{ccc}\hfill y\left(s\right)& =& h\left(s\right)-\mathrm{u}\left(s\right)y^{\alpha }\left(\eta \left(s\right)\right)\ge h\left(s\right)-\mathrm{u}\left(s\right)h^{\alpha }\left(\eta \left(s\right)\right)\hfill \\ & =& h\left(s\right)-\mathrm{u}\left(s\right)h^{\alpha }\left(s\right)=\left(1-\mathrm{u}\left(s\right)h^{\alpha -1}\left(s\right)\right)h\left(s\right).\hfill \end{array}$$

(6)

Since $0<\alpha \le 1$ and $h\left(s\right)\ge c_1,$ then

$$h^{\alpha -1}\left(s\right)\le c_1^{\alpha -1}={\displaystyle \frac{1}{c_1^{1-\alpha }}}.$$

Putting the last inequality into (6), we have

$$y\left(s\right)\ge \left(1-{\displaystyle \frac{\mathrm{u}\left(s\right)}{c_1^{1-\alpha }}}\right)h\left(s\right).$$

From (1), we have

$$\begin{array}{ccc}\hfill {\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }& =& -\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right)y^{\beta }\left({\mu }_i\left(s\right)\right)\hfill \\ & \le & -\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-{\displaystyle \frac{\mathrm{u}\left({\mu }_i\left(s\right)\right)}{c_1^{1-\alpha }}}\right)}^{\beta }{h}^{\beta }\left({\mu }_i\left(s\right)\right).\hfill \end{array}$$

Since $h^{\prime }\left(s\right)>0$ and ${\mu }_i\left(s\right)\le s,$ then

$${\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le -h^{\beta }\left(\mu \left(s\right)\right)\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-{\displaystyle \frac{\mathrm{u}\left({\mu }_i\left(s\right)\right)}{c_1^{1-\alpha }}}\right)}^{\beta }=-\tilde{\mathrm{k}}\left(s\right)h^{\beta }\left(\mu \left(s\right)\right),$$

and thus,

$${\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le -\tilde{\mathrm{k}}\left(s\right)h^{\beta }\left(\mu \left(s\right)\right).$$

□

Lemma 5. 

If $h^{\prime }\left(s\right)<0$, then

$$y\left(s\right)\ge h\left(s\right)\left[1-\mathrm{u}\left(s\right){\displaystyle \frac{{\pi }^{\alpha }\left(\eta \left(s\right)\right)}{c_0^{1-\alpha }\pi \left(s\right)}}\right],$$

(7)

and

$${\left(\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\ge \widehat{\mathrm{k}}\left(s\right)h^{\beta }\left(s\right).$$

(8)

Proof. 

Since $h^{\prime }\left(s\right)<0,$ then

$$-h\left(s\right)\le {\int }_s^{\infty }{h}^{\prime }\left(\mathsf{\ell }\right)\mathrm{d}\mathsf{\ell }={\int }_s^{\infty }{\displaystyle \frac{{\mathfrak{a}}^{1/\gamma }\left(\mathsf{\ell }\right)h^{\prime }\left(\mathsf{\ell }\right)}{{\mathfrak{a}}^{1/\gamma }\left(\mathsf{\ell }\right)}}\mathrm{d}\mathsf{\ell }\le {\mathfrak{a}}^{1/\gamma }\left(s\right)h^{\prime }\left(s\right)\pi \left(s\right).$$

Thus,

$$h\left(s\right)+{\mathfrak{a}}^{1/\gamma }\left(s\right)h^{\prime }\left(s\right)\pi \left(s\right)\ge 0.$$

(9)

So, we obtain

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}s}}\left({\displaystyle \frac{h\left(s\right)}{\pi \left(s\right)}}\right)\ge 0.$$

(10)

Then,

$$\begin{array}{ccc}\hfill y\left(s\right)& \ge & h\left(s\right)-\mathrm{u}\left(s\right)h^{\alpha }\left(\eta \left(s\right)\right)\hfill \\ & \ge & h\left(s\right)-\mathrm{u}\left(s\right){\left({\displaystyle \frac{\pi \left(\eta \left(s\right)\right)}{\pi \left(s\right)}}\right)}^{\alpha }{h}^{\alpha }\left(s\right)\hfill \\ & =& \left[1-\mathrm{u}\left(s\right){\left({\displaystyle \frac{\pi \left(\eta \left(s\right)\right)}{\pi \left(s\right)}}\right)}^{\alpha }{h}^{\alpha -1}\left(s\right)\right]h\left(s\right).\hfill \end{array}$$

Since the function $h/\pi$ is positive and increasing, then there are an $s_1\ge s_0$ and a positive constant $c_0$ such that $h\left(s\right)/\pi \left(s\right)\ge c_0$ for $s\ge s_1.$ Hence, $h^{\alpha -1}\left(s\right)\le c_0^{\alpha -1}{\pi }^{\alpha -1}\left(s\right)$, and so

$$y\left(s\right)\ge \left[1-\mathrm{u}\left(s\right){\displaystyle \frac{{\pi }^{\alpha }\left(\eta \left(s\right)\right)}{c_0^{1-\alpha }\pi \left(s\right)}}\right]h\left(s\right).$$

From (1), we have

$$\begin{array}{ccc}\hfill {\left(\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }& =& \sum _{i=1}^n{\mathfrak{q}}_i\left(s\right)y^{\beta }\left({\mu }_i\left(s\right)\right)\hfill \\ & \ge & \sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-\mathrm{u}\left({\mu }_i\left(s\right)\right){\displaystyle \frac{{\pi }^{\alpha }\left(\eta \left({\mu }_i\left(s\right)\right)\right)}{c_0^{1-\alpha }\pi \left({\mu }_i\left(s\right)\right)}}\right)}^{\beta }{h}^{\beta }\left({\mu }_i\left(s\right)\right).\hfill \end{array}$$

Since $h^{\prime }\left(s\right)<0,$ and ${\widehat{\mu }}_i\left(s\right)\le s,$ then

$$\begin{array}{ccc}\hfill {\left(\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }& \ge & h^{\beta }\left(\widehat{\mu }\left(s\right)\right)\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-\mathrm{u}\left({\mu }_i\left(s\right)\right){\displaystyle \frac{{\pi }^{\alpha }\left(\eta \left({\mu }_i\left(s\right)\right)\right)}{c_0^{1-\alpha }\pi \left({\mu }_i\left(s\right)\right)}}\right)}^{\beta }\hfill \\ & =& \widehat{\mathrm{k}}\left(s\right)h^{\beta }\left(\widehat{\mu }\left(s\right)\right)\ge \widehat{\mathrm{k}}\left(s\right)h^{\beta }\left(s\right).\hfill \end{array}$$

□

3. Main Results

This section presents a collection of theorems that establish different criteria for the oscillation of solutions to (1) in both canonical and noncanonical cases. Each criterion is meticulously derived using an appropriate Riccati transformation, ensuring a comprehensive analysis of the oscillatory behavior.

3.1. Canonical Case

In this subsection, we consider the canonical case, characterized by the condition

$${\int }_{s_0}^s{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\alpha }\left(\mathsf{\ell }\right)}}\mathrm{d}\mathsf{\ell }\to \infty \mathrm{as}s\to \infty .$$

(11)

Theorem 1. 

Assume that (11) holds. If $\beta \ge \gamma$ and there is a nondecreasing function $\varrho \in C^1([s_0,\infty ),\left(0,\infty \right))$ such that

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left(\varrho \left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right)-{\displaystyle \frac{1}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mu \left(\mathsf{\ell }\right)\right){\left({\varrho }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\gamma +1}}{{\left(c_2\varrho \left(\mathsf{\ell }\right){\mu }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\gamma }}}\right)\mathrm{d}\mathsf{\ell }=\infty$$

(12)

holds for every $c_1,c_2>0,$ then (1) is oscillatory.

Proof. 

We suppose for contradiction that (1) has an eventually positive solution $y\left(s\right)$. From (11), we find that (1) fulfills case ${\mathrm{C}}_1.$ Now, we define the function $\omega \left(s\right)$ by

$$\omega \left(s\right):=\varrho \left(s\right){\displaystyle \frac{\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }}{h^{\beta }\left(\mu \left(s\right)\right)}},$$

(13)

which yields $\omega \left(s\right)>0,$ $s\ge s_1,$

$${\omega }^{\prime }\left(s\right)={\varrho }^{\prime }\left(s\right){\displaystyle \frac{\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }}{h^{\beta }\left(\mu \left(s\right)\right)}}+\varrho \left(s\right){\displaystyle \frac{{\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }}{h^{\beta }\left(\mu \left(s\right)\right)}}-\beta \varrho \left(s\right){\mu }^{\prime }\left(s\right){\displaystyle \frac{\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }{h}^{\prime }\left(\mu \left(s\right)\right)}{h^{\beta +1}\left(\mu \left(s\right)\right)}}.$$

(14)

We see from (5), (13), and (14) that

$${\omega }^{\prime }\left(s\right)\le {\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)-\beta {\mu }^{\prime }\left(s\right)\omega \left(s\right){\displaystyle \frac{h^{\prime }\left(\mu \left(s\right)\right)}{h\left(\mu \left(s\right)\right)}}.$$

(15)

Since $\mu \left(s\right)\le s$ and ${\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le 0,$ then

$${\mathfrak{a}}^{1/\gamma }\left(s\right)h^{\prime }\left(s\right)\le {\mathfrak{a}}^{1/\gamma }\left(\mu \left(s\right)\right)h^{\prime }\left(\mu \left(s\right)\right).$$

(16)

Substituting (16) into (15), we obtain

$${\omega }^{\prime }\left(s\right)\le {\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)-{\displaystyle \frac{\beta {\mu }^{\prime }\left(s\right)}{{\left(\varrho \left(s\right)\mathfrak{a}\left(\mu \left(s\right)\right)\right)}^{1/\gamma }}}{\left[h\left(\mu \left(s\right)\right)\right]}^{\beta /\gamma -1}{\omega }^{1/\gamma +1}\left(s\right).$$

(17)

Because $h^{\prime }>0$ and $\beta \ge \gamma$, there exist constants $c_2>0$ and $s_2\ge s_1$ such that

$$h^{\beta /\gamma -1}\left(\mu \left(s\right)\right)\ge c_2,s\ge s_2.$$

(18)

Thus, inequality (17) gives

$${\omega }^{\prime }\left(s\right)\le -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-{\displaystyle \frac{\gamma c_2{\mu }^{\prime }\left(s\right)}{{\left[\varrho \left(s\right)\mathfrak{a}\left(\mu \left(s\right)\right)\right]}^{1/\gamma }}}{\omega }^{1/\gamma +1}\left(s\right).$$

(19)

Using Lemma 1, where we define $B={\varrho }^{\prime }\left(s\right)/\varrho \left(s\right),$ $\mathfrak{a}=\gamma c_2{\mu }^{\prime }\left(s\right)/{\left[\varrho \left(s\right)\mathfrak{a}\left(\mu \left(s\right)\right)\right]}^{1/\gamma },$ and $u\left(s\right)=\omega \left(s\right),$ we can derive the following inequality:

$${\omega }^{\prime }\left(s\right)\le -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{1}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mu \left(s\right)\right){\left({\varrho }^{\prime }\left(s\right)\right)}^{\gamma +1}}{{\left(c_2\varrho \left(s\right){\mu }^{\prime }\left(s\right)\right)}^{\gamma }}}.$$

(20)

Integrating (20) from $s_3$ to $s,$ one arrives at

$${\int }_{s_3}^s\left(\varrho \left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right)-{\displaystyle \frac{1}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mu \left(\mathsf{\ell }\right)\right){\left({\varrho }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\gamma +1}}{{\left(c_2\varrho \left(\mathsf{\ell }\right){\mu }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\gamma }}}\right)\mathrm{d}\mathsf{\ell }\le \omega \left(s_3\right),$$

which contradicts (12) as $s\to \infty$. This completes the proof. □

Theorem 2. 

Assume that (11) holds. If $\beta \ge \gamma$ and there is a nondecreasing function ${\varrho }_1\in C^1([s_0,\infty ),\left(0,\infty \right))$ such that

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left(M_1^{\beta -\gamma }{\varrho }_1\left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right){\left({\displaystyle \frac{\mu \left(\mathsf{\ell }\right)}{\mathsf{\ell }}}\right)}^{\beta /ϵ}-{\displaystyle \frac{1}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mathsf{\ell }\right){\left({\varrho }_1^{\prime }\left(\mathsf{\ell }\right)\right)}^{\gamma +1}}{{\varrho }_1^{\gamma }\left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }=\infty$$

(21)

holds for every $c_1,c_2>0,$ then (1) is oscillatory.

Proof. 

We suppose for contradiction that (1) has an eventually positive solution $y\left(s\right)$. Now, we define the function ${\omega }_1\left(s\right)$ by

$${\omega }_1\left(s\right):={\varrho }_1\left(s\right){\displaystyle \frac{\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }}{h^{\gamma }\left(s\right)}},$$

(22)

which yields ${\omega }_1\left(s\right)>0,$ and

$${\omega }_1^{\prime }\left(s\right)={\varrho }_1\left(s\right){\displaystyle \frac{\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }}{h^{\gamma }\left(s\right)}}+{\varrho }_1\left(s\right){\displaystyle \frac{{\left(\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }}{h^{\gamma }\left(s\right)}}-\gamma {\varrho }_1\left(s\right){\displaystyle \frac{\mathfrak{a}\left(s\right){\left(h^{\prime }\left(s\right)\right)}^{\gamma }{h}^{\prime }\left(s\right)}{h^{\gamma +1}\left(s\right)}}.$$

(23)

We see from (5), (22), and (23) that

$${\omega }_1^{\prime }\left(s\right)\le -{\varrho }_1\left(s\right)\tilde{\mathrm{k}}\left(s\right){\displaystyle \frac{h^{\beta }\left(\mu \left(s\right)\right)}{h^{\gamma }\left(s\right)}}+{\displaystyle \frac{{\varrho }_1^{\prime }\left(s\right)}{{\varrho }_1\left(s\right)}}{\omega }_1\left(s\right)-{\displaystyle \frac{\gamma }{{\left({\varrho }_1\left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\gamma }}}{\omega }_1^{1/\gamma +1}\left(s\right).$$

(24)

From Lemma 2, it follows that

$$h\left(s\right)\ge ϵs{h}^{\prime }\left(s\right),$$

which implies

$${\displaystyle \frac{h\left(\mu \left(s\right)\right)}{h\left(s\right)}}\ge {\left({\displaystyle \frac{\mu \left(s\right)}{s}}\right)}^{1/ϵ}.$$

(25)

Substituting (25) into (24), we obtain

$${\omega }_1^{\prime }\left(s\right)\le -{\varrho }_1\left(s\right)\tilde{\mathrm{k}}\left(s\right){\displaystyle \frac{h^{\beta }\left(s\right)}{h^{\gamma }\left(s\right)}}{\left({\displaystyle \frac{\mu \left(s\right)}{s}}\right)}^{\beta /ϵ}+{\displaystyle \frac{{\varrho }_1^{\prime }\left(s\right)}{{\varrho }_1\left(s\right)}}{\omega }_1\left(s\right)-{\displaystyle \frac{\gamma }{{\left({\varrho }_1\left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\gamma }}}{\omega }_1^{1/\gamma +1}\left(s\right).$$

(26)

Since $h^{\prime }\left(s\right)>0$ and $\beta \ge \gamma$, there exist an $s_1\ge s_0$ and a constant $M_1>0$ such that

$$h^{\beta -\gamma }\left(s\right)>M_1^{\beta -\gamma }.$$

(27)

Thus, inequality (26) gives

$${\omega }_1^{\prime }\left(s\right)\le -M_1^{\beta -\gamma }{\varrho }_1\left(s\right)\tilde{\mathrm{k}}\left(s\right){\left({\displaystyle \frac{\mu \left(s\right)}{s}}\right)}^{\beta /ϵ}+{\displaystyle \frac{{\varrho }_1^{\prime }\left(s\right)}{{\varrho }_1\left(s\right)}}{\omega }_1\left(s\right)-{\displaystyle \frac{\gamma }{{\left({\varrho }_1\left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\gamma }}}{\omega }_1^{\left(1+\gamma \right)/\gamma }\left(s\right).$$

(28)

Using Lemma 1, where we define $B={\varrho }_1^{\prime }\left(s\right)/{\varrho }_1\left(s\right),$ $\mathfrak{a}=\gamma /{\left({\varrho }_1\left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\gamma },$ and $u\left(s\right)={\omega }_1\left(s\right),$ we can derive the following inequality:

$${\omega }_1^{\prime }\left(s\right)\le -M_1^{\beta -\gamma }{\varrho }_1\left(s\right)\tilde{\mathrm{k}}\left(s\right){\left({\displaystyle \frac{\mu \left(s\right)}{s}}\right)}^{\beta /ϵ}+{\displaystyle \frac{1}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left({\varrho }_1^{\prime }\left(s\right)\right)}^{\gamma +1}}{{\varrho }_1^{\gamma }\left(s\right)}}.$$

Integrating (28) from $s_2$ to $s,$ one arrives at

$${\int }_{s_2}^s\left(M_1^{\beta -\gamma }{\varrho }_1\left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right){\left({\displaystyle \frac{\mu \left(\mathsf{\ell }\right)}{\mathsf{\ell }}}\right)}^{\beta /ϵ}-{\displaystyle \frac{1}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mathsf{\ell }\right){\left({\varrho }_1^{\prime }\left(\mathsf{\ell }\right)\right)}^{\gamma +1}}{{\varrho }_1^{\gamma }\left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }\le {\omega }_1\left(s_2\right),$$

which contradicts (12) as $s\to \infty$. This completes the proof. □

Example 1. 

Consider the following NDE:

$${\left(s{\left({\left(y\left(s\right)+{\displaystyle \frac{1}{s}}{y}^{1/3}\left({\eta }_{0}s\right)\right)}^{\prime }\right)}^5\right)}^{\prime }+\sum _{i=1}^{10}{\mathfrak{q}}_i{y}^5\left({\mu }_{i}s\right)=0,s\ge 1,$$

(29)

where ${\eta }_0,$ ${\mu }_i\in \left(0,1\right)$, and ${\mathfrak{q}}_i>0$ for $i=1,2,...,10.$ By comparing (1) and (29), we can deduce that $n=10,$ $\alpha =1/3,$ $\beta =\gamma =5,$ $\mathfrak{a}\left(s\right)=s,$ $\eta \left(s\right)={\eta }_{0}s,$ ${\mu }_i\left(s\right)={\mu }_{i}s,$ $\mathrm{u}\left(s\right)=1/s,$ ${\mathfrak{q}}_i\left(s\right)={\mathfrak{q}}_i.$

Consequently, we find that

$$\pi \left(s\right)={\displaystyle \frac{5}{4}}{s}^{4/5},$$

and

$$\tilde{\mathrm{k}}\left(s\right)=\sum _{i=1}^{10}{\mathfrak{q}}_i{\left(1-{\displaystyle \frac{1}{c_1^{2/3}{\mu }_{i}s}}\right)}^5.$$

By setting $\varrho \left(s\right)={\varrho }_1\left(s\right)=1$, it becomes clear that (29) exhibits oscillatory behavior, as stated by Theorems 1 and 2.

Theorem 3. 

Assume that (11) holds. If $0<\beta <\gamma$ and there is a nondecreasing function $\varrho \in C^1([s_0,\infty ),\left(0,\infty \right))$ such that

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left(\varrho \left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right)-{\displaystyle \frac{1}{{\left(\beta +1\right)}^{\beta +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mathsf{\ell }\right){\left({\varrho }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\beta +1}}{{\left(c_3\varrho \left(\mathsf{\ell }\right){\mu }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\beta }}}\right)\mathrm{d}\mathsf{\ell }=\infty$$

(30)

holds for every $c_1,c_3>0,$ then (1) is oscillatory.

Proof. 

We suppose for contradiction that (1) has an eventually positive solution $y\left(s\right)$. As in the proof of Lemma 4, we have (4) and (5). The function $\omega \left(s\right)$ is defined in (13), and then (14) holds. By (5), (13), and (14), we conclude that

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(s\right)& \le & -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-\beta {\mu }^{\prime }\left(s\right)\omega \left(s\right){\displaystyle \frac{h^{\prime }\left(\mu \left(s\right)\right)}{h\left(\mu \left(s\right)\right)}}\hfill \\ & \le & -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-\beta {\mu }^{\prime }\left(s\right)\omega \left(s\right){\displaystyle \frac{h^{\prime }\left(s\right)}{h\left(\mu \left(s\right)\right)}}\hfill \end{array}$$

and

$${\omega }^{\prime }\left(s\right)\le -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-{\displaystyle \frac{\beta {\mu }^{\prime }\left(s\right)}{{\left(\varrho \left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\beta }}}{\left[h^{\prime }\left(s\right)\right]}^{1-\gamma /\beta }{\omega }^{1/\beta +1}\left(s\right).$$

(31)

Given that $0<\beta <\gamma$ and condition (${\mathrm{C}}_1$) is satisfied, and since ${\mathfrak{a}}^{\prime }\left(s\right)\ge 0$, it follows that $h^{″}\left(s\right)\le 0$. This implies that $h^{\prime }\left(s\right)$ is nonincreasing. Consequently, there exist constants $c_3>0$ and $s_3\ge s_2$ such that

$${\left(h^{\prime }\left(s\right)\right)}^{1-\gamma /\beta }\ge c_3,s\ge s_3.$$

(32)

From (31) and (32), it follows that

$$\omega \left(s\right)\le -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{{\varrho }^{\prime }\left(s\right)}{\varrho \left(s\right)}}\omega \left(s\right)-{\displaystyle \frac{c_3\beta {\mu }^{\prime }\left(s\right)}{{\left(\varrho \left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\beta }}}{\omega }^{\left(\beta +1\right)/\beta }\left(s\right).$$

(33)

Using Lemma 1, where we define $B={\varrho }^{\prime }\left(s\right)/\varrho \left(s\right),$ $\mathfrak{a}=c_3\beta {\mu }^{\prime }\left(s\right)/{\left(\varrho \left(s\right)\mathfrak{a}\left(s\right)\right)}^{1/\beta },$ and $u\left(s\right)=\omega \left(s\right)$, we can deduce from (33) that

$$\omega \left(s\right)\le -\varrho \left(s\right)\tilde{\mathrm{k}}\left(s\right)+{\displaystyle \frac{1}{{\left(\beta +1\right)}^{\beta +1}}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left({\varrho }^{\prime }\left(s\right)\right)}^{\beta +1}}{{\left(c_3\varrho \left(s\right){\mu }^{\prime }\left(s\right)\right)}^{\beta }}}.$$

(34)

By performing the integration of (34) over the interval $\left[s_4,s\right]$, it follows that

$${\int }_{s_4}^s\left(\varrho \left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right)-{\displaystyle \frac{1}{{\left(\beta +1\right)}^{\beta +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mathsf{\ell }\right){\left({\varrho }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\beta +1}}{{\left(c_3\varrho \left(\mathsf{\ell }\right){\mu }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\beta }}}\right)\mathrm{d}\mathsf{\ell }\le \omega \left(s\right),$$

which contradicts (30) as $s\to \infty$. This completes the proof. □

Example 2. 

Consider

$${\left(s^2{\left({\left(y\left(s\right)+{\displaystyle \frac{1}{5s^2}}{y}^{1/3}\left({\displaystyle \frac{1}{3}}s\right)\right)}^{\prime }\right)}^5\right)}^{\prime }+s^3\left(y\left({\displaystyle \frac{1}{2}}s\right)+y\left({\displaystyle \frac{1}{3}}s\right)+y\left({\displaystyle \frac{1}{4}}s\right)\right)=0,s\ge 1,$$

(35)

where $n=3,$ $\alpha =1/3,$ $\gamma =5,$ $\beta =1,$ $\mathfrak{a}\left(s\right)=s^2,$ $\eta \left(s\right)=s/3,$ ${\mu }_1\left(s\right)=s/2,$ ${\mu }_2\left(s\right)=s/3,$ ${\mu }_3\left(s\right)=s/4,$ $\mathrm{u}\left(s\right)=1/5s^2,$ ${\mathfrak{q}}_i\left(s\right)=s^3.$

Thus, we find

$$\pi \left(s\right)={\displaystyle \frac{5}{3}}{s}^{3/5}.$$

Let $\varrho \left(s\right)=s$. Then,

$$\begin{array}{ccc}\hfill \tilde{\mathrm{k}}\left(s\right)& =& \sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-{\displaystyle \frac{\mathrm{u}\left({\mu }_i\left(s\right)\right)}{c_1^{1-\alpha }}}\right)}^{\beta }\hfill \\ & =& s^3\left(1-{\displaystyle \frac{4}{5c_1^{2/3}{s}^2}}+1-{\displaystyle \frac{9}{5c_1^{2/3}{s}^2}}+1-{\displaystyle \frac{16}{5c_1^{2/3}{s}^2}}\right)\hfill \\ & =& s^3\left(3-{\displaystyle \frac{4}{5c_1^{2/3}{s}^2}}-{\displaystyle \frac{9}{5c_1^{2/3}{s}^2}}-{\displaystyle \frac{16}{5c_1^{2/3}{s}^2}}\right)\hfill \\ & =& s^3\left(3-{\displaystyle \frac{29}{5c_1^{2/3}{s}^2}}\right)>0,\mathrm{as}s>{\left({\displaystyle \frac{29}{15}}\right)}^{1/2}{\displaystyle \frac{1}{c_1^{1/3}}}.\hfill \end{array}$$

Therefore, condition (30) becomes

$$\begin{array}{ccc}& & \underset{s\to \infty }{limsup}{\int }_{s_0}^s\left(\varrho \left(\mathsf{\ell }\right)\tilde{\mathrm{k}}\left(\mathsf{\ell }\right)-{\displaystyle \frac{1}{{\left(\beta +1\right)}^{\beta +1}}}{\displaystyle \frac{\mathfrak{a}\left(\mathsf{\ell }\right){\left({\varrho }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\beta +1}}{{\left(c_3\varrho \left(\mathsf{\ell }\right){\mu }^{\prime }\left(\mathsf{\ell }\right)\right)}^{\beta }}}\right)\mathrm{d}\mathsf{\ell }\hfill \\ & =& \underset{s\to \infty }{limsup}{\int }_{s_0}^s\left(\left(3-{\displaystyle \frac{29}{5c_1^{2/3}{\mathsf{\ell }}^2}}\right){\mathsf{\ell }}^4-{\displaystyle \frac{\mathsf{\ell }}{c_3}}\right)\mathrm{d}\mathsf{\ell }=\infty .\hfill \end{array}$$

Thus, by Theorem 3, it follows that every solution of (35) exhibits oscillatory behavior.

3.2. Noncanonical Case

In this subsection, we consider the noncanonical case, characterized by the condition

$${\int }_{s_0}^{\infty }{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\alpha }\left(\mathsf{\ell }\right)}}\mathrm{d}\mathsf{\ell }<\infty .$$

(36)

Theorem 4. 

Assume that (12), (36), and

$$max\left\{{\displaystyle \frac{\mathrm{u}\left({\mu }_i\left(s\right)\right)}{c_1^{1-\alpha }}},\mathrm{u}\left(s\right){\displaystyle \frac{{\pi }^{\alpha }\left(\eta \left({\mu }_i\left(s\right)\right)\right)}{c_0^{1-\alpha }\pi \left({\mu }_i\left(s\right)\right)}}\right\}<1,\mathit{for}i=1,2,\cdots ,n,$$

(37)

hold for every $c_0,$ $c_1,$ $c_2>0.$ If $\beta \ge \gamma ,$ and

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left({\pi }^{\beta }\left(\mathsf{\ell }\right)\widehat{\mathrm{k}}\left(\mathsf{\ell }\right)-{\left({\displaystyle \frac{\beta }{\beta +1}}\right)}^{\beta +1}{\displaystyle \frac{1}{c_4^{\beta }{\mathfrak{a}}^{1/\gamma }\left(\mathsf{\ell }\right)\pi \left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }=\infty$$

(38)

holds for $c_0,$ $c_4>0,$ then (1) is oscillatory.

Proof. 

We suppose for contradiction that (1) has an eventually positive solution $y\left(s\right)$. Based on Lemma 3, the behavior of h and its derivatives can be classified into two distinct cases: $\left({\mathrm{C}}_1\right)$ and $\left({\mathrm{C}}_2\right).$

Consider the case where$\left({\mathrm{C}}_1\right)$ holds. Similar to the proof of Theorem 1, we arrive at a contradiction with (12).

Consider the case where$\left({\mathrm{C}}_2\right)$ holds. Define

$$\phi \left(s\right):={\displaystyle \frac{\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }}{h^{\beta }\left(s\right)}},s\ge s_1.$$

(39)

Consequently, $\phi \left(s\right)>0.$ Using (9), we see that

$$h\left(s\right)\ge {\mathfrak{a}}^{1/\gamma }\left(s\right)\left(-h^{\prime }\left(s\right)\right)\pi \left(s\right).$$

(40)

This implies

$${\left[\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right]}^{1-\beta /\gamma }\ge {\pi }^{\beta }\left(s\right)\phi \left(s\right)>0.$$

Given that ${\left[\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right]}^{1-\beta /\gamma }$ is a nonincreasing function for $\beta \ge \gamma$, there exist constants $L_1>0$ and $s_2\ge s_1$ such that

$${\left[\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right]}^{1-\beta /\gamma }\le L_1.$$

Therefore, we have

$$0<{\pi }^{\beta }\left(s\right)\phi \left(s\right)<L_1,s\ge s_2.$$

(41)

Taking the derivative of Equation (39), we obtain

$${\phi }^{\prime }\left(s\right)={\displaystyle \frac{{\left(\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }}{h^{\beta }\left(s\right)}}+\beta {\displaystyle \frac{\mathfrak{a}\left(s\right){\left(-h^{\prime }\left(s\right)\right)}^{\gamma +1}}{h^{\beta +1}\left(s\right)}}.$$

(42)

Using (8) and (39), we conclude that

$${\phi }^{\prime }\left(s\right)\ge \widehat{\mathrm{k}}\left(s\right)+{\displaystyle \frac{\beta }{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}{\left[{\mathfrak{a}}^{1/\gamma }\left(s\right)\left(-h^{\prime }\left(s\right)\right)\right]}^{1-\gamma /\beta }{\phi }^{1/\beta +1}\left(s\right).$$

Given that ${\left[{\mathfrak{a}}^{1/\gamma }\left(s\right)\left(-h^{\prime }\left(s\right)\right)\right]}^{1-\gamma /\beta }$ increases monotonically for $\beta \ge \gamma ,$ it follows that there exist constants $c_4>0$ and $s_3\ge s_2$ (when $\beta =\gamma ,$ $c_4=1$), such that

$${\left[{\mathfrak{a}}^{1/\gamma }\left(s\right)\left(-h^{\prime }\left(s\right)\right)\right]}^{1-\gamma /\beta }>c_4,s\ge s_3.$$

As a result,

$${\phi }^{\prime }\left(s\right)\ge \widehat{\mathrm{k}}\left(s\right)+{\displaystyle \frac{\beta c_4}{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}{\phi }^{1/\beta +1}\left(s\right).$$

(43)

Multiplying (43) by ${\pi }^{\beta }\left(s\right)$ and then integrating the resulting inequality from $s_4$ to s, we arrive at

$${\int }_{s_4}^s{\pi }^{\beta }\left(\mathsf{\ell }\right)\widehat{\mathrm{k}}\left(\mathsf{\ell }\right)\mathrm{d}\mathsf{\ell }\le {\int }_{s_4}^s\beta {\pi }^{\beta -1}\left(\mathsf{\ell }\right){\mathfrak{a}}^{-1/\gamma }\left(\mathsf{\ell }\right)\left[\phi \left(\mathsf{\ell }\right)-c_4\pi \left(\mathsf{\ell }\right){\phi }^{1/\beta +1}\left(\mathsf{\ell }\right)\right]+{\pi }^{\beta }\left(s\right)\phi \left(s\right)-{\pi }^{\beta }\left(s_4\right)\phi \left(s_4\right).$$

By employing (41) and Lemma 1 where $B=1,$ $\mathfrak{a}=c_4\pi \left(s\right),$ and $u\left(s\right)=\phi \left(s\right),$ we have

$${\int }_{s_4}^s\left({\pi }^{\beta }\left(\mathsf{\ell }\right)\widehat{\mathrm{k}}\left(\mathsf{\ell }\right)-{\left({\displaystyle \frac{\beta }{\beta +1}}\right)}^{\beta +1}{\displaystyle \frac{1}{c_4^{\beta }{\mathfrak{a}}^{1/\gamma }\left(\mathsf{\ell }\right)\pi \left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }\le L_1-{\pi }^{\beta }\left(s_4\right)\phi \left(s_4\right).$$

This contradicts condition (38) as $s\to \infty$. Hence, the proof is concluded. □

By setting $\alpha =1$ and $\beta =\gamma$ in (1) and considering Theorem 4, we immediately derive the following result.

Corollary 1. 

Let $\alpha =1,$ $\beta =\gamma ,$ and

$$max\left\{\mathrm{u}\left({\mu }_i\left(s\right)\right),\mathrm{u}\left({\mu }_i\left(s\right)\right){\displaystyle \frac{\pi \left(\eta \left({\mu }_i\left(s\right)\right)\right)}{\pi \left({\mu }_i\left(s\right)\right)}}\right\}<1,\mathit{for}i=1,2,...,n.$$

(44)

If (36),

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-\mathrm{u}\left({\mu }_i\left(s\right)\right)\right)}^{\beta }\mathrm{d}\mathsf{\ell }=\infty ,$$

(45)

and

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left({\pi }^{\beta }\left(\mathsf{\ell }\right)\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-\mathrm{u}\left({\mu }_i(\mathsf{\ell })\right){\displaystyle \frac{\pi \left(\eta \left({\mu }_i(\mathsf{\ell })\right)\right)}{\pi \left({\mu }_i(\mathsf{\ell })\right)}}\right)}^{\beta }-{\left({\displaystyle \frac{\beta }{\beta +1}}\right)}^{\beta +1}{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\beta }\left(\mathsf{\ell }\right)\pi \left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }=\infty$$

(46)

hold, then (1) is oscillatory.

Example 3. 

Consider

$${\left(s^6{\left[{\left(y\left(s\right)+{\displaystyle \frac{1}{4}}y\left({\displaystyle \frac{s}{2}}\right)\right)}^{\prime }\right]}^3\right)}^{\prime }+{\mathfrak{q}}_0{s}^2\left(y^3\left({\displaystyle \frac{1}{3}}s\right)+y^3\left({\displaystyle \frac{1}{4}}s\right)+y^3\left({\displaystyle \frac{1}{5}}s\right)\right)=0,s\ge 1.$$

(47)

We verify that Equation (47) satisfies the conditions of Corollary 1. Note that, in (47), $n=3,$ $\alpha =1,$ $\beta =\gamma =3,$ $\mathfrak{a}\left(s\right)=s^6,$ $\mathrm{u}\left(s\right)=1/4,$ $\mathfrak{q}\left(s\right)={\mathfrak{q}}_0{s}^2,$ ${\mathfrak{q}}_0>0,$ $\eta \left(s\right)=s/2,$ and $\widehat{\mu }\left(s\right)=s/3$. Consequently, we find

$$\pi \left(s\right)={\displaystyle \frac{1}{s}}.$$

Thus,

$${\displaystyle \frac{\mathrm{u}\left({\mu }_i\left(s\right)\right)\pi \left(\eta \left({\mu }_i\left(s\right)\right)\right)}{\pi \left({\mu }_i\left(s\right)\right)}}={\displaystyle \frac{1}{2}},i=1,2,3,$$

Condition (44) is satisfied as follows:

$$max\left\{\mathrm{u}\left({\mu }_i\left(s\right)\right),\mathrm{u}\left({\mu }_i\left(s\right)\right){\displaystyle \frac{\pi \left(\eta \left({\mu }_i\left(s\right)\right)\right)}{\pi \left({\mu }_i\left(s\right)\right)}}\right\}=max\left\{{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}}\right\}={\displaystyle \frac{1}{2}}<1,$$

condition (45) is satisfied because

$$\begin{array}{ccc}\hfill \underset{s\to \infty }{limsup}{\int }_{s_0}^s\sum _{i=1}^n{\mathfrak{q}}_i\left(s\right){\left(1-\mathrm{u}\left({\mu }_i\left(s\right)\right)\right)}^{\beta }\mathrm{d}\mathsf{\ell }& =& \underset{s\to \infty }{limsup}{\int }_{s_0}^s\sum _{i=1}^3{\mathfrak{q}}_0{s}^2{\left(1-{\displaystyle \frac{1}{4}}\right)}^3\mathrm{d}\mathsf{\ell }\hfill \\ & =& \underset{s\to \infty }{limsup}{\int }_{s_0}^s\sum _{i=1}^3{\displaystyle \frac{3^3{\mathfrak{q}}_0}{4^3}}{s}^2\mathrm{d}\mathsf{\ell }\hfill \\ & =& \underset{s\to \infty }{limsup}{\int }_{s_0}^s{\displaystyle \frac{3^4{\mathfrak{q}}_0}{4^3}}{s}^2\mathrm{d}\mathsf{\ell }=\infty ,\hfill \end{array}$$

and condition (46) is satisfied as follows:

$$\begin{array}{ccc}& & \underset{s\to \infty }{limsup}{\int }_{s_0}^s\left({\pi }^{\beta }\left(\mathsf{\ell }\right)\sum _{i=1}^n{\mathfrak{q}}_i(\mathsf{\ell }){\left(1-\mathrm{u}\left({\mu }_i(\mathsf{\ell })\right){\displaystyle \frac{\pi \left(\eta \left({\mu }_i(\mathsf{\ell })\right)\right)}{\pi \left({\mu }_i(\mathsf{\ell })\right)}}\right)}^{\beta }-{\left({\displaystyle \frac{\beta }{\beta +1}}\right)}^{\beta +1}{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\beta }\left(\mathsf{\ell }\right)\pi \left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }\hfill \\ & & \underset{s\to \infty }{=limsup}{\int }_1^s\left({\displaystyle \frac{1}{{\mathsf{\ell }}^3}}\sum _{i=1}^3{\mathfrak{q}}_0{\mathsf{\ell }}^2{\left(1-{\displaystyle \frac{1}{2}}\right)}^3-{\left({\displaystyle \frac{3}{4}}\right)}^4{\displaystyle \frac{1}{\mathsf{\ell }}}\right)\mathrm{d}\mathsf{\ell }\hfill \\ & =& \left({\displaystyle \frac{3}{8}}{\mathfrak{q}}_0-{\left({\displaystyle \frac{3}{4}}\right)}^4\right)\underset{s\to \infty }{limsup}{\int }_1^s{\displaystyle \frac{1}{\mathsf{\ell }}}\mathrm{d}\mathsf{\ell }=\infty ,\hfill \end{array}$$

where

$${\mathfrak{q}}_0>{\displaystyle \frac{27}{32}}.$$

Thus, according to Corollary 1, Equation (47) exhibits oscillatory behavior when ${\mathfrak{q}}_0>27/32.$

Theorem 5. 

Let $0<\beta <\gamma ,$ (36), (37), and (30) hold for every $c_0,$ $c_1,$ $c_2>0$. If

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left({\pi }^{\gamma }\left(\mathsf{\ell }\right)\widehat{\mathrm{k}}\left(\mathsf{\ell }\right)-{\left({\displaystyle \frac{\gamma }{\gamma +1}}\right)}^{\gamma +1}{\left({\displaystyle \frac{\gamma }{\beta }}\right)}^{\gamma }{\displaystyle \frac{1}{c_5^{\gamma }{\mathfrak{a}}^{1/\gamma }\left(\mathsf{\ell }\right)\pi \left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }=\infty$$

(48)

holds for every $c_0,$ $c_5>0,$ then (1) is oscillatory.

Proof. 

We suppose for contradiction that (1) has an eventually positive solution $y\left(s\right)$. Based on Lemma 3, the behavior of h and its derivatives can be classified into two distinct cases: $\left({\mathrm{C}}_1\right)$ and $\left({\mathrm{C}}_2\right).$

Consider the case where$\left({\mathrm{C}}_1\right)$ holds. Similar to the proof of Theorem 3, we arrive at a contradiction with (30).

Consider the case where$\left({\mathrm{C}}_2\right)$ holds. By the definition of $\phi \left(s\right)$, together with (40), we find that

$$h^{\gamma -\beta }\left(s\right)\ge {\pi }^{\gamma }\left(s\right)\phi \left(s\right),s\ge s_1\ge s_0.$$

Given that $h^{\gamma -\beta }\left(s\right)$ is a nonincreasing function for $0<\beta <\gamma ,$ there exist constants $L_2>0$ and $s_2\ge s_1$ such that

$$h^{\gamma -\beta }\left(s\right)\le L_2,$$

and thus,

$$0<{\pi }^{\gamma }\left(s\right)\phi \left(s\right)\le L_2,s\ge s_2.$$

(49)

Additionally, using (8) and (39) in (42), we obtain

$${\phi }^{\prime }\left(s\right)\ge \widehat{\mathrm{k}}\left(s\right)+{\displaystyle \frac{\beta }{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}{h}^{\beta /\gamma -1}\left(s\right){\phi }^{1/\gamma +1}\left(s\right)$$

(50)

Given that $h^{\beta /\gamma -1}\left(s\right)$ increases monotonically for $0<\beta <\gamma ,$ it follows that there exist constants $c_5>0$ and $s_3\ge s_2$ such that

$$h^{\beta /\gamma -1}\left(s\right)>c_5,s\ge s_3.$$

Thus, we obtain

$${\phi }^{\prime }\left(s\right)\ge \widehat{\mathrm{k}}\left(s\right)+{\displaystyle \frac{\beta c_5}{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}{\phi }^{1/\gamma +1}\left(s\right),s\ge s_3.$$

(51)

Multiplying (51) by ${\pi }^{\gamma }\left(s\right)$ and then integrating the resulting inequality from $s_4$ to s, we arrive at

$${\int }_{s_4}^s{\pi }^{\gamma }\left(\mathsf{\ell }\right)\widehat{\mathrm{k}}\left(\mathsf{\ell }\right)\mathrm{d}\mathsf{\ell }\le {\int }_{s_4}^s{\pi }^{\gamma -1}\left(\mathsf{\ell }\right){\mathfrak{a}}^{-1/\gamma }\left(\mathsf{\ell }\right)\left[\gamma \phi \left(\mathsf{\ell }\right)-c_5\beta \pi \left(\mathsf{\ell }\right){\phi }^{1/\gamma +1}\left(\mathsf{\ell }\right)\right]+{\pi }^{\gamma }\left(s\right)\phi \left(s\right)-{\pi }^{\gamma }\left(s_4\right)\phi \left(s_4\right).$$

By employing (41) and Lemma 1, where $B=\gamma ,$ $\mathfrak{a}=c_5\beta \pi \left(\mathsf{\ell }\right),$ and $u\left(s\right)=\phi \left(s\right),$ we have

$${\int }_{s_4}^s\left({\pi }^{\gamma }\left(\mathsf{\ell }\right)\widehat{\mathrm{k}}\left(\mathsf{\ell }\right)-{\left({\displaystyle \frac{\gamma }{\gamma +1}}\right)}^{\gamma +1}{\left({\displaystyle \frac{\gamma }{\beta }}\right)}^{\gamma }{\displaystyle \frac{1}{c_5^{\gamma }{\mathfrak{a}}^{1/\gamma }\left(\mathsf{\ell }\right)\pi \left(\mathsf{\ell }\right)}}\right)\mathrm{d}\mathsf{\ell }\le L_1-{\pi }^{\beta }\left(s_4\right)\phi \left(s_4\right).$$

This contradicts condition (48) as $s\to \infty$. Hence, the proof is concluded. □

Remark 1. 

The following table, Table 1, summarizes the key differences among our five proposed theorems, highlighting specific conditions for and γ. We introduce some special cases that demonstrate how our results serve as a generalization and improvement of numerous previous works, reinforcing the significance of our contributions to the field of oscillatory behavior in differential equations.

Table 1. Comparison of our theorems based on conditions, covered cases, special cases, and novelty.

Theorems	$\mathit{\beta }$ and $\mathit{\gamma }$	Comparison with Literature Works		Cases
		Special Case	Novelty
Theorem 1	$\beta \ge \gamma$	$\beta =\gamma =1$	Extends Theorem 2.1 in [31]	canonical
		$\gamma$ = 1	Extends Theorem 1 in [32]
Theorem 2	$\beta \ge \gamma$	n = 1	Improves Theorem 2.1 in [33]
Theorem 3	$0<\beta <\gamma$	$\beta <1$, $\gamma \ne 1$	Improves Theorems in [31,34]
Theorem 4	$\beta \ge \gamma$	$\beta =\gamma =1$	Extended Theorem 2.2 in [31]	noncanonical
Theorem 5	$0<\beta <\gamma$	$\beta <1$, $\gamma \ne 1$	Improves Theorem 2.2 in [31]

Remark 2. 

All the special cases presented in Table 1 require $n=1$, as previous works compared in this study overlooked the impact of multiple delays or multiple coefficients associated with the dependent variable y.

Remark 3. 

In addition to Theorem 2 improving Theorem 2.1 in [33] when $n=1$, the remaining theorems, Theorems 1, 3–5, extend Theorems 2.2–2.6 in [33] when $n=2,3,...$

4. Conclusions

This research has thoroughly investigated the oscillatory properties of second-order Emden–Fowler differential equations with a sublinear neutral term, including both canonical and noncanonical cases. By proving several theorems, we have established criteria for the fluctuation of all solutions of the studied equation. The Riccati technique was employed in diverse ways to derive these results. Our study broadens the scope of previous research, thereby enriching the cognitive understanding of this mathematical domain. It is noteworthy that our findings were obtained under the stringent condition ${lim}_{s\to \infty }$u$\left(s\right)=0$. This opens a pathway for future studies to explore the possibility of achieving similar results without the necessity of adhering to this condition. Additionally, an intriguing direction for future research would be to apply the same methodology and expand the study to include higher-order equations using the Riccati technique. This would not only validate the robustness of our approach but also potentially unveil new insights into the oscillatory behavior of more complex differential equations.