General Neutral Functional Differential Equations of Third Order: Enhanced Oscillation Criteria

Abstract

This study aims to establish new oscillation criteria for solutions of a specific class of functional differential equations. Our findings extend and refine the recently developed criteria for this type of equation by various authors and also encompass classical criteria for related problems. Our approach relies on the Riccati technique to derive conditions that preclude the possibility of non-oscillatory solutions. The inherent symmetry of these solutions plays a key role in formulating the new criteria presented here. By applying techniques from the theory of symmetric differential equations and leveraging symmetric functions, we are able to establish precise conditions for oscillation. To enhance practical applicability, we propose multiple distinct criteria while minimizing the constraints typically imposed. Several examples are provided to illustrate the accuracy, applicability, and versatility of the new criteria.

1. Introduction

In this article, we establish sufficient conditions for the oscillatory behavior of solutions to the equation

$${\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }+\sum _{i=1}^k{q}_i\left(\top \right){\left(x\left({\tau }_i\left(\top \right)\right)\right)}^d=0,\mathrm{for}\top \ge {\top }_0>0,$$

(1)

where $k\ge 1$ is an integer and

$$\varrho \left(\top \right)=x\left(\top \right)+r\left(\top \right)x\left(g\left(\top \right)\right).$$

(2)

Here, we assume that

(I₁)

$d>0$ is a quotient of odd positive integer;

(I₂)

$ɩ\in C\left(\left[{\top }_0,\infty \right)\right), ɩ\left(\top \right)>0$ and

$${\int }_{{\top }_0}^{\infty }{\displaystyle \frac{1}{{ɩ}^{1/d}\left(s\right)}}\mathrm{d}s=\infty ;$$

(I₃)

$r,g\in C\left({\top }_0,\infty \right),$${lim}_{\top \to \infty }g\left(\top \right)=\infty ,$ $g\left(\top \right)\le \top$,

$$g^{\prime }\left(\top \right)\ge g_0>0,$$

(3)

$$g\circ {\tau }_i={\tau }_i\circ g,$$

(4)

and

$$0\le r\left(\top \right)\le r_0<\infty ;$$

(5)

(I₄)

$q_i\in C\left({\top }_0,\infty \right),{\tau }_i\in C^1\left({\top }_0,\infty \right),{lim}_{\top \to \infty }{\tau }_i\left(\top \right)=\infty$ and ${\tau }_i\left(\top \right)\le \tau \left(\top \right)\le \top .$ Furthermore, $q_i\left(\top \right)\ge 0$ is not identically zero.

Definition 1.

A solution of (1) is a function $x\in C\left([{\top }_x,\infty ),[0,\infty )\right)$ with ${\top }_x\ge {\top }_0$, and $ɩ{\left({\varrho }^{″}\right)}^d\in C\left([{\top }_x,\infty ),[0,\infty )\right)$, satisfying Equation (1) on $[{\top }_x,\infty ).$ We focus on solutions of (1) that exist on $[{\top }_x,\infty )$ which satisfy

$$sup\{\left|x\left(\top \right)\right|:{\top }_x\le \top <\infty \}>0.$$

Furthermore, if the solution x does not eventually become either positive or negative, it is classified as an oscillatory solution. Otherwise, it is called a non-oscillatory solution.

A neutral delay differential equation is a subclass of delay differential equations characterized by the presence of delayed derivatives of the dependent variable. This feature increases the complexity of neutral delay differential equations compared to standard delay differential equations, as the system’s evolution depends not only on past states but also on past derivatives. Neutral delay differential equations have significant applications in fields such as control systems, biological modelling, and population dynamics, where delayed feedback influences system behavior. The analysis of neutral delay differential equations requires advanced mathematical techniques, including spectral methods and Lyapunov functionals, to assess stability. The inclusion of a neutral term introduces additional oscillatory behavior, potentially leading to higher frequencies or instability [1].

The investigation of oscillatory dynamics in differential equations constitutes a pivotal research domain, driven by its broad applicability across scientific, engineering, and technological disciplines. Oscillatory phenomena manifest in diverse systems, including mechanical vibrations, electrical circuits, ecological interactions, and biological rhythms. Elucidating the conditions governing oscillatory solutions enables researchers to predict, regulate, and optimize processes in fields ranging from biomedical engineering to sustainable energy systems.

Recent progress in neutral differential equations (NDEs) has provided transformative insights into delay-dominated systems. In biological contexts, NDE frameworks have advanced the modeling of gene regulatory networks with delayed feedback mechanisms, such as CRISPR-Cas9 dynamics, and have elucidated circadian rhythm oscillations by incorporating coupled mRNA-protein synthesis delays [1,2,3]. Epidemiological models employing NDEs now integrate pathogen latency periods (e.g., SARS-CoV-2 variant emergence) to refine outbreak predictions, while neural system analyses leverage synaptic transmission delays to decode pathological oscillations in Parkinsonian dynamics. Engineering applications of NDEs address critical real-world challenges: autonomous vehicle navigation systems utilize delay-compensated motion planning to counteract sensor-to-actuator lags, smart grids deploy NDE-based load-frequency controllers to stabilize power networks amid renewable energy intermittency and communication delays, and robotic swarms achieve synchronized motion protocols despite distributed computational latencies. By explicitly accounting for delays in both state variables and their derivatives, NDEs offer a unified methodology to analyze and optimize systems where temporal lags fundamentally govern dynamics—from subcellular processes to large-scale infrastructure networks [4,5,6]. Oscillatory behavior remains equally foundational in electrical engineering, underpinning the design of resonant circuits, oscillators, and frequency-selective filters. The Hodgkin-Huxley equations, which describe action potential generation in neural circuits, exemplify oscillations critical to neurophysiological processes. Similarly, cardiac electrophysiology models rely on oscillatory mechanisms to characterize arrhythmias and pacemaker dynamics, while ecological models employ oscillation analysis to predict predator-prey population cycles. These interdisciplinary applications underscore the centrality of oscillatory phenomena in both natural and engineered systems, with theoretical advances continually informing technological innovation and biological discovery [7,8,9,10].

Study of oscillatory behavior in solutions of differential equations is a critical area of research due to its extensive applications in science, engineering, and technology. Oscillations are observed in phenomena ranging from mechanical vibrations and electrical circuits to biological systems and population dynamics. Understanding the conditions under which solutions exhibit oscillatory behavior allows researchers to predict, control, and optimize processes across diverse fields. Recent advances in neutral differential equations (NDEs) have unlocked critical insights into delay-dominated systems across biology and engineering. In biological contexts, NDEs model gene regulatory networks with delayed feedback, such as CRISPR-Cas9 timing dynamics, and predict oscillatory patterns in circadian rhythms by accounting for coupled mRNA-protein synthesis delays. They also refine epidemiological forecasts by incorporating pathogen latency periods (e.g., emerging SARS-CoV-2 variants) and decode pathological neural oscillations in Parkinson’s disease through synaptic transmission delays. Meanwhile, engineering applications leverage NDEs to address real-world challenges: autonomous vehicles utilize delay-compensated motion planning to mitigate sensor-to-actuator lags, smart grids employ NDE-based load-frequency control to stabilize renewable energy integration amid communication delays, and robotic swarms achieve synchronized collective behavior despite distributed computation lags. By explicitly addressing delays in both state variables and their derivatives, NDEs provide a unified framework to dissect and optimize systems where temporal lags fundamentally shape dynamics—from cellular processes to large-scale engineered networks [1,2,3,4,5,6,11]. Similarly, in electrical engineering, oscillatory behavior is fundamental to the operation of activity, and ecological interactions. Oscillations also play a central role in the Hodgkin-Huxley equations describing action potentials in neurosc circuits, including oscillators and filters. In biology and medicine, oscillatory phenomena emerge in models of cardiac rhythms, neuronalience.

The interplay between oscillation theory and solution symmetry offers profound insights into the dynamics of complex systems. Symmetries inherent in governing equations often constrain the oscillatory behavior of solutions, leading to synchronized patterns or phase-locked oscillations. For instance, in systems with rotational or translational symmetry, oscillatory modes may emerge as symmetry-breaking phenomena, while conserved quantities can dictate amplitude or frequency relationships.

Analyzing oscillatory behavior typically involves comparison theorems, Lyapunov functional techniques, fixed-point theory, asymptotic properties, or integral averaging methods. These approaches help identify conditions under which solutions oscillate or remain bounded, providing insights into system stability and dynamics. This analysis is particularly important in physics, biology, and engineering, where oscillatory phenomena often play a central role in system behavior and control [12,13].

There is a long history of research into the oscillatory behavior of solutions to third-order differential equations, reflecting its significance in both mathematical analysis and applications. Early investigations into the subject focused on establishing qualitative properties of solutions, including conditions under which oscillations occur, motivated by problems in physics and engineering. Researchers such as Nehari and Kamenev contributed foundational results in the mid-20th century, establishing criteria for oscillation and non-oscillation using comparison theorems and variational methods. Over time, advancements in nonlinear analysis and the development of sophisticated mathematical tools expanded the scope of these studies. Modern research emphasises the interplay between oscillatory behavior and properties. Recent advancements in nonlinear analysis and computational techniques have expanded the scope of these studies to include damping, forcing terms, and variable coefficients inspired by real-world phenomena modeled by third-order differential equations in mechanics, control systems, and fluid dynamics [14,15,16,17].

Grace et al. [18] investigated the oscillation of third-order nonlinear functional differential equations of the form

$${\left(ɩ\left(\top \right){\left(x^{″}\left(\top \right)\right)}^d\right)}^{\prime }+q\left(\top \right)f\left(x\left(\tau \left(\top \right)\right)\right)=0,$$

(6)

where $f:\mathbb{R}\to \mathbb{R}$, $xf\left(x\right)>0.$ They used comparison theory to establish some sufficient conditions for oscillation of (6).

Zhong et al. [19] extended the Equation (6) to the neutral differential equation

$${\left(ɩ\left(\top \right){\left({\left(x\left(\top \right)+r\left(\top \right)x\left(g\left(\top \right)\right)\right)}^{″}\right)}^d\right)}^{\prime }+q\left(\top \right)f\left(x\left(\tau \left(\top \right)\right)\right)=0,$$

(7)

where

$$0\le r\left(\top \right)\le 1.$$

(8)

Subsequent works, including those of Candan [20,21] established different oscillation criteria for

$${\left(ɩ\left(\top \right){\left(x\left(\top \right)+r\left(\top \right)x\left(g\left(\top \right)\right)\right)}^{″}\right)}^{\prime }+q\left(\top \right)x\left(\tau \left(\top \right)\right)=0,\mathrm{for}\top \ge {\top }_0,$$

(9)

under the condition

$$0\le r\left(\top \right)\le r_0<1.$$

(10)

In addition, the authors in [5] studied the oscillation of solutions to Equation (7) under the assumptions:

$$-1<r\left(\top \right)<1$$

(11)

and

$${ɩ}^{\prime }\left(\top \right)\ge 0.$$

(12)

Karpuz et al. [22] considered the oscillation for odd-order delay differential equations of the form

$${\left(x\left(\top \right)+r\left(\top \right)x\left(g\left(\top \right)\right)\right)}^{\left(n\right)}+q\left(\top \right)x\left(\tau \left(\top \right)\right)=0,$$

assuming that (11) holds. On the other hand, the authors in [23] analyzed the third-order neutral differential equation

$${\left(\left({\left(x\left(\top \right)+r\left(\top \right)x\left(g\left(\top \right)\right)\right)}^{″}\right)\right)}^{\prime }+q\left(\top \right)f\left(x\left(\tau \left(\top \right)\right)\right)=0,$$

where

$$-1\le r\left(\top \right)\le 0.$$

(13)

The main aim of this work is to reduce the constraints imposed on such equations and their special cases. We employ the Riccati technique to provide multiple criteria that eliminate the possibility of nonoscillatory solutions, thereby guaranteeing the oscillatory behaviour of the equation considered.

The following lemma is classic [22] and will be used throughout the paper.

Lemma 1.

Let $y_1$, $y_2$, $y_3\in C\left([{\top }_0,\infty ),\mathbb{R}\right)$, $y_3\left(\top \right)\le \top$ for all $\top \in [{\top }_0,\infty )$ and ${lim}_{\top \to \infty }{y}_3\left(\top \right)=\infty .$ Additionally, assume there exists a function $h\in C\left([{\top }_{*},\infty ),{\mathbb{R}}^{+}\right),$ where ${\top }_{*}:={min}_{\top \in [{\top }_0,\infty )}\left\{y_3\left(\top \right)\right\},$ such that for all $\top \in [{\top }_0,\infty ),$

$$h\left(\top \right)=y_1\left(\top \right)-y_2\left(\top \right)h\left(y_3\left(\top \right)\right).$$

Assume further that

$$\underset{\top \to \infty }{lim\; inf}{y}_2\left(\top \right)>-1,$$

and that ${lim}_{\top \to \infty }{y}_1\left(\top \right)$ exists. If ${lim\; sup}_{\top \to \infty }h\left(\top \right)>0,$ then it follows that ${lim}_{\top \to \infty }{y}_1\left(\top \right)>0.$

2. Main Results

We introduce the following notation for the sake of brevity

$$\varpi \left(\top \right)={\displaystyle \frac{1}{ɩ\left(g\left(\top \right)\right)}}{\int }_{\top }^{\infty }\tilde{q}\left(s\right)\mathrm{d}s,$$

where

$$\tilde{q}\left(\top \right):=\underset{i=1,2,...,k}{min}\{q_i\left(\top \right),q_i\left(g\left(\top \right)\right)\},\top \in \left[{\top }_0,\infty \right).$$

(14)

Additionally, the following result is the basis on which our study is based, since in order to ensure that Equation (1) is oscillatory we exclude non-oscillatory solutions of types (15) and (16). In most of the work, case (15) is excluded, while in other parts condition (18) is used to guarantee the non-existence of case (16).

Lemma 2.

Let $x\left(\top \right)>0$ be a solution of (1). Then

$${\varrho }^{″}\left(\top \right)>0,{\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\le 0,$$

for $\top \ge {\top }_1,$ with ${\top }_1$ sufficiently large. Moreover, only the following two cases can occur:

$$\varrho \left(\top \right)>0,{\varrho }^{\prime }\left(\top \right)>0$$

(15)

or

$$\varrho \left(\top \right)>0,{\varrho }^{\prime }\left(\top \right)<0.$$

(16)

Proof. 

Assume that $x\left(\top \right)$ is a positive solution of (1) on $[{\top }_0,\infty )$. Then, it follows from (I₃) that $x\left(g\left(\top \right)\right)>0.$ Since $r\left(\top \right)>0$ and

$$\varrho \left(\top \right)=x\left(\top \right)+r\left(\top \right)x\left(g\left(\top \right)\right),$$

we see that $\varrho \left(\top \right)>0.$ Take ${\top }_{*}\in [{\top }_0,\infty )$, ${\top }_{*}>{\top }_0$ such that $x\left({\tau }_i\left(\top \right)\right)>0$ on $[{\top }_{*},\infty )$. From (1), we see that

$${\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }=-\sum _{i=1}^k{q}_i\left(\top \right){\left(x\left({\tau }_i\left(\top \right)\right)\right)}^d<0,$$

for $\top \ge {\top }_{*}.$ That is, $ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d$ is of one sign and nonincreasing. Thus, ${\varrho }^{″}\left(\top \right)$ is also of one sign. Now, we proceed by contradiction. If we assume that ${\varrho }^{″}\left(\top \right)\le 0$ then there exists a constant $K<0$ such that

$$ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\le K<0\mathrm{for}\top \ge {\top }_1\ge {\top }_{*}.$$

Integrating this inequality from ${\top }_1$ to ⊤, we have

$${\varrho }^{\prime }\left(\top \right)\le {\varrho }^{\prime }\left({\top }_1\right)+K^{1/d}{\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{{ɩ}^{1/d}\left(s\right)}}\mathrm{ds}.$$

Letting $\top \to \infty$, it follows that ${\varrho }^{\prime }\left(\top \right)\to -\infty ,$ and hence $\varrho \left(\top \right)\to -\infty ,$ which contradicts that $\varrho \left(\top \right)>0.$ Therefore, ${\varrho }^{″}\left(\top \right)>0$ for $\top \in [{\top }_1,\infty ),$ which completes the proof. □

In the following lemmas, we obtain a useful relationships, which will be used later.

Lemma 3.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2$ be positive functions on $[0$,$\infty )$. The following inequalities hold:

• $\left(I_1\right)$−${\left({\mathcal{F}}_1+{\mathcal{F}}_2\right)}^{\gamma }\le 2^{\gamma -1}\left({\mathcal{F}}_1^{\gamma }+{\mathcal{F}}_2^{\gamma }\right),$ for $\gamma \in [1,\infty );$
• $\left(I_2\right)$−${\left({\mathcal{F}}_1+{\mathcal{F}}_2\right)}^{\gamma }\le \left({\mathcal{F}}_1^{\gamma }+{\mathcal{F}}_2^{\gamma }\right),$ for $\gamma \in (0,1].$

Proof. 

Let assume that ${\mathcal{F}}_1,{\mathcal{F}}_2>0$ and define the function U by

$$U\left(\mathcal{F}\right)={\mathcal{F}}^{\gamma },0<\mathcal{F}<\infty .$$

(17)

Then,

$$U^{″}\left(\mathcal{F}\right)=\gamma \left(\gamma -1\right){\mathcal{F}}^{\gamma -2}\ge 0,\mathrm{for}\mathcal{F}>0,$$

and U is a convex function. Therefore,

$$U\left({\displaystyle \frac{{\mathcal{F}}_1+{\mathcal{F}}_2}{2}}\right)-{\displaystyle \frac{U\left({\mathcal{F}}_1\right)+U\left({\mathcal{F}}_2\right)}{2}}\le 0,$$

which according to the definition in (17) results in $\left({\mathrm{I}}_1\right)$

$$2^{\gamma -1}\left({\mathcal{F}}_1^{\gamma }+{\mathcal{F}}_2^{\gamma }\right)\ge {\left({\mathcal{F}}_1+{\mathcal{F}}_2\right)}^{\gamma }.$$

Now, define

$$V\left({\mathcal{F}}_1,{\mathcal{F}}_2\right)={\mathcal{F}}_1^{\gamma }+{\mathcal{F}}_2^{\gamma }-{\left({\mathcal{F}}_1+{\mathcal{F}}_2\right)}^{\gamma },$$

Fixing ${\mathcal{F}}_1$, and noting that for $\gamma \in (0,1]$ it is

$${\displaystyle \frac{\partial V\left({\mathcal{F}}_1,{\mathcal{F}}_2\right)}{\partial {\mathcal{F}}_2}}=\gamma \left({\mathcal{F}}_2^{\gamma -1}-{\left({\mathcal{F}}_1+{\mathcal{F}}_2\right)}^{\gamma -1}\right)\ge 0,$$

it follows that $V\left({\mathcal{F}}_1,{\mathcal{F}}_2\right)>0.$ The proof is completed. □

Lemma 4.

Suppose that $x\left(\top \right)>0$ is a solution of Equation (1), and ${lim}_{\top \to \infty }x\left(\top \right)\ne 0.$ If

$${\int }_{{\top }_0}^{\infty }{\int }_v^{\infty }{\varpi }^{1/d}\left(u\right)\mathrm{d}u\mathrm{d}v=\infty ,$$

(18)

then (15) is satisfied.

Proof. 

Let $x>0$ be a solution of (1). We will prove the case when $d\ge 1$, and the case $0<d\le 1$ follows similarly. Proceeding by contradiction, we suppose that ${\varrho }^{\prime }\left(\top \right)<0$. Set

$$\widehat{W}\left(\top \right)=ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d.$$

From (1), we get

$$\begin{array}{ccc}\hfill {\widehat{W}}^{\prime }\left(\top \right)+{\displaystyle \frac{r_0^d}{g_0}}{\widehat{W}}^{\prime }\left(g\left(\top \right)\right)& \le & -\left(\sum _{i=1}^k{q}_i\left(\top \right)x^d\left({\tau }_i\left(\top \right)\right)+{\displaystyle \frac{r_0^d}{g_0}}\sum _{i=1}^k{q}_i\left(g\left(\top \right)\right)x^d\left({\tau }_i\left(g\left(\top \right)\right)\right)\right)\hfill \\ & \le & -\tilde{q}\left(\top \right)\sum _{i=1}^k\left(x^d\left({\tau }_i\left(\top \right)\right)+{\displaystyle \frac{r_0^d}{g_0}}{x}^d\left({\tau }_i\left(g\left(\top \right)\right)\right)\right).\hfill \end{array}$$

According to $\left({\mathrm{I}}_1\right)$ and (14), we have

$${\widehat{W}}^{\prime }\left(\top \right)+{\displaystyle \frac{r_0^d}{g_0}}{\widehat{W}}^{\prime }\left(g\left(\top \right)\right)\le -{\displaystyle \frac{\tilde{q}\left(\top \right)}{2^{d-1}}}\sum _{i=1}^k{\varrho }^d\left({\tau }_i\left(\top \right)\right).$$

(19)

Integrating (19) from ⊤ to ∞, we obtain

$$\widehat{W}\left(\top \right)+{\displaystyle \frac{r_0^d}{g_0}}\widehat{W}\left(g\left(\top \right)\right)\ge {\displaystyle \frac{{\int }_{\top }^{\infty }\tilde{q}\left(s\right){\sum }_{i=1}^k{\varrho }^d\left({\tau }_i\left(s\right)\right)\mathrm{d}s}{2^{d-1}}}.$$

(20)

In view of (2), we see that $\varrho \left(\top \right)\ge x\left(\top \right)$ and

$${\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }=-\sum _{i=1}^k{q}_i\left(\top \right)x^d\left({\tau }_i\left(\top \right)\right)\le 0.$$

(21)

Since $g\left(\top \right)\le \top$ and using (21), we find

$$\widehat{W}\left(g\left(\top \right)\right)\ge \widehat{W}\left(\top \right),$$

which setting into (20) yields

$$\widehat{W}\left(g\left(\top \right)\right)\ge {\displaystyle \frac{2^{1-d}}{\dot{E}}}{\int }_{\top }^{\infty }\tilde{q}\left(s\right)\sum _{i=1}^k{\varrho }^d\left({\tau }_i\left(s\right)\right)\mathrm{d}s.$$

where $\dot{E}=\left(g_0+r_0^d\right)/g_0.$ From Lemma 1, since ${lim}_{\top \to \infty }x\left(\top \right)\ne 0$, ${lim}_{\top \to \infty }\varrho \left(\top \right)=L>0$, L is a positive constant, and ${\varrho }^d\left(\tau \left(\top \right)\right)\ge L$. Then, we see that

$${\varrho }^{″}\left(g\left(\top \right)\right)-kL{\left({\displaystyle \frac{2^{1-d}}{\dot{E}}}\right)}^{{\displaystyle \frac{1}{d}}}{\left({ɩ}^{-1}\left(g\left(u\right)\right)\right)}^{{\displaystyle \frac{1}{d}}}{\left({\int }_{\top }^{\infty }\tilde{q}\left(s\right)\mathrm{d}s\right)}^{{\displaystyle \frac{1}{d}}}\ge 0.$$

(22)

Integrating (22) from ⊤ to ∞, we obtain

$$-g_0^{-1}{\varrho }^{\prime }\left(g\left(\top \right)\right)-kL{\displaystyle \frac{2^{\frac{1-d}{d}}}{{\dot{E}}^{\frac{1}{d}}}}{\int }_{\top }^{\infty }{\left({ɩ}^{-1}\left(g\left(u\right)\right)\right)}^{{\displaystyle \frac{1}{d}}}{\left({\int }_{\top }^{\infty }\tilde{q}\left(s\right)\mathrm{d}s\right)}^{{\displaystyle \frac{1}{d}}}\mathrm{d}u\ge 0.$$

Integrating again from ${\top }_0$ to ∞, we get

$$-g_0^{-2}\varrho \left(g\left({\top }_0\right)\right)-kL{\displaystyle \frac{2^{\frac{1-d}{d}}}{{\dot{E}}^{\frac{1}{d}}}}{\int }_{{\top }_0}^{\infty }{\int }_v^{\infty }{\left({ɩ}^{-1}\left(g\left(u\right)\right)\right)}^{{\displaystyle \frac{1}{d}}}{\left({\int }_u^{\infty }\tilde{q}\left(s\right)\mathrm{d}s\right)}^{{\displaystyle \frac{1}{d}}}\mathrm{d}u\mathrm{d}v\ge 0,$$

this contradicts (18). Thus ${\varrho }^{\prime }\left(\top \right)$ is positive, and the proof is complete. □

Lemma 5.

Suppose that ϱ satisfies (15) for $\top \ge {\top }_1$. Then

$${\displaystyle \frac{{\varrho }^{\prime }\left(\top \right)}{{\varrho }^{″}\left(\top \right)}}\ge {\Pi }_1\left(\top ,{\top }_1\right){ɩ}^{1/d}\left(\top \right)$$

(23)

and

$${\displaystyle \frac{\varrho \left(\top \right)}{{\varrho }^{″}\left(\top \right)}}\ge {\Pi }_2\left(\top ,{\top }_1\right){ɩ}^{1/d}\left(\top \right),$$

(24)

where

$${\Pi }_1\left(\top ,{\top }_1\right)={\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{{ɩ}^{1/d}\left(s\right)}}\mathrm{d}s,\mathit{and}{\Pi }_2\left(\top ,{\top }_1\right)={\int }_{{\top }_1}^{\top }{\Pi }_1\left(s,{\top }_1\right)\mathrm{d}s.$$

Proof. 

Since ${\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\le 0$, it follows that

$$\begin{array}{ccc}\hfill {\varrho }^{\prime }\left(\top \right)& \ge & {\int }_{{\top }_1}^{\top }{ɩ}^{-1/d}\left(s\right){\left(ɩ\left(s\right){\left({\varrho }^{″}\left(s\right)\right)}^d\right)}^{{\displaystyle \frac{1}{d}}}\mathrm{d}s\hfill \\ & \ge & {ɩ}^{1/d}\left(\top \right)\left({\varrho }^{″}\left(\top \right)\right){\Pi }_1\left(\top ,{\top }_1\right).\hfill \end{array}$$

This implies

$$\begin{array}{ccc}\hfill \varrho \left(\top \right)& \ge & {ɩ}^{{\displaystyle \frac{1}{d}}}\left(\top \right)\left({\varrho }^{″}\left(\top \right)\right){\int }_{{\top }_1}^{\top }{\int }_{{\top }_1}^s{ɩ}^{-1/d}\left(u\right)\mathrm{d}u\mathrm{d}s\hfill \\ & =& {ɩ}^{{\displaystyle \frac{1}{d}}}\left(\top \right)\left({\varrho }^{″}\left(\top \right)\right){\Pi }_2\left(\top ,{\top }_1\right).\hfill \end{array}$$

This completes the proof. □

2.1. Riccati Techniques-Oscillation Criteria

In this section, by employing the Riccati method under distinct hypotheses that accommodate various potential models, we establish the oscillation criteria of solutions of the differential equation.

Theorem 1.

Let $1\le d<\infty ,$${\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0,$ for which

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left[k{2}^{1-d}\Theta \left(s\right)\tilde{q}\left(s\right)-\dot{E}{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^{d+1}}{{\left(\Theta \left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right)\right)}^d}}\right]\mathrm{d}s=\infty ,$$

(25)

where $\dot{E}=\left(g_0+r_0^d\right)/g_0$. Then any solution x of (1) is oscillatory or satisfies

$$x\left(\top \right)\to 0\mathrm{as}\top \to \infty .$$

(26)

Proof. 

Let $x>0$ be a solution of (1), which does not approach zero. From (15) and (19) and from Lemma 5, we get (23). Let define

$$\omega \left(\top \right)={\displaystyle \frac{\Theta \left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d}{{\varrho }^d\left(\tau \left(\top \right)\right)}}$$

(27)

and

$$\upsilon \left(\top \right)={\displaystyle \frac{\Theta \left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d}{{\varrho }^d\left(\tau \left(\top \right)\right)}}.$$

(28)

Differentiating $\omega \left(\top \right)$, we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\top \right)& =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d+\Theta \left(\top \right){\left({\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\hfill \\ & =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d+{\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\hfill \\ & & -d\Theta \left(\top \right)ɩ\left(\top \right){\tau }^{\prime }\left(\top \right)\left({\displaystyle \frac{{\left({\varrho }^{″}\left(\top \right)\right)}^d{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)}{{\varrho }^{2d}\left(\tau \left(\top \right)\right)}}\right).\hfill \end{array}$$

(29)

From (15), (23) and $\tau \left(\top \right)\le \top ,$ it follows that

$$\begin{array}{ccc}\hfill {\varrho }^{\prime }\left(\tau \left(\top \right)\right)& \ge & {\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)\left({ɩ}^{1/d}\left(\tau \left(\top \right)\right){\varrho }^{″}\left(\tau \left(\top \right)\right)\right)\hfill \\ & \ge & {\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)\left({ɩ}^{1/d}\left(\top \right){\varrho }^{″}\left(\top \right)\right).\hfill \end{array}$$

Since $\tau \left(\top \right)<\top$, and from (27) and (29), we get that

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\top \right)& \le & \Theta \left(\top \right){\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }{\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}+\omega \left(\top \right){\displaystyle \frac{{\Theta }^{\prime }\left(\top \right)}{\Theta \left(\top \right)}}\hfill \\ & & -{\omega }^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}.\hfill \end{array}$$

(30)

Also, according to Lemma 4, we find

$$\begin{array}{ccc}\hfill {\upsilon }^{\prime }\left(\top \right)& =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d+\Theta \left(\top \right){\left({\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }\hfill \\ & =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d+{\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }\hfill \\ & & -{\displaystyle \frac{1}{{\varrho }^{2d}\left(\tau \left(\top \right)\right)}}d\Theta \left(\top \right)ɩ\left(g\left(\top \right)\right){\tau }^{\prime }\left(\top \right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right).\hfill \end{array}$$

(31)

From $\tau \left(\top \right)\le g\left(\top \right)$, (15) and (23), we have

$$\begin{array}{ccc}\hfill {\varrho }^{\prime }\left(\tau \left(\top \right)\right)& \ge & {\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)\left({ɩ}^{1/d}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)\right)\hfill \\ & \ge & {\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)\left({ɩ}^{1/d}\left(g\left(\top \right)\right){\varrho }^{″}\left(g\left(\top \right)\right)\right).\hfill \end{array}$$

Using (28) and (31), we get

$$\begin{array}{ccc}\hfill {\upsilon }^{\prime }\left(\top \right)& \le & {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }+\upsilon \left(\top \right){\displaystyle \frac{{\Theta }^{\prime }\left(\top \right)}{\Theta \left(\top \right)}}\hfill \\ & & -{\upsilon }^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}.\hfill \end{array}$$

(32)

Let define

$$\mathcal{L}\left(\top \right)={\omega }^{\prime }\left(\top \right)+{\displaystyle \frac{r_0^d}{g_0}}{\upsilon }^{\prime }\left(\top \right).$$

(33)

It follows from (30) and (32) that

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right)[{\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }+{\displaystyle \frac{r_0^d}{g_0}}{\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }]\hfill \\ & & +\omega \left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\omega }^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}\left[\upsilon \left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\upsilon }^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}\right].\hfill \end{array}$$

(34)

From (19), we see that

$${\displaystyle \frac{{\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }+\frac{r_0^d}{g_0}{\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\le -k{2}^{1-d}\tilde{q}\left(\top \right).$$

(35)

That is, from (34), we have

$${\displaystyle \frac{[{\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }+\frac{r_0^d}{g_0}{\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }]}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\le -k{2}^{1-d}\Theta \left(\top \right)\tilde{q}\left(\top \right)$$

(36)

Using (36) and (34), we get

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & -k{2}^{1-d}\Theta \left(\top \right)\tilde{q}\left(s\right)+\omega \left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\omega }^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}\left(\upsilon \left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\upsilon }^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}\right).\hfill \end{array}$$

(37)

Now, using (37) in the inequality [24]

$$N_{1}u-N_2{u}^{\left(d+1\right)/d}\le {\displaystyle \frac{d^d{N}_1^{d+1}}{{\left(d+1\right)}^{d+1}{N}_2^d}},N_2>0,$$

(38)

we arrive at

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & -k{2}^{1-d}\Theta \left(\top \right)\tilde{q}\left(s\right)+{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}\right)}^{d+1}}{{\left({\tau }^{\prime }\left(s\right)\Theta \left(\top \right){\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)\right)}^d}}\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left(\left({\Theta }^{\prime }\left(\top \right)\right)+\right)}^{d+1}}{{\left(\Theta \left(\top \right){\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right)\right)}^d}}.\hfill \end{array}$$

(39)

Integrating both sides from ${\top }_2$ to ⊤, we see that

$$\mathcal{L}\left({\top }_2\right)\ge {\int }_{{\top }_2}^{\top }\left(k{2}^{1-d}\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{\dot{E}{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^{d+1}}{{\left({\tau }^{\prime }\left(s\right)\Theta \left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right)\right)}^d}}\right)\mathrm{d}s,$$

which contradicts (25), and the proof is finished. □

Theorem 2.

Let $0<d\le 1,$${\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left[k\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{\dot{E}{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^{d+1}}{{\left(\Theta \left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right)\right)}^d}}\right]\mathrm{d}s=\infty ,$$

(40)

Then, any solution x of (1) is oscillatory or satisfies (26).

Proof. 

By Lemma 3, similar to the proof of Theorem 1, we obtain (40). The proof is complete. □

Theorem 3.

Let $1\le d<\infty ,$${\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left[k{2}^{1-d}\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{\dot{E}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^2}{4{\tau }^{\prime }\left(s\right)d\Theta \left(s\right)\left({\Pi }_2{\left(\tau \left(s\right),{\top }_1\right)}^{d-1}{\Pi }_1\left(\tau \left(s\right),{\top }_1\right)\right)}}\right]\mathrm{d}s=\infty ,$$

(41)

where $\dot{E}=\left(g_0+r_0^d\right)/g_0$. Then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

Let $x>0$ be a solution of (1), which does not approach zero. From (15) and, (19) in Lemma 4 and Lemma 5, we obtain (23) and (24).

Using the functions in (27) and (28), as in the proof of Theorem 1, we get (29) and (31), which from (29) yields

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\top \right)& =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d+{\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\hfill \\ & & -{\displaystyle \frac{d{\left(\Theta \left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^2{\tau }^{\prime }\left(\top \right)}{{\varrho }^{2d}\left(\tau \left(\top \right)\right)}}{\displaystyle \frac{{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)}{\Theta \left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d}}.\hfill \end{array}$$

(42)

From (15), (23), (24) and $\tau \left(\top \right)\le \top$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d}}{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)& =& {\displaystyle \frac{1}{ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d}}{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)\hfill \\ & \ge & {\displaystyle \frac{{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\left({ɩ}^{\frac{1}{d}}\left(\tau \left(\top \right)\right){\varrho }^{″}\left(\tau \left(\top \right)\right)\right)}^d}{{\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d}}\hfill \\ & \ge & {\displaystyle \frac{{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)}{{\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}}}\hfill \end{array}$$

(43)

Using (43) in (42), and in view of (29), we get

$${\omega }^{\prime }\left(\top \right)\le {\displaystyle \frac{\Theta \left(\top \right){\left[ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right]}^{\prime }}{{\varrho }^d\left(\tau \left(\top \right)\right)}}+{\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}\omega \left(\top \right)-{\displaystyle \frac{d{\tau }^{\prime }\left(\top ,s\right){\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)}{\Theta \left(\top \right){\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}}}{\omega }^2\left(\top \right).$$

(44)

Using (31), we see that

$$\begin{array}{ccc}\hfill {\upsilon }^{\prime }\left(\top \right)& =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d+{\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }\hfill \\ & & -{\displaystyle \frac{1}{{\varrho }^{2d}\left(\tau \left(\top \right)\right)}}{\displaystyle \frac{d{\left(\Theta \left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^2{\tau }^{\prime }\left(\top \right){\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)}{ɩ\left(g\left(\top \right)\right)\Theta \left(\top \right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d}}\hfill \end{array}$$

(45)

In view of (15), (23), (24) and $\tau \left(\top \right)\le g\left(\top \right)$, we see that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d}}{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)& =& {\displaystyle \frac{1}{ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d}}{\varrho }^{d-1}\left(\tau \left(\top \right)\right){\varrho }^{\prime }\left(\tau \left(\top \right)\right)\hfill \\ & \ge & {\displaystyle \frac{{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\left({ɩ}^{\frac{1}{d}}\left(\tau \left(\top \right)\right){\varrho }^{″}\left(\tau \left(\top \right)\right)\right)}^d}{{\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d}}\hfill \\ & \ge & {\displaystyle \frac{{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)}{{\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}}}.\hfill \end{array}$$

(46)

Combining (46) and (45), and applying (31), we get

$$\begin{array}{ccc}\hfill {\upsilon }^{\prime }\left(\top \right)& \le & {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }+\upsilon \left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}\hfill \\ & & -{\upsilon }^2\left(\top \right){\displaystyle \frac{{\tau }^{\prime }\left(\top ,s\right)d{\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{d-1}{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)}{\Theta \left(\top \right)}}.\hfill \end{array}$$

(47)

From (33), (44) and (47), we have

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & \Theta \left(\top \right){\displaystyle \frac{{\left[ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right]}^{\prime }}{{\varrho }^d\left(\tau \left(\top \right)\right)}}+{\displaystyle \frac{\frac{r_0^d}{g_0}{\left[ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right]}^{\prime }}{{\varrho }^d\left(\tau \left(\top \right)\right)}}\hfill \\ & & +{\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}\omega \left(\top \right)-{\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right)}{\Theta \left(\top \right){\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}}}{\omega }^2\left(\top \right)\hfill \\ & & +{\displaystyle \frac{r_0^d\left(\top \right)}{g_0}}\left[{\displaystyle \frac{\left({\Theta }^{\prime }\left(\top \right)\right)\upsilon \left(\top \right)}{\Theta \left(\top \right)}}-{\displaystyle \frac{d{\Pi }_1\left(\tau \left(\top \right),{\top }_1\right){\tau }^{\prime }\left(\top \right){\upsilon }^2\left(\top \right)}{\Theta \left(\top \right){\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}}}\right].\hfill \end{array}$$

(48)

Using (35), (48) and applying the inequality [24]

$$Bu-A{u}^2-{\displaystyle \frac{1}{4}}{B}^2{A}^{-1}\le 0,A>0,$$

we obtain

$$\mathcal{L}\left(\top \right)\le -k{2}^{1-d}\Theta \left(\top \right)\tilde{q}\left(s\right)+{\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^{1-d}{\left({\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}\right)}^2}{4{\tau }^{\prime }\left(s\right)d\Theta \left(\top \right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right)}}.$$

(49)

Integrating (49) from ${\top }_2$ to ⊤, we find

$${\int }_{{\top }_2}^{\top }\left({\displaystyle \frac{k\Theta \left(s\right)\tilde{q}\left(s\right)}{2^{d-1}}}-{\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(\tau \left(s\right),{\top }_1\right)\right)}^{1-d}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^2}{4d\Theta \left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right)}}\right)\mathrm{d}s\le \mathcal{L}\left({\top }_2\right),$$

which is a contradiction with (41). This completes the proof. □

Theorem 4.

Let $0<d\le 1$, ${\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left[k\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(\tau \left(s\right),{\top }_1\right)\right)}^{1-d}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^2}{4d\Theta \left(s\right){\tau }^{\prime }\left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right)}}\right]\mathrm{d}s=\infty .$$

(50)

Then, any solution x of (1) is oscillatory or satisfies (26).

Proof. 

From Lemma 3, similar to the proof of Theorem 3, we obtain (50). The proof is complete. □

Theorem 5.

Let $1\le d<\infty$ and $\tau \left(\top \right)\ge g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left[2^{1-d}k\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}\dot{E}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^{d+1}}{{\left(\Theta \left(s\right){\Pi }_1\left(g\left(s\right),{\top }_1\right)g^{\prime }\left(s\right)\right)}^d}}\right]\mathrm{d}s=\infty ,$$

(51)

where $\dot{E}=\left(g_0+r_0^d\right)/g_0$. Then, any solution x of (1) is oscillatory or satisfies (26).

Proof. 

Let $x>0$ be a solution of (1), which does not approach zero. From (15) and (19) in Lemma 4 and Lemma 5, we get (23). Define

$$\tilde{\omega }\left(\top \right)={\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d>0.$$

(52)

Differentiating, we get

$$\begin{array}{ccc}\hfill {\tilde{\omega }}^{\prime }\left(\top \right)& =& {\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d+\Theta \left(\top \right){\left({\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\hfill \\ & =& {\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d+{\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }\hfill \\ & & -{\displaystyle \frac{d\Theta \left(\top \right)ɩ\left(\top \right)g^{\prime }\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d{\varrho }^{d-1}g\left(\top \right){\varrho }^{\prime }\left(g\left(\top \right)\right)}{{\varrho }^{2d}\left(g\left(\top \right)\right)}}.\hfill \end{array}$$

(53)

By (15), (23) and $g\left(\top \right)\le \top ,$ we get

$$\begin{array}{ccc}\hfill {\varrho }^{\prime }\left(g\left(\top \right)\right)& \ge & {\Pi }_1\left(g\left(\top \right),{\top }_1\right)\left({ɩ}^{1/d}\left(g\left(\top \right)\right){\varrho }^{″}\left(g\left(\top \right)\right)\right)\hfill \\ & \ge & {\Pi }_1\left(g\left(\top \right),{\top }_1\right)\left({ɩ}^{1/d}\left(g\left(\top \right)\right){\varrho }^{″}\left(\top \right)\right).\hfill \end{array}$$

Using (52) and (53) yields

$$\begin{array}{ccc}\hfill {\tilde{\omega }}^{\prime }\left(\top \right)& \le & {\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }+\tilde{\omega }\left(\top \right){\displaystyle \frac{{\Theta }^{\prime }\left(\top \right)}{\Theta \left(\top \right)}}\hfill \\ & & -{\tilde{\omega }}^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(g\left(\top \right),{\top }_1\right)g^{\prime }\left(\top \right)}{{\Theta }^{\frac{1}{d}}\left(\top \right)}}.\hfill \end{array}$$

(54)

Now, set

$$\tilde{v}\left(\top \right)={\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d.$$

(55)

Differentiating, we obtain

$$\begin{array}{ccc}\hfill {\tilde{v}}^{\prime }\left(\top \right)& =& {\displaystyle \frac{1}{{\varrho }^d\left(\tau \left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\hfill \\ & & +\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\displaystyle \frac{{\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d}{{\varrho }^d\left(g\left(\top \right)\right)}}\right)}^{\prime }\hfill \\ & =& {\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}{\Theta }^{\prime }\left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\hfill \\ & & +{\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }\hfill \\ & & -{\displaystyle \frac{d\Theta \left(\top \right)ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d{\varrho }^{d-1}\left(g\left(\top \right)\right){\varrho }^{\prime }\left(g\left(\top \right)\right)g^{\prime }\left(\top \right)}{{\varrho }^{2d}\left(g\left(\top \right)\right)}}.\hfill \end{array}$$

(56)

From (15) and (23), we have

$${\varrho }^{\prime }\left(g\left(\top \right)\right)\ge {\Pi }_1\left(g\left(s\right),{\top }_1\right){ɩ}^{1/d}\left(g\left(\top \right)\right){\varrho }^{″}\left(g\left(\top \right)\right).$$

Using (55) and (56), we get

$$\begin{array}{ccc}\hfill {\tilde{v}}^{\prime }\left(\top \right)& \le & {\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}g\left(\top \right)\right)}^d\right)}^{\prime }+\tilde{v}\left(\top \right){\displaystyle \frac{{\Theta }^{\prime }\left(\top \right)}{\Theta \left(\top \right)}}\hfill \\ & & -{\tilde{v}}^{\left(d+1\right)/d}\left(\top \right)d{\Pi }_1\left(g\left(\top \right),{\top }_1\right)g^{\prime }\left(\top \right){\displaystyle \frac{1}{{\Theta }^{1/d}\left(\top \right)}},\hfill \end{array}$$

(57)

From (33), (54) and (57), we obtain that

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & {\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}\Theta \left(\top \right){\left(ɩ\left(\top \right){\left({\varrho }^{″}\left(\top \right)\right)}^d\right)}^{\prime }+{\displaystyle \frac{r_0^d}{g_0}}{\displaystyle \frac{1}{{\varrho }^d\left(g\left(\top \right)\right)}}{\left(ɩ\left(g\left(\top \right)\right){\left({\varrho }^{″}\left(g\left(\top \right)\right)\right)}^d\right)}^{\prime }\hfill \\ & & +\tilde{\omega }\left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\tilde{\omega }}^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{\Pi }_1\left(g\left(\top \right),{\top }_1\right)g^{\prime }\left(\top \right)}{{\Theta }^{1/d}\left(\top \right)}}\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}\left(\upsilon \left(\top \right){\displaystyle \frac{\left({\Theta }^{\prime }\left(\top \right)\right)+}{\Theta \left(\top \right)}}-{\tilde{v}}^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{g^{\prime }\left(\top \right)d{\Pi }_1\left(g\left(\top \right),{\top }_1\right)}{{\Theta }^{1/d}\left(\top \right)}}\right).\hfill \end{array}$$

(58)

According to (15), (35), (58) and $\tau (\top )\ge g(\top ),$ we obtain

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & -k{2}^{1-d}\Theta \left(\top \right)\tilde{q}\left(s\right)+\tilde{\omega }\left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\tilde{\omega }}^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{g}^{\prime }\left(\top \right){\Pi }_1\left(g\left(\top \right),{\top }_1\right)}{{\Theta }^{1/d}\left(\top \right)}}\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}\left(\upsilon \left(\top \right){\displaystyle \frac{{\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}}{\Theta \left(\top \right)}}-{\tilde{v}}^{\left(d+1\right)/d}\left(\top \right){\displaystyle \frac{d{g}^{\prime }\left(\top \right){\Pi }_1\left(g\left(\top \right),{\top }_1\right)}{{\Theta }^{1/d}\left(\top \right)}}\right).\hfill \end{array}$$

(59)

Using (59) and (38), we get

$$\begin{array}{ccc}\hfill \mathcal{L}\left(\top \right)& \le & -k{2}^{1-d}\Theta \left(\top \right)\tilde{q}\left(s\right)+{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}\right)}^{d+1}}{{\left(\Theta \left(\top \right){\Pi }_1\left(g\left(\top \right),{\top }_1\right)g^{\prime }\left(\top \right)\right)}^d}}\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}\right)}^{d+1}}{{\left(\Theta \left(\top \right){\Pi }_1\left(g\left(\top \right),{\top }_1\right)g^{\prime }\left(\top \right)\right)}^d}}.\hfill \end{array}$$

(60)

Integrating (60) from ${\top }_2$ to ⊤ yields

$${\int }_{{\top }_2}^{\top }\left(k{2}^{1-d}\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}\dot{E}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^{d+1}}{{\left(\Theta \left(s\right)g^{\prime }\left(s\right){\Pi }_1\left(g\left(s\right),{\top }_1\right)\right)}^d}}\right)\mathrm{d}s\le \mathcal{L}\left({\top }_2\right),$$

which is a contradiction with (51), and the proof is completed. □

Theorem 6.

Let $0<d\le 1$ and $\tau \left(\top \right)\ge g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left(k\Theta \left(s\right)\tilde{q}\left(s\right)-{\displaystyle \frac{\dot{E}{\left(d+1\right)}^{-\left(d+1\right)}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^{d+1}}{{\left({\Pi }_1\left(g\left(s\right),{\top }_1\right)\Theta \left(s\right)g^{\prime }\left(s\right)\right)}^d}}\right)\mathrm{d}s=\infty .$$

(61)

Thus, any solution x of (1) is either oscillatory or satisfying (26).

Proof. 

We can obtain (61) from Lemma 3, just like we did in the proof of Theorem 5. □

Theorem 7.

Let $1\le d<\infty$ and $\tau \left(\top \right)\ge g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left[k{2}^{1-d}\Theta \left(s\right)\tilde{q}\left(s\right)-\dot{E}{\displaystyle \frac{{\left({\Pi }_2\left(g\left(s\right),{\top }_1\right)\right)}^{1-d}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^2}{4{\Pi }_1\left(g\left(s\right),{\top }_1\right)d\Theta \left(s\right)g^{\prime }\left(s\right)}}\right]\mathrm{d}s=\infty ,$$

(62)

where $\dot{E}=\left(g_0+r_0^d\right)/g_0$. Then, any x of (1) is oscillatory or satisfies (26).

Proof. 

By (53) and (56), as in proof of Theorem 3, we obtain (62). The proof is complete. □

Theorem 8.

Let $0<d\le 1$ and $\tau \left(\top \right)\ge g\left(\top \right).$ Moreover, assume that (18) is satisfied and that there exists a function $\Theta \in C^1\left(\left[{\top }_0,\infty \right),\left(0,\infty \right)\right),$ for ${\top }_2>{\top }_1\ge {\top }_0$, such that

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }\left(k\Theta \left(s\right)\tilde{q}\left(s\right)-\dot{E}{\displaystyle \frac{{\left({\Pi }_2\left(g\left(s\right),{\top }_1\right)\right)}^{1-d}{\left({\left({\Theta }^{\prime }\left(s\right)\right)}_{+}\right)}^2}{4{\Pi }_1\left(g\left(s\right),{\top }_1\right)d\Theta \left(s\right)g^{\prime }\left(s\right)}}\right)\mathrm{d}s=\infty ,$$

(63)

where $\dot{E}=\left(g_0+r_0^d\right)/g_0$. Then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

According to Lemma 3 and Theorem 7, as in the proof of Theorem 3, we get (63). The proof is complete. □

Remark 1.

From Theorems 1–8, by choosing different choices of $\Theta ,$ we obtain different oscillation criteria for (1).

2.2. Philos-Type Oscillation Criteria

Now, we will present some Philos oscillation results for (1).

Definition 2.

Let denote $D_0=\left\{\left(\top ,s\right):\top >s\ge {\top }_0\right\}$ and $D=\left\{\left(\top ,s\right):\top \ge s\ge {\top }_0\right\}$. We say that Q $\in H=C\left(D,\mathbb{R}\right)$ possesses the property R if it holds

(I)

$Q\left(\top ,\top \right)=0,$ for $\top \ge {\top }_0,$ and $Q\left(\top ,s\right)>0$ for $\left(\top ,s\right)\in D_0;$

(II)

The partial derivative of Q with respect to the second variable is continuous and non-positive in $D_0$.

For convenience, given a function $\Theta \in C^1\left([{\top }_0,\infty \right)\left(0,\infty \right))$, we define the function $h\in C\left(D_0,\mathbb{R}\right)$ as

$$h\left(\top ,s\right)={\displaystyle \frac{\Theta \left(s\right)}{{\left(Q\left(\top ,s\right)\right)}^{\frac{d}{d+1}}}}\left(-{\displaystyle \frac{\partial }{\partial s}}Q\left(\top ,s\right)-{\displaystyle \frac{{\Theta }^{\prime }\left(s\right)}{\Theta \left(s\right)}}Q\left(\top ,s\right)\right),\left(\top ,s\right)\in D_0,$$

(64)

which implies

$$h\left(\top ,s\right){\displaystyle \frac{{\left(Q\left(\top ,s\right)\right)}^{\frac{d}{d+1}}}{\Theta \left(s\right)}}=-{\displaystyle \frac{\partial }{\partial s}}Q\left(\top ,s\right)-{\displaystyle \frac{{\Theta }^{\prime }\left(s\right)}{\Theta \left(s\right)}}Q\left(\top ,s\right).$$

Also, we will assume that

$$h_{-}\left(\top ,s\right):=max\{0,-h\left(\top ,s\right)\}.$$

Theorem 9.

Let $1\le d<\infty ,{\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (64) holds. If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_2}^{\top }\left(2^{1-d}Q\left(\top ,s\right)\Theta \left(s\right)\tilde{q}\left(s\right)-G_1\left(\top ,s\right)\right)\mathrm{d}s=\infty ,$$

(65)

where

$$G_1\left(\top ,s\right)=\dot{E}{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left(h_{-}\left(\top ,s\right)\right)}^{d+1}}{{\left(\Theta \left(s\right){\tau }^{\prime }\left(\top \right){\Pi }_2\left(\tau \left(\top \right),{\top }_1\right)\right)}^d}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

Let $x>0$ be a solution of (1), which does not approach zero. As in Theorem 1, set $\tilde{\omega }$ and $\upsilon$ yielding (37). Replacing ${\left({\Theta }^{\prime }\left(\top \right)\right)}_{+}$ by ${\Theta }^{\prime }\left(\top \right)$ and using (37), we get

$$\begin{array}{ccc}\hfill {\tilde{G}}_1\left(\top \right)& \le & -{\omega }^{\prime }\left(\top \right)-{\upsilon }^{\prime }\left(\top \right){\displaystyle \frac{r_0^d}{g_0}}+\omega \left(\top \right){\displaystyle \frac{{\Theta }^{\prime }\left(\top \right)}{\Theta \left(\top \right)}}\hfill \\ & & -{\omega }^{{\displaystyle \frac{d+1}{d}}}\left(\top \right)d{\Theta }^{-{\displaystyle \frac{1}{d}}}\left(\top \right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right)\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}\left({\displaystyle \frac{{\Theta }^{\prime }\left(\top \right)\upsilon \left(\top \right)}{\Theta \left(\top \right)}}-{\displaystyle \frac{{\upsilon }^{\frac{d+1}{d}}\left(\top \right)d{\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right)}{{\Theta }^{\frac{1}{d}}\left(\top \right)}}\right).\hfill \end{array}$$

(66)

This implies

$$\begin{array}{ccc}\hfill {\int }_{{\top }_2}^{\top }{\tilde{G}}_1\left(s\right)Q\left(\top ,s\right)\mathrm{d}s& \le & -{\int }_{{\top }_2}^{\top }Q\left(\top ,s\right){\omega }^{\prime }\left(s\right)\mathrm{d}s+{\int }_{{\top }_2}^{\top }{\displaystyle \frac{1}{\Theta \left(s\right)}}Q\left(\top ,s\right){\Theta }^{\prime }\left(s\right)\omega \left(s\right)\mathrm{d}s\hfill \\ & & -{\displaystyle \frac{r_0^d}{g_0}}\left({\int }_{{\top }_2}^{\top }Q\left(\top ,s\right){\upsilon }^{\prime }\left(s\right)\mathrm{d}s+{\int }_{{\top }_2}^{\top }{\displaystyle \frac{1}{\Theta \left(s\right)}}Q\left(\top ,s\right){\Theta }^{\prime }\left(s\right)\upsilon \left(s\right)\mathrm{d}s\right)\hfill \\ & & -{\int }_{{\top }_2}^{\top }d{\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\Theta }^{-{\displaystyle \frac{1}{d}}}\left(s\right)Q\left(\top ,s\right){\tau }^{\prime }\left(s\right){\omega }^{{\displaystyle \frac{d+1}{d}}}\left(\top \right)\mathrm{d}s\hfill \\ & & -{\displaystyle \frac{r_0^d}{g_0}}{\int }_{{\top }_2}^{\top }dQ\left(\top ,s\right){\Theta }^{-{\displaystyle \frac{1}{d}}}\left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right){\upsilon }^{{\displaystyle \frac{d+1}{d}}}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill {\int }_{{\top }_2}^{\top }{\tilde{G}}_1\left(s\right)Q\left(\top ,s\right)\mathrm{d}s& \le & Q\left(\top ,{\top }_2\right)\omega \left({\top }_2\right)-{\int }_{{\top }_2}^{\top }\left(-{\displaystyle \frac{\partial Q\left(\top ,s\right)}{\partial \left(s\right)}}-{\displaystyle \frac{Q\left(\top ,s\right)}{\Theta \left(s\right)}}{\Theta }^{\prime }\left(s\right)\right)\omega \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\top }_2}^{\top }Q\left(\top ,s\right)d{\Theta }^{-{\displaystyle \frac{1}{d}}}\left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right){\omega }^{{\displaystyle \frac{d+1}{d}}}\left(s\right)\mathrm{d}s\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}Q\left(\top ,{\top }_2\right)\upsilon \left({\top }_2\right)\hfill \\ & & -{\displaystyle \frac{r_0^d}{g_0}}{\int }_{{\top }_2}^{\top }\left[-{\displaystyle \frac{\partial Q\left(\top ,s\right)}{\partial \left(s\right)}}-{\displaystyle \frac{Q\left(\top ,s\right)}{\Theta \left(s\right)}}{\Theta }^{\prime }\left(s\right)\right]\upsilon \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\top }_2}^{\top }d{\displaystyle \frac{{\Pi }_1\left(\tau \left(s\right),{\top }_1\right)Q\left(\top ,s\right){\upsilon }^{\frac{d+1}{d}}\left(s\right)}{{\Theta }^{\frac{1}{d}}\left(s\right){\left({\tau }^{\prime }\left(s\right)\right)}^{-1}}}\mathrm{d}s,\hfill \end{array}$$

Using (64),we get

$$\begin{array}{ccc}\hfill {\int }_{{\top }_2}^{\top }{\tilde{G}}_1\left(s\right)Q\left(\top ,s\right)\mathrm{d}s& \le & Q\left(\top ,{\top }_2\right)\omega \left({\top }_2\right)+{\displaystyle \frac{r_0^d}{g_0}}Q\left(\top ,{\top }_2\right)\upsilon \left({\top }_2\right)\hfill \\ & & +{\int }_{{\top }_2}^{\top }\left({\displaystyle \frac{h_{-}\left(\top ,s\right)\omega \left(s\right)}{{\left(Q\left(\top ,s\right)\right)}^{\frac{-d}{d+1}}\Theta \left(s\right)}}-{\displaystyle \frac{d{\Pi }_1\left(\tau \left(s\right),{\top }_1\right)Q\left(\top ,s\right){\omega }^{\frac{d+1}{d}}\left(s\right)}{{\Theta }^{\frac{1}{d}}\left(s\right){\left({\tau }^{\prime }\left(\top \right)\right)}^{-1}}}\right)\mathrm{d}s\hfill \\ & & +{\displaystyle \frac{r_0^d}{g_0}}{\int }_{{\top }_2}^{\top }\left({\displaystyle \frac{h\left(\top ,s\right)\upsilon \left(s\right)}{{\left(Q\left(\top ,s\right)\right)}^{\frac{-d}{d+1}}\Theta \left(s\right)}}-{\displaystyle \frac{d{\Pi }_1\left(\tau \left(s\right),{\top }_1\right)Q\left(\top ,s\right){\upsilon }^{\frac{d+1}{d}}\left(s\right)}{{\Theta }^{\frac{1}{d}}\left(s\right){\left({\tau }^{\prime }\left(\top \right)\right)}^{-1}}}\right)\mathrm{d}s.\hfill \end{array}$$

(67)

Using (33), (67) and (38), we find

$$\mathcal{L}\left({\top }_2\right)\ge Q^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_2}^{\top }\left[{\displaystyle \frac{Q\left(\top ,s\right)\Theta \left(s\right)\tilde{q}\left(s\right)}{2^{d-1}}}-\dot{E}{\displaystyle \frac{{\left(d+1\right)}^{-\left(d+1\right)}{\left(h_{-}\left(\top ,s\right)\right)}^{d+1}{\left(Q\left(\top ,s\right)\right)}^{d+1}}{{\left({\tau }^{\prime }\left(s\right)\Theta \left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right)\right)}^d}}\right]\mathrm{d}s,$$

which contradicts (65). This ends the proof. □

Theorem 10.

Let $0<d\le 1,{\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (64) holds. If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_1}^{\top }Q\left(\top ,s\right)\Theta \left(s\right)F_1\left(\top ,s\right)\mathrm{d}s=\infty ,$$

(68)

where

$$F_1\left(\top ,s\right):=\tilde{q}\left(s\right)-\dot{E}{\displaystyle \frac{{\left(h_{-}\left(\top ,s\right)\right)}^{d+1}}{{\left(d+1\right)}^{d+1}{\left(\Theta \left(s\right){\Pi }_1\left(\tau \left(s\right),{\top }_1\right){\tau }^{\prime }\left(s\right)\right)}^d}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

According to Theorem 2, as in the proof of Theorem 9, we see that (68) holds. □

Theorem 11.

Let $1\le d<\infty ,{\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that

$${\displaystyle \frac{h\left(\top ,s\right){\left(Q\left(\top ,s\right)\right)}^{1/2}}{\Theta \left(s\right)}}=-{\displaystyle \frac{\partial }{\partial s}}Q\left(\top ,s\right)-{\displaystyle \frac{{\Theta }^{\prime }\left(s\right)Q\left(\top ,s\right)}{\Theta \left(s\right)}},\left(\top ,s\right)\in D_0.$$

(69)

If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_1}^{\top }\left({\displaystyle \frac{1}{2^{d-1}}}Q\left(\top ,s\right)\Theta \left(s\right)\tilde{q}\left(s\right)-G_2\left(\top ,s\right)\right)\mathrm{d}s=\infty ,$$

(70)

where

$$G_2\left(\top ,s\right):={\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(\tau \left(s\right),{\top }_1\right)\right)}^{1-d}{\left(h_{-}\left(\top ,s\right)\right)}^2}{4{\Pi }_1\left(\tau \left(s\right),{\top }_1\right)d\Theta \left(s\right){\tau }^{\prime }\left(s\right)}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

By using (19) and (48) in Theorem 3, as in proof of Theorem 9, we note that (70) holds. □

Theorem 12.

Let $0<d\le 1,{\tau }^{\prime }\left(\top \right)>0$ and $\tau \left(\top \right)\le g\left(\top \right).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (69) holds. If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_2}^{\top }\left(\Theta \left(s\right)\tilde{q}\left(s\right)Q\left(\top ,s\right)-F_2\left(\top ,s\right)\right)\mathrm{d}s=\infty ,$$

(71)

where

$$F_2\left(\top ,s\right)={\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(\tau \left(s\right),{\top }_1\right)\right)}^{1-d}{\left(h_{-}\left(\top ,s\right)\right)}^2}{4d{\Pi }_1\left(\tau \left(s\right),{\top }_1\right)\Theta \left(s\right){\tau }^{\prime }\left(s\right)}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

By Theorem 4, similar to the proof of Theorem 9, we obtain (71). □

Theorem 13.

Let $1\le d<\infty$ and $\tau (\top )\ge g(\top ).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (64) holds. If

$$\underset{\top \to \infty }{lim\; sup}{\displaystyle \frac{1}{Q\left(\top ,{\top }_2\right)}}{\int }_{{\top }_2}^{\top }\left({\displaystyle \frac{1}{2^{d-1}}}\Theta \left(s\right)Q\left(\top ,s\right)\tilde{q}\left(s\right)-G_3\left(\top ,s\right)\mathrm{d}s\right)=\infty ,$$

(72)

where

$$G_3\left(\top ,s\right)={\displaystyle \frac{\dot{E}{\left(d+1\right)}^{-\left(d+1\right)}{\left(h_{-}\left(\top ,s\right)\right)}^{d+1}}{{\left(\Theta \left(s\right){\Pi }_1\left(g\left(s\right),{\top }_1\right)g^{\prime }\left(s\right)\right)}^d}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

According to (59) in Theorem 5, as in the proof of that of Theorem 9, we get (72). □

Theorem 14.

Let $0<d\le 1$ and $\tau (\top )\ge g(\top ).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (64) holds. If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_2}^{\top }\left(\Theta \left(s\right)\tilde{q}\left(s\right)Q\left(\top ,s\right)-F_3\left(\top ,s\right)\right)\mathrm{d}s=\infty ,$$

(73)

where

$$F_3\left(\top ,s\right)={\displaystyle \frac{\dot{E}{\left(d+1\right)}^{-\left(d+1\right)}{\left(h_{-}\left(\top ,s\right)\right)}^{d+1}}{{\left({\Pi }_1\left(g\left(s\right),{\top }_1\right)\Theta \left(s\right)g^{\prime }\left(s\right)\right)}^d}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

According to Theorem 6, as in the proof of that of Theorem 9, we see that (73) holds. □

Theorem 15.

Let $1\le d<\infty$ and $\tau (\top )\ge g(\top ).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (69) holds. If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_2}^{\top }\left({\displaystyle \frac{1}{2^{d-1}}}Q\left(\top ,s\right)\Theta \left(s\right)\tilde{q}\left(s\right)-G_4\left(\top ,s\right)\right)\mathrm{d}s=\infty ,$$

(74)

where

$$G_4\left(\top ,s\right)={\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(g\left(s\right),{\top }_1\right)\right)}^{1-d}{\left(h_{-}\left(\top ,s\right)\right)}^2}{4d{\Pi }_1\left(g\left(s\right),{\top }_1\right)\Theta \left(s\right)g^{\prime }\left(s\right)}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

According to Theorem 7 as in the proof of that of Theorem 9, we get (74). □

Theorem 16.

Let $0<d\le 1$ and $\tau (\top )\ge g(\top ).$ Assume that (18) is satisfied, and that $Q\in H$ has the property R such that (69) holds. If

$$\underset{\top \to \infty }{lim\; sup}{Q}^{-1}\left(\top ,{\top }_2\right){\int }_{{\top }_2}^{\top }\left(2^{1-d}Q\left(\top ,s\right)\Theta \left(s\right)\tilde{q}\left(s\right)-F_4\left(\top ,s\right)\right)\mathrm{d}s=\infty ,$$

(75)

where

$$F_4\left(\top ,s\right)={\displaystyle \frac{\dot{E}{\left({\Pi }_2\left(g\left(s\right),{\top }_1\right)\right)}^{1-d}{\left(h_{-}\left(\top ,s\right)\right)}^2}{4d{\Pi }_1\left(g\left(s\right),{\top }_1\right)\Theta \left(s\right)g^{\prime }\left(s\right)}},$$

with $\dot{E}=\left(g_0+r_0^d\right)/g_0$, then any solution x of (1) is oscillatory or satisfies (26).

Proof. 

According to Theorem 8, as in the proof of that of Theorem 9, we obtain (75). □

Remark 2.

From Theorems 9–16, by choosing different choices of Θ and $Q,$ we obtain different oscillation criteria for Equation (1).

3. Examples

Example 1.

Consider the equation

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\top }}\left(\top {\left({\displaystyle \frac{{\mathrm{d}}^2}{{\mathrm{d}}^2\top }}\left(x\left(\top \right)+r_{0}x\left(0.5\top \right)\right)\right)}^3\right)+\left({\top }^{-5}\lambda \right)x^3\left(0.5\top \right)+\left({\top }^{-6}\lambda \right)x^3\left(0.5\top \right)=0,$$

(76)

where $\lambda >0,$ $r_0>0.$ Let $d=3,ɩ\left(\top \right)=\top ,\tau \left(\top \right)=g\left(\top \right)=0.5\top ,$ $g_0=0.5$ and

$$\tilde{q}\left(\top \right)=min\{q\left(\top \right),q\left(g\left(\top \right)\right)\}=\lambda {\top }^{-6}.$$

Then, we have

$${\Pi }_1\left(\top ,{\top }_1\right):={\int }_{{\top }_1}^{\top }ɩ{\left(s\right)}^{-1/3}\mathrm{d}s={\int }_{{\top }_1}^{\top }{s}^{-1/3}\mathrm{d}s\ge {\top }^{2/3},$$

and thus, (18) holds. Choosing $\Theta \left(\top \right)={\top }^5,$ from Theorem 1, we have

$${\displaystyle \frac{1}{4}}\left(\lambda -5^4{2}^5{4}^{-3}\left(2r_0^3+1\right)\right)\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }{\displaystyle \frac{1}{s}}\mathrm{d}s=\infty ,$$

Hence, any solution of (76) is oscillatory or satisfies (26) provided that

$$r_0^3<{\displaystyle \frac{4^3\lambda }{2^6{5}^4}}-{\displaystyle \frac{1}{2}}.$$

(77)

Since

$${\Pi }_2\left(\top ,{\top }_1\right)\ge {\displaystyle \frac{0.4}{{\top }^{\frac{-5}{3}}}},$$

from Theorem 2 we obtain that

$$\left({\displaystyle \frac{\lambda }{2^2}}-{\displaystyle \frac{\left(1+2r_0^3\right)}{\left(48\right){\left(5\right)}^{-4}{\left(2\right)}^{-5}}}\right)\underset{\top \to \infty }{lim}{\int }_{{\top }_2}^{\top }{s}^{-1}\mathrm{d}s=\infty .$$

Therefore, any solution of (76) is oscillatory or satisfies (26) provided that

$$r_0^3<{\displaystyle \frac{12\lambda }{2^6·5^4}}-{\displaystyle \frac{1}{2}}.$$

(78)

Now, to test the strength of the criteria for Equation (76), see Figure 1.

Remark 3.

From the previous Example 1, we observe the following:

• 1—Our results improve results of [5], [Example 1], where the authors of [5], [Example 1], obtained that

  $$\lambda >{\displaystyle \frac{729}{2}}.$$
  If we choose $r_0=1/3.$ From (77), we conclude that

  $$5{\left({\displaystyle \frac{5}{4}}\right)}^3{\left(2\right)}^5\left(1+2r_0^3\right)<{\displaystyle \frac{729}{2}}.$$
• 2—Theorem 1 gives a better result than Theorem 3.

Example 2.

Consider the equation

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\top }}\left(\top {\left({\displaystyle \frac{{\mathrm{d}}^2}{{\mathrm{d}}^2\top }}\left(x\left(\top \right)+r_{0}x\left(0.5\top \right)\right)\right)}^3\right)+\lambda {\top }^{-6}{x}^3\left(1.5\top \right)=0,$$

(79)

where $d=3,$ λ, $r_0>0.$ Let $ɩ\left(\top \right)=\top ,\tau \left(\top \right)=1.5\top ,g\left(\top \right)=0.5\top ,$ and $q\left(\top \right)=\lambda {\top }^{-6}$. We have $g^{\prime }\left(\top \right)=g_0=0.5,$ and we note that $1.5\top =\tau \left(\top \right)\ge g\left(\top \right)=0.5\top ,$

$${\Pi }_1\left(\top ,{\top }_1\right):={\int }_{{\top }_1}^{\top }ɩ{\left(s\right)}^{-1/3}\mathrm{d}s={\int }_{{\top }_1}^{\top }{s}^{-1/3}\mathrm{d}s\ge {\top }^{2/3},$$

$$\tilde{q}\left(\top \right)=min\{q\left(\top \right),q\left(g\left(\top \right)\right)\}=\lambda {\top }^{-6},$$

and

$${\Pi }_2\left(\top ,{\top }_1\right):={\int }_{{\top }_1}^{\top }{\int }_{{\top }_1}^sɩ{\left(u\right)}^{-1/3}\mathrm{d}u\mathrm{d}s\ge 0.4{\top }^{5/3}.$$

Choosing $\Theta \left(\top \right)={\top }^5$, from Theorem 7 we obtain that

$$\left[{\displaystyle \frac{1}{12}}\left(3\lambda -{\left(2\right)}^3.{\left(25\right)}^2\left(1+2r_0^3\right)\right)\right]\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }{s}^{-1}\mathrm{d}s=\infty .$$

So, any solution of (79) is oscillatory or satisfies (26) if

$$r_0^3<{\displaystyle \frac{3\lambda }{{\left(25\right)}^2.2^4}}-{\displaystyle \frac{1}{2}}.$$

Applying Theorem 5 we obtain that

$$\left({\displaystyle \frac{1}{4}}\lambda -\left(2r_0^3+1\right){\displaystyle \frac{2^5.5^4}{4^4}}\right)\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_2}^{\top }{s}^{-1}\mathrm{d}s=\infty .$$

Therefore, if

$$r_0^3<{\displaystyle \frac{4^3\lambda }{\left(2^6\right)\left(5^4\right)}}-{\displaystyle \frac{1}{2}}.$$

then any solution of (79) is oscillatory or satisfies (26).

Now, to test the strength of criteria for Equation (76), see Figure 2.

Remark 4.

For Equation (79) at $a=0.3$, Figure 2 illustrates the minimum permissible values of parameter λ required to satisfy conditions of Theorems 5 and 7. Notably, these conditions are mutually distinct over the interval $r_0\in (0,1.2)$, with none being a subset of another.

Remark 5.

For Example 2, Theorem 5 gives a better result than Theorem 7.

4. Conclusions

In this work, we have introduced new theorems on the oscillatory behavior of solutions to Equation (1). All the criteria we derived are based on condition (5), whereas most previous studies relied on condition (8). The conditions proposed in this paper are less restrictive compared to earlier studies, For example, we do not need restrictions (8) in [19], (10) in [20,21], (11) in [5], and (13) in [23]. We obtained diverse results based on the different Riccati-type assumptions employed during the proofs. As a result, these findings can be applied to a broader range of models.

Our findings have a significant impact on the advancement and refinement of oscillation theory, as they provide more flexible conditions than those found in previous studies.

Exploring alternative methods to eliminate restriction (4) would be a worthwhile direction for future research.