Introduction

This work explores a second-order differential equation with neutral terms of a sublinear and superlinear form, which can be expressed as follows:

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ \left( \varkappa \left( \textrm{s}\right) + \textrm{p}\left( \textrm{s}\right) \varkappa ^{\alpha }\left( \tau \left( \textrm{s}\right) \right) +\textrm{g}\left( \textrm{s}\right) \varkappa ^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) \right) ^{\prime }\right] ^{\gamma }\right) ^{\prime }+\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{ s})\varkappa ^{\beta }(\sigma _{j}(\textrm{s}))=0,\ \textrm{s}\ge \textrm{s}_{0}. \end{aligned}$$
(1)

For the purposes of this paper, it is assumed that:

\(\left( {\mathcal {A}}_{1}\right)\):

\(\gamma ,\) \(\beta ,\) \(\alpha\) and \(\delta\) are ratios of two positive odd integers, with \(\alpha >1\) and \(\delta <1\);

\(\left( {\mathcal {A}}_{2}\right)\):

\(\tau ,\varsigma \mathfrak {,}\sigma _{j}\in C^{1}([\textrm{s}_{0},\infty ), {\mathbb {R}} ),\) \(\tau (\textrm{s})\le \textrm{s},\) \(\varsigma (\textrm{s})\le \textrm{s },\) \(\sigma _{j}\left( \textrm{s}\right) \le \textrm{s},\) \(\sigma _{j}^{\prime }(\textrm{s})>0,\) and\(\ \lim _{\textrm{s}\rightarrow \infty }\tau \left( \textrm{s}\right) =\lim _{\textrm{s}\rightarrow \infty }\sigma _{j}\left( \textrm{s}\right) =\infty ,\) \(j=1,2,\ldots ,m;\)

\(\left( {\mathcal {A}}_{3}\right)\):

p, g\(, {\mathfrak {f}}_{j}\in C([\textrm{s}_{0},\infty ),\left( 0,\infty \right) ),\) and \({\mathfrak {f}}_{j}(\textrm{s})\) is not eventually zero on \([\textrm{s} ^{*},\infty )\) for \(\textrm{s}^{*}\ge \textrm{s}_{0},\) \(j=1,2,\ldots ,m\) ;

\(\left( {\mathcal {A}}_{4}\right)\):

\(\hslash \in C^{1}([\textrm{s} _{0},\infty ),\left( 0,\infty \right) ),\) \(\hslash ^{\prime }\left( \textrm{s }\right) \ge 0,\) and satisfies

$$\begin{aligned} \int _{\textrm{s}_{0}}^{\infty }\frac{1}{\hslash ^{1/\gamma }\left( \nu \right) }\textrm{d}\nu <\infty . \end{aligned}$$
(2)

It is useful to define \({\mathcal {U}}\left( \textrm{s}\right)\) in the following manner:

$$\begin{aligned} {\mathcal {U}}\left( \textrm{s}\right) =\varkappa \left( \textrm{s}\right) + \textrm{p}\left( \textrm{s}\right) \varkappa ^{\alpha }\left( \tau \left( \textrm{s}\right) \right) +\textrm{g}\left( \textrm{s}\right) \varkappa ^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) . \end{aligned}$$
(3)

A function \(\varkappa \in C^{1}([\textrm{s}_{\varkappa },\infty ), {\mathbb {R}} )\), where \(\textrm{s}_{\varkappa }\geqslant \textrm{s}_{0},\) is considered a solution of the Eq. (1) if it satisfies the condition \(\hslash ( \textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right] ^{\gamma }\in C^{1}[\textrm{s}_{\varkappa },\infty )\), and it fulfills the Eq. (1) for all \(\textrm{s}\in [\textrm{s}_{\varkappa },\infty )\). We focus only on solutions \(\varkappa\) of Eq. (1) that are defined on the half-line \([\textrm{s}_{\varkappa },\infty )\) and satisfy the condition:

$$\begin{aligned} \sup \{|\varkappa (\textrm{s})|:\textrm{s}\geqslant \textrm{S}\}>0,\text { for all }\textrm{S}\ge \textrm{s}_{\varkappa }. \end{aligned}$$

A solution of the Eq. (1) is called oscillatory if it neither eventually remains positive nor eventually remains negative. On the other hand, if the solution eventually becomes positive or negative, it is considered non-oscillatory. The Eq. (1) is considered oscillatory if all of its solutions exhibit oscillatory behavior.

Differential equations (DEs) are essential tools for understanding and analyzing complex natural phenomena in many scientific and engineering fields. They provide an effective mathematical means of describing changes in systems over time or distance, making them central to many applications such as physics, chemistry, economics and engineering. The importance of these equations is clearly demonstrated by their use in studying phenomena such as motion, heat transfer, and population growth, which enhances the deep understanding of human and natural societies (see1,2,3).

These equations are widely used in the mathematical modeling of engineering and physical systems. For example, they can be used in electrical systems involving lossless transmission lines, as well as in studying the vibrations of mechanical masses connected by elastic strings. They also play an important role in solving time delay problems in environmental and economic applications. Pioneering studies such as Jadlovská4, Bohner et al.5, Baculíková and Džurina6, Tamilvanan et al.7, Almarri et al.8, Alnafisah et al.9, Tunç et al.10, and Braverman et al.11 have shown the importance of these equations in improving models and providing effective solutions.

The study of the behavior of oscillatory solutions of DEs of different orders is one of the fundamental areas for understanding dynamical systems affected by oscillations. Studies focus on analyzing the oscillatory properties of solutions, where the results reveal that the behavior of these solutions is closely affected by the order of the equation and the nature of the associated coefficients. These studies are of particular importance in determining the conditions necessary for the emergence and continuity of oscillations, which contributes to improving mathematical models of physical phenomena, electrical and mechanical systems, in addition to environmental and economic applications. Many studies such as Zhang et al.12, Chatzarakis et al.13, Graef et al.14, Alqahtani et al.15, Masood et al.16, and Batiha et al.17 have contributed to enhancing our understanding of this field by providing new analytical methods and advanced criteria for understanding oscillations in differential equations. Here are some previous related studies:Grace and Lalli18 and Xu and Xia19 have proven the oscillation of the equation

$$\begin{aligned} \left( \hslash (\textrm{s})\left( \varkappa \left( \textrm{s}\right) + \textrm{p}\left( \textrm{s}\right) \varkappa \left( \textrm{s}-\tau \right) \right) ^{\prime }\right) ^{\prime }+{\mathfrak {f}}(\textrm{s})Q\left( \varkappa (\textrm{s}-\sigma ))\right) =0, \end{aligned}$$

under the condition \(Q\left( x\right) /x\ge k,\) and

$$\begin{aligned} \int _{\textrm{s}_{0}}^{\infty }\frac{1}{\hslash \left( \nu \right) }\textrm{d }\nu =\infty . \end{aligned}$$

Baculíková and Dazrina20 developed powerful criteria for the oscillation of second-order NDEs

$$\begin{aligned} \left( \hslash (\textrm{s})\left( \varkappa \left( \textrm{s}\right) + \textrm{p}\left( \textrm{s}\right) \varkappa \left( \tau \left( \textrm{s} \right) \right) \right) ^{\prime }\right) ^{\prime }+{\mathfrak {f}}(\textrm{s} )Q\left( \varkappa (\sigma \left( \textrm{s}\right) ))\right) =0, \end{aligned}$$

effectively reducing the problem to first-order cases using comparison theorems.

Sun et al.21, and Jadlovská22 have investigated the oscillatory properties of solutions to half-linear NDEs

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ \left( \varkappa \left( \textrm{s}\right) + \textrm{p}\left( \textrm{s}\right) \varkappa \left( \tau \left( \textrm{s} \right) \right) \right) ^{\prime }\right] ^{\gamma }\right) ^{\prime }+ {\mathfrak {f}}(\textrm{s})\varkappa ^{\gamma }(\sigma (\textrm{s}))=0, \end{aligned}$$
(4)

under the condition

$$\begin{aligned} \int _{\textrm{s}_{0}}^{\infty }\frac{1}{\hslash ^{1/\gamma }\left( \nu \right) }\textrm{d}\nu =\infty . \end{aligned}$$

Various methods have been employed to derive criteria for oscillation. In addition, the Eq. (4) under the assumption (2) has been thoroughly examined in studies by Han et al.23, and Aldiaiji et al.24. Baculíková and Džurina25, Agarwal et al.26, Li and Rogovchenko27 , and Masood et al.28 have addressed the importance of the oscillation behavior of NDEs

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ \left( \varkappa \left( \textrm{s}\right) + \textrm{p}\left( \textrm{s}\right) \varkappa \left( \tau \left( \textrm{s} \right) \right) \right) ^{\prime }\right] ^{\gamma }\right) ^{\prime }+ {\mathfrak {f}}(\textrm{s})\varkappa ^{\beta }(\sigma (\textrm{s}))=0. \end{aligned}$$
(5)

They conducted a comprehensive study and obtained criteria for oscillation.

In recent years, Agarwal et al.29, Džurina et al.30 and Wu et al.31 have investigated oscillation criteria for differential equations involving a sublinear neutral term, given by

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ \left( \varkappa \left( \textrm{s}\right) + \textrm{g}\left( \textrm{s}\right) \varkappa ^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) \right) ^{\prime }\right] ^{\gamma }\right) ^{\prime }+{\mathfrak {f}}(\textrm{s})\varkappa ^{\beta }(\sigma (\textrm{s} ))=0. \end{aligned}$$
(6)

The results for this equation were established under the assumption (2).

Previous studies focus on differential equations with a linear neutral term, as in (4) and (5), or those containing sublinear terms, as in (6). In this study, the scope of analysis is extended to the differential Eq. (1), which includes both superlinear and sublinear terms, adding a new dimension to the understanding of oscillatory behavior. The methodology relies on the use of the Riccati technique supported by innovative substitutions to derive new and accurate criteria that define the oscillation properties. In addition, applied examples are presented that demonstrate the effectiveness of the derived criteria when treating equations containing mixed terms. This study contributes to enhancing the theoretical understanding of NDEs and expanding their horizons to include advanced applications in dynamic systems, control, and mathematical modeling of physical phenomena.

Preliminary results

In this section, we present some of the lemmas and results containing monotonic properties essential to achieving the main results. We begin by defining the following concepts:

$$\begin{aligned} & \sigma _{\min }\left( \textrm{s}\right) :=\min \left\{ \sigma _{j}\left( \textrm{s}\right) ,\ j=1,2,\ldots ,m\right\} , \\ & \sigma _{\max }\left( \textrm{s}\right) :=\max \left\{ \sigma _{j}\left( \textrm{s}\right) ,\ j=1,2,\ldots ,m\right\} , \\ & \pi \left( \textrm{s}\right) =\int _{\textrm{s}}^{\infty }\frac{1}{\hslash ^{1/\gamma }\left( \nu \right) }\textrm{d}\nu , \end{aligned}$$

and

$$\begin{aligned} \pi (\textrm{s},\textrm{s}_{0}):=\int _{\textrm{s}_{0}}^{\textrm{s}}\frac{1}{ \hslash ^{1/\gamma }\left( \nu \right) }\textrm{d}\nu \text {.} \end{aligned}$$

Lemma 1

32 Suppose that A and B are constants and that \(\gamma\) is the ratio of two odd positive numbers. Then

$$\begin{aligned} Bu-Au^{\left( \gamma +1\right) /\gamma }\le \frac{\gamma ^{\gamma }}{\left( \gamma +1\right) ^{\gamma +1}}\frac{B^{\gamma +1}}{A^{\gamma }},\ A>0. \end{aligned}$$
(7)

Lemma 2

33 Let \(\varkappa (\textrm{s})\) be an eventually positive solution of Eq. (1). Then, the associated function \({\mathcal {U}}\) eventually satisfies one of the following cases for \(\textrm{s}\geqslant \textrm{s}_{1}\geqslant \textrm{s}_{0}\):

$$\begin{aligned} (\textrm{C}_{1})&:&{\mathcal {U}}\left( \textrm{s}\right)>0,\ {\mathcal {U}}^{\prime }\left( \textrm{s}\right)>0,\ \left( \hslash \left( \textrm{ s}\right) \left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right) ^{\prime }<0, \\ (\textrm{C}_{2})&:&{\mathcal {U}}\left( \textrm{s}\right) >0,\ {\mathcal {U}}^{\prime }\left( \textrm{s}\right)<0,\ \left( \hslash \left( \textrm{ s}\right) \left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right) ^{\prime }<0. \end{aligned}$$

The symbol \(\textrm{C}_{i}\) refers to the set of all solutions that eventually become positive and satisfy condition (\(\textrm{C}_{i}\)) for \(i=1,2.\)

Remark 1

It is clear that if \(\varkappa \left( \textrm{s}\right)\) is a solution of Eq. ( 1), then \(-\varkappa \left( \textrm{s}\right)\) is also a solution of Eq. ( 1). Therefore, we can restrict our attention to only the positive solutions of the studied equation.

Main results

This section presents essential results concerning the monotonic behavior of solutions to Eq. (1) aWe create specific conditions that eliminate the possibility of positive solutions, considering scenarios \(( \textrm{C}_{1})\) and \((\textrm{C}_{2})\) separately and with a clear technique.

Category \(\textrm{C}_{1}\)

Lemma 3

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{1}\). Assume that

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha -1}(\textrm{s},\textrm{s}_{1})=0\text {, and }\lim _{\textrm{s} \rightarrow \infty }\textrm{g}\left( \textrm{s}\right) =0. \end{aligned}$$
(8)

Then, eventually:

\(\left( \textrm{a}_{1}\right)\) \({\mathcal {U}}(\textrm{s})\geqslant \hslash ^{1/\gamma }(\textrm{s}){\mathcal {U}}^{\prime }(\textrm{s})\pi (\textrm{s}, \textrm{s}_{1});\)

\(\left( \textrm{b}_{1}\right)\) \({\mathcal {U}}(\textrm{s})/\pi (\textrm{s}, \textrm{s}_{0})\) is decreasing; 

\(\left( \textrm{c}_{1}\right)\) \(\varkappa \left( \textrm{s}\right) >L {\mathcal {U}}\left( \textrm{s}\right) ,\) where \(L\in \left( 0,1\right) ;\)

\(\left( \textrm{d}_{1}\right)\) \(\left( \hslash (\textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right] ^{\gamma }\right) ^{\prime }\le -L^{\beta }{\mathcal {U}}^{\beta }(\sigma _{\min }(\textrm{s}))\sum _{j=1}^{m} {\mathfrak {f}}_{j}(\textrm{s}).\)

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{1}\). Then, there exists a value \(\textrm{s}_{1}\ge \textrm{s}_{0}\) such that for \(\textrm{s} \ge \textrm{s}_{1}\), we have \(\varkappa \left( \textrm{s}\right) >0,\) \(\varkappa \left( \tau \left( \textrm{s}\right) \right) >0,\) \(\varkappa \left( \varsigma \left( \textrm{s}\right) \right) >0\) and \(\varkappa \left( \sigma _{j}\left( \textrm{s}\right) \right) >0,\) \(j=1,2,...m.\) \(\left( \textrm{a}_{1}\right)\) Since \(\hslash ^{1/\gamma }\left( \textrm{s} \right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right)\) is decreasing, we deduce that

$$\begin{aligned} {\mathcal {U}}\left( \textrm{s}\right)= & {\mathcal {U}}\left( \textrm{s} _{1}\right) +\int _{\textrm{s}_{1}}^{\textrm{s}}\frac{\hslash ^{1/\gamma }\left( \nu \right) {\mathcal {U}}^{\prime }\left( \nu \right) }{\hslash ^{1/\gamma }\left( \nu \right) }\textrm{d}\nu \\\ge & {\mathcal {U}}\left( \textrm{s}_{1}\right) +\hslash ^{1/\gamma }\left( \textrm{s}\right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \int _{ \textrm{s}_{1}}^{\textrm{s}}\frac{1}{\hslash ^{1/\gamma }\left( \nu \right) } \textrm{d}\nu \\\ge & \hslash ^{1/\gamma }\left( \textrm{s}\right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \pi \left( \textrm{s},\textrm{s}_{1}\right) ,\ \textrm{s}\ge \textrm{s}_{1}. \end{aligned}$$

\(\left( \textrm{b}_{1}\right)\) Based on the inequality above, we determine that

$$\begin{aligned} \left( \frac{{\mathcal {U}}(\textrm{s})}{\pi (\textrm{s},\textrm{s}_{1})} \right) ^{\prime }=\frac{\hslash ^{1/\gamma }(\textrm{s}){\mathcal {U}}^{\prime }(\textrm{s})\pi (\textrm{s},\textrm{s}_{1})-{\mathcal {U}}(\textrm{s})}{ \hslash ^{1/\gamma }(\textrm{s})\pi ^{2}(\textrm{s},\textrm{s}_{1})}\le 0. \end{aligned}$$

\(\left( \textrm{c}_{1}\right)\) Using (3), it follows that \({\mathcal {U}}\left( \textrm{s}\right) \ge \varkappa \left( \textrm{s}\right)\). Consequently

$$\begin{aligned} \varkappa \left( \textrm{s}\right)= & {\mathcal {U}}\left( \textrm{s}\right) - \textrm{p}\left( \textrm{s}\right) \varkappa ^{\alpha }\left( \tau \left( \textrm{s}\right) \right) -\textrm{g}\left( \textrm{s}\right) \varkappa ^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) \\\ge & {\mathcal {U}}\left( \textrm{s}\right) -\textrm{p}\left( \textrm{s} \right) {\mathcal {U}}^{\alpha }\left( \tau \left( \textrm{s}\right) \right) - \textrm{g}\left( \textrm{s}\right) {\mathcal {U}}^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) . \end{aligned}$$

Now, Because \(\alpha >1,\) and \(\delta <1,\) and given that \({\mathcal {U}}( \textrm{s})/\pi (\textrm{s},\textrm{s}_{1})\) is decreasing and positive, while \({\mathcal {U}}(\textrm{s})\left( \textrm{s}\right)\) is increasing, we can conclude that there exist two constants \(l_{1}\) and \(l_{2}\) such that

$$\begin{aligned} \varkappa \left( \textrm{s}\right)\ge & {\mathcal {U}}\left( \textrm{s} \right) -\textrm{p}\left( \textrm{s}\right) {\mathcal {U}}^{\alpha }\left( \textrm{s}\right) -\textrm{g}\left( \textrm{s}\right) {\mathcal {U}}^{\delta }\left( \textrm{s}\right) \\= & \left[ 1-\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha -1}(\textrm{s}, \textrm{s}_{1})\left( \frac{{\mathcal {U}}\left( \textrm{s}\right) }{\pi ( \textrm{s},\textrm{s}_{1})}\right) ^{\alpha -1}-\textrm{g}\left( \textrm{s} \right) {\mathcal {U}}^{\delta -1}\left( \textrm{s}\right) \right] {\mathcal {U}} \left( \textrm{s}\right) \\\ge & \left[ 1-l_{1}\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha -1}( \textrm{s},\textrm{s}_{1})-l_{2}\textrm{g}\left( \textrm{s}\right) \right] {\mathcal {U}}\left( \textrm{s}\right) . \end{aligned}$$

By (8), it is possible to choose \(L\in \left( 0,1\right)\) such that

$$\begin{aligned} \varkappa \left( \textrm{s}\right) \ge L{\mathcal {U}}\left( \textrm{s}\right) . \end{aligned}$$

\(\left( \textrm{d}_{1}\right)\) Using (1) and (\(\textrm{c}_{1}\)), we see that

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right] ^{\gamma }\right) ^{\prime }=-\sum _{j=1}^{m}{\mathfrak {f}}_{j}( \textrm{s})\varkappa ^{\beta }(\sigma _{j}(\textrm{s}))\le -L^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s}){\mathcal {U}}^{\beta }(\sigma _{j}( \textrm{s})). \end{aligned}$$

Since \({\mathcal {U}}^{\prime }>0,\,\)and \(\sigma _{j}(\textrm{s})\ge \sigma _{\min }(\textrm{s}),\) for \(j=1,2,\ldots ,m,\) then

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right] ^{\gamma }\right) ^{\prime }\le -L^{\beta }{\mathcal {U}} ^{\beta }(\sigma _{\min }(\textrm{s}))\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{ s}). \end{aligned}$$

Thus, the proof is complete. \(\square\)

Lemma 4

Suppose that the condition (8) is satisfied. If \(\beta \ge \gamma\) and there is a nondecreasing function \(\rho \in C^{1}([\textrm{ s}_{0},\infty ),\left( 0,\infty \right) )\)

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( L^{\beta }\rho \left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\frac{1}{\left( \gamma +1\right) ^{\gamma +1}}\frac{\hslash \left( \sigma _{\min }\left( \nu \right) \right) \left( \rho ^{\prime }\left( \nu \right) \right) ^{\gamma +1}}{\left( c_{1}\rho \left( \nu \right) \sigma _{\min }^{\prime }\left( \nu \right) \right) ^{\gamma }} \right) \textrm{d}\nu =\infty , \end{aligned}$$
(9)

holds for every \(c_{1}>0\) and \(L\in \left( 0,1\right) ,\) then \(\textrm{C} _{1}=\varnothing\).

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{1}\). Now, define a function \(\omega \left( \textrm{s}\right)\) by

$$\begin{aligned} \omega \left( \textrm{s}\right) :=\rho \left( \textrm{s}\right) \frac{ \hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }}{{\mathcal {U}}^{\beta }\left( \sigma _{\min }\left( \textrm{ s}\right) \right) }, \end{aligned}$$
(10)

which yields \(\omega \left( \textrm{s}\right) >0,\) \(\textrm{s}\ge \textrm{s} _{1},\)

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) =\rho ^{\prime }\left( \textrm{s} \right) \frac{\hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{ s}\right) \right) ^{\gamma }}{{\mathcal {U}}^{\beta }\left( \sigma _{\min }\left( \textrm{s}\right) \right) }+\rho \left( \textrm{s}\right) \frac{ \left( \hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right) ^{\gamma }\right) ^{\prime }}{{\mathcal {U}}^{\beta }\left( \sigma _{\min }\left( \textrm{s}\right) \right) }-\beta \rho \left( \textrm{s }\right) \sigma _{\min }^{\prime }\left( \textrm{s}\right) \frac{\hslash ( \textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }{\mathcal {U}}^{\prime }\left( \sigma _{\min }\left( \textrm{s} \right) \right) }{{\mathcal {U}}^{\beta +1}\left( \sigma _{\min }\left( \textrm{ s}\right) \right) }. \end{aligned}$$
(11)

Using \(\left( \textrm{d}_{1}\right)\), (10), and (11), we get

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) \le \frac{\rho ^{\prime }\left( \textrm{s}\right) }{\rho \left( \textrm{s}\right) }\omega \left( \textrm{s} \right) -L^{\beta }\rho \left( \textrm{s}\right) \sum _{j=1}^{m}{\mathfrak {f}} _{j}(\textrm{s})-\beta \sigma _{\min }^{\prime }\left( \textrm{s}\right) \frac{{\mathcal {U}}^{\prime }\left( \sigma _{\min }\left( \textrm{s}\right) \right) }{{\mathcal {U}}\left( \sigma _{\min }\left( \textrm{s}\right) \right) } \omega \left( \textrm{s}\right) . \end{aligned}$$
(12)

Since \(\sigma_{\min } \left( \textrm{s}\right) \le \textrm{s},\) and \(\left( \hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }(\textrm{s})\right) ^{\gamma }\right) ^{\prime }<0,\) then

$$\begin{aligned} \hslash ^{1/\gamma }\left( \textrm{s}\right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \le \hslash ^{1/\gamma }\left( \sigma _{\min }\left( \textrm{s}\right) \right) {\mathcal {U}}^{\prime }\left( \sigma _{\min }\left( \textrm{s}\right) \right) . \end{aligned}$$
(13)

Substituting (13) into (12), we obtain

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) \le \frac{\rho ^{\prime }\left( \textrm{s}\right) }{\rho \left( \textrm{s}\right) }\omega \left( \textrm{s} \right) -L^{\beta }\rho \left( \textrm{s}\right) \sum _{j=1}^{m}{\mathfrak {f}} _{j}(\textrm{s})-\frac{\beta \sigma _{\min }^{\prime }\left( \textrm{s} \right) }{\left( \rho \left( \textrm{s}\right) \hslash \left( \sigma _{\min }\left( \textrm{s}\right) \right) \right) ^{1/\gamma }}\left[ {\mathcal {U}} \left( \sigma _{\min }\left( \textrm{s}\right) \right) \right] ^{\beta /\gamma -1}\omega ^{1/\gamma +1}\left( \textrm{s}\right) . \end{aligned}$$
(14)

Due to the fact that \({\mathcal {U}}^{\prime }>0\) and \(\beta \ge \gamma\), there are constants \(c_{1}>0\) and \(\textrm{s}_{2}\ge \textrm{s}_{1}\) such that

$$\begin{aligned} {\mathcal {U}}^{\beta /\gamma -1}\left( \sigma _{\min }\left( \textrm{s}\right) \right) \ge c_{1}, \textrm{s}\ge \textrm{s}_{2}. \end{aligned}$$
(15)

Thus, the inequality (14) gives

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) \le -L^{\beta }\rho \left( \textrm{s}\right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\rho ^{\prime }\left( \textrm{s}\right) }{\rho \left( \textrm{s}\right) }\omega \left( \textrm{s}\right) -\frac{\gamma c_{1}\sigma _{\min }^{\prime }\left( \textrm{s}\right) }{\left[ \rho \left( \textrm{s}\right) \hslash \left( \sigma _{\min }\left( \textrm{s}\right) \right) \right] ^{1/\gamma }}\omega ^{1/\gamma +1}\left( \textrm{s}\right) . \end{aligned}$$
(16)

By applying Lemma 1, where we defin \(B=\rho ^{\prime }\left( \textrm{ s}\right) /\rho \left( \textrm{s}\right) ,\) \(A=\gamma c_{1}\sigma _{\min }^{\prime }\left( \textrm{s}\right) /\left[ \rho \left( \textrm{s}\right) \hslash \left( \sigma _{\min }\left( \textrm{s}\right) \right) \right] ^{1/\gamma },\) and \(u\left( \textrm{s}\right) =\omega \left( \textrm{s} \right) ,\) the inequality derived is as follows

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) \le -L^{\beta }\rho \left( \textrm{s}\right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{1}{\left( \gamma +1\right) ^{\gamma +1}}\frac{\hslash \left( \sigma _{\min }\left( \textrm{s}\right) \right) \left( \rho ^{\prime }\left( \textrm{s}\right) \right) ^{\gamma +1}}{\left( c_{1}\rho \left( \textrm{s}\right) \sigma _{\min }^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }}. \end{aligned}$$
(17)

Integrating (17) from \(\textrm{s}_{3}\) to \(\textrm{s},\) we can conclude that

$$\begin{aligned} \int _{\textrm{s}_{3}}^{\textrm{s}}\left( L^{\beta }\rho \left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\frac{1}{\left( \gamma +1\right) ^{\gamma +1}}\frac{\hslash \left( \sigma _{\min }\left( \nu \right) \right) \left( \rho ^{\prime }\left( \nu \right) \right) ^{\gamma +1}}{\left( c_{1}\rho \left( \nu \right) \sigma _{\min }^{\prime }\left( \nu \right) \right) ^{\gamma }}\right) \textrm{d}\nu \le \omega \left( \textrm{s} _{3}\right) , \end{aligned}$$

this contradicts (9) as \(\textrm{s}\rightarrow \infty\). \(\square\)

By setting \(\rho \left( \textrm{s}\right) =1\) in (9), we obtain the following corollary:

Corollary 1

Let \(\beta \ge \gamma .\) Assume that (8) holds. If

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )\textrm{d}\nu =\infty , \end{aligned}$$
(18)

then \(\textrm{C}_{1}=\varnothing\).

Lemma 5

Suppose that the condition (8) is satisfied. If \(\beta \ge \gamma\) and there is a nondecreasing function \(\rho _{1}\in C^{1}([ \textrm{s}_{0},\infty ),\left( 0,\infty \right) )\) such that

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( L_{1}\rho _{1}\left( \nu \right) \left( \frac{\pi (\sigma _{\min }(\nu ),\textrm{s}_{0})}{\pi (\nu ,\textrm{s}_{0})}\right) ^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\frac{1}{\left( \gamma +1\right) ^{\gamma +1}}\frac{\hslash \left( \nu \right) \left( \rho _{1}^{\prime }\left( \nu \right) \right) ^{\gamma +1}}{\rho _{1}^{\gamma }\left( \nu \right) }\right) \textrm{d}\nu =\infty , \end{aligned}$$
(19)

holds for every \(c_{2}>0,\) then (1) is oscillatory, where \(L_{1}:=c_{2}L^{\beta }\).

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{1}\). Let us define

$$\begin{aligned} \omega _{1}\left( \textrm{s}\right) :=\rho _{1}\left( \textrm{s}\right) \frac{\hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right) ^{\gamma }}{{\mathcal {U}}^{\gamma }\left( \textrm{s}\right) } >0. \end{aligned}$$
(20)

Then

$$\begin{aligned} \omega _{1}^{\prime }\left( \textrm{s}\right) =\rho _{1}^{\prime }\left( \textrm{s} \right) \frac{\hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{ s}\right) \right) ^{\gamma }}{{\mathcal {U}}^{\gamma }\left( \textrm{s}\right) } +\rho _{1}\left( \textrm{s}\right) \frac{\left( \hslash (\textrm{s})\left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right) ^{\prime }}{{\mathcal {U}}^{\gamma }\left( \textrm{s}\right) }-\gamma \rho _{1}\left( \textrm{s}\right) \frac{\hslash (\textrm{s})\left( {\mathcal {U}} ^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }{\mathcal {U}}^{\prime }\left( \textrm{s}\right) }{{\mathcal {U}}^{\gamma +1}\left( \textrm{s}\right) } . \end{aligned}$$
(21)

We see from \(\left( \textrm{d}_{1}\right)\), 20, and (21) that

$$\begin{aligned} \omega _{1}^{\prime }\left( \textrm{s}\right) \le -L^{\beta }\rho _{1}\left( \textrm{s}\right) \frac{{\mathcal {U}}^{\beta }\left( \sigma _{\min }(\textrm{s})\right) }{{\mathcal {U}}^{\gamma }\left( \textrm{s}\right) } \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\rho _{1}^{\prime }\left( \textrm{s}\right) }{\rho _{1}\left( \textrm{s}\right) }\omega _{1}\left( \textrm{s}\right) -\frac{\gamma }{\left( \rho _{1}\left( \textrm{s}\right) \hslash (\textrm{s})\right) ^{1/\gamma }}\omega _{1}^{1/\gamma +1}\left( \textrm{s}\right) . \end{aligned}$$
(22)

From Lemma 3, we see that \({\mathcal {U}}(\textrm{s})/\pi (\textrm{s}, \textrm{s}_{0})\) is decreasing, i.e.,

$$\begin{aligned} {\mathcal {U}}(\sigma _{\min }(\textrm{s}))\ge \frac{\pi (\sigma _{\min }( \textrm{s}),\textrm{s}_{0})}{\pi (\textrm{s},\textrm{s}_{0})}{\mathcal {U}}( \textrm{s}). \end{aligned}$$
(23)

Using (23) in (22), we get

$$\begin{aligned} \omega _{1}^{\prime }\left( \textrm{s}\right) \le -L^{\beta }\rho _{1}\left( \textrm{s}\right) {\mathcal {U}}^{\beta -\gamma }\left( \textrm{s} \right) \left( \frac{\pi (\sigma _{\min }(\textrm{s}),\textrm{s}_{0})}{\pi ( \textrm{s},\textrm{s}_{0})}\right) ^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}( \textrm{s})+\frac{\rho _{1}^{\prime }\left( \textrm{s}\right) }{\rho _{1}\left( \textrm{s}\right) }\omega _{1}\left( \textrm{s}\right) -\frac{ \gamma }{\left( \rho _{1}\left( \textrm{s}\right) \hslash (\textrm{s} )\right) ^{1/\gamma }}\omega _{1}^{1/\gamma +1}\left( \textrm{s}\right) . \end{aligned}$$
(24)

Given that \({\mathcal {U}}^{\prime }\left( \textrm{s}\right) >0\) and \(\beta \ge \gamma\), there exist constants \(\textrm{s}_{1}\ge \textrm{s}_{0}\) and \(c_{2}>0\) such that

$$\begin{aligned} {\mathcal {U}}^{\beta -\gamma }\left( \textrm{s}\right) >c_{2}. \end{aligned}$$
(25)

This leads to the following inequality

$$\begin{aligned} \omega _{1}^{\prime }\left( \textrm{s}\right) \le -c_{2}L^{\beta }\rho _{1}\left( \textrm{s}\right) \left( \frac{\pi (\sigma _{\min }(\textrm{s}), \textrm{s}_{0})}{\pi (\textrm{s},\textrm{s}_{0})}\right) ^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\rho _{1}^{\prime }\left( \textrm{s}\right) }{\rho _{1}\left( \textrm{s}\right) }\omega _{1}\left( \textrm{s}\right) -\frac{\gamma }{\left( \rho _{1}\left( \textrm{s}\right) \hslash (\textrm{s})\right) ^{1/\gamma }}\omega _{1}^{\left( 1+\gamma \right) /\gamma }\left( \textrm{s}\right) . \end{aligned}$$
(26)

By applying Lemma 1, where we defin \(B=\rho _{1}^{\prime }\left( \textrm{s}\right) /\rho _{1}\left( \textrm{s}\right) ,\) \(\hslash =\gamma /\left( \rho _{1}\left( \textrm{s}\right) \hslash (\textrm{s})\right) ^{1/\gamma },\) and \({\mathcal {U}}\left( \textrm{s}\right) =\omega _{1}\left( \textrm{s}\right) ,\) we get

$$\begin{aligned} \omega _{1}^{\prime }\left( \textrm{s}\right) \le -L_{1}\rho _{1}\left( \textrm{s}\right) \left( \frac{\pi (\sigma _{\min }(\textrm{s}),\textrm{s} _{0})}{\pi (\textrm{s},\textrm{s}_{0})}\right) ^{\beta }\sum _{j=1}^{m} {\mathfrak {f}}_{j}(\textrm{s})+\frac{1}{\left( \gamma +1\right) ^{\gamma +1}} \frac{\hslash \left( \textrm{s}\right) \left( \rho _{1}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma +1}}{\rho _{1}^{\gamma }\left( \textrm{s} \right) }. \end{aligned}$$

IUpon integrating inequality (26) from \(\textrm{s}_{2}\) to \(\textrm{s} ,\) we get

$$\begin{aligned} \int _{\textrm{s}_{2}}^{\textrm{s}}\left( L_{1}\rho _{1}\left( \nu \right) \left( \frac{\pi (\sigma _{\min }(\nu ),\textrm{s}_{0})}{\pi (\nu ,\textrm{s} _{0})}\right) ^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\frac{1}{\left( \gamma +1\right) ^{\gamma +1}}\frac{\hslash \left( \nu \right) \left( \rho _{1}^{\prime }\left( \nu \right) \right) ^{\gamma +1}}{\rho _{1}^{\gamma }\left( \nu \right) }\right) \textrm{d}\nu \le \omega _{1}\left( \textrm{s} _{2}\right) , \end{aligned}$$

This inequality leads to a contradiction with condition (19) as \(\textrm{s}\rightarrow \infty\).Therefore, the proof is concluded. \(\square\)

Corollary 2

Let \(\beta \ge \gamma .\) Assume that (8)hold. If

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( \frac{\pi (\sigma _{\min }(\nu ),\textrm{s}_{0})}{\pi (\nu ,\textrm{s}_{0})}\right) ^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu ) \textrm{d}\nu =\infty , \end{aligned}$$
(27)

then \(\textrm{C}_{1}=\varnothing\).

Lemma 6

Let \(0<\beta <\gamma\). Assume that (8) holds. If there is a nondecreasing function \(\rho \in C^{1}([\textrm{s}_{0},\infty ),\left( 0,\infty \right) )\) such that

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( L^{\beta }\rho \left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\frac{1}{\left( \beta +1\right) ^{\beta +1}}\frac{\hslash \left( \nu \right) \left( \rho ^{\prime }\left( \nu \right) \right) ^{\beta +1}}{ \left( c_{3}\rho \left( \nu \right) \sigma _{\min }^{\prime }\left( \nu \right) \right) ^{\beta }}\right) \textrm{d}\nu =\infty , \end{aligned}$$
(28)

holds for every \(c_{3}>0,\) and \(\epsilon \in \left( 0,1\right) ,\) then \(\textrm{C}_{1}=\varnothing .\)

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{1}\). As in the proof of 9 in Lemma 4. The function \(\omega \left( \textrm{s} \right)\) is defined in (10), then (11) holds. By (12), we conclude that

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) \le -L^{\beta }\rho \left( \textrm{s}\right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\rho ^{\prime }\left( \textrm{s}\right) }{\rho \left( \textrm{s}\right) }\omega \left( \textrm{s}\right) -\beta \sigma _{\min }^{\prime }\left( \textrm{s} \right) \frac{{\mathcal {U}}^{\prime }\left( \sigma _{\min }\left( \textrm{s} \right) \right) }{{\mathcal {U}}\left( \sigma _{\min }\left( \textrm{s}\right) \right) }\omega \left( \textrm{s}\right) . \end{aligned}$$

By using (13), we see that

$$\begin{aligned} \omega ^{\prime }\left( \textrm{s}\right) \le -L^{\beta }\rho \left( \textrm{s}\right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\rho ^{\prime }\left( \textrm{s}\right) }{\rho \left( \textrm{s}\right) }\omega \left( \textrm{s}\right) -\frac{\beta \sigma _{\min }^{\prime }\left( \textrm{s}\right) }{\left( \rho \left( \textrm{s}\right) \hslash \left( \textrm{s}\right) \right) ^{1/\beta }}\left[ {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right] ^{\left( \beta -\gamma \right) /\beta }\omega ^{1/\beta +1}\left( \textrm{s}\right) . \end{aligned}$$
(29)

Since \(\hslash ^{\prime }\left( \textrm{s}\right) \ge 0\), then \({\mathcal {U}} ^{\prime \prime }\left( \textrm{s}\right) \le 0\), this easily shows that \({\mathcal {U}}^{\prime }\left( \textrm{s}\right)\) is nonincreasing. Then there are \(c_{3}>0\) and \(\textrm{s}_{3}\ge \textrm{s}_{2}\) such that

$$\begin{aligned} \left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\left( \beta -\gamma \right) /\beta }\ge c_{3},\ \textrm{s}\ge \textrm{s}_{3}. \end{aligned}$$
(30)

From (29) and (30) it follows that

$$\begin{aligned} \omega \left( \textrm{s}\right) \le -L^{\beta }\rho \left( \textrm{s} \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\rho ^{\prime }\left( \textrm{s}\right) }{\rho \left( \textrm{s}\right) }\omega \left( \textrm{s}\right) -\frac{c_{3}\beta \sigma _{\min }^{\prime }\left( \textrm{s }\right) }{\left( \rho \left( \textrm{s}\right) \hslash \left( \textrm{s} \right) \right) ^{1/\beta }}\omega ^{\left( \beta +1\right) /\beta }\left( \textrm{s}\right) . \end{aligned}$$
(31)

Using Lemma 1, where we define \(B=\rho ^{\prime }\left( \textrm{s} \right) /\rho \left( \textrm{s}\right) ,\) \(\hslash =c_{3}\beta \sigma _{\min }^{\prime }\left( \textrm{s}\right) /\left( \rho \left( \textrm{s}\right) \hslash \left( \textrm{s}\right) \right) ^{1/\beta },\) and \({\mathcal {U}} \left( \textrm{s}\right) =\omega \left( \textrm{s}\right) ,\) the inequality ( 31) leads to

$$\begin{aligned} \omega \left( \textrm{s}\right) \le -L^{\beta }\rho \left( \textrm{s} \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{1}{\left( \beta +1\right) ^{\beta +1}}\frac{\hslash \left( \textrm{s}\right) \left( \rho ^{\prime }\left( \textrm{s}\right) \right) ^{\beta +1}}{\left( c_{3}\rho \left( \textrm{s}\right) \sigma _{\min }^{\prime }\left( \textrm{s}\right) \right) ^{\beta }}. \end{aligned}$$
(32)

Integration of (32) over \(\left[ \textrm{s}_{4},\textrm{s}\right]\), we get

$$\begin{aligned} \int _{\textrm{s}_{4}}^{\textrm{s}}\left( L^{\beta }\rho \left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\frac{1}{\left( \beta +1\right) ^{\beta +1}}\frac{\hslash \left( \nu \right) \left( \rho ^{\prime }\left( \nu \right) \right) ^{\beta +1}}{\left( c_{3}\rho \left( \nu \right) \sigma _{\min }^{\prime }\left( \nu \right) \right) ^{\beta }}\right) \textrm{d}\nu \le \omega \left( \textrm{s}_{4}\right) , \end{aligned}$$

which contradicts (28) as \(\textrm{s}\rightarrow \infty\). As a result, the proof is complete. \(\square\)

Setting \(\rho \left( \textrm{s}\right) =1\) in (28), we arrive at the same condition (18) as stated in Corollary 2.

Category \(\textrm{C}_{2}\)

Lemma 7

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{2}\). Assume that

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \left( \frac{\pi (\tau \left( \textrm{s}\right) )}{\pi (\textrm{s})}\right) ^{\alpha }=0,\text { and }\lim _{\textrm{s}\rightarrow \infty }\textrm{g} \left( \textrm{s}\right) \frac{\pi ^{\delta }(\varsigma \left( \textrm{s} \right) )}{\pi \left( \textrm{s}\right) }=0. \end{aligned}$$
(33)

\(\left( \textrm{a}_{2}\right)\) \({\mathcal {U}}(\textrm{s})\geqslant -\hslash ^{1/\gamma }(\textrm{s}){\mathcal {U}}^{\prime }(\textrm{s})\pi (\textrm{s});\)

\(\left( \textrm{b}_{2}\right)\) \({\mathcal {U}}(\textrm{s})/\pi (\textrm{s})\) is increasing; 

\(\left( \textrm{c}_{2}\right)\) \(\varkappa \left( \textrm{s}\right) > {\widetilde{L}}{\mathcal {U}}\left( \textrm{s}\right) ,\) where \({\widetilde{L}}\in \left( 0,1\right) ;\)

\(\left( \textrm{d}_{2}\right)\) \(\left( \hslash (\textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right] ^{\gamma }\right) ^{\prime }\le -{\widetilde{L}}^{\beta }{\mathcal {U}}^{\beta }(\sigma _{\max }(\textrm{s} ))\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s}).\)

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{2}\). Then, there exists a value \(\textrm{s}_{1}\ge \textrm{s}_{0}\) such that for \(\textrm{s} \ge \textrm{s}_{1}\), we have \(\varkappa \left( \textrm{s}\right) >0,\) \(\varkappa \left( \pi \left( \textrm{s}\right) \right) >0,\) \(\varkappa \left( \varsigma \left( \textrm{s}\right) \right) >0\) and \(\varkappa \left( \sigma _{j}\left( \textrm{s}\right) \right) >0,\) for \(j=1,2,...m\)

 \(\left( \textrm{a}_{2}\right)\) Since \(\left( \hslash \left( \nu \right) \left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right) ^{\prime }<0\), we get

$$\begin{aligned} \hslash \left( \nu \right) \left( {\mathcal {U}}^{\prime }\left( \nu \right) \right) ^{\gamma }\le \hslash \left( \textrm{s}\right) \left( {\mathcal {U}} ^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\text { for }\nu \ge \textrm{s}\ge \textrm{s}_{1}, \end{aligned}$$

or equivalently

$$\begin{aligned} {\mathcal {U}}^{\prime }\left( \nu \right) \le \frac{1}{\hslash ^{1/\gamma }\left( \nu \right) }\hslash ^{1/\gamma }\left( \textrm{s}\right) {\mathcal {U}} ^{\prime }\left( \textrm{s}\right) . \end{aligned}$$
(34)

By integrating (34) from \(\textrm{s}\) to \(\infty\), we can conclude that

$$\begin{aligned} -{\mathcal {U}}\left( \textrm{s}\right) \le \hslash ^{1/\gamma }\left( \textrm{ s}\right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \int _{\textrm{s} }^{\infty }\frac{1}{\hslash ^{1/\gamma }\left( \nu \right) }\textrm{d}\nu =\hslash ^{1/\gamma }\left( \textrm{s}\right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \pi \left( \textrm{s}\right) . \end{aligned}$$

That is,

$$\begin{aligned} {\mathcal {U}}\left( \textrm{s}\right) \ge -\hslash ^{1/\gamma }\left( \textrm{ s}\right) {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \pi \left( \textrm{s} \right) . \end{aligned}$$

\(\left( \textrm{b}_{2}\right)\) Based on the inequality above, we determine that

$$\begin{aligned} \left( \frac{{\mathcal {U}}(\textrm{s})}{\pi (\textrm{s})}\right) ^{\prime }= \frac{\hslash ^{1/\gamma }(\textrm{s}){\mathcal {U}}^{\prime }(\textrm{s})\pi ( \textrm{s})+{\mathcal {U}}(\textrm{s})}{\hslash ^{1/\gamma }(\textrm{s})\pi ^{2}(\textrm{s})}\ge 0. \end{aligned}$$

\(\left( \textrm{c}_{2}\right)\) Using (3), it follows that \({\mathcal {U}}\left( \textrm{s}\right) \ge \varkappa \left( \textrm{s}\right)\). Consequently

$$\begin{aligned} \varkappa \left( \textrm{s}\right) \ge {\mathcal {U}}\left( \textrm{s}\right) - \textrm{p}\left( \textrm{s}\right) {\mathcal {U}}^{\alpha }\left( \tau \left( \textrm{s}\right) \right) -\textrm{g}\left( \textrm{s}\right) {\mathcal {U}} ^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) . \end{aligned}$$

Now, Because \(\alpha >1,\) and \(\delta <1,\) and given that \({\mathcal {U}}( \textrm{s})/\pi (\textrm{s})\) is increasing and positive, while \({\mathcal {U}} ^{\prime }<0\), we can conclude that there exist two constants \(l_{3}\) and \(l_{4}\) such that

$$\begin{aligned} \varkappa \left( \textrm{s}\right)\ge & {\mathcal {U}}\left( \textrm{s} \right) -\textrm{p}\left( \textrm{s}\right) {\mathcal {U}}^{\alpha }\left( \textrm{s}\right) -\textrm{g}\left( \textrm{s}\right) {\mathcal {U}}^{\delta }\left( \textrm{s}\right) \\= & {\mathcal {U}}\left( \textrm{s}\right) -\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha }(\tau \left( \textrm{s}\right) )\frac{{\mathcal {U}}^{\alpha }\left( \tau \left( \textrm{s}\right) \right) }{\pi ^{\alpha }(\tau \left( \textrm{s}\right) )}-\textrm{g}\left( \textrm{s}\right) \pi ^{\delta }(\varsigma \left( \textrm{s}\right) )\frac{{\mathcal {U}}^{\delta }\left( \varsigma \left( \textrm{s}\right) \right) }{\pi ^{\delta }(\varsigma \left( \textrm{s}\right) )} \\\ge & {\mathcal {U}}\left( \textrm{s}\right) -\textrm{p}\left( \textrm{s} \right) \pi ^{\alpha }(\tau \left( \textrm{s}\right) )\frac{{\mathcal {U}} ^{\alpha }\left( \textrm{s}\right) }{\pi ^{\alpha }(\textrm{s})}-\textrm{g} \left( \textrm{s}\right) \pi ^{\delta }(\varsigma \left( \textrm{s}\right) ) \frac{{\mathcal {U}}^{\delta }\left( \textrm{s}\right) }{\pi ^{\delta }(\textrm{ s})} \\= & \left[ {\mathcal {U}}\left( \textrm{s}\right) -\textrm{p}\left( \textrm{s} \right) \left( \frac{\pi (\tau \left( \textrm{s}\right) )}{\pi (\textrm{s})} \right) ^{\alpha }{\mathcal {U}}^{\alpha -1}\left( \textrm{s}\right) -\textrm{g} \left( \textrm{s}\right) \frac{\pi ^{\delta }(\varsigma \left( \textrm{s} \right) )}{\pi \left( \textrm{s}\right) }\left( \frac{{\mathcal {U}}\left( \textrm{s}\right) }{\pi (\textrm{s})}\right) ^{\delta -1}\right] {\mathcal {U}} \left( \textrm{s}\right) \\\ge & \left[ 1-l_{3}\textrm{p}\left( \textrm{s}\right) \left( \frac{\pi (\tau \left( \textrm{s}\right) )}{\pi (\textrm{s})}\right) ^{\alpha }-l_{4} \textrm{g}\left( \textrm{s}\right) \frac{\pi ^{\delta }(\varsigma \left( \textrm{s}\right) )}{\pi \left( \textrm{s}\right) }\right] {\mathcal {U}}\left( \textrm{s}\right) \end{aligned}$$

By (33), we can choose \({\widetilde{L}}\in \left( 0,1\right)\) such that

$$\begin{aligned} \varkappa \left( \textrm{s}\right) \ge {\widetilde{L}}{\mathcal {U}}\left( \textrm{s}\right) . \end{aligned}$$

\(\left( \textrm{d}_{2}\right)\) Using (1) and (\(\textrm{c}_{2}\)), we see that

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right] ^{\gamma }\right) ^{\prime }=-\sum _{j=1}^{m}{\mathfrak {f}}_{j}( \textrm{s})\varkappa ^{\beta }(\sigma _{j}(\textrm{s}))\le -{\widetilde{L}} ^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s}){\mathcal {U}}^{\beta }(\sigma _{j}(\textrm{s})). \end{aligned}$$

Since \({\mathcal {U}}^{\prime }<0,\,\)and \(\ \sigma _{j}(\textrm{s})\le \sigma _{\max }(\textrm{s}),\) for \(j=1,2,\ldots ,m,\) then

$$\begin{aligned} \left( \hslash (\textrm{s})\left[ {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right] ^{\gamma }\right) ^{\prime }\le -{\widetilde{L}}^{\beta } {\mathcal {U}}^{\beta }(\sigma _{\max }(\textrm{s}))\sum _{j=1}^{m}{\mathfrak {f}} _{j}(\textrm{s}). \end{aligned}$$

Thus, the proof is complete.As a result, the proof is complete. \(\square\)

Lemma 8

Let \(\beta \ge \gamma .\) Assume that (33) holds. If

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( {\widetilde{L}}^{\beta }\pi ^{\beta }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\left( \frac{\beta }{\beta +1}\right) ^{\beta +1}\frac{1}{c_{4}^{\beta }\hslash ^{1/\gamma }\left( \nu \right) \pi \left( \nu \right) }\right) \textrm{d}\nu =\infty , \end{aligned}$$
(35)

is satisfied for every \(c_{4}>0\), then \(\textrm{C}_{2}=\varnothing .\)

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{2}\). Define

$$\begin{aligned} \Psi \left( \textrm{s}\right) :=\frac{\hslash \left( \textrm{s}\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }}{ {\mathcal {U}}^{\beta }\left( \textrm{s}\right) }>0,\ \textrm{s}\ge \textrm{s}_{1}. \end{aligned}$$
(36)

Using \(\left( \textrm{a}_{2}\right)\), we see that

$$\begin{aligned} {\mathcal {U}}\left( \textrm{s}\right) \ge \hslash ^{1/\gamma }\left( \textrm{s }\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) \, \pi \left( \textrm{s}\right) . \end{aligned}$$
(37)

This implies

$$\begin{aligned} \left[ \hslash \left( \textrm{s}\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right] ^{1-\beta /\gamma }\ge \, \pi ^{\beta }\left( \textrm{s}\right) \Psi \left( \textrm{s}\right) >0. \end{aligned}$$

Because \(\left[ \hslash \left( \textrm{s}\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right] ^{1-\beta /\gamma }\) is a non-increasing function for \(\beta \ge \gamma\), there exists a constant \(L_{2}>0\) and \(\textrm{s}_{2}\ge \textrm{s}_{1}\) such that

$$\begin{aligned} \left[ \hslash \left( \textrm{s}\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{\gamma }\right] ^{1-\beta /\gamma }\le L_{2}. \end{aligned}$$

Thus, we conclude that

$$\begin{aligned} 0<\pi ^{\beta }\left( \textrm{s}\right) \Psi \left( \textrm{s}\right) <L_{2},\ \textrm{s}\ge \textrm{s}_{2}. \end{aligned}$$
(38)

Taking the derivative of Eq. (36), we get

$$\begin{aligned} \Psi ^{\prime }\left( \textrm{s}\right) =\frac{\left( \hslash \left( - {\mathcal {U}}^{\prime }\right) ^{\gamma }\right) ^{\prime }}{{\mathcal {U}} ^{\beta }}+\beta \frac{\hslash \left( -{\mathcal {U}}^{\prime }\right) ^{\gamma +1}}{{\mathcal {U}}^{\beta +1}}. \end{aligned}$$
(39)

Using \(\left( \textrm{d}_{2}\right)\) and (36), we conclude that

$$\begin{aligned} \Psi ^{\prime }\left( \textrm{s}\right)\ge & {\widetilde{L}}^{\beta }\frac{ {\mathcal {U}}^{\beta }(\sigma _{\max }(\textrm{s}))}{{\mathcal {U}}^{\beta }\left( \textrm{s}\right) }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{ \beta }{\hslash ^{1/\gamma }\left( \textrm{s}\right) }\left[ \hslash ^{1/\gamma }\left( \textrm{s}\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) \right] ^{1-\gamma /\beta }\Psi ^{1/\beta +1}\left( \textrm{s}\right) \\\ge & {\widetilde{L}}^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+ \frac{\beta }{\hslash ^{1/\gamma }\left( \textrm{s}\right) }\left[ \hslash ^{1/\gamma }\left( \textrm{s}\right) \left( -{\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) \right] ^{1-\gamma /\beta }\Psi ^{1/\beta +1}\left( \textrm{s}\right) . \end{aligned}$$

Since \(\left[ \hslash ^{1/\gamma }\left( -{\mathcal {U}}^{\prime }\right) \right] ^{1-\gamma /\beta }\) is monotonically increasing for \(\beta \ge \gamma ,\) we can find constants \(c_{4}>0,\) and \(\textrm{s}_{3}\ge \textrm{s} _{2}\) (with \(\beta =\gamma ,\) \(c_{4}=1\)), such that

$$\begin{aligned} \left[ \hslash ^{1/\gamma }\left( \textrm{s}\right) \left( -{\mathcal {U}} ^{\prime }\left( \textrm{s}\right) \right) \right] ^{1-\gamma /\beta }>c_{4}, \textrm{s}\ge \textrm{s}_{3}. \end{aligned}$$

Thus, we have

$$\begin{aligned} \Psi ^{\prime }\left( \textrm{s}\right) \ge {\widetilde{L}}^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\beta c_{4}}{\hslash ^{1/\gamma }\left( \textrm{s}\right) }\Psi ^{1/\beta +1}\left( \textrm{s} \right) . \end{aligned}$$
(40)

Now, multiplying Eq. (39) by \(\pi ^{\beta }\left( \textrm{s} \right)\) and integrating the resulting inequality from \(\textrm{s}_{4}\) to \(\textrm{s}\), we arrive at

$$\begin{aligned} \int _{\textrm{s}_{4}}^{\textrm{s}}{\widetilde{L}}^{\beta }\pi ^{\beta }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )\textrm{d}\nu\le & \int _{ \textrm{s}_{4}}^{\textrm{s}}\beta \pi ^{\beta -1}\left( \nu \right) \hslash ^{-1/\gamma }\left( \nu \right) \left[ \Psi \left( \nu \right) -c_{4}\pi \left( \nu \right) \Psi ^{1/\beta +1}\left( \nu \right) \right] \\ & +\pi ^{\beta }\left( \textrm{s}\right) \Psi \left( \textrm{s}\right) -\pi ^{\beta }\left( \textrm{s}_{4}\right) \Psi \left( \textrm{s}_{4}\right) . \end{aligned}$$

By utilizing (38) and Lemma 1 where \(B=1,\) \(A=c_{3}\pi \left( \textrm{s}\right) ,\) and \({\mathcal {U}}\left( \textrm{s}\right) =\Psi \left( \textrm{s}\right) ,\) we obtain

$$\begin{aligned} \int _{\textrm{s}_{4}}^{\textrm{s}}\left( {\widetilde{L}}^{\beta }\pi ^{\beta }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\left( \frac{\beta }{\beta +1}\right) ^{\beta +1}\frac{1}{c_{4}^{\beta }\hslash ^{1/\gamma }\left( \nu \right) \pi \left( \nu \right) }\right) \textrm{d}\nu \le L_{2}-\pi ^{\beta }\left( \textrm{s}_{4}\right) \Psi \left( \textrm{s} _{4}\right) . \end{aligned}$$

This leads to a contradiction with condition (35) as \(\textrm{s} \rightarrow \infty\), thus concluding the proof. \(\square\)

Lemma 9

Let \(0<\beta <\gamma .\) Assume that (33) holds. If

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( {\widetilde{L}}^{\beta }\pi ^{\gamma }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\left( \frac{\gamma }{\gamma +1}\right) ^{\gamma +1}\left( \frac{\gamma }{\beta }\right) ^{\gamma }\frac{1}{ c_{5}\hslash ^{1/\gamma }\left( \nu \right) \pi \left( \nu \right) }\right) \textrm{d}\nu =\infty , \end{aligned}$$
(41)

is satisfied for every \(c_{5}>0\), then \(\textrm{C}_{2}=\varnothing .\)

Proof

Let \(\varkappa \left( \textrm{s}\right) \in \textrm{C}_{2}\). Using the definition of \(\Psi \left( \textrm{s}\right)\) in conjunction with Eq. ( 37), we obtain the following result:

$$\begin{aligned} {\mathcal {U}}^{\gamma -\beta }\left( \textrm{s}\right) \ge \pi ^{\gamma }\left( \textrm{s}\right) \Psi \left( \textrm{s}\right) , \textrm{s} \ge \textrm{s}_{1}\ge \textrm{s}_{0}. \end{aligned}$$

Given that \({\mathcal {U}}^{\gamma -\beta }\left( \textrm{s}\right)\) is a non-increasing function for \(0<\beta <\gamma ,\) there exists a constant \(L_{3}>0\) and \(\textrm{s}_{2}\ge \textrm{s}_{1}\) such that

$$\begin{aligned} {\mathcal {U}}^{\gamma -\beta }\left( \textrm{s}\right) \le L_{3}, \end{aligned}$$

thus

$$\begin{aligned} 0<\pi ^{\gamma }\left( \textrm{s}\right) \Psi \left( \textrm{s}\right) \le L_{3}, \textrm{s}\ge \textrm{s}_{2}. \end{aligned}$$
(42)

Additionally, using \(\left( \textrm{d}_{2}\right)\) and (36) in (39), we obtain

$$\begin{aligned} \Psi ^{\prime }\left( \textrm{s}\right) \ge {\widetilde{L}}^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\beta }{\hslash ^{1/\gamma }\left( \textrm{s}\right) }{\mathcal {U}}^{\beta /\gamma -1}\left( \textrm{s} \right) \Psi ^{1/\gamma +1}\left( \textrm{s}\right) \end{aligned}$$
(43)

As \({\mathcal {U}}^{\beta /\gamma -1}\left( \textrm{s}\right)\) increasing for \(0<\beta <\gamma ,\) we can deduce that there are constants\(c_{4}>0,\) and \(\textrm{s}_{3}\ge \textrm{s}_{2}\), such that

$$\begin{aligned} {\mathcal {U}}^{\beta /\gamma -1}\left( \textrm{s}\right) >c_{5}, \textrm{s}\ge \textrm{s}_{3}. \end{aligned}$$

Thus, we get

$$\begin{aligned} \Psi ^{\prime }\left( \textrm{s}\right) \ge {\widetilde{L}}^{\beta }\sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})+\frac{\beta c_{5}}{\hslash ^{1/\gamma }\left( \textrm{s}\right) }\Psi ^{1/\gamma +1}\left( \textrm{s} \right) ,\ \textrm{s}\ge \textrm{s}_{3}. \end{aligned}$$
(44)

Multiplying Eq. (39) by \(\pi ^{\gamma }\left( \textrm{s}\right)\) and integrating the resulting inequality from \(\textrm{s}_{4}\) to \(\textrm{s}\), we arrive at

$$\begin{aligned} \int _{\textrm{s}_{4}}^{\textrm{s}}{\widetilde{L}}^{\beta }\pi ^{\gamma }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )\textrm{d}\nu\le & \int _{ \textrm{s}_{4}}^{\textrm{s}}\pi ^{\gamma -1}\left( \nu \right) \hslash ^{-1/\gamma }\left( \nu \right) \left[ \gamma \Psi \left( \nu \right) -c_{5}\beta \pi \left( \nu \right) \Psi ^{1/\gamma +1}\left( \nu \right) \right] \\ & +\pi ^{\gamma }\left( \textrm{s}\right) \Psi \left( \textrm{s}\right) -\pi ^{\gamma }\left( \textrm{s}_{4}\right) \Psi \left( \textrm{s}_{4}\right) . \end{aligned}$$

By employing (38) and Lemma 1 where \(B=\gamma ,\) \(\hslash =c_{5}\beta \pi \left( \nu \right) ,\) and \({\mathcal {U}}\left( \nu \right) =\Psi \left( \nu \right) ,\) we have

$$\begin{aligned} \int _{\textrm{s}_{4}}^{\textrm{s}}\left( {\widetilde{L}}^{\beta }\pi ^{\gamma }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\left( \frac{\gamma }{\beta }\right) ^{\gamma }\left( \frac{\gamma }{\gamma +1}\right) ^{\gamma +1}\frac{1}{c_{5}\hslash ^{1/\gamma }\left( \nu \right) \pi \left( \nu \right) }\right) \textrm{d}\nu \le L_{3}-\pi ^{\beta }\left( \textrm{s} _{4}\right) \Psi \left( \textrm{s}_{4}\right) . \end{aligned}$$

This contradicts condition (41) as \(\textrm{s}\rightarrow \infty\). As a result, the proof is complete. \(\square\)

Setting \(\rho \left( \textrm{s}\right) =1\) in (28), we obtain the following corollary:

Corollary 3

Let \(0<\beta <\gamma .\) Assume that (33) holds. If (18) and (41) are satisfied, then \(\textrm{C}_{2}=\varnothing .\)

Oscillation theorems and applications

This section gives a comprehensive set of theorems for establishing oscillation criteria, which are immediately obtained by merging the key results.

Theorem 4

If (8) and (33) hold, and (9) together with (35) are satisfied, then Eq. (1) is oscillatory.

Proof

Let us assume, for the sake of contradiction, that \(\varkappa \left( \textrm{ s}\right)\) is an eventually positive solution of Eq. (1). According to Lemma 2, we can deduce that two possible scenarios exist regarding the behavior of \({\mathcal {U}}\left( \textrm{s}\right)\) and its derivatives. By applying Lemma 4 and Lemma 8, we observe that the conditions (9) and (35) imply the non-existence of solutions to Eq. (1) that satisfy the requirements of cases \(( \textrm{C}_{1})\) and \((\textrm{C}_{2})\), respectively. Consequently, we conclude that our initial assumption must be incorrect, thereby proving that the solutions to Eq. (1) are indeed oscillatory. Thus, the proof is complete. \(\square\)

Theorem 5

Let (8) and (33) hold. If (18) together with (35) are satisfied, then Eq. (1) is oscillatory.

Proof

Let us assume, for the sake of contradiction, that \(\varkappa \left( \textrm{ s}\right)\) is an eventually positive solution of Eq. (1). According to Lemma 2, we can deduce that two possible scenarios exist regarding the behavior of \({\mathcal {U}}\left( \textrm{s}\right)\) and its derivatives. By applying Corollary 1 and Lemma 8, we observe that the conditions (18) and (35) imply the non-existence of solutions to Eq. (1) that satisfy the requirements of cases \(( \textrm{C}_{1})\) and \((\textrm{C}_{2})\), respectively. Consequently, we conclude that our initial assumption must be incorrect, thereby proving that the solutions to Eq. (1) are indeed oscillatory. Thus, the proof is complete. \(\square\)

Remark 6

The proofs of the remaining theorems follow the same methodology and approach as the proofs of the two theorems above. Therefore, the proofs have been omitted.

Theorem 7

Let (8) and (33) hold. If (19) together with (35) are satisfied, then Eq. (1) is oscillatory.

Theorem 8

Let (8) and (33) hold. If (27) together with (35) are satisfied, then Eq. (1) is oscillatory.

Theorem 9

Let (8) and (33) hold. If (28) together with (41) are satisfied, then Eq. (1) is oscillatory.

Theorem 10

Let (8) and (33) hold. If (18) together with (41) are satisfied, then Eq. (1) is oscillatory.

Remark 11

It is important to note that Theorem 4 represents a significant generalization of the results in the literature. Specifically, when \(\left[ \alpha =\delta =1,\, \textrm{p}\left( \textrm{s}\right) =0,\text { and }j=1 \right]\) Theorem 4 becomes the result in29, Theorem 2.1. Furthermore, by setting \(\left[ \alpha =1\text { and }\textrm{p}\left( \textrm{s}\right) =0\text { and }j=1\right]\) Theorem 4 simplifies to the result in7, Theorem 1. In the special case where \(\left[ \textrm{ p}\left( \textrm{s}\right) =0\text { and }j=1\right] ,\) Theorem 4 reduces to the result in31, Theorem 2.4.

Remark 12

Theorem 7 presents new results that differ from those found in7, Theorem 1,29, Theorem 2.1 and31, Theorem 2.5, which only addressed specific cases of Eq. (1).

Remark 13

Theorem 9 improves upon the results obtained in 5, and 8 which addressed only specific cases of Eq. (1), particularly when \(\gamma \ge \beta .\) In contrast, Theorem 9 extends these results to cover the case where \(0<\beta <\gamma\).

Examples

To ensure the importance of our findings, we present below some illustrative examples that help highlight the different aspects of the findings.

Example 14

Consider the equation:

$$\begin{aligned} \left( \textrm{s}^{10}\left[ \left( \varkappa \left( \textrm{s}\right) + \frac{1}{\textrm{s}-2}\varkappa ^{5}\left( \tau _{0}\textrm{s}\right) +\frac{ 1}{\textrm{s}-3}\varkappa ^{1/5}\left( \varsigma _{0}\textrm{s}\right) \right) ^{\prime }\right] ^{5}\right) ^{\prime }+\sum _{j=1}^{10}\textrm{s} ^{4}{\mathfrak {f}}_{0}\varkappa ^{5}\left( \sigma _{j}\textrm{s}\right) =0, \textrm{s}\ge 1, \end{aligned}$$
(45)

where \(\tau _{0},\varsigma _{0},\sigma _{j}\in \left( 0,1\right) ,\) \(j=1,2,\ldots ,10.\) Comparing Eq. (45) with Eq. (1), we determine the following parameters:

$$\begin{aligned} m= & 10, \gamma =\beta =\alpha =5, \delta =\frac{1}{5}, \hslash \left( \textrm{s}\right) =\textrm{s}^{10}, \sigma _{\min }\left( \textrm{s}\right) =\sigma _{\min }\textrm{s}, \sigma _{\max }\left( \textrm{s}\right) =\sigma _{\max }\textrm{s,}\ \tau \left( \textrm{s}\right) =\tau _{0}\textrm{s}, \\ \varsigma \left( \textrm{s}\right)= & \varsigma _{0}\textrm{s},\ \textrm{p}\left( \textrm{s}\right) =\frac{1}{\textrm{s}-2},\ \textrm{g} \left( \textrm{s}\right) =\frac{1}{\textrm{s}-3},\text { and }\sum _{j=1}^{m} {\mathfrak {f}}_{j}(\textrm{s})=\sum _{j=1}^{10}{\mathfrak {f}}_{0}\textrm{s}^{4}=10 {\mathfrak {f}}_{0}\textrm{s}^{4}. \end{aligned}$$

In addition, the following relations hold:

$$\begin{aligned} \pi \left( \textrm{s}\right) =\frac{1}{\textrm{s}}\text { and }\pi \left( \textrm{s},\textrm{s}_{1}\right) =\frac{\textrm{s}-\textrm{s}_{1}}{\textrm{s} }. \end{aligned}$$

Using (8), we calculate the following limits:

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha -1}(\textrm{s},\textrm{s}_{1})=\lim _{\textrm{s}\rightarrow \infty } \frac{1}{\left( \textrm{s}-2\right) }\cdot \left( \frac{\textrm{s}-\textrm{s} _{1}}{\textrm{s}}\right) ^{4}=0\text {, and }\lim _{\textrm{s}\rightarrow \infty }\textrm{g}\left( \textrm{s}\right) =\lim _{\textrm{s}\rightarrow \infty }\frac{1}{\textrm{s}-3}=0. \end{aligned}$$

From (33), we obtain:

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \left( \frac{\pi (\tau \left( \textrm{s}\right) )}{\pi (\textrm{s})}\right) ^{\alpha }=\lim _{\textrm{s}\rightarrow \infty }\frac{1}{\textrm{s}-2}\cdot \left( \frac{1}{\tau _{0}}\right) ^{5}=0,\text { and }\lim _{\textrm{s} \rightarrow \infty }\textrm{g}\left( \textrm{s}\right) \frac{\pi ^{\delta }(\varsigma \left( \textrm{s}\right) )}{\pi \left( \textrm{s}\right) }=\lim _{ \textrm{s}\rightarrow \infty }\frac{1}{\textrm{s}-3}\cdot \frac{\textrm{s} ^{4/5}}{\varsigma _{0}^{1/5}}=0. \end{aligned}$$
Fig. 1
figure 1

Asymptotic Conditions for Eq. (45)

Full size image

As shown in Fig. 1, the asymptotic conditions clearly hold under the given assumptions.

The conditions in (18) and (27) are evidently satisfied. Condition (35) simplifies as follows:

$$\begin{aligned} & \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{\textrm{s}_{0}}^{ \textrm{s}}\left( {\widetilde{L}}^{\beta }\pi ^{\beta }\left( \nu \right) \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\nu )-\left( \frac{\beta }{\beta +1}\right) ^{\beta +1}\frac{1}{c_{4}^{\beta }\hslash ^{1/\gamma }\left( \nu \right) \pi \left( \nu \right) }\right) \textrm{d}\nu \\= & \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{1}^{\textrm{s} }\left( {\widetilde{L}}^{5}\frac{1}{\nu ^{5}}10{\mathfrak {f}}_{0}\nu ^{4}-\left( \frac{5}{6}\right) ^{6}\frac{\nu }{c_{4}^{5}\nu ^{2}}\right) \textrm{d}\nu \\= & \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{1}^{\textrm{s} }\left( 10{\widetilde{L}}^{5}{\mathfrak {f}}_{0}-\left( \frac{5}{6}\right) ^{6} \frac{1}{c_{4}^{5}}\right) \frac{1}{\nu }\textrm{d}\nu \\= & \left( 10{\widetilde{L}}^{5}{\mathfrak {f}}_{0}-\left( \frac{5}{6}\right) ^{6} \frac{1}{c_{4}^{5}}\right) \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\ln \textrm{s}=\infty , \end{aligned}$$

which holds when:

$$\begin{aligned} {\mathfrak {f}}_{0}>\frac{1}{10\left( c_{4}{\widetilde{L}}\right) ^{5}}\left( \frac{5}{6}\right) ^{6}, \ c_{4}>0,\ {\widetilde{L}}\in \left( 0,1\right) . \end{aligned}$$

Therefore, according to Theorems 5, and 8, Eq. (45 ) exhibits oscillatory behavior when the above condition is satisfied.

Example 15

Consider

$$\begin{aligned} \left( \textrm{s}^{14}\left( {\mathcal {U}}^{\prime }\left( \textrm{s}\right) \right) ^{7}\right) ^{\prime }+\textrm{s}^{6}{\mathfrak {f}}_{0}\left[ \varkappa ^{3}\left( \frac{1}{4}\textrm{s}\right) +\varkappa ^{3}\left( \frac{1}{5}\textrm{s}\right) +\varkappa ^{3}\left( \frac{1}{6}\textrm{s} \right) \right] =0, \textrm{s}\ge 1, \end{aligned}$$
(46)

where

$$\begin{aligned} {\mathcal {U}}\left( \textrm{s}\right) =\varkappa \left( \textrm{s}\right) + \frac{1}{\textrm{s}^{2}}\varkappa ^{3}\left( \frac{1}{3}\textrm{s}\right) + \frac{1}{\textrm{s}^{3}}\varkappa ^{1/3}\left( \frac{1}{4}\textrm{s}\right) . \end{aligned}$$

Clearly:

$$\begin{aligned} m= & 3, \gamma =7, \beta =3, \alpha =3, \delta = \frac{1}{3}, \hslash \left( \textrm{s}\right) =\textrm{s}^{14}, \sigma _{\min }\left( \textrm{s}\right) =\frac{1}{6}\textrm{s}, \sigma _{\max }\left( \textrm{s}\right) =\frac{1}{4}\textrm{s}, \\ \tau \left( \textrm{s}\right)= & \frac{1}{3}\textrm{s}, \varsigma \left( \textrm{s}\right) =\frac{1}{4}\textrm{s}, \textrm{p} \left( \textrm{s}\right) =\frac{1}{\textrm{s}^{2}}, \textrm{g}\left( \textrm{s}\right) =\frac{1}{\textrm{s}^{3}}, \sum _{j=1}^{m}\mathfrak {f }_{j}(\textrm{s})=\sum _{j=1}^{3}\textrm{s}^{6}{\mathfrak {f}}_{0}=3\textrm{s} ^{6}{\mathfrak {f}}_{0}. \end{aligned}$$

Consequently, we find that:

$$\begin{aligned} \pi \left( \textrm{s}\right) =\frac{1}{\textrm{s}}, \pi \left( \textrm{s},\textrm{s}_{1}\right) =\frac{\textrm{s}-\textrm{s}_{1}}{\textrm{s} }. \end{aligned}$$

Using (8), we compute:

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha -1}(\textrm{s},\textrm{s}_{1})=\lim _{\textrm{s}\rightarrow \infty } \frac{1}{\textrm{s}^{2}}\cdot \left( \frac{\textrm{s}-\textrm{s}_{1}}{ \textrm{s}}\right) ^{2}=0,\text { and }\lim _{\textrm{s}\rightarrow \infty } \textrm{g}\left( \textrm{s}\right) =\lim _{\textrm{s}\rightarrow \infty } \frac{1}{\textrm{s}^{3}}=0. \end{aligned}$$

From (33), we also derive

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \left( \frac{\pi (\tau \left( \textrm{s}\right) )}{\pi (\textrm{s})}\right) ^{\alpha }=\lim _{\textrm{s}\rightarrow \infty }\frac{1}{\textrm{s}^{2}}\cdot \left( 3\right) ^{3}=0,\text { and }\lim _{\textrm{s}\rightarrow \infty } \textrm{g}\left( \textrm{s}\right) \frac{\pi ^{\delta }(\varsigma \left( \textrm{s}\right) )}{\pi \left( \textrm{s}\right) }=\lim _{\textrm{s} \rightarrow \infty }\frac{1}{\textrm{s}^{3}}\cdot 4^{1/3}\textrm{s}^{2/3}=0. \end{aligned}$$
Fig. 2
figure 2

Asymptotic Conditions for Eq. (46).

Full size image

Figure 2 clearly illustrates that the asymptotic conditions are satisfied under the assumed hypotheses.

Now, if we let \(\rho \left( \textrm{s}\right) =1,\) conditions (18) and (28) are clearly satisfied.Additionally, condition (41) leads to:

$$\begin{aligned} \underset{\textrm{s}\rightarrow \infty }{\lim \sup }\int _{1}^{\textrm{s} }\left( {\widetilde{L}}^{3}\cdot \frac{1}{\nu ^{7}}\cdot 3\nu ^{6}{\mathfrak {f}} _{0}-\left( \frac{7}{8}\right) ^{8}\cdot \left( \frac{7}{3}\right) ^{7}\cdot \frac{\nu }{c_{5}\nu ^{2}}\right) \textrm{d}\nu= & \left( 3{\widetilde{L}}^{3} {\mathfrak {f}}_{0}-\frac{129.39}{c_{5}}\right) \underset{\textrm{s}\rightarrow \infty }{\cdot \lim \sup }\int _{1}^{\textrm{s}}\frac{1}{\nu }\textrm{d}\nu \\= & \left( 3{\widetilde{L}}^{3}{\mathfrak {f}}_{0}-\frac{129.39}{c_{5}}\right) \underset{\textrm{s}\rightarrow \infty }{\cdot \lim \sup }\ln s=\infty , \end{aligned}$$

This inequality holds when:

$$\begin{aligned} {\mathfrak {f}}_{0}>\frac{43.13}{c_{5}{\widetilde{L}}^{3}},\text { where }c_{5}>0, \text { and }{\widetilde{L}}\in \left( 0,1\right) . \end{aligned}$$
Fig. 3
figure 3

Oscillation Regions for Different \(c_{5}\) Values.

Full size image

Figure 3 displays the oscillation regions corresponding to different values of the parameter \(c_{5}\).

By Theorems 9 and 10, it follows that every solution of Eq. (46) xhibits oscillatory behavior under the given conditions.

Choosing \(c_{5}=1.5\) and \({\widetilde{L}}=0.8,\) we find that Eq. (46) exhibits oscillatory behavior for all \({\mathfrak {f}}_{0}>56.159.\)

Example 17

Consider

$$\begin{aligned} \left( \textrm{e}^{3\textrm{s}}\left( {\mathcal {U}}^{\prime }\left( \textrm{s} \right) \right) ^{3}\right) ^{\prime }+\textrm{e}^{4\textrm{s}}{\mathfrak {f}} _{0}\left[ \varkappa ^{3}\left( \frac{1}{2}\textrm{s}\right) +\varkappa ^{3}\left( \frac{1}{3}\textrm{s}\right) +\varkappa ^{3}\left( \frac{1}{4} \textrm{s}\right) \right] =0, \end{aligned}$$
(47)

where

$$\begin{aligned} {\mathcal {U}}\left( \textrm{s}\right) =\chi \left( \textrm{s}\right) +\frac{1}{ \textrm{e}^{4\textrm{s}}}\chi ^{3}\left( \frac{1}{3}\textrm{s}\right) +\frac{ 1}{\textrm{e}^{3\textrm{s}}}\chi ^{1/3}\left( \frac{1}{2}\textrm{s}\right) . \end{aligned}$$

Clearly:

$$\begin{aligned} m= & 3, \gamma =\beta =3, \alpha =3, \delta =\frac{1}{3} , \hslash \left( \textrm{s}\right) =\textrm{e}^{3\textrm{s}}, \sigma _{\min }\left( \textrm{s}\right) =\frac{1}{4}\textrm{s,}\ \sigma _{\max }\left( \textrm{s}\right) =\frac{1}{2}\textrm{s}, \\ \tau \left( \textrm{s}\right)= & \frac{1}{3}\textrm{s}, \varsigma \left( \textrm{s}\right) =\frac{1}{2}\textrm{s},\textrm{p}\left( \textrm{s}\right) =\frac{1}{\textrm{e}^{4\textrm{s}}}, \textrm{g} \left( \textrm{s}\right) =\frac{1}{\textrm{e}^{3\textrm{s}}}, \sum _{j=1}^{m}{\mathfrak {f}}_{j}(\textrm{s})=\sum _{j=1}^{3}\textrm{e}^{4 \textrm{s}}{\mathfrak {f}}_{0}=3\textrm{e}^{4\textrm{s}}{\mathfrak {f}}_{0}. \end{aligned}$$

Consequently, we find that:

$$\begin{aligned} \pi \left( \textrm{s}\right) =\frac{1}{\textrm{e}^{\textrm{s}}}, \pi \left( \textrm{s},\textrm{s}_{1}\right) =\textrm{e}^{-\textrm{s}_{1}}- \textrm{e}^{-\textrm{s}}. \end{aligned}$$

Using (8), we compute:

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \pi ^{\alpha -1}(\textrm{s},\textrm{s}_{1})=\lim _{\textrm{s}\rightarrow \infty } \frac{\left( \textrm{e}^{-\textrm{s}_{1}}-\textrm{e}^{-\textrm{s}}\right) ^{2}}{\textrm{e}^{4\textrm{s}}}=0,\text { and }\lim _{\textrm{s}\rightarrow \infty }\textrm{g}\left( \textrm{s}\right) =\lim _{\textrm{s}\rightarrow \infty }\frac{1}{\textrm{e}^{3\textrm{s}}}=0. \end{aligned}$$

From (33), we also derive

$$\begin{aligned} \lim _{\textrm{s}\rightarrow \infty }\textrm{p}\left( \textrm{s}\right) \left( \frac{\pi (\tau \left( \textrm{s}\right) )}{\pi (\textrm{s})}\right) ^{\alpha }=\lim _{\textrm{s}\rightarrow \infty }\frac{1}{\textrm{e}^{4\textrm{ s}}}\left( \frac{\textrm{e}^{\textrm{s}}}{\textrm{e}^{\frac{\textrm{s}}{3}}} \right) ^{3}=0,\text { and }\lim _{\textrm{s}\rightarrow \infty }\textrm{g} \left( \textrm{s}\right) \frac{\pi ^{\delta }(\varsigma \left( \textrm{s} \right) )}{\pi \left( \textrm{s}\right) }=\lim _{\textrm{s}\rightarrow \infty }\lim _{\textrm{s}\rightarrow \infty }\frac{1}{\textrm{e}^{3\textrm{s}}}\frac{ \textrm{e}^{\textrm{s}}}{\textrm{e}^{\frac{\textrm{s}}{12}}}=0. \end{aligned}$$
Fig. 4
figure 4

Asymptotic Conditions for Eq. (47).

Full size image

Figure 4 clearly demonstrates that the asymptotic conditions are satisfied under the stated assumptions.

It is straightforward to verify that all the conditions of Theorems 4- 8 are satisfied. Therefore, according to these theorems, Eq. ( 47) exhibits oscillatory behavior.

Conclusion

In this work, the oscillatory properties and asymptotic behavior of a class of NDEs with multiple delays, involving both superlinear and sublinear terms, are analyzed within the framework of the non-canonical case. New oscillation criteria are developed using the Riccati technique with different substitutions, representing a qualitative contribution and a significant expansion in the scientific literature related to this field. The obtained results are not only an extension of previous results but also constitute a qualitative leap toward a more in-depth understanding of the dynamics of this class of equations. The obtained results open up broad horizons for future studies. The interesting aspect is that the approach we adopted in this study can be used to obtain the same results without having to rely on the conditions (8) and (33) imposed in our current study. This may contribute to simplifying the models and expanding the range of potential applications for this type of equation. Moreover, the proposed methodology opens the door to the study of higher-order differential equations, allowing for a deeper study of the nonlinear and oscillatory properties of more complex systems.