On the Oscillatory Behavior of a Class of Mixed Fractional-Order Nonlinear Differential Equations

Abstract

This paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations incorporating both the Liouville right-sided and conformable fractional derivatives. Symmetry plays a key role in understanding the oscillatory behavior of these systems. The motivation behind this study arises from the need for a more generalized framework to analyze oscillatory behavior in fractional differential equations, bridging the gap in the existing literature. By employing the generalized Riccati technique and the integral averaging method, we establish new oscillation criteria that extend and refine previous results. Illustrative examples are provided to validate the theoretical findings and highlight the effectiveness of the proposed methods.

1. Introduction

The theory and applications of fractional differential equations are contained in many monographs and articles [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. In recent years, fractional-order differential equations have proven to be valuable and effective tools in modeling various phenomena in science and engineering. In fact, numerous applications can be found in the design of fractional control systems. The electrical properties of nerve cell membranes and the propagation of electrical signals are characterized by fractional-order derivatives. The fractional advective–dispersive equation has been the model basis for the simulation of transport in porous media. Fundamental explorations of the mechanical, electrical, and thermal constitutive relations of various engineering materials, such as viscoelastic polymers, have been successfully modeled. In financial markets, fractional-order models have recently been used to describe the probability distribution of log prices in the long-time limit, helping to characterize the natural variability in prices over the long term. See, for example, [1,9,10,13,14,18,21,23,24,28].

Recent progress in fractional calculus has introduced advanced analytical and numerical methods for the solution of complex systems governed by non-integer-order derivatives. Several studies have explored the stability, control, and optimization of fractional-order dynamical systems. In biomedical applications, fractional models have been utilized to describe anomalous diffusion in tissue and drug delivery processes. In electrical engineering, fractional-order circuits have demonstrated enhanced frequency response characteristics compared to traditional integer-order circuits. Moreover, integrating machine learning and artificial intelligence with fractional models has significantly improved their predictive accuracy, making them a valuable tool for future research and technological advancements.

Fractional differentials and integrals provide more accurate models of the aforementioned system. Several definitions exist for fractional derivatives and integrals, such as the Riemann–Liouville definition, the Caputo definition, and the Liouville right-sided definition on the half-axis R+. These definitions are based on integrals with singular kernels and exhibit non-local behavior, failing to satisfy the product, quotient, and chain rules. In contrast, in 2014, Khalil et al. introduced a limit-based definition analogous to that of standard derivatives (see [2,3,4,12,20,22]).

In recent years, increasing interest has been shown in obtaining sufficient conditions for the oscillatory and non-oscillatory behavior of different classes of fractional differential equations. Symmetry is fundamental in both classical and fractional oscillatory systems, influencing their stability, conservation laws, and qualitative behavior. In the case of fractional oscillators governed by the Liouville right-sided fractional derivative, symmetry plays a distinct role in shaping system dynamics. The oscillation theory of fractional differential equations with the Liouville right-sided definition has been studied by many authors, including Chen [8], Xu [25], Han [29], and Pan [16], while the damping term has been investigated by other authors, such as Qi [19] and Zheng [26,27].

In 2013, Xu [25] investigated the oscillatory behavior of a class of nonlinear fractional differential equations of the following form:

$$\begin{array}{c}\hfill {\left(a\left(t\right){\left[\left(r\left(t\right)g{\left(D_{-}^{\beta }y\left(t\right)\right)}^{\prime }\right]\right]}^{\eta }\right)}^{\prime }-F\left(t,{\int }_t^{\infty }{(\nu -t)}^{-\beta }y\left(\nu \right)d\nu \right)=0fort\ge t_0>0.\end{array}$$

In the same year, Zheng and Feng [27] discussed the oscillatory behavior of the following equation:

$$\begin{array}{cc}\hfill {\left(a\left(t\right){\left[\left(r\left(t\right){\left(D_{-}^{\beta }y\left(t\right)\right)}^{\prime }\right]\right]}^{\gamma }\right)}^{\prime }& +p\left(t\right){\left[\left(r\left(t\right){\left(D_{-}^{\beta }y\left(t\right)\right)}^{\prime }\right]\right]}^{\gamma }\hfill \\ & -q\left(t\right)f\left({\int }_t^{\infty }{(\xi -t)}^{-\beta }y\left(\xi \right)d\xi \right)=0\hfill \end{array}$$

for $t\in [t_0,\infty )$, $0<\beta <1.$

Our study extends the work of [30] by analyzing oscillatory behavior in $2+\beta$-order equations using both Liouville right-sided and conformable fractional derivatives, unlike their $2\beta$-order approach with the Riemann–Liouville left-sided derivative. We employ generalized Riccati techniques and integral averaging methods, leading to improved oscillation criteria and a more flexible mathematical framework. Furthermore, our results have broader real-world applications in engineering, physics, and finance, making them more versatile than previous findings. These advances establish our study as a significant improvement in both theory and application.

In 2017, Pavithra and Muthulakshmi [15] studied the oscillatory behavior of a class of nonlinear fractional differential equations with a damping term of the following form:

$$\begin{array}{cc}\hfill {\left(a\left(t\right){\left[\left(r\left(t\right)g{\left(D_{-}^{\beta }y\left(t\right)\right)}^{\prime }\right]\right]}^{\eta }\right)}^{\prime }& +p\left(t\right){\left[\left(r\left(t\right)g{\left(D_{-}^{\beta }y\left(t\right)\right)}^{\prime }\right]\right]}^{\eta }\hfill \\ & -F\left(t,{\int }_t^{\infty }{(\nu -t)}^{-\beta }y\left(\nu \right)d\nu \right)=0fort\ge t_0>0.\hfill \end{array}$$

In 2020, Chatzarakis et al. [7], studied the oscillatory properties of a specific class of mixed fractional differential equations involving both the conformable fractional derivative and the Riemann–Liouville left-sided fractional derivative. We initiated our work [7] in the year 2020. Since then, several papers related to this work have been published in various fields, including chemical sciences, medical applications, and existence theory. Please refer to the references for further details [31,32,33]. From the literature quoted above, we have observed that the Liouville right-sided fractional derivative, together with the classical integer-order derivative, is used for $(2+\beta )$-order nonlinear differential equations. To the best of our knowledge, it seems that no work has been conducted using conformable and Liouville right-sided derivatives in fractional-order differential equations.

Motivated by this gap, we have initiated a study of the following oscillation problem for a class of mixed fractional-order nonlinear differential equations of the form

$$\begin{array}{cc}\hfill & T_{{\beta }_3}\left[b_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)\right]+t^{1-{\beta }_3}P\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)\hfill \\ & -F\left(t,{\int }_t^{\infty }{(\nu -t)}^{-{\beta }_1}y\left(\nu \right)d\nu \right)=0,t\ge t_0>0,0<{\beta }_i<1,i=1,2,3\hfill \end{array}$$

(1)

where $D_{-}^{{\beta }_1}$ denotes the Liouville right-sided fractional derivative, and $T_{{\beta }_2},T_{{\beta }_3}$ denotes the conformable fractional derivatives.

Throughout this paper, we assume that the following conditions hold:

• $\left(A_1\right)$ $R\left(t\right)={\int }_{t_0}^t{\displaystyle \frac{P\left(s\right)}{b_2\left(s\right)}}ds,P\left(t\right)\in C([t_0,\infty ),{\mathbb{R}}_{+}))$;
• $\left(A_2\right)$ $b_2\left(t\right)\in C^{{\beta }_3}([t_0,\infty ),{\mathbb{R}}_{+})),b_1\left(t\right)\in C^{{\beta }_2+{\beta }_3}([t_0,\infty ),{\mathbb{R}}_{+})),{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]}}{d}_{{\beta }_2}s=\infty ;$
• $\left(A_3\right)$ $\left(a\right) g_i\in C^{{\beta }_2+{\beta }_3}(\mathbb{R},\mathbb{R})$ is an increasing and odd function and there exist positive constants ${\delta }_i$ such that ${\displaystyle \frac{x}{g_i\left(x\right)}}\le {\delta }_i>0$ for $x{g}_i\left(x\right)\ne 0,$ where $i=1,2$ and let $\delta ={\delta }_1{\delta }_2$;
  $\left(b\right)$ $g_i^{-1}\in C(\mathbb{R},\mathbb{R})$ with $u{g}_i^{-1}\left(u\right)>0$ for $u\ne 0,$ and there exist some positive constants ${\lambda }_i$ such that $g_i^{-1}\left(uv\right)\ge {\lambda }_i{g}_i^{-1}\left(u\right)g_i^{-1}\left(v\right)$ for $uv\ne 0,i=1,2$, where $\lambda ={\lambda }_1{\lambda }_2$;
• $\left(A_4\right)$ $F(t,K)\in C^1([t_0,\infty )\times \mathbb{R},{\mathbb{R}}_{+}))$ there exists a function $Q\left(t\right)\in C^1([t_0,\infty ),{\mathbb{R}}_{+}))$ such that ${\displaystyle \frac{F(t,K)}{g_2\left(K\right)}}\ge Q\left(t\right)$ for $K\ne 0$ $t\ge t_0$.

By a solution of (1), we mean a function $y\left(t\right)\in C({\mathbb{R}}_{+},\mathbb{R})$ such that ${\int }_t^{\infty }{(\nu -t)}^{-{\beta }_1}$$y\left(\nu \right)d\nu \in C^1({\mathbb{R}}_{+},\mathbb{R}),b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\in C^{{\beta }_2+{\beta }_3}({\mathbb{R}}_{+},\mathbb{R})$ and satisfies (1) on $[t_0,\infty ).$

A nontrivial solution of Equation (1) is termed oscillatory if it has infinitely many zeros, and it is called non-oscillatory otherwise. Equation (1) is classified as oscillatory if every solution of the equation is oscillatory.

Motivated by the gap in the existing literature, this paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations involving both the Liouville right-sided fractional derivative and the conformable fractional derivative. Unlike previous studies [8,15,16,19,25,26,27,29], which primarily focused on single types of fractional derivatives, our work integrates these two fractional operators, leading to a more generalized and flexible framework for the study of oscillatory behavior. By employing the generalized Riccati technique in Theorems 1 and 2 and the integral averaging method in Theorem 3, we establish new and improved oscillation criteria that extend and refine existing results. The significance and effectiveness of our theoretical findings are further demonstrated through illustrative examples. This study not only broadens the applicability of fractional differential equations in mathematical modeling but also provides a unified approach that enhances the understanding of oscillatory dynamics in fractional-order systems. Future studies may focus on the experimental validation of these results through numerical simulations and real-world applications in engineering and applied sciences.

The remainder of this paper is organized as follows. In Section 2, we present preliminary definitions and essential lemmas related to Liouville right-sided and conformable fractional derivatives. In Section 3, we establish new oscillation results for the given class of fractional differential equations. In Theorems 1 and 2, we solve the problem using the generalized Riccati technique, while Theorem 3 applies the integral averaging method. The above results are obtained using two types of fractional derivatives: the Liouville right-sided derivative and the conformable fractional derivative. In Section 4, illustrative examples are provided to validate the theoretical findings. Finally, the paper concludes with a summary of the key results.

2. Preliminaries

Before proceeding with our analysis of Equation (1), it is important to clarify the meaning of the operators $D_{-}^{\beta }y\left(t\right)$ and $T_{\beta }\left(f\right)\left(t\right)$. To ensure clarity, we will briefly review the key concepts of fractional calculus, specifically the Liouville right-sided approach and Khalil’s conformable fractional derivative. First, we will define the Liouville right-sided operator.

Definition 1 

([14]). The Liouville right-sided fractional derivative of order β of $y\left(t\right)$ is defined by

$$\begin{array}{c}\hfill D_{-}^{\beta }y\left(t\right)=-{\displaystyle \frac{1}{\mathsf{\Gamma }(1-\beta )}}{\displaystyle \frac{d}{dt}}{\int }_t^{\infty }{(\nu -t)}^{-\beta }y\left(\nu \right)d\nu ,t\in {\mathbb{R}}_{+}=(0,\infty ),\end{array}$$

where $\mathsf{\Gamma }(.)$ is the gamma function defined by $\mathsf{\Gamma }\left(t\right)={\int }_0^{\infty }{e}^{-s}{s}^{t-1}ds,t\in {\mathbb{R}}_{+}.$

Lemma 1 

([30]). Let y(t) be a solution of (1) and

$$\begin{array}{c}\hfill K\left(t\right)={\int }_t^{\infty }{(\nu -t)}^{-\beta }y\left(\nu \right)d\nu .\end{array}$$

(2)

Then,

$$\begin{array}{c}\hfill K^{\prime }\left(t\right)=-\mathsf{\Gamma }(1-\beta )D_{-}^{\beta }y\left(t\right).\end{array}$$

(3)

Next, we give the definition of the conformable fractional derivative proposed by Khalil et al. [12].

Definition 2. 

Given a function $f:[0,\infty )\to \mathbb{R}.$ Then, the conformable fractional derivative of f of order β is defined by

$$\begin{array}{c}\hfill T_{\beta }\left(f\right)\left(t\right)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{f(t+ϵt^{1-\beta })-f\left(t\right)}{ϵ}}\end{array}$$

for all $t>0,\beta \in (0,1).$ If f is β-differentiable in some $(0,a),a>0,and{lim}_{t\to 0^{+}}{f}^{\left(\beta \right)}\left(t\right)$ exists, then define

$$\begin{array}{c}\hfill f^{\left(\beta \right)}\left(0\right)=\underset{t\to 0^{+}}{lim}{f}^{\left(\beta \right)}\left(t\right).\end{array}$$

We will sometimes write $f^{\left(\beta \right)}\left(t\right)$ for $T_{\beta }\left(f\right)\left(t\right),$ to denote the conformable fractional derivatives of f of order $\beta .$

For a detailed discussion of the properties of the conformable fractional derivative, refer to Khalil’s [12] paper.

Definition 3 

([22]). Let $\beta \in (0,1]$ and $0\le a<b.$ A function $f:[a,b]\to \mathbb{R}$ is β-fractional integrable on [a,b] if the integral

$$\begin{array}{c}\hfill {\int }_a^{b}f\left(x\right)d_{\beta }x:={\int }_a^{b}f\left(x\right)x^{\beta -1}dx\end{array}$$

exists and is finite.

Lemma 2 

([2]). Let $f:(a,b)\to \mathbb{R}$ be differentiable and $0<\beta \le 1$. Then, for all $t>a$, we have

$$\begin{array}{c}\hfill I_{\beta }^a{T}_{\beta }^a\left(f\right)\left(t\right)=f\left(t\right)-f\left(a\right).\end{array}$$

(4)

The following inequality, which is used in the subsequent sections, is taken from Hardy et al. [11].

Lemma 3. 

If X and Y are nonnegative, then

$$\begin{array}{c}\hfill mX{Y}^{m-1}-X^m\le (m-1)Y^m,m>1.\end{array}$$

(5)

3. Main Results

In this section, we will introduce new oscillation criteria for Equation (1).

Lemma 4. 

Assume that $y\left(t\right)$ is an eventually positive solution of (1). If

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(s\right)}}\right)ds=\infty \end{array}$$

(6)

and

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right)d_{{\beta }_2}\xi \right)d\tau =\infty ,\end{array}$$

(7)

then there exists a sufficiently large T such that $T_{{\beta }_2}\left(b_1\left(s\right)g_1\left(D_{-}^{{\beta }_1}y\left(s\right)\right)\right)<0$ on $[T,\infty ),$ and one of the following two conditions holds:

(i) 

$D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[T,\infty )$

(ii) 

$D_{-}^{{\beta }_1}y\left(t\right)>0$ on $[T,\infty ),$ and ${lim}_{t\to \infty }K\left(t\right)=0.$

Proof. 

Let $t_1\ge t_0$ be such that $y\left(t\right)>0$ on $[t_1,\infty )$ and so $K\left(t\right)>0$ on $[t_1,\infty ).$ Thus, by (1) and $\left(A_4\right),$ we have

$$\begin{array}{c}\hfill T_{{\beta }_3}\left[e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)\right]\ge e^{R\left(t\right)}Q\left(t\right)g_2\left(K\left(t\right)\right)>0,t\in [t_1,\infty ),\end{array}$$

(8)

which means that $e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)$ is strictly increasing on $[t_1,\infty )$. Consequently, we can conclude that $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)$ is eventually of one sign. We claim that $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)<0$ on $[t_2,\infty ),$ where $t_2>t_1$ is sufficiently large. Otherwise, there exists a sufficiently large $t_3>t_2$ such that $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)>0$ on $[t_3,\infty ).$ Then, for $t\in [t_3,\infty )$ and using Lemma 2, we obtain

$$\begin{array}{cc}\hfill b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)-& b_1\left(t_3\right)g_1\left(D_{-}^{{\beta }_1}y\left(t_3\right)\right)\hfill \\ & ={\int }_{t_3}^t{\displaystyle \frac{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]T_{{\beta }_2}\left(b_1\left(s\right)g_1\left(D_{-}^{{\beta }_1}y\left(s\right)\right)\right)}{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]}}{d}_{{\beta }_2}s,\hfill \end{array}$$

$$\begin{array}{cc}\hfill b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)-& b_1\left(t_3\right)g_1\left(D_{-}^{{\beta }_1}y\left(t_3\right)\right)\hfill \\ & \ge g_2^{-1}\left[e^{R\left(t_3\right)}{b}_2\left(t_3\right)\right]T_{{\beta }_2}\left(b_1\left(t_3\right)g_1\left(D_{-}^{{\beta }_1}y\left(t_3\right)\right)\right)\hfill \\ & \times {\int }_{t_3}^t{\displaystyle \frac{1}{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]}}{d}_{{\beta }_2}s.\hfill \end{array}$$

(9)

From $\left(A_2\right)$, we have ${lim}_{t\to \infty }{b}_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)=\infty ,$ which implies that, for some sufficiently large $t_4>t_3,b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)>0.$ Thus, it is obvious that

$$\begin{array}{c}\hfill b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\ge b_1\left(t_4\right)g_1\left(D_{-}^{{\beta }_1}y\left(t_4\right)\right):=c>0,t\in [t_4,\infty ).\end{array}$$

(10)

From $\left(A_3\right),$ we have

$$\begin{array}{c}\hfill -{\displaystyle \frac{K^{\prime }\left(t\right)}{\mathsf{\Gamma }(1-{\beta }_1)}}=D_{-}^{{\beta }_1}y\left(t\right)\ge g_1^{-1}\left[{\displaystyle \frac{c}{b_1\left(t\right)}}\right]\ge {\lambda }_1{g}_1^{-1}\left(c\right)g_1^{-1}\left({\displaystyle \frac{1}{b_1\left(t\right)}}\right),t\in [t_4,\infty )\end{array}$$

and therefore

$$\begin{array}{c}\hfill g_1^{-1}\left({\displaystyle \frac{1}{b_1\left(t\right)}}\right)\le -{\displaystyle \frac{K^{\prime }\left(t\right)}{{\lambda }_1\mathsf{\Gamma }(1-{\beta }_1)g_1^{-1}\left(c\right)}},t\in [t_4,\infty ).\end{array}$$

(11)

Integrating (11) from $t_4$ to t, we obtain

$$\begin{array}{c}\hfill {\int }_{t_4}^t{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(s\right)}}\right)ds\le {\displaystyle \frac{K\left(t_4\right)}{{\lambda }_1\mathsf{\Gamma }(1-{\beta }_1)g_1^{-1}\left(c\right)}},t\in [t_4,\infty ).\end{array}$$

Letting $t\to \infty ,$ it follows that

$$\begin{array}{c}\hfill {\int }_{t_4}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(s\right)}}\right)ds<\infty ,t\in [t_4,\infty ),\end{array}$$

which contradicts (6). Therefore, $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)<0$ on $[t_2,\infty ).$

From $\left(A_3\right),$ we find that $D_{-}^{{\beta }_1}y\left(t\right)$ is eventually one sign. Consequently, there are two possibilities: (i) $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[T,\infty )$ or (ii) $D_{-}^{{\beta }_1}y\left(t\right)>0$ on $[T,\infty )$ for sufficiently large T. Suppose that $D_{-}^{{\beta }_1}y\left(t\right)>0$ for $t\in [T,\infty ),$ for sufficiently large $T>t_2.$ Thus, $K^{\prime }\left(t\right)<0,t\in [T,\infty ),$ and we have ${lim}_{t\to \infty }K\left(t\right)=l.$ Now, we claim that $l=0.$ Otherwise, assuming $l>0$, then $K\left(t\right)\ge l$ on $[T,\infty ).$ By (8), we have

$$\begin{array}{c}\hfill T_{{\beta }_3}\left[e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)\right]\ge e^{R\left(t\right)}Q\left(t\right)g_2\left(l\right),\end{array}$$

for $t\in [T,\infty ).$

${\beta }_3$—integrating the above inequality from t to ∞ and using Lemma 2, we can derive

$$\begin{array}{c}\hfill -e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)\ge g_2\left(l\right){\int }_t^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s.\end{array}$$

From $\left(A_3\right),$ we have that

$$\begin{array}{c}\hfill T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\le -{\lambda }_{2}l{g}_2^{-1}\left[{\displaystyle \frac{1}{e^{R\left(t\right)}{b}_2\left(t\right)}}{\int }_t^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right],\end{array}$$

(12)

for $t\in [T,\infty ).$

${\beta }_2$—integrating both sides of (12) from t to $\infty ,$ we obtain

$$\begin{array}{c}\hfill -b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\le -{\lambda }_{2}l{\int }_t^{\infty }{g}_2^{-1}\left[{\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right]d_{{\beta }_2}\xi ,\end{array}$$

for $t\in [T,\infty ).$

$$\begin{array}{c}\hfill D_{-}^{{\beta }_1}y\left(t\right)\ge g_1^{-1}\left({\displaystyle \frac{{\lambda }_{2}l}{b_1\left(t\right)}}{\int }_t^{\infty }{g}_2^{-1}\left[{\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right]d_{{\beta }_2}\xi \right).\end{array}$$

Applying Lemma 1 and $\left(A_3\right)\left(b\right),$ we have

$$\begin{array}{c}\hfill -{\displaystyle \frac{K^{\prime }\left(t\right)}{\mathsf{\Gamma }(1-{\beta }_1)}}\ge \lambda g_1^{-1}\left(l\right)g_1^{-1}\left({\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_t^{\infty }{g}_2^{-1}\left[{\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right]d_{{\beta }_2}\xi \right).\end{array}$$

(13)

Integrating both sides of (13) from T to t, we obtain

$$\begin{array}{cc}\hfill & K\left(t\right)\le K\left(T\right)\hfill \\ & -\lambda \mathsf{\Gamma }(1-{\beta }_1)g_1^{-1}\left(l\right){\int }_T^t{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left[{\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right]d_{{\beta }_2}\xi \right)d\tau .\hfill \end{array}$$

(14)

Letting $t\to \infty$ and using (7), we obtain ${lim}_{t\to \infty }K\left(t\right)=-\infty$. This contradicts $K\left(t\right)>0.$ The proof of the lemma is complete. □

Lemma 5. 

Suppose that $y\left(t\right)$ is an eventually positive solution of (1) such that $T_{{\beta }_2}$$\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)<0,D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_1,\infty ),$ where $t_1>t_0$ is sufficiently large. Then,

$$\begin{array}{c}\hfill K^{\prime }\left(t\right)\ge {\displaystyle \frac{-\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{b_1\left(t\right)}}\end{array}$$

(15)

and

$$\begin{array}{c}\hfill K\left(t\right)\ge -\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right),\end{array}$$

(16)

where $R_1(t_1,t)={\int }_{t_1}^t{\displaystyle \frac{1}{e^{R\left(s\right)}{b}_2\left(s\right)}}{d}_{{\beta }_2}s,R_2(t_1,t)={\int }_{t_1}^t{\displaystyle \frac{R_1(T,s)}{b_1\left(s\right)}}ds.$

Proof. 

As in Lemma 4, we deduce that $e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)$ is strictly increasing on $[t_1,\infty )$. So, we have

$$\begin{array}{cc}\hfill b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)& \le {\int }_{t_1}^t{\displaystyle \frac{e^{R\left(s\right)}{b}_2\left(s\right)T_{{\beta }_2}\left(b_1\left(s\right)g_1\left(D_{-}^{{\beta }_1}y\left(s\right)\right)\right)}{e^{R\left(s\right)}{b}_2\left(s\right)}}{d}_{{\beta }_2}s\hfill \\ & \le e^{R\left(t\right)}{b}_2\left(t\right)T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right){\int }_{t_1}^t{\displaystyle \frac{1}{e^{R\left(s\right)}{b}_2\left(s\right)}}{d}_{{\beta }_2}s\hfill \\ & =R_1(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right).\hfill \end{array}$$

From $\left(A_3\right),$ we obtain

$$\begin{array}{cc}\hfill b_1\left(t\right)\left({\displaystyle \frac{D_{-}^{{\beta }_1}y\left(t\right)}{{\delta }_1}}\right)& \le b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\hfill \\ & \le R_1(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill K^{\prime }\left(t\right)\ge -{\displaystyle \frac{\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{b_1\left(t\right)}}.\end{array}$$

Integrating the above inequality from $t_1$ to t, we obtain

$$\begin{array}{cc}\hfill K\left(t\right)-K\left(t_1\right)& \ge {\int }_{t_1}^t-{\displaystyle \frac{\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,s)e^{R\left(s\right)}{b}_2\left(s\right)g_2\left(T_{{\beta }_2}\left(b_1\left(s\right)g_1\left(D_{-}^{{\beta }_1}y\left(s\right)\right)\right)\right)}{b_1\left(s\right)}}ds\hfill \\ & \ge -\delta \mathsf{\Gamma }(1-{\beta }_1)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right){\int }_{t_1}^t{\displaystyle \frac{R_1(t_1,s)}{b_1\left(s\right)}}ds.\hfill \end{array}$$

Consequently,

$$\begin{array}{c}\hfill K\left(t\right)\ge -\delta \mathsf{\Gamma }(1-{\beta }_1)R_2(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right).\end{array}$$

The proof of the lemma is complete. □

Theorem 1. 

Assume that (6), (7) hold and suppose that $g_2^{\prime }\left(v\right)$ exists such that $g_2^{\prime }\left(v\right)\ge \mu$ for some $\mu >0$ and for all $v\ne 0.$ If there exist two functions $\varphi \left(t\right)\in C^{{\beta }_3}\left([t_0,\infty ),{\mathbb{R}}_{+}\right),\eta \left(t\right)\in C^{{\beta }_3}\left([t_0,\infty ),[0,\infty )\right)$ such that

$$\begin{array}{cc}\hfill {\int }_T^{\infty }& (\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)+{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)-\hfill \\ & {\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}})ds=\infty ,\hfill \end{array}$$

(17)

for sufficiently large T, where $R_1(T,s)$ is defined in Lemma 5, then every solution of (1) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

Proof. 

Suppose that (1) has a nonoscillatory solution $y\left(t\right)$ on $[t_0,\infty ).$ Without loss of generality, we may assume that $y\left(t\right)>0$ on $[t_1,\infty )$ for $t_1>t_0.$ By Lemma 4, we have $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)<0$, $t\in [t_2,\infty )$ for some sufficiently large $t_2>t_1$ and either $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_2,\infty )$ or ${lim}_{t\to \infty }K\left(t\right)=0.$

First, suppose that $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_2,\infty )$. We define the generalized Riccati function

$$\begin{array}{c}\hfill w\left(t\right)=\varphi \left(t\right)\left(-{\displaystyle \frac{e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{g_2\left(K\left(t\right)\right)}}+\eta \left(t\right)\right),\end{array}$$

(18)

when $w\left(t\right)>0$ on $[t_2,\infty ).$

Now, differentiating (18) ${\beta }_3$ times with respect to t for $t\in [t_2,\infty ),$

$$\begin{array}{cc}\hfill T_{{\beta }_3}w\left(t\right)=& T_{{\beta }_3}\varphi \left(t\right)\left(-{\displaystyle \frac{e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{g_2\left(K\left(t\right)\right)}}+\eta \left(t\right)\right)\hfill \\ & +\varphi \left(t\right)T_{{\beta }_3}\left(-{\displaystyle \frac{e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{g_2\left(K\left(t\right)\right)}}\right)+\varphi \left(t\right)T_{{\beta }_3}\eta \left(t\right).\hfill \end{array}$$

Then, making use of $\left(1\right),\left(8\right),\left(A_3\right)$, and $\left(15\right)$, it follows that

$$\begin{array}{cc}\hfill w^{\prime }\left(t\right)\le & {\displaystyle \frac{{\varphi }^{\prime }\left(t\right)}{\varphi \left(t\right)}}w\left(t\right)-\varphi \left(t\right)e^{R\left(t\right)}Q\left(t\right)t^{{\beta }_3-1}+\varphi \left(t\right){\eta }^{\prime }\left(t\right)\hfill \\ & +\varphi \left(t\right)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)t^{1-{\beta }_3}\mu \hfill \\ & \times {\displaystyle \frac{-\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{b_1\left(t\right){\left(g_2\left(K\left(t\right)\right)\right)}^2}}\hfill \\ \hfill \le & {\displaystyle \frac{{\varphi }^{\prime }\left(t\right)}{\varphi \left(t\right)}}w\left(t\right)-\varphi \left(t\right)e^{R\left(t\right)}Q\left(t\right)t^{{\beta }_3-1}+\varphi \left(t\right){\eta }^{\prime }\left(t\right)\hfill \\ & -{\displaystyle \frac{\varphi \left(t\right)\mu t^{1-{\beta }_3}\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)}{b_1\left(t\right)}}{\left({\displaystyle \frac{w\left(t\right)}{\varphi \left(t\right)}}-\eta \left(t\right)\right)}^2,\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill w^{\prime }\left(t\right)\le & -\varphi \left(t\right)e^{R\left(t\right)}Q\left(t\right)t^{{\beta }_3-1}+\varphi \left(t\right){\eta }^{\prime }\left(t\right)-{\displaystyle \frac{\varphi \left(t\right)}{b_1\left(t\right)}}{t}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t){\eta }^2\left(t\right)\hfill \\ & +{\displaystyle \frac{{\left[2\eta \left(t\right)\varphi \left(t\right)t^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)+{\varphi }^{\prime }\left(t\right)b_1\left(t\right)\right]}^2}{4b_1\left(t\right)\varphi \left(t\right)t^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_1,t)}}.\hfill \end{array}$$

(19)

Integrating the above inequality from $t_2$ to t, we obtain

$$\begin{array}{cc}\hfill {\int }_{t_2}^t& (\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)+{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}})ds\le w\left(t_2\right)-w\left(t\right)\le w\left(t_2\right)\hfill \end{array}$$

and, letting $t\to \infty ,$ we obtain a contradiction to (17). The proof of the theorem is complete. □

Theorem 2. 

Assume that (6) and (7) hold. If there exist two functions $\varphi \left(t\right)\in C^{{\beta }_3}\left([t_0,\infty ),{\mathbb{R}}_{+}\right)$, $\eta \left(t\right)\in C^{{\beta }_3}\left([t_0,\infty ),[0,\infty )\right)$, such that

$$\begin{array}{cc}\hfill {\int }_T^{\infty }& ({\displaystyle \frac{\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}}{{\delta }_2}}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)+{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{\delta }_1\mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}})ds=\infty \hfill \end{array}$$

(20)

for sufficiently large T, where $R_1(T,s)$ is defined in Lemma 5, then every solution of (1) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

Proof. 

Suppose that (1) has a nonoscillatory solution $y\left(t\right)$ on $[t_0,\infty ).$ Without loss of generality, we may assume that $y\left(t\right)>0$ on $[t_1,\infty )$ for $t_1>t_0.$ By Lemma 4, we have $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)<0$, $t\in [t_2,\infty )$ for some sufficiently large $t_2>t_1$ and either $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_2,\infty )$ or ${lim}_{t\to \infty }K\left(t\right)=0.$

Assume that $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_2,\infty )$. Let us define the generalized Riccati function as follows

$$\begin{array}{c}\hfill w\left(t\right)=\varphi \left(t\right)\left(-{\displaystyle \frac{e^{R\left(t\right)}{b}_2\left(t\right)g_2\left(T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)\right)}{K\left(t\right)}}+\eta \left(t\right)\right)\end{array}$$

(21)

when $w\left(t\right)>0$ on $[t_2,\infty ).$ The rest of the proof is similar to that of Theorem 1. □

Next, we discuss some new oscillation criteria for (1) by using the integral averaging method.

Theorem 3. 

Let ${\mathbb{D}}_0=\{(t,s):t>s\ge t_0\}$ and $\mathbb{D}=\{(t,s):t\ge s\ge t_0\}.$

Assume that (6), (7) hold and there exists a function $H\in C^{\prime }(\mathbb{D};\mathbb{R})$ that is said to belong to the class $\mathbb{P}$ if

• $\left(T_1\right)$ $H(t,t)=0$ for $t\ge t_0,H(t,s)>0$ on ${\mathbb{D}}_0$;
• $\left(T_2\right)$ H has a continuous and non-positive partial derivative on ${\mathbb{D}}_0$ with respect to the second variable and

  $$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}& {\displaystyle \frac{1}{H(t,t_0)}}{\int }_{t_0}^{t}H(t,s)[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}}]ds=\infty \hfill \end{array}$$

  (22)

  for all sufficiently large T, where $\varphi ,\eta$ are defined as in Theorem 1. Then, every solution of (1) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

Proof. 

Suppose that (1) has a nonoscillatory solution $y\left(t\right)$ on $[t_0,\infty ).$ Without loss of generality, we may suppose that $y\left(t\right)>0$ on $[t_1,\infty )$ for some large $t_1>t_0.$ By Lemma 4, we have $T_{{\beta }_2}\left(b_1\left(t\right)g_1\left(D_{-}^{{\beta }_1}y\left(t\right)\right)\right)<0$, $t\in [t_2,\infty )$ for some sufficiently large $t_2>t_1$ and either $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_2,\infty )$ or ${lim}_{t\to \infty }K\left(t\right)=0.$

Now, we assume $D_{-}^{{\beta }_1}y\left(t\right)<0$ on $[t_2,\infty )$ for some sufficiently large $t_2>t_1.$ Let $w\left(t\right)$ be defined as in Theorem 1. By (19), we have

$$\begin{array}{cc}\hfill & \varphi \left(t\right)e^{R\left(t\right)}Q\left(t\right)t^{{\beta }_3-1}-\varphi \left(t\right){\eta }^{\prime }\left(t\right)+{\displaystyle \frac{\varphi \left(t\right)}{b_1\left(t\right)}}{t}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,t){\eta }^2\left(t\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(t\right)\varphi \left(t\right)t^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,t)+{\varphi }^{\prime }\left(t\right)b_1\left(t\right)\right]}^2}{4b_1\left(t\right)\varphi \left(t\right)t^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,t)}}\le -w^{\prime }\left(t\right).\hfill \end{array}$$

(23)

Multiplying both sides by $H(t,s)$ and then integrating it with respect to s from $t_2$ to t yields

$$\begin{array}{cc}\hfill {\int }_{t_2}^t& H(t,s)[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)}}]ds\le -{\int }_{t_2}^{t}H(t,s)w^{\prime }\left(s\right)ds\hfill \\ & \le H(t,t_2)w\left(t_2\right)\le H(t,t_0)w\left(t_2\right).\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill {\int }_{t_0}^t& H(t,s)[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)}}]ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & H(t,t_0){\int }_{t_0}^{t_2}|\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)}}|ds+H(t,t_0)w\left(t_2\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}& {\displaystyle \frac{1}{H(t,t_0)}}{\int }_{t_0}^{t}H(t,s)[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)+\hfill \\ & {\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s){\eta }^2\left(s\right)-\hfill \\ & {\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(t_2,s)}}]ds<\infty ,\hfill \end{array}$$

which contradicts (22). The proof of the theorem is complete. □

In this theorem, if we take $H(t,s)$ for some special functions such as ${(t-s)}^m$ or $log\left({\displaystyle \frac{t}{s}}\right),$ then we can obtain some corollaries as follows.

Corollary 1. 

Assume that (6), (7) hold and

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}& {\displaystyle \frac{1}{{(t-t_0)}^m}}{\int }_{t_0}^t{(t-s)}^m[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}}]ds=\infty \hfill \end{array}$$

for sufficiently large T. Then, every solution of (1) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

Corollary 2. 

Assume that (6), (7) hold and

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}& {\displaystyle \frac{1}{logt-log{t}_0}}{\int }_{t_0}^t(logt-logs)[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}}]ds=\infty \hfill \end{array}$$

for sufficiently large T. Then, every solution of (1) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

4. Examples

In this section, we give some examples to illustrate our main results.

Example 1. 

Consider the fractional differential equation

$$\begin{array}{cc}\hfill & T_{{\displaystyle \frac{5}{7}}}\left[T_{{\displaystyle \frac{1}{3}}}\left(D_{-}^{{\displaystyle \frac{1}{2}}}y\left(t\right)\right)\right]+t^{1-{\displaystyle \frac{5}{7}}}{t}^{-{\displaystyle \frac{16}{7}}}\left(T_{{\displaystyle \frac{1}{3}}}\left(D_{-}^{{\displaystyle \frac{1}{2}}}y\left(t\right)\right)\right)\hfill \\ & -{\displaystyle \frac{M}{t^2}}\left({\int }_t^{\infty }{(\nu -t)}^{-{\displaystyle \frac{1}{2}}}y\left(\nu \right)d\nu \right)=0fort\ge 1.\hfill \end{array}$$

(24)

This corresponds to (1) with $b_2\left(t\right)=b_1\left(t\right)=1,{\beta }_1={\displaystyle \frac{1}{2}},{\beta }_2={\displaystyle \frac{1}{3}},{\beta }_3={\displaystyle \frac{5}{7}}$, $g_1\left(y\right)=y,{\displaystyle \frac{y}{g_1\left(y\right)}}\le {\delta }_1=1,g_2\left(y\right)=y,P\left(t\right)=t^{-{\displaystyle \frac{16}{7}}},F(t,K)=Q\left(t\right)\left({\int }_t^{\infty }{(\nu -t)}^{-{\beta }_1}y\left(\nu \right)d\nu \right)$, where ${\displaystyle \frac{M}{t^2}}=Q\left(t\right).$ Then, we have

$$\begin{array}{c}\hfill R\left(t\right)={\int }_{t_0}^t{\displaystyle \frac{P\left(s\right)}{b_2\left(s\right)}}ds=-{\displaystyle \frac{7}{9}}[t^{-{\displaystyle \frac{9}{7}}}-1]\le 1,\end{array}$$

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{\displaystyle \frac{1}{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]}}{d}_{{\beta }_2}s={\int }_1^{\infty }{\displaystyle \frac{s^{\frac{1}{3}-1}}{e}}ds=\infty \end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(s\right)}}\right)ds={\int }_1^{\infty }ds=\infty .\end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill & {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right)d_{{\beta }_2}\xi \right)d\tau \hfill \\ & ={\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}ds\right){\xi }^{{\beta }_2-1}d\xi \right)d\tau \hfill \\ & ={\int }_1^{\infty }{\int }_{\tau }^{\infty }\left[{\displaystyle \frac{1}{e}}{\int }_{\xi }^{\infty }{\displaystyle \frac{eM}{s^2}}{s}^{{\displaystyle \frac{5}{7}}-1}ds\right]{\xi }^{{\displaystyle \frac{1}{3}}-1}d\xi d\tau ={\displaystyle \frac{7M}{9}}{\int }_1^{\infty }{\int }_{\tau }^{\infty }{\displaystyle \frac{1}{{\xi }^{\frac{9}{7}}}}{\xi }^{{\displaystyle \frac{1}{3}}-1}d\xi d\tau \hfill \\ & ={\displaystyle \frac{7M}{9}}{\int }_1^{\infty }{\int }_{\tau }^{\infty }{\xi }^{-{\displaystyle \frac{41}{21}}}d\xi d\tau ={\displaystyle \frac{49M}{60}}{\int }_1^{\infty }{\displaystyle \frac{1}{{\tau }^{\frac{20}{21}}}}d\tau =\infty .\hfill \end{array}$$

On the other hand, for sufficiently large T, we obtain

$$\begin{array}{c}\hfill R_1(T,t)={\int }_T^t{\displaystyle \frac{1}{e^{R\left(s\right)}{b}_2\left(s\right)}}{d}_{{\beta }_2}s={\int }_T^t{\displaystyle \frac{1}{e}}{s}^{{\beta }_2-1}ds={\displaystyle \frac{1}{e}}{\int }_T^t{s}^{{\displaystyle \frac{1}{3}}-1}ds\to \infty (t\to \infty ).\end{array}$$

Thus, we can take $T^{*}>T$ such that $R_1(T,t)>1$ for $t\in [T^{*},\infty )$.

Letting $\varphi \left(s\right)=s^{{\displaystyle \frac{9}{7}}},\eta \left(s\right)=0,\mu =1,\delta =1.$

$$\begin{array}{cc}\hfill {\int }_T^{\infty }& \varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)+{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)-\hfill \\ & {\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}}ds\hfill \\ & \ge {\int }_T^{\infty }\left[s^{{\displaystyle \frac{9}{7}}}eM{s}^{-2}{s}^{-{\displaystyle \frac{2}{7}}}-{\displaystyle \frac{1}{4\sqrt{\pi }}}{\left({\displaystyle \frac{9}{7}}\right)}^2{s}^{{\displaystyle \frac{-9}{7}}}{s}^{{\displaystyle \frac{4}{7}}}{s}^{{\displaystyle \frac{-2}{7}}}\right]ds\hfill \\ & ={\int }_T^{\infty }{\displaystyle \frac{1}{s}}\left[eM-{\displaystyle \frac{81}{196\sqrt{\pi }}}\right]ds\hfill \\ & ={\int }_T^{T*}{\displaystyle \frac{1}{s}}\left[eM-{\displaystyle \frac{81}{196\sqrt{\pi }}}\right]ds+{\int }_{T*}^{\infty }{\displaystyle \frac{1}{s}}\left[eM-{\displaystyle \frac{81}{196\sqrt{\pi }}}\right]ds\to \infty ,\hfill \end{array}$$

provided that $M>{\displaystyle \frac{81}{196e\sqrt{\pi }}}.$ Hence, all conditions of Theorem 1 are satisfied. Therefore, every solution of (24) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

Example 2. 

Consider the fractional differential equation

$$\begin{array}{cc}\hfill & T_{{\displaystyle \frac{5}{7}}}\left[(2+cost)\left(T_{{\displaystyle \frac{3}{7}}}\left(D_{-}^{{\displaystyle \frac{1}{2}}}y\left(t\right)\right)\right)\right]+t^{1-{\displaystyle \frac{5}{7}}}{t}^{-3}\left(T_{{\displaystyle \frac{3}{7}}}\left(D_{-}^{{\displaystyle \frac{1}{2}}}y\left(t\right)\right)\right)\hfill \\ & -{\displaystyle \frac{M}{t^{\frac{5}{7}}}}\left({\int }_t^{\infty }{(\nu -t)}^{-{\displaystyle \frac{1}{2}}}y\left(\nu \right)d\nu \right)=0fort\ge 1.\hfill \end{array}$$

(25)

This corresponds to (1) with $b_2\left(t\right)=2+cost,b_1\left(t\right)=1,{\beta }_1={\displaystyle \frac{1}{2}},{\beta }_2={\displaystyle \frac{3}{7}},{\beta }_3={\displaystyle \frac{5}{7}}$, $g_1\left(y\right)=y,{\displaystyle \frac{y}{g_1\left(y\right)}}\le {\delta }_1=1,g_2\left(y\right)=y,P\left(t\right)=t^{-3},F(t,K)=Q\left(t\right)\left({\int }_t^{\infty }{(\nu -t)}^{-{\beta }_1}y\left(\nu \right)d\nu \right)$, where ${\displaystyle \frac{M}{t^{\frac{5}{7}}}}=Q\left(t\right).$ Then, we have

$$\begin{array}{c}\hfill R\left(t\right)={\int }_{t_0}^t{\displaystyle \frac{P\left(s\right)}{b_2\left(s\right)}}ds={\int }_1^t{\displaystyle \frac{s^{-3}}{2+coss}}ds\le {\displaystyle \frac{1}{2}}\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{\displaystyle \frac{1}{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]}}{d}_{{\beta }_2}s={\int }_1^{\infty }{\displaystyle \frac{s^{\frac{3}{7}-1}}{\sqrt{e}(2+coss)}}ds\ge {\displaystyle \frac{1}{3\sqrt{e}}}{\int }_1^{\infty }{s}^{{\displaystyle \frac{3}{7}}-1}ds=\infty .\end{array}$$

In addition, we can obtain

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(s\right)}}\right)ds={\int }_1^{\infty }ds=\infty .\end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill & {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right)d_{{\beta }_2}\xi \right)d\tau \hfill \\ & ={\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}ds\right){\xi }^{{\beta }_2-1}d\xi \right)d\tau \hfill \\ & ={\int }_1^{\infty }{\int }_{\tau }^{\infty }\left[{\displaystyle \frac{1}{\sqrt{e}(2+cos\xi )}}{\int }_{\xi }^{\infty }{\displaystyle \frac{\sqrt{e}M}{s^{\frac{5}{7}}}}{s}^{{\displaystyle \frac{5}{7}}-1}ds\right]{\xi }^{{\displaystyle \frac{3}{7}}-1}d\xi d\tau \hfill \\ & \ge {\displaystyle \frac{M}{3}}{\int }_1^{\infty }{\int }_{\tau }^{\infty }\left({\int }_{\xi }^{\infty }{\displaystyle \frac{1}{s}}ds\right){\xi }^{{\displaystyle \frac{3}{7}}-1}d\xi d\tau =\infty .\hfill \end{array}$$

On the other hand, for sufficiently large T, we obtain

$$\begin{array}{c}\hfill R_1(T,t)={\int }_T^t{\displaystyle \frac{1}{e^{R\left(s\right)}{b}_2\left(s\right)}}{d}_{{\beta }_2}s={\int }_T^t{\displaystyle \frac{1}{\sqrt{e}(2+coss)}}{s}^{{\displaystyle \frac{3}{7}}-1}ds\to \infty (t\to \infty ).\end{array}$$

Thus, we can take $T^{*}>T$ such that $R_1(T,t)>1$ for $t\in [T^{*},\infty )$.

Letting $\varphi \left(s\right)=1,\eta \left(s\right)=0,\mu =1,{\delta }_2=\delta =1.$

$$\begin{array}{cc}\hfill {\int }_T^{\infty }& ({\displaystyle \frac{\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}}{{\delta }_2}}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)+{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{\delta }_1\mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)\delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}})ds\hfill \\ & \ge {\int }_T^{\infty }\sqrt{e}M{s}^{-{\displaystyle \frac{5}{7}}}{s}^{-{\displaystyle \frac{2}{7}}}ds={\int }_T^{\infty }{\displaystyle \frac{1}{s}}ds\to \infty ,\hfill \end{array}$$

Hence, all conditions of Theorem 2 are satisfied. Therefore, every solution of (25) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

Example 3. 

Consider the fractional differential equation

$$\begin{array}{cc}\hfill & T_{{\displaystyle \frac{5}{7}}}\left[(2+cost)\left(T_{{\displaystyle \frac{1}{4}}}\left(D_{-}^{{\displaystyle \frac{1}{2}}}y\left(t\right)\right)\right)\right]+t^{1-{\displaystyle \frac{5}{7}}}{t}^{-3}\left(T_{{\displaystyle \frac{1}{4}}}\left(D_{-}^{{\displaystyle \frac{1}{2}}}y\left(t\right)\right)\right)\hfill \\ & -{\displaystyle \frac{M}{t^2}}\left({\int }_t^{\infty }{(\nu -t)}^{-{\displaystyle \frac{1}{2}}}y\left(\nu \right)d\nu \right)=0fort\ge 1.\hfill \end{array}$$

(26)

This corresponds to (1) with $b_2\left(t\right)=2+cost,b_1\left(t\right)=1,{\beta }_1={\displaystyle \frac{1}{2}},{\beta }_2={\displaystyle \frac{1}{4}},{\beta }_3={\displaystyle \frac{5}{7}},g_1\left(y\right)=y$, ${\displaystyle \frac{y}{g_1\left(y\right)}}\le {\delta }_1=1,g_2\left(y\right)=y,P\left(t\right)=t^{-3},F(t,K)=Q\left(t\right)\left({\int }_t^{\infty }{(\nu -t)}^{-{\beta }_1}y\left(\nu \right)d\nu \right)$, where ${\displaystyle \frac{M}{t^2}}=Q\left(t\right).$

Then, we have

$$\begin{array}{c}\hfill R\left(t\right)={\int }_{t_0}^t{\displaystyle \frac{P\left(s\right)}{b_2\left(s\right)}}ds={\int }_1^t{\displaystyle \frac{s^{-3}}{2+coss}}ds\le {\displaystyle \frac{1}{2}}\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{\displaystyle \frac{1}{g_2^{-1}\left[e^{R\left(s\right)}{b}_2\left(s\right)\right]}}{d}_{{\beta }_2}s={\int }_1^{\infty }{\displaystyle \frac{s^{\frac{1}{4}-1}}{\sqrt{e}(2+coss)}}ds\ge {\displaystyle \frac{1}{3\sqrt{e}}}{\int }_1^{\infty }{s}^{{\displaystyle \frac{1}{4}}-1}ds=\infty .\end{array}$$

In addition, we can obtain

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(s\right)}}\right)ds={\int }_1^{\infty }ds=\infty .\end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill & {\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)d_{{\beta }_3}s\right)d_{{\beta }_2}\xi \right)d\tau \hfill \\ & ={\int }_{t_0}^{\infty }{g}_1^{-1}\left({\displaystyle \frac{1}{b_1\left(\tau \right)}}{\int }_{\tau }^{\infty }{g}_2^{-1}\left({\displaystyle \frac{1}{e^{R\left(\xi \right)}{b}_2\left(\xi \right)}}{\int }_{\xi }^{\infty }{e}^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}ds\right){\xi }^{{\beta }_2-1}d\xi \right)d\tau \hfill \\ & ={\int }_1^{\infty }{\int }_{\tau }^{\infty }\left[{\displaystyle \frac{1}{\sqrt{e}(2+cos\xi )}}{\int }_{\xi }^{\infty }{\displaystyle \frac{\sqrt{e}M}{s^{\frac{5}{7}}}}{s}^{{\displaystyle \frac{5}{7}}-1}ds\right]{\xi }^{{\displaystyle \frac{1}{4}}-1}d\xi d\tau \hfill \\ & \ge {\displaystyle \frac{M}{3}}{\int }_1^{\infty }{\int }_{\tau }^{\infty }\left({\int }_{\xi }^{\infty }{\displaystyle \frac{1}{s}}ds\right){\xi }^{{\displaystyle \frac{1}{4}}-1}d\xi d\tau =\infty .\hfill \end{array}$$

On the other hand, for sufficiently large T, we obtain

$$\begin{array}{c}\hfill R_1(T,t)={\int }_T^t{\displaystyle \frac{1}{e^{R\left(s\right)}{b}_2\left(s\right)}}{d}_{{\beta }_2}s={\int }_T^t{\displaystyle \frac{1}{\sqrt{e}(2+coss)}}{s}^{{\displaystyle \frac{1}{4}}-1}ds\to \infty (t\to \infty ).\end{array}$$

Thus, we can take $T^{*}>T$ such that $R_1(T,t)>1$ for $t\in [T^{*},\infty )$.

Letting $\varphi \left(s\right)=s^{{\displaystyle \frac{9}{7}}},\eta \left(s\right)=0,\mu =1,\delta =1andH(t,s)=log\left({\displaystyle \frac{t}{s}}\right).$

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}& {\displaystyle \frac{1}{H(t,t_0)}}{\int }_{t_0}^{t}H(t,s)[\varphi \left(s\right)e^{R\left(s\right)}Q\left(s\right)s^{{\beta }_3-1}-\varphi \left(s\right){\eta }^{\prime }\left(s\right)\hfill \\ & +{\displaystyle \frac{\varphi \left(s\right)}{b_1\left(s\right)}}{s}^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s){\eta }^2\left(s\right)\hfill \\ & -{\displaystyle \frac{{\left[2\eta \left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)+{\varphi }^{\prime }\left(s\right)b_1\left(s\right)\right]}^2}{4b_1\left(s\right)\varphi \left(s\right)s^{1-{\beta }_3}\mu \delta \mathsf{\Gamma }(1-{\beta }_1)R_1(T,s)}}]ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge \underset{t\to \infty }{lim\; sup}{\displaystyle \frac{1}{logt}}\left({\int }_T^{\infty }log{\displaystyle \frac{t}{s}}\left[s^{{\displaystyle \frac{9}{7}}}\sqrt{e}M{s}^{-2}{s}^{-{\displaystyle \frac{2}{7}}}-{\displaystyle \frac{1}{4\sqrt{\pi }}}{\left({\displaystyle \frac{9}{7}}\right)}^2{s}^{{\displaystyle \frac{-9}{7}}}{s}^{{\displaystyle \frac{4}{7}}}{s}^{{\displaystyle \frac{-2}{7}}}\right]ds\right)\hfill \\ & =\underset{t\to \infty }{lim\; sup}{\displaystyle \frac{1}{logt}}\left({\int }_T^{\infty }log{\displaystyle \frac{t}{s}}\left[\sqrt{e}M-{\displaystyle \frac{81}{196\sqrt{\pi }}}\right]{\displaystyle \frac{1}{s}}ds\right)\hfill \\ & =\underset{t\to \infty }{lim\; sup}{\displaystyle \frac{1}{logt}}({\int }_T^{T*}log{\displaystyle \frac{t}{s}}\left[\sqrt{e}M-{\displaystyle \frac{81}{196\sqrt{\pi }}}\right]{\displaystyle \frac{1}{s}}ds\hfill \\ & +{\int }_{T*}^{\infty }log{\displaystyle \frac{t}{s}}\left[\sqrt{e}M-{\displaystyle \frac{81}{196\sqrt{\pi }}}\right]{\displaystyle \frac{1}{s}}ds)\to \infty ,\hfill \end{array}$$

provided that $M>{\displaystyle \frac{81}{196e\sqrt{e\pi }}}.$ Hence, all conditions of Theorem 3 are satisfied. Therefore, every solution of (26) is oscillatory or ${lim}_{t\to \infty }K\left(t\right)=0.$

5. Conclusions

This paper derived new oscillation results for a class of mixed fractional-order nonlinear differential equations involving the conformable fractional derivative and the Liouville right-sided fractional derivative. The Liouville right-sided fractional derivative is especially valuable for the modeling of systems with memory effects, time asymmetry, and long-range interactions. We applied the generalized Riccati technique and the integral averaging method to achieve these results. This work extended and generalized several findings from the existing literature [9,15,16,29] to mixed fractional differential equations. Additionally, illustrative examples were provided to demonstrate the effectiveness of the newly established results. In future work, we aim to compute the analytic solutions of the proposed equations and develop numerical methods for validation. Additionally, graphical representations of oscillatory behavior will be explored to gain deeper insights. Further studies will focus on practical applications in engineering, biomedical sciences, and finance, integrating experimental data for real-world validation. Numerical simulations and computational models will be employed to strengthen the theoretical findings.