Forced-Perturbed Fractional Differential Equations of Higher Order: Asymptotic Properties of Non-Oscillatory Solutions

Abstract

This study investigates the asymptotic behavior of non-oscillatory solutions to forced-perturbed fractional differential equations with the Caputo fractional derivative. The main aim is to unify the Beta Integral Lemma (Lemma 2) and the Gamma Integral Lemma (Lemma 3) into a single framework. By combining these two powerful tools, we propose new criteria that effectively characterize the asymptotic behavior of non-oscillatory solutions to the given equations. The analysis of such solutions has significant implications in the fields of oscillation and stability theory. Notably, our findings extend prior work by exploring a wider range of equations with more general functions and coefficients, thereby broadening the applicability and deepening the understanding of both asymptotic and oscillatory behaviors. Moreover, the criteria we introduce offer improvements over previous approaches, as demonstrated by the example provided, which highlights the advantages of our results in comparison to earlier methods.

1. Introduction

Fractional differential equations have recently attracted attention due to their potential applications in a variety of domains, including as engineering, mechanics, physics, chemistry, aerodynamics, mathematical biology, electrodynamics, and others. We direct the reader to the monographs [1,2,3,4] for further information. For particular conclusions from mathematical biology and physics, see especially the works [5,6,7,8,9], where the models are developed using differential equations with forces idealized by nonlocal and/or taxis-driven factors. In the literature, oscillation and other asymptotic conclusions for solutions of these kinds of equations are comparatively rare. The importance of studying the asymptotic behavior of differential equations lies in the possibility of benefiting from this study in many branches of qualitative theories such as oscillation theory (see [10,11,12,13,14]), stability theory (see [15,16,17,18]), and Existence and Uniqueness theory (see [19,20,21,22,23]).

In the work, we consider the following forced fractional differential equation

$${}^{C}D_c^{\alpha }\psi \left(s\right)-f(s,\varphi \left(s\right))=e\left(s\right)+h(s,\varphi \left(s\right)),c>1,\alpha \in (0,1),$$

(1)

where $\delta \ge 1$ is the ratio of positive odd integers, $\psi \left(s\right)={\left(a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\right)}^{(n-1)}$ with $n\in \mathbb{N}$ and ${}^{C}D_c^{\alpha }\psi \left(s\right)$ is the Caputo fractional derivative [24] of $\psi \left(s\right)$ of order $0<\alpha <1$ defined as

$${}^{C}D_c^{\alpha }\psi \left(s\right):={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_c^s{(s-\nu )}^{-\alpha }{\psi }^{\prime }\left(\nu \right)\mathrm{d}\nu ,c>1,$$

where

$$\Gamma \left(\varphi \right)={\int }_0^{\infty }{\nu }^{\varphi -1}{e}^{-\nu }\mathrm{d}\nu ,\varphi >0.$$

By a solution of (1), we mean a function $\varphi \left(s\right)\in C^1([c,\infty ],\mathbb{R})$ with $a{\left({\varphi }^{\prime }\right)}^{\delta }\in {\mathrm{C}}^{n-1}\left(s\right)$$\left(\right[c,\infty ),\mathbb{R})$ and satisfies (1) for $s\ge s_1>s_0$. A solution is said to be oscillatory if it is neither eventually positive nor eventually negative, and it is non-oscillatory otherwise.

Throughout, we always assume that

(A₁)

$e\in \mathrm{C}\left([c,\infty ),\mathbb{R}\right)$ and $a\in \mathrm{C}\left([c,\infty ),(0,\infty )\right)$ such that

$${\int }_c^{\infty }{a}^{-{\displaystyle \frac{1}{\delta }}}\left(\nu \right)\mathrm{d}\nu =\infty ;$$

(A₂)

$f,h\in \mathrm{C}\left([c,\infty )\times \mathbb{R},\mathbb{R}\right)$ and that there exist $p^{*},q^{*}\in \mathrm{C}\left([c,\infty ),(0,\infty )\right)$ and positive real numbers $\beta$, $\gamma$ and $\tau$ such that

$$\varphi f(\varphi ,s)\le p^{*}\left(s\right)s^{\tau -1}\mid \varphi {\mid }^{\beta +1}\mathrm{and}\varphi h(s,\varphi )\le q^{*}\left(s\right)\mid \varphi {\mid }^{\gamma +1}\mathrm{for}\varphi \ne 0\mathrm{and}s\ge c.$$

Due to its numerous applications in fields including engineering, chemical processes, computational biology, electrodynamics, and more, differential equations of fractional order have garnered a lot of attention; see the references [5,25,26,27,28] for further details and in-depth discussions. In particular, Grace [29] investigated the forced differential equations of the form

$${}^{C}D_c^{\alpha }\psi \left(s\right)=e\left(s\right)+h(s,\varphi \left(s\right)),c>1,\alpha \in (0,1),$$

(2)

where $\psi \left(s\right)={\left(a\left(s\right)\left({\varphi }^{\prime }\left(s\right)\right)\right)}^{\prime }$. In this direction, we refer the readers to some of the work [30]. Similarly, Grace [31] examined the forced fractional differential equations of the form

$${}^{C}D_c^{\alpha }\psi \left(s\right)=e\left(s\right)+h(s,\varphi \left(s\right)),c>1,\alpha \in (0,1),$$

(3)

where $\delta \ge 1$ and $\psi \left(s\right)=(a\left(s\right){\left({\varphi }^{\prime \delta }\right)}^{\prime }$. Additional generalizations and enhancements of these findings are available in [32,33].

These works allow us to note the following:

• If we let $\delta =1$, $h(s,\varphi (s\left)\right)=0$ and $n=2$, then (1) is the same as (2). We may also note that in [29], only Lemma 2 (below) is used to established two sufficient conditions for determining the asymptotic behavior of the solutions to (2).
• If we let $f(s,\varphi (s\left)\right)=0$ and $n=2$, then (1) is the same as (2). We may also note that in [31], only Lemma 3 (below) is used to established four sufficient conditions governing the asymptotic behavior of solutions to (3).

Previous studies have significantly advanced the understanding of the asymptotic and oscillatory behavior of fractional equations. Grace et al. [29,31] provided sufficient conditions to analyze the asymptotic properties of solutions for coercive equations. Additionally, works such as [32,33] extended these findings to the specific case of n = 2. However, these investigations were restricted to particular equation forms and developed their results independently for Lemma 2 or Lemma 3, leaving unresolved the challenge of formulating oscillatory criteria and asymptotic properties for equations involving higher-order derivatives. For more interesting work on the qualitative properties of solutions to fractional differential equations, see [34,35,36,37,38]

Inspired by the studies referenced above, the primary aim of this paper is to employ the two Lemmas 2 and 3 in a single result and develop new criteria that determine the asymptotic behavior of non-oscillatory solutions to Equation (1). It is important to note that Equation (1) can be equivalently expressed as a Volterra-type equation of the form

$$\psi \left(s\right)=c_0+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_c^s{(s-\nu )}^{\alpha -1}[e\left(\nu \right)+f(\nu ,\varphi \left(\nu \right))+h(\nu ,\varphi \left(\nu \right))]\mathrm{d}\nu ,$$

(4)

where $c>1$, $0<\alpha <1$ and $c_0=\psi \left(c\right)$ is a real constant.

The reminder of this paper is organized as follows. Section 2 contains some lemmas that will play a crucial role in proving our main oscillation results. Section 3 contains our main theorems and one example to illustrate them. Finally, we conclude our findings and discuss future research directions.

2. Basic Lemmas

To obtain our main results, we need the following lemmas.

Lemma 1

([39]). Let ϕ, ψ be non-negative and $m>1$ such that ${\displaystyle \frac{1}{m}}+{\displaystyle \frac{1}{m^{*}}}=1$. Then

$$\varphi \psi \le {\displaystyle \frac{1}{m}}{\varphi }^m+{\displaystyle \frac{1}{m^{*}}}{\psi }^{m^{*}}$$

(5)

equality holds if and only if $\psi ={\varphi }^{m-1}.$

Lemma 2

([40]). Let β, γ and p be positive constants such that $p(\beta -1)+1>0$ and $p(\gamma -1)+1>0$. Then

$${\int }_0^s{(s-\nu )}^{p(\beta -1)}{\nu }^{p(\gamma -1)}\mathrm{d}\nu =s^{\theta }B,s\ge 0,$$

where $\theta =p(\beta +\gamma -2)+1$, $B:=B\left[p\right(\gamma -1)+1,p(\beta -1)+1]$ and $B[\xi ,\eta ]$ is the beta function defined by

$$B[\xi ,\eta ]={\int }_0^1{\nu }^{\xi -1}{(1-\nu )}^{\eta -1}\mathrm{d}\nu for\xi ,\eta >0.$$

Lemma 3

([25]). Let $\alpha ,p>0$ such that $p(\alpha -1)+1>0$. Then

$${\int }_0^s{(s-\nu )}^{p(\alpha -1)}{e}^{p\nu }\mathrm{d}\nu \le Q{e}^{ps}fors\ge 0,$$

where $Q={\displaystyle \frac{\Gamma (1+p(\alpha -1\left)\right)}{p^{1+p(\alpha -1)}}}$.

3. Main Results

In what follows, for $b_1,b_2\in \mathrm{C}\left([c,\infty ),(0,\infty )]\right)$, we define

$$g_1\left(s\right)=(\delta -\gamma ){\left({\displaystyle \frac{{\gamma }^{\gamma }}{{\delta }^{\delta }}}\right)}^{{\displaystyle \frac{1}{\delta -\gamma }}}{\left({\displaystyle \frac{q^{*}\left(s\right)}{b_1\left(s\right)}}\right)}^{{\displaystyle \frac{\delta }{\delta -\gamma }}}\mathrm{and}{g}_2\left(s\right)=(\delta -\beta ){\left({\displaystyle \frac{{\beta }^{\beta }}{{\delta }^{\delta }}}\right)}^{{\displaystyle \frac{1}{\delta -\beta }}}{\left({\displaystyle \frac{p^{*}\left(s\right)}{b_2\left(s\right)}}\right)}^{{\displaystyle \frac{\delta }{\delta -\beta }}}.$$

In view of $\left(A_1\right)$ and for $s\ge c>1$, we use the following notation:

$$R(s,c)={\int }_c^s{a}^{-{\displaystyle \frac{1}{\delta }}}\left(\nu \right)\mathrm{d}\nu \to \infty ass\to \infty .$$

(6)

The following first theorem provides sufficient conditions under which any non-oscillatory solution of Equation (1) will satisfy

$$\left|\varphi \left(s\right)\right|=\mathrm{O}\left(s^{{\displaystyle \frac{(n-1)}{\delta }}}R(s,c)\right)\mathrm{as}s\to \infty .$$

Theorem 1.

Let $\left(A_1\right)$ and $\left(A_2\right)$ hold. Suppose that $p>1$, $q={\displaystyle \frac{p}{p-1}}$, $p(\alpha -1)+1>0$ with $0<\alpha <1$ and $p(\tau -1)+1>0$ with $\tau =2-\alpha -{\displaystyle \frac{1}{p}}$. If there exist $b_1,b_2\in \mathrm{C}\left([c,\infty ),(0,\infty )]\right)$ and two constants $a_1,a_2>0$ such that

$${\int }_{s_1}^{\infty }\left[a_1{e}^{q(s-\nu )}{b}_1^q\left(\nu \right)+a_2{b}_2^q\left(\nu \right)\right]{\left({\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q\mathrm{d}\nu <\infty ,$$

(7)

$$\underset{s\to \infty }{\lim }{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}\left|e\left(\nu \right)\right|\mathrm{d}\nu <\infty ,$$

(8)

$$\underset{s\to \infty }{\lim }{\int }_c^s{(s-\nu )}^{\alpha -1}{g}_1\left(\nu \right)\mathrm{d}\nu <\infty ,$$

(9)

$${\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{g}_2\left(\nu \right)\mathrm{d}\nu <\infty ,$$

(10)

then every non-oscillatory solution of Equation (1) satisfies

$$\underset{s\to \infty }{\lim \sup }{\displaystyle \frac{\left|\varphi \right(s\left)\right|}{s^{\frac{(n-1)}{\delta }}R(s,c)}}<\infty .$$

(11)

Proof. 

Let $\varphi$ be an eventually positive solution of Equation (1). We may assume that $\varphi \left(s\right)>0$ for $s\ge s_1$ for some $s_1\ge c$. In view of $\left(A_2\right)$, we let $F(s,\varphi (s\left)\right)=f(s,\varphi (s\left)\right)+h(s,\varphi (s\left)\right)$, and then (4) may be written as

$$\begin{array}{cc}\hfill {\left(a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\right)}^{(n-1)}& \le c_0+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_c^s{(s-\nu )}^{\alpha -1}[e\left(\nu \right)+F(\nu ,\varphi \left(\nu \right))]\mathrm{d}\nu \hfill \\ \hfill & \le |c_0|+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_c^s{(s-\nu )}^{\alpha -1}\left|e\left(\nu \right)\right|\mathrm{d}\nu +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_c^s{(s-\nu )}^{\alpha -1}\left|F(\nu ,\varphi \left(\nu \right))\right|\mathrm{d}\nu \hfill \\ \hfill & \begin{array}{c}\le |c_0|+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}[{\int }_c^{s_1}{(s-\nu )}^{\alpha -1}\left|F(\nu ,x\left(\nu \right))\right|d\nu \hfill \\ +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}[p^{*}\left(\nu \right)x^{\beta }\left(\nu \right)-b_2\left(\nu \right)x^{\delta }\left(\nu \right)]d\nu \hfill \\ +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right)x^{\delta }\left(\nu \right)d\nu \hfill \\ +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}[q^{*}\left(\nu \right)x^{\gamma }\left(\nu \right)-b_1\left(\nu \right)x^{\delta }\left(\nu \right)]d\nu \hfill \\ +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right)x^{\delta }\left(\nu \right)d\nu +{\int }_c^s{(s-\nu )}^{\alpha -1}\left|e\left(\nu \right)\right|d\nu ].\hfill \end{array}\hfill \end{array}$$

Applying (5) of Lemma 1 to $[q^{*}\left(\nu \right){\varphi }^{\gamma }\left(\nu \right)-b_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)]$ with $n={\displaystyle \frac{\delta }{\gamma }}>1$, $\varphi ={\varphi }^{\gamma }\left(\nu \right)$, $\psi ={\displaystyle \frac{\gamma }{\delta }}\left({\displaystyle \frac{q^{*}\left(\nu \right)}{b_1\left(\nu \right)}}\right)$, and $m={\displaystyle \frac{\delta }{\delta -\gamma }}$, we obtain

$$\begin{array}{cc}\hfill q^{*}\left(\nu \right){\varphi }^{\gamma }\left(\nu \right)-b_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)& ={\displaystyle \frac{\delta }{\gamma }}{b}_1\left(\nu \right)\left[{\varphi }^{\gamma }\left(\nu \right){\displaystyle \frac{\gamma }{\delta }}{\displaystyle \frac{q^{*}\left(\nu \right)}{b_1\left(\nu \right)}}-{\displaystyle \frac{\gamma }{\delta }}{\left({\varphi }^{\gamma }\left(\nu \right)\right)}^{{\displaystyle \frac{\delta }{\gamma }}}\right]\hfill \\ \hfill & ={\displaystyle \frac{\delta }{\gamma }}{b}_1\left(\nu \right)\left[\varphi \psi -{\displaystyle \frac{1}{n}}{\varphi }^n\right]\hfill \\ \hfill & \le {\displaystyle \frac{\delta }{\gamma }}{b}_1\left(\nu \right)\left({\displaystyle \frac{1}{m}}{\psi }^m\right)\hfill \\ \hfill & =\left({\displaystyle \frac{\delta -\gamma }{\gamma }}\right){\left[{\displaystyle \frac{\gamma }{\delta }}{q}^{*}\left(\nu \right))\right]}^{{\displaystyle \frac{\delta }{\delta -\gamma }}}{\left(b_1\left(\nu \right)\right)}^{{\displaystyle \frac{\delta }{\gamma -\delta }}}\hfill \\ \hfill & =(\delta -\gamma ){\left({\displaystyle \frac{{\gamma }^{\gamma }}{{\delta }^{\delta }}}\right)}^{{\displaystyle \frac{1}{\delta -\gamma }}}{\left({\displaystyle \frac{{\left(q^{*}\left(\nu \right)\right)}^{\delta }}{b_1^{\delta }\left(\nu \right)}}\right)}^{{\displaystyle \frac{1}{\delta -\gamma }}}=:g_1\left(\nu \right).\hfill \end{array}$$

Similarly, we see that

$$\begin{array}{cc}\hfill p^{*}\left(\nu \right){\varphi }^{\beta }\left(\nu \right)-b_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)& \le \left({\displaystyle \frac{\delta -\beta }{\beta }}\right){\left[{\displaystyle \frac{\beta }{\delta }}{p}^{*}\left(\nu \right))\right]}^{{\displaystyle \frac{\delta }{\delta -\beta }}}{\left(b_2\left(\nu \right)\right)}^{{\displaystyle \frac{\delta }{\beta -\delta }}}\hfill \\ \hfill & =(\delta -\beta ){\left({\displaystyle \frac{{\beta }^{\beta }}{{\delta }^{\delta }}}\right)}^{{\displaystyle \frac{1}{\delta -\beta }}}{\left({\displaystyle \frac{{\left(p^{*}\left(\nu \right)\right)}^{\delta }}{b_2^{\delta }\left(\nu \right)}}\right)}^{{\displaystyle \frac{1}{\delta -\beta }}}=:g_2\left(\nu \right).\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}{\left(a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\right)}^{(n-1)}\le \left|c_0\right|+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}[\hfill & {\int }_c^s{(s-\nu )}^{\alpha -1}\left|e\left(\nu \right)\right|d\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{g}_1\left(\nu \right)d\nu \hfill \\ & +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{g}_2\left(\nu \right)d\nu +{\int }_c^{s_1}{(s-\nu )}^{\alpha -1}\left|F(\nu ,x\left(\nu \right))\right|d\nu \hfill \\ & +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right)x^{\delta }\left(\nu \right)d\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right)x^{\delta }\left(\nu \right)d\nu ],\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill & {\left(a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\right)}^{(n-1)}\hfill \\ \hfill & \le C^{*}+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left[{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu \right]:=w\left(s\right),\hfill \end{array}$$

(12)

where in view of (8), (9) and (10), $C^{*}$ is the upper bound of the function

$$\begin{array}{c}|c_0|+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left[{\int }_c^s{(s-\nu )}^{\alpha -1}\left|e\left(\nu \right)\right|\mathrm{d}\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{g}_1\left(\nu \right)\mathrm{d}\nu \right.\hfill \\ \hfill \left.+{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{g}_2\left(\nu \right)\mathrm{d}\nu +{\int }_c^{s_1}{(s-\nu )}^{\alpha -1}\left|F(\nu ,\varphi \left(\nu \right))\right|\mathrm{d}\nu \right].\end{array}$$

Integrating $(n-1)$ times the inequality (12) from $s_1$ to s and then interchanging the order of integration, we have

$$\begin{array}{cc}\hfill a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\le \sum _{\phi =0}^{n-1}{K}_{\phi }{\displaystyle \frac{{(s-s_1)}^{\phi }}{\phi !}}& +{\displaystyle \frac{1}{\Gamma (\alpha +n-1)}}{\int }_{s_1}^s{(s-\nu )}^{\alpha +n-2}{b}_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\alpha +n-1)}}{\int }_{s_1}^s{(s-\nu )}^{\alpha +n-2}{\nu }^{\tau -1}{b}_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu ,\hfill \end{array}$$

where $K_{\phi }:\left|{\left(a{\left({\varphi }^{\prime }\right)}^{\delta }\right)}^{\left(\phi \right)}\left(s_1\right)\right|>0$ for all $0\le \phi <n-1$. As a result,

$$\begin{array}{c}a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\hfill \\ \begin{array}{c}\le s^{n-1}[\sum _{\phi =0}^{n-1}{K}_{\phi }{\displaystyle \frac{s^{\phi -n+1}}{\phi !}}\hfill \\ +{\displaystyle \frac{1}{\Gamma (\alpha +n-1)}}\left({\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right)x^{\delta }\left(\nu \right)d\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right)x^{\delta }\left(\nu \right)d\nu \right)]\hfill \end{array}\hfill \\ \begin{array}{c}\le [\sum _{\phi =0}^{n-1}{K}_{\phi }{\displaystyle \frac{s^{\varphi }}{\phi !}}\hfill \\ +{\displaystyle \frac{s^{n-1}}{\Gamma (\alpha +n-1)}}\left({\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right)x^{\delta }\left(\nu \right)d\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right)x^{\delta }\left(\nu \right)d\nu \right)].\hfill \end{array}\hfill \end{array}$$

Thus, we have

$$a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\le C_1+C_2{s}^{n-1}w\left(s\right):=W\left(s\right),$$

where ${\displaystyle C_1=\sum _{\phi =0}^{n-1}{K}_{\phi }\frac{1}{\phi !}}$ and $C_2={\displaystyle \frac{1}{\Gamma (\alpha +n-1)}}$. Thus,

$${\varphi }^{\prime }\left(s\right)\le {\left({\displaystyle \frac{W\left(s\right)}{a\left(s\right)}}\right)}^{{\displaystyle \frac{1}{\delta }}}.$$

(13)

Integrating (13) from $s_1$ to s and noting that $W\left(s\right)$ is an increasing function, we see that there exists a positive constant $\widehat{c}$ such that

$$\varphi \left(s\right)\le \widehat{c}+{\int }_c^s{\left({\displaystyle \frac{W\left(\nu \right)}{a\left(\nu \right)}}\right)}^{{\displaystyle \frac{1}{\delta }}}\mathrm{d}\nu \le \widehat{c}+W^{{\displaystyle \frac{1}{\delta }}}\left(s\right){\int }_c^s{a}^{-{\displaystyle \frac{1}{\delta }}}\left(\nu \right)\mathrm{d}\nu =:\widehat{c}+R(s,c)W^{{\displaystyle \frac{1}{\delta }}}\left(s\right).$$

Using the assumption on $R(s,c)$, we have

$$\varphi \left(s\right)\le R(s,c)\left(\widehat{c}+W^{{\displaystyle \frac{1}{\delta }}}\left(s\right)\right)\mathrm{for}c>0,$$

or,

$${\displaystyle \frac{\varphi \left(s\right)}{R(s,c)}}\le \left(\widehat{c}+W^{{\displaystyle \frac{1}{\delta }}}\left(s\right)\right)\mathrm{for}s\ge c>1.$$

(14)

By using the inequality ${(A+B)}^{\alpha }\le 2^{\alpha -1}(A^{\alpha }+B^{\alpha })$ for $A,B>0$, we find

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\varphi \left(s\right)}{R(s,c)}}\right)}^{\delta }& \le 2^{\delta -1}{\widehat{c}}^{\delta }+2^{\delta -1}W\left(s\right)\hfill \\ \hfill & \le 2^{\delta -1}{\widehat{c}}^{\delta }+2^{\delta -1}(C_1+C_2{s}^{n-1}w\left(s\right))\hfill \end{array}$$

and so, we obtain

$${\left({\displaystyle \frac{\varphi \left(s\right)}{s^{\frac{(n-1)}{\delta }}R(s,c)}}\right)}^{\delta }\le {\displaystyle \frac{2^{\delta -1}{\widehat{c}}^{\delta }}{s^{n-1}}}+2^{\delta -1}\left({\displaystyle \frac{C_1}{s^{n-1}}}+C_{2}w\left(s\right)\right)\le c^{*}+2^{\delta -1}{C}_{2}w\left(s\right),$$

(15)

where $c^{*}$ is the upper bound for the function ${\displaystyle \frac{2^{\delta -1}{\widehat{c}}^{\delta }}{s^{n-1}}}+{\displaystyle \frac{C_1{2}^{\delta -1}}{s^{n-1}}}$. Also,

$$\begin{array}{cc}{\left({\displaystyle \frac{\varphi \left(s\right)}{s^{\frac{(n-1)}{\delta }}R(s,c)}}\right)}^{\delta }\le c^{*}+2^{\delta -1}{C}_2(C^{*}\hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}({\int }_{t_1}^t{(t-s)}^{\alpha -1}{b}_1\left(s\right)x^{\delta }\left(s\right)ds\hfill \\ & +{\int }_{t_1}^t{(t-s)}^{\alpha -1}{s}^{\tau -1}{b}_2\left(s\right)x^{\delta }\left(s\right)ds\left)\right),\hfill \end{array}$$

or,

$${\left({\displaystyle \frac{\varphi \left(s\right)}{s^{\frac{(n-1)}{\delta }}R(s,c)}}\right)}^{\delta }\le c_1^{*}+c_2^{*}\left({\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu +{\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu \right),$$

(16)

where $c_1^{*}=c^{*}+2^{\delta -1}{C}_2{C}^{*}$ and $c_2^{*}={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{2}^{\delta -1}{C}_2$.

An application of Holder’s inequality and Lemma 3 to ${\displaystyle {\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu }$ yields

$$\begin{array}{cc}\hfill {\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{b}_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu & ={\int }_{s_1}^s\left({(s-\nu )}^{\alpha -1}{e}^{\nu }\right)\left(e^{-\nu }{b}_1\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\right)\mathrm{d}\nu \hfill \\ \hfill & \le {\left({\int }_{s_1}^s{(s-\nu )}^{p(\alpha -1)}{e}^{p\nu })\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{e}^{-q\nu }{b}_1^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \le {\left(Q{e}^{ps}\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{e}^{-q\nu }{b}_1^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & =Q^{{\displaystyle \frac{1}{p}}}{e}^s{\left({\int }_{s_1}^s{e}^{-q\nu }{b}_1^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

(17)

where $Q>0$ is given in Lemma 3. Similarly, an application of Holder’s inequality and Lemma 2 to ${\displaystyle {\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu }$ yields

$$\begin{array}{cc}\hfill {\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu & \le {\left({\int }_{s_1}^s{(s-\nu )}^{p(\alpha -1)}{\nu }^{p(\tau -1)}\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{b}_2^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \le {\left(B{s}^{\theta }\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{b}_2^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

where $B=B\left[p\right(\tau -1)+1,p(\alpha -1)+1]$ and $\theta =p(\alpha +\tau -2)+1=0$. As a result,

$${\int }_{s_1}^s{(s-\nu )}^{\alpha -1}{\nu }^{\tau -1}{b}_2\left(\nu \right){\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu \le B^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{b}_2^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}.$$

(18)

Using (17) and (18) in (16), it follows that

$$\begin{array}{cc}\hfill z\left(s\right)& :={\left({\displaystyle \frac{\varphi \left(s\right)}{s^{\frac{(n-1)}{\delta }}R(s,c)}}\right)}^{\delta }\hfill \\ \hfill & \le 1+c_1^{*}+c_2^{*}\left[Q^{{\displaystyle \frac{1}{p}}}{e}^s{\left({\int }_{s_1}^s{e}^{-q\nu }{b}_1^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}+B^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{b}_2^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}\right],\hfill \end{array}$$

that is,

$$\begin{array}{cc}z\left(s\right)\le (1+c_1^{*})+c_2^{*}\times [\hfill & Q^{{\displaystyle \frac{1}{p}}}{e}^t{\left({\int }_{t_1}^t{e}^{-qs}{b}_1^q\left(s\right){\left(s^{n-1}{R}^{\delta }(s,c)\right)}^q{z}^q\left(s\right)ds\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +B^{{\displaystyle \frac{1}{p}}}{\left({\int }_{t_1}^t{b}_2^q\left(s\right){\left(s^{n-1}{R}^{\delta }(s,c)\right)}^q{z}^q\left(s\right)ds\right)}^{{\displaystyle \frac{1}{q}}}].\hfill \end{array}$$

(19)

Using the Jensen’s inequality for $\varphi ,\psi ,w\ge 0$, that is, ${(\varphi +\psi +w)}^q\le 3^{q-1}({\varphi }^q+{\psi }^q+w^q)$ and $q>1$ to (19), we obtain

$$\begin{array}{cc}{z}^q\left(s\right)\le 3^{q-1}[{(1+c_1^{*})}^q\hfill & +{\left(c_2^{*}{Q}^{{\displaystyle \frac{1}{p}}}{e}^t\right)}^q\left({\int }_{t_1}^t{e}^{-qs}{b}_1^q\left(s\right){\left(s^{n-1}{R}^{\delta }(s,c)\right)}^q{z}^q\left(s\right)ds\right)\hfill \\ & +{\left(c_2^{*}\right)}^q{B}^{{\displaystyle \frac{q}{p}}}\left({\int }_{t_1}^t{b}_2^q\left(s\right){\left(s^{n-1}{R}^{\delta }(s,c)\right)}^q{z}^q\left(s\right)ds\right)],\hfill \end{array}$$

or,

$$z^q\left(s\right)\le 3^{q-1}\left({\left(1+c_1^{*}\right)}^q+{\int }_{s_1}^s\left[Q^{{\displaystyle \frac{q}{p}}}{e}^{-q(\nu -s)}{b}_1^q\left(\nu \right)+B^{{\displaystyle \frac{q}{p}}}{b}_2^q\left(\nu \right)\right]{\left(c_2^{*}{\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q{z}^q\left(\nu \right)\mathrm{d}\nu \right).$$

(20)

Setting $\widehat{Q}\left(s\right)=\left[Q^{{\displaystyle \frac{q}{p}}}{e}^{-q(\nu -s)}{b}_1^q\left(\nu \right)+B^{{\displaystyle \frac{q}{p}}}{b}_2^q\left(\nu \right)\right]{\left({\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q$, $u\left(s\right)=z^q\left(s\right)$, i.e., $z\left(s\right)=u^{{\displaystyle \frac{1}{q}}}\left(s\right)$, $P=3^{q-1}{(1+c_1^{*})}^q$ and $P_1=3^{q-1}{\left(c_2^{*}\right)}^q$ in (20), we have

$$u\left(s\right)\le P+P_1{\int }_{s_1}^s\widehat{Q}\left(\nu \right)u\left(\nu \right)\mathrm{d}\nu \mathrm{for}s\ge s_1\ge c.$$

(21)

The conclusion follows from Gronwall’s inequality and (7) that

$$\underset{s\to \infty }{lim\; sup}{\displaystyle \frac{\left|\varphi \right(s\left)\right|}{s^{\frac{(n-1)}{\delta }}R(s,c)}}<\infty .$$

This completes the proof. □

Next, we give sufficient conditions under which any non-oscillatory solution of (1) satisfying

$$\left|\varphi \left(s\right)\right|=\mathrm{O}\left(s^{{\displaystyle \frac{(n-1)}{\delta }}R(s,c)}{e}^{{\displaystyle \frac{s}{\delta }}}\right)\mathrm{as}s\to \infty .$$

Theorem 2.

Let condition (7) in Theorem 1 be replaced by

$${\int }_{s_1}^{\infty }\left[a_2{b}_2^q\left(\nu \right)+a_1{b}_1^q\left(\nu \right)\right]{\left({\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q\mathrm{d}\nu <\infty \mathrm{for}s\ge s_1\ge c,$$

(22)

where $a_1$ and $a_2$ are positive constants. Then, every non-oscillatory solution of Equation (1) satisfies

$$\underset{s\to \infty }{\lim \sup }{\displaystyle \frac{e^{\frac{-s}{\delta }}\left|\varphi \left(s\right)\right|}{s^{\frac{(n-1)}{\delta }R(s,c)}}}<\infty .$$

Proof. 

We proceed as in the proof of Theorem 1 to obtain (20) and which will take the form

$$\begin{array}{cc}\hfill z\left(s\right)& :={\left({\displaystyle \frac{e^{-\frac{s}{\delta }}\varphi \left(s\right)}{s^{\frac{n-1}{\delta }}R(s,c)}}\right)}^{\delta }\hfill \\ \hfill & :=1+c_1^{*}{e}^{-s}\begin{array}{cc}& [c_2^{*}{Q}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{t_1}^t{e}^{-qs}{b}_1^q\left(s\right){\left({\int }_{t_1}^t{e}^{-s}{s}^{n-1}{R}^{\delta }(s,c)\right)}^q{z}^q\left(s\right)ds\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +B^{{\displaystyle \frac{1}{p}}}{\left({\int }_{t_1}^t{e}^{-qs}{b}_2^q\left(s\right){\left({\int }_{t_1}^t{e}^{-s}{s}^{n-1}{R}^{\delta }(s,c)\right)}^q{z}^q\left(s\right)ds\right)}^{{\displaystyle \frac{1}{q}}}].\hfill \end{array}\hfill \end{array}$$

The rest of the proof is the same and hence is omitted. □

The following example is illustrative.

Example 1.

Consider the nth order integro-differential equation

$$\begin{array}{c}{\left(s{\left({\varphi }^{\prime }\left(s\right)\right)}^3\right)}^{(n-1)}={\int }_c^s{(s-\nu )}^{{\displaystyle \frac{-1}{2}}}{e}^{-\nu }\sin 3\nu \mathrm{d}\nu \hfill \\ \hfill +{\int }_c^s{(s-\nu )}^{{\displaystyle \frac{-1}{2}}}{e}^{-4\nu }{\nu }^{{\displaystyle \frac{-1}{6}}}{\varphi }^{{\displaystyle \frac{5}{3}}}\left(\nu \right)\mathrm{d}\nu +{\int }_c^s{(s-\nu )}^{{\displaystyle \frac{-1}{2}}}{e}^{-4\nu }{\varphi }^{{\displaystyle \frac{5}{3}}}\left(\nu \right)\mathrm{d}\nu ,\end{array}$$

(23)

where $c=8$ and n is a positive integer. Let $p={\displaystyle \frac{3}{2}}$ and $q=3$, $\alpha ={\displaystyle \frac{1}{2}}$. Clearly, $p(\alpha -1)+1={\displaystyle \frac{1}{4}}>0$, $\tau =2-\alpha -{\displaystyle \frac{1}{p}}={\displaystyle \frac{5}{6}}$ and $p(\tau -1)+1={\displaystyle \frac{3}{4}}$. Let $a\left(s\right)=s$, $e\left(s\right)=e^{-4s}\sin s$ and $p^{*}\left(s\right)=q^{*}\left(s\right)=b_1\left(s\right)=b_2\left(s\right)=e^{-4s}$. Now,

$$R(s,c)=R(s,8)={\int }_8^s{\nu }^{{\displaystyle \frac{-1}{3}}}\mathrm{d}\nu ={\displaystyle \frac{3}{2}}(s^{{\displaystyle \frac{2}{3}}}-4).$$

It is easy to check that the hypothesis of Theorem 1 or Theorem 2 is fulfilled. So, applying Theorem 1, we conclude that every non-oscillatory solution of Equation (23) satisfies

$$\underset{s\to \infty }{\lim \sup }{\displaystyle \frac{\left|\varphi \right(s\left)\right|}{s^{\frac{(n-1)}{3}}{(s^{\frac{2}{3}}-4)}^3}}<\infty .$$

Also, by applying Theorem 2, we conclude that

$$\underset{s\to \infty }{\lim \sup }{\displaystyle \frac{e^{\frac{-s}{3}}\left|\varphi \left(s\right)\right|}{s^{\frac{(n-1)}{3}}{(s^{\frac{2}{3}}-4)}^3}}<\infty .$$

Remark 1.

We may note that our results in [31,33] do not apply to Equation (23), since they only applied for $n=2$.

In Equation (1), if $h(s,\varphi )=0$, one can easily obtain the following immediate result.

Theorem 3.

Let the hypotheses of Theorem 1 hold with $h(s,\varphi )=0$. Assume that condition (7) is replaced by

$${\int }_{s_1}^{\infty }{b}_2^q\left(\nu \right){\left({\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q\mathrm{d}\nu <\infty .$$

(24)

Then, the conclusion of Theorem 1 holds.

Next, we let $\tau =1$ and give the following theorem as another sufficient condition under any non-oscillatory solution of (1) to satisfy

$$\left|\varphi \left(s\right)\right|=\mathrm{O}\left(s^{{\displaystyle \frac{(n-1)}{\delta }}R(s,c)}{e}^{{\displaystyle \frac{s}{\delta }}}\right)\mathrm{as}s\to \infty .$$

Theorem 4.

Let $\left(A_1\right)$ and $\left(A_2\right)$ with $\tau =1$ hold. Suppose that $p>1$, $q={\displaystyle \frac{p}{p-1}}$ and $p(\alpha -1)+1>0$ with $0<\alpha <1$. If conditions (10) and (11) hold, and for some positive constants $a_1$ and $a_2$

$${\int }_c^{\infty }{e}^{-q\nu }{\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]}^q{\left({\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q\mathrm{d}\nu <\infty ,$$

(25)

then every non-oscillatory solution of Equation (1) satisfies

$$\underset{s\to \infty }{lim\; sup}{\displaystyle \frac{e^{\frac{-s}{\delta }}\left|\varphi \left(s\right)\right|}{s^{\frac{(n-1)}{\delta }R(s,c)}}}<\infty .$$

Proof. 

Let $\varphi$ be an eventually positive solution of equation (1). Proceeding exactly as in the proof of Theorem 1, we arrive at (16) with $\tau =1$, namely

$${\left({\displaystyle \frac{\varphi \left(s\right)}{s^{\frac{(n-1)}{\delta }}R(s,c)}}\right)}^{\delta }\le c_1^{*}+c_2^{*}\left({\int }_{s_1}^s{(s-\nu )}^{\alpha -1}\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]{\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu \right).$$

By applying Holder’s inequality and Lemma 3 to the preceding right-hand side integral, we obtain

$$\begin{array}{cc}\hfill & {\int }_{s_1}^s{(s-\nu )}^{\alpha -1}\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]{\varphi }^{\delta }\left(\nu \right)\mathrm{d}\nu \hfill \\ \hfill & ={\int }_{s_1}^s\left({(s-\nu )}^{\alpha -1}{e}^{\nu }\right)\left(e^{-\nu }\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]{\varphi }^{\delta }\left(\nu \right)\right)\mathrm{d}\nu \hfill \\ \hfill & \le {\left({\int }_{s_1}^s\left({(s-\nu )}^{p(\alpha -1)}{e}^{p\nu }\right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{e}^{-q\nu }{\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]}^q{\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \le {\left(Q{e}^{ps}\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{s_1}^s{e}^{-q\nu }{b}_2^q\left(\nu \right){\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & =Q^{{\displaystyle \frac{1}{p}}}{e}^s{\left({\int }_{s_1}^s{e}^{-q\nu }{\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]}^q{\varphi }^{\delta q}\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

where $Q>0$ is given in Lemma 3. As a result,

$$z\left(s\right):={\left({\displaystyle \frac{e^{\frac{-s}{\delta }}\left|\varphi \left(s\right)\right|}{s^{\frac{(n-1)}{\delta }R(s,c)}}}\right)}^{\delta }\le (1+c_1^{*})+c_2^{*}{\left({\int }_0^s{\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]}^q{e}^{-q\nu }{R}^{\delta q}(\nu ,c)z^q\left(\nu \right)\mathrm{d}\nu \right)}^{{\displaystyle \frac{1}{q}}}.$$

Next, following the line of proof of Theorem 1 and using the Jensen’s inequality for $\varphi ,\psi \ge 0$ and $q>1$, we obtain

$$z^q\left(s\right)\le 2^{q-1}\left({(1+c_1^{*})}^q+{\left(c_2^{*}\right)}^q{\int }_{s_1}^s{e}^{-q\nu }{\left[b_1\left(\nu \right)+b_2\left(\nu \right)\right]}^q{\left({\nu }^{n-1}{R}^{\delta }(\nu ,c)\right)}^q{z}^q\left(\nu \right)\mathrm{d}\nu \right).$$

The rest of the proof is also similar to that of Theorem 1 and hence is omitted. □

4. Conclusions

The study of fractional differential equations has been a fertile and active field in recent times. The study of the properties of the solutions of these equations is one of the fields rich in interesting analytical and applied issues. This study uses the Caputo fractional derivative to analyze the asymptotic behavior of the non-oscillatory solutions of some forced-perturbed fractional differential equations. We obtain new results through which we can test the asymptotic behavior of the non-oscillatory solutions of the equation under study. The results are new, enhancing and expanding on a number of earlier findings in the literature. To illustrate the significance of our recent finding, one example is provided. Finally, we propose the following possible next directions for this research.

• It will be interesting to investigate Equation (1) in the context of condition

  $$\begin{array}{c}\hfill {\int }_c^{\infty }{a}^{-{\displaystyle \frac{1}{\delta }}}\left(\nu \right)\mathrm{d}\nu <\infty ;\end{array}$$

  (26)
• It will be interesting to extends these results to the more general equations of the following form under both conditions (6) and (26):

  $${}^{C}D_c^{\alpha }\psi \left(s\right)-f(s,\varphi \left(s\right))=e\left(s\right)+g\left(s\right){\varphi }^{\beta }\left(s\right)+h(s,\varphi \left(s\right)),c>1,\alpha \in (0,1),$$

  where $\delta \ge 1$ and $\beta >0$ represent the ratios of positive odd integers, $\psi \left(s\right)={\left(a\left(s\right){\left({\varphi }^{\prime }\left(s\right)\right)}^{\delta }\right)}^{(n-1)}$ with $n\in \mathbb{N}$.