Useful Results for the Qualitative Analysis of Generalized Hattaf Mixed Fractional Differential Equations with Applications to Medicine

Abstract

Most solutions of fractional differential equations (FDEs) that model real-world phenomena in various fields of science, industry, and engineering are complex and cannot be solved analytically. This paper mainly aims to present some useful results for studying the qualitative properties of solutions of FDEs involving the new generalized Hattaf mixed (GHM) fractional derivative, which encompasses many types of fractional operators with both singular and non-singular kernels. In addition, this study also aims to unify and generalize existing results under a broader operator. Furthermore, the obtained results are applied to some linear systems arising from medicine.

1. Introduction

Fractional differential equations (FDEs) have become a powerful tool for modeling complex systems that exhibit nonlocal dynamics, memory effects, hereditary properties, and anomalous behavior, which cannot be accurately described using classical ordinary differential equations (ODEs). For example, FDEs have been used in biology to model biological complexity, from subcellular processes to ecosystem dynamics. Their ability to capture memory, heterogeneity, and anomalous transport makes them a significant contributor to the development of predictive biology, precision medicine, and bioengineering.

In the literature, FDEs have been widely applied across many fields, and they have been formulated by means of various fractional operators, such as the Caputo fractional derivative [1], the Caputo–Fabrizio (CF) fractional derivative [2], the Atangana–Baleanu (AB) fractional derivative [3], the weighted AB fractional derivative [4], the generalized Hattaf fractional (GHF) derivative [5], the Hattaf mixed fractional derivative [6], the Hadamard fractional derivative [7,8] and the Katugampola fractional derivative [9]. More recently, a new generalized Hattaf mixed (GHM) fractional derivative was introduced in [10] to include all the above-cited fractional derivatives [1,2,3,4,5,6,7,8,9] as well as other types, such as the new weighted fractional derivative with respect to another function [11], the generalized AB fractional derivative, the derivative involving the generalized Mittag-Leffler function [12], the AB fractional derivative with respect to another function [13], the weighted CF fractional derivative with respect to another function [14], the power fractional derivative [15], as well as the modified fractional derivative [16].

Several researchers have studied the qualitative analysis of FDEs based on various inequalities. For instance, Aguila-Camacho et al. [17] established a useful inequality for the Caputo fractional derivative of the quadratic Lyapunov function to prove the stability of numerous fractional systems. This useful inequality was extended to investigate the stability of FDEs with the AB fractional derivative [18], the Hadamard fractional derivative [19], the GHF derivative [20], as well as the generalized proportional Caputo fractional derivative [21]. Another idea proposed by Vargas-De-León [22] aimed to extend the Volterra-type Lyapunov function to fractional-order epidemic systems via an inequality, allowing one to estimate the Caputo fractional derivative of this function. Similarly, the last inequality was extended to the Caputo fractional derivative with respect to another function in [23], to the GHF derivative in [20], and to the AB fractional derivative in [18].

On the other hand, the stability analysis of fractional nonlinear systems involving the Caputo fractional derivative was studied by Delavari et al. [24] by means of Bihari and Bellman–Gronwall inequalities [25,26]. An extension of the comparison principle to the Hadamard fractional derivative was used by Wang et al. [27] to analyze the stability of a class of nonlinear Hadamard-type fractional differential systems. A new version of the fractional comparison principle was introduced in [28] to discuss the qualitative properties of FDEs with the GHF derivative, such as stability, asymptotic stability, and Mittag-Leffler stability. This new version extends the results presented in [29] for the Caputo fractional derivative and the one in [24] for the Riemann–Liouville fractional derivative.

In recent years, FDEs have received growing interest in pharmacokinetics and other medical-related applications due to their ability to capture nonlocal dynamics, long-term memory effects, and anomalous behaviors, making them a powerful tool for modeling complex biomedical systems. Their flexibility also makes FDEs particularly suitable for pharmacokinetics, biomedical signal processing, tumor growth, and disease modeling, enabling more accurate predictions, improved data fitting, and enhanced clinical decision-making. In [30], the authors investigated the application of FDEs to analyze datasets related to various drug processes exhibiting anomalous kinetics. Awadalla et al. [31] modeled drug concentration in human blood using the Psi-Caputo fractional derivative. The dynamics of a tumor growth model involving the Caputo fractional derivative were analyzed in [32]. In 2024, Wanassi and Torres [33] proposed a fractional model for blood alcohol concentration based on the same Psi-Caputo derivative, aiming to better capture the long-range dependencies and memory effects that play an important role in modeling blood alcohol concentration. The role of fractional derivatives in pharmacokinetic/pharmacodynamic anesthesia model using bispectral index scale (BIS) data was more recently investigated by Vellappandi and Lee [34].

The contributions of this study are as follows: (i) to extend the aforementioned inequalities, and (ii) to establish significant and practical results for investigating the qualitative properties of FDEs involving the GHM fractional derivative, aiming to unify and generalize a broad class of fractional operators with both singular and non-singular kernels. The choice of this fractional operator is motivated by its nonlocal nature and high flexibility. It allows for the adjustment of a wide range of parameters, making it particularly suitable for fitting real data and accurately capturing complex real-world dynamics. Moreover, the GHM fractional derivative incorporates a kernel with multiple tunable parameters, enabling it to represent various types of memory effects, including exponential, power-law, and Mittag-Leffler kernels, within a unified framework. To achieve the objectives of this study, the remainder of this paper is organized as follows: Section 2 introduces key preliminary concepts and results required for the subsequent analysis. Section 3 is devoted to the main results of the study, including generalizations of several known inequalities and the extension of the comparison principle to the GHM fractional derivative. Section 4 presents an application of our main analytical results to linear systems in the field of medicine. Finally, Section 5 concludes this paper, summarizing the main results, comparing them with related studies, highlighting their potential for extension to other applications, and outlining directions for future research.

2. Preliminary Concepts and Results

This section recalls the concepts related to the new GMH fractional derivative and presents some preliminary results necessary for the present study.

Definition 1

([10]). Assume that $(\alpha ,⋏)\in {[0,1]}^2$, $Re(\ell )>0$, $r>0$, $m>0$, $\wp \in \mathbb{R}$, $\hslash \in {\mathbb{R}}^{*}$, and $\vartheta \in {\mathcal{H}}^1(a,b)$. The GHM fractional derivative of the function $\vartheta \left(t\right)$ of order α in the Caputo sense with the weight function $\varpi \left(t\right)$ and with respect to another function $\chi \left(t\right)$ is defined as follows:

$${}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\vartheta \left(t\right)={\displaystyle \frac{\aleph (p+⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\ell -1}{E}_{r,\ell }^{\wp }[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]{\left(\varpi \vartheta \right)}^{\prime }\left(\xi \right)d\xi ,$$

(1)

where $\aleph (.)$ is a normalization function, satisfying $\aleph \left(0\right)=\aleph \left(1\right)=1$, $\varpi ,\chi \in C^1(a,b)$ with $\varpi ,{\chi }^{\prime }>0$ on the interval $[a,b]$; ${\displaystyle E_{r,\ell }^{\wp }\left(t\right)=\sum _{k=0}^{+\infty }\frac{{(\wp )}_k{t}^k}{k!\Gamma (rk+\ell )}}$ is the generalized Mittag-Leffler function with three parameters [35], with ${(\wp )}_0=1$ and ${(\wp )}_k=\wp (\wp +1)\dots (\wp +k-1)$ denoting the Pochhammer symbol; and ${\lambda }_{\alpha ,⋏}^{\hslash }={\displaystyle \frac{\hslash (\alpha +⋏-1)}{2-\alpha -⋏}}$.

It is important to note that the GHM fractional derivative given by Definition 1 includes numerous fractional derivatives with singular and non-singular kernels available in the literature. As examples, it becomes the Caputo fractional derivative [1] when $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)=t$; the CF fractional derivative [2] when $r=m=⋏=\hslash =\ell =\wp =1$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)=t$; the AB fractional derivative [3] when $r=m=\alpha$, $⋏=\hslash =\ell =\wp =1$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)=t$; the weighted AB fractional derivative [4] when $r=m=\alpha$, $⋏=\hslash =\ell =\wp =1$, and $\chi \left(t\right)=t$; the GHF derivative [5] when $⋏=\hslash =\ell =\wp =1$ and $\chi \left(t\right)=t$; the Hattaf mixed fractional derivative [6] when $\ell =⋏$, $\wp =1$, and $\chi \left(t\right)=t$; the Hadamard fractional derivative [7,8] when $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=ln\left(t\right)$; the Katugampola fractional derivative [9] when $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)={\displaystyle \frac{t^{\rho }}{\rho }}$ with $\rho >0$; the new weighted fractional derivative with respect to another function [11] when $⋏=\hslash =1$; the generalized AB fractional derivative with the generalized Mittag-Leffler function [12] when $r=m=\alpha$, $⋏=\hslash =1$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)=t$; the AB fractional derivative with respect to another function [13] when $r=m=\alpha$, $⋏=\hslash =\ell =\wp =1$, and $\varpi \left(t\right)=1$; the weighted CF fractional derivative with respect to another function [14] when $r=m=⋏=\hslash =\ell =\wp =1$; the CF fractional derivative with respect to another function [14] when $r=m=⋏=\hslash =\ell =\wp =1$ and $\varpi \left(t\right)=1$; the power fractional derivative [15] when $\ell =⋏=\wp =1$, $m=r$, $\hslash =ln\left(\overline{p}\right)$ (with $\overline{p}>0$) and $\chi \left(t\right)=t$; the modified fractional derivative [16] when $\ell =2-⋏$, $\wp =\hslash =1$, $m=r=\alpha$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)=t$; as well as the fractional derivative introduced in [36] when $\ell =⋏$, $\wp =\hslash =1$, $m=r=\alpha$, $\varpi \left(t\right)=1$, and $\chi \left(t\right)=t$.

Now, we recall the fractional integral associated with the GHM fractional derivative.

Definition 2

([10]). When $m=r$, the fractional integral corresponding to the GHM fractional derivative is given by

$${\displaystyle I_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,\ell }\vartheta \left(t\right)=\sum _{k=0}^{+\infty }{(}_k^{\wp })\frac{{\hslash }^k{(\alpha +⋏-1)}^k}{{(2-\alpha -⋏)}^{k-1}\aleph (\alpha +⋏-1)}{}^R\mathcal{I}_{a,\varpi ,\chi }^{kr-\ell +1}\vartheta \left(t\right),}$$

(2)

where ${\displaystyle {}^R\mathcal{I}_{a,\varpi ,\chi }^{\beta }\vartheta \left(t\right)}$ is the weighted Riemann–Liouville fractional integral of the the function $\vartheta \left(t\right)$ with respect to another function $\chi \left(t\right)$ [37] defined by

$${\displaystyle {}^R\mathcal{I}_{a,\varpi ,\chi }^{\beta }\vartheta \left(t\right)=\frac{1}{\Gamma \left(\beta \right)\varpi \left(t\right)}{\int }_a^t{\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\beta -1}\left(\varpi \vartheta \right)\left(\xi \right)d\xi ,}$$

(3)

for $\beta >0$ and ${\displaystyle {}^R\mathcal{I}_{a,\varpi ,\chi }^0\vartheta \left(t\right)=\vartheta \left(t\right)}$.

The generalized Hattaf fractional integral introduced in Definition 2 covers many forms of fractional integrals, including the generalized weighted fractional integral with respect to another function [11], the GHF integral [5], the fractional integral corresponding to the generalized AB fractional derivative with the generalized Mittag-Leffler function [12], the weighted AB fractional integral [4], the AB fractional integral with respect to another function [13], the AB fractional integral [3], the weighted CF fractional integral with respect to another function [14], the CF fractional derivative [2], the fractional integral corresponding to the new mixed fractional derivative [6], the power fractional integral [15], the fractional integral introduced in [36], the modified fractional integral [16], the weighted Riemann–Liouville fractional integral with respect to another function [37], the Hadamard fractional integral [38,39], the Katugampola fractional integral [40], the Riemann–Liouville fractional integral [41], the Riemann–Liouville fractional integral with respect to another function [41,42,43], and the tempered fractional integral [44,45].

Next, we recall the following fundamental result, which extends the Newton–Leibniz formula presented in [41] to the Caputo fractional derivative with a singular kernel, in [6] for the mixed fractional derivative, and in [46] to the AB fractional derivative in the Caputo sense.

Lemma 1.

Let $(\alpha ,⋏)\in {[0,1]}^2$, $r>0$, $Re(\ell )>0$, $\wp \in \mathbb{R}$, $\hslash \in {\mathbb{R}}^{*}$ and $f\in {\mathcal{H}}^1(a,b)$. Then

$${\displaystyle I_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,\ell }\left({}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\vartheta \right)\left(t\right)=\vartheta \left(t\right)-\frac{\varpi \left(a\right)\vartheta \left(a\right)}{\varpi \left(t\right)}.}$$

Based on Theorem 2.6 of [10], we deduce the following result.

Lemma 2.

The weighted Laplace transform with respect to another function χ of the GHM fractional derivative is given as follows:

$${\mathcal{L}}_{\varpi ,\chi }{\{}^C{D}_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\vartheta \left(t\right)\}\left(s\right)={\displaystyle \frac{\aleph (\alpha +⋏-1)}{2-\alpha -⋏}}\sum _{k=0}^{+\infty }{\displaystyle \frac{{(\wp )}_k{(-{\lambda }_{\alpha ,⋏}^{\hslash })}^k}{k!s^{km+\ell }}}[s{\mathcal{L}}_{\varpi ,\chi }\left\{\vartheta \left(t\right)\right\}\left(s\right)-\left(\varpi \vartheta \right)\left(a\right)],$$

where

$${\mathcal{L}}_{\varpi ,\chi }\left\{\vartheta \left(t\right)\right\}\left(s\right)={\int }_a^{+\infty }{\chi }^{\prime }\left(t\right)e^{-s\left(\chi \left(t\right)-\chi \left(a\right)\right)}\varpi \left(t\right)\vartheta \left(t\right)dt.$$

3. Main Results

Let g be a continuous function and u be a continuously differentiable function. For any constant $\lambda \in \mathbb{R}$, we consider the following function:

$$G\left(t\right)={\displaystyle \frac{1}{\varpi \left(t\right)}}{\int }_{\lambda }^{\varpi \left(t\right)u\left(t\right)}g\left(x\right)dx.$$

(4)

Hence, we have the following:

$$\begin{array}{ccc}\hfill {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }G\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\ell -1}{E}_{r,\ell }^{\wp }[-{\lambda }_{\alpha ,⋏}^{\hslash }(\chi \left(t\right)\hfill \\ & & -\chi \left(\xi \right){)}^m]g\left(\varpi \left(\xi \right)u\left(\xi \right)\right){\left(\varpi u\right)}^{\prime }\left(\xi \right)d\xi .\hfill \end{array}$$

We define

$${\displaystyle g_1\left(t\right)={}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }G\left(t\right)-g\left(\varpi \left(t\right)u\left(t\right)\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

Thus,

$$g_1\left(t\right)={\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\ell -1}{E}_{r,\ell }^{\wp }[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_1^{\prime }\left(\xi \right)d\xi ,$$

where

$$v_1\left(\xi \right)=g\left(\varpi \left(t\right)u\left(t\right)\right)\left(\varpi \left(t\right)u\left(t\right)-\varpi \left(\xi \right)u\left(\xi \right)\right)+{\int }_{\varpi \left(t\right)u\left(t\right)}^{\varpi \left(\xi \right)u\left(\xi \right)}g\left(x\right)dx.$$

Theorem 1.

For any constant $\lambda \in \mathbb{R}$, the GHM fractional derivative of the function $G\left(t\right)$ defined by (4) satisfies the following inequalities:

(i) 

When g is an increasing function and $\ell =\wp =1$ or $\ell =⋏=1-\alpha$, we have

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }G\left(t\right)\le g\left(\varpi \left(t\right)u\left(t\right)\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

(5)

(ii) 

When g is a decreasing function and $\ell =\wp =1$ or $\ell =⋏=1-\alpha$, we have

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }G\left(t\right)\ge g\left(\varpi \left(t\right)u\left(t\right)\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

(6)

Proof. 

From integration by parts and using ${\displaystyle \frac{d}{dt}}{E}_{r,\ell }^{\wp }\left(t\right)=\wp E_{r,r+\ell }^{\wp +1}\left(t\right)$, with $\ell =\wp =1$, we obtain the following:

$$\begin{array}{ccc}\hfill g_1\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}{E}_{r,1}^1[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_1\left(\xi \right){|}_{\xi =a}^{\xi =t}\hfill \\ & & -{\displaystyle \frac{m\aleph (\alpha +⋏-1){\lambda }_{\alpha ,⋏}^{\hslash }}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{v}_1\left(\xi \right){\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{m-1}{E}_{r,r+1}^2[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]d\xi .\hfill \end{array}$$

Since $v_1\left(t\right)=0$, we have the following:

$$\begin{array}{ccc}\hfill g_1\left(t\right)& =& -{\displaystyle \frac{\aleph (\alpha +⋏-1)v_1\left(a\right)}{(2-\alpha -⋏)\varpi \left(t\right)}}{E}_r[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^m]\hfill \\ & & -{\displaystyle \frac{m\aleph (\alpha +⋏-1){\lambda }_{\alpha ,⋏}^{\hslash }}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{v}_1\left(\xi \right){\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{m-1}{E}_{r,r+1}^2[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]d\xi .\hfill \end{array}$$

We define the following function:

$${\psi }_{\lambda ,g}\left(\xi \right)=g\left(\lambda \right)(\lambda -\xi )+{\int }_{\lambda }^{\xi }g\left(x\right)dx.$$

Obviously, ${\psi }_{\lambda ,g}^{\prime }\left(\xi \right)=g\left(\xi \right)-g\left(\lambda \right)$. When g is an increasing function, the function ${\psi }_{\lambda ,g}\left(\xi \right)$ is decreasing on the interval $(-\infty ,\lambda ]$ and increasing on $[\lambda ,+\infty )$, with ${\psi }_{\lambda ,g}\left(\lambda \right)=0$. Thus, ${\psi }_{\lambda ,g}\left(\xi \right)$ has the global minimum at $\xi =\lambda$. Hence,

$${\psi }_{\lambda ,g}\left(\xi \right)\ge 0,\mathrm{for}\mathrm{all}(\lambda ,\xi )\in {\mathbb{R}}^2.$$

As $v_1\left(\xi \right)={\psi }_{\varpi \left(t\right)u\left(t\right),g}\left(\varpi \left(\xi \right)u\left(\xi \right)\right)$, we obtain $v_1\left(\xi \right)\ge 0$ for all $\xi \in \mathbb{R}$. Similarly, we can easily prove that $v_1\left(\xi \right)\le 0$ for all $\xi \in \mathbb{R}$ when g is a decreasing function. This proves $\left(i\right)$ when $\ell =\wp =1$.

For $\ell =⋏=1-\alpha$, the expression of $g_1\left(t\right)$ is as follows:

$$\begin{array}{ccc}\hfill g_1\left(t\right)& =& {\displaystyle \frac{1}{\Gamma (1-\alpha )\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha }{v}_1^{\prime }\left(\xi \right)d\xi \hfill \\ & =& {\displaystyle \frac{1}{\Gamma (1-\alpha )\varpi \left(t\right)}}{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha }{v}_1{\left(\xi \right)|}_{\xi =a}^{\xi =t}\hfill \\ & & -{\displaystyle \frac{\alpha }{\Gamma (1-\alpha )\varpi \left(t\right)}}{\int }_a^t{v}_1\left(\xi \right){\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha -1}d\xi .\hfill \end{array}$$

Since $v_1^{\prime }\left(t\right)=0$, we have ${\displaystyle \underset{\xi \to t}{lim}{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha }{v}_1\left(\xi \right)=0}$. Hence,

$$\begin{array}{ccc}\hfill g_1\left(t\right)& =& -{\displaystyle \frac{v_1\left(a\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{-\alpha }}{\Gamma (1-\alpha )\varpi \left(t\right)}}-{\displaystyle \frac{\alpha }{\Gamma (1-\alpha )\varpi \left(t\right)}}{\int }_a^t{v}_1\left(\xi \right){\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha -1}d\xi .\hfill \end{array}$$

This proves $\left(ii\right)$ for $\ell =⋏=1-\alpha$. □

Corollary 1.

Let $u\left(t\right)\in \mathbb{R}$ be a continuously differentiable function. Then, at any time $t\ge a$, for $\ell =\wp =1$ or $\ell =⋏=1-\alpha$, we have the following:

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\varpi \left(t\right)u^2\left(t\right)\right)\le 2\varpi \left(t\right)u\left(t\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

(7)

Proof. 

By applying Theorem 1 (i) to the function $g\left(x\right)=2x$, we have the following:

$$\begin{array}{ccc}\hfill {\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\varpi \left(t\right)u^2\left(t\right)\right)}& =& {\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\frac{1}{\varpi \left(t\right)}{\int }_{\lambda }^{u\left(t\right)\varpi \left(t\right)}2xdx\right)}\hfill \\ & \le & {\displaystyle 2\varpi \left(t\right)u\left(t\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}\hfill \end{array}$$

This completes the proof of Corollary 1. □

Remark 1.

Corollary 1 generalizes and extends various results existing in the literature. For instance,

• When $\ell =\wp =⋏=\hslash =1$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, we get Corollary 1 of [20] for the GHF derivative.
• When $\ell =\wp =⋏=\hslash =1$, $r=m=\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, we obtain Lemma 3.1 of [18] for the AB fractional derivative.
• When $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, we get Lemma 1 of [17] for the Caputo fractional derivative.
• When $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=ln\left(t\right)$, we get Remark 3 of [19] for the Caputo fractional derivative.
• When $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=e^{-{\displaystyle \frac{\rho -1}{\rho }}t}$ and $\chi \left(t\right)={\displaystyle \frac{t}{\rho }}$ with $\rho \in (0,1]$, we obtain Lemma 1 of [21] for the generalized proportional Caputo fractional derivative [47,48].

Corollary 2.

Let $u\left(t\right)\in {\mathbb{R}}^{+}$ be a continuously differentiable function and $u^{*}>0$. Then, at any time $t\ge a$, for $\ell =\wp =1$ or $\ell =⋏=1-\alpha$, we have the following:

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left[u\left(t\right)-\frac{u^{*}}{\varpi \left(t\right)}-\frac{u^{*}}{\varpi \left(t\right)}ln\left(\frac{u\left(t\right)\varpi \left(t\right)}{u^{*}}\right)\right]\le \left(1-\frac{u^{*}}{\varpi \left(t\right)u\left(t\right)}\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

(8)

Proof. 

For $g\left(x\right)={\displaystyle \frac{1}{x}}$, we have $G\left(t\right)={\displaystyle \frac{1}{\varpi \left(t\right)}}ln\left({\displaystyle \frac{u\left(t\right)\varpi \left(t\right)}{u^{*}}}\right)$.

Since $x\mapsto {\displaystyle \frac{1}{x}}$ is a decreasing function on $(0,+\infty )$, it follows from Theorem 1 (ii) that

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }G\left(t\right)\ge g\left(\varpi \left(t\right)u\left(t\right)\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right),}$$

which implies that

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left[\frac{1}{\varpi \left(t\right)}ln\left(\frac{u\left(t\right)\varpi \left(t\right)}{u^{*}}\right)\right]\ge \frac{1}{\varpi \left(t\right)u\left(t\right)}{}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

Then,

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left[u\left(t\right)-\frac{u^{*}}{\varpi \left(t\right)}ln\left(\frac{u\left(t\right)\varpi \left(t\right)}{u^{*}}\right)\right]\le \left(1-\frac{u^{*}}{\varpi \left(t\right)u\left(t\right)}\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

Since ${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\frac{u^{*}}{\varpi \left(t\right)}\right)=0}$, we have the following:

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left[u\left(t\right)-\frac{u^{*}}{\varpi \left(t\right)}-\frac{u^{*}}{\varpi \left(t\right)}ln\left(\frac{u\left(t\right)\varpi \left(t\right)}{u^{*}}\right)\right]\le \left(1-\frac{u^{*}}{\varpi \left(t\right)u\left(t\right)}\right){}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

This completes the proof of Corollary 2. □

Remark 2.

Corollary 2 generalizes and extends many inequalities used to establish the global stability of FDEs. For example,

• When $\ell =⋏=1-\alpha$ and $\varpi \left(t\right)=1$, we obtain the recent result presented in Theorem 5 of [23].
• When $\ell =\wp =⋏=\hslash =1$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, we get Corollary 2 of [20] for GHF derivative.
• When $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, we get Lemma 3.1 of [22] for the Caputo fractional derivative.
• When $\ell =\wp =⋏=\hslash =1$, $r=m=\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, we obtain Lemma 3.2 of [18] for the AB fractional derivative.

Theorem 2.

Let $u\left(t\right)\in {\mathbb{R}}^n$ be a continuously differentiable function and $P\in {\mathbb{R}}^{n\times n}$ be a symmetric positive definite matrix. Then, at any time $t\ge a$, for $\ell =\wp =1$ or $\ell =⋏=1-\alpha$, we have the following:

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\varpi \left(t\right)u{\left(t\right)}^{T}Pu\left(t\right)\right)\le 2\varpi \left(t\right)u{\left(t\right)}^{T}P{}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

(9)

Proof. 

Consider the following function:

$${\displaystyle g_2\left(t\right)={}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\varpi \left(t\right)u{\left(t\right)}^{T}Pu\left(t\right)\right)-2\varpi \left(t\right)u{\left(t\right)}^{T}P{}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

(10)

Then

$$\begin{array}{ccc}\hfill g_2\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\ell -1}{E}_{r,\ell }^{\wp }[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]\left(2v_2{\left(\xi \right)}^{T}P\dot{v_2}\left(\xi \right)\right)d\xi ,\hfill \end{array}$$

where $v_2\left(\xi \right)=\varpi \left(\xi \right)u\left(\xi \right)-\varpi \left(t\right)u\left(t\right)$. Hence,

$$\begin{array}{ccc}\hfill g_2\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\ell -1}{E}_{r,\ell }^{\wp }[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]{\left(v_2{\left(\xi \right)}^{T}P\dot{v_2}\left(\xi \right)\right)}^{\prime }d\xi ,\hfill \end{array}$$

Integrating by parts, for $\ell =\wp =1$, we obtain the following:

$$\begin{array}{ccc}\hfill g_2\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}\{E_{r,1}^1[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right){|}_{\xi =a}^{\xi =t}\hfill \\ & & -m{\lambda }_{\alpha ,⋏}^{\hslash }{\int }_a^t{\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{m-1}{E}_{r,r+1}^2[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)d\xi \}.\hfill \end{array}$$

Since ${\displaystyle \underset{\xi \to t}{lim}{E}_{r,1}^1[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)=v_2{\left(t\right)}^{T}P{v}_2\left(t\right)=0}$, we have the following:

$$\begin{array}{ccc}\hfill g_2\left(t\right)& =& -{\displaystyle \frac{\aleph (\alpha +⋏-1)}{(2-\alpha -⋏)\varpi \left(t\right)}}\{E_{r,1}^1[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_2{\left(a\right)}^{T}P{v}_2\left(a\right)\hfill \\ & & +m{\lambda }_{\alpha ,⋏}^{\hslash }{\int }_a^t{\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{m-1}{E}_{r,r+1}^2[-{\lambda }_{\alpha ,⋏}^{\hslash }{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^m]v_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)d\xi \}.\hfill \end{array}$$

On the other hand, the expression of $g_2\left(t\right)$ for $\ell =⋏=1-\alpha$ is as follows:

$$\begin{array}{ccc}\hfill g_2\left(t\right)& =& {\displaystyle \frac{1}{\Gamma (1-\alpha )\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha }2v_2{\left(\xi \right)}^{T}P\dot{v_2}\left(\xi \right)d\xi ,\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (1-\alpha )\varpi \left(t\right)}}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha }{\left(v_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)\right)}^{\prime }d\xi ,\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (1-\alpha )\varpi \left(t\right)}}\{{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha }{v}_2{\left(\xi \right)}^{T}P{v}_2{\left(\xi \right)|}_{\xi =a}^{\xi =t}\hfill \\ & & -\alpha {\int }_a^t{\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha -1}{v}_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)d\xi \}.\hfill \end{array}$$

Since $\ell =1-\alpha >0$, we have $\alpha <1$. It follows from the Hospital’s rule that

$${\displaystyle \underset{\xi \to t}{lim}\frac{v_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)}{{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\alpha }}=\underset{\xi \to t}{lim}\frac{2v_2{\left(\xi \right)}^{T}P{\dot{v}}_2\left(\xi \right)}{-\alpha {\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\alpha -1}}=0.}$$

Hence,

$$\begin{array}{ccc}\hfill g_2\left(t\right)& =& {\displaystyle \frac{-{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{-\alpha }}{\Gamma (1-\alpha )\varpi \left(t\right)}}{v}_2{\left(a\right)}^{T}P{v}_2\left(a\right)\hfill \\ & & -{\displaystyle \frac{\alpha }{\Gamma (1-\alpha )\varpi \left(t\right)}}{\int }_a^t{\chi }^{\prime }\left(\xi \right){\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{-\alpha -1}{v}_2{\left(\xi \right)}^{T}P{v}_2\left(\xi \right)d\xi .\hfill \end{array}$$

Therefore, $g_2\left(t\right)\le 0$ for all $t\ge a$ when $\ell =\wp =1$ or $\ell =⋏=1-\alpha$, which implies that

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }\left(\varpi \left(t\right)u{\left(t\right)}^{T}Pu\left(t\right)\right)\le 2\varpi \left(t\right)u{\left(t\right)}^{T}P{}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right).}$$

The proof is completed. □

Remark 3.

Theorem 2 extends and generalizes many recent results reported in previous studies. More precisely, Theorem 2 coincides with

(i) 

Lemma 2.5 of [49] when $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)={\displaystyle \frac{t^{\rho }}{\rho }}$ with $\rho >0$;

(ii) 

Corollary 1 of [23] when $\ell =⋏=1-\alpha$, $\varpi \left(t\right)=1$ and $P=I_n$;

(iii) 

Lemma 1 of [50] when $\ell =\wp =⋏=\hslash =1$, $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t$, which covers the results presented in [18,20,51].

Theorem 3.

(Fractional comparison principle). Let $u\left(t\right)$ and $v\left(t\right)$ be two functions defined on the interval $[t_0,+\infty )$ with ${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right)\ge {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }v\left(t\right)}$ and $u\left(t_0\right)\ge v\left(t_0\right)$. Then $u\left(t\right)\ge v\left(t\right)$ for all $t\ge t_0$.

Proof. 

Since ${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }u\left(t\right)\ge {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,m,\ell }v\left(t\right)}$, it follows from Lemma 1 that

$$u\left(t\right)-{\displaystyle \frac{u\left(t_0\right)\varpi \left(t_0\right)}{\varpi \left(t\right)}}\ge v\left(t\right)-{\displaystyle \frac{v\left(t_0\right)\varpi \left(t_0\right)}{\varpi \left(t\right)}},$$

which leads to

$$u\left(t\right)\ge v\left(t\right)+{\displaystyle \frac{\varpi \left(t_0\right)\left(u\left(t_0\right)-v(t_0\right)}{\varpi \left(t\right)}}.$$

As $u\left(t_0\right)\ge v\left(t_0\right)$, we have $u\left(t\right)\ge v\left(t\right)$, for all $t\ge t_0$. □

Remark 4.

Theorem 3 extends three results, as follows: The first is in Lemma 6.1 of [29] involving the Caputo fractional derivative; the second is in Theorem 2.4 of [24] involving the Riemann–Liouville fractional derivative, and the third is in Lemma 1 of [28] involving the GHF derivative.

As in [37], the weighted convolution of functions f and g is defined by

$$\begin{array}{ccc}\hfill \left(f{\ast }_{\varpi ,\chi }g\right)\left(t\right)& =& {\displaystyle \frac{1}{\varpi \left(t\right)}}{\int }_a^t\varpi \left(\chi \left(t\right)+\chi \left(a\right)-\chi \left(\xi \right)\right)f\left({\chi }^{-1}\left(\chi \left(t\right)+\chi \left(a\right)-\chi \left(\xi \right)\right)\right)\varpi \left(\xi \right)g\left(\xi \right){\chi }^{\prime }\left(\xi \right)d\xi .\hfill \end{array}$$

Obviously, we have

$${\mathcal{L}}_{\varpi ,\chi }\left\{f{\ast }_{\varpi ,\chi }g\right\}={\mathcal{L}}_{\varpi ,\chi }\left\{f\right\}{\mathcal{L}}_{\varpi ,\chi }\left\{g\right\}.$$

Theorem 4.

Let κ be a constant, $y\left(t\right)$ and $u\left(t\right)$ be two functions such that

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }y\left(t\right)=\kappa y\left(t\right)+u\left(t\right).}$$

(11)

(i) 

If $\ell =\wp =1$, then

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)y\left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}u\left(t\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{a_{\alpha ,⋏}^2}}\left({\displaystyle \frac{E_{r,r}\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-r}}}{\ast }_{\varpi ,\chi }u\left(t\right)\right),\hfill \end{array}$$

where $a_{\alpha ,⋏}=\aleph (\alpha +⋏-1)-\kappa (2-\alpha -⋏)\ne 0$ and $\kappa \ne 0$.

(ii) 

If $\ell =⋏=1-\alpha$, then

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle \frac{\varpi \left(a\right)y\left(a\right)}{\varpi \left(t\right)}}{E}_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]+{\displaystyle \frac{E_{\alpha ,\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-\alpha }}}{\ast }_{\varpi ,\chi }u\left(t\right).\hfill \end{array}$$

Proof. 

By applying the weighed Laplace transform ${\mathcal{L}}_{\varpi ,\chi }$ to both sides of (11) and using Lemma (2), for $\ell =\wp =1$, we obtain the following:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\varpi ,\chi }\left\{y\left(t\right)\right\}\left(s\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)y\left(a\right)s^{r-1}}{[\aleph (\alpha +⋏-1)-\kappa (2-\alpha -⋏)]s^r-\hslash (\alpha +⋏-1)\kappa }}\hfill \\ & & +{\displaystyle \frac{[(2-\alpha -⋏)s^r+\hslash (\alpha +⋏-1)]{\mathcal{L}}_{\varpi ,\chi }\left\{u\left(t\right)\right\}\left(s\right)}{[\aleph (\alpha +⋏-1)-\kappa (2-\alpha -⋏)]s^r-\hslash (\alpha +⋏-1)\kappa }}.\hfill \end{array}$$

Since $a_{\alpha ,⋏}=\aleph (\alpha +⋏-1)-\kappa (2-\alpha -⋏)\ne 0$, we have

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\varpi ,\chi }\left\{y\left(t\right)\right\}\left(s\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)y\left(a\right)}{a_{\alpha ,⋏}}}{\displaystyle \frac{s^{r-1}}{s^r-\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}}+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}{\mathcal{L}}_{\varpi ,\chi }\left\{u\left(t\right)\right\}\left(s\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{a_{p,⋏}^2}}{\displaystyle \frac{1}{s^r-\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}},\hfill \\ & =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)y\left(a\right)}{a_{\alpha ,⋏}}}{\mathcal{L}}_{\varpi ,\chi }\left\{{\varpi }^{-1}\left(t\right)E_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]\right\}\left(s\right)\hfill \\ & & +{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}{\mathcal{L}}_{\varpi ,\chi }\left\{u\left(t\right)\right\}\left(s\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{a_{\alpha ,⋏}^2}}{\mathcal{L}}_{\varpi ,\chi }\{\hfill \\ & & {\displaystyle \frac{E_{r,r}\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-r}}}\}\left(s\right){\mathcal{L}}_{\varpi ,\chi }\left\{u\left(t\right)\right\}\left(s\right).\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)y\left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}u\left(t\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{a_{\alpha ,⋏}^2}}\left({\displaystyle \frac{E_{r,r}\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-r}}}{\ast }_{\varpi ,\chi }u\left(t\right)\right),\hfill \end{array}$$

This proves (i). For $\ell =⋏=1-\alpha$, we have

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\varpi ,\chi }\left\{y\left(t\right)\right\}\left(s\right)& =& \varpi \left(a\right)y\left(a\right){\displaystyle \frac{s^{\alpha -1}}{s^{\alpha }-\kappa }}+{\displaystyle \frac{1}{s^{\alpha }-\kappa }}{\mathcal{L}}_{\varpi ,\chi }\left\{u\left(t\right)\right\}\left(s\right),\hfill \\ & =& \varpi \left(a\right)y\left(a\right){\mathcal{L}}_{\varpi ,\chi }\left\{{\varpi }^{-1}\left(t\right)E_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]\right\}\left(s\right)\hfill \\ & & +{\mathcal{L}}_{\varpi ,\chi }\left\{{\varpi }^{-1}\left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]\right\}{\mathcal{L}}_{\varpi ,\chi }\left\{u\left(t\right)\right\},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle \frac{\varpi \left(a\right)y\left(a\right)}{\varpi \left(t\right)}}{E}_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]+{\displaystyle \frac{E_{\alpha ,\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-\alpha }}}{\ast }_{\varpi ,\chi }u\left(t\right).\hfill \end{array}$$

This proves (ii). □

Corollary 3.

Let $\kappa >0$ and $\vartheta \left(t\right)$ be a function such that

$${\displaystyle {}^{C}D_{0,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\vartheta \left(t\right)\le -\kappa \vartheta \left(t\right).}$$

(12)

(i) 

If $\ell =\wp =1$, then

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & \vartheta \left(0\right)E_r\left({\displaystyle \frac{-\hslash \kappa (\alpha +⋏-1){\left(\chi \left(t\right)-\chi \left(0\right)\right)}^r}{\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏)}}\right).\hfill \end{array}$$

(ii) 

If $\ell =⋏=1-\alpha$, then

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & \vartheta \left(0\right)E_{\alpha }\left(\kappa {\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right).\hfill \end{array}$$

Proof. 

It follows from (12) that there exists a nonnegative function $u\left(t\right)$ such that

$${\displaystyle {}^{C}D_{0,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\vartheta \left(t\right)=-\kappa \vartheta \left(t\right)-u\left(t\right).}$$

According to Theorem 4, for $\ell =\wp =1$, we have the following:

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(0\right)\vartheta \left(0\right)}{\varpi \left(t\right)(\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏\left)\right)}}{E}_r\left({\displaystyle \frac{-\hslash \kappa (\alpha +⋏-1){\left(\chi \left(t\right)-\chi \left(0\right)\right)}^r}{\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏)}}\right)\hfill \\ & & -{\displaystyle \frac{2-\alpha -⋏}{\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏)}}u\left(t\right)\hfill \\ & & -{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{{(\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏))}^2}}({\displaystyle \frac{E_{r,r}[-\frac{\hslash \kappa (\alpha +⋏-1)}{\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏)}{\left(\chi \left(t\right)-\chi \left(0\right)\right)}^r]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{1-r}}}\hfill \\ & & {\ast }_{\varpi ,\chi }u\left(t\right)).\hfill \end{array}$$

Since $u\left(t\right)\ge 0$ and $\varpi \left(0\right)\le \varpi \left(t\right)$ for $t\ge 0$, we get

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & \vartheta \left(0\right)E_r\left({\displaystyle \frac{-\hslash \kappa (\alpha +⋏-1){\left(\chi \left(t\right)-\chi \left(0\right)\right)}^r}{\aleph (\alpha +⋏-1)+\kappa (2-\alpha -⋏)}}\right).\hfill \end{array}$$

Similarly, for $\ell =⋏=1-\alpha$, we obtain the following:

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& =& {\displaystyle \frac{\varpi \left(0\right)\vartheta \left(0\right)}{\varpi \left(t\right)}}{E}_{\alpha }\left(\kappa {\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right)-{\displaystyle \frac{E_{\alpha ,\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-\alpha }}}{\ast }_{\varpi ,\chi }u\left(t\right).\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & \vartheta \left(0\right)E_{\alpha }\left(\kappa {\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right).\hfill \end{array}$$

□

Remark 5.

Corollary 3 includes the result given in [50]; it suffices to take $⋏=\hslash =1$ and $\chi \left(t\right)=t$ in $\left(i\right)$.

Corollary 4.

Let $\kappa \in {\mathbb{R}}^{+}$, $\vartheta \left(t\right)$ and $u\left(t\right)$ be two nonnegative functions such that

$${\displaystyle {}^{C}D_{a,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\vartheta \left(t\right)\le \kappa \vartheta \left(t\right)+u\left(t\right).}$$

(13)

(i) 

If $\ell =\wp =1$, then

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)\vartheta \left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}u\left(t\right)\hfill \\ & & {\displaystyle +\frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{\varpi \left(t\right)a_{\alpha ,⋏}^2}\Gamma \left(r\right)E_{r,r}{\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]}^R{\mathcal{I}}_{a,\varpi ,\chi }^{r}u\left(t\right).}\hfill \end{array}$$

(ii) 

If $\ell =⋏=1-\alpha$, then

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & {\displaystyle \frac{\varpi \left(a\right)\vartheta \left(a\right)}{\varpi \left(t\right)}{E}_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]+\Gamma \left(\alpha \right)E_{\alpha ,\alpha }{\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]}^R{\mathcal{I}}_{a,\varpi ,\chi }^{\alpha }u\left(t\right).}\hfill \end{array}$$

Proof. 

Let ${\displaystyle \tilde{u}\left(t\right)={}^{C}D_{0,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\vartheta \left(t\right)-\kappa \vartheta \left(t\right)}$. By applying Theorem 4, for $\ell =\wp =1$, we have the following:

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)\vartheta \left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}\tilde{u}\left(t\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{a_{\alpha ,⋏}^2}}\left({\displaystyle \frac{E_{r,r}\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-r}}}{\ast }_{\varpi ,\chi }\tilde{u}\left(t\right)\right),\hfill \\ & =& {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)\vartheta \left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}\tilde{u}\left(t\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{\varpi \left(t\right)a_{\alpha ,⋏}^2}}{\int }_a^t{\displaystyle \frac{E_{r,r}\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^r\right]}{{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{1-r}}}\varpi \left(\xi \right)\tilde{u}\left(\xi \right){\chi }^{\prime }\left(\xi \right)d\xi .\hfill \end{array}$$

Since $\tilde{u}\left(t\right)\le u\left(t\right)$, we have

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)\vartheta \left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}u\left(t\right)\hfill \\ & & +{\displaystyle \frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{\varpi \left(t\right)a_{\alpha ,⋏}^2}}{\int }_a^t{\displaystyle \frac{E_{r,r}\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^r\right]}{{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{1-r}}}\varpi \left(\xi \right)u\left(\xi \right){\chi }^{\prime }\left(\xi \right)d\xi ,\hfill \\ & \le & {\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(a\right)\vartheta \left(a\right)}{a_{\alpha ,⋏}\varpi \left(t\right)}}{E}_r\left[{\displaystyle \frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]+{\displaystyle \frac{2-\alpha -⋏}{a_{\alpha ,⋏}}}u\left(t\right)\hfill \\ & & {\displaystyle +\frac{\hslash (\alpha +⋏-1)\aleph (\alpha +⋏-1)}{\varpi \left(t\right)a_{\alpha ,⋏}^2}\Gamma \left(r\right)E_{r,r}{\left[\frac{\hslash \kappa (\alpha +⋏-1)}{a_{\alpha ,⋏}}{\left(\chi \left(t\right)-\chi \left(a\right)\right)}^r\right]}^R{\mathcal{I}}_{a,\varpi ,\chi }^{r}u\left(t\right).}\hfill \end{array}$$

In a similar way, for $\ell =⋏=1-\alpha$, we obtain the following:

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& =& {\displaystyle \frac{\varpi \left(a\right)\vartheta \left(a\right)}{\varpi \left(t\right)}}{E}_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]+{\displaystyle \frac{E_{\alpha ,\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]}{\varpi \left(t\right){\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{1-\alpha }}}{\ast }_{\varpi ,\chi }\tilde{u}\left(t\right),\hfill \\ & =& {\displaystyle \frac{\varpi \left(a\right)\vartheta \left(a\right)}{\varpi \left(t\right)}}{E}_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]+{\displaystyle \frac{1}{\varpi \left(t\right)}}{\int }_a^t{\displaystyle \frac{E_{\alpha ,\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\alpha }\right]}{{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{1-\alpha }}}\varpi \left(\xi \right)\tilde{u}\left(\xi \right){\chi }^{\prime }\left(\xi \right)d\xi ,\hfill \\ & \le & {\displaystyle \frac{\varpi \left(a\right)\vartheta \left(a\right)}{\varpi \left(t\right)}{E}_{\alpha }\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]+\Gamma \left(\alpha \right)E_{\alpha ,\alpha }{\left[\kappa {\left(\chi \left(t\right)-\chi \left(a\right)\right)}^{\alpha }\right]}^R{\mathcal{I}}_{a,\varpi ,\chi }^{\alpha }u\left(t\right).}\hfill \end{array}$$

This completes the proof. □

Remark 6.

Corollary 4 covers the χ-Caputo Bellman–Gronwall inequality presented in Theorem 3 of [23]; it suffices to take $\varpi \left(t\right)=1$ in $\left(ii\right)$.

4. Application

This section presents an application of the main results obtained to investigate a problem in pharmacokinetics, which is a branch of medicine that studies the absorption, distribution, metabolism, and elimination of drugs in a living body.

To describe the dynamics of a drug concentration in a living body, we propose the following mathematical model with linear FDEs involving the GHM fractional derivative:

$$\left\{\begin{array}{c}{\displaystyle {}^{C}D_{0,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }y\left(t\right)=-dy\left(t\right),}\hfill \\ y\left(0\right)=y_0,\hfill \end{array}\right.$$

(14)

where $y\left(t\right)$ represents the drug concentration in the body at time t, d is a positive constant that can be experimentally determined for each drug, and $y_0$ denotes the initial drug dose administered.

The drug concentration model presented by system (14) captures the time evolution of a drug within the body, governed by absorption, distribution, metabolism, and elimination. The GHM fractional derivative used in system (14) accounts for delayed drug absorption, heterogeneous tissue diffusion, and long-term retention, which are commonly observed in real biological systems.

By applying Theorem 4 for the case $\ell =⋏=1-\alpha$, the solution of (14) is given by

$$y\left(t\right)={\displaystyle \frac{\varpi \left(0\right)y_0}{\varpi \left(t\right)}}{E}_{\alpha }\left(-d{\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right).$$

(15)

In this situation, the pharmacokinetics model presented in 2022 by Awadalla et al. [31] for predicting drug concentration levels in human blood over time is a special case of (14). Furthermore, the solution of (14) given by (15) is reduced to that presented in [31] by choosing $\varpi \left(t\right)=1$.

For the situation $\ell =\wp =1$, the application of Theorem 4 yields the solution of (14) as follows:

$$y\left(t\right)={\displaystyle \frac{\aleph (\alpha +⋏-1)\varpi \left(0\right)y_0}{[\aleph (\alpha +⋏-1)+d(2-\alpha -⋏\left)\right]\varpi \left(t\right)}}{E}_r\left({\displaystyle \frac{-d\hslash (\alpha +⋏-1)}{\aleph (\alpha +⋏-1)+d(2-\alpha -⋏)}}{\left(\chi \left(t\right)-\chi \left(0\right)\right)}^r\right).$$

(16)

Let $d=0.5459$, $\varpi \left(t\right)=1$, $\chi \left(t\right)=t+1$, and $\aleph \left(\alpha \right)=1$. Figure 1 and Figure 2 present the graphs of solutions of the pharmacokinetics model (14) for different values of order $\alpha$.

Based on Figure 1 and Figure 2, we observe that as the fractional order $\alpha$ decreases, the memory effect becomes more pronounced, leading to a slower decay of drug concentration. This behavior reflects the nonlocal and history-dependent nature of fractional-order models, which more accurately capture the dynamics of real biological systems compared to classical models. Additionally, the slower decay of drug concentration at lower fractional orders may correspond to slow diffusion or delayed drug absorption due to complex bio-macromolecular interactions.

In medicine, various hormone therapies are employed to either lower estrogen levels or balance estrogen with other hormones. One such drug, Tamoxifen, is primarily used to treat hormone-dependent breast cancer by blocking estrogen receptors, which prevents cancer cells from utilizing estrogen for growth. Apart from cancer treatment, Tamoxifen is also applied in managing other conditions associated with excess estrogen. To investigate the impact of this drug on estrogen dynamics, we consider the following model:

$$\left\{\begin{array}{c}{\displaystyle {}^{C}D_{0,\wp ,\hslash ,\varpi ,\chi }^{\alpha ,⋏,r,r,\ell }\mathcal{E}\left(t\right)=(1-k)s-\nu \mathcal{E}\left(t\right),}\hfill \\ \mathcal{E}\left(0\right)={\mathcal{E}}_0,\hfill \end{array}\right.$$

(17)

where $\mathcal{E}\left(t\right)$ represents the estrogen levels at time t. The parameter s is the source rate of estrogen, while $\nu$ denotes the rate at which estrogen is washed out from the body system. Furthermore, the parameter k represents the drug efficacy.

It follows from Theorem 4 for the case $\ell =⋏=1-\alpha$ that the solution of (17) is given by

$$\mathcal{E}\left(t\right)={\displaystyle \frac{\varpi \left(0\right){\mathcal{E}}_0}{\varpi \left(t\right)}}{E}_{\alpha }\left(-\nu {\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right)+{\displaystyle \frac{(1-k)s}{\varpi \left(t\right)}}{\int }_a^t{\displaystyle \frac{E_{\alpha ,\alpha }[-\nu {\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{\alpha }]}{{\left(\chi \left(t\right)-\chi \left(\xi \right)\right)}^{1-\alpha }}}\varpi \left(\xi \right)\left(\xi \right){\chi }^{\prime }\left(\xi \right)d\xi ,$$

(18)

When the weight function $\varpi \left(t\right)$ is constant, (18) is as follows

$$\mathcal{E}\left(t\right)={\displaystyle \frac{(1-k)s}{\nu }}+\left({\mathcal{E}}_0-{\displaystyle \frac{(1-k)s}{\nu }}\right)E_{\alpha }\left(-\nu {\left(\chi \left(t\right)-\chi \left(0\right)\right)}^{\alpha }\right).$$

(19)

Similar to the first application example, let $\varpi \left(t\right)=1$ and $\chi \left(t\right)=t+1$. It is obvious that ${\mathcal{E}}^{*}={\displaystyle \frac{(1-k)s}{\nu }}$ is the unique globally stable steady state of model (17), which represents the total amount of estrogen in the human body. However, normal estrogen levels vary by sex and life stage. In premenopausal women, levels fluctuate during the menstrual cycle, typically ranging from 15 to 350 pg/mL, while postmenopausal women have significantly lower levels, often below 10 pg/mL. In men, estrogen levels remain relatively stable throughout life (10–40 pg/mL) [52,53,54]. In addition, the washout rate of estrogen by the body is equal to $0.97$ day⁻¹ [55]. Therefore, we obtain s between 0 and $339.5$ pg/mL/day. A complete list of parameters and their estimated values for model (17) is presented in

Table 1. Parameters and values of model (17).

Parameters	Meaning	Value	Source
s	Source rate of estrogen	0–339.5 pg/mL/day	Calculated
$\nu$	Washout rate of estrogen	$0.97$ day−1	[55]
	by the body
k	Drug efficacy	0–1	Variable

.

Figure 3 shows that hormone therapy has a significant impact on estrogen dynamics, allowing high estrogen levels to be reduced to below 100 pg/mL when its efficacy exceeds $60\%$.

5. Conclusions

In this work, we proposed an analytical framework for studying the qualitative behavior of solutions of FDEs based on the GHM fractional derivative, which includes a broad class of fractional operators with both singular and non-singular kernels. We established a series of comparison principles and generalized Lyapunov-type inequalities that extend numerous results from previous studies. The research results were applied to some linear pharmacokinetic models in the medical field.

Based on the above results and due to the core advantages associated with the GHM fractional derivative, future research directions will focus on the stability analysis and the existence of solutions for FDEs involving the novel GHM fractional derivative, as well as the extension of application to modeling phenomena arising from ecology [56,57] and other medical fields, such as tumor growth modeling, physiological signal analysis, and infectious disease dynamics.