Mittag–Leffler Stability of a Switched Fractional Gene Regulatory Network Model with a Short Memory

Abstract

A model of gene regulatory networks with generalized Caputo fractional derivatives with respect to another function is set up in this paper. The main characteristic of the model is the presence of a switching rule, which changes at certain times at both the lower limit of the applied fractional derivative and the right-side part of the equations. This gives the opportunity for better and more adequate modeling of the problem. Mittag–Leffler-type stability is defined for the model and studied with two types of Lyapunov functions. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are given, and two types of sufficient conditions are obtained. An example is provided to illustrate our theoretical results and the influences of the type of fractional derivative as well the switching rule on the stability behavior of the equilibrium.

1. Introduction

Switched systems for integer-order differential equations have been studied by many authors (see, for example, [1,2,3,4,5]), and several qualitative properties for linear ordinary differential systems have been studied, such as, optimal control ([6]), common linear co-positive Lyapunov functions and their existence ([7]), stability ([8,9,10]), and finite time boundedness ([11]). Also, recently, some discrete nonlinear switched systems have been studied (see, for example, [12,13]).

Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering for modeling some physical phenomena having memory and genetic characteristics (see, for example, [14]). Also, fractional derivatives and fractional differential equations have been applied in modeling switched processes (see, for example, [15,16]). At the same time, the application of fractional derivatives to dynamic models is more complicated than the ordinary derivative, and this is mainly connected with the memory property of the fractional derivative and the presence of a lower limit in them (see, for example, the classical books [17,18,19]). In some papers, the lower limit of the applied Caputo fractional derivative in the switched system is fixed at the initial time (see, for example, [20,21], and, unfortunately, the integral presentation used is incorrect). In [22,23], the sequence of switching times is initially given, and Caputo-type fractional derivatives are applied with a changeable lower limit at the switching times (so-called short memory). The concept of short memory was proposed and applied in [24,25] and subsequently adopted by other authors. In [26] a short-memory nonlinear switched system and a Markov switching rule are considered.

In our paper, we adopt the idea for short memory in switched systems with a generalized Caputo fractional derivative with respect to another function (CFDF) (see [23]) and apply it to study stability properties of a model for a gene regulatory network. The gene regulatory network consists of a number of genes interacting with proteins. Mathematical models of gene regulatory networks are described and studied in several papers (see, for example, for fractional models [27]). By a switched system, we mean a dynamic system consisting of a family of continuous-time subsystems of fractional differential equations and a rule that orchestrates the switching among them. The switching rule is given initially, and when it is activated, an appropriate subsystem from the given family is chosen. We assume that one, and only one, subsystem is active at each time instant. We consider the case of short memory, i.e., the lower limit of the considered fractional derivative is changed and equal to the time when the switching rule is activated. First, we describe in detail the presence of the switching rule in the studied model and describe the solution. Then, we define an equilibrium of the model. We focus on the stability of the equilibrium and use two types of Lyapunov functions—with squares and with absolute values. These results are the basis of the defined two types of generalized Mittag–Leffler stability, and we obtain two types of sufficient conditions. An example illustrates the obtained results.

The main contributions of this paper could be summarized as follows:

-

A generalized Caputo fractional derivative with respect to another function is used to model the dynamics of gene regulatory networks;

-

A piecewise constant switching rule is applied to model the changing of dynamics at some certain points;

-

Switching times in the model could be finite or infinite;

-

Short memories are considered, i.e., the lower limits of the applied fractional derivative are changing at any switching time;

-

An equilibrium of the given switched model is defined;

-

Two types of Mittag–Leffler stability of the equilibrium are defined depending significantly on the type of norm used;

-

Mittag–Leffler-type stability is studied with two types of Lyapunov functions—a quadratic one and one with absolute values.

2. Fractional Integrals and Derivatives with Respect to Another Function

Let $a,T:0\le a<T\le \infty$. We will use the following set of functions:

$$\mathcal{P}(a,T)=\{\psi \in C^1([a,T],[0,\infty )):{\psi }^{\prime }\left(t\right)>0,t\in (a,T]\}.$$

Definition 1

(Definition 1 [28]). Let $q>0,\rho >0,$ and the function $\psi \in \mathcal{P}(a,T)$. Where the integral exists, the generalized fractional integral with respect to another function (FIF) of the function $\upsilon :[a,T]\to \mathbb{R}$ is defined by

$${}^{\rho }I_a^{q,\psi }\upsilon \left(t\right)={\displaystyle \frac{{\rho }^{1-q}}{\Gamma \left(q\right)}}{\int }_a^t{\left(\psi \right(s\left)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right){\displaystyle \frac{\upsilon \left(s\right)}{{(\psi {\left(t\right)}^{\rho }-\psi {\left(s\right)}^{\rho })}^{1-q}}}ds,t\in (a,T].$$

Definition 2

([28]). Let $q\in (0,1),\rho >0$, and the function $\psi \in \mathcal{P}(a,T)$. Where the integral exists, the generalized Caputo fractional derivative with respect to another function (CFDF) of the function $\upsilon :[a,T]\to \mathbb{R}$ is defined by

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_a^{q,\psi }\upsilon \left(t\right)={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{{\upsilon }^{\prime }\left(s\right)}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}ds,t\in (a,T].\hfill \end{array}$$

(1)

In connection with Definition 2, we will introduce the following set of functions:

$$\begin{array}{cc}& C^q((a,T],\mathbb{R},\psi ,\rho )=\{\upsilon \in C((a,T],\mathbb{R}):{\upsilon }^{\prime }\mathrm{exists} \mathrm{almost} \mathrm{everywhere} \mathrm{in}(a,T],\hfill \\ & \mathrm{and} \mathrm{there} \mathrm{exists} \mathrm{CFDF} {}_C{}^{\rho }I_a^{q,\psi }\upsilon \left(t\right)fort\in (a,T]\}.\hfill \end{array}$$

Remark 1.

The above-defined CFDF generalizes the following fractional derivatives:

-

The Caputo fractional derivative if $\rho =1,\psi \left(t\right)\equiv t$;

-

The Caputo fractional derivative with respect to another function if $\rho =1$;

-

The Katugampola-type generalized fractional derivative if $\psi \left(t\right)\equiv t$.

Lemma 1

([23]). Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$, and $\lambda \in \mathbb{R}$ be a constant. The solution of the scalar linear fractional initial value problem with CFDF

$${}_{\rho }{}^{C}D_a^{q,\psi }\upsilon \left(t\right)=\lambda \upsilon \left(t\right),fort\in (a,T],and\upsilon \left(a\right)={\upsilon }_0$$

is the function

$$\upsilon \left(t\right)={\upsilon }_0{E}_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right),t\in [a,T].$$

(2)

Lemma 2

([23]). Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$, $\upsilon :[a,T]\to \mathbb{R}$, $a<T\le \infty$, and there exists a point $\xi \in (a,T]$ such that $\upsilon \left(\xi \right)=0$ and $\upsilon \left(t\right)<0$ for $t\in [a,\xi )$, and the CFDF ${}_{\rho }{}^{C}D_a^{q,\psi }{\upsilon \left(t\right)|}_{t=\xi }$ exists. Then, if ${lim}_{s\to \xi -}\left({\displaystyle \frac{v\left(s\right)}{{({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}\right)=0$, we have ${}_{\rho }{}^{C}D_a^{q,\psi }{\upsilon \left(t\right)|}_{t=\xi }\ge 0$.

Remark 2.

In Lemma 2, if $\upsilon \in C^1([a,T],\mathbb{R})$ then L’Hopital’s rule guarantees that

$$\underset{s\to \xi -}{lim}\left({\displaystyle \frac{\upsilon \left(s\right)}{{({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}\right)=\underset{s\to \xi -}{lim}{\upsilon }^{\prime }\left(s\right){\displaystyle \frac{{({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{1-q}}{-q\rho {\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)}}=0.$$

Note that from $\xi >a$ and ${\psi }^{\prime }\left(s\right)>0,s\in [a.\xi ]$, it follows that $\psi \left(\xi \right)>0$.

Of course, one could also put other conditions (other than $\upsilon \in C^1([a,T],\mathbb{R})$); for example, instead, one could assume ${lim}_{s\to \xi -}{\upsilon }^{\prime }\left(s\right)$ exists and is a real number to guarantee that this limit is zero.

Lemma 3.

Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$, and $\lambda \in \mathbb{R}$ be a constant. The function $\upsilon \in C^q((a,T),\mathbb{R},\psi ,\rho )$ satisfies the scalar linear fractional differential inequalities with CFDF

$${}_{\rho }{}^{C}D_a^{q,\psi }\upsilon \left(t\right)<\lambda \upsilon \left(t\right),fort\in (a,T],and\upsilon \left(a\right)<{\upsilon }_0.$$

(3)

Then,

$$\upsilon \left(t\right)<{\upsilon }_0{E}_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right),t\in [a,T].$$

(4)

Proof. 

Let ${\omega }^{*}\left(t\right)={\upsilon }_0{E}_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right),t\in [a,T]$. Note that ${\omega }^{*}\in C^q((a,T),\mathbb{R},\psi ,\rho )$, and $\upsilon \left(a\right)\le {\upsilon }_0={\omega }^{*}\left(a\right)$.

We will prove the inequality

$$\upsilon \left(t\right)<{\omega }^{*}\left(t\right)\mathrm{for}t\in [a,T].$$

(5)

Assume the contrary, i.e., there exists a point $\eta \in (a,T]$ such that $\upsilon \left(t\right)<{\omega }^{*}\left(t\right)$ on $[a,\eta )$ and $\upsilon \left(\eta \right)={\omega }^{*}\left(\eta \right)$.

Consider the function $y^{*}\left(t\right)=\upsilon \left(t\right)-{\omega }^{*}\left(t\right)\in C^q((a,T),\mathbb{R},\psi ,\rho )$. Then, $y^{*}\left(\eta \right)=0$, $y^{*}\left(t\right)<0$ for $t\in [a,\eta )$, and the CFDF ${}_{\rho }{}^{C}D_a^{q,\psi }{y}^{*}{\left(t\right)|}_{t=\eta }$ exists. All the conditions in Lemma 2 are satisfied for the point $\xi =\eta$ and the function $y^{*}\left(t\right),t\in [a,\eta ]$. According to Lemma 2 and Remark 2 applied to the function $y^{*}(.)\in C^1((a,T],\mathbb{R})$ we have

$${}_{\rho }{}^{C}D_0^{q,\psi }(\upsilon \left(t\right)-{\omega }^{*}\left(t\right)){|}_{t=\eta }\ge 0.$$

(6)

From (3), we have

$${}_{\rho }{}^{C}D_0^{q,\psi }(\upsilon \left(t\right)-{\omega }^{*}\left(t\right)){|}_{t=\eta }={}_{\rho }{}^{C}D_0^{q,\psi }\upsilon \left(t\right){{|}_{t=\eta }-{}_{\rho }{}^{C}D_0^{q,\psi }{\omega }^{*}\left(t\right)|}_{t=\eta }<\lambda \upsilon \left(\eta \right)-\lambda {\omega }^{*}\left(\eta \right)=0.$$

(7)

Inequality (7) contradicts (6). The obtained contradiction proves inequality (5) on $[a,T]$. □

Lemma 4.

Let $\psi \in \mathcal{P}(a,T)$, with $a,T\in \mathbb{R},T\le \infty$ (if $T=\infty$ then the interval is open) and $q\in (0,1),\rho >0$ be two reals, and the function $\varpi \in {\bigcup }_{k=0}^m{C}^q(({\tau }_k,{\tau }_{k+1}],\mathbb{R},\psi ,\rho )$ satisfies the nonlinear scalar fractional differential inequalities with changeable right-hand sides

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }\varpi \left(t\right)<{\lambda }_k\varpi \left(t\right),fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m\hfill \\ & \varpi ({\tau }_k-0)=\varpi ({\tau }_k+0),k=1,2,\dots ,m,\hfill \\ & \varpi \left({\tau }_0\right)<A,\hfill \end{array}$$

(8)

where ${\left\{{\lambda }_k\right\}}_{k=0}^m$ is a given sequence of real constants and $A\in \mathbb{R}$.

Then,

$$\begin{array}{cc}& \varpi \left(t\right)<A\left(\prod _{i=0}^{k-1}{E}_q\left({\lambda }_i{\left({\displaystyle \frac{\psi {\left({\tau }_{i+1}\right)}^{\rho }-{\left(\psi \left({\tau }_i\right)\right)}^{\rho }}{\rho }}\right)}^q\right)\right)E_q\left({\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q\right)\hfill \\ & fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m.\hfill \end{array}$$

(9)

Proof. 

We will use induction to prove the claim.

Case 1.

Consider the interval $[{\tau }_0,{\tau }_1]$. Define the function ${\omega }_0^{*}\left(t\right)=A{E}_q\left({\lambda }_0{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_0\right)\right)}^{\rho }}{\rho }}\right)}^q\right)$ for $t\in [{\tau }_0,{\tau }_1]$. Note that ${\omega }_0^{*}\in C^q(({\tau }_0,{\tau }_1),\mathbb{R},\psi ,\rho )$, $\varpi \left({\tau }_0\right)<A={\omega }_0^{*}\left({\tau }_0\right)$ and ${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }{\omega }_0^{*}\left(t\right)={\lambda }_0{\omega }_0^{*}\left(t\right),t\in ({\tau }_0,{\tau }_1]$.

We will prove the inequality

$$\varpi \left(t\right)<{\omega }_0^{*}\left(t\right)fort\in [{\tau }_0,{\tau }_1].$$

(10)

Assume the contrary, i.e., there exists a point $\eta \in ({\tau }_0,{\tau }_1]$ such that $\varpi \left(t\right)<{\omega }^{*}\left(t\right)$ on $[{\tau }_0,\eta )$ and $\varpi \left(\eta \right)={\omega }^{*}\left(\eta \right)$.

Consider the function $y^{*}\left(t\right)=\varpi \left(t\right)-{\omega }_0^{*}\left(t\right)\in C^q(({\tau }_0,{\tau }_1),\mathbb{R},\psi ,\rho )$. Then, $y^{*}\left(\eta \right)=0$, $y^{*}\left(t\right)<0$ for $t\in [{\tau }_0,\eta )$, and the CFDF ${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }{y}^{*}{\left(t\right)|}_{t=\eta }$ exists. All the conditions in Lemma 2 are satisfied for the points $\eta ,a={\tau }_0$ and the function $y^{*}\left(t\right),t\in [{\tau }_0,\eta ]$. According to Lemma 2 and Remark 2 applied to the function $y^{*}(.)\in C^1((t_0.t_1],\mathbb{R})$ with $a={\tau }_0,T={\tau }_1$, we have

$${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }(\varpi \left(t\right)-{\omega }_0^{*}\left(t\right)){|}_{t=\eta }\ge 0.$$

(11)

From the first inequality (8), for $t=\eta$ and $k=0$, we have

$${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }(\varpi \left(t\right)-{\omega }_0^{*}\left(t\right)){|}_{t=\eta }<{\lambda }_0\varpi \left(\eta \right)-{\lambda }_0{\omega }_0^{*}\left(\eta \right)=0.$$

(12)

Inequality (12) contradicts (11). The obtained contradiction proves inequality (10) on $[{\tau }_0,{\tau }_1]$; therefore,

$$\varpi \left(t\right)<A{E}_q\left({\lambda }_0{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_0\right)\right)}^{\rho }}{\rho }}\right)}^q\right),t\in [{\tau }_0,{\tau }_1],$$

(13)

i.e., inequality (9) holds on $[{\tau }_0,{\tau }_1]$.

Case 2.

Consider the interval $({\tau }_1,{\tau }_2]$.

Define the function ${\omega }_1^{*}\left(t\right)=A{E}_q\left({\lambda }_0{\left({\displaystyle \frac{\psi {\left({\tau }_1\right)}^{\rho }-{\left(\psi \left({\tau }_0\right)\right)}^{\rho }}{\rho }}\right)}^q\right)E_q\left({\lambda }_1{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_1\right)\right)}^{\rho }}{\rho }}\right)}^q\right)$ for $t\in [{\tau }_1,{\tau }_2]$. Then, ${\omega }_1^{*}\in C^q(({\tau }_1,{\tau }_2),\mathbb{R},\psi ,\rho )$, and from inequality (13) with $t={\tau }_1$, we get $\varpi \left({\tau }_1\right)=\varpi ({\tau }_1+0)=\varpi ({\tau }_1-0)<A{E}_q\left({\lambda }_0{\left({\displaystyle \frac{\psi {\left({\tau }_1\right)}^{\rho }-{\left(\psi \left({\tau }_0\right)\right)}^{\rho }}{\rho }}\right)}^q\right)={\omega }_1^{*}\left({\tau }_1\right).$ From the definition of the function ${\omega }_1^{*}(.)$ and the properties of the Mittag–Leffler function, we have ${}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }{\omega }_1^{*}\left(t\right)={\lambda }_{1}A{E}_q\left({\lambda }_0{\left({\displaystyle \frac{\psi {\left({\tau }_1\right)}^{\rho }-{\left(\psi \left({\tau }_0\right)\right)}^{\rho }}{\rho }}\right)}^q\right)E_q\left({\lambda }_1{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_1\right)\right)}^{\rho }}{\rho }}\right)}^q\right)$$={\lambda }_1{\omega }_1^{*}\left(t\right)$.

We will prove the inequality

$$\varpi \left(t\right)<{\omega }_1^{*}\left(t\right)\mathrm{for}t\in [{\tau }_1,{\tau }_2].$$

(14)

Assume the contrary, i.e., there exists a point $\eta \in ({\tau }_1,{\tau }_2]$ such that $\varpi \left(t\right)<{\omega }_1^{*}\left(t\right)$ on $[{\tau }_1,\eta )$ and $\varpi \left(\eta \right)={\omega }_1^{*}\left(\eta \right)$.

From Lemma 2 applied to the function $\varpi (.)-{\omega }_1^{*}(.)$ with $a={\tau }_1,T={\tau }_2$, and Remark 2, we have ${}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }(\varpi \left(t\right)-{\omega }^{*}\left(t\right)){|}_{t=\eta }\ge 0$.

According to the first inequality (8), for $t=\eta$ and $k=1$, we have

$${}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }(\varpi \left(t\right)-{\omega }_1^{*}\left(t\right)){|}_{t=\eta }={}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }\varpi \left(t\right){{|}_{t=\eta }-{}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }{\omega }_1^{*}\left(t\right)|}_{t=\eta }<{\lambda }_1\varpi \left(\eta \right)-{\lambda }_1{\omega }_1^{*}\left(\eta \right)=0.$$

The obtained contradiction proves inequality (14) on $[{\tau }_1,{\tau }_2]$, i.e., the inequality (9) holds for $k=1$.

Case 3.

Assume that there exists an integer $j\ge 2$ such that inequality (9) holds for $k=j$.

Consider the interval $[{\tau }_{j+1},{\tau }_{j+2}].$

Consider the function

$$\begin{array}{cc}{\omega }_{j+1}^{*}\left(t\right)\hfill & =A\left(\prod _{i=0}^j{E}_q\left({\lambda }_i{\left({\displaystyle \frac{\psi {\left({\tau }_{i+1}\right)}^{\rho }-{\left(\psi \left({\tau }_i\right)\right)}^{\rho }}{\rho }}\right)}^q\right)\right)E_q\left({\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_{j+1}\right)\right)}^{\rho }}{\rho }}\right)}^q\right),\hfill \\ \hfill & t\in [{\tau }_{j+1},{\tau }_{j+2}].\hfill \end{array}$$

Then, ${\omega }_{j+1}^{*}\in C^q(({\tau }_{j+1},{\tau }_{j+2}),\mathbb{R},\psi ,\rho )$, and from the assumption about inequality (9) with $t={\tau }_{j+1}$, we get $\varpi \left({\tau }_{j+1}\right)<A\left({\prod }_{i=0}^j{E}_q\left({\lambda }_i{\left({\displaystyle \frac{\psi {\left({\tau }_{i+1}\right)}^{\rho }-{\left(\psi \left({\tau }_i\right)\right)}^{\rho }}{\rho }}\right)}^q\right)\right)={\omega }_{j+1}^{*}\left({\tau }_{j+1}\right).$ From the definition of the function ${\omega }_{j+1}^{*}(.)$ and the properties of the Mittag–Leffler function, we have ${}_{\rho }{}^{C}D_{{\tau }_{j+1}}^{q,\psi }{\omega }_{j+1}^{*}\left(t\right)={\lambda }_{j+1}{\omega }_{j+1}^{*}\left(t\right),t\in [{\tau }_{j+1},{\tau }_{j+2}]$.

We will prove the inequality

$$\varpi \left(t\right)<{\omega }_{j+1}^{*}\left(t\right)\mathrm{for}t\in [{\tau }_{j+1},{\tau }_{j+2}].$$

(15)

Assume the contrary, i.e., there exists a point $\eta \in ({\tau }_{j+1},{\tau }_{j+2}]$ such that $\varpi \left(t\right)$$<{\omega }_{j+1}^{*}\left(t\right)$ on $[{\tau }_{j+1},\eta )$ and $\varpi \left(\eta \right)={\omega }_{j+1}^{*}\left(\eta \right)$.

From Lemma 2 applied to the function $\varpi (.)-{\omega }_{j+1}^{*}(.)$ with $a={\tau }_{j+1},T={\tau }_{j+2}$, and Remark 2, we have ${}_{\rho }{}^{C}D_{{\tau }_{j+1}}^{q,\psi }(\varpi \left(t\right)-{\omega }_{j+1}^{*}\left(t\right)){|}_{t=\eta }\ge 0.$

According to (8), for $t=\eta$ and $k=j+1$, we have

$${}_{\rho }{}^{C}D_{{\tau }_{j+1}}^{q,\psi }(\varpi \left(t\right)-{\omega }_{j+1}^{*}\left(t\right)){|}_{t=\eta }<{\lambda }_{j+1}\varpi \left(\eta \right)-{\lambda }_{j+1}{\omega }_{j+1}^{*}\left(\eta \right)=0.$$

The obtained contradiction proves inequality (15) on $[{\tau }_{j+1},{\tau }_{j+2}]$, i.e., the inequality (9) holds for $k=j+1$.

□

Lemma 5.

Let $q\in (0,1),\rho >0$ be two reals, and $\psi \in \mathcal{P}(a,T)$. Let the function $u\in C^q([a,T],\mathbb{R},\psi ,\rho )$ with $a,T\in \mathbb{R},T\le \infty$ (if $T=\infty$ then the interval is open) be such that $u^2\in C^q((a,T],\mathbb{R},\psi ,\rho )$.

Then,

$${}_{\rho }{}^{C}D_a^{q,\psi }{u}^2\left(t\right)\le 2u\left(t\right){}_{\rho }{}^{C}D_a^{q,\psi }u\left(t\right),t\in (a,T].$$

(16)

Proof. 

From (1) and integration by parts, we have that for any $t\in (a,T]$,

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_a^{q,\psi }{u}^2\left(t\right)-2u\left(t\right)\left({}_{\rho }{}^{C}D_a^{q,\psi }u\left(t\right)\right)\hfill \\ & ={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{2u\left(s\right)u^{\prime }\left(s\right)}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}ds-2u\left(t\right){\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{u^{\prime }\left(s\right)}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}ds\hfill \\ & =2{\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{(u\left(s\right)-u\left(t\right))u^{\prime }\left(s\right)}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}ds={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{1}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}d{(u\left(s\right)-u\left(t\right))}^2\hfill \\ & ={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\displaystyle \frac{{(u\left(s\right)-u\left(t\right))}^2}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}{|}_{s=a}^{s=t}-{\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{(u\left(s\right)-u\left(t\right))}^{2}d{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{-q}.\hfill \end{array}$$

(17)

Using L’Hopital’s rule, we obtain the following:

Case 1.

Let $\rho \in (0,1]$. Then,

$$\begin{array}{cc}& \underset{s\to t^{-}}{lim}{\displaystyle \frac{{[u\left(s\right)-u\left(t\right)]}^2}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}\hfill \\ & =\underset{s\to t^{-}}{lim}{\displaystyle \frac{2u^{\prime }\left(s\right)(u\left(s\right)-u\left(t\right))}{q\rho {\psi }^{\prime }\left(s\right)}}{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{1-q}{\left(\psi \left(s\right)\right)}^{1-\rho }\hfill \\ & =0.\hfill \end{array}$$

(18)

Case 2.

Let $\rho >1$. Then, similar to (18), we get

$$\begin{array}{cc}& \underset{s\to t^{-}}{lim}{\displaystyle \frac{{[u\left(s\right)-u\left(t\right)]}^2}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}=\underset{s\to t^{-}}{lim}{\displaystyle \frac{2u^{\prime }\left(s\right)(u\left(s\right)-u\left(t\right))}{q\rho {({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{q-1}{\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)}}\hfill \\ & =\underset{s\to t^{-}}{lim}{\displaystyle \frac{2u^{\prime }\left(s\right)(u\left(s\right)-u\left(t\right))}{q\rho {\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)}}{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{1-q}\hfill \\ & =0.\hfill \end{array}$$

(19)

Applying equalities (18) and (19) to (17), we obtain

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_a^{q,\psi }{u}^2\left(t\right)-2u\left(t\right)\left({}_{\rho }{}^{C}D_a^{q,\psi }u\left(t\right)\right)\hfill \\ & =-{\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\displaystyle \frac{{(u\left(a\right)-u\left(t\right))}^2}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho })}^q}}-{\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{(u\left(s\right)-u\left(t\right))}^{2}d{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{-q}\hfill \\ & \le -{\displaystyle \frac{q{\rho }^{q+1}}{\Gamma (1-q)}}{\int }_a^t{(u\left(s\right)-u\left(t\right))}^2{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{-q-1}{\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

(20)

The integral in (20) has a singularity at the upper limit t, but it is a removable singularity because by L’Hopital’s rule, we obtain the following:

$$\begin{array}{cc}& \underset{s\to t^{-}}{lim}{\displaystyle \frac{{(u\left(s\right)-u\left(t\right))}^2}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{q+1}}}=\underset{s\to t^{-}}{lim}{\displaystyle \frac{2u^{\prime }\left(s\right)(u\left(s\right)-u\left(t\right))}{(q+1)\rho {({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q{\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)}}\hfill \end{array}$$

and

$$\begin{array}{cc}& \underset{s\to t^{-}}{lim}{\displaystyle \frac{2u^{\prime }\left(s\right)(u\left(s\right)-u\left(t\right))}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}=\underset{s\to t^{-}}{lim}{\displaystyle \frac{2\frac{d}{ds}\left(u^{\prime }\left(s\right)(u\left(s\right)-u\left(t\right))\right)}{-q{\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)}}{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^{1-q}=0.\hfill \end{array}$$

Thus, from (20), we obtain inequality (16). □

Lemma 6.

Let $q\in (0,1),\rho >0$ be two reals $q\in (0,1),\rho >0$ be two reals, and $\psi \in \mathcal{P}(a,T)$. The function $u\in C^q([a,T],\mathbb{R},\psi ,\rho )$ has a constant sign on $[a,T]$ with $a,T\in \mathbb{R},T\le \infty$ (if $T=\infty$ then the interval is open).

Then, for any $t\in [a,T]$, the equality

$${}_{\rho }{}^{C}D_a^{q,\psi }|u\left(t\right)|=\left(signu\left(t\right)\right){}_{\rho }{}^{C}D_a^{q,\psi }u\left(t\right)$$

(21)

holds.

Proof. 

For any $t\in [a,T]$, we have ${|u\left(x\right)|}^{\prime }=sign\left(u\left(x\right)\right)u^{\prime }\left(x\right)$. Then, from (1), and noting that the sign of the function $u(.)$ is not changeable on the interval $[a,T]$, we have

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_a^{q,\psi }|u\left(t\right)|={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{sign\left(u\left(s\right)\right)u^{\prime }\left(s\right)}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}ds\hfill \\ & =sign\left(u\left(t\right)\right){\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{u^{\prime }\left(s\right)}{{({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}ds=sign{\left(u\left(t\right)\right)}_{\rho }^C{D}_a^{q,\psi }u\left(t\right).\hfill \end{array}$$

(22)

□

Remark 3.

Note that in Lemma 6, the conditions for the sign of the considered function ($u(.)$) are very important, and they are true for various types of applied fractional derivatives, such as Caputo fractional derivatives, proportional fractional derivatives, and CFDF. If they are not satisfied then the claim and inequality (21) might not be true on the whole interval.

3. Description of the Switched Fractional Gene Regulatory Network Model with CFDF

In this paper, we will use the following notation ${\parallel u\parallel }_n=\sqrt{{\sum }_{i=1}^n{u}_i^2}$, where $u\in {\mathbb{R}}^n,$$u=(u_1,u_2,\dots ,u_n).$

By a switched system, we mean a dynamic system consisting of a family of continuous-time subsystems of fractional differential equations with CFDF, initially given switching times (times at which the systems will be changed eventually from one to another), and a given switching rule that orchestrates the switching among them. In this paper, the dynamic system consists of a family of fractional differential subsystems with CFDF. The switching rule plays an exceedingly crucial role in determining

-

The switching time of the system;

-

Whether the subsystem becomes active after each switching;

-

Whether the lower limit of the applied CFDF is equal to the switching time.

Under the influence of the switching rule, the system will exhibit more intriguing dynamic behavior.

We will assume that the following important condition for the switching system is given:

(H) The piecewise constant switching rule ($\sigma \left(t\right)=C_k$ for $t\in ({\tau }_k,{\tau }_{k+1}],$$k=0,1,2,\dots ,m,$) is given, where $m\le \infty$ is an integer, $C_k,k=0,1,2,\dots ,m$ are non-negative integers, and the points ${\tau }_k,k=1,2,\dots ,m+1$ are given such that ${\tau }_0=0,$${\tau }_k<{\tau }_{k+1},k=0,1,2,\dots ,m$, and if $m=\infty$ then ${lim}_{k\to \infty }{\tau }_k=\infty$.

Remark 4.

In $m<\infty$, we denote ${\tau }_{m+1}=\infty$.

Define the set of all the values of the switching rule ($\sigma (.)$) by

$${\mathcal{M}}_{\sigma }=\{C_j:j=1,2,3,\dots ,m\}.$$

Remark 5.

Note that the switching rule is activated only at its points of discontinuity. If for an integer number $j=0,1,2,\dots ,m-1$, the equality $C_j=C_{j+1}$, i.e., the switching is continuous at the point ${\tau }_{j+1}$ then the switching rule is not activated at time ${\tau }_{j+1}$.

If $m<\infty$ then the switching rule is activated a finite number of times and for the time greater or equal to ${\tau }_m$, the given system is a regular system without any switching.

In this paper, we will assume the switching rule $\sigma (.)$, satisfying the condition (H), is initially given, and we will consider a class of switched fractional-order gene regulatory networks modeled by CFDF (SFGM) as follows:

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{x}_i\left(t\right)=-d_i{x}_i\left(t\right)+\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)+I_i,\hfill \\ & {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{y}_i\left(t\right)=-b_i{y}_i\left(t\right)+c_i{x}_i\left(t\right),\hfill \\ & \mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],i=1,2,\dots ,N,k=0,1,2,\dots ,m,\hfill \\ & x_i\left(0\right)=x_i^0,y_i\left(0\right)=y_i^0,i=1,2,\dots ,N,\hfill \end{array}$$

(23)

where $0<q<1$; $\rho \in (0,1]$; N is the number of nodes; $x_j^0,y_j^0\in \mathbb{R}$; $x_j\left(t\right),y_j\left(t\right),$$j=1,2,\dots ,N,$ denote the concentrations of messenger ribonucleic acid (mRNA) and protein of the jth node at time t, respectively; $d_j>0$ and $b_j>0$ are degradation velocities of mRNA and protein, respectively; $c_j$ is the translation rate; the functions $h_j^{\sigma \left(t\right)}\in C(\mathbb{R},\mathbb{R}),j=1,2,\dots ,N,$ represent the activator initiates of protein and mRNA; and the coupling matrix of the network ($A=\left\{a_{jp}\right\}\in {\mathbb{R}}^{N\times N}$) is described by

$$a_{jp}=\{\begin{array}{cc}-{\gamma }_{jp}\hfill & l\mathrm{is}\mathrm{a}\mathrm{repressor}\mathrm{of}\mathrm{gene}j\hfill \\ 0\hfill & p\mathrm{does}\mathrm{not}\mathrm{regulate}\mathrm{gene}j\hfill \\ {\gamma }_{jp}\hfill & p\mathrm{is}\mathrm{a}\mathrm{initiator}\mathrm{of}\mathrm{gene}j.\hfill \end{array}$$

Let $Z^{\left(0\right)}=(x_1^0,x_2^0,\dots ,x_N^0,y_1^0,y_2^0,\dots ,y_N^0)$.

We will describe the solution of SGFM (23).

Let $t\in ({\tau }_0,{\tau }_1]$. Then, SGFM (23) is reduced to

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }{x}_i\left(t\right)=-d_i{x}_i\left(t\right)+\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_0\right)}\left(y_j\left(t\right)\right)+I_i,\hfill \\ & {}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }{y}_i\left(t\right)=-b_i{y}_i\left(t\right)+c_i{x}_i\left(t\right),\mathrm{for}t\in ({\tau }_0,{\tau }_1],i=1,2,\dots ,N,\hfill \\ & x_i\left(0\right)=x_i^0,y_i\left(0\right)=y_i^0,i=1,2,\dots ,N.\hfill \end{array}$$

(24)

Denote the solution of (24) by

$$U^{\left(0\right)}\left(t\right)=(U_1^{\left(0\right)}\left(t\right),U_2^{\left(0\right)}\left(t\right),\dots ,U_N^{\left(0\right)}\left(t\right),U_{N+1}^{\left(0\right)}\left(t\right),U_{N+2}^{\left(0\right)}\left(t\right),\dots ,U_{2N}^{\left(0\right)}\left(t\right))\in R^{2N},t\in [{\tau }_0,{\tau }_1]$$

, with $U_i^{\left(0\right)}\left(t\right)=x_i\left(t\right)$ and $U_{N+i}^{\left(0\right)}\left(t\right)=y_{N+i}\left(t\right)$, where the vector function $(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right),y_1\left(t\right),y_2\left(t\right),\dots ,y_N\left(t\right))$ is a solution of (24) on the interval $[{\tau }_0,{\tau }_1]$. Note that $U_i^{\left(0\right)}\in C^q(({\tau }_0,{\tau }_1],\mathbb{R},\psi ,\rho ),i=1,2,3,\dots ,2N,$ and $U^{\left(0\right)}\left(0\right)=Z^{\left(0\right)}$.

Let $C_0\ne C_1$. Then, at time ${\tau }_1$, the switching rule ($\sigma \left(t\right)=C_1$) is activated, and the system of equations is changed, i.e., consider the IVP for the system of fractional differential equations with CFDF:

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }{x}_i\left(t\right)=-d_i{x}_i\left(t\right)+\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_1\right)}\left(y_j\left(t\right)\right)+I_i,\hfill \\ & {}_{\rho }{}^{C}D_{{\tau }_1}^{q,\psi }{y}_i\left(t\right)=-b_i{y}_i\left(t\right)+c_i{x}_i\left(t\right),\mathrm{for}t\in ({\tau }_1,{\tau }_2],i=1,2,\dots ,N,\hfill \\ & x_i\left({\tau }_1\right)=U_i^{\left(0\right)}\left({\tau }_1\right),y_i\left({\tau }_1\right)=U_{N+i}^{\left(0\right)}\left({\tau }_1\right),i=1,2,\dots ,N.\hfill \end{array}$$

(25)

Denote the solution of (25) by $U^{\left(1\right)}\left(t\right)=(U_1^{\left(1\right)}\left(t\right),U_2^{\left(1\right)}\left(t\right),\dots ,U_N^{\left(1\right)}\left(t\right),U_{N+1}^{\left(1\right)}\left(t\right),U_{N+2}^{\left(1\right)}\left(t\right),\dots ,U_{2N}^{\left(1\right)}\left(t\right))\in {\mathbb{R}}^{2N},t\in [{\tau }_1,{\tau }_2]$, with $U_i^{\left(1\right)}\left(t\right)=x_i\left(t\right)$ and $U_{N+i}^{\left(1\right)}\left(t\right)=y_{N+i}\left(t\right)$, where the vector function $(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right),y_1\left(t\right),y_2\left(t\right),\dots ,y_N\left(t\right))$ is a solution of (25) on the interval $[{\tau }_1,{\tau }_2]$. Note that $U_i^{\left(1\right)}\in C^q(({\tau }_1,{\tau }_2],\mathbb{R},\psi ,\rho ),i=1,2,3,\dots ,2N,$ and $U^{\left(1\right)}\left({\tau }_1\right)=U^{\left(0\right)}\left({\tau }_1\right)$.

Let $k:2\le k\le m$ be an integer and $C_k\ne C_{k-1}$. Then, at time ${\tau }_k$, the switching rule ($\sigma \left(t\right)=C_k$) is activated, and the system of equations is changed, i.e., consider the IVP for the system of fractional differential equations with CFDF:

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{x}_i\left(t\right)=-d_i{x}_i\left(t\right)+\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)+I_i,\hfill \\ & {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{y}_i\left(t\right)=-b_i{y}_i\left(t\right)+c_i{x}_i\left(t\right),\mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],i=1,2,\dots ,N,\hfill \\ & x_i\left({\tau }_k\right)=U_i^{(k-1)}\left({\tau }_k\right),y_i\left({\tau }_k\right)=U_{N+i}^{(k-1)}\left({\tau }_k\right),i=1,2,\dots ,N.\hfill \end{array}$$

(26)

Denote the solution of (26) by

$$U^{\left(k\right)}\left(t\right)=(U_1^{\left(k\right)}\left(t\right),U_2^{\left(k\right)}\left(t\right),\dots ,U_N^{\left(k\right)}\left(t\right),U_{N+1}^{\left(k\right)}\left(t\right),U_{N+2}^{\left(k\right)}\left(t\right),\dots ,U_{2N}^{\left(k\right)}\left(t\right))\in R^{2N},t\in [{\tau }_k,{\tau }_{k+1}],$$

with $U_i^{\left(k\right)}\left(t\right)=x_i\left(t\right)$ and $U_{N+i}^{\left(k\right)}\left(t\right)=y_{N+i}\left(t\right)$, where $(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right),y_1\left(t\right),y_2\left(t\right),\dots ,$$y_N\left(t\right))$ is a solution of (26) on the interval $[{\tau }_k,{\tau }_{k+1}]$. Note that $U_i^{\left(k\right)}\in C^q(({\tau }_k,{\tau }_{k+1}],\mathbb{R},\psi ,\rho ),$$i=1,2,3,\dots ,2N,$ and $U^{\left(k\right)}\left({\tau }_k\right)=U^{(k-1)}\left({\tau }_k\right)$.

If $m=\infty$, we use the above procedure infinite times and define the solution of (23) by the equalities

$$\begin{array}{c}\hfill U\left(t\right)=\{\begin{array}{c}{U}^{\left(0\right)}\left(t\right),\mathrm{for}t\in [{\tau }_0,{\tau }_1],\hfill \\ U^{\left(1\right)}\left(t\right),\mathrm{for}t\in ({\tau }_1,{\tau }_2]\hfill \\ U^{\left(2\right)}\left(t\right),\mathrm{for}t\in ({\tau }_2,{\tau }_3]\hfill \\ \dots \dots \dots \dots \dots \hfill \end{array}\end{array}$$

(27)

If $m<\infty$ then the last eventual switching time is ${\tau }_m$, and we consider the last system similar to (26), with $k=m$ on interval $[{\tau }_m,\infty )$. Then, the solution of (23) is given by

$$\begin{array}{c}\hfill U\left(t\right)=\{\begin{array}{c}{U}^{\left(0\right)}\left(t\right),\mathrm{for}t\in [{\tau }_0,{\tau }_1],\hfill \\ U^{\left(1\right)}\left(t\right),\mathrm{for}t\in ({\tau }_1,{\tau }_2]\hfill \\ U^{\left(2\right)}\left(t\right),\mathrm{for}t\in ({\tau }_2,{\tau }_3]\hfill \\ \dots \dots \dots \dots \dots \hfill \\ U^{\left(m\right)}\left(t\right),\mathrm{for}t\in ({\tau }_m,\infty ).\hfill \end{array}\end{array}$$

(28)

Remark 6.

The solution ($U\left(t\right)$) of SGFM (23) is continuous at any switching point ${\tau }_k,k=1,2,3,$$\dots ,m$.

In the case of $m=\infty$, the point ${\tau }_{m+1}=\infty$, and the last interval ($({\tau }_m,{\tau }_{m+1}]$) is open.

Remark 7.

Commonly, the activator functions ($h_j^{\left(k\right)}(·),j=1,2,\dots ,N,k\in {\mathcal{M}}_{\sigma },$) are indicated in the Hill form ($h\left(s\right)={\displaystyle \frac{s^{\beta }}{{\alpha }^{\beta }+s^{\beta }}},s\in \mathbb{R}$), where ${\beta }_j$ are the Hill coefficients, and $\alpha \ge 0$ is a constant. Changing the constant α reflects on the behavior of the concentrations of the messenger ribonucleic acid (mRNA) and protein. The constant α will determine the activator function for the switching rule.

Remark 8.

Note that the model (23) without any switching rule and Caputo fractional derivative is studied in [27,29,30], and several sufficient conditions for stability are obtained by the application of absolute-value Lyapunov functions without any assumption for the sign of the functions describing the concentrations of mRNA and protein (see Remark 3).

Remark 9.

We study SFGM (23) with fractional derivative CFDF defined by (1). Therefore, any solution of (23) has to have a derivative defined by (1) on the intervals $({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m$, i.e., it has to be a function from the set ${\cup }_{k=0}^m{C}^q(({\tau }_k,{\tau }_{k+1}),{\mathbb{R}}^{2N},\psi ,\rho )$. If for an integer $k\ge 0$, a function $U(.)$ is not from the set $C^q(({\tau }_k,{\tau }_{k+1}),{\mathbb{R}}^{2N},\psi ,\rho )$ then it could not have a fractional derivative (${}_C{}^{\rho }D_{{\tau }_k}^{q,\psi }U\left(t\right),t\in ({\tau }_k,{\tau }_{k+1}]$) defined by (1), and it will not satisfy (23).

Now, we will define an equilibrium of the studied switched model.

Definition 3.

An equilibrium of SGFM (23) on the interval $[{\tau }_k,{\tau }_{k+1}]$ is the constant vector $W^{\left(k\right)}=(Q_1^{\left(k\right)},Q_2^{\left(k\right)},\dots ,Q_n^{\left(k\right)},P_1^{\left(k\right)},P_2^{\left(k\right)},\dots ,P_N^{\left(k\right)})\in {\mathbb{R}}^{2N}$ with non-negative components satisfying the algebraic system

$$\begin{array}{cc}& 0=-d_i{Q}_i^{\left(k\right)}+\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right)+I_i,\hfill \\ & 0=-b_i{P}_i^{\left(k\right)}+c_i{Q}_i^{\left(k\right)},i=1,2,\dots ,N.\hfill \end{array}$$

(29)

Definition 4.

An equilibrium of SGFM (23) is called the function

$$\begin{array}{c}\hfill W=\{\begin{array}{c}{W}^{\left(0\right)},fort\in [{\tau }_0,{\tau }_1],\hfill \\ W^{\left(1\right)},fort\in ({\tau }_1,{\tau }_2]\hfill \\ W^{\left(2\right)},fort\in ({\tau }_2,{\tau }_3]\hfill \\ \dots \dots \dots \dots \dots \hfill \\ W^{\left(m\right)},fort\in ({\tau }_m,{\tau }_{m+1}],\hfill \end{array}\end{array}$$

(30)

where $W^{\left(k\right)}\in {\mathbb{R}}^{2N}$ is the equilibrium of SGFM (23) on the interval $[{\tau }_k,{\tau }_{k+1}]$, $k=0,1,2,\dots ,m$.

Remark 10.

Note that if $m=\infty$ then the last interval in (30) is open.

Remark 11.

If $h_j^{\left(k\right)}\left(0\right)=0,j=1,2,\dots ,N,k=0,1,2,\dots ,m,$ and $I_i=0,i=1,2,\dots ,N$ then the model (23) has a zero equilibrium.

3.1. Quadratic Lyapunov Function

We now introduce the following assumptions:

(A1)

For any $k=0,1,2,\dots ,m$, the activator functions are $h_j^{\left(k\right)}(·)\in C(\mathbb{R},\mathbb{R}),j=1,2,\dots ,N$, and there exist constants ${\gamma }_j^{\left(k\right)}>0$ such that for any $u,v\in \mathbb{R}$, the inequalities $|h_j^{\left(k\right)}\left(u\right)$$-h_j^{\left(k\right)}\left(v\right)|\le {\gamma }_j^{\left(k\right)}|u-v|,j=1,2,\dots ,N$ hold.

(A2)

There exist positive constants (${\mu }_j,{\mu }_{N+j},j=1,2,\dots ,\mathbb{N}$) such that the coefficients in (23) satisfy the inequalities

$${\displaystyle \frac{{\mu }_{N+i}}{{\mu }_i}}|c_i|+\sum _{j=1}^N{\gamma }_j^{\left(C_k\right)}|a_{ij}|<2d_i,i=1,2,\dots ,N,k=0,1,\dots ,m,$$

$$|c_i|+\sum _{j=1}^N{\displaystyle \frac{{\mu }_j}{{\mu }_{N+i}}}|a_{ji}|{\gamma }_i^{\left(C_k\right)}<2b_i,i=1,2,\dots ,N,k=0,1,\dots ,m.$$

(A3)

SFGM (23) has an equilibrium ($W\in {\mathbb{R}}^{2N}$) defined by (30).

(A4)

For any initial values, SFGM has a solution ($U\left(t\right),t\in [0,\infty ),$) defined by (27) (or (28)) such that ${\left(U_i^{\left(k\right)}\right)}^2\in C^q([{\tau }_k,{\tau }_{k+1}],\mathbb{R},\psi ,\rho )$ for $k=1,2,\dots ,m$ and $i=1,2,3,\dots ,2N$.

Definition 5.

The equilibrium ($W\in {\mathbb{R}}^{2N}$) of the model (23), defined by (30), is generalized Mittag–Leffler stable in squares if there exist constants $M_k,{\lambda }_k>0,k=0,1,2,\dots ,m$ such that

$$\begin{array}{cc}& \parallel U^{\left(k\right)}\left(t\right)-W^{\left(k\right)}{\parallel }_{2N}\le M_k{\parallel Z^{\left(0\right)}-W^{\left(0\right)}\parallel }_{2N}\sqrt{E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)}\hfill \\ & fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m,\hfill \end{array}$$

(31)

where the function $U\in C^q(({\tau }_0,\infty ),{\mathbb{R}}^{2N},\psi ,\rho )$, defined by (27) (or (28)), is a solution of (23).

Theorem 1.

Let assumptions (H) and (A1)–(A4) be satisfied. Then, the equilibrium ($W\in {\mathbb{R}}^{2N}$) of the model (23) is generalized Mittag–Leffler stable in squares.

Proof. 

Let $W\in {\mathbb{R}}^{2N}$, defined by (30), with $W^{\left(k\right)}=(Q_1^{\left(k\right)},Q_2^{\left(k\right)},\dots ,Q_n^{\left(k\right)},P_1^{\left(k\right)},P_2^{\left(k\right)},\dots ,P_N^{\left(k\right)})$, be the equilibrium of the model (23).

According to condition (A2), there exist constants ${\mu }_k,{\mu }_{N+k},k=1,2,\dots ,N$.

For any $k=0,1,2,\dots ,m$, we consider the Lyapunov function ($V_k:{\mathbb{R}}^{2N}\to [0,\infty )$) defined by

$$V_k\left(U\right)=\sum _{i=1}^N{\mu }_i{(u_i-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N{\mu }_{N+k}{(v_i-P_i^{\left(k\right)})}^2,U=(u_1,u_2,\dots ,u_N,v_1,v_2,\dots ,v_N).$$

Let the function $U\in C^q(({\tau }_0,\infty ),{\mathbb{R}}^{2N},\psi ,\rho )$, defined by (27) (or (28)), be a solution of (23), with $U^{\left(k\right)}\left(t\right)=(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right),y_1\left(t\right),y_2\left(t\right),\dots ,y_N\left(t\right))\in R^{2N}$ being a solution of (23) on the interval $({\tau }_k,{\tau }_{k+1}]$.

Let $k,k=0,1,2,\dots ,m$ be a fixed number, and consider the interval $({\tau }_k,{\tau }_{k+1}]$.

According to Lemma 5, with $a={\tau }_k,T={\tau }_{k+1}$, and $u\left(t\right)=x_i\left(t\right)-Q_i^{\left(k\right)},i=1,2,\dots ,n$, or $u\left(t\right)=y_i\left(t\right)-P_i^{\left(k\right)},i=1,2,\dots ,n$, we obtain

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{V}_k\left(U\left(t\right)\right)=\sum _{i=1}^N{\mu }_i{}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N{\mu }_{N+i}{}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{(y_i\left(t\right)-P_i^{\left(k\right)})}^2\hfill \\ & \le 2\sum _{i=1}^N{\mu }_i(x_i\left(t\right)-Q_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }(x_i\left(t\right)-Q_i^{\left(k\right)})\hfill \\ & +2\sum _{i=1}^N{\mu }_{N+i}(y_i\left(t\right)-P_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }(y_i\left(t\right)-P_i^{\left(k\right)})\hfill \\ & =2\sum _{i=1}^N{\mu }_i(x_i\left(t\right)-Q_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{x}_i\left(t\right)+2\sum _{i=1}^N{\mu }_{N+i}(y_i\left(t\right)-P_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{y}_i\left(t\right)\hfill \\ & =-2\sum _{i=1}^N{\mu }_i{d}_i(x_i\left(t\right)-Q_i^{\left(k\right)})x_i\left(t\right)+2\sum _{i=1}^N\sum _{j=1}^N{\mu }_i{a}_{ij}(x_i\left(t\right)-Q_i^{\left(k\right)})h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)\hfill \\ & -2\sum _{i=1}^N{\mu }_{N+i}{b}_i(y_i\left(t\right)-P_i^{\left(k\right)})y_i\left(t\right)+2\sum _{i=1}^N{\mu }_{N+i}{c}_i(y_i\left(t\right)-P_i^{\left(k\right)})x_i\left(t\right).\hfill \end{array}$$

(32)

From the first equation in (29), we have

$$0=-{\mu }_i{d}_i{Q}_i^{\left(k\right)}(x_i\left(t\right)-Q_i^{\left(k\right)})+{\mu }_i(x_i\left(t\right)-Q_i^{\left(k\right)})\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right),$$

i.e.,

$$0=-2\sum _{i=1}^N{\mu }_i{d}_i{Q}_i^{\left(k\right)}(x_i\left(t\right)-Q_i^{\left(k\right)})+2\sum _{i=1}^N{\mu }_i(x_i\left(t\right)-Q_i^{\left(k\right)})\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right).$$

(33)

From the second equation in (29), we have

$$0=-{\mu }_{N+i}{b}_i{P}_i^{\left(k\right)}(y_i\left(t\right)-P_i^{\left(k\right)})+c_i{\mu }_{N+i}{Q}_i^{\left(k\right)}(y_i\left(t\right)-P_i^{\left(k\right)}),$$

i.e.,

$$0=-2\sum _{i=1}^N{\mu }_{N+i}{b}_i{P}_i^{\left(k\right)}(y_i\left(t\right)-P_i^{\left(k\right)})+2\sum _{i=1}^N{c}_i{Q}_i^{\left(k\right)}(y_i\left(t\right)-P_i^{\left(k\right)}).$$

(34)

Applying equalities (33) and (34) to (32), we obtain

$$\begin{array}{cc}& {}_{{\tau }_k}{}^C\mathcal{D}^{q,\rho }{}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{V}_k\left(U\left(t\right)\right)\hfill \\ & \le -2\sum _{i=1}^N{\mu }_i{d}_i{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2+2\sum _{i=1}^N\sum _{j=1}^N{\mu }_i{a}_{ij}(x_i\left(t\right)-Q_i^{\left(k\right)})(h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)-h_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right))\hfill \\ & -2\sum _{i=1}^N{\mu }_{N+i}{b}_i{(y_i\left(t\right)-P_i^{\left(k\right)})}^2+2\sum _{i=1}^N{\mu }_{N+i}{c}_i(y_i\left(t\right)-P_i^{\left(k\right)})(x_i\left(t\right)-Q_i^{\left(k\right)})\hfill \\ & \le -2\sum _{i=1}^N{\mu }_i{d}_i{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2+2\sum _{i=1}^N\sum _{j=1}^N{\mu }_i|a_{ij}||x_i\left(t\right)-Q_i^{\left(k\right)}||h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)-h_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right)|\hfill \\ & -2\sum _{i=1}^N{\mu }_{N+i}{b}_i{(y_i\left(t\right)-P_i^{\left(k\right)})}^2+\sum _{i=1}^N{\mu }_{N+i}|c_i|({(y_i\left(t\right)-P_i^{\left(k\right)})}^2+{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2).\hfill \end{array}$$

(35)

Since $C_k\in {\mathcal{M}}_{\sigma }$, according to condition (A1), there exists a number ${\gamma }_j^{\left(C_k\right)}>0$ such that $|h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)-h_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right)| \le {\gamma }_j^{\left(C_k\right)}|y_j\left(t\right)-P_j^{\left(k\right)}|,j=1,2,\dots ,N,$ and from (35), we get

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{V}_k\left(U\left(t\right)\right)\le -2\sum _{i=1}^N{\mu }_i{d}_i{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2-2\sum _{i=1}^N{\mu }_{N+i}{b}_i{(y_i\left(t\right)-P_i^{\left(k\right)})}^2\hfill \\ & +\sum _{i=1}^N{\mu }_{N+i}|c_i|({(y_i\left(t\right)-P_i^{\left(k\right)})}^2+{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2)\hfill \\ & +2\sum _{i=1}^N\sum _{j=1}^N{\mu }_i|a_{ij}||x_i\left(t\right)-Q_i^{\left(k\right)}|{\gamma }_j^{\left(C_k\right)}|y_j\left(t\right)-P_j^{\left(k\right)}|\hfill \\ & \le \sum _{i=1}^N(-2{\mu }_i{d}_i+{\mu }_{N+i}|c_i|){(x_i\left(t\right)-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N\sum _{j=1}^N{\mu }_i{\gamma }_j^{\left(C_k\right)}|a_{ij}|{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2\hfill \\ & +\sum _{i=1}^N(-2{\mu }_{N+i}{b}_i+{\mu }_{N+i}|c_i|){(y_i\left(t\right)-P_i^{\left(k\right)})}^2+\sum _{i=1}^N\sum _{j=1}^N{\mu }_i|a_{ij}|{\gamma }_j^{\left(C_k\right)}{(y_j\left(t\right)-P_j^{\left(k\right)})}^2\hfill \\ & \le \left(\sum _{i=1}^N(-2{\mu }_i{d}_i+{\mu }_{N+i}|c_i|+\sum _{j=1}^N{\mu }_i{\gamma }_j^{\left(C_k\right)}|a_{ij}|){(x_i\left(t\right)-Q_i^{\left(k\right)})}^2\right)\hfill \\ & +\left(\sum _{i=1}^N(-2{\mu }_{N+i}{b}_i+{\mu }_{N+i}|c_i|+\sum _{j=1}^N{\mu }_j|a_{ji}|{\gamma }_i^{\left(C_k\right)}){(y_i\left(t\right)-P_i^{\left(k\right)})}^2\right)\hfill \\ & \le {\eta }_k\sum _{i=1}^N{\mu }_i{(x_i\left(t\right)-Q_i^{\left(k\right)})}^2+{\varsigma }_k\sum _{i=1}^N{\mu }_{N+i}{(y_i\left(t\right)-P_i^{\left(k\right)})}^2\hfill \\ & \le -{\lambda }_k{V}_k\left(U\left(t\right)\right),\hfill \end{array}$$

(36)

where

$${\eta }_k=\underset{i=1,2,\dots ,N}{max}(-2d_i+{\displaystyle \frac{{\mu }_{N+i}}{{\mu }_i}}|c_i|+\sum _{j=1}^N{\gamma }_j^{\left(C_k\right)}|a_{ij}|)<0,$$

$${\varsigma }_k=\underset{i=1,2,\dots ,N}{max}(-2b_i+|c_i|+\sum _{j=1}^N{\displaystyle \frac{{\mu }_j}{{\mu }_{N+i}}}|a_{ji}|{\gamma }_i^{\left(C_k\right)})<0,$$

$${\lambda }_k=-max\{{\eta }_k,{\varsigma }_k\}>0.$$

Lemma 4 is applied to the function $y\left(t\right)=V_k\left(U\left(t\right)\right)$, and $A=V_k\left(U\left(0\right)\right)+\epsilon ={\sum }_{i=1}^N{\mu }_i{(x_i^0-Q_i^{\left(k\right)})}^2+{\sum }_{i=1}^N{\mu }_{N+k}{(y_i^0-P_i^{\left(k\right)})}^2+\epsilon$, where $\epsilon >0$ is an arbitrary positive number. Therefore, the inequality

$$\begin{array}{cc}& V_k\left(U\left(t\right)\right)<A{L}_k^2{E}_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m,\hfill \end{array}$$

(37)

holds, with

$$L_k=\sqrt{\prod _{i=1}^{k-1}{E}_q(-{\lambda }_i{\left({\displaystyle \frac{\psi {\left({\tau }_{i+1}\right)}^{\rho }-{\left(\psi \left({\tau }_i\right)\right)}^{\rho }}{\rho }}\right)}^q)}.$$

Taking the limit as $\epsilon \to 0$ in inequality (37), we get

$$\begin{array}{cc}& V_k\left(U\left(t\right)\right)\le L_k^2{V}_k\left(U\left(0\right)\right)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\hfill \\ & \mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m.\hfill \end{array}$$

(38)

Let

$${\mu }_{min}=\underset{i=1,2,\dots ,N}{min}\{{\mu }_k,{\mu }_{N+i}\}\mathrm{and}{\mu }_{max}=\underset{i=1,2,\dots ,N}{max}\{{\mu }_i,{\mu }_{N+i}\}.$$

From the definition of the Lyapunov function ($V_k(.)$) and inequality (38) we obtain

$$\begin{array}{cc}& {\mu }_{min}{\parallel U^{\left(k\right)}\left(t\right)-W^{\left(k\right)}\parallel }_{2N}^2={\mu }_{min}(\sum _{i=1}^N{(u_i\left(t\right)-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N{(v_i\left(t\right)-P_i^{\left(k\right)})}^2)\hfill \\ & \le \sum _{i=1}^N{\mu }_k{(u_i\left(t\right)-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N{\mu }_{N+i}{(v_i\left(t\right)-P_i^{\left(k\right)})}^2\hfill \\ & =V_k\left(U\left(t\right)\right)\le L_k^2{V}_k\left(U\left(0\right)\right)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\hfill \\ & =L_k^2(\sum _{i=1}^N{\mu }_k{(u_i^{\left(0\right)}-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N{\mu }_{N+i}{(v_i^{\left(0\right)}-P_i^{\left(k\right)})}^2)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\hfill \\ & \le L_k^2{\mu }_{max}(\sum _{i=1}^N{(u_i^{\left(0\right)}-Q_i^{\left(k\right)})}^2+\sum _{i=1}^N{(v_i^{\left(0\right)}-P_i^{\left(k\right)})}^2)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q).\hfill \end{array}$$

(39)

Inequality (39) establishes inequality (31), with $M_k=\sqrt{L_k{\displaystyle \frac{{\mu }_{max}}{{\mu }_{min}}}}.$ □

Corollary 1.

Let the conditions of Theorem 1 be satisfied. Then, the inequality

$$\parallel U^{\left(k\right)}\left(t\right)-W^{\left(k\right)}{\parallel }_{2N}\le \sqrt{{\displaystyle \frac{{\mu }_{max}}{{\mu }_{min}}}}{\parallel Z^{\left(0\right)}-W^{\left(0\right)}\parallel }_{2N},t\in ({\tau }_k,{\tau }_{k+1}]$$

holds.

The proof follows from inequality (31), and $E_q\left(z\right)\le 1$ for $z\le 0$.

3.2. Absolute Values in Lyapunov Functions

We introduce the following assumptions:

(A5)

There exist positive constants ${\mu }_j,{\mu }_{N+j},j=1,2,\dots ,\mathbb{N}$ such that the coefficients in (23) satisfy the inequalities

$${\displaystyle \frac{{\mu }_{N+i}}{{\mu }_i}}|c_i|<d_i,i=1,2,\dots ,N,k=0,1,\dots ,m$$

and

$${\displaystyle \frac{{\mu }_i}{{\mu }_{N+i}}}\sum _{j=1}^N|a_{ij}|{\gamma }_j^{\left(C_k\right)}<b_i,i=1,2,\dots ,N,k=0,1,\dots ,m.$$

(A6)

For any positive initial values, SFGM has a solution ($U\left(t\right),t\in [0,\infty ),$) defined by (27) (or (28)) such that for any $k=1,2,\dots ,m$ and $i=1,2,3,\dots ,2N$, the function $U_i^{\left(k\right)}(.)$ is positive on the interval $[{\tau }_k,{\tau }_{k+1}]$.

Remark 12.

Positive solutions in fractional-order gene regulatory networks are crucial because biological concentrations cannot be negative. Therefore, condition (A6) is very important.

One of the techniques used to prove the positivity of the solutions, particularly when starting with positive initial conditions, is the comparison method. It is successfully applied in [31] to study Caputo-fractional-order gene regulatory networks.

Using the ideas of [31], we will prove that condition (A6) is satisfied in the special case of the activation initiatives of proteins of mRNA.

We will use the following notations:

${\mathcal{B}}_i=\{j:j\mathrm{corresponds} \mathrm{to} \mathrm{the} \mathrm{transcription} \mathrm{factor}\left(j\right)\mathrm{that} \mathrm{is} a \mathrm{repressor} \mathrm{of} \mathrm{gene}i\}$;

${\mathcal{A}}_i=\{j:j\mathrm{corresponds} \mathrm{to} \mathrm{the} \mathrm{transcription} \mathrm{factor}\left(j\right)\mathrm{that} \mathrm{is} \mathrm{an} \mathrm{activator} \mathrm{of} \mathrm{gene}i\}$.

Lemma 7.

Let the following conditions be satisfied:

1.

The perturbations ($I_i={\sum }_{j\in {\mathcal{B}}_i}{\gamma }_{ij}$), with ${\gamma }_{i,j}>0,i=1,2,\dots ,N$;

2.

The activation initiatives of proteins of mRNA satisfy $h_j^{\left(C_k\right)}\left(0\right)=0$ and $h_j^{\left(C_k\right)}\left(x\right)>0$ for any $x>0,j=1,2,\dots ,N$ and $h_j^{\left(C_k\right)}\left(x\right)<1$ for $j\in {\mathcal{B}}_i,i=1,2,\dots ,N$.

Then, for any positive initial values (${\mathcal{Z}}_0>0$), the solution ($U\left(t\right)>0$) of SFGM (23) is positive.

Proof. 

We will use induction to prove this.

Consider the interval $[{\tau }_0,{\tau }_1]$. Since the initial values ${\mathcal{Z}}_0>0$ are positive, there exists a point $\eta \in ({\tau }_0,{\tau }_1]$ such that

-

All the concentrations of messenger ribonucleic acid (mRNA) are $x_i\left(t\right)>0$, and all the concentrations of protein are $y_i\left(t\right)>0$ for the ith node ($i=1,2,\dots ,N$) for $t\in [{\tau }_0,\eta )$;

-

$x_i\left(\eta \right)\ge 0$ and $y_i\left(\eta \right)\ge 0$ for $i=1,2,\dots ,N$.

Case 1. Suppose that there exists $m\in \{1,2,\dots ,N\}$ such that $x_m\left(\eta \right)=0$.

From Lemma 2 applied to the function $-x_m(.)$ with $a={\tau }_0,T={\tau }_1$, and Remark 2, we have ${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }(-x_m\left(t\right)){|}_{t=\eta }\ge 0$. This, with the first equality in (23), gives

$$0\ge {}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }{x}_m{\left(t\right)|}_{t=\eta }=\sum _{j=1}^N{a}_{mj}{h}_j^{\left(C_0\right)}\left(y_j\left(\eta \right)\right)+\sum _{j\in {\mathcal{B}}_m}{\gamma }_{mj}.$$

On the other hand, from the definition of $a_{ij}$, we get

$$\begin{array}{cc}& \sum _{j=1}^N{a}_{mj}{h}_j^{\left(C_0\right)}\left(y_j\left(\eta \right)\right)+\sum _{j\in {\mathcal{B}}_m}{\gamma }_{mj}\hfill \\ & =\sum _{j\in {\mathcal{A}}_m}^N{\gamma }_{mj}{h}_j^{\left(C_0\right)}\left(y_j\left(\eta \right)\right)+\sum _{j\in {\mathcal{B}}_m}^N{\gamma }_{mj}(1-h_j^{\left(C_0\right)}\left(y_j\left(\eta \right)\right))\ge \sum _{j\in {\mathcal{B}}_m}^N{\gamma }_{mj}>0.\hfill \end{array}$$

(40)

The obtained contradiction proves that the point $\eta$ does not exist, i.e., $x_m\left(t\right)>0$ for $t\in [{\tau }_0,{\tau }_1]$. Thus, for $i=1,2,\dots ,N$, we have $x_i\left(t\right)>0,t\in [{\tau }_0,{\tau }_1]$.

Case 2. Suppose that there exists $m\in \{1,2,\dots ,N\}$ such that $y_m\left(\eta \right)=0$.

From Lemma 2 applied to the function $-y_m(.)$ with $a={\tau }_0,T={\tau }_1$, and Remark 2, we have ${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }(-y_m\left(t\right)){|}_{t=\eta }\ge 0$. This, with the second equality in (23), gives us

$${}_{\rho }{}^{C}D_{{\tau }_0}^{q,\psi }{y}_m{\left(t\right)|}_{t=\eta }=c_m{x}_m\left(\eta \right)>0.$$

The obtained contradiction proves that the point $\eta$ does not exist, i.e., $y_m\left(t\right)>0$ for $t\in [{\tau }_0,{\tau }_1]$. Thus, for $i=1,2,\dots ,N$, we have $y_i\left(t\right)>0,t\in [{\tau }_0,{\tau }_1]$.

Therefore, the solution ($U\left(t\right)>0$) of SFGM (23) is positive on $[{\tau }_0,{\tau }_1]$.

Following the same procedure for the rest of the intervals ($[{\tau }_k,{\tau }_{k+1}],k=1,2,\dots ,m$), we prove the claim. □

Remark 13.

Let $h_j^{\left(C_k\right)}\left(x\right)={\displaystyle \frac{x^{{\beta }_k}}{C_k+x^{{\beta }_k}}}$, in which ${\beta }_k$ are the so-called Hill coefficients. Then, the functions $h_j^{\left(C_k\right)}(.)$ satisfy the conditions of Lemma 7 (see Remark 7).

Definition 6.

The equilibrium $W\in {\mathbb{R}}^{2N}$ of the model (23), defined by (30), is generalized Mittag–Leffler stable in absolute values if there exist constants $M_k,{\lambda }_k>0,k=0,1,2,\dots ,m,$ such that

$$\begin{array}{cc}& \parallel |U^{\left(k\right)}\left(t\right)-W^{\left(k\right)}{\parallel |}_{2N}\le M_k\parallel |Z^{\left(0\right)}-W^{\left(0\right)}{\parallel |}_{2N}{E}_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\hfill \\ & fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m,\hfill \end{array}$$

(41)

where the function $U^{\left(k\right)}\left(t\right)=(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right),y_1\left(t\right),y_2\left(t\right),\dots ,y_N\left(t\right)),k=0,1,2,\dots ,m$ is a solution of (23) on the interval $({\tau }_k,{\tau }_{k+1}]$.

Theorem 2.

Let assumptions (H), (A1), (A3), (A5), and (A6) be satisfied. Then, the equilibrium ($W\in {\mathbb{R}}^{2N}$) of the model (23) is generalized Mittag–Leffler stable in absolute values.

Proof. 

Let $W\in {\mathbb{R}}^{2N}$, defined by (30), with $W^{\left(k\right)}=(Q_1^{\left(k\right)},Q_2^{\left(k\right)},\dots ,Q_n^{\left(k\right)},P_1^{\left(k\right)},P_2^{\left(k\right)},\dots ,P_N^{\left(k\right)})$, be the equilibrium of the model (23) (it exists according to condition (A3)).

According to condition (A5), there exist constants ${\mu }_k,{\mu }_{N+k},k=1,2,\dots ,N$.

For any $k=0,1,2,\dots ,m$, we consider the Lyapunov function ($V_k:{\mathbb{R}}^{2N}\to [0,\infty )$) defined by

$$V_k\left(U\right)=\sum _{i=1}^N{\mu }_i|u_i-Q_i^{\left(k\right)}|+\sum _{i=1}^N{\mu }_{N+k}|v_i-P_i^{\left(k\right)}|,U=(u_1,u_2,\dots ,u_N,v_1,v_2,\dots ,v_N)\in {\mathbb{R}}^{2N}.$$

Let $k=0,1,2,\dots ,m$ be a fixed number, and consider the interval $({\tau }_k,{\tau }_{k+1}]$. Let $U^{\left(k\right)}\left(t\right)$$=(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right),y_1\left(t\right),y_2\left(t\right),\dots ,y_N\left(t\right))$ be a solution of (23) on the interval $({\tau }_k,{\tau }_{k+1}]$.

According to Lemma 6, with $a={\tau }_k,T={\tau }_{k+1}$, and $u\left(t\right)=x_i\left(t\right)-Q_i^{\left(k\right)},i=1,2,\dots ,n$, or $u\left(t\right)=y_i\left(t\right)-P_i^{\left(k\right)},i=1,2,\dots ,n$, we obtain

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{V}_k\left(U\left(t\right)\right)=\sum _{i=1}^N{\mu }_i{}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }|x_i\left(t\right)-Q_i^{\left(k\right)}|+\sum _{i=1}^N{\mu }_{N+i}{}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }|y_i\left(t\right)-P_i^{\left(k\right)}|\hfill \\ & \le \sum _{i=1}^N{\mu }_{i}sign(x_i\left(t\right)-Q_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }(x_i\left(t\right)-Q_i^{\left(k\right)})\hfill \\ & +\sum _{i=1}^N{\mu }_{N+i}sign(y_i\left(t\right)-P_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }(y_i\left(t\right)-P_i^{\left(k\right)})\hfill \\ & =\sum _{i=1}^N{\mu }_{i}sign(x_i\left(t\right)-Q_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{x}_i\left(t\right)+\sum _{i=1}^N{\mu }_{N+i}sign(y_i\left(t\right)-P_i^{\left(k\right)}){}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{y}_i\left(t\right)\hfill \\ & =-\sum _{i=1}^N{\mu }_i{d}_{i}sign(x_i\left(t\right)-Q_i^{\left(k\right)})x_i\left(t\right)+\sum _{i=1}^N\sum _{j=1}^N{\mu }_i{a}_{ij}sign(x_i\left(t\right)-Q_i^{\left(k\right)})h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)\hfill \\ & -\sum _{i=1}^N{\mu }_{N+i}{b}_{i}sign(y_i\left(t\right)-P_i^{\left(k\right)})y_i\left(t\right)+\sum _{i=1}^N{\mu }_{N+i}{c}_{i}sign(y_i\left(t\right)-P_i^{\left(k\right)})x_i\left(t\right).\hfill \end{array}$$

(42)

From the first equation in (29), we have

$$0=-\sum _{i=1}^N{\mu }_i{d}_i{Q}_i^{\left(k\right)}sign(x_i\left(t\right)-Q_i^{\left(k\right)})+\sum _{i=1}^N{\mu }_{i}sign(x_i\left(t\right)-Q_i^{\left(k\right)})\sum _{j=1}^N{a}_{ij}{h}_j^{\left(C_k\right)}\left(P_j^{\left(k\right)}\right)$$

(43)

From the second equation in (29), we have

$$0=-\sum _{i=1}^N{\mu }_{N+i}{b}_i{P}_i^{\left(k\right)}sign(y_i\left(t\right)-P_i^{\left(k\right)})+\sum _{i=1}^N{c}_i{Q}_i^{\left(k\right)}sign(y_i\left(t\right)-P_i^{\left(k\right)}).$$

(44)

Applying equalities (43) and (44) to (42), we obtain

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{V}_k\left(U\left(t\right)\right)\hfill \\ & \le -\sum _{i=1}^N{\mu }_i{d}_i|x_i\left(t\right)-Q_i^{\left(k\right)}|+\sum _{i=1}^N\sum _{j=1}^N{\mu }_i|a_{ij}||h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)-h_j^{\left(C_k\right)}(P_j^{\left(k\right)})|\hfill \\ & -\sum _{i=1}^N{\mu }_{N+i}{b}_i|y_i\left(t\right)-P_i^{\left(k\right)}|+\sum _{i=1}^N{\mu }_{N+i}|c_i||x_i\left(t\right)-Q_i^{\left(k\right)}|.\hfill \end{array}$$

(45)

Since $C_k\in {\mathcal{M}}_{\sigma }$, according to condition (A1), there exist numbers ${\gamma }_j^{\left(C_k\right)}>0$ such that $|h_j^{\left(C_k\right)}\left(y_j\left(t\right)\right)-h_j^{\left(C_k\right)}(P_j^{\left(k\right)})| \le {\gamma }_j^{\left(C_k\right)}|y_j\left(t\right)-P_j^{\left(k\right)}|,j=1,2,\dots ,N,$ and from (45), we get

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }{V}_k\left(U\left(t\right)\right)\le \sum _{i=1}^N(-{\mu }_i{d}_i+{\mu }_{N+i}|c_i|)|x_i\left(t\right)-Q_i^{\left(k\right)}|\hfill \\ & +\sum _{i=1}^N(-{\mu }_{N+i}{b}_i+\sum _{j=1}^N{\mu }_i|a_{ij}|{\gamma }_j^{\left(C_k\right)})|y_i\left(t\right)-P_i^{\left(k\right)}|\hfill \\ & \le {\eta }_k\sum _{i=1}^N{\mu }_i|x_i\left(t\right)-Q_i^{\left(k\right)}|+{\varsigma }_k\sum _{i=1}^N{\mu }_{N+i}|y_i\left(t\right)-P_i^{\left(k\right)}|\hfill \\ & \le -{\lambda }_k{V}_k\left(U\left(t\right)\right),\hfill \end{array}$$

(46)

where ${\eta }_k={max}_{i=1,2,\dots ,N}(-d_i+{\displaystyle \frac{{\mu }_{N+i}}{{\mu }_i}}|c_i|)<0,{\varsigma }_k={max}_{i=1,2,\dots ,N}(-b_i+{\displaystyle \frac{{\mu }_i}{{\mu }_{N+i}}}{\sum }_{j=1}^N|a_{ij}|{\gamma }_j^{\left(C_k\right)})<0,$ ${\lambda }_k=-max\{{\eta }_k,{\varsigma }_k\}>0.$

Lemma 4 is applied to the function $y\left(t\right)=V_k\left(U\left(t\right)\right)$, and $A=V_k\left(U\left(0\right)\right)+\epsilon ={\sum }_{i=1}^N{\mu }_i|x_i^0-Q_i^{\left(k\right)}| +{\sum }_{i=1}^N{\mu }_{N+k}|y_i^0-P_i^{\left(k\right)}|+\epsilon$, where $\epsilon >0$ is an arbitrary positive number. Therefore, the inequality

$$\begin{array}{cc}& V_k\left(U\left(t\right)\right)<A{L}_k{E}_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,m,\hfill \end{array}$$

(47)

holds, with

$$L_k=\prod _{i=0}^{k-1}{E}_q(-{\lambda }_i{\left({\displaystyle \frac{\psi {\left({\tau }_{i+1}\right)}^{\rho }-{\left(\psi \left({\tau }_i\right)\right)}^{\rho }}{\rho }}\right)}^q).$$

Taking the limit as $\epsilon \to 0$ in inequality (47), we get

$$\begin{array}{cc}& V_k\left(U\left(t\right)\right)\le L_k{V}_k\left(U\left(0\right)\right)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q),t\in ({\tau }_k,{\tau }_{k+1}].\hfill \end{array}$$

(48)

Let

$${\mu }_{min}=\underset{i=1,2,\dots ,N}{min}\{{\mu }_k,{\mu }_{N+i}\}\mathrm{and}{\mu }_{max}=\underset{i=1,2,\dots ,N}{max}\{{\mu }_i,{\mu }_{N+i}\}.$$

From the definition of the Lyapunov function ($V_k(.)$) and inequality (48), we obtain

$$\begin{array}{cc}& {\mu }_{min}\parallel |U^{\left(k\right)}\left(t\right)-W^{\left(k\right)}{\parallel |}_{2N}={\mu }_{min}(\sum _{i=1}^N|u_i\left(t\right)-Q_i^{\left(k\right)}|+\sum _{i=1}^N|v_i\left(t\right)-P_i^{\left(k\right)}|)\hfill \\ & \le \sum _{i=1}^N{\mu }_k|u_i\left(t\right)-Q_i^{\left(k\right)}|+\sum _{i=1}^N{\mu }_{N+i}|v_i\left(t\right)-P_i^{\left(k\right)}|\hfill \\ & =V_k\left(U\left(t\right)\right)\le L_k{V}_k\left(U\left(0\right)\right)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\hfill \\ & =M_k(\sum _{i=1}^N{\mu }_k|u_i^{\left(0\right)}-Q_i^{\left(k\right)}|+\sum _{i=1}^N{\mu }_{N+i}|v_i^{\left(0\right)}-P_i^{\left(k\right)}|)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q)\hfill \\ & \le L_k{\mu }_{max}(\sum _{i=1}^N|u_i^{\left(0\right)}-Q_i^{\left(k\right)}|+\sum _{i=1}^N|v_i^{\left(0\right)}-P_i^{\left(k\right)}|)E_q(-{\lambda }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\tau }_k\right)\right)}^{\rho }}{\rho }}\right)}^q).\hfill \end{array}$$

(49)

Inequality (49) establishes inequality (41), with $M_k=L_k{\displaystyle \frac{{\mu }_{max}}{{\mu }_{min}}}.$ □

Corollary 2.

Let the conditions of Theorem 2 be satisfied. Then, the inequality

$$\parallel |U^{\left(k\right)}\left(t\right)-W^{\left(k\right)}{\parallel |}_{2N}\le {\displaystyle \frac{{\mu }_{max}}{{\mu }_{min}}}\parallel |Z^{\left(0\right)}-W^{\left(0\right)}{\parallel |}_{2N},t\in ({\tau }_k,{\tau }_{k+1}]$$

holds.

4. Applications

We will consider the model of three repressor protein concentrations, $p_i$, and their corresponding mRNA concentrations, $m_i,i=1,2,3$, which are defined and studied in [32], where the kinetics of the system are determined by ordinary differential equations. To have a more appropriate model, we will adopt this model, but we will generalize it by the application of CGFDF and a switching rule reflection of the activation function.

Let the switching rule be given by $\sigma \left(t\right)=C_k$ for $t\in [k,k+1),k=0,1,2,\dots ,m$, where $C_k=k+1$, and $m\le \infty$. This function has discontinuities at points ${\tau }_k=k$, which are the activation points of the switching rule of the model.

Consider the model

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_k^{q,\psi }{m}_i\left(t\right)=-m_i+\alpha {\displaystyle \frac{p_i^n}{(C_k+p_i^n)}}+{\alpha }_0\hfill \\ & {}_{\rho }{}^{C}D_k^{q,\psi }{p}_i\left(t\right)=-\beta (p_i-m_i),t\in (k,k+1],i=1,2,3,k=0,1,2,\dots ,m,\hfill \end{array}$$

(50)

where (see [32])

-

$\beta$ is the ratio of the protein decay rate to the mRNA decay rate;

-

n is a Hill coefficient.

System (50) is similar to (23), with $d_i=1,b_i=c_i=\beta$, $a_{ii}=\alpha$, and $a_{ij}=0$ for $j\ne i$, $I_i={\alpha }_0$ and $h_i^k\left(u\right)={\displaystyle \frac{u^n}{(C_k+u^n)}},i=1,2,3$.

Let the Hill coefficient ($n=2$), ${\mu }_i={\mu }_{3+i},i=1,2,3$, $\beta =0.9$, $\alpha =1.3$, and ${\alpha }_0=1$.

Case 1. Let $m=2<\infty$. Then, the switching rule will act only two times at times $t=1,2$, and the model (50) will be reduced to

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_0^{q,\psi }{m}_i\left(t\right)=-m_i+{\displaystyle \frac{\alpha p_i^n}{(1+p_i^n)}}+1,{}_{\rho }{}^{C}D_0^{q,\psi }{p}_i\left(t\right)=-0.9(p_i-m_i),t\in (0,1],i=1,2,3,\hfill \\ & {}_{\rho }{}^{C}D_1^{q,\psi }{m}_i\left(t\right)=-m_i+{\displaystyle \frac{\alpha p_i^n}{(2+p_i^n)}}+1,{}_{\rho }{}^{C}D_1^{q,\psi }{p}_i\left(t\right)=--0.9(p_i-m_i),t\in (1,2],i=1,2,3,\hfill \\ & {}_{\rho }{}^{C}D_2^{q,\psi }{m}_i\left(t\right)=-m_i+{\displaystyle \frac{\alpha p_i^n}{(3+p_i^n)}}+1,{}_{\rho }{}^{C}D_2^{q,\psi }{p}_i\left(t\right)=-0.9(p_i-m_i),t\in (2,\infty ),i=1,2,3.\hfill \end{array}$$

(51)

Then, $|h_i^k\left(u\right)-h_i^k\left(v\right)| = |{\displaystyle \frac{u^2}{(k+1+u^2)}}-{\displaystyle \frac{v^2}{(k+1+v^2)}}| = |u-v||{\displaystyle \frac{(k+1)(u+v)}{(k+1+u^2)(k+1+v^2)}}|,k=0,1,2$. Note that ${max}_{k=0,1,2}{\displaystyle \frac{(k+1)(u+v)}{(k+1+u^2)(k+1+v^2)}}{|}_{x=y=\sqrt{{\displaystyle \frac{k+1}{3}}}}={\displaystyle \frac{3\sqrt{3}}{8}}$, i.e., condition (A1) is satisfied, with ${\gamma }_j^{\left(k\right)}={\displaystyle \frac{3\sqrt{3}}{8}}$, $j=1,2,3,k=0,1,2$.

Also,

$${\displaystyle \frac{{\mu }_{3+i}}{{\mu }_i}}|c_i| = |c_i| =\beta =0.9<d_i=1,i=1,2,3,k=0,1,2,$$

and

$${\displaystyle \frac{{\mu }_{3+i}}{{\mu }_i}}\sum _{j=1}^3|a_{ij}|{\gamma }_j^{\left(k\right)}|=\alpha {\gamma }_i^{\left(k\right)}=1.3{\displaystyle \frac{3\sqrt{3}}{8}}\approx 0.844375<b_i=\beta =0.9,i=1,2,3,k=0,1,2,$$

i.e., condition (A5) is satisfied.

According to Lemma 7 and Remark 13, condition (A6) is satisfied.

The equilibrium ($W^{\left(k\right)}=(Q_1^{\left(k\right)},Q_2^{\left(k\right)},Q_3^{\left(k\right)},P_1^{\left(k\right)},P_2^{\left(k\right)},P_3^{\left(k\right)})\in {\mathbb{R}}^6,P_i^{\left(k\right)},Q_i^{\left(k\right)}\ge 0,$) of the model (51) on the interval $[k,k+1]$ for $k=0,1,2$ is a solution of the system

$$Q_i^{\left(k\right)}=P_i^{\left(k\right)},Q_i^{\left(k\right)}=\alpha {\displaystyle \frac{{\left(Q_i^{\left(k\right)}\right)}^n}{k+1+{\left(Q_i^{\left(k\right)}\right)}^n}}+1,i=1,2,3.$$

(52)

Therefore, $Q_i^{\left(0\right)},P_i^{\left(0\right)}\approx 2.05015,$$Q_i^{\left(1\right)},P_i^{\left(1\right)}\approx 1.80578,$$Q_i^{\left(2\right)},P_i^{\left(2\right)}\approx 1.59759,i=1,2,3,$ i.e., condition (A3) is satisfied.

According to Theorem 2, the equilibrium is generalized Mittag–Leffler stable in absolute values. The constant of the stability is $M_k=L_k{\displaystyle \frac{{\mu }_{max}}{{\mu }_{min}}}=L_k$, where

$$L_k=\prod _{i=0}^{k-1}{E}_q(-{\lambda }_i{\left({\displaystyle \frac{\psi {\left({\tau }_{i+1}\right)}^{\rho }-{\left(\psi \left({\tau }_i\right)\right)}^{\rho }}{\rho }}\right)}^q),$$

${\eta }_k={max}_{i=1,2,3}(-d_i+{\displaystyle \frac{{\mu }_{3+i}}{{\mu }_i}}|c_i|)--1+0.9=-0.1<0,{\varsigma }_k={max}_{i=1,2,3}-b_i+{\displaystyle \frac{{\mu }_i}{{\mu }_{N+i}}}{\sum }_{j=1}^3|a_{ij}|{\gamma }_j^{\left(C_k\right)}<0,$${\lambda }_k=-max\{{\eta }_k,{\varsigma }_k\}>0.$

This constant depends significantly on the fractional order of the applied derivative as well as on the used function ($\psi$) and the parameter ($\rho$) in it.

For example, let the fractional order be $q=0.3$.

Suppose that the applied function is $\psi \left(t\right)=t$ and the parameter is $\rho =1$ (Caputo fractional derivative). Then,

$$L_0=1,L_1=E_{0.3}(-{\lambda }_0)=E_{0.3}(-0.0556252)\approx 0.941312,L_2=E_{0.3}(-{\lambda }_0)E_{0.3}(-{\lambda }_1)\approx 0.886069,$$

where ${\eta }_0={\eta }_1,{\eta }_2=-1+0.9=-0.1<0,{\varsigma }_0={\varsigma }_1={\varsigma }_2=-0.9+1.3{\displaystyle \frac{3\sqrt{3}}{8}}\approx -0.0556252<0,{\lambda }_j=-max\{{\eta }_k,{\varsigma }_k\}\approx 0.0556252>0.$

Therefore, the inequalities

$$\begin{array}{cc}& |m_i\left(t\right)-2.05015|\le |m_i\left(0\right)-2.05015|E_{0.3}(-0.0556252t^{0.3}),t\in [0,1]\hfill \\ & |m_i\left(t\right)-1.80578|\le 0.941312|m_i\left(0\right)-2.05015|E_{0.3}(-0.0556252{(t-1)}^{0.3}),t\in (1,2]\hfill \\ & |m_i\left(t\right)-1.59759|\le 0.886069|m_i\left(0\right)-2.05015|E_{0.3}(-0.0556252{(t-2)}^{0.3}),t>2\hfill \end{array}$$

(53)

hold.

Case 2. Let $m=\infty$, i.e., the switching rule will act at any time $t=k,k=0,1,2,\dots$.

Then, $|h_i^k\left(u\right)-h_i^k\left(v\right)| = |{\displaystyle \frac{u^2}{(k+1+u^2)}}-{\displaystyle \frac{v^2}{(k+1+v^2)}}| = |u-v||{\displaystyle \frac{(k+1)(u+v)}{(k+1+u^2)(k+1+v^2)}}|,k=0,1,2,\dots$. Note that ${max}_{k=0,1,2,\dots }{\displaystyle \frac{(k+1)(u+v)}{(k+1+u^2)(k+1+v^2)}}{|}_{x=y=\sqrt{{\displaystyle \frac{k+1}{3}}}}={\displaystyle \frac{3\sqrt{3}}{8}}$, i.e., condition (A1) is satisfied, with ${\gamma }_j^{\left(k\right)}={\displaystyle \frac{3\sqrt{3}}{8}}$, $j=1,2,3,k=0,1,2,\dots$.

Also,

$${\displaystyle \frac{{\mu }_{3+i}}{{\mu }_i}}|c_i| = |c_i| =\beta =0.9<d_i=1,i=1,2,3,k=0,1,2,\dots$$

and

$${\displaystyle \frac{{\mu }_{3+i}}{{\mu }_i}}\sum _{j=1}^3|a_{ij}|{\gamma }_j^{\left(k\right)}|=\alpha {\gamma }_i^{\left(k\right)}=1.3{\displaystyle \frac{3\sqrt{3}}{8}}\approx 0.844375<b_i=\beta =0.9,i=1,2,3,k=0,1,2,\dots ,$$

i.e., condition (A5) is satisfied.

According to Lemma 7 and Remark 13, condition (A6) is satisfied.

The equilibrium ($W^{\left(k\right)}=(Q_1^{\left(k\right)},Q_2^{\left(k\right)},Q_3^{\left(k\right)},P_1^{\left(k\right)},P_2^{\left(k\right)},P_3^{\left(k\right)})\in {\mathbb{R}}^6,P_i^{\left(k\right)},Q_i^{\left(k\right)}\ge 0,$) of the model (51) on the interval $[k,k+1]$ for $k=0,1,2,\dots$ is a solution of the system

$$Q_i^{\left(k\right)}=P_i^{\left(k\right)},Q_i^{\left(k\right)}=\alpha {\displaystyle \frac{{\left(Q_i^{\left(k\right)}\right)}^n}{k+1+{\left(Q_i^{\left(k\right)}\right)}^n}}+1,i=1,2,3.$$

(54)

The system (52) has solutions for any integers ($k\ge 0$), and these solutions form a decreasing sequence approaching 1. For example, $Q_i^{\left(0\right)},P_i^{\left(0\right)}\approx 2.05015,$$Q_i^{\left(1\right)},P_i^{\left(1\right)}\approx 1.80578,$ etc. Therefore, condition (A3) is satisfied.

According to Theorem 2, the equilibrium is generalized Mittag–Leffler stable in absolute values, and according to Definition 6 and Equation (41), we can conclude that the difference between any component of the solution of the model (50) and the corresponding component of the equilibrium does not exceed the difference between the initial values.

Let, for example, $q=0.3$, $\psi \left(t\right)=t$, and $\rho =1$ (Caputo fractional derivative). Then, the constant of the stability is $M_k=L_k{\displaystyle \frac{{\mu }_{max}}{{\mu }_{min}}}=L_k$, where

$$L_k=\prod _{i=0}^{k-1}{E}_q(-0.0556252{(t-k)}^q),$$

${\eta }_k=-0.1<0,{\varsigma }_k\approx -0.0556252<0,$ and ${\lambda }_k\approx 0.0556252>0.$

5. Conclusions

In this paper, gene regulatory networks are studied in the case where the changeable dynamics are described by a given switching rule and by a generalized Caputo fractional derivative with respect to another function. We study the case of short memories, i.e., the lower limit of the applied fractional derivative is changed at any time of the activation of the switching rule. Two types of Lyapunov functions—a quadratic one and one with absolute values—are considered. Initially, some properties of their generalized Caputo fractional derivatives with respect to another function are studied. These properties are useful not only for the study in this paper but also to study various types of stabilities of differential equations and models with this type of fractional derivative. In this paper, the proved properties of both Lyapunov functions are the basis of the Mittag–Leffler-type stability of the considered switched model.