Finite-Time Stability in Nonhomogeneous Delay Differential Equations of Fractional Hilfer Type

Abstract

In the current contribution, integral representations of the solutions of homogeneous and nonhomogeneous delay differential equation of a fractional Hilfer derivative are established in terms of the delayed Mittag-Leffler-type matrix function of two parameters. By using the method of variation of constants, the solution representations are represented. Finite-time stability of the solutions is examined with provision of appropriate sufficient conditions. Finally, an illustrated numerical example is introduced to apply the theoretical results.

1. Introduction

Equations involving fractional derivatives and time delays are so-called fractional delay differential equations (FDDEs). The term time delays in FDDEs expresses the history of an earlier state. A time delay is very often encountered in different technical branches of science, e.g., physics, chemistry, control systems, electro-chemistry, bioengineering, population dynamics and many other areas [1,2,3,4,5,6,7]. In more complex interconnected systems, time delays may also be introduced, since changes in one variable may affect other variables with some lags. FDDEs have drawn great interest due to their potentiality to constitute complex phenomena [8,9,10]. Hence, the study of fractional delay differential equations has widespread interest. The analysis and application of models incorporating fractional delay differential equations (FDDEs) rely on a theoretical basis that still presents some not completely clear aspects [11].

Stability analysis has extremely significant problems for the control systems. A stable system may involve an unacceptable transient performance at one fixed point in experimental implementations. Asymptotic stability, Lyapunov stability and exponential stability are possibilities, with the behavior of systems through an infinite time interval. These lead to the inevitability of consideration of the boundedness of the trajectory of a system over determining a finite-time interval from the engineering point of view, or as an alternative to a given long-time asymptotical behavior for the trajectory of a system from the mathematics point of view. Consequently, the term finite-time stability (the boundedness of system trajectory) is shown to describe the behavior of dynamical systems [12].

As an alternative to classical stability analysis, Dorato [13] introduced, for the first time, the concept of finite-time stability (FTS) to delay differential equations, which has demonstrated to be more appropriate for equations whose variables must lie within explicit bounds. FTS is regarded as the attitude of the solution of a system with finite intervals. It has need of specific bounds on system variables. This means the concept of FTS diverges from classical stability.

It is worth mentioning that Lazarevic [14] examined the FTS of linear fractional delay differential equations wicontrols. He established some sufficient conditions to investigate FTS. Thenceforth, there have been many contributions [15,16,17,18,19,20] studying the FTS of fractional differential equations with the aid of Gronwall inequality and linear-matrix inequality.

Firstly, Khusainov and Shukin [21] developed the term of an exponential matrix $e^{Bt}$ for linear differential equation $y^{\prime }\left(x\right)=By\left(x\right)$ for a delayed exponential matrix $e_{\tau }^{Bt}$ for the delay differential equation of the form

$$\begin{array}{ccc}\hfill y^{{}^{\prime }}\left(x\right)& =By(x-\tau ),\hfill & \hfill x\in (0,\infty ),\\ \hfill y\left(x\right)& =\psi \left(x\right),\hfill & \hfill x\in [-\tau ,0]\end{array}$$

where B is $n\times n$ constant matrix and $y:[-\tau ,\infty )⟶{\mathbb{R}}^n$ and $\psi :[-\tau ,0]\to \mathbb{R}$ are continuous functions. For the discrete system, Diblik and Khusainov [22] transferred this idea to seek a solution by establishing a matrix with a discrete delayed exponential. For more continued contributions, several authors established new definitions for delayed Mittag-Leffler-type matrix functions (DMLMF2P). Li and Wang [23] introduced DMLMF1P ${\mathbb{E}}_{\tau }^{B{x}^{\alpha }}$, which has been used to provide a representation to the solution of the equation

$$\begin{array}{ccc}\hfill {(}^c{\mathbb{D}}_{0+}^{\delta }y)\left(t\right)& =By(t-\tau ),\hfill & \hfill t\in [0,T],\tau >0,\\ \hfill y\left(t\right)& =\psi \left(t\right),\hfill & \hfill -\tau \le t\le 0\end{array}$$

where ${}^c{\mathbb{D}}_{0+}^{\delta }$ denotes the Caputo derivative of order $\delta \in (0,1)$. They completed their work by introducing the concept of DMLMF2P ${\mathbb{E}}_{\tau ,\beta }^{B{x}^{\alpha }}$ to be a representation of the solution to the nonhomogeneous fractional delay differential equation [24]

$$\begin{array}{ccc}\hfill {(}^c{\mathbb{D}}_{0+}^{\delta }y)\left(t\right)& =By(t-\tau )+g\left(t\right),\hfill & \hfill t\in [0,T],\tau >0,\\ \hfill y\left(t\right)& =\psi \left(t\right),\hfill & \hfill -\tau \le t\le 0.\end{array}$$

In addition, they investigated the FTS results. Instead of the Caputo derivative, Yang et al. [25] studied the previous equations with a fractional derivative according to Hadamard. For more about DMLMF2P, we refer the reader to [26,27,28].

Fractional calculus is a standout amidst the utmost accurate devices to redescribe natural phenomena. Indeed, the utilization of a fractional differential equation for modeling is more effectual than integer derivatives, which can easily demonstrate originally properties and memory [29,30,31,32]. It is worth mentioning that the topic of fractional calculus has drawn the attentionof a huge number of contributors, due to its important role and applications in various scientific fields [33,34,35,36,37].

It is remarkable that the fractional Hilfer derivative is considered as a generalization of both fractional derivatives according to Caputo and Riemann-Liouville. This is because it depends on a parameter that takes its value from zero to one. If this parameter is close to one, it converts to a Caputo type and if it approaches zero, it converts to a Riemann-Liouville type. For this reason and inspired by the works above, we introduce an integral representation to the solution of the homogeneous system of a Hilfer type

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill & \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }y\right)\left(x\right)=By(x-\tau ),\hfill & \hfill B\in {\mathbb{R}}^{n\times n},x\in [0,T],\tau >0,\\ \hfill & y\left(x\right)=\varphi \left(x\right),\hfill & \hfill \varphi \left(x\right)\in {\mathbb{R}}^n,-\tau <x\le 0,\\ \hfill & \underset{x\to -{\tau }^{+}}{lim}\left({\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }y\right)\left(x\right)=b,\hfill & \hfill b\in {\mathbb{R}}^n\end{array}\right.\end{array}$$

(1)

where ${\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }$ is the fractional Hilfer derivative of type $\beta \in [0,1]$ and order $\alpha \in (0,1)$, ${\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }y$ denotes the R-L fractional integral of order $1-\gamma$ such that $\gamma =\alpha +\beta -\alpha \beta$, $T=k\tau$, $k\in \mathbb{N}$, $\tau$ is a fixed moment, and $\varphi (·)$ is an arbitrary Riemann-Liouville differentiable function. That is: ${\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi$ exists and $\varphi \in C((-\tau ,0],{\mathbb{R}}^n)$. It is clear that $0<\gamma <1$, $\gamma \ge \alpha$ and $\gamma \ge \beta$.

Since the study of nonhomogeneous equations is more interesting and the results of homogeneous system can be exploited by applying the method of variation of constants, we provide an integral representation to the solution of the following nonhomogeneous system, of a Hilfer type

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill & \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }y\right)\left(x\right)=By(x-\tau )+h\left(x\right),\hfill & \hfill x\in [0,T],\tau >0,\\ \hfill & y\left(x\right)=\varphi \left(x\right),\hfill & \hfill \varphi \left(x\right)\in {\mathbb{R}}^n,-\tau <x\le 0,\\ \hfill & \underset{x\to -{\tau }^{+}}{lim}\left({\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }y\right)\left(x\right)=b,\hfill & \hfill b\in {\mathbb{R}}^n\end{array}\right.\end{array}$$

(2)

where $h\left(x\right)\in C([0,T],{\mathbb{R}}^n)$. For both systems, we investigate the FTS results under appropriate conditions.

The presented manuscript is organized as follows: The next section introduces the basic definitions and concepts of an R-L fractional integral, Hilfer derivative and FTS. Section 3 is regarded to include the solutions of homogeneous and nonhomogeneous fractional differential equations. The finite-time stability results are stated in Section 4. In Section 5, we give an example to demonstrate the validity of our theoretical results. We end our work by including a conclusion section.

2. Basic Definitions

The current section is devoted to stating the basic definitions and concepts of an R-L fractional integral, Hilfer derivative and FTS. Throughout this paper, let $\Theta$ and I denote zero and the standard identity matrices, respectively. Furthermore, let $t,r\in \mathbb{R}$, $t<r$ and $C((t,r],{\mathbb{R}}^n)$ denote a Banach space of continuous vector-valued functions.

Definition 1

([8]). The left-side R-L integral of order $\delta >0$ for a function $y:[t,\infty )\to \mathbb{R}$ can be written as

$$\left({\mathbb{I}}_{t^{+}}^{\delta }y\right)\left(x\right)=\frac{1}{\Gamma \left(\delta \right)}{\int }_t^x{(x-s)}^{\delta -1}y\left(s\right)ds.$$

Definition 2

([8]). The R-L fractional derivative of order $m-1\le \delta <m,m\in \mathbb{N}$ for a function $y:[t,\infty )\to \mathbb{R}$ can be written as

$$\begin{array}{ccc}\hfill \left({\mathbb{D}}_{t^{+}}^{\delta }y\right)\left(x\right)& =& \left({\mathbb{D}}_{t^{+}}^m{\mathbb{I}}_{t^{+}}^{m-\delta }y\right)\left(x\right)\hfill \\ & =& \frac{1}{\Gamma (m-\delta )}\frac{d^m}{d{x}^m}{\int }_t^x{(x-s)}^{m-\delta -1}y\left(s\right)ds.\hfill \end{array}$$

Definition 3

([38]). The derivative of order $m-1\le \delta <m,m\in \mathbb{N}$, due to Caputo, for a function $y:[t,\infty )\to \mathbb{R}$ can be written as

$$\begin{array}{ccc}\hfill {(}^c{\mathbb{D}}_{t^{+}}^{\delta }y)\left(x\right)& =& \left({\mathbb{D}}_{t^{+}}^{\delta }y\right)\left(x\right)-\sum _{k=0}^{m-1}\frac{y^{\left(k\right)\left(t\right)}}{\Gamma (k-\delta +1)}{(x-t)}^{k-\delta },x>t\hfill \end{array}$$

Lemma 1.

Let $\delta >0$, $\lambda >0$ and $0<\gamma <1$. Then,

$$\begin{array}{cc}\hfill {\mathbb{I}}_{t^{+}}^{\delta }{(x-t)}^{\lambda -1}& =\frac{\Gamma \left(\lambda \right)}{\Gamma (\lambda +\delta )}{(x-t)}^{\lambda +\delta -1},\hfill \\ \hfill {\mathbb{D}}_{t^{+}}^{\gamma }{(x-t)}^{\lambda -1}& =\left\{\begin{array}{ccc}& \frac{\Gamma \left(\lambda \right)}{\Gamma (\lambda -\gamma )}{(x-t)}^{\lambda -\gamma -1},\hfill & \hfill \lambda \ne \gamma ,\\ \hfill & 0,\hfill & \hfill \lambda =\gamma \end{array}\right.\hfill \\ \hfill {\mathbb{D}}_{t^{+}}^{\gamma }{\mathbb{I}}_{t^{+}}^{\gamma }f\left(x\right)& =f\left(x\right),\hfill \\ \hfill {\mathbb{I}}_{t^{+}}^{\gamma }{\mathbb{D}}_{t^{+}}^{\gamma }f\left(x\right)& =f\left(x\right)-\frac{{(x-t)}^{\gamma -1}}{\Gamma \left(\gamma \right)}{\mathbb{I}}_{t^{+}}^{1-\gamma }f\left(t\right).\hfill \end{array}$$

Definition 4

([39]). Let $\beta \in [0,1]$ and $m-1<\delta <m$, $m\in \mathbb{N}$. The fractional derivative due to the Hilfer of type β and order α is presented as

$${\mathbb{D}}_{t^{+}}^{\alpha ,\beta }f\left(x\right)={\mathbb{I}}_{t^{+}}^{\beta (m-\alpha )}{\mathbb{D}}_{t^{+}}^m{\mathbb{I}}_{t^{+}}^{(1-\beta )(m-\alpha )}f\left(x\right),x>t.$$

It is not difficult to show the properties.

Lemma 2.

Let $\beta \in [0,1]$, $0<\alpha <1$, $\lambda >0$ and $\gamma =\alpha +\beta -\alpha \beta$. Then,

$$\begin{array}{cc}\hfill {\mathbb{I}}_{t^{+}}^{\alpha }{\mathbb{D}}_{t^{+}}^{\alpha ,\beta }f\left(x\right)& ={\mathbb{I}}_{t^{+}}^{\gamma }{\mathbb{D}}_{t^{+}}^{\gamma }f\left(x\right)=f\left(x\right)-\frac{{(x-t)}^{\gamma -1}}{\Gamma \left(\gamma \right)}{\mathbb{I}}_{t^{+}}^{1-\gamma }f\left(t\right),\hfill \\ \hfill {\mathbb{D}}_{t^{+}}^{\alpha ,\beta }{(x-t)}^{\lambda -1}& =\left\{\begin{array}{ccc}& \frac{\Gamma \left(\lambda \right)}{\Gamma (\lambda -\alpha )}{(x-t)}^{\lambda -\alpha -1},\hfill & \hfill \lambda \ne \gamma ,\\ \hfill & 0,\hfill & \hfill \lambda =\gamma \end{array}\right..\hfill \end{array}$$

Definition 5

([15]). The system given by (1) is FTS with respect to $\{0,I,\gamma ,\epsilon \}$ if and only if $\Vert \varphi \Vert <\gamma$ implies $\Vert y\left(x\right)\Vert <\epsilon$ for all $x\in [0,a],a>0$, γ, ε are real positive numbers and $\varphi \left(x\right)(-\tau <x\le 0)$ is the initial function of observation.

Definition 6

([27]). Let γ and σ be two complex parameters with $\Re \left(\gamma \right)>0$. The Mittag-Leffler function ${\mathbb{E}}_{\gamma ,\sigma }$ is defined by

$${\mathbb{E}}_{\gamma ,\sigma }\left(z\right)=\sum _{r=0}^{\infty }\frac{z^r}{\Gamma (\gamma r+\sigma )},z\in \mathbb{C}$$

Next, we provide the concept of DMLMF2P.

Definition 7

([24]). The DMLMF2P ${\mathbb{E}}_{\tau ,\beta }^{B{x}^{\alpha }}:\mathbb{R}⟶{\mathbb{R}}^{n\times n}$ is represented as

$$\begin{array}{c}\hfill {\mathbb{E}}_{\tau ,\beta }^{B{x}^{\alpha }}=\left\{\begin{array}{ccc}\hfill & \Theta ,\hfill & \hfill -\infty <x\le -\tau ,\\ \hfill & I\frac{{(x+\tau )}^{\beta -1}}{\Gamma \left(\beta \right)},\hfill & \hfill -\tau <x\le 0,\\ \hfill & I\frac{{(x+\tau )}^{\beta -1}}{\Gamma \left(\beta \right)}+B\frac{x^{\alpha +\beta -1}}{\Gamma (\alpha +\beta )}+\cdots \hfill \\ \hfill & +B^k\frac{{(x-(k-1)\tau )}^{k\alpha +\beta -1}}{\Gamma (k\alpha +\beta )},\hfill & \hfill (k-1)\tau <x\le k\tau ,k\in \mathbb{N}.\end{array}\right.\end{array}$$

In a similar way to Lemma 2.4 in [23] and Lemma 2.10 in [24], we can present the next lemma.

Lemma 3.

Let $k\in \mathbb{N}$, $x\in \left[\right(k-1)\tau ,k\tau ]$, $0<\alpha <1$, $0\le \beta \le 1$ with $\alpha +\beta \ge 1$ and $f:(-\tau ,0]\to {\mathbb{R}}^n$ be a continuous function. Then,

$$\begin{array}{cc}\hfill & ∥{\mathbb{E}}_{\tau ,\beta }^{B{x}^{\alpha }}∥\le {(x+\tau )}^{\beta -1}{\mathbb{E}}_{\alpha ,\beta }\left({\parallel B\parallel (x+\tau )}^{\alpha }\right)\le {\tau }^{\beta -1}{\mathbb{E}}_{\alpha ,\beta }\left(\parallel B\parallel x^{\alpha }\right),\hfill \\ & \\ \hfill & ∥{\int }_{-\tau }^0{\mathbb{E}}_{\tau ,\beta }^{B{(x-\tau -s)}^{\alpha }}f\left(s\right)ds∥\le {\mathbb{E}}_{\alpha ,\beta }\left(\parallel B\parallel x^{\alpha }\right){\int }_{-\tau }^0{(x-s)}^{\beta -1}\parallel f\left(s\right)\parallel ds.\hfill \end{array}$$

By employing Lemma 2.19 in [24], we can present the next lemma.

Lemma 4.

Let $k\in \mathbb{N}$, $x\in \left[\right(k-1)\tau ,k\tau ]$, $0\le \beta \le 1$, $0<\alpha <1$ with $\alpha +\beta \ge 1$ and $f:(-\tau ,0]\to {\mathbb{R}}^n$ be a continuous function. Then,

$$\begin{array}{c}\hfill ∥{\int }_0^x{\mathbb{E}}_{\tau ,\beta }^{B{(x-\tau -s)}^{\alpha }}f\left(s\right)ds∥\le {\mathbb{E}}_{\alpha ,\beta }\left(\parallel B\parallel x^{\alpha }\right){\int }_0^x{(x-s)}^{\beta -1}\parallel f\left(s\right)\parallel ds.\end{array}$$

3. Representation of the Solutions

This section is regarded to include the solutions of homogeneous and nonhomogeneous fractional differential equations through utilizing the system of symbols ${\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}$ for DMLMF2P.

Theorem 1.

Let ${\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}:\mathbb{R}⟶{\mathbb{R}}^{n\times n}$ be the DMLMF2P defined in Definition 7. Then,

$$\left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{\mathbb{E}}_{\tau ,\gamma }^{B{t}^{\alpha }}\right)\left(x\right)=B{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau )}^{\alpha }}.$$

(3)

Proof. 

We divide the real line into three intervals $(-\infty ,-\tau ]$, $(-\tau ,0]$ and $(0,\infty )={\bigcup }_{k=1}^{\infty }((k-1)\tau ,k\tau ]$ and show that Equation (3) is satisfied in each interval.

Interval $(-\infty ,-\tau ]$.

For arbitrary $x\in (-\infty ,-\tau ]$,

$${\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}={\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau )}^{\alpha }}=\Theta .$$

Interval $(-\tau ,0]$.

For arbitrary $x\in (-\tau ,0]$ , the DMLMF2P is defined as

$${\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}=I\frac{{(x+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}.$$

By applying Hilfer fractional derivative and using Lemma 2, we obtain

$$\begin{array}{c}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{\mathbb{E}}_{\tau ,\gamma }^{B{t}^{\alpha }}\right)\left(x\right)=\left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }\left(I\frac{{(t+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}\right)\right)\left(x\right)=\Theta =B{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau )}^{\alpha }}.\end{array}$$

Interval $\left(\right(k-1)\tau ,k\tau ]$, $k\in \mathbb{N}$.

Suppose that $x\in \left(\right(k-1)\tau ,k\tau ]$ and $k\in \mathbb{N}$. The mathematical induction is used to demonstrate our results.

1.

For $k=1$, $0<x\le \tau$, the DMLMF2P is defined as

$${\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}=I\frac{{(x+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}+B\frac{x^{\alpha +\gamma -1}}{\Gamma (\alpha +\gamma )}.$$

Operating by Hilfer derivative with using Lemma 2, we obtain

$$\begin{array}{ccc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{\mathbb{E}}_{\tau ,\gamma }^{B{t}^{\alpha }}\right)\left(x\right)& =& \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }\left(I\frac{{(t+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}+B\frac{t^{\alpha +\gamma -1}}{\Gamma (\alpha +\gamma )}\right)\right)\left(x\right)\hfill \\ & =& \Theta +B\left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }\frac{t^{\alpha +\gamma -1}}{\Gamma (\alpha +\gamma )}\right)\left(x\right)=B\frac{x^{\gamma -1}}{\Gamma \left(\gamma \right)}\hfill \\ & =& B\left(I\frac{{((x-\tau )+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}\right)=B{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau )}^{\alpha }}\hfill \end{array}$$

which shows that Equation (3) is true for $k=1$.

2.

For $k=N\in \mathbb{N}$ and $(N-1)\tau <x\le N\tau$. Assume that the relation below is verified

$$\begin{array}{cc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{\mathbb{E}}_{\tau ,\gamma }^{B{t}^{\alpha }}\right)\left(x\right)& =B\frac{x^{\gamma -1}}{\Gamma \left(\gamma \right)}+B^2\frac{{(x-\tau )}^{\alpha +\gamma -1}}{\Gamma (\alpha +\gamma )}+\cdots \hfill \\ \hfill & +B^N\frac{{(x-(N-2)\tau )}^{(N-1)\alpha +\gamma -1}}{\Gamma \left(\right(N-1)\alpha +\gamma )}=B{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau )}^{\alpha }}.\hfill \end{array}$$

3.

For $k=N+1$ and $N\tau <x\le (N+1)\tau$, the DMLMF2P is defined as

$$\begin{array}{ccc}\hfill {\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}& =& I\frac{{(x+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}+B\frac{{\left(x\right)}^{\alpha +\gamma -1}}{\Gamma (\alpha +\gamma )}+\cdots \hfill \\ & +& B^N\frac{{(x-(N-1)\tau )}^{N\alpha +\gamma -1}}{\Gamma (N\alpha +\gamma )}+B^{N+1}\frac{{(x-N\tau )}^{(N+1)\alpha +\gamma -1}}{\Gamma \left(\right(N+1)\alpha +\gamma )}.\hfill \end{array}$$

Operating by Hilfer derivative to obtain

$$\begin{array}{cc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{\mathbb{E}}_{\tau ,\gamma }^{B{t}^{\alpha }}\right)\left(x\right)& =B\frac{x^{\gamma -1}}{\Gamma \left(\gamma \right)}+B^2\frac{{(x-\tau )}^{\alpha +\gamma -1}}{\Gamma (\alpha +\gamma )}+\cdots \hfill \\ \hfill & +B^N\frac{{(x-(N-2)\tau )}^{(N-1)\alpha +\gamma -1}}{\Gamma \left(\right(N-1)\alpha +\gamma )}+B^{N+1}\frac{{(x-(N-1)\tau )}^{N\alpha +\gamma -1}}{\Gamma (N\alpha +\gamma )}\hfill \\ \hfill & =B{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau )}^{\alpha }}\hfill \end{array}$$

These conclude that relation (3) is satisfied for all $(k-1)\tau <x\le k\tau$ and $k\in \mathbb{N}$. □

The following theorem is concerned with establishing a representation to the homogeneous system (1).

Theorem 2.

The solution of problem (1) can be expressed as

$$y\left(x\right)={\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}b+{\int }_{-\tau }^0{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(s\right)ds.$$

(4)

Proof. 

Assume that $W_0\left(x\right)={\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}$ satisfies equation (3). Then, the solution of (1) should be

$$y\left(x\right)=W_0\left(x\right)C+{\int }_{-\tau }^0{W}_0(x-\tau -s)z\left(s\right)ds$$

(5)

where $C\in {\mathbb{R}}^n$ is an obscure constant vector and $z(·)$ is an obscure R-L differentiable function. Substituting by Equation (5) into the second condition in (1) using Definition 1, we have

$$\begin{array}{cc}\hfill b& =\underset{x\to -{\tau }^{+}}{lim}\left({\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }y\right)\left(x\right)\hfill \\ \hfill & =\underset{x\to -{\tau }^{+}}{lim}\left(\frac{1}{\Gamma (1-\gamma )}{\int }_{-\tau }^x{(x-t)}^{-\gamma }\left[W_0\left(t\right)C+{\int }_{-\tau }^0{W}_0(t-\tau -s)z\left(s\right)ds\right]dt\right).\hfill \end{array}$$

Letting $x⟶-{\tau }^{+}$, using Definition 7, we obtain $W_0(t-\tau -s)=\Theta$ for $-\tau <s\le 0$. Hence, we have

$$\begin{array}{cc}\hfill b& =\underset{x\to -{\tau }^{+}}{lim}\left(\frac{1}{\Gamma (1-\gamma )}{\int }_{-\tau }^x{(x-t)}^{-\gamma }{W}_0\left(t\right)Cdt\right)\hfill \\ \hfill & =\underset{x\to -{\tau }^{+}}{lim}\left(\frac{C}{\Gamma (1-\gamma )\Gamma \left(\gamma \right)}{\int }_{-\tau }^x{(x-t)}^{-\gamma }{(t+\tau )}^{\gamma -1}dt\right)\hfill \\ \hfill & =\underset{x\to -{\tau }^{+}}{lim}\left(\frac{C}{\Gamma (1-\gamma )\Gamma \left(\gamma \right)}\mathbb{B}[1-\gamma ,\gamma ]\right)=C.\hfill \end{array}$$

It indicates that Equation (5) has the form

$$y\left(x\right)={\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}b+{\int }_{-\tau }^0{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}z\left(s\right)ds.$$

When $-\tau <x\le 0$, one has to divide the previous interval into two subintervals

Interval I when$-\tau <s\le x$.

In this case, we obtain $-\tau <x-\tau -s\le x$, which gives

$${\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}=I\frac{{(x-s)}^{\gamma -1}}{\Gamma \left(\gamma \right)},-\tau <s\le x.$$

Interval II when$x\le s\le 0$.

In this case, we obtain $x-\tau \le x-\tau -s\le -\tau$, which gives

$${\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}=\Theta ,x\le s\le 0.$$

Then, according to problem (1) with the two cases above, the previous integral may take the form

$$\begin{array}{cc}\hfill \varphi \left(x\right)& =\frac{{(x+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}b+{\int }_{-\tau }^x\frac{{(x-s)}^{\gamma -1}}{\Gamma \left(\gamma \right)}z\left(s\right)ds=\frac{{(x+\tau )}^{\gamma -1}}{\Gamma \left(\gamma \right)}b+{\mathbb{I}}_{-{\tau }^{+}}^{\gamma }z\left(x\right).\hfill \end{array}$$

By operating the R-L fractional derivatives ${\mathbb{D}}_{-{\tau }^{+}}^{\gamma }$ on both sides of the previous equation using Lemma 1, we can obtain

$$\begin{array}{ccc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(x\right)& =& z\left(x\right).\hfill \end{array}$$

This ends the proof. □

In the following theorem, we are seeking to provide a particular solution corresponding to the nonhomogeneous term $h\left(x\right)$ in system (2) by means of using the method of variation of constants. We apply the same idea used in [24].

Theorem 3.

Let $y_p\left(x\right)$ be a particular solution of problem (2) with $y_p\left(x\right)=0$ for all $x\in [-\tau ,0]$. Then,

$$y_p\left(x\right)={\int }_0^x{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}\left({\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\right)\left(s\right)ds.$$

(6)

Proof. 

According to the variation of variables method, the solution of problem (2) should be

$$y_p\left(x\right)={\int }_0^x{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}a\left(s\right)ds$$

(7)

where $a\left(s\right)$ is an unknown vector function. Since the particular solution satisfies problem (2), for $k\tau <x\le (k+1)\tau$ and $k\in {\mathbb{N}}_0$, we obtain

$$\begin{array}{cc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{y}_p\right)\left(x\right)& =B{y}_p(x-\tau )+h\left(x\right)\hfill \\ \hfill & =B{\int }_0^{x-\tau }{\mathbb{E}}_{\tau ,\gamma }^{B{(x-2\tau -s)}^{\alpha }}a\left(s\right)ds+h\left(x\right).\hfill \end{array}$$

(8)

Operating Hilfer derivative on both sides of (7) and using the result of Theorem 3.1 in [24], we have

$$\begin{array}{ccc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }{y}_p\right)\left(x\right)& =& \left({\mathbb{D}}_{0^{+}}^{\alpha ,\beta }{y}_p\right)\left(x\right)\hfill \\ & =& \left({\mathbb{I}}_{0^{+}}^{\gamma -\alpha }{\mathbb{D}}_{0^{+}}^{\gamma }{y}_p\right)\left(x\right)\hfill \\ & =& {\mathbb{I}}_{0^{+}}^{\gamma -\alpha }\left[a\left(x\right)+B{\int }_0^{x-\tau }{\mathbb{E}}_{\tau ,\gamma }^{B{(x-2\tau -s)}^{\alpha }}a\left(s\right)ds\right]\hfill \\ & =& {\mathbb{I}}_{0^{+}}^{\gamma -\alpha }a\left(x\right)+\frac{B}{\Gamma (\gamma -\alpha )}{\int }_0^x{(x-t)}^{\gamma -\alpha -1}\left({\int }_0^{t-\tau }{\mathbb{E}}_{\tau ,\gamma }^{B{(t-2\tau -s)}^{\alpha }}a\left(s\right)ds\right)dt\hfill \\ & =& {\mathbb{I}}_{0^{+}}^{\gamma -\alpha }a\left(x\right)+\frac{B}{\Gamma (\gamma -\alpha )}{\int }_0^{x-\tau }a\left(s\right)\left({\int }_{s+\tau }^x{(x-t)}^{\gamma -\alpha -1}{\mathbb{E}}_{\tau ,\gamma }^{B{(t-2\tau -s)}^{\alpha }}dt\right)ds\hfill \\ & =& {\mathbb{I}}_{0^{+}}^{\gamma -\alpha }a\left(x\right)+B{\int }_0^{x-\tau }{\mathbb{E}}_{\tau ,\gamma }^{B{(x-2\tau -s)}^{\alpha }}a\left(s\right)ds.\hfill \end{array}$$

Equating with Equation (8) leads to

$$h\left(x\right)={\mathbb{I}}_{0^{+}}^{\gamma -\alpha }a\left(x\right).$$

Operating by R-L derivative using the third statement in Lemma 1, we find that

$$a\left(x\right)={\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right).$$

This completes the proof. □

Outlining the previous two theorems, we can formulate the representation of the solution to problem (2). It is known that the general solution $y\left(x\right)$ of problem (2) can be represented as a sum of the complementary solution for the homogeneous case obtained in Theorem 2 and the particular solution for the nonhomogeneous case obtained in Theorem 3. Therefore, we can derive the following theorem.

Theorem 4.

The solution of problem (2) can be represented as

$$\begin{array}{cc}\hfill y\left(x\right)& ={\mathbb{E}}_{\tau ,\gamma }^{B{x}^{\alpha }}b+{\int }_{-\tau }^0{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(s\right)ds\hfill \\ \hfill & +{\int }_0^x{\mathbb{E}}_{\tau ,\gamma }^{B{(x-\tau -s)}^{\alpha }}\left({\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(s\right)\right)ds.\hfill \end{array}$$

(9)

4. Finite-time Stability Results

The current section is presented to give the FTS results by utilizing DMLMF2P. Before proving the following theorems, we introduce the following hypotheses.

(R₁) 

The functions (${\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)\in C([0,T],{\mathbb{R}}^n)$ and $\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(x\right)\in C((-\tau ,0],{\mathbb{R}}^n)$.

(R₂) 

There exists a positive function $\psi \left(x\right)\in C([0,T],{\mathbb{R}}^{+})$, such that $\parallel {\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)\parallel \le \psi \left(x\right)$.

(R₃) 

There exists a positive function $\phi \left(x\right)\in C((-\tau ,0],{\mathbb{R}}^{+})$, such that $\parallel \left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(x\right)\parallel \le \phi \left(x\right)$.

(R₄) 

There exists a positive function $\eta \left(x\right)\in L^q([0,T],{\mathbb{R}}^{+})$ with $p>1$, $q>1$ and $\frac{1}{p}+\frac{1}{q}=1$, such that $\parallel {\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)\parallel \le \eta \left(x\right)$ and

$$\begin{array}{ccc}\hfill Q\left(x\right)& =& {\left({\int }_0^x{\left(\eta \left(t\right)\right)}^{q}dt\right)}^{\frac{1}{q}}<\infty .\hfill \end{array}$$

(R₅) 

There exists a positive constant M, such that

$$\begin{array}{ccc}\hfill M& =& {\left({\int }_{-\tau }^0{∥\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(s\right)∥}^{q}ds\right)}^{\frac{1}{q}}<\infty .\hfill \end{array}$$

Theorem 5.

Assume that the assumptions $\left(R_1\right)-\left({\mathrm{R}}_3\right)$ hold. If $\alpha \ge (1-\beta )/(2-\beta )$, $\Vert \varphi \left(x\right)\Vert <\delta$ and

$$\begin{array}{c}\hfill {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })<\frac{\gamma \epsilon }{\gamma \parallel b\parallel {\tau }^{\gamma -1}+{\tau }^{\gamma }\phi +T^{\gamma }\psi }\end{array}$$

for all $x\in [0,T]$ where $\phi ={sup}_{x\in (-\tau ,0]}\left\{\phi \left(x\right)\right\}$ and $\psi ={sup}_{x\in [0,T]}\left\{\psi \left(x\right)\right\}$. Then, the problem (2) is a finite time stable with respect to $\{0,[0,T],\tau ,\delta ,\epsilon \}$.

Proof. 

According to the solution of (2) which is expressed in form (9) using the properties of the norm and the results obtained in Lemmas 3 and 4, we have

$$\begin{array}{ccc}\hfill \Vert y\left(x\right)\Vert & \le & {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+\parallel \phi \left(x\right)\parallel {\int }_{-\tau }^0{(x-s)}^{\gamma -1}ds+\parallel \psi \left(x\right)\parallel {\int }_0^x{(x-s)}^{\gamma -1}ds\right\}\hfill \\ & =& {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+\frac{{(x+\tau )}^{\gamma }-x^{\gamma }}{\gamma }\parallel \phi \left(x\right)\parallel +\frac{x^{\gamma }}{\gamma }\parallel \psi \left(x\right)\parallel \right\}.\hfill \end{array}$$

It is easy to see that the function $x\mapsto {(x+\tau )}^{\gamma }-x^{\gamma }$ is decreasing on $[0,T]$ for all $\tau >0$ and $0<\gamma <1$, which implies that ${(x+\tau )}^{\gamma }-x^{\gamma }\le {\tau }^{\gamma }$; therefore, we have

$$\begin{array}{ccc}\hfill \Vert y\left(x\right)\Vert & \le & {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+\frac{{\tau }^{\gamma }}{\gamma }\phi +\frac{T^{\gamma }}{\gamma }\psi \right\}<\epsilon ,x\in [0,T].\hfill \end{array}$$

This completes the proof. □

Corollary 1.

Assume that the assumptions $\left({\mathrm{R}}_1\right)$ and $\left({\mathrm{R}}_3\right)$ hold. If $\alpha \ge (1-\beta )/(2-\beta )$, $\Vert \varphi \left(x\right)\Vert <\delta$ and

$$\begin{array}{c}\hfill {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })<\frac{\gamma \epsilon {\tau }^{1-\gamma }}{\gamma \parallel b\parallel +\tau \phi }\end{array}$$

for all $x\in [0,T]$ where $\phi ={sup}_{x\in (-\tau ,0]}\left\{\phi \left(x\right)\right\}$, then problem (1) is a finite time stable with respect to $\{0,[0,T],\tau ,\delta ,\epsilon \}$.

Theorem 6.

Assume that the assumptions $\left({\mathrm{R}}_1\right)$, $\left({\mathrm{R}}_4\right)$ and $\left({\mathrm{R}}_5\right)$ hold. If $\alpha \ge (1-\beta )/(2-\beta )$, $\gamma >1-\frac{1}{p}=\frac{1}{q}$, $\Vert \varphi \left(x\right)\Vert <\delta$ and

$${\mathbb{E}}_{\alpha ,\gamma }\left(\left|\left|B\right|\right|x^{\alpha }\right)\left[\parallel b\parallel {\tau }^{\gamma -1}+\frac{M{\tau }^{\gamma -\frac{1}{q}}+Q{T}^{\gamma -\frac{1}{q}}}{{(1-p(1-\gamma ))}^{\frac{1}{p}}}\right]<\epsilon$$

for all $x\in [0,T]$ where $Q={sup}_{0\le x\le T}Q\left(x\right)$, then problem (2) is finite time stable with respect to $\{0,[0,T],\tau ,\delta ,\epsilon \}$.

Proof. 

By the properties of the norm, we have

$$\begin{array}{ccc}\hfill \Vert y\left(x\right)\Vert & \le & {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+{\int }_{-\tau }^0{(x-s)}^{\gamma -1}\parallel \left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(s\right)\parallel ds\right.\hfill \\ & +& \left.{\int }_0^x{(x-s)}^{\gamma -1}\parallel \left({\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\right)\left(s\right)\parallel ds\right\}\hfill \\ & \le & {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+{\left({\int }_{-\tau }^0{(x-s)}^{p(\gamma -1)}ds\right)}^{\frac{1}{p}}{\left({\int }_{-\tau }^0{∥\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(s\right)∥}^{q}ds\right)}^{\frac{1}{q}}\right.\hfill \\ & +& \left.{\left({\int }_0^x{(x-s)}^{p(\gamma -1)}ds\right)}^{\frac{1}{p}}{\left({\int }_0^x{∥\left({\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\right)\left(s\right)∥}^{q}ds\right)}^{\frac{1}{q}}\right\}\hfill \\ & \le & {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+M{\left(\frac{{(x+\tau )}^{1-p(1-\gamma )}-x^{1-p(1-\gamma )}}{1-p(1-\gamma )}\right)}^{\frac{1}{p}}\right.\hfill \\ & +& \left.Q\left(x\right){\left(\frac{x^{1-p(1-\gamma )}}{1-p(1-\gamma )}\right)}^{\frac{1}{p}}\right\}.\hfill \end{array}$$

Since $p>1$ and $\gamma <1$, then $p(\gamma -1)+1<1$ and so, as above, we obtain

$$\begin{array}{ccc}\hfill \Vert y\left(x\right)\Vert & \le & {\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })\left\{\parallel b\parallel {\tau }^{\gamma -1}+M{\left(\frac{{\tau }^{1-p(1-\gamma )}}{p(\gamma -1)+1}\right)}^{\frac{1}{p}}\right.\hfill \\ & +& \left.Q{\left(\frac{x^{1-p(1-\gamma )}}{1-p(1-\gamma )}\right)}^{\frac{1}{p}}\right\}<\epsilon .\hfill \end{array}$$

This ends the proof. □

Corollary 2.

Assume that the assumptions $\left({\mathrm{R}}_1\right)$ and $\left({\mathrm{R}}_s\right)$ hold. If $\alpha \ge (1-\beta )/(2-\beta )$, $\gamma >1-\frac{1}{p}=\frac{1}{q}$, $\Vert \varphi \left(x\right)\Vert <\delta$ and

$${\mathbb{E}}_{\alpha ,\gamma }\left(\left|\left|B\right|\right|x^{\alpha }\right)\left[\parallel b\parallel {\tau }^{\gamma -1}+\frac{M{\tau }^{\gamma -\frac{1}{q}}}{{(1-p(1-\gamma ))}^{\frac{1}{p}}}\right]<\epsilon$$

for all $x\in [0,T]$, then problem (1) is finite time stable with respect to $\{0,[0,T],\tau ,\delta ,\epsilon \}$.

5. A Numerical Example

This section is presented to give an example to demonstrate the validity of our theoretical results. Here, we use $\parallel y\parallel ={\sum }_{i=1}^n\left|y_n\right|$ and $\parallel B\parallel ={max}_{1\le j\le n}{\sum }_{i=1}^n\left|b_{ij}\right|$, where $y=(y_1,y_2,\cdots ,y_n)\in {\mathbb{R}}^n$ is a vector and $A\in {\mathbb{R}}^{n\times n}$ with elements $b_{ij}\in \mathbb{R}$, which are a rectilinear norm or ${\ell }_1$-norm and matrix norm, respectively.

Example 1.

Consider the problem

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill & \left({\mathbb{D}}_{-2^{+}}^{\frac{1}{2},\frac{1}{4}}y\right)\left(x\right)=By(x-2)+h\left(x\right),\hfill & \hfill x\in [0,5]\\ \hfill & y\left(x\right)=\varphi \left(x\right)={\left(\frac{1}{4}{(x+2)}^2,\frac{1}{4}{(x+2)}^2\right)}^T,\hfill & \hfill -2<x\le 0\\ \hfill & \underset{x\to -{\tau }^{+}}{lim}\left({\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }y\right)\left(x\right)=b\hfill \end{array}\right.\end{array}$$

where $\alpha =0.5,\beta =0.25,\gamma =0.625,\tau =2,T=5,k=4,p=2$, $q=2$, $y\left(x\right)={(y_1\left(x\right),y_2\left(x\right))}^T$ and

$$\begin{array}{c}\hfill B=\left(\begin{array}{ccc}\hfill & 0.3\hfill & \hfill 0\\ \hfill & 0\hfill & \hfill -0.2\end{array}\right),h\left(x\right)=\frac{1}{100}\left(\begin{array}{cc}\hfill & x^3-2x^{2\gamma }\hfill \\ \hfill & x^2-x^{\gamma }\hfill \end{array}\right)\end{array}$$

It is easy to see that $\parallel B\parallel =0.3$, $\alpha =0.5>(1-\beta )(2-\beta )=3/7$ and $\gamma =0.625>1/q=1/2$. By carrying this out on Mathematica software, we can find that

$$\underset{0\le x\le 5}{sup}{\mathbb{E}}_{\alpha ,\gamma }(\Vert B\Vert x^{\alpha })={\mathbb{E}}_{0.5,0.625}(0.3\times 5^{0.5})\sim 2.43241$$

It is clear that the function

$$\varphi \left(x\right)={\left(\frac{1}{4}{(x+2)}^2,\frac{1}{4}{(x+2)}^2\right)}^T$$

is continuous on $(-2,0]$ and $\parallel \varphi \parallel ={sup}_{-2<x\le 0}\frac{1}{2}{(x+2)}^2=2$, which determines that $\delta =2$. According to Lemma 1, we obtain

$$\begin{array}{ccc}\hfill {\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }\varphi \left(x\right)& =& \frac{1}{2}{\left(\frac{{(x+2)}^{3-\gamma }}{\Gamma (4-\gamma )},\frac{{(x+2)}^{3-\gamma }}{\Gamma (4-\gamma )}\right)}^T.\hfill \end{array}$$

Since $y\left(x\right)=\varphi \left(x\right)$ for all $-\tau <x\le 0$, then

$$\begin{array}{cc}\hfill b& =\underset{x\to -{\tau }^{+}}{lim}{\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }y\left(x\right)=\underset{x\to -{\tau }^{+}}{lim}{\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }\varphi \left(x\right)=\left(\begin{array}{cc}& 0\hfill \\ & 0\hfill \end{array}\right)\hfill \end{array}$$

which implies that $\parallel b\parallel =0$. In addition, we can obtain

$$\begin{array}{cc}\hfill \left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(x\right)& =\frac{d}{dx}{\mathbb{I}}_{-{\tau }^{+}}^{1-\gamma }\varphi \left(x\right)=\frac{1}{2}\frac{{(x+2)}^{2-\gamma }}{\Gamma (3-\gamma )}\left(\begin{array}{cc}& 1\hfill \\ & 1\hfill \end{array}\right),\hfill \\ \hfill \parallel \left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(x\right)\parallel & =\frac{{(x+2)}^{2-\gamma }}{\Gamma (3-\gamma )}\hfill \end{array}$$

which leads to $\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(x\right)\in C((-\tau ,0],{\mathbb{R}}^2)$ and the hypothesis (${\mathrm{R}}_1$) is satisfied. In addition, we can calculate ${\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)$ as

$$\begin{array}{cc}\hfill {\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)& =\frac{1}{100}\left(\begin{array}{cc}\hfill & \frac{6x^{3-\gamma +\alpha }}{\Gamma (4-\gamma +\alpha )}-\frac{2\Gamma (2\gamma +1)x^{\gamma +\alpha }}{\Gamma (1+\gamma +\alpha )}\hfill \\ \hfill & \frac{2x^{2-\gamma +\alpha }}{\Gamma (3-\gamma +\alpha )}-\frac{\Gamma (\gamma +1)x^{\alpha }}{\Gamma (1+\alpha )}\hfill \end{array}\right)\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill \parallel {\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)\parallel & =\frac{1}{100}\left|\frac{6x^{3-\gamma +\alpha }}{\Gamma (4-\gamma +\alpha )}-\frac{2\Gamma (2\gamma +1)x^{\gamma +\alpha }}{\Gamma (1+\gamma +\alpha )}\right|+\frac{1}{100}\left|\frac{2x^{2-\gamma +\alpha }}{\Gamma (3-\gamma +\alpha )}-\frac{\Gamma (\gamma +1)x^{\alpha }}{\Gamma (1+\alpha )}\right|\hfill \\ \hfill & \le \frac{1}{100}\left(\frac{6x^{3-\gamma +\alpha }}{\Gamma (4-\gamma +\alpha )}+\frac{2\Gamma (2\gamma +1)x^{\gamma +\alpha }}{\Gamma (1+\gamma +\alpha )}+\frac{2x^{2-\gamma +\alpha }}{\Gamma (3-\gamma +\alpha )}+\frac{\Gamma (\gamma +1)x^{\alpha }}{\Gamma (1+\alpha )}\right).\hfill \end{array}$$

This leads to ${\mathbb{D}}_{0^{+}}^{\gamma -\alpha }h\left(x\right)\in C((-\tau ,0],{\mathbb{R}}^2)$ and the hypothesis (${\mathrm{R}}_1$) is satisfied.

Application to Theorem 5 

According to assumption (${\mathrm{R}}_2$), we can take

$$\begin{array}{c}\hfill \psi \left(x\right)=\frac{1}{100}\left(\frac{6x^{3-\gamma +\alpha }}{\Gamma (4-\gamma +\alpha )}+\frac{2\Gamma (2\gamma +1)x^{\gamma +\alpha }}{\Gamma (1+\gamma +\alpha )}+\frac{2x^{2-\gamma +\alpha }}{\Gamma (3-\gamma +\alpha )}+\frac{\Gamma (\gamma +1)x^{\alpha }}{\Gamma (1+\alpha )}\right).\end{array}$$

which determines that $\psi \left(x\right)\in C([0,5],{\mathbb{R}}^{+})$ and

$$\begin{array}{c}\hfill \psi =\underset{0\le x\le 5}{sup}\psi \left(x\right)\sim 1.57543.\end{array}$$

According to assumption (${\mathrm{R}}_3$), we can take

$$\phi \left(x\right)=\frac{{(x+2)}^{2-\gamma }}{\Gamma (3-\gamma )}$$

which determines that

$$\phi =\underset{-\tau <x\le 0}{sup}\phi \left(x\right)=\frac{2^{2-\gamma }}{\Gamma (3-\gamma )}\sim 2.12204.$$

In view of our calculations and the results of Theorem 5, we have to take $ϵ>29.5019$, which determines that the system given by (2) is finite stable with respect to $\{0,[0,5],2,30\}$.

Application to Theorem 6 

In view of hypothesis (${\mathrm{R}}_4$), we can take

$$\begin{array}{c}\hfill \eta \left(x\right)=\frac{1}{100}\left(\frac{6x^{3-\gamma +\alpha }}{\Gamma (4-\gamma +\alpha )}+\frac{2\Gamma (2\gamma +1)x^{\gamma +\alpha }}{\Gamma (1+\gamma +\alpha )}+\frac{2x^{2-\gamma +\alpha }}{\Gamma (3-\gamma +\alpha )}+\frac{\Gamma (\gamma +1)x^{\alpha }}{\Gamma (1+\alpha )}\right).\end{array}$$

which determines that $\eta \left(x\right)\in L^2([0,5],{\mathbb{R}}^{+})$ and

$$\begin{array}{cc}\hfill Q\left(x\right)& =\frac{1}{100}{\left({\int }_0^x{\left(\eta \left(t\right)\right)}^{2}dt\right)}^{\frac{1}{2}}\hfill \\ \hfill & =\frac{1}{100}{\left({\int }_0^x{\left(\frac{6x^{3-\gamma +\alpha }}{\Gamma (4-\gamma +\alpha )}+\frac{2\Gamma (2\gamma +1)x^{\gamma +\alpha }}{\Gamma (1+\gamma +\alpha )}+\frac{2x^{2-\gamma +\alpha }}{\Gamma (3-\gamma +\alpha )}+\frac{\Gamma (\gamma +1)x^{\alpha }}{\Gamma (1+\alpha )}\right)}^{2}dt\right)}^{\frac{1}{2}}.\hfill \end{array}$$

This leads to $Q\sim 1.44476$. In view of hypothesis (${\mathrm{R}}_5$), we find that

$$\begin{array}{ccc}\hfill M& =& {\left({\int }_{-\tau }^0{∥\left({\mathbb{D}}_{-{\tau }^{+}}^{\gamma }\varphi \right)\left(s\right)∥}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ & \le & \frac{1}{\Gamma (3-\gamma )}{\left({\int }_{-\tau }^0{(s+\tau )}^{4-2\gamma }ds\right)}^{\frac{1}{2}}\hfill \\ & =& \frac{{\tau }^{\frac{5}{2}-\gamma }}{{(5-2\gamma )}^{\frac{1}{2}}\Gamma (3-\gamma )}\sim 1.54972.\hfill \end{array}$$

In view of our calculations and the results of Theorem 6, we have to take $ϵ>16.8162$ which determines that the system given by (2) is finite stable with respect to $\{0,[0,5],2,17\}$.

6. Conclusions

Integrals representations to the solutions for linear homogeneous and nonhomogeneous fractional delay differential equations of a Hilfer type have been obtained in terms of DMLMF2P as defined in Section 3. The particular solution corresponding to a nonhomogeneous system is provided via using the method of variation of constants. Under appropriate assumptions, we presented a variety of results to finite-time stability in Section 3. Due to these results and applying the values in Example 1, we can take $ϵ=29.502$ in Theorem 5 and $ϵ=16.8163$ in Theorem 6. Comparing the values of $ϵ$, we find an optimal threshold $ϵ=16.8163$ such that $\parallel y\left(x\right)\parallel$ does not exceed it on $[0,5]$. Our fractional approach here is due to Hilfer ${\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }$ when $0<\alpha <1$. Future work will take the same approach due to Hilfer with $1<\alpha <2$ or the operator ${\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }\left({\mathbb{D}}_{-{\tau }^{+}}^{\alpha ,\beta }\right)$ with $0<\alpha <1$. In both cases, we will need to define two forms of DMLMF2P.