The Strict Stability of Impulsive Differential Equations with a Caputo Fractional Derivative with Respect to Other Functions

Abstract

The aim of this paper is to study a nonlinear system of impulsive fractional differential equations and Caputo fractional derivatives with respect to another function (CFF). The main characteristics of these fractional derivatives are two-fold: first, the lower limit of CFF equals the impulsive time of the considered interval; second, the applied function in CFF is changeable at each interval without impulses. An auxiliary system of two linear scalar impulsive fractional differential equations with CFF is considered, and strict stability in a couple is defined. The behavior of its solutions is illustrated with several examples. Also, we use appropriate Lyapunov functions to obtain sufficient conditions for the strict stability of the studied system. These sufficient conditions depend significantly on the type of impulsive function.

1. Introduction

Stability is a very important property of the solutions of any type of differential equation. In the literature, various types of stability are defined, studied, and applied to different types of differential equations. These research areas include strict stability (see, for example, [1]), systems of differential equations with impulses [2], delay differential equations with impulses [3,4], dynamic systems on a time scale [5,6], differential equations with supremum [7], Caputo fractional differential equations [8], and fractional equations with non-instantaneous impulses [9]). Strict stability provides some information about the rate of decay of the solutions, and it is different from asymptotic stability since it does not allow us to approach zero as we move toward infinity.

In this paper, we focus on strict stability for a nonlinear fractional differential equation with two main characteristics: first, impulses are involved in the equations, and second, the applied fractional derivative is a Caputo fractional derivative with respect to another function (CFF).

Note that the presence of any type of impulse, deterministic (see, for example, [2,3,4]) or stochastic (see, for example, [10]), plays a very important role in the behavior of the solutions, and it requires a deep theoretical study.

Fractional derivatives of various types are successfully applied in fractional models in physics [11], bioengineering [12], viscoelasticy [13], ecology [14], and disease [15,16,17]. The CFF applied in this paper has a lower limit equal to the corresponding impulsive time, and the applied function is also changeable in each interval without impulses, so it makes the applied fractional derivative more general.

In this paper, several comparison results are established for scalar impulsive fractional differential equations with CFF. Strict stability in a couple is defined for an auxiliary system of two scalar impulsive fractional differential equations with CFF, and strict stability is studied using appropriate Lyapunov functions. The definition and several of the theoretical results are illustrated with examples.

Note that if there is no impulse in the equation, we obtain results regarding the strict stability of fractional differential equations with CFF considered in [18] as a partial case of this paper.

The contributions in this paper can be summarized as follows:

-

Initially, points of impulses are applied to the nonlinear fractional differential equation;

-

The applied CFF has two main characteristics: first, the lower limit of the CFF equals the impulsive time of the considered interval; second, the applied function in the CFF is changeable in each interval without impulses.

-

Strict stability is defined, and the dependence of this type of stability on the behavior of the solutions is illustrated with several examples;

-

Several auxiliary results for scalar impulsive fractional equations with the CFF are established;

-

Several types of sufficient conditions based on appropriate Lyapunov functions are obtained;

-

All sufficient conditions depend significantly on the impulsive functions in the considered equations;

-

Theoretical results are illustrated with several examples.

2. Preliminary Results for Fractional Differintegrals

Let $A,B:0\le A<B\le \infty$. Consider the set of functions

$$C^1((A,B],{\mathbb{R}}^n)=\{y\in C((A,B],{\mathbb{R}}^n):y^{\prime }existsanditiscontinuouson(A,B]\}.$$

Definition 1 

([19,20,21]). Let $\rho >0$, $\psi \in C^1([A,B],[0,\infty ))$ and ${\psi }^{\prime }\left(s\right)>0,s\in [A,B]$. The Riemann fractional integral with respect to another function (FIF) of the function $\upsilon :[A,B]\to \mathbb{R}$ is defined by the following (where the integral exists):

$$I_A^{\rho ,\psi }\upsilon \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\rho \right)}}{\int }_A^t{\psi }^{\prime }\left(s\right){\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\rho -1}\upsilon \left(s\right)ds,t\in (A,B].$$

Definition 2 

([19,20,21]). Let $\rho \in (0,1)$, and the function $\psi \in C^1([A,B],[0,\infty ))$ and ${\psi }^{\prime }\left(s\right)>0,s\in [A,B]$. The Caputo fractional derivative with respect to another function (CFF1) of the function $\upsilon :[A,B]\to \mathbb{R}$ is defined by the following (where the integral exists):

$$\begin{array}{c}\hfill {}^{C}D_A^{\rho ,\psi }\upsilon \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\rho )}}{\int }_A^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\rho }{\upsilon }^{\prime }\left(s\right)ds,t\in (A,B].\end{array}$$

In the case of a vector function, the integral IFF and the derivative CFF1 are defined component-wise.

In the case $B=\infty$, the intervals are half-opened.

We will use the following set of functions ($0\le A<B$):

$$\begin{array}{cc}& C^{\rho }((A,B],{\mathbb{R}}^n,\psi )=\{U\in C((A,B],{\mathbb{R}}^n):U^{\prime }\mathrm{exists}\mathrm{almost}\mathrm{everywhere}\mathrm{in}(A,B]\hfill \\ & \mathrm{and}\mathrm{there}\mathrm{exists}\mathrm{CFF}1{}^{C}D_A^{\rho ,\psi }U\left(t\right)\mathrm{for}t\in (A,B]\}.\hfill \end{array}$$

Remark 1. 

For a given $t\in (A,B]$ in Definition 2 for the CFF1 ${}^{C}D_A^{\rho ,\psi }U\left(t\right)$ (and throughout this paper when we use it), we assume the functions $U\in C([A,t],{\mathbb{R}}^n),n\ge 1,$ and $U^{\prime }$ exist almost everywhere on $[A,t]$.

Remark 2. 

In the case $\psi \left(t\right)=t$, the above-defined FIF and CFF1 coincide with classical definitions of the Riemann fractional integral and the Caputo fractional derivative [20].

We will use the following results for CFF1:

Lemma 1 

([18]). Let $\upsilon :[A,B]\to \mathbb{R}$ and $0\le A<B\le \infty$. Here, there exists a point $\varsigma \in (A,B]$, such that $\upsilon \left(\varsigma \right)=0$ and $\upsilon \left(s\right)<0$ for $s\in [A,\varsigma )$, and the CFF1 ${}^{C}D_A^{\rho ,\psi }{\upsilon \left(t\right)|}_{t=\varsigma }$ exists with $\psi \in C^1([A,B],[0,\infty ))$ and ${\psi }^{\prime }\left(s\right)>0,s\in [A,B]$. Then, if ${lim}_{\sigma \to \varsigma -}\left({\displaystyle \frac{v\left(\sigma \right)}{{(\psi \left(\varsigma \right)-\psi \left(\sigma \right))}^{\rho }}}\right)=0$, we have ${}^{C}D_A^{\rho ,\psi }{\upsilon \left(t\right)|}_{t=\varsigma }\ge 0$.

Corollary 1 

([18]). Let $\upsilon :[A,B]\to \mathbb{R}$ and $0\le A<B\le \infty$. Here, there exists a point $\varsigma \in (A,B]$, such that $\upsilon \left(\varsigma \right)=0$ and $\upsilon \left(s\right)>0$ for $s\in [A,\varsigma )$, and the CFF1 ${}^{C}D_A^{\rho ,\psi }{\upsilon \left(t\right)|}_{t=\varsigma }$ exists with $\psi \in C^1([A,B],[0,\infty ))$ and ${\psi }^{\prime }\left(s\right)>0,s\in [A,B]$. Then, if ${lim}_{\sigma \to \varsigma -}\left({\displaystyle \frac{v\left(\sigma \right)}{{(\psi \left(\varsigma \right)-\psi \left(\sigma \right))}^{\rho }}}\right)=0$, we have ${}^{C}D_A^{\rho ,\psi }{\upsilon \left(t\right)|}_{t=\varsigma }leq0$.

Remark 3. 

Note the condition ${lim}_{\sigma \to \varsigma -}\left({\displaystyle \frac{U\left(\sigma \right)}{{\left(\psi \varsigma \right)-\psi \left(\sigma \right))}^{\rho }}}\right)=0$ in Lemma 1 and Corollary 1 is automatically true if $U\in C^1((A,B],\mathbb{R})$ (because of L’Hôpital’s rule). Also, this condition regarding the zero limit is also true if ${lim}_{\sigma \to \varsigma -}{U}^{\prime }\left(\sigma \right)$ exists and is a real number.

Lemma 2 

([19]). Let $0\le A<B\le \infty$, the function $\psi \in C^1([A,B],[0,\infty ))$, and ${\psi }^{\prime }\left(s\right)>0,s\in [A,B]$. The solution of the scalar linear fractional initial value problem with CFF1,

$${}^{C}D_A^{\rho ,\psi }U\left(t\right)=\lambda U\left(t\right)fort\in (A,B]andU\left(A\right)=U_0,$$

is the function

$$U\left(t\right)=U_0{E}_{\rho }\left(\lambda {(\psi \left(t\right)-\psi \left(A\right))}^{\rho }\right),t\in [A,B],$$

where $E_{\rho }\left(t\right)$ is the Mittag–Leffler function of one parameter, $\lambda \in \mathbb{R}$.

3. Some Preliminary Results for Impulsive Fractional Initial Value Problems

Throughout this paper, we will assume $\rho \in (0,1)$.

Let the sequence of points ${\left\{{\tau }_i\right\}}_{i=0}^{\infty }:0={\tau }_0<{\tau }_1<{\tau }_i<{\tau }_{i+1},i=2,3,\dots ,$ with ${lim}_{i\to \infty }{\tau }_i=\infty ,$ be given.

Let the sequence of functions ${\left\{\psi {(.)}_m\right\}}_{m=0}^{\infty }$ be such that ${\psi }_k\in C^1([{\tau }_k,{\tau }_{k+1}],[0,\infty ))$ and ${\psi }_k^{\prime }\left(s\right)>0,s\in [{\tau }_k,{\tau }_{k+1}]$.

For any $k=0,1,2,\dots$ on the interval $({\tau }_k,{\tau }_{k+1}]$, we will consider the following CFF1 defined in Definition 2 with $A={\tau }_k,B={\tau }_{k+1}$ and $\psi (.)\equiv {\psi }_k(.)$, i.e.,

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}\upsilon \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\rho )}}{\int }_{{\tau }_k}^t{\left({\psi }_k\left(t\right)-{\psi }_k\left(s\right)\right)}^{-\rho }{\upsilon }^{\prime }\left(s\right)ds,t\in ({\tau }_k,{\tau }_{k+1}].$$

Definition 3. 

The Caputo fractional derivative with respect to other functions of $U(.)$ defined on $[0,\infty )$ (CFF) will be the set of all fractional derivatives ${}^{C}D_{{\tau }_k}^{q,{\psi }_k}U\left(t\right),({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots$

We will use the following sets of functions:

$$\begin{array}{cc}& P{C}^{\rho }\left({\mathbb{R}}^n\right)=\{U:[0,\infty )\to {\mathbb{R}}^n,U\in C^{\rho }(({\tau }_i,{\tau }_{i+1}],{\mathbb{R}}^n,{\psi }_i),i=0,1,2,\dots \hfill \\ & \mathrm{and}\underset{s\to {\tau }_i-0}{lim}U\left(s\right)=U\left({\tau }_i\right)<\infty ,\underset{s\to {\tau }_i+0}{lim}U\left(s\right)=U({\tau }_i+0)<\infty \mathrm{for}i=1,2,\dots \},\hfill \\ & P{C}^1\left({\mathbb{R}}^n\right)=\{U:[0,\infty )\to {\mathbb{R}}^n,u\in C^1(({\tau }_i,{\tau }_{i+1}],{\mathbb{R}}^n)\mathrm{for}i=0,1,2,\dots ,\hfill \\ & \mathrm{and}\underset{s\to {\tau }_i-0}{lim}{U}^{\prime }\left(s\right)=U^{\prime }\left({\tau }_i\right)<\infty ,\underset{s\to {\tau }_i+0}{lim}{U}^{\prime }\left(s\right)=U^{\prime }({\tau }_i+0)<\infty \mathrm{for}i=1,2,\dots \}.\hfill \end{array}$$

Let $U\in P{C}^{\rho }\left({\mathbb{R}}^n\right)$. Then, we define the norm ${||U||}_{PC}={sup}_{t\ge 0}||U\left(t\right)||$, where $||.||$ is a norm in ${\mathbb{R}}^n$.

In this paper, we will study the initial value problem (IVP) for the impulsive fractional differential equations with CFF for $\rho \in (0,1)$:

$$\begin{array}{cc}& {}^{C}D_{{\tau }_i}^{\rho ,{\psi }_i}y\left(t\right)=f(t,y\left(t\right)),t\in ({\tau }_i,{\tau }_{i+1}],i=0,1,2,\dots ,\hfill \\ & y({\tau }_i+0)={\mathcal{I}}_i\left(y\left({\tau }_i\right)\right),i=1,2,\dots ,\hfill \\ & y\left(0\right)=y_0,\hfill \end{array}$$

(1)

where $y_0\in {\mathbb{R}}^n,f\in C([0,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}^n)$, $f(t,0)\equiv 0$, $t\ge 0$, and ${\mathcal{I}}_i:{\mathbb{R}}^n\to {\mathbb{R}}^n,i=1,2,\dots$

We will assume that the function $f\in C([0,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}^n)$ is such that for any initial value $y_0\in {\mathbb{R}}^n$, the IVP (1) has a solution $y(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)$.

In the case of IVP for a linear impulsive fractional differential equation with CFF, we obtain an explicit form of the solution.

Lemma 3. 

Let $\rho \in (0,1)$ and

$$\begin{array}{cc}& {}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}U\left(t\right)={\lambda }_{k}Ufo\mathbb{R}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\hfill \\ & U({\tau }_k+0)=b_{k}U\left({\tau }_k\right),k=1,2,\dots ,\hfill \\ & U\left(0\right)=U_0,\hfill \end{array}$$

(2)

where $U_0\in \mathbb{R}$, $b_k\in \mathbb{R},k=1,2,\dots$, and ${\lambda }_k\in \mathbb{R},k=0,1,2,\dots$ are given constants.

Then, the solution of (2) is the following function:

$$\begin{array}{cc}& U\left(t\right)=U_0\left(\prod _{i=0}^{k-1}{b}_{i+1}{E}_{\rho }\left({\lambda }_i{({\psi }_i\left({\tau }_{i+1}\right)-{\psi }_i\left({\tau }_i\right))}^{\rho }\right)\right)E_q\left({\lambda }_k{({\psi }_k\left(t\right)-{\psi }_k\left({\tau }_k\right))}^{\rho }\right),\hfill \\ & t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,3,\dots .\hfill \end{array}$$

(3)

Proof. 

The proof follows from induction with respect to the intervals $({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots$ and Lemma 2 with $\lambda ={\lambda }_k,A={\tau }_k,B={\tau }_{k+1},\psi (.)={\psi }_k(.)$. □

We now introduce the $\mathrm{\Lambda }$ class of Lyapunov-like functions, which will be used to investigate the strict stability of the system of differential equations with CFF (1).

Definition 4. 

Let $\Delta \subset {\mathbb{R}}^n,0\in \Delta$. The function $V(.):\Delta \to [0,\infty )$ is from the $\mathrm{\Lambda }(\Delta )$ class if $V(.)\in C(\Delta ,[0,\infty \left)\right)$ is continuously differentiable in Δ.

Remark 4. 

The functions $V\left(x\right)={\sum }_{i=1}^n{x}_i^2,V\left(x\right)={\sum }_{i=1}^n|x_i|,V\left(x\right)={\sum }_{i=1}^n{e}^{x_i},V\left(x\right)={\sum }_{i=1}^n{e}^{-x_i},V\left(x\right)=e^{{\sum }_{i=1}^n{x}_i},V\left(x\right)=e^{-{\sum }_{i=1}^n{x}_i}$, for example, are from the $\mathrm{\Lambda }(\Delta )$ class, where $x\in {\mathbb{R}}^n,x=(x_1,x_2,\dots ,x_n)$.

Consider IVP for the nonlinear scalar impulsive fractional differential equation with CFF.

$$\begin{array}{cc}& {}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}U\left(t\right)=g(t,U)+\eta ,\mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\hfill \\ & U({\tau }_k+0)=J_k\left(U\left({\tau }_k\right)\right)+\eta ,k=1,2,\dots ,\hfill \\ & U\left(0\right)=U_0+\eta ,\hfill \end{array}$$

(4)

where $U_0\in \mathbb{R},g\in C([0,\infty )\times \mathbb{R},\mathbb{R})$, $\eta \in \mathbb{R}$, and $J_k\in C(\mathbb{R},\mathbb{R}),k=1,2,\dots$.

We will say condition (A) is satisfied if the following is true:

(A).

For any point $U_0\in \mathbb{R}$, the IVP (4) with $\eta \in \mathbb{R}:|\eta |<L$ has an unique solution $U_{\eta }(.)\in P{C}^{\rho }\left(\mathbb{R}\right)\cap P{C}^1\left(\mathbb{R}\right)$, where $L>0$ is a small enough number $L>0$.

Lemma 4. 

Assume the following:

 1. 

Condition (A) is fulfilled.

 2. 

Functions $g\in C\left(\right[0,\infty )\times \mathbb{R},\mathbb{R})$ and $J_k\in C(\mathbb{R},\mathbb{R}),k=1,2,\dots$ are non-decreasing.

 3. 

The function $\omega \in P{C}^q\left(\mathbb{R}\right)\cap P{C}^1\left(\mathbb{R}\right)$ satisfies the inequalities

$$\begin{array}{cc}& {}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}\omega \left(t\right)\le (\ge )g(t,\omega ),fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \omega ({\tau }_k+0)\le (\ge )J_k\left(\omega \left({\tau }_k\right)\right),k=1,2,\dots ,\hfill \\ & \omega \left(0\right)={\omega }_0,\hfill \end{array}$$

(5)

where ${\omega }_0\in \mathbb{R}$.

Then, the inequality ${\omega }_0\le (\ge )U_0$ implies $\omega \left(s\right)\le (\ge )U^{*}\left(s\right)$ for $s\ge 0,$ where $U^{*}(.)\in P{C}^{\rho }\left(\mathbb{R}\right)\cap P{C}^1\left(\mathbb{R}\right)$ is the unique solution of (4) with $\eta =0$.

Case 1. In (5), inequalities ≤ are satisfied and ${\omega }_0\le U_0$.

Let $\eta \in (0,L)$ be an arbitrary number. According to condition 1, there exists a unique solution $U_{\eta }\left(t\right)$ to the initial value problem (4). We will prove the following:

$$\omega \left(t\right)<U_{\eta }\left(t\right)\mathrm{for}t>0.$$

(6)

Using induction with respect to the intervals defined by the impulsive times ${\tau }_k,k=1,2,\dots$, we prove inequality (6).

Case 1.1 Consider the interval $[0,{\tau }_1]$.

Note that $\omega \left(0\right)={\omega }_0\le U_0<U_0+\eta =U_{\eta }\left(0\right)$.

We now prove the inequality

$$asfollows:\omega \left(t\right)<U_{\eta }\left(t\right)\mathrm{for}t\in [0,{\tau }_1].$$

(7)

Now, assume the contrary, i.e., there exists a point $\xi \in (0,{\tau }_1)$ such that $\omega \left(t\right)<U_{\eta }\left(t\right)$ on $[0,\xi )$ and $\omega \left(\xi \right)=U_{\eta }\left(\xi \right)$.

All the conditions in Lemma 1 are satisfied for the point $\varsigma =\xi \in (0,{\tau }_1)$, $A=0,B={\tau }_1$ and the function $\omega \left(t\right)-U_{\eta }\left(t\right),t\in [0,\xi ]$. According to Lemma 1 applied to the function $\omega (.)-U_{\eta }(.)$ with $A=0,B={\tau }_1$ and Remark 3, we obtain the following: ${}^{C}D_0^{\rho ,{\psi }_0}\left(\omega \left(t\right)-U_{\eta }\left(t\right)\right){|}_{t=\xi }\ge 0$, i.e.,

$${}^{C}D_0^{\rho ,{\psi }_0}\omega \left(t\right){{|}_{t=\xi }\ge {}^{C}D_0^{\rho ,{\psi }_0}{U}_{\eta }\left(t\right)|}_{t=\xi }=g(\xi ,U_{\eta }\left(\xi \right))+\eta >g(\xi ,U_{\eta }\left(\xi \right))=g(\xi ,\omega \left(\xi \right)).$$

According to the first inequality of (5) for $t=\xi$ and $k=0,{\tau }_0=0$, we have ${}^{C}D_0^{\rho ,{\psi }_0}{\omega \left(t\right)|}_{t=\xi }\le g(\xi ,\omega \left(\xi \right))$. The obtained contradiction proves inequality (7).

Case 1.2. Consider the interval $({\tau }_1,{\tau }_2]$.

According to the definition of the set $P{C}^{\rho }\left(\mathbb{R}\right)$, conditions 1 and 3, and $\omega (.),U_{\eta }(.)\in P{C}^q\left(\mathbb{R}\right)$, the inequalities $U_{\eta }({\tau }_1+0)<\infty ,\omega ({\tau }_1+0)<\infty$ hold. Therefore, we could consider the functions $\omega (.),U_{\eta }(.)$ on the whole interval $[{\tau }_1,{\tau }_2]$, assuming that at point ${\tau }_i$, they are defined as $\omega ({\tau }_1+0)$ and $U_{\eta }({\tau }_1+0)$, respectively. From the impulsive conditions in (4) and (5), the proved inequality $\omega \left({\tau }_1\right)\le U_{\eta }\left({\tau }_1\right)$, and the monotonicity of $J_1(.)$, we have $\omega ({\tau }_1+0)\le J_1\left(\omega \left({\tau }_1\right)\right)\le J_1\left(U_{\eta }\left({\tau }_1\right)\right)<J_1\left(U_{\eta }\left({\tau }_1\right)\right)+\eta =U_{\eta }\left({\tau }_1\right)$.

We now prove the following inequality

$$:\omega \left(t\right)<U_{\eta }\left(t\right)\mathrm{for}t\in ({\tau }_1,{\tau }_2].$$

(8)

Now, assume the contrary, i.e., there exists a point $\xi \in ({\tau }_1,{\tau }_2)$ such that $\omega \left(t\right)<U_{\eta }\left(t\right)$ on $[{\tau }_1+0,\xi )$ and $\omega \left(\xi \right)=U_{\eta }\left(\xi \right)$.

All the conditions in Lemma 1 are satisfied for points $\xi \in ({\tau }_1,{\tau }_2)$ and $A={\tau }_1+0,B={\tau }_2$ and the function $\omega \left(t\right)-U_{\eta }\left(t\right),t\in [{\tau }_1+0,\xi ]$. According to Lemma 1 applied to the function $\omega (.)-U_{\eta }(.)$ with $A={\tau }_1,B={\tau }_2$ and Remark 3, we obtain ${}^{C}D_{{\tau }_1}^{\rho ,{\psi }_1}\left(\omega \left(t\right)-U_{\eta }\left(t\right)\right){|}_{t=\xi }\ge 0$, i.e.,

$${}^{C}D_{{\tau }_1}^{\rho ,{\psi }_1}\omega \left(t\right){{|}_{t=\xi }\ge {}^{C}D_{{\tau }_1}^{\rho ,{\psi }_1}{U}_{\eta }\left(t\right)|}_{t=\xi }=g(\xi ,U_{\eta }\left(\xi \right))+\eta >g(\xi ,U_{\eta }\left(\xi \right))=g(\xi ,\omega \left(\xi \right)).$$

According to the first inequality of (5) for $t=\xi$ and $k=1$, we have ${}^{C}D_{{\tau }_1}^{\rho ,{\psi }_1}{\omega \left(t\right)|}_{t=\xi }\le g(\xi ,\omega \left(\xi \right))$. The obtained contradiction proves inequality (8).

Case 1.3. Assume inequality (6) holds for $t\in [0,{\tau }_j],j\ge 2,$ and consider the interval $({\tau }_j,{\tau }_{j+1}]$.

The proof of inequality (6) for $t\in ({\tau }_j,{\tau }_{j+1}]$ is similar to that of Case 1.2, with the replacement of point ${\tau }_1$ and function $J_1(.)$ by point ${\tau }_j$ and function $J_j(.)$, respectively.

Take the limit as $\eta \to 0$ in (6) and obtain $\omega \left(t\right)\le U_{\eta }{\left(t\right)|}_{\eta =0}=U^{*}\left(t\right),t\ge 0.$

Case 2. Suppose in (5) the inequalities ≥ hold and ${\omega }_0\ge U_0$. Then, similarly to Case 1 with $\eta \in (-L,0]$ and Corollary 1 applied instead of Lemma 1, we obtain the claim.

According to Lemma 1 [19] with $\beta =1+\rho ,\alpha =\rho$, we obtain ${}^{C}D_A^{\rho ,\psi }{(\psi \left(t\right)-\psi \left(A\right))}^q=\mathrm{\Gamma }(1+\rho )$. Then, the solution of the initial value problem for the scalar fractional differential equation with CFF ${}^{C}D_A^{\rho ,\psi }U\left(t\right)=\eta ,U\left(A\right)=\eta$ is $U\left(t\right)=\eta +{(\psi \left(t\right)-\psi \left(A\right))}^q{\displaystyle \frac{\eta }{\mathrm{\Gamma }(1+\rho )}}$.

Let $g(.,.)\equiv 0$ and $J_k(.)\equiv 0,k=1,2,\dots ,$ in (4). Therefore, for any $\eta \in \mathbb{R}$, the solution of (4) is $U\left(t\right)=\eta \left(1+{\displaystyle \frac{{({\psi }_k\left(t\right)-{\psi }_k\left({\tau }_k\right))}^q}{\mathrm{\Gamma }(1+\rho )}}\right),t\in ({\tau }_k,{\tau }_{k+1}],$ i.e., condition (A) is satisfied, and as a partial case of Lemma 4, we obtain the following.

Corollary 2. 

Let $\omega \in P{C}^{\rho }\left(\mathbb{R}\right)\cap P{C}^1\left(\mathbb{R}\right)$ satisfy the following inequalities:

$$\begin{array}{cc}& {}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}\omega \left(t\right)\le (\ge )0,fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \omega ({\tau }_k+0)\le (\ge )0,k=1,2,\dots ,\hfill \\ & \omega \left(0\right)={\omega }_0,\hfill \end{array}$$

(9)

where ${\omega }_0\in \mathbb{R}$.

Then, the inequality ${\omega }_0\le (\ge )0$ implies $\omega \left(t\right)\le (\ge )0,t\ge 0$.

4. Strict Stability

The aim of this paper is to study the strict stability of impulsive fractional differential equations with CFF (1).

Following the definitions for strict stability for ordinary differential equations (see, for example, [1]), we provide the following definition.

Definition 5. 

The zero solution of the fractional system with CFF (1) is strictly stable if, for a given ${ϵ}_1>0$, there exists ${\delta }_1={\delta }_1\left({ϵ}_1\right)>0$ and for any ${\delta }_2={\delta }_2\left({ϵ}_1\right),{\delta }_2\in (0,{\delta }_1)$ there exists ${ϵ}_2={ϵ}_2\left({\delta }_2\right)\in (0,{\delta }_2)$ such that for any initial value $y_0\in {\mathbb{R}}^n:{\delta }_2<||y_0||<{\delta }_1$, the inequality ${ϵ}_2<||y(t;y_0){||}_{PC}<{ϵ}_1$ holds for $t\ge 0$, where $y(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ is a solution of (1).

We will now illustrate the importance of the applied function $\psi (.)$ and the impulses on the strict stability properties. For this purpose, we will compare the case of the absence of impulses (see Example 1 [18]) and the case of the presence of impulses.

Example 1.  (Strict stability)

 Case 1. 

(No impulses) (see Example 1 [18]). Consider the scalar fractional differential equation with CFF

$${}^{C}D_0^{0.3,\psi }x\left(t\right)=\lambda x,fort>0andx\left(0\right)=x_0$$

(10)

with a solution (see Lemma 2) $x\left(t\right)=x_0{E}_{\alpha }\left(\lambda {(\psi \left(t\right)-\psi \left(0\right))}^{0.3}\right)$ for $t\ge 0$ (the function $\psi (.)$ will be defined later).

 Case 1.1. 

Let the function $\psi (.)$ used in CFF be such that $\psi \left(t\right)\le L,t\ge 0$ with $L>0$. For example, consider $\psi \left(t\right)={\displaystyle \frac{t+1}{t+2}},t\ge 0$ and $L=1$. The inequalities $1=E_{0.3}\left(0\right)\le E_{0.3}\left({({\displaystyle \frac{t+1}{t+2}}-0.5)}^{0.3}\right)<E_{0.3}(0.5^{0.3})$ hold for $t\ge 0$.

 Case 1.1.1. 

Let $\lambda =1$.

For any ${\epsilon }_1>0$, if $|x_0|<{\delta }_1={\displaystyle \frac{{\epsilon }_1}{E_{0.3}(0.5^{0.3})}}$, then $|x\left(t\right)|<{\epsilon }_1$, and for any ${\delta }_2\in (0,{\delta }_1)=\left(0,{\displaystyle \frac{{\epsilon }_1}{E_{0.3}(0.5^{0.3})}}\right)$ there exists ${\epsilon }_2=0.5{\delta }_2<{\delta }_2$ such that the inequality $|x_0|>{\delta }_2$ implies $|x\left(t\right)|>{\delta }_2>{\epsilon }_2$. Therefore, the zero solution of the scalar linear fractional differential equation with CFF (10) is strictly stable.

 Case 1.1.2. 

Let $\lambda =-1$.

For any ${\epsilon }_1>0$, if $|x_0|<{\delta }_1={\epsilon }_1$, then $|x\left(t\right)|<{\epsilon }_1$, and for any ${\delta }_2\in (0,{\delta }_1)=(0,{\epsilon }_1)$ there exists ${\epsilon }_2=E_{0.3}(-0.5^{0.3}){\delta }_2<{\delta }_2$ such that the inequality $|x_0|>{\delta }_2$ implies $|x\left(t\right)|>{\delta }_2{E}_{0.3}(-0.5^{0.3})={\epsilon }_2$. Therefore, the zero solution of the scalar fractional differential equation with CFF (10) is strictly stable.

 Case 1.2. 

Let the function $\psi (.)$ used in CFF be such that ${lim}_{t\to \infty }\psi \left(t\right)=\infty$. For example, consider $\psi \left(t\right)=t,t\ge 0$.

 Case 1.2.1. 

Let $\lambda =1$.

Then, the solution is $x\left(t\right)=x_0{E}_{0.3}\left(t^{0.3}\right)$. Since $E_{0.3}(\left(t^{0.3}\right)$ is an increasing unbounded function, the zero solution of (10) is not strictly stable.

 Case 1.2.2. 

Let $\lambda =-1$.

Then, the solution of (10) is $x\left(t\right)=x_0{E}_{0.3}(-t^{0.3})$. Since $E_{0.3}(-t^{0.3})\le E_{0.3}\left(0\right)=1,t\ge 0$ is a decreasing function approaching zero, the solution of (10) approaches zero at infinity, and the zero solution is not strictly stable.

 Case 2. 

(Impulsive case). Consider the scalar fractional differential equation with CFF

$$\begin{array}{cc}& {}^{C}D_k^{0.3,\psi }x\left(t\right)=\lambda x,fort\in (k,k+1],k=0,1,2,\dots ,\hfill \\ & x(k+0)=b_{k}x\left(k\right),k=1,2,\dots ,\hfill \\ & x\left(0\right)=x_0\hfill \end{array}$$

(11)

with a solution (see Lemma 3)

$$x\left(t\right)=x_0\left(\prod _{i=1}^k{b}_i{E}_{0.3}\left(\lambda {(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\right)\right)E_{0.3}\left(\lambda {(\psi \left(t\right)-\psi \left(k\right))}^{0.3}\right).$$

The function $\psi (.)$ will be defined later.

 Case 2.1. 

Let the function $\psi (.)$ used in CFF be such that $\psi \left(t\right)\le L,t\ge 0$ with $L>0$. For example, as in Case 1.1, we consider functions $\psi \left(t\right)={\displaystyle \frac{t+1}{t+2}},t\ge 0$ and $L=1$. We have $\psi \left(t_i\right)-\psi \left(t_{i-1}\right)={\displaystyle \frac{i+1}{i+2}}-{\displaystyle \frac{i}{i+1}}={\displaystyle \frac{i^2+i+1}{(i+1)(i+2)}}\in (0.45,1)$. Then, $E_{0.3}(0.{45}^{0.3})\le E_{0.3}\left({(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\right)<E_{0.3}\left(1\right)$ for $i=1,2,\dots .$

 Case 2.1.1. 

Let $\lambda =1$.

The solution of (11) is

$$x\left(t\right)=x_0\left(\prod _{i=1}^k{b}_i{E}_{0.3}\left({(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\right)\right)E_{0.3}\left({(\psi \left(t\right)-\psi \left(k\right))}^{0.3}\right).$$

-

If $b_i={\displaystyle \frac{1}{E_{0.3}\left({\left(\frac{i^2+i+1}{(i+1)(i+2)}\right)}^{0.3}\right)}},$ then for any $i=1,2,\dots$, we obtain $b_i{E}_{0.3}\left({(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\right)=1$.

For any ${\epsilon }_1>0$ and ${\delta }_1={\displaystyle \frac{{\epsilon }_1}{E_{0.3}\left(1\right)}}$, if $|x_0|<{\delta }_1$, then $|x\left(t\right)|=|x_0|E_{0.3}\left({(\psi \left(t\right)-\psi \left(k\right))}^{0.3}\right)<{\delta }_1{E}_{0.3}\left(1\right)={\epsilon }_1$, and for any ${\delta }_2\in (0,{\delta }_1)=\left(0,{\displaystyle \frac{{\epsilon }_1}{E_{0.3}\left(1\right)}}\right)$ there exists ${\epsilon }_2={\delta }_2{E}_{0.3}(0.{45}^{0.3})$ such that the inequality $|x_0|>{\delta }_2$ implies $|x\left(t\right)|=|x_0|E_{0.3}\left({(\psi \left(t\right)-\psi \left(k\right))}^{0.3}\right)>{\delta }_2{E}_{0.3}(0.{45}^{0.3})={\epsilon }_2$.

Therefore, the zero solution of the scalar impulsive linear fractional differential equation with CFF (10) is strictly stable.

-

If $b_i=0.5,$ then $|x\left(t\right)|\le |x_0|{\left(0.5E_{0.3}\left(1\right)\right)}^k{E}_{0.3}\left(1\right)$.

From ${lim}_{k\to \infty }{\left(0.5E_{0.3}\left(1\right)\right)}^k=\infty$, it follows that the solution of (11) increases without any bound, and the zero solution is not strictly stable.

Compare this with case 1.1.1 and note that the impulses can change the strict stability property of the zero solution.

 Case 2.1.2. 

Let $\lambda =-1$. Then, $-{(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\in (-1,0)$.

The solution of (11) is

$$x\left(t\right)=x_0\left(\prod _{i=1}^k{b}_i{E}_{0.3}(-{(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3})\right)E_{0.3}(-{(\psi \left(t\right)-\psi \left(k\right))}^{0.3}).$$

-

If $b_i={\displaystyle \frac{1}{E_{0.3}(-{\left(\frac{i^2+i+1}{(i+1)(i+2)}\right)}^{0.3})}},$ then for any $i=1,2,\dots$, we obtain $b_i{E}_{0.3}(-{(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3})=1$.

For any ${\epsilon }_1>0$ and ${\delta }_1={\epsilon }_1$, if $|x_0|<{\delta }_1$, then $|x\left(t\right)|=|x_0|E_{0.3}(-{(\psi \left(t\right)-\psi \left(k\right))}^{0.3})<{\delta }_1{E}_{0.3}\left(0\right)={\epsilon }_1$, and for any ${\delta }_2\in (0,{\delta }_1)=\left(0,{\epsilon }_1\right)$ there exists ${\epsilon }_2={\delta }_2{E}_{0.3}(-1)$ such that the inequality $|x_0|>{\delta }_2$ implies $|x\left(t\right)|=|x_0|E_{0.3}(-{(\psi \left(t\right)-\psi \left(k\right))}^{0.3})>{\delta }_2{E}_{0.3}(-1)={\epsilon }_2$.

Therefore, the zero solution of the scalar impulsive linear fractional differential equation with CFF (10) is strictly stable.

-

If $b_i=5,$ then $|x\left(t\right)|\ge |x_0|{\left(5E_{0.3}(-1)\right)}^k{E}_{0.3}(-{(\psi \left(t\right)-\psi \left(k\right))}^{0.3})$ for $t\in (k,k+1]$.

From ${lim}_{k\to \infty }{\left(5E_{0.3}(-1)\right)}^k=\infty$, it follows that the solution of (11) increases without any bound, and the zero solution is not strictly stable.

Compare this with case 1.1.2 and note that the impulses can change the strict stability property of the zero solution.

 Case 2.2. 

Let the function $\psi (.)$ used in CFF be such that ${lim}_{t\to \infty }\psi \left(t\right)=\infty$. As in Case 1.2, we consider the function $\psi \left(t\right)=t,t\ge 0$. Then, $\psi \left(i\right)-\psi \left(i-1\right)=1$.

 Case 2.2.1. 

Let $\lambda =1$.

Then, the solution of (11) is $x\left(t\right)=x_0\left({\prod }_{i=1}^k{b}_i{E}_{0.3}\left(1\right)\right)E_{0.3}\left({(t-k)}^{0.3}\right).$

If $b_i={\displaystyle \frac{1}{E_{0.3}\left(1\right)}},$ then we have $|x\left(t\right)|=|x_0|E_{0.3}\left({(t-k)}^{0.3}\right)$ for $t\in (k,k+1]$ and $|x_0|\le |x\left(t\right)|\le |x_0|E_{0.3}\left(1\right)$.

For any ${\epsilon }_1>0$ and ${\delta }_1={\displaystyle \frac{{\epsilon }_1}{E_{0.3}\left(1\right)}}$, if $|x_0|<{\delta }_1$, then $|x\left(t\right)|<{\epsilon }_1$, and for any ${\delta }_2\in (0,{\delta }_1)=\left(0,{\displaystyle \frac{{\epsilon }_1}{E_{0.3}\left(1\right)}}\right)$ there exists ${\epsilon }_2={\delta }_2$ such that the inequality $|x_0|>{\delta }_2$ implies $|x\left(t\right)|>{\epsilon }_2$.

Therefore, the zero solution of the scalar impulsive linear fractional differential equation with CFF (11) is strictly stable (compare this with case 1.2.1 and note that the impulses can change the strict stability property of the zero solution).

 Case 2.2.2. 

Let $\lambda =-1$.

Then, the solution of (10) is $x\left(t\right)=x_0\left({\prod }_{i=1}^m{b}_i{E}_{0.3}(-1)\right)E_{0.3}(-{(t-m)}^{0.3}),t\in (m,m+1].$

If $b_i={\displaystyle \frac{1}{E_{0.3}(-1)}},$ then we have $|x\left(t\right)|=|x_0|E_{0.3}(-{(t-i)}^{0.3})$ for $t\in (i,i+1]$ and $|x_0|E_{0.3}(-1)\le |x\left(t\right)|\le |x_0|$.

For any ${\epsilon }_1>0$ and ${\delta }_1={\epsilon }_1$, if $|x_0|<{\delta }_1$, then $|x\left(t\right)|<{\epsilon }_1$, and for any ${\delta }_2\in (0,{\delta }_1)=\left(0,{\epsilon }_1\right)$ there exists ${\epsilon }_2={\delta }_2{E}_{0.3}(-1)$ such that the inequality $|x_0|>{\delta }_2$ implies $|x\left(t\right)|>{\epsilon }_2$.

Therefore, the zero solution of the scalar impulsive linear fractional differential equation with CFF (10) is strictly stable (compare this with Case 1.2.2 and note that the impulses can change the strict stability property of the zero solution).

Remark 5. 

There are some important points in the application of strict stability to fractional differential equations with CFF:

-

In the case without impulses, the function applied to CFF has an influence on the strict stability properties of the solutions (see Case 1 of Example 1 and [18]);

-

In the impulsive case, the function applied to CFF is not as important but the type of impulsive functions has an influence on the strict stability property of the solutions (see Case 2 of Example 1).

We now obtain sufficient conditions for the strict stability of (1), which depend significantly on the impulsive functions.

We consider the following couple of scalar impulsive fractional differential equations with CFF:

$$\begin{array}{cc}& {}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}U\left(t\right)={\Theta }_1\left(t,U\left(t\right)\right),{}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}\omega \left(t\right)={\Theta }_2\left(t,\omega \left(t\right)\right),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\hfill \\ & U({\tau }_k+0)=J_k^{\left(1\right)}\left(U\left({\tau }_k\right)\right),\omega ({\tau }_k+0)=J_k^{\left(2\right)}\left(\omega \left({\tau }_k\right)\right),k=1,2,\dots ,\hfill \\ & U\left(0\right)=U_0,\omega \left(0\right)={\omega }_0,\hfill \end{array}$$

(12)

where $U_0,{\omega }_0\in \mathbb{R}$, ${\Theta }_i,:{\mathbb{R}}_{+}\times \mathbb{R}\to \mathbb{R}$, ${\Theta }_i(t,0)\equiv 0$, and $J_k^{\left(i\right)}:\mathbb{R}\to \mathbb{R},J_k^{\left(i\right)}\left(0\right)=0,k=1,2,\dots ,i=1,2$.

Definition 6. 

The zero solution of the couple of scalar equations with CFF (12) is strictly stable in each couple if for a given ${ϵ}_1>0$ there exists ${\delta }_1={\delta }_1\left({ϵ}_1\right)>0$ and for any ${\delta }_2={\delta }_2\left({ϵ}_1\right)\in (0,{\delta }_1)$ there exists ${ϵ}_2={ϵ}_2\left({\delta }_2\right)\in (0,{\delta }_2)$ such that the inequalities $|U_0|<{\delta }_1$ and ${\delta }_2<|{\omega }_0|$ imply $|U(t;U_0)|<{ϵ}_1$ and ${ϵ}_2<|\omega (t;{\omega }_0)|$ for $t\ge 0$, where the couple of functions $\left(U(t;U_0),\omega (t;{\omega }_0)\right)$ is a solution of (12).

Example 2. 

Consider the couple of impulsive fractional differential equations with CFF:

$$\begin{array}{cc}& {}^{C}D_a^{\rho ,{\psi }_k}U=\lambda U,{}^{C}D_a^{\rho ,{\psi }_k}\omega =\mu \omega fort\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\hfill \\ & U({\tau }_k+0)={\gamma }_{k}U\left({\tau }_k\right),\omega ({\tau }_k+0)={\nu }_k\omega \left({\tau }_k\right),k=1,2,\dots ,\hfill \\ & U\left(0\right)=U_0,\omega \left(0\right)={\omega }_0,\hfill \end{array}$$

(13)

where $U_0,{\omega }_0\in \mathbb{R}$ and $\lambda ,\mu ,{\gamma }_k,{\nu }_k,k=1,2,\dots ,$ are constants.

According to Lemma 3, for the couple $\left(U\right(t),\omega (t\left)\right)$,

$$\begin{array}{cc}& U\left(t\right)=U_0\left(\prod _{i=0}^{k-1}{\gamma }_{i+1}{E}_{\rho }\left(\lambda {({\psi }_i\left({\tau }_{i+1}\right)-{\psi }_i\left({\tau }_i\right))}^{\rho }\right)\right)E_{\rho }\left(\lambda {({\psi }_k\left(t\right)-{\psi }_k\left({\tau }_k\right))}^{\rho }\right),\hfill \\ & \omega \left(t\right)={\omega }_0\left(\prod _{i=0}^{k-1}{\nu }_{i+1}{E}_{\rho }\left(\mu {({\psi }_i\left({\tau }_{i+1}\right)-{\psi }_i\left({\tau }_i\right))}^{\rho }\right)\right)E_{\rho }\left(\mu {({\psi }_k\left(t\right)-{\psi }_k\left({\tau }_k\right))}^{\rho }\right),\hfill \\ & t\in ({\tau }_k,{\tau }_{k+1}]k=0,1,2,3,\dots \hfill \end{array}$$

(14)

is a solution of (3).

 Case 1. 

Let $\lambda \le 0,\mu \ge 0$ and $|{\gamma }_k|\le 1,|{\nu }_k|\ge 1.k=1,2,\dots .$

From the properties of Mittag–Leffler function, $0<E_{\rho }\left(z\right)\le 1$ for $z\le 0$ and $E_{\rho }\left(z\right)\ge 1$ for $z\ge 0$. It follows that $|U\left(t\right)|\le |U_0|\left({\prod }_{i=1}^k|{\gamma }_i|\right)\le |U_0|$ and $|\omega \left(t\right)|\ge |{\omega }_0|\left({\prod }_{i=1}^k|{\nu }_i|\right)\ge |{\omega }_0|$, i.e., the zero solution of the couple of (13) is strictly stable in the couple.

 Case 2. 

Let $\lambda >0,\mu <0$ and $|{\gamma }_k|\ge 1,|{\nu }_k|\le 1,k=1,2,\dots .$ Then, exchange functions U and ω and obtain Case 1.

 Case 3. 

Let $\lambda ,\mu >0$.

 Case 3.1. 

Let $|{\gamma }_k|>1,|{\nu }_k|>1,k=1,2,\dots .$ Then, functions $|U\left(t\right)|\ge |U_0|\left({\prod }_{i=1}^k|{\gamma }_i|\right)$ and $|\omega \left(t\right)|\ge |{\omega }_0|\left({\prod }_{i=1}^k|{\nu }_i|\right)$ increase without any bound, and the zero solution of (13) is not strictly stable in the couple.

 Case 3.2. 

Let $|{\gamma }_k|\le {\displaystyle \frac{1}{E_{\rho }\left(\lambda {({\psi }_{k-1}\left({\tau }_k\right)-{\psi }_{k-1}\left({\tau }_{k-1}\right))}^{\rho }\right)}}<1$ and $|{\nu }_k|\ge {\displaystyle \frac{1}{E_{\rho }\left(\mu {({\psi }_{k-1}\left({\tau }_k\right)-{\psi }_{k-1}\left({\tau }_{k-1}\right))}^{\rho }\right)}},$$k=1,2,\dots .$

Then, $|U\left(t\right)|\le |U_0|E_{\rho }\left(\lambda {({\psi }_k\left({\tau }_{k+1}\right)-{\psi }_k\left({\tau }_k\right))}^{\rho }\right)$ and $|\omega \left(t\right)|\ge |{\omega }_0|E_{\rho }$$\left(\mu {({\psi }_k\left(t\right)-{\psi }_k\left({\tau }_k\right))}^{\rho }\right)>|{\omega }_0|,t\in ({\tau }_k,{\tau }_{k+1}]$. This behavior depends on the applied functions in CFF. If there exists a constant $L>0$ such that for any $k=0,1,2,\dots$ the inequality ${\psi }_k\left({\tau }_{k+1}\right)-{\psi }_k\left({\tau }_k\right){)}^q)\le L$ holds, then the zero solution of the couple of (13) is strictly stable in the couple.

 Case 4. 

Let $\lambda ,\mu <0$.

 Case 4.1. 

Let $|{\gamma }_k|\le \gamma <1,|{\nu }_k|\le \nu <1,k=1,2,\dots .$ Then, ${lim}_{k\to \infty }{\gamma }^k=0$ and ${lim}_{k\to \infty }{\nu }^k=0$. Since $|U\left(t\right)|\le |U_0|\left({\prod }_{i=1}^k|{\gamma }_i|\right)\le |U_0|{\gamma }^k$ and $|\omega \left(t\right)|\le |{\omega }_0|\left({\prod }_{i=1}^k|{\nu }_i|\right)\le |{\omega }_0|{\nu }^k$, the zero solution of (13) is not strictly stable in the couple.

 Case 4.2. 

Let $|{\gamma }_k|<1$ and $|{\nu }_k|\ge {\displaystyle \frac{1}{E_{\rho }\left(\mu {({\psi }_{k-1}\left({\tau }_k\right)-{\psi }_{k-1}\left({\tau }_{k-1}\right))}^{\rho }\right)}},k=1,2,\dots .$

Then, $|U\left(t\right)|<|U_0|$ and $|\omega \left(t\right)|\ge |{\omega }_0|E_{\rho }\left(\mu {({\psi }_k\left(t\right)-{\psi }_k\left({\tau }_k\right))}^{\rho }\right),t\in ({\tau }_k,{\tau }_{k+1}]$. This behavior depends on the applied functions in CFF. If there exists a constant $M>0$ such that for any $k=0,1,2,\dots$ the inequality $M\le {\psi }_k\left({\tau }_{k+1}\right)-{\psi }_k\left({\tau }_k\right){)}^{\rho })$ holds, then $|\omega \left(t\right)|\ge |{\omega }_0|E_{\rho }\left(\mu M^{\rho }\right),t\in ({\tau }_k,{\tau }_{k+1}]$, and the zero solution of the couple of (13) is strictly stable in the couple.

5. Main Results

We now obtain some sufficient conditions for the strict stability of the system with CFF (1).

In this section, we will consider in CFF a function $\psi (.)$ such that $\psi \in C^1([0,\infty ),[0,\infty ))$ and ${\psi }^{\prime }\left(t\right)>0,t\ge 0$.

We will use the following set:

$$\begin{array}{ccc}& & \mathcal{K}=\{a\in C[{\mathbb{R}}_{+},{\mathbb{R}}_{+}]:a\mathrm{is}\mathrm{strictly}\mathrm{increasing}\mathrm{and}a\left(0\right)=0\}.\hfill \end{array}$$

Theorem 1. 

Let the following conditions be satisfied:

 1. 

The functions ${\Theta }_i\in C([0,\infty )\times \mathbb{R},\mathbb{R})$, ${\Theta }_i(t,0)\equiv 0,t\ge 0,i=1,2$, and $J_k^{\left(i\right)}:\mathbb{R}\to \mathbb{R}:J_k^{\left(i\right)}\left(0\right)=0,k=1,2,\dots ,i=1,2$ are non-decreasing.

 2. 

Condition (A) holds for functions $g(.,.)={\Theta }_i(.,.),$ and $J_k(.)=J_k^{\left(i\right)}(.),k=1,2,\dots ,i=1,2\dots$.

 3. 

There exists a function $V_1\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right),$ such that $V_1\left(0\right)=0$ and

(i) 

$\alpha (||x||)\le V_1\left(x\right)$ for $x\in {\mathbb{R}}^n,$ where $\alpha \in \mathcal{K}$.

(ii) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_1\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_1\left({\Psi }^{*}\left(t\right)\right)\le {\Theta }_1(t,V_1\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots$$

hold.

(iii) 

For any $k=1,2,\dots$ and any $x\in {\mathbb{R}}^n$, the inequalities

$$V_1\left({\mathcal{I}}_k\left(x\right)\right)\le J_k^{\left(1\right)}\left(V_1\left(x\right)\right)$$

hold.

 4. 

There exists a function $V_2\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right)$ such that

(iv) 

$\gamma (||x||)\le V_2\left(x\right)\le \beta (||x||)$ for $x\in {\mathbb{R}}^n,$ where $\beta ,\gamma \in \mathcal{K}$.

(v) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (4), the composite function $V_2\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_2\left({\Psi }^{*}\left(t\right)\right)\ge {\Theta }_2(t,V_2\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold.

(vi) 

For any $k=1,2,\dots$ and any $x\in {\mathbb{R}}^n$, the inequalities

$$V_2\left({\mathcal{I}}_k\left(x\right)\right)\ge J_k^{\left(2\right)}\left(V_2\left(x\right)\right)$$

hold.

 5. 

The zero solution of (12) is strictly stable in a couple.

Then, the zero solution of the impulsive fractional differential equation with CFF (1) is strictly stable.

Proof. 

Let ${\epsilon }_1>0$ be an arbitrary number.

Note that for any $U_0,{\omega }_0\in \mathbb{R}$ according to condition (A) with $\eta =0$, $g(.,.)={\Theta }_i(.,.)$, and $J_k(.)=J_k^{\left(i\right)}(.),i=1,2,k=1,2,\dots$ there exists a unique couple of functions $(U(t;U_0),\omega (t;{\omega }_0))$, which is a solution of the couple (12).

From condition 5 of Theorem 1 and Definition 6, the following two properties are satisfied:

(P1).

For the number $\alpha \left({\epsilon }_1\right)>0$, there exists ${\delta }_3={\delta }_3\left({\epsilon }_1\right)>0$ such that if $|U_0|<{\delta }_3$, then the first component $U(t;U_0)$ of the corresponding solution of (12) satisfies

$$|U(t;U_0)|<\alpha \left({\epsilon }_1\right)fort\ge 0.$$

(15)

(P2).

For any ${\delta }_4\in (0,{\delta }_3)$, there exists a number ${\epsilon }_3\in (0,{\delta }_4)$ such that if $|{\omega }_0|>{\delta }_4$, then the second component $\omega (t;{\omega }_0)$ of the corresponding solution of (12) satisfies

$$|\omega (t;{\omega }_0)|>{\epsilon }_{3}fort\ge 0.$$

(16)

Since $V_1\left(0\right)=0$, there exists ${\delta }_1={\delta }_1\left({\epsilon }_1\right)\in (0,{\delta }_3)$ such that $V_1\left(x\right)<{\delta }_3$ for $x\in {\mathbb{R}}^n:||x||<{\delta }_1$.

Let ${\delta }_2\in (0,{\delta }_1)$. Choose $y_0\in {\mathbb{R}}^n$:

$${\delta }_2<||y_0||<{\delta }_1,$$

(17)

and let ${\Psi }^{*}\left(t\right)=\Psi (t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ be a solution of (1) for the initial value $y_0$.

Choose the couple of initial values $(U_0^{*},{\omega }_0^{*})$ in (12) such that $U_0=V_1\left(y_0\right)\ge 0$ and ${\omega }_0=V_2\left(y_0\right)\ge 0$, and denote the corresponding unique solution of (12) by $\left(U^{*}(t;U_0^{*}),{\omega }^{*}(t;{\omega }_0^{*})\right)$.

From the inequality $||y_0||<{\delta }_1$ and the choice of the initial value $U_0$, it follows that $U_0=V_1\left(y_0\right)<{\delta }_3$, and according to property (P1) and (15), the inequality

$$|U^{*}(t;U_0)|<\alpha \left({\epsilon }_1\right),t\ge 0$$

(18)

holds.

According to condition (A) with $g(.,.)={\Theta }_1(.,.)$, $J_k(.)=J_k^{\left(1\right)}(.),k=1,2,\dots ,$$U\left(0\right)=U_0^{*}+\eta$ and conditions 3(ii) and 3(iii), all the conditions of Lemma 4 are satisfied for $\omega \left(t\right)=V_1\left({\Psi }^{*}\left(t\right)\right)$ and $U^{*}\left(t\right)=U^{*}(t;U_0)$. According to Lemma 4, the inequality $V_1\left({\Psi }^{*}\left(t\right)\right)\le U^{*}(t;U_0),t\ge 0,$ holds.

From condition 3(i) and inequality (18), we obtain the following:

$$\alpha (||{\Psi }^{*}\left(t\right)||)\le V_1\left({\Psi }^{*}\left(t\right)\right)\le U^{*}(t;U_0)<\alpha \left({\epsilon }_1\right),t\ge 0,$$

or

$$||{\Psi }^{*}\left(t\right)||<{\epsilon }_1,t\ge 0.$$

(19)

According to condition (A) with $g(.,.)={\Theta }_2(.,.)$, $J_k(.)=J_k^{\left(2\right)}(.),k=1,2,\dots ,$$\omega \left(0\right)={\omega }_0^{*}-\eta$ and conditions 3(ii) and 3(iii), all the conditions of Lemma 4 are satisfied for $\omega \left(t\right)=V_2\left({\Psi }^{*}\left(t\right)\right)$ and $U^{*}\left(t\right)={\omega }^{*}(t;{\omega }_0)$. According to Lemma 4, the inequality $V_2\left({\Psi }^{*}\left(t\right)\right)\ge {\omega }^{*}(t;{\omega }_0),t\ge 0,$ holds.

From condition 4(iv), we obtain the following inequality:

$$\beta (||{\Psi }^{*}\left(t\right)||)\ge V_2\left({\Psi }^{*}\left(t\right)\right)\ge {\omega }^{*}(t;{\omega }_0),t\ge 0.$$

(20)

From $\gamma \in \mathcal{K}$, it follows that there exists a number ${\tilde{\delta }}_4\in (0,{\delta }_3)$ such that $\gamma \left({\delta }_2\right)>{\tilde{\delta }}_4$.

From condition 4(iv) and inequality $||y_0||>{\delta }_2$ (see (17)), it follows that ${\omega }_0=V_2\left(y_0\right)\ge \gamma (||y_0||)>\gamma \left({\delta }_2\right)>{\tilde{\delta }}_4$.

Now, choose ${\delta }_4={\tilde{\delta }}_4\in (0,{\delta }_3)$ in property (P2).

According to property (P2), there exists a number ${\tilde{\epsilon }}_3\in (0,{\tilde{\delta }}_4)$ such that the inequality $|{\omega }_0|>{\tilde{\delta }}_4$ implies

$$|{\omega }^{*}(t;{\omega }_0)|>{\tilde{\epsilon }}_{3}fort\ge 0.$$

(21)

Denote ${\epsilon }_2={\beta }^{-1}\left({\tilde{\epsilon }}_3\right)$.

From inequalities (20) and (21), we obtain the following:

$$\beta (||{\Psi }^{*}\left(t\right)||)\ge V_2\left({\Psi }^{*}\left(t\right)\right)\ge {\omega }^{*}(t;{\omega }_0)|>{\tilde{\epsilon }}_3=\beta \left({\epsilon }_2\right),t\ge 0,$$

(22)

or

$$||{\Psi }^{*}\left(t\right)||>{\epsilon }_2,t\ge 0.$$

(23)

From the choice of the initial value $y_0$ (see (17)) and inequalities (19) and (23), it follows that the zero solution of the system with CFF (1) is strictly stable. □

Theorem 2. 

Let the following conditions be satisfied:

1. 

Functions ${\Theta }_i\in C([0,\infty )\times \mathbb{R},\mathbb{R})$, ${\Theta }_i(t,0)\equiv 0,i=1,2$, ${\Theta }_1(t,u)\ge {\Theta }_2(t,u)$ for $t\ge 0,u\in \mathbb{R}$, and $J_k^{\left(i\right)}(.),J_k^{\left(i\right)}\left(0\right)=0,k=1,2,\dots ,i=1,2,$ are non-decreasing, and condition (A) is satisfied for functions $g(.,.)={\Theta }_i(.,.),i=1,2$ and $J_k(.)=J_k^{\left(i\right)}(.),k=1,2,\dots ,i=1,2$.

2. 

There exists a function $V\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right)$ such that

(i) 

$\alpha (||x||)\le V\left(x\right)\le \beta (||x||)$ for $x\in {\mathbb{R}}^n,$ where $\alpha ,\beta \in \mathcal{K}$.

(ii) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequalities

$${\Theta }_2(t,V\left({\Psi }^{*}\left(t\right)\right))\le {}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}V\left({\Psi }^{*}\left(t\right)\right)\le {\Theta }_1(t,V\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold;

(iii) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$J_k^{\left(2\right)}\left(V\left(x\right)\right)\le V\left({\mathcal{I}}_k\left(x\right)\right)\le J_k^{\left(1\right)}\left(V\left(x\right)\right)$$

hold.

3. 

The zero solution of the couple of system scalar equations with CFF (12) is strictly stable in the couple.

Then, the zero solution of the system with CFF (1) is strictly stable.

The result of Theorem 2 is a special case of Theorem 1.

From Theorem 1, we obtain the following sufficient conditions for the strict stability of (1) in the special case of linear inequalities for Lyapunov functions.

Theorem 3. 

Let the following conditions be satisfied:

1. 

There exists a function $V_1\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right),$ such that $V_1\left(0\right)=0$.

(i) 

$\alpha (||x||)\le V_1\left(x\right)$ for $x\in {\mathbb{R}}^n,$ where $\alpha \in \mathcal{K}$;

(ii) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_1\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_1\left({\Psi }^{*}\left(t\right)\right)\le \lambda V_1\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold, where $\lambda \le 0$;

(iii) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$V_1\left({\mathcal{I}}_k\left(x\right)\right)\le {\gamma }_k{V}_1\left(x\right)$$

hold, where $0<{\gamma }_k\le 1,k=1,2,\dots$.

2. 

There exists a function $V_2\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right)$ such that

(iv) 

$\kappa (||x||)\le V_2\left(x\right)\le \beta (||x||)$ for $x\in {\mathbb{R}}^n,$ where $\beta ,\kappa \in \mathcal{K}$;

(v) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_2\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_2\left({\Psi }^{*}\left(t\right)\right)\ge \mu V_2\left({\Psi }^{*}\left(t\right)\right),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold with $\mu \ge 0$;

(vi) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$V_2\left({\mathcal{I}}_k\left(x\right)\right)\ge {\nu }_k{V}_2\left(x\right)$$

hold, where ${\nu }_k>1,k=1,2,\dots$.

Then, the zero solution of the system with CFF (1) is strictly stable.

Proof. 

From Lemma 3 applied to system (12) with $g_1(t,U)=\lambda U,g_2(t,U)=\mu U$ and $J_k^{\left(1\right)}\left(U\right)={\gamma }_{k}U,J_k^{\left(2\right)}\left(U\right)={\nu }_{k}U$, it follows that conditions 1,2 of Theorem 1 are satisfied. Also, from Case 1 of Example 2, it follows that condition 5 of Theorem 1 is satisfied. □

In Theorem 3, the main constants $\lambda ,\mu$ in the linear conditions for Lyapunov functions in Theorem 3 have different signs. In the case when both constants are positive, based on Example 2 and Case 3.2, we obtain the following results with a restriction about the applied functions in CFF.

Theorem 4. 

Let the following conditions be satisfied:

1. 

There exists a constant $\mathcal{L}>0$ such that ${\psi }_k\left({\tau }_{k+1}\right)-{\psi }_k\left({\tau }_k\right)\le \mathcal{L}$ for $k=0,1,2,\dots$.

2. 

There exists a function $V_1\in \mathrm{\Lambda },V_1\left(0\right)=0,$ such that

(i) 

$\alpha (||x||)\le V_1\left(x\right)$ for $x\in {\mathbb{R}}^n,$ where $\alpha \in \mathcal{K}$;

(ii) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_1\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_1\left({\Psi }^{*}\left(t\right)\right)\le \lambda V_1\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold, where $\lambda >0$;

(iii) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$V_1\left({\mathcal{I}}_k\left(x\right)\right)\le {\gamma }_k{V}_1\left(x\right)$$

hold, where $0<{\gamma }_k{E}_q\left(\lambda {\mathcal{L}}^q\right)\le 1,k=1,2,\dots$.

3. 

There exists a function $V_2\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right)$ such that

(iv) 

$\gamma (|||x|)\le V_2\left(x\right)\le \beta (||x||)$ for $x\in {\mathbb{R}}^n,$ where $\beta \in \mathcal{K}$;

(v) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_2\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_2\left({\Psi }^{*}\left(t\right)\right)\ge \mu V_2\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold with $\mu >0$;

(vi) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$V_2\left({\mathcal{I}}_k\left(x\right)\right)\le {\nu }_k{V}_2\left(x\right)$$

hold, where ${\nu }_k{E}_q\left(\mu {({\psi }_{k-1}\left({\tau }_k\right)-{\psi }_{k-1}\left({\tau }_{k-1}\right))}^q\right)\ge 1,k=1,2,\dots$.

Then, the zero solution of the system with CFF (1) is strictly stable.

Proof. 

Consider the couple of impulsive fractional differential equations with CFF (13) with a solution provided by (14). According to Case 3.2 of Example 2, the zero solution of the couple (13) is strictly stable in the couple, i.e., condition 5 of Theorem 1 is satisfied.

According to Theorem 1 with ${\Theta }_1(t,U)=\lambda U,{\Theta }_2(t,U)=\mu U$ and $J_k^{\left(1\right)}\left(U\right)={\gamma }_{k}U,J_k^{\left(2\right)}\left(U\right)={\nu }_{k}U,k=1,2,\dots ,$ we deduce the claim of Theorem 4. □

Remark 6. 

If ${\psi }_k\left(t\right)={\displaystyle \frac{t}{t+1}},k=0,1,2,\dots$ and ${\tau }_k=k,k=1,2,\dots$, then ${\psi }_k(k+1)-{\psi }_k\left(k\right)={\displaystyle \frac{k+1}{k+2}}-{\displaystyle \frac{k}{k+1}}\le 0.5$, and the constant $\mathcal{L}=0.5$ in condition 1.

If ${\psi }_k\left(t\right)=t^2,k=0,1,2,\dots$ and ${\tau }_k=k,k=1,2,\dots ,$, then ${lim}_{k\to \infty }({\psi }_k(k+1)-{\psi }_k\left(k\right))={lim}_{k\to \infty }({(k+1)}^2-k^2)={lim}_{k\to \infty }(2k+1)=\infty$, and the constant $\mathcal{L}$ in condition 1 does not exist.

If ${\psi }_k\left(t\right)=t,k=0,1,2,\dots$ and ${\tau }_k=k,k=1,2,\dots ,$, then ${\psi }_k(k+1)-{\psi }_k\left(k\right)=1$, and the constant $L=1$ in condition 1.

We now illustrate the usefulness of Theorem 4.

Example 3. 

Consider the initial value problem for the scalar linear impulsive fractional differential equation with CFF (11) with $\lambda =1>0$. Its solution is

$$U\left(t\right)=U_0\left(\prod _{i=1}^k{b}_i{E}_{0.3}\left({(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\right)\right)E_{0.3}\left({(\psi \left(t\right)-\psi \left(k\right))}^{0.3}\right)$$

with $U_0>0$, and $b_k={\displaystyle \frac{1}{E_{0.3}\left({(\psi \left(k\right)-\psi \left(k-1\right))}^{0.3}\right)}},k=1,2,\dots$.

As in Case 2.2.1 of Example 1, it can be seen that the solution $U\left(t\right)$ of (22) is strictly stable.

Now, we check the conditions of Theorem 3.

Consider $V_1\left(x\right)=x^2$. Then, we obtain ${}^{C}D_{{\tau }_k}^{0.3,\psi }{U}^2\left(t\right)\le 2U\left(t\right){}^{C}D_{{\tau }_k}^{0.3,\psi }U\left(t\right)=2V\left(U\left(t\right)\right)$ for $t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,3,\dots .$, i.e., condition 1(ii) is satisfied with $\lambda =2>0$, and ${\psi }_k(.)\equiv \psi (.),k=0,1,2,\dots$.

Also, for any $y\in {\mathbb{R}}^n$ we obtain $V_1\left(b_{k}y\right)=b_k^2{y}^2={\displaystyle \frac{1}{E_{0.3}\left({(\psi \left(k\right)-\psi \left(k-1\right))}^{0.3}\right)}}{V}_1\left(y\right)$, i.e., condition 1(iii) is satisfied with ${\gamma }_k={\displaystyle \frac{1}{E_{0.3}\left({(\psi \left(k\right)-\psi \left(k-1\right))}^{0.3}\right)}}\in (0,1]$.

Consider the Lyapunov function $V_2\left(x\right)=e^x,x\in \mathbb{R}.$

Then, we obtain ${\left(e^{U\left(t\right)}\right)}^{\prime }=e^{U\left(t\right)}{U}^{\prime }\left(t\right)=e^{U\left(t\right)}{U}_0\left({\prod }_{i=1}^k{b}_i{E}_{0.3}\left(\lambda {(\psi \left(i\right)-\psi \left(i-1\right))}^{0.3}\right)\right)$${\displaystyle \frac{d}{dt}}{E}_{0.3}\left(\lambda {(\psi \left(t\right)-\psi \left(k\right))}^{0.3}\right)>0$. According to Definition 2, the inequality ${}^{C}D_{{\tau }_k}^{0.3,\psi }{e}^{U\left(t\right)}>0,t\in ({\tau }_k,{\tau }_{k+1}]$ holds, i.e., condition 2(v) of Theorem 3 is satisfied with $\mu =0$ and ${\psi }_k(.)\equiv \psi (.),k=0,1,2,\dots$.

Also, for any $y\in {\mathbb{R}}^n$ we obtain $V_1\left(b_{k}y\right)=e^{b_{k}y}=e^{{\displaystyle \frac{1}{E_{0.3}\left(1\right)}}}{e}^y=e^{{\displaystyle \frac{1}{E_{0.3}\left(1\right)}}}{V}_1\left(y\right)$, i.e., condition 2(vi) is satisfied with ${\nu }_k=e^{{\displaystyle \frac{1}{E_{0.3}^2\left(1\right)}}}>1$.

Therefore, according to Theorem 3, the solution of (22) is strictly stable.

In the case when both constants in the linear conditions for Lyapunov functions are negative, based on Example 2, Case 4.2, we obtain the following results with a restriction regarding the applied functions in CFF.

Theorem 5. 

Let the following conditions be satisfied:

1. 

There exists a constant $M>0$ such that $M\le {\psi }_k\left({\tau }_{k+1}\right)-{\psi }_k\left({\tau }_k\right)$ for $k=0,1,2,\dots$.

2. 

There exists a function $V_1\in \mathrm{\Lambda },V_1\left(0\right)=0,$ such that

(i) 

$\alpha (||x||)\le V_1\left(x\right)$ for $x\in {\mathbb{R}}^n,$ where $\alpha \in \mathcal{K}$;

(ii) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_1\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_1\left({\Psi }^{*}\left(t\right)\right)\le \lambda V_1\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold, where $\lambda <0$;

(iii) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$V_1\left({\mathcal{I}}_k\left(x\right)\right)\le {\gamma }_k{V}_1\left(x\right)$$

hold, where $0<{\gamma }_k\le 1,k=1,2,\dots$.

3. 

There exists a function $V_2\in \mathrm{\Lambda }\left({\mathbb{R}}^n\right)$ such that

(iv) 

$\gamma (||x||)\le V_2\left(x\right)\le \beta (||x||)$ for $x\in {\mathbb{R}}^n,$ where $\beta \in \mathcal{K}$;

(v) 

For any solution ${\Psi }^{*}\left(t\right)={\Psi }^{*}(t;y_0)\in P{C}^{\rho }\left({\mathbb{R}}^n\right)\cap P{C}^1\left({\mathbb{R}}^n\right)$ of (1), the composite function $V_2\left({\Psi }^{*}(.)\right)\in P{C}^{\rho }\left(\mathbb{R}\right)$ and the inequality

$${}^{C}D_{{\tau }_k}^{\rho ,{\psi }_k}{V}_2\left({\Psi }^{*}\left(t\right)\right)\ge \mu V_2\left({\Psi }^{*}\left(t\right)\right)),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,$$

hold with $\mu <0$;

(vi) 

For any $k=1,2,\dots$ and $x\in {\mathbb{R}}^n$, the inequalities

$$V_2\left({\mathcal{I}}_k\left(x\right)\right)\le {\nu }_k{V}_2\left(x\right)$$

hold, where ${\nu }_k{E}_q\left(\mu {({\psi }_{k-1}\left({\tau }_k\right)-{\psi }_{k-1}\left({\tau }_{k-1}\right))}^q\right)\ge 1,k=1,2,\dots$

Then, the zero solution of the system with CFF (1) is strictly stable.

Proof. 

Consider the couple of impulsive fractional differential equations with CFF (13) with a solution provided by (14). According to Case 4.2 of Example 2, the zero solution of the couple (13) is strictly stable in the couple, i.e., condition 5 of Theorem 1 is satisfied.

According to Theorem 1 with ${\Theta }_1(t,U)=\lambda U,{\Theta }_2(t,U)=\mu U$ and $J_k^{\left(1\right)}\left(U\right)={\gamma }_{k}U,J_k^{\left(2\right)}\left(U\right)={\nu }_{k}U,k=1,2,\dots ,$ we deduce the claim of Theorem 5. □

Remark 7. 

If ${\psi }_k\left(t\right)={\displaystyle \frac{t}{t+1}},k=0,1,2,\dots$, and ${\tau }_k=k,k=1,2,\dots ,$, then ${lim}_{k\to \infty }{\psi }_k(k+1)-{\psi }_k\left(k\right)={lim}_{k\to \infty }{\displaystyle \frac{2k+1}{k(k+1)}}=0$, and the constant M in condition 1 does not exist.

If ${\psi }_k\left(t\right)=t^2,k=0,1,2,\dots$ and ${\tau }_k=k,k=1,2,\dots ,$, then ${\psi }_k(k+1)-{\psi }_k\left(k\right)={(k+1)}^2-k^2=2k+1\ge 1$, and the constant $M=1$ in condition 1.

Remark 8. 

In the case ${\tau }_k=0,k=1,2,\dots$, the main problem (1) coincides with the initial value problem in [18]. From the obtained results in this paper, we have the results in [18]. In the case ${\tau }_k=0,k=1,2,\dots$ and $\psi \left(t\right)\equiv t$, the results of [8] follow from the above results.

6. Conclusions

We considered nonlinear impulsive fractional differential equations with CFF. The main aim of the paper was to study strict stability. We illustrated the influence of both the function applied in CFF and the impulsive functions on the strict stability properties of the solutions using several examples. For the main purpose of the paper, we proved some comparison results. Several types of sufficient conditions were obtained via Lyapunov functions. We note that all of our results depend significantly on impulsive functions. The results obtained are new and will hopefully be applied to study the strict stability of states in real models.