The Stability of Set-Valued Differential Equations with Different Initial Time in the Sense of Fractional-like Hukuhara Derivatives

Abstract

This paper investigates set-valued differential equations with fractional-like Hukuhara derivatives. Firstly, a novel comparison principle is given by introducing the upper quasi-monotone increasing functions. Then, the stability criteria of Lipschitz stability and practical stability of such equations with different initial time are obtained via the new comparison principle and vector Lyapunov functions.

1. Introduction

Set-valued differential equations has attracted extensive attention of scholars because of its important applications in system identification, signal processing, optics, thermal, rheology and many other fields. There are some interesting results such as set-valued differential equations [1,2,3,4,5,6,7,8], set-valued functional differential equations [9,10,11,12,13] as well as stochastic set-valued differential equations [14,15,16]. The literature [17,18] systematically summarized the research results of this kind of problems.

Lyapunov’s stability theory is a very important issue not only in theory but also in application. Recently, some scholars have discussed the Lipschitz stability and practical stability of fractional differential systems (see [19,20,21,22,23,24,25]). We notice that most of the existing research results are concerned the perturbation of space variables at the same initial time. However, in a real problem, We also need to consider the change of different initial time. Up till now, there are some results of fractional differential equations in this case, we can find it in [26,27,28,29]. there are few studies of set-valued differential equations with different initial time in the sense of fractional-like Hukuhara derivatives (see [30]).

Based on the above analysis, we investigate set-valued differential equations in sense of fractional-like Hukuhara derivatives. we first give a novel comparison principle via the upper quasi-monotone increasing of vector functions. In addition, the Lipschitz stability and practical stability of such equations at different initial time in the sense of fractional-like Hukuhara derivatives are obtained by using the new comparison theorem and the vector Lyapunov function method. The results obtained extend the existing research results.

2. Preliminaries

Denote $K_c({\mathbb{R}}^n)$ compact and convex nonempty subsets of ${\mathbb{R}}^n$. We consider the following Hausdorff metric

$$H[A,B]=max\{\underset{x\in A}{sup}inf[d(x,y):y\in B],\underset{y\in B}{sup}inf[d(y,x):x\in A]\},$$

in which $A,B\in K_c({\mathbb{R}}^n)$. In particular, $H[X,\theta ]=\underset{x\in X}{sup}d(x,\theta )$, where $\theta ={(0,0,0···0)}^T\in K_c({\mathbb{R}}^n)$.

The properties of Hausdorff metric can be included in literature [18].

Definition 1

(see [23]). A set $Z\in K_c({\mathbb{R}}^n)$ is called the Hausdorff difference, if for $X,Y\in K_c({\mathbb{R}}^n)$, $X=Y+Z$ is true, and can be represented as $X-Y$.

Remark 1.

Under general conditions $A-B\ne A+(-1)B$. For example, $A=[0,1]$, then $(-1)A=[-1,0]$, $A+(-1)A=[0,1]+[-1,0]=[-1,1]\ne A-A$.

Fractional calculus unified and generalized the concepts of general derivatives and integrals, their properties can be found in [23].

For a mapping $X:J\to K_c({\mathbb{R}}^n)$ and $J=[t_0,\infty )$, the Hukuhara derivative of order q can be expressed as

$${\mathcal{D}}_H^{q}X(t_0)=\underset{\varpi \to 0^{+}}{lim}\frac{X(t_0+\varpi {(t-t_0)}^{1-q})-X(t_0)}{\varpi }=\underset{\varpi \to 0^{+}}{lim}\frac{X(t_0)-X(t_0-\varpi {(t-t_0)}^{1-q})}{\varpi }$$

where $t>t_0$, $0<q\le 1$.

Denote $C^q(J,K_c({\mathbb{R}}^n))$ the set of set-valued mappings X that are $q—$differentiable $0<q\le 1$.

The relationship between fractional Hukuhara derivative and integer Hukuhara derivative is as follows. For a differentiable set-valued mapping of order q, $X:J\to K_c({\mathbb{R}}^n)$ and $J=[t_0,\infty )$, we have (see [23])

$${\mathcal{D}}_H^{q}X(t_0)={(t-t_0)}^{1-q}{D}_{H}X(t_0),$$

where $0<q\le 1$.

We first give the following function classes and the concept of upper quasi-monotone increasing on vector functions.

$\mathcal{M}(J)=\{m\in C[J,{\mathbb{R}}_{+}]|$$m(u)$ is strictly monotonically increasing, $m(0)=0$, and there exist function $q_m\in C(J,{\mathbb{R}}_{+})$: for $\alpha \ge 0$ and $q_m(\alpha )\ge 1$ such that $m^{-1}(\alpha u)\le u{q}_m(\alpha )$};

$\mathcal{K}(J)=\{p\in C[J,{\mathbb{R}}_{+}]|$$p(u)$ is strictly monotonically increasing, $p(0)=0$, and there is a constant $K_p>0$ satisfying $p(u)\le K_{p}u$};

$S(r)=\{X\in K_c({\mathbb{R}}^n):H[X,\theta ]\le r,r>0$ is a constant}.

Remark 2.

We can give the examples for above function classes. For example, the function $p(u)=K_{1}u\in \mathcal{K}({\mathbb{R}}_{+})$ for $K_p=K_1>0$, $m(u)=K_2{u}^2\in \mathcal{M}[0,1]$ for $K_2\in (0,1]$.

Definition 2.

A function $f\in C[J\times {\mathbb{R}}_{+}^m,{\mathbb{R}}^m],1\le m<n$, is called upper quasi-monotone increasing in x, for a given $x,w,z\in {\mathbb{R}}_{+}^m$, $x\le \underset{1\le i\le m}{max}{w}_{i}z$ implies $f(t,x)\le \underset{1\le i\le m}{max}{f}_i(t,w)z$, in which $z={(z_1,z_2,···,z_m)}^T$, $z_i=1$, $i=1,2,···,m$.

Definition 3.

A function $V\in C[J\times K_c({\mathbb{R}}^n),{\mathbb{R}}_{+}^m]$, $V(t,\theta )=0$, is called the vector Lyapunov function, if the following inequality

$$V(t,X)-V(t,Y)\le LH[X,Y]\epsilon ,t\in J$$

holds, where $X,Y\in K_c({\mathbb{R}}^n)$, $L>0$ is a constant, $\epsilon ={({\epsilon }_1,{\epsilon }_2,···,{\epsilon }_m)}^T,{\epsilon }_i=1$, $i=1,···,m.$

Consider the set-valued differential equations at different initial time in the sense of fractional-like Hukuhara derivatives

$$\begin{array}{cc}\hfill & {\mathcal{D}}_H^{q}X(t)=F(t,X(t)),X(t_0)=X_0,t\ge t_0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & {\mathcal{D}}_H^{q}Z(t)=F(t,Z(t)),Z({\tau }_0)=Z_0,t\ge {\tau }_0,\hfill \end{array}$$

(2)

where $F\in C^q[J\times K_c({\mathbb{R}}^n),K_c({\mathbb{R}}^n)]$.

We can define the Dini derivatives of $V(t,X)$ relative to (1) and (2) as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_{+}^{q}V(t,Z(t)-X(t-\zeta ))& =\underset{h\to 0^{+}}{lim\; sup}\frac{1}{h^q}[V(t,Z(t)-X(t-\zeta )+h^q(F(t,Z(t))-F(t,X(t-\zeta ))))\hfill \\ \hfill & -V(t-h,Z(t)-X(t-\zeta ))],\hfill \end{array}$$

where $\zeta ={\tau }_0-t_0$.

3. The Stability with Different Initial Time and Comparison Principle

We first present the definitions of some stability with different initial time.

Definition 4

(see [26]). The solution $Z(t,{\tau }_0,Z_0)$ of Equation (2) relative to the solution $X(t-\zeta ,t_0,X_0)$ of Equation (1) is said to be

$(B_1)$ Lipschitz stable if there are constants $\delta =\delta ({\tau }_0)>0$, $\overline{\delta }=\overline{\delta }({\tau }_0)>0$, $M=M({\tau }_0)>1$, such that for $|{\tau }_0-t_0|<\overline{\delta }$ and $H[Z_0-X_0,\theta ]\le \delta$, the equality $H[Z(t,{\tau }_0,Z({\tau }_0))-X(t-\zeta ,t_0,X_0),\theta ]\le MH[Z_0-X_0,\theta ]$ is true;

$(B_2)$ Uniformly Lipschitz stable if M of $(B_1)$ is independent of ${\tau }_0$.

Definition 5

(see [27]). If for a given $(\mu ,B)$ with $0<\mu <B$, the solution $Z(t,{\tau }_0,Z_0)$ of Equation (2) relative to the solution $X(t-\zeta ,t_0,X_0)$ of Equation (1) is called to be

$(B_3)$ Practical stable if there is a ${\delta }_1={\delta }_1(t_0,\mu ,B)>0$, and for $|{\tau }_0-t_0|<{\delta }_1$ and $H[Z_0-X_0,\theta ]<\mu$, the inequality $H[Z(t,{\tau }_0,Z_0)-X(t-\zeta ,t_0,X_0),\theta ]<B$ is true, $t\ge t_0$;

$(B_4)$ Uniformly practically stable, if ${\delta }_1$ in $(B_3)$ is independent of $t_0$, $(B_3)$ is established.

Let’s start with a new comparison principle.

Lemma 1.

Assume that

$(C_1)$ V is a vector Lyapunov function, and

$${\mathcal{D}}_{+}^{q}V(t,Z(t)-X(t-\zeta ))\le f(t,V(t,Z(t)-X(t-\zeta ))),$$

where $V\in C[J\times K_c({\mathbb{R}}^n),{\mathbb{R}}_{+}^m]$;

$(C_2)$$f\in C[J\times {\mathbb{R}}_{+}^m,{\mathbb{R}}^m]$ is upper quasi-monotone increasing in x, and the system

$$\begin{array}{c}\hfill {\mathcal{D}}_H^{q}x(t)=f(t,x),x({\tau }_0)=x_0\ge 0\end{array}$$

(3)

has a solution $x(t)$.

Then $V(t_0,Z_0-X_0)\le \underset{1\le i\le m}{max}{x}_i({\tau }_0)\epsilon$ implies

$$V(t,Z(t,{\tau }_0,Z_0)-X(t-\zeta ,t_0,X_0))\le \underset{1\le i\le m}{max}{x}_i(t)\epsilon ,$$

where $\epsilon ={({\epsilon }_1,···,{\epsilon }_m)}^T,{\epsilon }_i=1,i=1,···,m.$

Proof. 

Set $v(t)=V(t,Z(t)-X(t-\zeta ))$, by $(C_1)$, for small $h=\varpi {(t-t_0)}^{1-q}>0$, $0<q\le 1$, $\varpi \to 0_{+}$, we can have

$$\begin{array}{cc}\hfill v(t)-v(t-h)=& V(t,Z(t)-X(t-\zeta ))-V(t-h,Z(t-h)-X(t-\zeta -h))\hfill \\ \hfill \le & V(t,Z(t)-X(t-\zeta ))\hfill \\ \hfill & -V(t-h,Z(t)-X(t-\zeta )+h^q(F(t,Z(t))-F(t,X(t-\zeta ))))\hfill \\ \hfill & +LH[Z(t)-X(t-\zeta )\hfill \\ \hfill & +h^q(F(t,Z(t))-F(t,X(t-\zeta ))),Z(t-h)-X(t-\zeta -h)]\epsilon .\hfill \end{array}$$

Furthermore, we can obtain

$$\begin{array}{cc}\hfill {\mathcal{D}}_{+}^{q}v(t)=& \underset{h\to 0^{+}}{lim\; sup}\frac{1}{h^q}[v(t)-v(t-h)]\hfill \\ \hfill \le & {\mathcal{D}}_{+}^{q}V(t,Z(t)-X(t-\zeta ))+L\underset{h\to 0^{+}}{lim\; sup}\frac{1}{h^q}H[Z(t)-X(t-\zeta )\hfill \\ \hfill & +h^q(F(t,Z(t))-F(t,X(t-\zeta ))),Z(t-h)-X(t-\zeta -h)]\epsilon .\hfill \end{array}$$

Let

$$Z(t-h)-X(t-\zeta -h)=Z(t)-X(t-\zeta )-\overline{Z}(t,h)$$

in which $\overline{Z}(t,h)$ is the Hukuhara difference of $Z(t-h)-X(t-\zeta -h)$ and $Z(t)-X(t-\zeta )$.

Then, we can obtain

$$\begin{array}{c}\frac{1}{h^q}H[Z(t)-X(t-\zeta )+h^q(F(t,Z(t))-F(t,X(t-\zeta ))),Z(t-h)-X(t-\zeta -h)]\hfill \\ \hspace{1em}\hspace{1em}=\frac{1}{h^q}H[Z(t)-X(t-\zeta )+h^q(F(t,Z(t))-F(t,X(t-\zeta ))),Z(t)-X(t-\zeta )-\overline{Z}(t,h)]\hfill \\ \hspace{1em}\hspace{1em}=\frac{1}{h^q}H[h^q(F(t,Z(t))-F(t,X(t-\zeta ))),Z(t)-X(t-\zeta )-(\overline{Z}(t-h)-X(t-\zeta -h))]\hfill \\ \hspace{1em}\hspace{1em}=H[F(t,Z(t))-F(t,X(t-\zeta )),\frac{1}{h^q}(Z(t)-X(t-\zeta )-(\overline{Z}(t-h)-X(t-\zeta -h)))],\hfill \end{array}$$

and

$$\begin{array}{c}\underset{h\to 0^{+}}{lim\; sup}\frac{1}{h^q}H[Z(t)-X(t-\zeta )+h^q(F(t,Z(t))-F(t,X(t-\zeta ))),\overline{Z}(t-h)-X(t-\zeta -h)]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=H[F(t,Z(t))-F(t,X(t-\zeta )),{\mathcal{D}}_H^{q}Z(t)-{\mathcal{D}}_H^{q}X(t-\zeta )]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=0.\hfill \end{array}$$

Thus

$${\mathcal{D}}_{+}^{q}v(t)\le f(t,V(t,Z(t)-X(t-\zeta ))).$$

Meanwhile, from the condition $(C_2)$, We find that

$${\mathcal{D}}_{+}^{q}v(t)\le \underset{1\le i\le m}{max}{f}_i(t,z(t))\epsilon ,V(t_0,Z_0-X_0)\le \underset{1\le i\le m}{max}{x}_i({\tau }_0)\epsilon .$$

Therefore $v(t)\le \underset{1\le i\le m}{max}{x}_i(t)\epsilon$, that is

$$V(t,Z(t)-X(t-\zeta ))\le \underset{1\le i\le m}{max}{x}_i(t)\epsilon .$$

This proves the claimed estimate of Lemma 1. □

Corollary 1.

If $f(t,V(t,Z(t,{\tau }_0,Z_0)-X(t,t_0,X_0)))=0$ is applicable in Lemma 1, then the following estimate

$$V(t,Z(t)-X(t-\zeta ))\le V(t_0,Z_0-X_0),t\ge t_0$$

is true.

4. Stability Criteria

Next, we discuss the stability of Lipschitz stability and practical stability at different initial time.

4.1. Lipschitz Stability with Different Initial Time

Theorem 1.

Assume that

$(M_1)$$V(t,X)$ is the vector Lyapunov function, and

$$\begin{array}{c}\hfill {\mathcal{D}}_{+}^{q}V(t,Z(t)-X(t-\zeta ))\le 0;\end{array}$$

(4)

$(M_2)$ the inequality

$$\begin{array}{c}\hfill b(H[Z(t)-X(t-\zeta ),\theta ])\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta )),b\in \mathcal{K}\end{array}$$

(5)

is true.

Then, the solution $Z(t)$ of Equation (2) is Lipschitz stable relative to the solution $X(t-\zeta )$ of Equation (1).

Proof. 

From the continuity of V and $V({\tau }_0,\theta )=0$, we can know that there exist $M=M({\tau }_0)>1$, ${\delta }_1>0$ and ${\delta }_2>0$, such that, for $H[Z_0-X_0,\theta ]\le {\delta }_1,|{\tau }_0-t_0|<{\delta }_2$, the inequality $\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)<b(MH[Z_0-X_0,\theta ])$ holds. Combined with Corollary 1 and the conditions $(M_1)$ and $(M_2)$, We obtain

$$\begin{array}{cc}\hfill b(H[Z(t)-X(t-\zeta ),\theta ])& \le \underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)\hfill \\ \hfill & <b(MH[Z_0-X_0,\theta ]).\hfill \end{array}$$

Furthermore, since $b\in \mathcal{K}$, we can obtain

$$H[Z(t)-X(t-\zeta ),\theta ]\le MH[Z_0-X_0,\theta ],t\ge {\tau }_0.$$

Thus, the conclusion of Theorem 1 is proved. □

Theorem 2.

Assume that $(M_1)$ of Theorem 1 holds and

$(M_3)$${\displaystyle b(H[Z(t)-X(t-\zeta ),\theta ])\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le a(H[Z(t)-X(t-\zeta ),\theta ]),}$ in which $a,b\in \mathcal{K}$.

Then, the solution $Z(t)$ of Equation (2) is uniformly Lipschitz stable relative to the solution $X(t-\zeta )$ of Equation (1).

Proof. 

Let $\delta =a^{-1}(b(MH[Z_0-X_0,\theta ]))$. Then, $MH[Z_0-X_0,\theta ]=b^{-1}(a(\delta ))$. According to Corollary 1 and the condition $(M_3)$, for $H[Z_0-X_0,\theta ]\le \delta$, it follows that

$$\begin{array}{cc}\hfill b(H[Z(t)-X(t-\zeta ),\theta ])& \le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\hfill \\ \hfill & \le \underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)\hfill \\ \hfill & \le a(H[Z_0-X_0,\theta ])\hfill \\ \hfill & \le a(\delta ).\hfill \end{array}$$

Furthermore, we obtain

$$H[Z(t)-X(t-\zeta ),\theta ]<b^{-1}(a(\delta ))=MH[Z_0-X_0,\theta ],t\ge {\tau }_0.$$

Thus, the result of Theorem 2 is proved. □

Theorem 3.

Assume that

$(D_1)$$V(t,X)$ is a vector Lyapunov function, and

$$\begin{array}{c}\hfill b(H[Z(t)-X(t-\zeta ),\theta ])\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\end{array}$$

(6)

holds, where $b\in \mathcal{K}$.

$(D_2)$ The inequality

$$\begin{array}{c}\hfill {\mathcal{D}}_{+}^{q}V(t,Z(t)-X(t-\zeta ))\le f(t,V(t,Z(t)-X(t-\zeta )))\end{array}$$

(7)

holds, in which $f\in C[J\times {\mathbb{R}}_{+}^m,{\mathbb{R}}^m]$.

$(D_3)$$f\in C[J\times {\mathbb{R}}_{+}^m,{\mathbb{R}}^m]$ is upper quasi-monotone increasing in x, and

$$\begin{array}{c}\hfill {\mathcal{D}}_H^{q}x(t)=f(t,x),x({\tau }_0)=x_0\ge 0,t\ge {\tau }_0\end{array}$$

(8)

is Lipschitz stable.

Then, the solution $Z(t)$ of Equation (2) is Lipschitz stable relative to the solution $X(t-\zeta )$ of the Equation (1).

Proof. 

From the condition $(D_3)$, there exist constants $M({\tau }_0)>1,{\delta }_1={\delta }_1({\tau }_0,M)$, for any $x_0\in {\mathbb{R}}_{+}^m,\underset{1\le i\le m}{max}{x}_i({\tau }_0)<{\delta }_1$, the inequality

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{x}_i(t)\le M\underset{1\le i\le m}{max}{x}_i({\tau }_0),t\ge {\tau }_0\end{array}$$

(9)

holds.

Since $V({\tau }_0,\theta )=0$, then there exists a ${\overline{\delta }}_1={\overline{\delta }}_1({\tau }_0,{\delta }_1)>0$, for $H[Z_0-X_0,\theta ]<{\overline{\delta }}_1$, $\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)<{\delta }_1$.

Choose $\delta =min\{{\delta }_1,{\overline{\delta }}_1\}$ and $\overline{M}>1$ such that $\overline{M}>ML$, we can assume $\tilde{M}=q_b(\overline{M})$.

Let $H[Z_0-X_0,\theta ]<\delta$. Consider a solution of the Equation (2), it follows that $\underset{1\le i\le m}{max}{x}_i({\tau }_0)=\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)<{\delta }_1$. Therefore, the function $\underset{1\le i\le m}{max}{x}_i(t)$ satisfies (9).

Using the condition $(D_2)$, we obtain

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le \underset{1\le i\le m}{max}{x}_i(t).\end{array}$$

(10)

Furthermore, using the condition $(D_1)$, we obtain

$$\begin{array}{cc}\hfill b(H[Z(t)-X(t-\zeta ),\theta ])& \le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\hfill \\ \hfill & \le \underset{1\le i\le m}{max}{x}_i(t)\le M\underset{1\le i\le m}{max}{x}_i({\tau }_0)\hfill \\ \hfill & =M\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)\hfill \\ \hfill & \le \overline{M}H[Z_0-X_0,\theta ].\hfill \end{array}$$

(11)

From the properties of $b\in \mathcal{K}$ and $b^{-1}(\overline{M}x)<\overline{M}{q}_b(x)$, we obtain

$$H[Z(t)-X(t-\zeta ),\theta ]\le b^{-1}(\overline{M}H[Z_0-X_0,\theta ])\le \tilde{M}H[Z_0-X_0,\theta ].$$

Therefore, the result of Theorem 3 hold. □

Corollary 2.

Assume that the conditions $(D_2)$ and $(D_3)$ in Theorem 3 hold, and the condition $(D_1)$ is replaced by

$(E_1)$$V(t,X)$ is a vector Lyapunov function, and there is a $K_1>0$ such that

$$\begin{array}{c}\hfill K_{1}H[Z(t)-X(t-\eta ),\theta ]\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta )).\end{array}$$

Then, the solution $Z(t)$ of Equation (2) is Lipschitz stable respect to the solution $X(t-\zeta )$ of Equation (1).

In the proof of Theorem 3, we only need to take $\overline{M}\ge 1,\overline{M}>M\frac{L}{K_1}$ and $\tilde{M}=\overline{M}$ to obtain the desired conclusion, we omit it here.

Theorem 4.

Assume that the condition $(D_2)$ in Theorem 3 holds and

$(D_5)$ There exist $a\in M([0,r])$, $b\in \mathcal{K}([0,r])$, $r>0$, such that

$$b(H[Z(t)-X(t-\zeta ),\theta ])\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le a(H[Z(t)-X(t-\zeta ),\theta ]).$$

(12)

$(D_6)$ The system

$$\begin{array}{c}\hfill {\mathcal{D}}_H^{q}f(t)=f(t,x),x({\tau }_0)=x_0\ge 0,t\ge {\tau }_0\end{array}$$

(13)

is uniformly Lipschitz stable, where $x\in {\mathbb{R}}_{+}^m,f\in C[J\times {\mathbb{R}}_{+}^m,{\mathbb{R}}^m]$.

Then, the solution $Z(t)$ of Equation (2) is uniformly Lipschitz stable relative to the solution $X(t-\zeta )$ of Equation (1).

Proof. 

According to the condition $(D_6)$, there exist $M>1$ and ${\delta }_1>0$, and for every $x_0\in {\mathbb{R}}_{+}^m$, $\underset{1\le i\le m}{max}{x}_i({\tau }_0)<{\delta }_1$, the inequality

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{x}_i(t)\le M\underset{1\le i\le m}{max}{x}_i({\tau }_0),t\ge {\tau }_0\end{array}$$

(14)

holds.

For $a\in M([0,r]),b\in \mathcal{K}([0,r])$, there are function $q_b(x)$ and constant $K_a>0$. Choosing $M_1\ge 1$ and $M_1>q_b(M)K_a,\delta \le \frac{r}{M_1}$. Therefore, ${\overline{\delta }}_1\le r$.

Let $\delta =min\{{\delta }_1,{\delta }_2,\frac{{\delta }_1}{K_a}\}$, Choosing the initial value $H[Z_0-X_0,\theta ]<\delta$ such that $H[Z_0-X_0,\theta ]<\delta \le {\overline{\delta }}_1\le \rho$. Consider the solution of Equation (2). Let

$$\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)=\underset{1\le i\le m}{max}{x}_i({\tau }_0).$$

According to the condition $(D_5)$, we obtain

$$\underset{1\le i\le m}{max}{x}_i({\tau }_0)\le a(H[Z_0-X_0,\theta ])\le K_{a}H[Z_0-X_0,\theta ]<K_a\delta <{\delta }_1.$$

Therefore, the function $\underset{1\le i\le m}{max}{x}_i(t)$ satisfies (14).

Next, We will prove

$$\begin{array}{c}\hfill H[Z(t,{\tau }_0,Z_0)-X(t-\zeta ,t_0,X_0),\theta ]\le M_{1}H[Z_0-X_0,\theta ],t\ge {\tau }_0.\end{array}$$

(15)

Suppose inequality (15) is false. Then, there exists a $T>{\tau }_0$, and

$$\begin{array}{cc}\hfill & H[Z(t,{\tau }_0,Z_0)-X(t-\zeta ,t_0,X_0),\theta ]\le M_{1}H[Z_0-X_0,\theta ],t\in [{\tau }_0,T],\hfill \\ \hfill & H[Z(T,{\tau }_0,Z_0)-X(T-\zeta ,t_0,X_0),\theta ]=M_{1}H[Z_0-X_0,\theta ],\hfill \\ \hfill & H[Z(t,{\tau }_0,Z_0)-X(t-\zeta ,t_0,X_0),\theta ]>M_{1}H[Z_0-X_0,\theta ],t\in (T,T+ϵ],ϵ>0.\hfill \end{array}$$

Then for $t\in [{\tau }_0,T]$, the inequality

$$H[Z(t,{\tau }_0,Z_0)-X(t-\zeta ,t_0,X_0),\theta ]\le M_{1}H[Z_0-X_0,\theta ]<M_1\delta \le M_1{\delta }_2\le r$$

holds.

Using the condition $(D_2)$, we obtain

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le \underset{1\le i\le m}{max}{x}_i(t).\end{array}$$

(16)

Furthermore, we obtain

$$\begin{array}{cc}\hfill M_{1}H[Z_0-X_0,\theta ]& =H[Z(T,{\tau }_0,Z_0)-X(T-\zeta ,t_0,X_0),\theta ]\hfill \\ \hfill & \le b^{-1}(\underset{1\le i\le m}{max}{V}_i(T,Z(T,{\tau }_0,Z_0)-X(T-\zeta ,t_0,X_0)))\hfill \\ \hfill & \le b^{-1}(\underset{1\le i\le m}{max}{x}_i(T))\le b^{-1}(M\underset{1\le i\le m}{max}{x}_i({\tau }_0))\hfill \\ \hfill & \le q_b(M\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0))\hfill \\ \hfill & <M_{1}H[Z_0-X_0,\theta ]\hfill \end{array}$$

(17)

The contradiction obtained proves the validity of (15).

Then, the result of Theorem 4 is true.

□

4.2. Practical Stability with Different Initial Time

Theorem 5.

Assume that $(H_1)$$V\in C[J\times K_c({\mathbb{R}}^n),{\mathbb{R}}_{+}^m]$ is the vector Lyapunov function, the inequality

$$b(H[Z(t)-X(t-\zeta ),\theta ])\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le a(H[Z(t)-X(t-\zeta ),\theta ])$$

(18)

holds, where $a,b\in \mathcal{K}$.

$(H_2)$ The inequality

$$\begin{array}{c}\hfill {\mathcal{D}}_{+}^{q}V(t,Z(t)-X(t-\zeta ))\le f(t,V(t,Z(t)-X(t-\zeta )))\end{array}$$

(19)

holds, where $f\in C[J\times {\mathbb{R}}_{+}^m,{\mathbb{R}}^m]$.

$(H_3)$ The system

$$\begin{array}{c}\hfill {\mathcal{D}}_H^{q}x(t)=f(t,x),x({\tau }_0)=x_0\ge 0,t\ge {\tau }_0\end{array}$$

(20)

is practical stable with different initial time with $(a(\mu ),b(B))$, where $\mu \in (0,B),a(\mu )<b(B)$, μ, $B\in \mathbb{R}$, $x\in {\mathbb{R}}_{+}^m$.

Then, the solution $Z(t)$ of Equation (2) is practical stable relative to the solution $X(t-\zeta )$ of Equation (1).

Proof. 

From the condition $(H_3)$, there is a $\delta =\delta (t_0,\mu ,B)>0$, and for $\underset{1\le i\le m}{max}{x}_i({\tau }_0)<a(\mu ),\zeta <\delta$, we have

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{x}_i(t)<b(B),t\ge {\tau }_0.\end{array}$$

(21)

Let $\underset{1\le i\le m}{max}{x}_i({\tau }_0)=\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)$. From the condition $(H_1)$, it follows that

$$\underset{1\le i\le m}{max}{x}_i({\tau }_0)=\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)\le a(H[Z_0-X_0,\theta ])<a(\mu ).$$

Therefore, the function $\underset{1\le i\le m}{max}{x}_i(t)$ satisfies inequality (21).

According to the conditions $(H_1)$ and $(H_2)$, we obtain

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le \underset{1\le i\le m}{max}{x}_i(t).\end{array}$$

(22)

Then from the condition $(H_1)$, inequalities (21) and (22), we obtain

$$b(H[Z(t)-X(t-\zeta ),\theta ])\le \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le \underset{1\le i\le m}{max}{x}_i(t)<b(B).$$

Furthermore, we obtain

$$H[Z(t)-X(t-\zeta ),\theta ]<B,t\ge t_0.$$

Then, the conclusion of Theorem 5 is proved. □

Theorem 6.

Assume that $(H_1)$ and $(H_2)$ in Theorem 5 hold, and

$(H_4)$ The system

$$\begin{array}{c}\hfill {\mathcal{D}}_H^{q}x(t)=f(t,x),x({\tau }_0)=x_0\ge 0,t\ge {\tau }_0\end{array}$$

(23)

is uniformly practical stable with different initial time with $(a(\mu ),b(B))$.

Then, the solution $Z(t)$ of Equation (2) is uniformly practical stable relative to the solution $X(t-\zeta )$ of Equation (1).

Proof. 

According to the condition $(H_4)$, for the given $(a(\mu ),b(B))$, there is a $\delta =\delta (\mu ,B)>0$, and for any $\zeta \in [-\delta ,\delta ]$, the inequality $\underset{1\le i\le m}{max}{x}_i({\tau }_0)<a(\mu )$ implies

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{x}_i(t)<b(B),t\ge {\tau }_0.\end{array}$$

(24)

Let $\underset{1\le i\le m}{max}{x}_i({\tau }_0)=\underset{1\le i\le m}{max}{V}_i({\tau }_0,Z_0-X_0)$. From the condition $(H_1)$ and $H[Z_0-X_0,\theta ]<\mu$, it follows that

$$\underset{1\le i\le m}{max}{x}_i({\tau }_0)<a(H[Z_0-X_0,\theta ])<a(\mu ).$$

Furthermore, we obtain

$$\begin{array}{c}\hfill H[Z(t)-X(t-\zeta ),\theta ]<B,t\ge t_0.\end{array}$$

(25)

Assume that (25) is false, i.e., there is a $t_1>t_0$, such that

$$H[Z(t_1)-X(t_1-\zeta ),\theta ]<B,H[Z(t)-X(t-\zeta ),\theta ]<B,t\in [t_0,t_1).$$

Using the condition $(H_2)$ and applying Lemma 1, we obtain

$$\begin{array}{c}\hfill \underset{1\le i\le m}{max}{V}_i(t,Z(t)-X(t-\zeta ))\le \underset{1\le i\le m}{max}{x}_i(t),t\in [t_0,t_1].\end{array}$$

(26)

From the choice of $t_1$, the condition $(H_1)$, inequalities (24) and (26), we obtain

$$\begin{array}{cc}\hfill b(A)& =b(H[Z(t_1)-X(t_1-\zeta ),\theta ])\hfill \\ \hfill & \le \underset{1\le i\le m}{max}{V}_i(t_1,Z(t_1)-X(t_1-\zeta ))\hfill \\ \hfill & \le \underset{1\le i\le m}{max}{x}_i(t_1)\hfill \\ \hfill & <b(B).\hfill \end{array}$$

The contradiction proves the true of inequality (25). The result in Theorem 6 is true. □

5. Conclusions

This paper studied fractional set-valued differential equations with different initial time. A new comparison principle is given via the upper quasi-monotone increasing and vector Lyapunov method. By using the comparison theorem, we obtain some new sufficient conditions ensuring Lipschitz stability and practical stability for such set-valued differential equations. In addition, when $n>1$, upper quasi-monotone increasing and quasi-monotone increasing do not include each other. Thus, the results obtained in this paper enrich the method of discriminating the stability of set-valued differential equations.