1 Introduction

The development of the theory of q-calculus can be dated back to the early 20th century in order to look for a better description of the phenomena having both discrete and continuous behaviors. The q-analog of fractional integrals and derivatives were first studied by Al-Salam [1–3] and then by Agrawal [4]. Recently, the q-fractional calculus has been payed more attention [5–8] because it serves as a bridge between fractional calculus and q-calculus.

In nonlinear systems, Lyapunov’s direct method provides an effective way to analyze the stability of a system without explicitly solving the differential equations. Motivated by the application of fractional calculus in nonlinear systems Li,Chen, and Podlubny [9, 10] proposed the Mittag-Leffler stability and Lyapunov direct method, and a considerable number results of stability analysis for fractional systems have been reported; see [11–21] and the references therein. However, to our knowledge, the q-Mittag-Leffler stability of q-fractional dynamic systems has not been studied. In this paper, we propose the q-Mittag-Leffler stability and the q-fractional Lyapunov direct method with a hope to enrich the knowledge of the theory of q-fractional calculus. We also present a simple Lyapunov function to get the q-Mittag-Leffler stability for many q-fractional-order systems and show that q-fractional-order dynamical systems also do not have to decay exponentially for the system to be stable in the Lyapunov sense.

2 Preliminaries

2.1 Definitions and properties of q-caculus

This section is devoted to recall some essential definitions and properties of q-calculus [1–4, 8].

If \(q\in R,0< q<1\), a subset A of R is called q-geometric if \(qx\in A\) whenever \(x\in A\). If a subset A of R is q-geometric, then it contains all geometric sequences \(\{xq^{n}\}_{n=0}^{\infty}\), \(x\in A\).

Definition 2.1

([8])

Let \(f(x)\) be a real function defined on a q-geometric set A. The q-derivative is defined by

$$\begin{aligned}& D_{q}f(x)=\frac{f(qx)-f(x)}{(q-1)x}, \quad x\in A\setminus\{0\}, \end{aligned}$$
(1)

and

$$\begin{aligned}& D_{q}f(x)\vert_{x=0}=\lim_{n\rightarrow\infty} \frac{f(q^{n})-f(0)}{q ^{n}}. \end{aligned}$$
(2)

Setting \(q\rightarrow1\), we have \(\lim_{q\rightarrow1}D_{q}f(x)=f ^{\prime}(x)\).

Also, the q-integral is given as

$$\begin{aligned}& \int_{0}^{x}f(t)\,d_{q}t=(1-q)x\sum _{n=0}^{\infty}q^{n}f \bigl(q^{n}x\bigr),\quad x\in A, \end{aligned}$$
(3)

and

$$\begin{aligned}& \int_{a}^{b}f(t)\,d_{q}t= \int_{0}^{b}f(t)\,d_{q}t- \int_{0}^{a}f(t)\,d_{q}t,\quad a,b\in A. \end{aligned}$$
(4)

We present here two basic properties concerning q-derivatives.

Property 1

([7])

$$\begin{aligned}& D_{q}(f \pm g) (x)=D_{q}f(x)\pm D_{q}g(x). \end{aligned}$$
(5)

Property 2

([7])

The q-Leibniz product rule is given by

$$\begin{aligned}& D_{q}\bigl[g(x)f(x)\bigr]=g(qx)D_{q}f(x)+f(x)D_{q}g(x), \end{aligned}$$
(6)

where \(D_{q}\) is the q-derivative.

The q-analogue of exponent \((s-t)^{(k)}\) is

$$(s-t)^{(0)}=1,\qquad (s-t)^{(k)}=\prod_{j=0}^{k-1} \bigl(x-yq^{j}\bigr),\quad k\in N,x,y \in R. $$

Definition 2.2

([7])

A q-analogue of the Riemann–Liouville fractional integral is defined as

$$\begin{aligned}& I_{q,a}^{\alpha}f(x)= \int_{0}^{x}\frac{(x-qs)^{(\alpha-1)}}{\Gamma _{q}(\alpha)}f(s)\,d_{q}s,\quad \alpha>0. \end{aligned}$$
(7)

If we let \(q\rightarrow1\), then the q-analogue of Riemann–Liouville fractional integral \({}_{q}I_{q,a}^{\alpha}f(x)\rightarrow I_{a}^{ \alpha}f(x)\).

Definition 2.3

([6])

The Riemann–Liouville type fractional q-derivative of a function \(f:(0, \infty)\rightarrow R\) is defined by

$$\begin{aligned}& \bigl(D_{q,a}^{\alpha}f\bigr) (x)= \textstyle\begin{cases} (I_{q,a}^{-\alpha}f)(x),& \alpha\leq0, \\ (D_{q,a}^{[\alpha]}I_{q,a}^{[\alpha]-\alpha}f)(x),& \alpha>0, \end{cases}\displaystyle \end{aligned}$$
(8)

where \([\alpha]\) denotes the smallest integer greater than or equal to α.

Definition 2.4

([6])

The Caputo type fractional q-derivative of a function \(f:(0, \infty)\rightarrow R\) is define by

$$\begin{aligned}& \bigl(^{C}D_{q,a}^{\alpha}f\bigr) (x)= \textstyle\begin{cases} (I_{q,a}^{-\alpha}f)(x),& \alpha\leq0, \\ (I_{q,a}^{[\alpha]-\alpha]}D_{q,a}^{[\alpha]}f)(x),& \alpha>0, \end{cases}\displaystyle \end{aligned}$$
(9)

where \([\alpha]\) denotes the smallest integer greater or equal to α.

2.2 q-Mittag-Leffler function

Similar to the Mittag-Leffler function frequently used in the solutions of fractional-order equations, the functions frequently used in the solutions of q-fractional-order equations are the q-analogues of Mittag-Leffler functions defined as

$$\begin{aligned}& e_{\alpha,\beta}(z,q)=\sum_{n=0}^{\infty} \frac{z^{n\alpha}}{ \Gamma_{q}(n\alpha+\beta)} \quad \bigl(\bigl\vert z(1-q)^{\alpha}\bigr\vert < 1\bigr) \end{aligned}$$
(10)

and

$$\begin{aligned}& E_{\alpha,\beta}(z,q)=\sum_{n=0}^{\infty} \frac{q^{\frac{ \alpha n(n-1)}{2}}z^{n\alpha}}{\Gamma_{q}(n\alpha+\beta)}\quad (z \in C), \end{aligned}$$
(11)

where \(\alpha>0\) and \(\beta\in\mathcal{C}\). When \(\beta=1\), the functions \(e_{\alpha,\beta}(z,q)\) and \(E_{\alpha,\beta}(z,q)\) are defined by

$$\begin{aligned}& e_{\alpha,1}(z,q)=\sum_{n=0}^{\infty} \frac{z^{n\alpha}}{ \Gamma_{q}(n\alpha+1)} \quad \bigl(\bigl\vert z(1-q)^{\alpha}\bigr\vert < 1\bigr) \end{aligned}$$
(12)

and

$$\begin{aligned}& E_{\alpha,1}(z,q)=\sum_{n=0}^{\infty} \frac{q^{ \frac{\alpha n(n-1)}{2}}z^{n\alpha}}{\Gamma_{q}(n\alpha+1)} \quad (z \in C). \end{aligned}$$
(13)

2.3 q-Laplace transform of fractional q-integrals, q-derivatives, and q-Mittag-Leffler functions

Theorem 2.5

([6])

If \(f\in\mathscr{L}_{q}^{1}[0,a]\) and \(\Phi(s)=_{q}L_{s}f(x)\), then

$$\begin{aligned}& _{q}L_{s}I_{q}^{\alpha}f(x)= \frac{(1-q)^{\alpha}}{s^{\alpha}} \Phi(s)\quad \textit{for } \alpha>0. \end{aligned}$$
(14)

If \(n-1<\alpha\leq n\) and \(I_{q}^{n-\alpha}f(x)\in C_{1}^{(n)}[0,a]\), then let \(\Phi(s)=_{q}L_{s}f(x)\). The q-Laplace transform of the Riemann–Liouville fractional and the Caputo fractional q-derivatives are given by

$$\begin{aligned}& _{q}L_{s}^{C}D_{q}^{\alpha}f(x)= \frac{s^{\alpha}}{(1-q)^{\alpha}}\Biggl( \Phi(s)-\sum_{r=0}^{n-1}D_{q}^{r}f \bigl(0^{+}\bigr)\frac{(1-q)^{r}}{s ^{r+1}}\Biggr) \end{aligned}$$
(15)

and

$$\begin{aligned}& _{q}L_{s}D_{q}^{\alpha}f(x)= \frac{s^{\alpha}}{(1-q)^{\alpha}} \Phi(s)-\sum_{m=1}^{n}D_{q}^{\alpha-m}f \bigl(0^{+}\bigr) \frac{s^{m-1}}{(1-q)^{m}}. \end{aligned}$$
(16)

Theorem 2.6

([6])

If \(\vert \frac{s}{1-q}\vert >\vert a\vert ^{\frac {1}{Re(\alpha)}}\), then

$$\begin{aligned}& _{q}L_{s}\bigl(x^{\beta-1}e_{\alpha,\beta}(ax;q) \bigr)=\frac{1}{1-q}\frac{( \frac{s}{1-q})^{\alpha-\beta}}{(\frac{s}{1-q})^{\alpha}-a}. \end{aligned}$$
(17)

Taking \(\beta=1\), we have

$$\begin{aligned}& _{q}L_{s}\bigl(e_{\alpha,1}(ax;q)\bigr)= \frac{1}{1-q}\frac{(\frac{s}{1-q})^{ \alpha-1}}{(\frac{s}{1-q})^{\alpha}-a}. \end{aligned}$$
(18)

3 q-Mittag-Leffler stability and Lyapunov direct method for differential systems with q-fractional order

Consider the Caputo fractional nonautonomous system q-Mittag-Leffler stability of solutions of the following system:

$$\begin{aligned}& \textstyle\begin{cases} ^{C}D_{q}^{\alpha}x(t)=f(t,x(t)), \\ x(t_{0})=x_{0}, \end{cases}\displaystyle \end{aligned}$$
(19)

where \(t\geq t_{0},t,t_{0}\in A,A=[t_{0}, t]_{q},0<\alpha<1\), and \(f:[t_{0}, t] \times R \rightarrow R\) is a function with \(f \in \mathscr{L}_{q,1}[t_{0}, t]\). Let \(f(t,0)=0\), for all \(t \in[t_{0}, t]_{q}\), so that system (19) admits the trivial solution.

Now we give some definitions that will be used in studying the q-Mittag-Leffler stability of (19).

Definition 3.1

The trivial solution \(x(t)=0\) of (19) is said to be asymptotically stable if for all \(\epsilon>0\) and \(t_{0}\in A\), there exists \(\delta=\delta(t_{0},\epsilon)\) such that if \(\Vert x_{0}\Vert <\delta\) implies that \(\lim_{t\rightarrow\infty} \Vert x(t)\Vert =0\).

Definition 3.2

(q-Mittag-Leffler stability)

The solution of (19) is said to be q-Mittag-Leffler stability if

$$\begin{aligned}& \bigl\Vert x(t)\bigr\Vert \leq\bigl\{ m\bigl[x(t_{0}) \bigr]e_{q,\alpha}\bigl(-\lambda(t-t_{0})^{\alpha}\bigr) \bigr\} ^{b}, \end{aligned}$$
(20)

where \(t_{q}\in A\) is the initial time, \(\alpha\in(0,1),\lambda \geq0,b>0,m(0)=0,m(x)\geq0\), and \(m(x)\) is locally Lipschitz on \(x\in B\subset R\) with Lipschitz constant \(m_{0}\). We further assume that \(t_{0}=0\).

Theorem 3.3

Let \(x=0\) be an equilibrium point for system (19), and let \(D\subset R\) be a domain containing origin. Let \(V(t, x(t)):[0,T] \times D\rightarrow R\) be a continuously differentiable function and locally Lipschitz with respect to x such that

$$\begin{aligned}& \beta_{1}\bigl\Vert x(t)\bigr\Vert ^{a}\leq V\bigl(t, x(t)\bigr)\leq\beta_{2}\bigl\Vert x(t)\bigr\Vert ^{ab}, \end{aligned}$$
(21)
$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr) \leq(-\beta_{3})\bigl\Vert x(t)\bigr\Vert ^{ab}, \end{aligned}$$
(22)

where \(t\in[0,T],t>0\), \(0<\alpha<1\), and \(\beta_{1}\), \(\beta_{2}\), \(\beta _{3}\), a, and b are arbitrary positive constants. Then \(x=0\) is q-Mittag-Leffler stable.

Proof

It follows from equations (19) and (20) that

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr)\leq-\frac{\beta_{3}}{\beta_{2}}V\bigl(t, x(t)\bigr). \end{aligned}$$
(23)

There exists a nonnegative function \(M(t)\) satisfying

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr)+M(t)=-\frac{\beta_{3}}{\beta_{2}}V\bigl(t, x(t)\bigr). \end{aligned}$$
(24)

Taking the q-Laplace transform of (24) gives

$$\begin{aligned}& \frac{s^{\alpha}}{(1-q)^{\alpha}}(V(s)-\frac {1}{s}V\bigl(0,x(0)\bigr)+M(s)=- \frac{ \beta_{3}}{\beta_{2}}V(s), \end{aligned}$$
(25)

where \(V(s)=_{q}L_{s}\{V(t,x(t))\}\). It then follows that

$$\begin{aligned} V(s) =&V(0,x(0))\frac{\frac{s^{\alpha-1}}{(1-q)^{\alpha}}}{\frac{s ^{\alpha}}{(1-q)^{\alpha}}+\frac{\beta_{3}}{\beta_{2}}}-\frac {M(s)}{\frac{s ^{\alpha}}{(1-q)^{\alpha}} +\frac{\beta_{3}}{\beta_{2}}} \\ =&V(0,x(0))\frac{1}{1-q}\frac{(\frac{s}{1-q})^{\alpha-1}}{( \frac{s}{1-q})^{\alpha}+\frac{\beta_{3}}{\beta_{2}}}-(1-q)M(s) \frac{1}{1-q} \frac{1}{\frac{s^{\alpha}}{(1-q)^{\alpha}}+\frac{\beta _{3}}{\beta_{2}}}. \end{aligned}$$
(26)

It follows from the inverse Laplace transform that the unique solution of (24) is

$$\begin{aligned}& V(t)=V\bigl(0,x(0)\bigr)e_{\alpha,1}\biggl(-\frac{\beta_{3}}{\beta_{2}}t;q\biggr)- \int_{0} ^{t}M(\tau) (t-q\tau)^{\alpha-1}e_{\alpha,\alpha} \biggl(-\frac{\beta _{3}}{\beta_{2}}(t-q\tau)^{\alpha};q\biggr)\,d\tau. \end{aligned}$$
(27)

Since \(0< q<1\), \(M(t)\geq0\), and \(e_{\alpha,\alpha}(-\frac{\beta _{3}}{\beta_{2}}(t-q\tau)^{\alpha};q)\) are nonnegative functions, we get

$$\begin{aligned}& V(t)\leq V\bigl(0,x(0)\bigr)e_{\alpha,1}\biggl(-\frac{\beta_{3}}{\beta_{2}}t;q \biggr). \end{aligned}$$
(28)

Substitution of (28) into (21) yields

$$\begin{aligned}& \bigl\Vert x(t)\bigr\Vert \leq\biggl[\frac{V(0,x(0))}{\beta_{1}}e_{\alpha,1} \biggl(-\frac{\beta _{3}}{\beta_{2}}t;q\biggr)\biggr]^{\frac{1}{a}}, \end{aligned}$$
(29)

where \(\frac{V(0,x(0))}{\beta_{1}}>0\) for \(x(0)\neq0\).

Let \(m=\frac{V(0,x(0))}{\beta_{1}}\geq0\). Then we have

$$\begin{aligned}& \bigl\Vert x(t)\bigr\Vert \leq\biggl[m e_{\alpha,1}\biggl(- \frac{\beta_{3}}{\beta_{2}}t;q\biggr)\biggr]^{ \frac{1}{a}}, \end{aligned}$$
(30)

where \(m=0\) if and only if \(x(0)=0\). Because \(V(t,x)\) is locally Lipschitz with respect to x and \(V(0,x(0))=0\) if and only if \(x(0)\), it follows that m is also Lipschitz with respect to \(x(0)\) and \(m(0)\), which implies the q-Mittag-Leffler stability.

In [8], an identity relation between the Caputo fractional q-derivative and the Riemann–Liouville fractional q-derivative is introduced:

$$\begin{aligned}& f(t)=_{t_{0}}D_{q}^{\alpha}f(t)-_{t_{0}}D_{q}^{\alpha} \Biggl(\sum_{k=0} ^{n-1}\frac{D_{q}^{k}f(0^{+})}{\Gamma_{q}(k+1)}x^{k} \Biggr), \end{aligned}$$
(31)

where \(\alpha>0\) and \(n=[\alpha]+1\). When \(0<\alpha<1\), we have

$$\begin{aligned}& ^{C}_{t_{0}}D_{q}^{\alpha}f(t)=_{t_{0}}D_{q}^{\alpha}f(t)- \frac{(t-t _{0})_{q}^{\alpha}}{\Gamma_{q}(1-\alpha)}f(t_{0}). \end{aligned}$$
(32)

 □

Theorem 3.4

If the assumptions in Theorem 3.3 are satisfied except replacing \({}^{C}_{t_{0}}D_{q}^{\alpha}\) by \({t_{0}}D_{q}^{ \alpha}\), then the trivial solution of (19) is q-Mittag-Leffler stable.

Proof

From (32) we have

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}V\bigl(t, x(t) \bigr)=_{0}D_{q}^{\alpha}V\bigl(t,x(t)\bigr)- \frac{t _{q}^{\alpha}}{\Gamma_{q}(1-\alpha)}V\bigl(0,x(0)\bigr)\quad \mbox{for } t \in[0,T], \end{aligned}$$
(33)

and since \(V(0,x(0))\geq0\) and \(\frac{t_{q}^{\alpha}}{\Gamma_{q}(1- \alpha)}\geq0\), we obtain the result.

Furthermore, if we extend the Lyapunov direct method to the case of q-fractional-order systems, then the asymptotic stability of the corresponding systems can be obtained. The following properties of the q-Mittag-Leffler function and the class-K functions are applied to analysis of the q-fractional Lyapunov direct method. □

Remark 3.5

Since

$$\begin{aligned}& D_{q}e_{{\alpha,1}}\bigl((-\lambda t;q)\bigr)=-\lambda t^{\alpha-1}e_{\alpha,\alpha-1}(-\lambda t;q), \end{aligned}$$
(34)

where \(t>0\), \(0<\alpha<1\), \(\lambda>0\), the q-Mittag-Leffler function \(e_{{\alpha,1}}(((-\lambda t)^{\alpha};q))\) is decreasing, so the q-Mittag-Leffler stability implies the asymptotic stability.

4 q-Mittag-Leffler stability of linear systems with q-fractional order

In this section, we present a new result that allows us to find Lyapunov candidate functions for demonstrating the q-Mittag-Leffler of many fractional-order systems using the results of the Lyapunov direct method in Theorem 3.3.

Theorem 4.1

Let \(x(t) \in R\) be defined in a suitable q-geometric set \(A=[0,a]_{q}\), \(D_{q}x(t)\in C_{q}[0,q]\) (where \(C_{q}[0,a]\) is the space of all continuous functions on the interval \([0, a]\)). Then, for any time \(t>0\), \(t\in A\),

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x^{2}(t) \leq\bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{\alpha }x(t),\quad 0< \alpha< 1. \end{aligned}$$
(35)

Proof

Proving expression (35) is equivalent to proving that

$$\begin{aligned}& \bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{\alpha}x(t)-^{C}_{0}D_{q}^{\alpha}x^{2}(t) \geq0. \end{aligned}$$
(36)

Using Definition 2.2 and Definition 2.4, \((x(t)+x(tq))^{C}_{0}D_{q}^{ \alpha}x(t)\) and \({}^{C}_{0}D_{q}^{\alpha}x^{2}(t)\) can be written as

$$\begin{aligned}& \bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{\alpha}x(t)= \bigl(x(t)+x(tq)\bigr)\frac{1}{\Gamma(1- \alpha)} \int_{0}^{t}(t-qs)^{-\alpha}D_{q}x(s)\,d_{q}s \end{aligned}$$
(37)

and

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x^{2}(t)= \frac{1}{\Gamma(1-\alpha)} \int_{0} ^{t}(t-qs)^{-\alpha}\bigl(x(s)+x(qs) \bigr)D_{q}x(s)\,d_{q}s. \end{aligned}$$
(38)

So, the left side of expression (36) can be written as

$$\begin{aligned}& \frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-qs)^{-\alpha }\bigl[ \bigl(x(t)-x(s)\bigr)+\bigl(x(tq)-x(sq)\bigr)\bigr]D _{q}x(s)\,d_{q}s. \end{aligned}$$
(39)

Now, let us define the axillary variable \(y(s)=x(t)-x(s)\), which implies that

$$\begin{aligned} D_{q}y^{2}(s) =&\bigl(y(s)+y(sq)\bigr)D_{q}y(s) \\ =&-\bigl[\bigl(x(t)-x(s)\bigr)+\bigl(x(tq)-x(sq)\bigr)\bigr]D_{q}x(s). \end{aligned}$$
(40)

In this way, expression (39) can be written as

$$\begin{aligned}& \frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-qs)^{-\alpha}\,d_{q}y^{2}(s)= -\frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-qs)^{-\alpha}\bigl[y(s)+y(sq) \bigr]D _{q}y(s)\,d_{q}s. \end{aligned}$$
(41)

Since \(x(t)\) is regular at zero, using the rule of q-integration by parts, expression (41) becomes

$$\begin{aligned} \int_{t_{0}}^{t}(t-qs)^{-\alpha}\,d_{q}y^{2}(s) =&y^{2}(t)(t-qt)^{- \alpha}-\Gamma(1-\alpha)y^{2}(0)t^{-\alpha} \\ &{}-\alpha q \int_{0}(t-qs)^{-\alpha-1}y^{2}(qs)\,d_{q}(s). \end{aligned}$$
(42)

Since \(y^{2}(t)=(x(t)-x(s))^{2}=0\), it follows that

$$\begin{aligned}& {}^{C}_{0}D_{q}^{\alpha}x(t)-^{C}_{0}D_{q}^{\alpha}x^{2}(t) \\& \quad =\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-qs)^{-\alpha }[(x(t)-x(s))+(x(tq)-x(sq))]D _{q}x(s)\,d_{q}s \\& \quad =-\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-qs)^{-\alpha}\,d_{q}y ^{2}(s) \\& \quad =\frac{1}{\Gamma(1-\alpha)}y^{2}(0)t^{-\alpha}+\frac{\alpha q}{ \Gamma(1-\alpha)}\int_{0}^{t}(t-qs)^{-\alpha-1}y^{2}(s)\,d_{q}(s) \\& \quad \geq0. \end{aligned}$$
(43)

This concludes the proof. □

Corollary 4.2

For the q-fractional-order system

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x(t)=f\bigl(t,x(t) \bigr), \end{aligned}$$
(44)

where \(\alpha\in(0,1)\), \(x=0\) is the equilibrium point, and \(D_{q}x(t)\in C_{q}[0,a]\), \(f(t,x(t))\in\pounds_{q}^{1}[0,a]\). If

$$\begin{aligned}& \bigl(x(t)+x(tq)\bigr)f\bigl(t,x(t)\bigr)\leq0, \quad \forall x\in A, \end{aligned}$$
(45)

then the origin of system (44) is q-Mittag-Leffler stable.

Proof

Let us propose the following Lyapunov candidate function:

$$\begin{aligned}& V\bigl(t,x(t)\bigr)=x^{2}. \end{aligned}$$
(46)

Applying Theorem 4.1 results in

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}V\bigl(t,x(t)\bigr) \leq\bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{ \alpha}x(t) \leq\bigl(x(t)+x(tq)\bigr)f\bigl(t,x(t)\bigr)\leq0, \end{aligned}$$
(47)

and thus the origin of system (44) is q-Mittag-Leffler stable. □

Proposition 4.3

For the system

$$\begin{aligned}& ^{C}_{0}D_{q}^{\alpha}x(t)=-x(t)-x(tq), \end{aligned}$$
(48)

where \(0<\alpha<1\) and \(D_{q}x(t)\in C_{q}[0,a]\), the origin of system (44) is q-Mittag-Leffler stable.

Proof

Let \(V(x(t))=x^{2}(t)\). Then

$$\begin{aligned} {}^{C}_{0}D_{q}^{\alpha}x^{2}(t) \leq& \bigl(x(t)+x(tq)\bigr)^{C}_{0}D_{q}^{ \alpha}x(t) \\ =& -\bigl(x(t)+x(tq)\bigr)^{2}\leq-\big\Vert x(t)\big\Vert ^{2}. \end{aligned}$$
(49)

So we can conclude that the trivial solution of system (48) is asymptotically stable.

Furthermore, from the expression of exact solution for (48) using two q-analogues of the Mittag-Leffler functions defined by (12) and (13),

$$\begin{aligned}& x(t)=c_{1}e_{(\alpha,1)}(-x,q)+c_{2}E_{(\alpha,1)}(-x,q), \end{aligned}$$
(50)

and the properties of these two functions the asymptotical stability can also be derived. □

5 Conclusions

In this paper, we studied the stability of systems with q-fractional order. We proposed the definition of q-Mittag-Leffler stability, presented sufficient criteria of q-Mittag-Leffler stability and the q-fractional Lyapunov direct method of nonlinear systems with q-fractional order. Meanwhile, the q-fractional Lyapunov candidate functions for demonstrating the q-Mittag-Leffler stability of many q-fractional-order systems were discussed. With the rapid development of advanced applied science, we believe that many other study subjects of the q-fractional calculus and q-fractional dynamical systems will attract more attention of researchers. In our following study, we will still focus on the stability problem of q-fractional differential equations in a variety of different forms.