Existence and Stability Results for Impulsive Fractional q-Difference Equation

Min Jiang^1*, Rengang Huang²

¹College of Computer and Information Engineering, Guizhou Minzu University, Guiyang, China.
²College of Business, Guizhou Minzu University, Guiyang, China.

DOI: 10.4236/jamp.2020.87107                

Abstract

In this paper, we study the boundary value problem for an impulsive fractional q-difference equation. Based on Banach’s contraction mapping principle, the existence and Hyers-Ulam stability of solutions for the equation which we considered are obtained. At last, an illustrative example is given for the main result.

Keywords

Impulsive Fractional q-Difference Equation, Hyers-Ulam Stability, Existence, q-Calculus

1. Introduction

The q-calculus or quantum calculus is an old subject that was initially developed by Jackson [1]; basic definitions and properties of q-calculus can be found in [2]. The fractional q-calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. But the definitions mentioned above about q-calculus can’t be applied to impulse points $t_k\mathrm{,}k\in \mathbb{Z}$, such that $t_k\in \left(qt\mathrm{,}t\right)$. In [5], the authors defined the concepts of fractional q-calculus by defining a q-shifting operator ${}_a\Phi {}_q\left(m\right)=qm+\left(1-q\right)a\mathrm{,}m\mathrm{,}a\in ℝ$. Using the q-shifting operator, the fractional impulsive q-difference equation was defined. In paper [5] [6] [7], the authors discussed the existence of solutions for the fractional impulsive q-difference equation with Riemann-Liouville and Caputo fractional derivatives respectively. Some other results about q-difference equations can be found in papers [8] - [16] and the references cited therein. Dumitru Baleanu et al. discussed the stability of non-autonomous systems with the q-Caputo fractional derivatives in reference [17]. However, the existence and stability of solutions for the fractional impulsive q-difference have not been yet studied.

Motivated greatly by the above mentioned excellent works, in this paper we investigate the following fractional impulsive q-difference equation with q-integral boundary conditions:

$\{\begin{array}{l}{}_{t_k}^c{D}_{q_k}^{{\alpha }_k}x\left(t\right)=f\left(t,x\left(t\right)\right),t\in J_k\subseteq J=\left[0,T\right],t\ne t_k,\hfill \\ \Delta x\left(t_k\right)=x\left(t_k^{+}\right)-x\left(t_k\right)={\phi }_k\left(x\left(t_k\right)\right),k=1,2,\cdots ,m,\hfill \\ {\eta }_{1}x\left(0\right)+{\eta }_{2}x\left(T\right)=\mu {\displaystyle \underset{k=0}{\overset{m}{\sum }}}{}_{t_k}{I}_{q_k}^{{\beta }_k}x\left(t_{k+1}\right).\hfill \end{array}$ (1)

where ${}_{t_k}^c{D}_{q_k}^{{\alpha }_k}$ is the fractional $q_k$ -derivative of the Caputo type of order ${\alpha }_k$ on $J_k$, $0<{\alpha }_k<1$, $0<q_k<1$, $J_0=\left[0,t_1\right]$, $J_0=\left[0,t_1\right]$, $k=1,2,\cdots ,m$, ${\phi }_k\in C\left(ℝ,ℝ\right)$, $f\in C\left(J\times ℝ,ℝ\right)$. ${}_{t_k}{I}_{q_k}^{{\beta }_k}$ denotes the Riemann-Liouville $q_k$ -fractional integral of order ${\beta }_k>0$ on $J_k,k=0,1,2,\cdots ,m$ and ${\eta }_1\mathrm{,}{\eta }_2\mathrm{,}\mu$ are three constants.

2. Preliminaries on q-Calculus and Lemmas

Here we recall some definitions and fundamental results on fractional q-integral and fractional q-derivative, for the full theory for which one is referred to [5] [6] [7].

For $q\in \left(\mathrm{0,1}\right)$, we define a q-shifting operator as ${}_a{\Phi }_q\left(m\right)\mathrm{<mo>\; =\; </mo>}qm+\left(1-q\right)a$. The new power of q-shifting operator is defined as ${}_a{\left(n-m\right)}_q^{\left(0\right)}=1$,

${}_a{\left(n-m\right)}_q^{\left(k\right)}={\displaystyle \underset{i=0}{\overset{k-1}{\prod }}}\left(n-{}_a{\Phi }_q^i\left(m\right)\right)$, $k\in \mathbb{N}\cup \left\{0\right\}$, $n\in ℝ$. If $\nu \in ℝ$, then ${}_a{\left(n-m\right)}_q^{\left(\nu \right)}=n^{\nu }{\displaystyle \underset{i=0}{\overset{\infty }{\prod }}}\frac{1-\frac{a}{n}{\Phi }_q^i\left(\frac{m}{n}\right)}{1-\frac{a}{n}{\Phi }_q^{i+\nu }\left(\frac{m}{n}\right)}$.

The q-derivative of a function $f$ on interval $\left[a\mathrm{,}b\right]$ is defined by

$\left({}_a{D}_{q}f\right)\left(t\right)=\frac{f\left(t\right)-f\left({}_a{\Phi }_q\left(t\right)\right)}{\left(1-q\right)\left(t-a\right)}\mathrm{,}t\ne a\mathrm{,}\left({}_a{D}_{q}f\right)\left(a\right)=\underset{t\to a}{lim}\left({}_a{D}_{q}f\right)\left(t\right)\mathrm{.}$

The q-integral of a function $f$ defined on the interval $\left[a\mathrm{,}b\right]$ is given by

$\left({}_a{I}_{q}f\right)\left(t\right)={\displaystyle {\int }_a^t}f\left(s\right){}_a\text{d}s=\left(1-q\right)\left(t-a\right){\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{q}^{i}f\left({}_a{\Phi }_{q^i}\left(t\right)\right)\mathrm{,}t\in \left[a\mathrm{,}b\right]\mathrm{.}$

Some results about operator ${}_a{D}_q$ and ${}_a{I}_q$ can be found in references [5]. Let us define fractional q-derivative and q-integral on interval $\left[a\mathrm{,}b\right]$ and outline some of their properties [5] [6] [7].

Definition 1 [5] The fractional q-derivative of Riemann-Liouville type of order $\nu \ge 0$ on interval $\left[a\mathrm{,}b\right]$ is defined by $\left({}_a{D}_q^{0}f\right)\left(t\right)=f\left(t\right)$ and

$\left({}_a{D}_q^{\nu }f\right)\left(t\right)=\left({}_a{D}_q^l{}_a{I}_q^{l-\nu }f\right)\left(t\right)\mathrm{,}\nu >\mathrm{0,}$

where l is the smallest integer greater than or equal to $\nu$.

Definition 2 [5] Let $\alpha \ge 0$ and $f$ be a function defined on $\left[a\mathrm{,}b\right]$. The fractional q-integral of Riemann-Liouville type is given by $\left({}_a{I}_q^{0}f\right)\left(t\right)=f\left(t\right)$ and

$\left({}_a{I}_q^{\alpha }f\right)\left(t\right)=\frac{1}{{\Gamma }_q\left(\alpha \right)}{\displaystyle {\int }_a^t}{}_a{\left(t-{}_a\Phi {}_q\left(s\right)\right)}_q^{\alpha -1}f\left(s\right){}_a{\text{d}}_{q}s\mathrm{,}\alpha >\mathrm{0,}t\in \left[a\mathrm{,}b\right]\mathrm{.}$

Lemma 1 [5] Let $\alpha \mathrm{,}\beta \in {ℝ}^{+}$ and $f$ be a continuous function on $\left[a\mathrm{,}b\right]\mathrm{,}a\ge 0$. The Riemann-Liouville fractional q-integral has the following semi-group property

${}_a{I}_q^{\beta }{}_a{I}_q^{\alpha }f\left(t\right)={}_a{I}_q^{\alpha }{}_a{I}_q^{\beta }f\left(t\right)={}_a{I}_q^{\alpha +\beta }f\left(t\right)\mathrm{.}$

Lemma 2 [5] Let $f$ be a q-integrable function on $\left[a\mathrm{,}b\right]$. Then the following equality holds

${}_a{D}_q^{\alpha }{}_a{I}_q^{\alpha }f\left(t\right)=f\left(t\right)\mathrm{,}\text{for}\alpha >\mathrm{0,}t\in \left[a\mathrm{,}b\right]\mathrm{.}$

Lemma 3 [5] Let $\alpha >0$ and p be a positive integer. Then for $t\in \left[a\mathrm{,}b\right]$ the following equality holds

${}_a{I}_q^{\alpha }{}_a{D}_q^{p}f\left(t\right)={}_a{D}_q^p{}_a{I}_q^{\alpha }f\left(t\right)-{\displaystyle \underset{k=0}{\overset{p-1}{\sum }}}\frac{{\left(t-a\right)}^{\alpha -p+k}}{{\Gamma }_q\left(\alpha +k-p+1\right)}{}_a{D}_q^{k}f\left(a\right)\mathrm{.}$

Definition 3 [7] The fractional q-derivative of Caputo type of order $\alpha \ge 0$ on interval $\left[a\mathrm{,}b\right]$ is defined by ${}_a^c{D}_q^{0}f\left(t\right)=f\left(t\right)$ and

$\left({}_a^c{D}_q^{\alpha }f\right)\left(t\right)=\left({}_a{I}_q^{n-\alpha }{}_a{D}_q^{n}f\right)\left(t\right)\mathrm{,}\alpha >\mathrm{0,}$

where $n$ is the smallest integer greater than or equal to $\alpha$.

Lemma 4 [7] Let $\alpha >0$ and n be the smallest integer great than or equal to $\alpha$. Then for $t\in \left[a\mathrm{,}b\right]$ the following equality holds

${}_a{I}_q^{\alpha }{}_a^c{D}_q^{\alpha }f\left(t\right)=f\left(t\right)-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}}\frac{{\left(t-a\right)}^k}{{\Gamma }_q\left(k+1\right)}{}_a{D}_q^{k}f\left(a\right)\mathrm{.}$

3. Main Results

In this section, we will give the main results of this paper.

Let $PC\left(J\mathrm{,}ℝ\right)=\{x\mathrm{<mo>\; :\; </mo>}J\to ℝ\mathrm{,}x\left(t\right)$ is continuous everywhere except for some $t_k$ at which $x\left(t_k^{+}\right)$ and $x\left(t_k^{-}\right)$ exist, and $x\left(t_k^{-}\right)=x\left(t_k\right)\mathrm{,}k=\mathrm{1,2,}\cdots \mathrm{,}m\}$. $PC\left(J\mathrm{,}ℝ\right)$ is a Banach space with the norm

$\Vert x\Vert =sup\left\{\left|x\left(t\right)\right|\mathrm{<mo>\; :\; </mo>}t\in J\right\}.$

First, for the sake of convenience, we introduce the following notations:

$\Lambda ={\eta }_1+{\eta }_2-\mu {\displaystyle \underset{i=0}{\overset{m}{\sum }}}{\Omega }_{{\beta }_i}\ne 0,{\Omega }_{{\sigma }_i}=\frac{{}_{t_i}{\left(t_{i+1}-t_i\right)}_{q_i}^{\left({\sigma }_i\right)}}{{\Gamma }_{q_i}\left({\sigma }_i+1\right)},$

where ${\sigma }_i\in \left\{{\alpha }_i,{\beta }_i,{\alpha }_i+{\beta }_i\right\},q_i\in \left(0,1\right),i=0,1,2,\cdots ,m$.

To obtain our main results, we need the following lemma.

Lemma 5 Let $\mu {\displaystyle \underset{i=0}{\overset{m}{\sum }}}{\Omega }_{{\beta }_i}\ne {\eta }_1+{\eta }_2$ and $h\left(t\right)\in C\left(J\mathrm{,}ℝ\right)$. Then for any $t\in J_k$, the solution of the following problem

$\{\begin{array}{l}{}_{t_k}^c{D}_{q_k}^{{\alpha }_k}x\left(t\right)=h\left(t\right),t\in J_k\subseteq J=\left[0,T\right],t\ne t_k,\hfill \\ \Delta x\left(t_k\right)=x\left(t_k^{+}\right)-x\left(t_k\right)={\phi }_k\left(x\left(t_k\right)\right),k=1,2,\cdots ,m,\hfill \\ {\eta }_{1}x\left(0\right)+{\eta }_{2}x\left(T\right)=\mu {\displaystyle \underset{k=0}{\overset{m}{\sum }}}{}_{t_k}{I}_{q_k}^{{\beta }_k}x\left(t_{k+1}\right)\hfill \end{array}$ (2)

is given by

$\begin{array}{c}x\left(t\right)=\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}h\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}h\left(t_{i+1}\right)\right)\underset{}{\overset{{}^{{}^{}}}{}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{\phi }_j\left(x\left(t_j\right)\right)+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}h\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{\phi }_i\left(x\left(t_i\right)\right)\right]\}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{\phi }_i\left(x\left(t_i\right)\right)+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}h\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}h\left(t\right).\end{array}$ (3)

Proof. Applying the operator ${}_{t_0}{I}_{q_0}^{{\alpha }_0}$ on both sides of the first equation of (2) for $t\in J_0$ and using Lemma 4, we have

$x\left(t\right)=x\left(t_0\right)+{}_{t_0}{I}_{q_0}^{{\alpha }_0}h\left(t\right).$

Then we get for $t=t_1$ that

$x\left(t_1\right)=x\left(t_0\right)+{}_{t_0}{I}_{q_0}^{{\alpha }_0}h\left(t_1\right).$ (4)

For $t\in J_1$, again taking the ${}_{t_1}{I}_{q_1}^{{\alpha }_1}$ to (4) and using the above process, we get

$x\left(t\right)=x\left(t_1^{+}\right)+{}_{t_1}{I}_{q_1}^{{\alpha }_1}h\left(t\right).$

Applying the impulsive condition $x\left(t_1^{+}\right)=x\left(t_1\right)+{\phi }_1\left(x\left(t_1\right)\right)$, we get

$x\left(t\right)=x\left(t_0\right)+{\phi }_1\left(x\left(t_1\right)\right)+{}_{t_0}{I}_{q_0}^{{\alpha }_0}h\left(t_1\right)+{}_{t_1}{I}_{q_1}^{{\alpha }_1}h\left(t\right).$

By the same way, for $t\in J_2$, we have

$x\left(t\right)=x\left(t_0\right)+{\phi }_1\left(x\left(t_1\right)\right)+{\phi }_2\left(x\left(t_2\right)\right)+{}_{t_0}{I}_{q_0}^{{\alpha }_0}h\left(t_1\right)+{}_{t_1}{I}_{q_1}^{{\alpha }_1}h\left(t_2\right)+{}_{t_2}{I}_{q_2}^{{\alpha }_2}h\left(t\right).$

Repeating the above process for $t\in J_k\subseteq J,k=0,1,2,\cdots ,m$, we get

$x\left(t\right)=x\left(t_0\right)+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{\phi }_i\left(x\left(t_i\right)\right)+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}h\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}h\left(t\right).$ (5)

From (5), we find that

$x\left(T\right)=x\left(t_0\right)+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{\phi }_i\left(x\left(t_i\right)\right)+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}h\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}h\left(T\right).$

From the boundary condition of (2), we get

$\begin{array}{c}x\left(t_0\right)=\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}h\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}h\left(t_{i+1}\right)\right)\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{\phi }_j\left(x\left(t_j\right)\right)+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}h\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{\phi }_i\left(x\left(t_i\right)\right)\right]\}.\end{array}$ (6)

Substituting (6) to (5), we obtain the solution (3). This completes the proof.

We define an operator $\mathcal{G}\mathrm{<mo>\; :\; </mo>}PC\left(J\mathrm{,}ℝ\right)\to PC\left(J\mathrm{,}ℝ\right)$ as follows:

$\begin{array}{c}\mathcal{G}x\left(t\right)=\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}f\left(s,x\right)\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}f\left(s,x\right)\left(t_{i+1}\right)\right)\underset{}{\overset{{}^{{}^{}}}{}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{\phi }_j\left(x\left(t_j\right)\right)+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}f\left(s,x\right)\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{\phi }_i\left(x\left(t_i\right)\right)\right]\}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{\phi }_i\left(x\left(t_i\right)\right)+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}f\left(s,x\right)\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}f\left(s,x\right)\left(t\right).\end{array}$ (7)

Then, the existence of solutions of system (1) is equivalent to the problem of fixed point of operator $\mathcal{G}$ in (7).

Theorem 1 Let $f\mathrm{<mo>\; :\; </mo>}J\times ℝ\to ℝ$ and ${\phi }_k:ℝ\to ℝ,k=1,2,\cdots ,m$ be continuous functions. Assume that $\mu {\displaystyle \underset{i=0}{\overset{m}{\sum }}}{\Omega }_{{\beta }_i}\ne {\eta }_1+{\eta }_2$ and the following conditions are satisfied:

(H₁) There exists a positive constant L such that $\left|{\phi }_k\left(x\right)-{\phi }_k\left(y\right)\right|\le L\left|x-y\right|$ for each $x\mathrm{,}y\in ℝ$ and $k=1,2,\cdots ,m$.

(H₂) There exists a function $M\left(t\right)\in C\left(J\mathrm{,}{ℝ}^{+}\right)$ such that

$\left|f\left(t\mathrm{,}x\right)-f\left(t\mathrm{,}y\right)\right|\le M\left(t\right)\left|x-y\right|\mathrm{,}\forall t\in J\mathrm{,}x\mathrm{,}y\in ℝ\mathrm{.}$

(H₃) $\Delta <1$.

Then problem (1) has a unique solution on J, where $M=\underset{t\in J}{sup}\left|M\left(t\right)\right|$ and

$\begin{array}{c}\Delta =\frac{1}{\Lambda }{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left(\mu M{\Omega }_{{\alpha }_i+{\beta }_i}+\left({\eta }_2+M\right){\Omega }_{{\alpha }_i}+\mu M{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{\Omega }_{{\alpha }_j}{\Omega }_{{\beta }_i}+\mu Li{\Omega }_{{\beta }_i}\right)\\ +\frac{1}{\Lambda }\left(\mu {\Omega }_{{\alpha }_0+{\beta }_0}+{\eta }_2{\Omega }_{{\alpha }_0}\right)+mL\left(\frac{1}{\Lambda }{\eta }_2+1\right).\end{array}$

Proof. The conclusion will follow once we have shown that the operator $\mathcal{G}$ defined (7) is a construction with respect to a suitable norm on $PC\left(J\mathrm{,}ℝ\right)$.

For any functions $x\mathrm{,}y\in PC\left(J\mathrm{,}ℝ\right)$, we have

$\begin{array}{l}\left|\left(\mathcal{G}x\right)\left(t\right)-\left(\mathcal{G}y\right)\left(t\right)\right|\\ \le \frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}\left|f\left(s,x\right)-f\left(s,y\right)\right|\left(t_{i+1}\right)+{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}\left|f\left(s,x\right)-f\left(s,y\right)\right|\left(t_{i+1}\right)\right)\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}\left|{\phi }_j\left(x\left(t_j\right)\right)-{\phi }_j\left(y\left(t_j\right)\right)\right|+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}\left|f\left(s,x\right)-f\left(s,y\right)\right|\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}\\ \underset{}{\overset{{}^{}}{}}+{\eta }_2\left|{\phi }_i\left(x\left(t_i\right)\right)-{\phi }_i\left(y\left(t_i\right)\right)\right|]\}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left|{\phi }_i\left(x\left(t_i\right)\right)-{\phi }_i\left(y\left(t_i\right)\right)\right|\\ +{\displaystyle \underset{i=0}{\overset{m-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}\left|f\left(s,x\right)-f\left(s,y\right)\right|\left(t_{i+1}\right)+{}_{t_m}{I}_{q_m}^{{\alpha }_m}\left|f\left(s,x\right)-f\left(s,y\right)\right|\left(t\right).\end{array}$

By conditions (H₁) and (H₂), we get

$\begin{array}{l}\left|\left(\mathcal{G}x\right)\left(t\right)-\left(\mathcal{G}y\right)\left(t\right)\right|\\ \le \frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}\left(M\Vert x-y\Vert \right)\left(t_{i+1}\right)+{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}\left(M\Vert x-y\Vert \right)\left(t_{i+1}\right)\right)\underset{}{\overset{{}^{}}{{}^{}}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}L\Vert x-y\Vert +{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}\left(M\Vert x-y\Vert \right)\right){\Omega }_{{\beta }_i}+{\eta }_{2}L\Vert x-y\Vert \right]\}\end{array}$

$\begin{array}{l}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}L\Vert x-y\Vert +{\displaystyle \underset{i=0}{\overset{m-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}\left(M\Vert x-y\Vert \right)\left(t_{i+1}\right)+{}_{t_m}{I}_{q_m}^{{\alpha }_m}\left(M\Vert x-y\Vert \right)\left(t_{m+1}\right)\\ \le \{\frac{1}{\Lambda }{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left(\mu M{\Omega }_{{\alpha }_i+{\beta }_i}+{\eta }_2{\Omega }_{{\alpha }_i}+\mu Li{\Omega }_{{\beta }_i}+\mu M{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{\Omega }_{{\alpha }_j}{\Omega }_{{\beta }_i}+M{\Omega }_{{\alpha }_i}\right)\\ +\frac{1}{\Lambda }\left(\mu {\Omega }_{{\alpha }_0+{\beta }_0}+{\eta }_2{\Omega }_{{\alpha }_0}\right)+mL\left(\frac{1}{\Lambda }{\eta }_2+1\right)\}\Vert x-y\Vert ,\end{array}$

which implies that

$\Vert \mathcal{G}x-\mathcal{G}y\Vert \le \Delta \Vert x-y\Vert \mathrm{.}$

Thus the operator $\mathcal{G}$ is a contraction in view of the condition (H₃). By Banach’s contraction mapping principle, the problem (1) has a unique solution on J. This completes the proof.

In the following, we study the Hyers-Ulam stability of impulsive fractional q-difference Equation (1). Let $\epsilon >0,ϵ>0$ and $\delta \mathrm{<mo>\; :\; </mo>}\left[\mathrm{0,}T\right]\to ℝ$ be a continuous function. Consider the inequalities:

$\{\begin{array}{l}\left|{}_{t_k}^c{D}_{q_k}^{{\alpha }_k}\bar{x}\left(t\right)-f\left(t,\bar{x}\left(t\right)\right)\right|\le \delta \left(t\right)\epsilon ,t\in J_k\subseteq J=\left[0,T\right],t\ne t_k,k=0,1,\cdots ,m,\hfill \\ \left|\Delta \bar{x}\left(t_k\right)-{\varphi }_k\left(\bar{x}\left(t_k\right)\right)\right|\le ϵ\epsilon ,k=1,2,\cdots ,m,\hfill \\ {\eta }_1\bar{x}\left(0\right)+{\eta }_2\bar{x}\left(T\right)=\mu {\displaystyle \underset{k=0}{\overset{m}{\sum }}}{}_{t_k}{I}_{q_k}^{{\beta }_k}\bar{x}\left(t_{k+1}\right).\hfill \end{array}$ (8)

Now, we give out the definition of Hyers-Ulam stability of system (1).

Definition 4 System (1) is Hyers-Ulam stable with respect to system (8), if there exists $A_f>0$ such that

$\left|\bar{x}-\tilde{x}\right|\le A_f\epsilon$

for all $t\in J$, where $\bar{x}$ is the solution of (8), and $\tilde{x}$ of the solution for system (1).

Theorem 2 Assume $f\mathrm{<mo>\; :\; </mo>}J\times ℝ\to ℝ$ satisfy assumption (H₂), ${\phi }_i:ℝ\to ℝ,i=1,2,\cdots ,m$ are continuous functions and satisfy assumption (H₁) and the condition (H₃) holds, $\underset{t\in J}{sup}\delta \left(t\right)\le 1$. Then the system (1) is Hyers-Ulam stable with respect to system (8).

Proof. Let ${}_{t_k}^c{D}_{q_k}^{{\alpha }_k}\bar{x}\left(t\right)=f\left(t,\bar{x}\left(t\right)\right)+g\left(t\right),k=0,1,\cdots ,m$ and $\Delta \bar{x}\left(t_k\right)={\phi }_k\left(\bar{x}\left(t_k\right)\right)+g_k,k=1,2,\cdots ,m$. Consider the system

$\{\begin{array}{l}{}_{t_k}^c{D}_{q_k}^{{\alpha }_k}\bar{x}\left(t\right)=f\left(t,\bar{x}\left(t\right)\right)+g\left(t\right),t\in J_k\subseteq J=\left[0,T\right],t\ne t_k,\hfill \\ \Delta \bar{x}\left(t_k\right)={\phi }_k\left(\bar{x}\left(t_k\right)\right)+g_k,k=1,2,\cdots ,m.\hfill \\ {\eta }_1\bar{x}\left(0\right)+{\eta }_2\bar{x}\left(T\right)=\mu {\displaystyle \underset{k=0}{\overset{m}{\sum }}}{}_{t_k}{I}_{q_k}^{{\beta }_k}\bar{x}\left(t_{k+1}\right).\hfill \end{array}$ (9)

Similarly to the system in Theorem 1, system (9) is equivalent to the following integral equation in Lemma 5.

$\begin{array}{c}\bar{x}\left(t\right)=\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}\left(f\left(s,\bar{x}\right)+g\left(s\right)\right)\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}\left(f\left(s,\bar{x}\right)+g\left(s\right)\right)\left(t_{i+1}\right)\right)\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}\left({\phi }_j\left(\bar{x}\left(t_j\right)\right)+g_j\right)+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}\left(f\left(s,\bar{x}\right)+g\left(s\right)\right)\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}\end{array}$

$\begin{array}{c}\begin{array}{c}\\ \end{array}-{\eta }_2\left({\phi }_i\left(\bar{x}\left(t_i\right)\right)+g_i\right)]\}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}\left({\phi }_i\left(\bar{x}\left(t_i\right)\right)+g_i\right)\\ +{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}\left(f\left(s,\bar{x}\right)+g\left(s\right)\right)\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}\left(f\left(t,\bar{x}\right)+g\left(t\right)\right)\end{array}$ (10)

Now, we define the operator $\tilde{\mathcal{G}}$ as following

$\begin{array}{c}\tilde{\mathcal{G}}x\left(t\right)=\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}f\left(s,x\right)\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}f\left(s,x\right)\left(t_{i+1}\right)\right)\underset{}{\overset{{}_{}{}^{}}{}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{\phi }_j\left(x\left(t_j\right)\right)+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}f\left(s,x\right)\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{\phi }_i\left(x\left(t_i\right)\right)\right]\}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{\phi }_i\left(x\left(t_i\right)\right)+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}f\left(s,x\right)\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}f\left(s,x\right)\left(t\right)+G\left(t\right)\\ =\mathcal{G}x+G\left(t\right).\end{array}$ (11)

where

$\begin{array}{c}G\left(t\right)=\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}g\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}g\left(t_{i+1}\right)\right)\underset{}{\overset{{}_{}{}^{}}{}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{g}_j+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}g\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{g}_i\right]\}\\ +{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{g}_i+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}g\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}g\left(t\right).\end{array}$ (12)

Note that

$\Vert \tilde{\mathcal{G}}x-\tilde{\mathcal{G}}y\Vert =\Vert \mathcal{G}x-\mathcal{G}y\Vert \mathrm{.}$

Then the existence of a solution of (1) implies the existence of a solution to (9), it follows from Theorem 1 that $\tilde{\mathcal{G}}$ is a contraction. Thus there is a unique fixed point $\bar{x}$ of $\tilde{\mathcal{G}}$, and respectively $\tilde{x}$ of $\mathcal{G}$.

Since $t\in \left[\mathrm{0,}T\right]$ and $\underset{t\in J}{sup}\delta \left(t\right)\le 1$, we obtain

$\begin{array}{c}\Vert G\Vert =\underset{t\in J}{\max }\left|G\left(t\right)\right|\\ =\underset{t\in J}{\max }|\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}g\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}g\left(t_{i+1}\right)\right)\underset{}{\overset{{}_{}{}^{}}{}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{g}_j+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}g\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{g}_i\right]\}\\ \underset{}{\overset{{}_{}{}^{}}{}}+{\displaystyle \underset{i=1}{\overset{k}{\sum }}}{g}_i+{\displaystyle \underset{i=0}{\overset{k-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}g\left(t_{i+1}\right)+{}_{t_k}{I}_{q_k}^{{\alpha }_k}g\left(t\right)|\end{array}$

$\begin{array}{l}\le \underset{t\in J}{\max }|\frac{1}{\Lambda }\{{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {}_{t_i}{I}_{q_i}^{{\alpha }_i+{\beta }_i}g\left(t_{i+1}\right)-{\eta }_2{}_{t_i}{I}_{q_i}^{{\alpha }_i}g\left(t_{i+1}\right)\right)\underset{}{\overset{{}_{}{}^{}}{}}\\ +{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left[\mu \left({\displaystyle \underset{j=1}{\overset{i}{\sum }}}{g}_j+{\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{}_{t_j}{I}_{q_j}^{{\alpha }_j}g\left(t_{j+1}\right)\right){\Omega }_{{\beta }_i}-{\eta }_2{g}_i\right]\}\end{array}$

$\begin{array}{l}\underset{}{\overset{{}_{}{}^{}}{}}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{g}_i+{\displaystyle \underset{i=0}{\overset{m-1}{\sum }}}{}_{t_i}{I}_{q_i}^{{\alpha }_i}g\left(t_{i+1}\right)+{}_{t_m}{I}_{q_m}^{{\alpha }_m}g\left(t\right)|\\ \le \{\frac{1}{\Lambda }{\displaystyle \underset{i=1}{\overset{m}{\sum }}}\left(\mu {\Omega }_{{\alpha }_i+{\beta }_i}+{\eta }_2{\Omega }_{{\alpha }_i}+\mu ϵi{\Omega }_{{\beta }_i}+\mu {\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{\Omega }_{{\alpha }_j}{\Omega }_{{\beta }_i}+{\Omega }_{{\alpha }_i}\right)\\ +\frac{1}{\Lambda }\left(\mu {\Omega }_{{\alpha }_0+{\beta }_0}+{\eta }_2{\Omega }_{{\alpha }_0}\right)+mϵ\left(\frac{1}{\Lambda }{\eta }_2+1\right)\}\epsilon .\end{array}$ (13)

Then, we get

$\begin{array}{c}\Vert \bar{x}-\tilde{x}\Vert =\Vert \tilde{\mathcal{G}}\bar{x}-\mathcal{G}\tilde{x}\Vert =\Vert \mathcal{G}\bar{x}-\mathcal{G}\tilde{x}+G\left(t\right)\Vert \le \Vert \mathcal{G}\bar{x}-\mathcal{G}\tilde{x}\Vert +\Vert G\Vert \\ \le \Delta \Vert \bar{x}-\tilde{x}\Vert +\{\frac{1}{\Lambda }{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {\Omega }_{{\alpha }_i+{\beta }_i}+{\eta }_2{\Omega }_{{\alpha }_i}+\mu ϵi{\Omega }_{{\beta }_i}+\mu {\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{\Omega }_{{\alpha }_j}{\Omega }_{{\beta }_i}+{\Omega }_{{\alpha }_i}\right)\\ +\frac{1}{\Lambda }\left(\mu {\Omega }_{{\alpha }_0+{\beta }_0}+{\eta }_2{\Omega }_{{\alpha }_0}\right)+mϵ\left(\frac{1}{\Lambda }{\eta }_2+1\right)\}\epsilon .\end{array}$ (14)

By condition (H₃), we have

$\begin{array}{c}\Vert \bar{x}-\tilde{x}\Vert \le {\left(1-\Delta \right)}^{-1}\{\frac{1}{\Lambda }{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {\Omega }_{{\alpha }_i+{\beta }_i}+{\eta }_2{\Omega }_{{\alpha }_i}+\mu ϵi{\Omega }_{{\beta }_i}+\mu {\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{\Omega }_{{\alpha }_j}{\Omega }_{{\beta }_i}+{\Omega }_{{\alpha }_i}\right)\\ +mϵ\left(\frac{1}{\Lambda }{\eta }_2+1\right)\}\epsilon .\end{array}$ (15)

Let $\begin{array}{l}{A}_f={\left(1-\Delta \right)}^{-1}\{\frac{1}{\Lambda }{\displaystyle \underset{i=0}{\overset{m}{\sum }}}\left(\mu {\Omega }_{{\alpha }_i+{\beta }_i}+{\eta }_2{\Omega }_{{\alpha }_i}+\mu ϵi{\Omega }_{{\beta }_i}+\mu {\displaystyle \underset{j=0}{\overset{i-1}{\sum }}}{\Omega }_{{\alpha }_j}{\Omega }_{{\beta }_i}+{\Omega }_{{\alpha }_i}\right)\\ +mϵ\left(\frac{1}{\Lambda }{\eta }_2+1\right)\}\end{array}$, then $\Vert \bar{x}-\tilde{x}\Vert \le A_f\epsilon \mathrm{.}$

This completes the proof.

Remark 1 Note that (1) has a very general form, as special instances results from (1), when, ${\eta }_1={\eta }_2=1,\mu =0$, (1) reduces to the antiperiodic boundary value problem of the impulsive fractional q-difference equation:

$\{\begin{array}{l}{}_{t_k}^c{D}_{q_k}^{{\alpha }_k}x\left(t\right)=f\left(t,x\left(t\right)\right),t\in J_k\subseteq J=\left[0,T\right],t\ne t_k,\hfill \\ \Delta x\left(t_k\right)=x\left(t_k^{+}\right)-x\left(t_k\right)={\phi }_k\left(x\left(t_k\right)\right),k=1,2,\cdots ,m,\hfill \\ x\left(0\right)+x\left(T\right)=0.\hfill \end{array}$

4. Example

Consider the following boundary value problem:

$\{\begin{array}{l}{}_{t_k}^c{D}_{\frac{3k+1}{4k+3}}^{\frac{k+1}{3k+2}}x\left(t\right)=\frac{{\sin }^{2}t}{t^2+50}\frac{2\left|x\left(t\right)\right|}{1+\left|x\left(t\right)\right|}+\frac{3t}{4},t\in \left[0,\frac{3}{2}\right]\backslash \left\{t_1,t_2\right\},\hfill \\ \Delta x\left(t_k\right)=\frac{1}{200k}\frac{x^2\left(t_k\right)+2\left|x\left(t_k\right)\right|}{1+\left|x\left(t_k\right)\right|}+\frac{k}{5},t_k=\frac{k}{2},k=1,2,\hfill \\ \frac{8}{3}x\left(0\right)+\frac{1}{6}x\left(\frac{3}{2}\right)=\frac{1}{2}{\displaystyle \underset{k=0}{\overset{2}{\sum }}}{}_{t_k}{I}_{\frac{3k+1}{4k+3}}^{\frac{k+1}{k^2+2}}x\left(t_{k+1}\right).\hfill \end{array}$ (16)

Corresponding to boundary value problem (1), one see that ${\alpha }_k=\frac{k+1}{3k+2}$, ${\beta }_k=\frac{k+1}{k^2+2}$, $q_k=\frac{3k+1}{4k+3}$, $t_k=\frac{k}{2}$, $f\left(t,x\right)=\frac{{\sin }^{2}t}{t^2+50}\frac{2\left|x\left(t\right)\right|}{1+\left|x\left(t\right)\right|}+\frac{3}{4}$, ${\phi }_k\left(x\left(t_k\right)\right)=\frac{1}{200k}\frac{x^2\left(t_k\right)+2\left|x\left(t_k\right)\right|}{1+\left|x\left(t_k\right)\right|}$. Through a simple calculation, we get

$\left|f\left(t,x\right)-f\left(t,y\right)\right|\le \frac{{\sin }^{2}t}{t^2+25}\left|x-y\right|,M\left(t\right)=\frac{{\sin }^{2}t}{t^2+25}\le \frac{1}{25}=M,$

$\left|{\phi }_k\left(x\right)-{\phi }_k\left(y\right)\right|\le \frac{1}{200k}\left|x-y\right|\le \frac{1}{200}\left|x-y\right|,L=\frac{1}{200},$

$\Lambda \doteq 1.7875>0,\Delta \doteq 0.4873<1.$

From Theorem 1, the problem (16) has a unique solution x on $\left[\mathrm{0,}\frac{3}{2}\right]$. Furthermore, the solution x is Hyers-Ulam stable with respect to the following system

$\{\begin{array}{l}\left|{}_{t_k}^c{D}_{\frac{3k+1}{4k+3}}^{\frac{k+1}{3k+2}}x\left(t\right)-\frac{{\sin }^{2}t}{t^2+50}\frac{2\left|x\left(t\right)\right|}{1+\left|x\left(t\right)\right|}-\frac{3t}{4}\right|\le \delta \left(t\right)\epsilon ,t\in \left[0,\frac{3}{2}\right]\backslash \left\{t_1,t_2\right\},\hfill \\ \left|\Delta x\left(t_k\right)-\frac{1}{200k}\frac{x^2\left(t_k\right)+2\left|x\left(t_k\right)\right|}{1+\left|x\left(t_k\right)\right|}-\frac{k}{5}\right|\le ϵ\epsilon ,t_k=\frac{k}{2},k=1,2,\hfill \\ \frac{8}{3}x\left(0\right)+\frac{1}{6}x\left(\frac{3}{2}\right)=\frac{1}{2}{\displaystyle \underset{k=0}{\overset{2}{\sum }}}{}_{t_k}{I}_{\frac{3k+1}{4k+3}}^{\frac{k+1}{k^2+2}}x\left(t_{k+1}\right),\hfill \end{array}$ (17)

where $ϵ>0,\epsilon >0$, $\underset{t\in \left[0,\frac{3}{2}\right]}{\sup }\delta \left(t\right)<1$.

5. Conclusion

In this paper, we study the existence and Hyers-Ulam stability of solutions for impulsive fractional q-difference equation. We obtain some results as following: 1) Using the q-shifting operator, the results of existence of solutions for impulsive fractional q-difference equation with q-integral boundary conditions are obtained. 2) The Hyers-Ulam stability of the nonlinear impulsive fractional q-difference equations was obtained.

Funding

This research was supported by Science and Technology Foundation of Guizhou Province (Grant No. [2016] 7075), by the Project for Young Talents Growth of Guizhou Provincial Department of Education under (Grant No. Ky [2017] 133), and by the project of Guizhou Minzu University under (Grant No.16yjrcxm002).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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