The q-calculus or quantum calculus is an old subject that was initially developed by Jackson ; basic definitions and properties of q-calculus can be found in . The fractional q-calculus had its origin in the works by Al-Salam  and Agarwal . But the definitions mentioned above about q-calculus can’t be applied to impulse points , such that . In , the authors defined the concepts of fractional q-calculus by defining a q-shifting operator . Using the q-shifting operator, the fractional impulsive q-difference equation was defined. In paper   , 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  -  and the references cited therein. Dumitru Baleanu et al. discussed the stability of non-autonomous systems with the q-Caputo fractional derivatives in reference . 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:
where is the fractional -derivative of the Caputo type of order on , , , , , , , . denotes the Riemann-Liouville -fractional integral of order on and 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   .
For , we define a q-shifting operator as . The new power of q-shifting operator is defined as ,
, , . If , then .
The q-derivative of a function on interval is defined by
The q-integral of a function defined on the interval is given by
Some results about operator and can be found in references . Let us define fractional q-derivative and q-integral on interval and outline some of their properties   .
Definition 1  The fractional q-derivative of Riemann-Liouville type of order on interval is defined by and
where l is the smallest integer greater than or equal to .
Definition 2  Let and be a function defined on . The fractional q-integral of Riemann-Liouville type is given by and
Lemma 1  Let and be a continuous function on . The Riemann-Liouville fractional q-integral has the following semi-group property
Lemma 2  Let be a q-integrable function on . Then the following equality holds
Lemma 3  Let and p be a positive integer. Then for the following equality holds
Definition 3  The fractional q-derivative of Caputo type of order on interval is defined by and
where is the smallest integer greater than or equal to .
Lemma 4  Let and n be the smallest integer great than or equal to . Then for the following equality holds
3. Main Results
In this section, we will give the main results of this paper.
Let is continuous everywhere except for some at which and exist, and . is a Banach space with the norm
First, for the sake of convenience, we introduce the following notations:
To obtain our main results, we need the following lemma.
Lemma 5 Let and . Then for any , the solution of the following problem
is given by
Proof. Applying the operator on both sides of the first equation of (2) for and using Lemma 4, we have
Then we get for that
For , again taking the to (4) and using the above process, we get
Applying the impulsive condition , we get
By the same way, for , we have
Repeating the above process for , we get
From (5), we find that
From the boundary condition of (2), we get
Substituting (6) to (5), we obtain the solution (3). This completes the proof.
We define an operator as follows:
Then, the existence of solutions of system (1) is equivalent to the problem of fixed point of operator in (7).
Theorem 1 Let and be continuous functions. Assume that and the following conditions are satisfied:
(H1) There exists a positive constant L such that for each and .
(H2) There exists a function such that
Then problem (1) has a unique solution on J, where and
Proof. The conclusion will follow once we have shown that the operator defined (7) is a construction with respect to a suitable norm on .
For any functions , we have
By conditions (H1) and (H2), we get
which implies that
Thus the operator 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 and be a continuous function. Consider the inequalities:
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 such that
for all , where is the solution of (8), and of the solution for system (1).
Theorem 2 Assume satisfy assumption (H2), are continuous functions and satisfy assumption (H1) and the condition (H3) holds, . Then the system (1) is Hyers-Ulam stable with respect to system (8).
Proof. Let and . Consider the system
Similarly to the system in Theorem 1, system (9) is equivalent to the following integral equation in Lemma 5.
Now, we define the operator as following
Then the existence of a solution of (1) implies the existence of a solution to (9), it follows from Theorem 1 that is a contraction. Thus there is a unique fixed point of , and respectively of .
Since and , we obtain
Then, we get
By condition (H3), we have
Let , then
This completes the proof.
Remark 1 Note that (1) has a very general form, as special instances results from (1), when, , (1) reduces to the antiperiodic boundary value problem of the impulsive fractional q-difference equation:
Consider the following boundary value problem:
Corresponding to boundary value problem (1), one see that , , , , , . Through a simple calculation, we get
From Theorem 1, the problem (16) has a unique solution x on . Furthermore, the solution x is Hyers-Ulam stable with respect to the following system
where , .
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.
This research was supported by Science and Technology Foundation of Guizhou Province (Grant No.  7075), by the Project for Young Talents Growth of Guizhou Provincial Department of Education under (Grant No. Ky  133), and by the project of Guizhou Minzu University under (Grant No.16yjrcxm002).
 Tariboom, T., Ntouvas, S.K. and Agarwal, R. (2015) New Concepts of Fractional Quantum Calculus and Applications to Impulsive Fractional q-Difference Equation. Advances in Difference Equations, 2015, Article No. 18.
 Tariboom, T. and Ntouvas, S.K. (2013) Quantum Calculus on Finite Intervals and Applications to Impulsive Difference Equations. Advances in Difference Equations, 2013, Article No. 282.
 Ahmad, B., Ntouyas, S.K., Tariboon, J., Alsaedi, A. and Alsulami, H.H. (2016) Impulsive Fractional q-Integro-Difference Equations with Separated Boundary Conditions. Applied Mathematics and Computation, 281, 199-213.
 Agarwal, R.P., Wang, G., Ahmad, B., Zhang, L., Hobiny, A. and Monaquel, S. (2015) On Existence of Solutions for Nonlinear q-Difference Equations with Nonlocal q-Integral Boundary Conditions. Mathematical Modelling and Analysis, 20, 604-618.
 Li, X.H., Han, Z.L., Sun, S.R. and Sun, L.Y. (2016) Eigenvalue Problems of Fractional q-Difference Equations with Generalized p-Laplacian. Applied Mathematics Letters, 57, 46-53.
 Ferreira, R. (2011) Positive Solutions for a Class of Boundary Value Problems with Fractional q-Differences. Computers & Mathematics with Applications, 61, 367-373.
 Samei, M.E. and Khalilzadeh Ranjbar, G. (2019) Some Theorems of Existence of Solutions for Fractional Hybrid q-Difference Inclusion. Journal of Advanced Mathematical Studies, 12, 63-76.
 Liang, S.H. and Samei, M.E. (2020) New Approach to Solutions of a Class of Singular Fractional q-Differential Problem via Quantum Calculus. Advances in Difference Equations, 2020, Article No. 14.
 Zhai, C.B. and Ren, J. (2018) The Unique Solution for a Fractional q-Difference Equation with Three-Point Boundary Conditions. Indagationes Mathematicae, 29, 948-961.