Existence and Stability Results for Impulsive Fractional q -Difference Equation

In this paper, we study the boundary value problem for an impulsive fractional q-difference equation. Based on Banach’s contraction mapping princi-ple, 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.


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 , k t k ∈  , such that ( ) , k t qt t ∈ . In [5], the authors defined the concepts of fractional q-calculus by defining a q-shifting operator ( ) ( ) 1 , , a q m qm q a m a Φ = + − ∈ . Using the q-shifting operator, the fractional sive 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:

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].
, we define a q-shifting operator as The new power of q-shifting operator is defined as ( Some results about operator a q D and a q I can be found in references [5].
Let us define fractional q-derivative and q-integral on interval [ ]  where l is the smallest integer greater than or equal to ν .
The fractional q-derivative of Caputo type of order where n is the smallest integer greater than or equal to α .
Lemma 4 [7] Let 0 α > and n be the smallest integer great than or equal to α . Then for the following equality holds

Main Results
In this section, we will give the main results of this paper. Let First, for the sake of convenience, we introduce the following notations: To obtain our main results, we need the following lemma.

Lemma 5 Let
is given by Proof. Applying the operator  (2) for 0 t J ∈ and using Lemma 4, we have Then we get for  x t x t x t ϕ By the same way, for Repeating the above process for , 0,1, 2, , From (5), we find that From the boundary condition of (2), we get Substituting (6) to (5) Then, the existence of solutions of system (1) is equivalent to the problem of fixed point of operator  in (7).
Thus the operator  is a contraction in view of the condition (H 3 ). 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 Now, we give out the definition of Hyers-Ulam stability of system (1). (1) is Hyers-Ulam stable with respect to system (8), if there exists Proof. Let

Definition 4 System
Similarly to the system in Theorem 1, system (9) is equivalent to the following integral equation in Lemma 5.
Note that 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 x of   , and respectively x  of  .   (16)