Abstract
In this paper we define new concepts of fractional quantum calculus by defining a new q-shifting operator. After giving the basic properties we define the q-derivative and q-integral. New definitions of Riemann-Liouville fractional q-integral and q-difference on an interval [a, b] are given and their basic properties are discussed. As applications of the new concepts, we prove existence and uniqueness results for first and second order initial value problems for impulsive fractional q-difference equations.
Introduction
The quantum calculus is known as the calculus without limits. It substitutes the classical derivative by a difference operator, which allows one to deal with sets of nondifferentiable functions. Quantum difference operators have an interest role due to their applications in several mathematical areas such as orthogonal polynomials, basic hypergeometric functions, combinatorics, the calculus of variations, mechanics and the theory of relativity. The book by Kac and Cheung [] covers many of the fundamental aspects of the quantum calculus.
Introduction
In recent years, the topic of q-calculus has attracted the attention of several researchers and a variety of new results can be found in the papers [-] and the references cited therein.
Introduction
In [] the notions of q k -derivative and q k -integral of a function f : J k := [t k , t k+ ] → R, have been introduced and their basic properties was proved. As applications existence and uniqueness results for initial value problems for first and second order impulsive q kdifference equations are proved. q-calculus analogs of some classical integral inequalities, such as the Hölder, Hermite-Hadamard, Trapezoid, Ostrowski, Cauchy-BunyakovskySchwarz, Grüss, and Grüss-Čebyšev ones are proved in [] .
Introduction
In this paper we define new concepts of fractional quantum calculus by defining a new q-shifting operator a q (m) = qm + ( -q)a. After giving the basic properties we define the q-derivative and q-integral. New definitions of Riemann-Liouville fractional q-integral and q-difference on an interval [a, b] are given and their basic properties are discussed. As applications of the new concepts, we prove existence and uniqueness results for first and second order initial value problems for impulsive fractional q-difference equations.
Introduction
In applications, as in constructing a q-Taylor formula or solving q-differential equations and inequalities of fractional order, it is interesting to allow nonzero lower limits of qintegration. In [, ] , the authors studied fractional q-integrals and q-derivatives which are based on Jackson sense on interval [a, b] . However, if the upper and lower limits of q-integration are b and a = bq n for some n ∈ N, respectively, then the infinite sum of the definition of q-integration is reduced to a finite sum, which is the restriction of definition of q-integral. From this point, our results generalized the classical definitions of fractional q-integrals and q-derivatives by shifting the point of the origin from zero to be a constant a ∈ R and applying the results to establish the impulsive fractional quantum difference equations.
Preliminaries
To make this paper self-contained, below we recall some known facts on fractional q-calculus. The presentation here can be found in, for example,
Preliminaries
The q-analog of the power function (n -m)
Preliminaries
We also use the natation  (γ ) =  for γ > . The q-gamma function is defined by
Preliminaries
The q-derivative of a function h is defined by
Preliminaries
and the q-derivatives of higher order are given by
Preliminaries
The q-integral of a function h defined on the interval [, b] is given by
Preliminaries
Similar to derivatives, an operator I k q is given by
Preliminaries
The fundamental theorem of calculus applies to these operators D q and I q , i.e.,
Preliminaries
and if h is continuous at t = , then
Preliminaries
For any s, t > , the q-beta function is defined by
Preliminaries
The expression of q-beta function in terms of the q-gamma function can be written as
Preliminaries
where l is the smallest integer greater than or equal to ν.
Lemma . []
Let α, β ≥  and f be a function defined in [, T] . Then the following formulas hold:
Lemma . []
Let α >  and n be a positive integer. Then the following equality holds:
Lemma . []
(.)
New concepts of fractional quantum calculus
Let us define a q-shifting operator as
New concepts of fractional quantum calculus
For any positive integer k, we have
New concepts of fractional quantum calculus
By computing directly, we get the following results.
New concepts of fractional quantum calculus
Property . For any m, n ∈ R and for all positive integer k, j, we have:
New concepts of fractional quantum calculus
The q-analog of the Pochhammer symbol is defined by
New concepts of fractional quantum calculus
We also define the new power of q-shifting operator as Property . For any γ , n, m ∈ R with n = a and k ∈ N ∪ {∞}, we have:
New concepts of fractional quantum calculus
Proof To prove (i), for any n, m ∈ R with n = a, it follows that
New concepts of fractional quantum calculus
Applying the method to prove (i) for (.), we have the results in (ii). Using the Pochhammer symbol, we obtain the last relation (ii) as requested. Substituting m = a k q (n) in (ii) and using Property .(v), we obtain the desired result in (iii).
New concepts of fractional quantum calculus
and the q-derivatives of higher order are given by
New concepts of fractional quantum calculus
The q-derivative of a product and ratio of functions f and
New concepts of fractional quantum calculus
Also similar to the derivative, an operator a I k q is given by
New concepts of fractional quantum calculus
The fundamental theorem of calculus applies to these operator a D q and a I q , i.e.,
New concepts of fractional quantum calculus
and if f is continuous at t = a, then
New concepts of fractional quantum calculus
The formula for q-integration by parts on an interval
New concepts of fractional quantum calculus
The reversing order of
New concepts of fractional quantum calculus
Then, from (.), the multiple q-integrals can be converted to a single 
New concepts of fractional quantum calculus
where l is the smallest integer greater than or equal to ν.
In [], we have the following formula for
It is easy to verify that
In [], we have the following formula for
where l is a positive integer. The next result gives the generalization of (.).
In [], we have the following formula for
Using (.) and applying Property .(iv), Property .(iii), it follows that
In [], we have the following formula for
Applying (.) for (.)-(.), we obtain the desired formula in (.).
In [], we have the following formula for
It follows from (.), (.), and Properties .(v) and .(iii) that the Riemann-Liouville fractional q-integral (.) can be written in the form of an infinite series as
In [], we have the following formula for
We recall the definition of the basic q-hypergeometric function as
In [], we have the following formula for
The Proof By taking into account of (.) and using Property ., we have
In [], we have the following formula for
Applying the substitution k = i + j, we obtain
In [], we have the following formula for
In (.) we interchange the order of the summations to get
In [], we have the following formula for
It is easy to verify that
In [], we have the following formula for
Consequently,
In [], we have the following formula for
Substituting (.) into (.), we obtain the series representation of a I α+β q f (t) and (.) holds. [a, b] . Then the following equality holds:
Lemma . Let f be a q-integrable function on
. For a positive noninteger α, n - < α < n, n ∈ N, by using Lemma ., we obtain
Lemma . For any t, s ∈ [a, b]. The following formulas hold:
( 
Lemma . For any t, s ∈ [a, b]. The following formulas hold:
To prove (ii), we use (.) with respect to s and Property .(v). We omit the details.
Lemma . Let α >  and p be a positive integer. Then for t ∈ [a, b] the following equality holds:
Proof Let α be a positive constant. Now, we will prove the formula (.) by using the mathematical induction. Suppose that p = . By Lemma ., we get
Lemma . Let α >  and p be a positive integer. Then for t ∈ [a, b] the following equality holds:
Therefore, by Lemma . and Property .(iii), we obtain
Lemma . Let α >  and p be a positive integer. Then for t ∈ [a, b] the following equality holds:
Next, suppose that (.) holds for p ∈ N. Then we have
Impulsive fractional q-difference equations
is continuous everywhere except for some t k at which x(t + k ) and x(t -k ) exist and
Impulsive fractional q-difference equations
Clearly PC γ is a Banach space.
Impulsive fractional q-difference equation of order 0 < α ≤ 1
In this subsection, we initiate the study of the existence and uniqueness of solutions for the following initial value problem for impulsive fractional q-difference equation of order
Impulsive fractional q-difference equation of order 0 < α ≤ 1
where
Impulsive fractional q-difference equation of order 0 < α ≤ 1
with < (·) = . The converse is also true.
Impulsive fractional q-difference equation of order 0 < α ≤ 1
Proof In view of Definition ., for t ∈ J  and t  = , it follows that
Impulsive fractional q-difference equation of order 0 < α ≤ 1
By Lemmas . and ., t ∈ J  , we have
Impulsive fractional q-difference equation of order 0 < α ≤ 1
The initial condition x() =  leads to c  =  which yields for t ∈ J  ,
Impulsive fractional q-difference equation of order 0 < α ≤ 1
The Riemann-Liouville fractional q  -integrating for order  -α for t = t  leads to
Impulsive fractional q-difference equation of order 0 < α ≤ 1
For t ∈ J  , taking the Riemann-Liouville fractional q  -integral of order α to (.) and using the above process, we have
Impulsive fractional q-difference equation of order 0 < α ≤ 1
Since
Impulsive fractional q-difference equation of order 0 < α ≤ 1
, it follows using (.) for t ∈ J  that
Impulsive fractional q-difference equation of order 0 < α ≤ 1
By computing directly, for t = t  , we obtain from (.)
Impulsive fractional q-difference equation of order 0 < α ≤ 1
Applying the Riemann-Liouville fractional q  -integrating of order α for (.) from t  to t, where t ∈ J  , then we have
Impulsive fractional q-difference equation of order 0 < α ≤ 1
Repeating the above process, for t ∈ J, we obtain (.).
Impulsive fractional q-difference equation of order 0 < α ≤ 1
On the other hand, assume that x is a solution of (.). Applying the Riemann-Liouville fractional q k -derivative of order α on (.) for t ∈ J k , k = , , , . . . , m and using () = ∞, it follows that
Impulsive fractional q-difference equation of order 0 < α ≤ 1
It is easy to verify that x(t k ) = ϕ k (x(t k )), k = , , . . . , m and x() = . This completes the proof. Proof We define an operator A :
Theorem . Assume that the following assumptions hold:
(H  ) f : J × R → R
is a continuous function and satisfies
f (t, x) -f (t, y) ≤ L|x -y|, L > , ∀t ∈ J, x, y ∈ R; (H  ) ϕ k : R → R, k = , , . .
. , m, are continuous functions and satisfy
with < (·) = . In addition, we define a ball B r = {x ∈ PC γ (J, R) : x PC γ ≤ r}. To show that Ax ∈ PC γ , we suppose τ  , τ  ∈ J k , and then
. , m, are continuous functions and satisfy
As
. , m, are continuous functions and satisfy
Therefore, we get Ax(t) ∈ PC γ . Now, we will show that AB r ⊂ B r . Assume that sup t∈J |f (t, )| = N  and max{|I k ()| : k = , , . . . , m} = N  , and setting
. , m, are continuous functions and satisfy
we choose a constant r such that
. , m, are continuous functions and satisfy
where δ ≤ ε < . For any x ∈ B r and for each t ∈ J k , we have
. , m, are continuous functions and satisfy
Multiplying both sides of the above inequality by (t -t k ) γ for each t ∈ J k , we have
. , m, are continuous functions and satisfy
This means that Ax PC γ ≤ r, which leads to AB r ⊂ B r . For x, y ∈ PC γ (J, R) and for each t ∈ J, we have
. , m, are continuous functions and satisfy
Multiplying both sides of the above inequality by (t -t k ) γ for each t ∈ J k , we have
It follows that
Ax -Ay PC γ ≤ x -y PC γ .
It follows that
As < , by the Banach contraction mapping principle, A is a contraction. Therefore, A has a fixed point which is a unique solution of (.) on J.
It follows that
Example . Consider the following impulsive fractional q-difference initial value problem: 
Impulsive fractional q-difference equation of order 1 < α ≤ 2
In this subsection, we investigate the initial value problem of impulsive fractional qdifference equation of order  < α ≤  the form
Impulsive fractional q-difference equation of order 1 < α ≤ 2
where Using the jump conditions of (.) with (.)-(.) for t ∈ J  , we get
x(t) .
Repeating the above process, for t ∈ J, we obtain (.) as required.
x(t) .
Next, we prove the existence and uniqueness of a solution to the initial value problem (.). We shall use the Banach fixed point theorem to accomplish this. 

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