Received 16 April 2016; accepted 26 June 2016; published 29 June 2016
Importance of fractional differential equations appears in many of the physical and engineering phenomena in the last two decades  -  . Problems with nonlocal conditions and related topics were studied in, for example  , and the nonlocal Cauchy problem  . The attention of researchers subject of q-difference equations appeared in recent years   . Initially, it was developed by Jackson   . Noted recently the attention of many researchers is in the field of fractional q-calculus   . Recently nonlocal fractional q-difference problems have aroused considerable attention   .
In this paper, we obtain the results of the existence and uniqueness of solutions for the Cauchy problem with nonlocal conditions for some fractional q-difference equations given by
Here, is the Caputo fractional q-derivative of order, and
are given continuous functions. It is worth mentioning that the nonlocal condition which can be applied effectively in physics is better than the classical Cauchy problem condition, see  .
Several authors have studied the semi-linear differential equations with nonlocal conditions in Banach space,   . In  , Dong et al. studied the existence and uniqueness of the solutions to the nonlocal problem for the fractional differential equation in Banach space. Motivated by these studied, we explore the Cauchy problem for nonlinear fractional q-difference equations according to the following hypotheses.
(H1) is jointly continuous.
(H3) is continuous and
The problem (1) is then devolved to the following formula
See reference  for more details.
2. Preliminaries on Fractional q-Calculus
Let and define
The q-analogue of the Pochhammer symbol was presented as follows
In general, if thereafter
The q-gamma function is defined by
The q-derivative of a function is here defined by
The q-integral of a function f defined in the interval is provided by
Now, it can be defined an operator, as follows
We can point to the basic formula which will be used at a later time,
where denotes the q-derivative with respect to variable s.
See reference  -  for more details.
Definition 2.1.  Let and f be a function defined on. The fractional q-integral of the Riemann-Liouville type is and
Definition 2.3.  The fractional q-derivative of the Caputo type of order is defined by
where is the smallest integer greater than or equal to.
Theorem 2.1.  Let and.Then, the following equality holds
Theorem 2.2.   (Krasnoselskii)
Let M be a closed convex non-empty subset of a Banach space. Suppose that A and B maps M into X, such that the following hypotheses are fulfilled:
1) for all;
2) A is continuous and AM is contained in a compact set;
3) B is a contraction mapping.
Then there exists such that
3. Main Results
Now, the obtained results are presented.
Let (H1)- (H3) hold, if and, the problem (1) has a unique solution.
Proof. Define by
Choose and let. So, we can prove that, where
. For it, let and. Consequently, we find that
This shows that therefore,.
Now, for, we obtain
Thus, by the Banach’s contraction mapping principle, we find that the problem (1) has a unique solution.
Our next results are based on Krasnoselskii’s fixed-point theorem.
Let (H1), (H2), (H3) with and (H4) hold, then the problem (1) has at least one solution on I.
Proof. Take, and consider
Let A and B the two operators defined on P by
respectively. Note that if then
By (H2), it is also clear that B is a contraction mapping.
Produced from Continuity of u, the operator is continuous in accordance with (H1). Also we observe that
Then A is uniformly bounded on P.
Now, let and That’s where f is bounded on the compact set it means
We will get
which is autonomous of u and head for zero as Consequently, A is equicontinuous. Thus, A is relatively compact on P. Therefore, according to the Arzela-Ascoli Theorem, A is compact on P. Thus, the problem (1) has at least one solution on I.
Example 4.1 Consider the following nonlocal problem
Let and Then we have
It is obviously that our assumptions in Theorem 3.1 holds with and for appropriate values of with and Indeed
Therefore the problem (3) has a unique solution on for values of and q sufficient stipulation (4). For illustration
・ If and then and
・ If and then and
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