Received 16 April 2016; accepted 26 June 2016; published 29 June 2016
![](//html.scirp.org/file/11-1100521x5.png)
1. Introduction
Importance of fractional differential equations appears in many of the physical and engineering phenomena in the last two decades [1] - [3] . Problems with nonlocal conditions and related topics were studied in, for example [4] , and the nonlocal Cauchy problem [5] . The attention of researchers subject of q-difference equations appeared in recent years [6] [7] . Initially, it was developed by Jackson [8] [9] . Noted recently the attention of many researchers is in the field of fractional q-calculus [10] [11] . Recently nonlocal fractional q-difference problems have aroused considerable attention [12] [13] .
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
(1)
Here,
is the Caputo fractional q-derivative of order
,
and ![](//html.scirp.org/file/11-1100521x10.png)
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 [14] .
Several authors have studied the semi-linear differential equations with nonlocal conditions in Banach space, [15] [16] . In [17] , 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.
(H2) ![](//html.scirp.org/file/11-1100521x14.png)
(H3)
is continuous and ![](//html.scirp.org/file/11-1100521x16.png)
(H4)
where ![](//html.scirp.org/file/11-1100521x18.png)
The problem (1) is then devolved to the following formula
(2)
See reference [18] 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
![]()
and satisfies ![]()
The q-derivative of a function
is here defined by
![]()
and
![]()
The q-integral of a function f defined in the interval
is provided by
![]()
Now, it can be defined an operator
, as follows
and ![]()
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 [7] - [10] for more details.
Definition 2.1. [19] Let
and f be a function defined on
. The fractional q-integral of the Riemann-Liouville type is
and
![]()
Definition 2.3. [19] 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. [20] Let
and
.Then, the following equality holds
![]()
Theorem 2.2. [18] [19] (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.
Theorem 3.1.
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
,
where ![]()
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.
Theorem 3.2.
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
![]()
and
![]()
respectively. Note that if
then
![]()
Thus ![]()
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
(3)
where ![]()
Set
![]()
and
![]()
Let
and
Then we have
![]()
and
![]()
It is obviously that our assumptions in Theorem 3.1 holds with
and for appropriate values of
with
and
Indeed
(4)
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 ![]()