Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional q-Difference Equation ()
1. Introduction
Considering the following boundary value problem of nonlinear fractional q-difference equation:
(P)
where 
is a nonnegative continuous function and 
is the fractional q-derivative of the Riemann-Liouville type.
Fractional differential calculus is a discipline to which many researchers are dedicating their time, perhaps because of its demonstrated applications in various fields of science and engineering [1]. Recently, there are many papers dealing with the boundary value problem of fractional differential equations, see [2-5] and references therein.
The q-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [6,7], and basic definitions and properties of q-difference calculus can be found in [8]. The fractional q-difference calculus had its origin in the works by Al-Salam [9] and Agarwal [10]. More recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional qdifference calculus were made, see [11,12].
The question of the existence of positive solutions for fractional q-difference boundary value problems is in its infancy, see [13-16]. No contributions exist, as far as we know, concerning the existence of positive solutions for problem (P).
This paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we discuss the existence of positive solutions for problem (P).
2. Preliminaries
Let 
and define
(2.1)
The q-analogue of the power function 
with 
is
(2.2)
More generally, if
, then
(2.3)
Note that, if 
then
. The q-gamma function is defined by
(2.4)
and satisfies 
The q-derivative of a function 
is here defined by
(2.5)
and q-derivative of higher order by
(2.6)
The q-integral of a function 
defined in the interval 
is given by
(2.7)
If 
and 
defined in the interval
, its integral from 
to 
is defined by
(2.8)
Remark 2.1. (see [17]) If 
and 
on
, then
Similarly as done for derivatives, an operator 
can be defined, namely,
(2.9)
The fundamental theorem of calculus applies to these operators 
and
, i.e.,
(2.10)
and if 
is continuous at
, then
(2.11)
Basic properties of the two operators can be found in [14]. We now point out three formulas that will be used later (
denotes the derivative with respect to variable
)
Remark 2.2. (see [14]) We note that if 
and
, then
(2.12)
Definition 2.3. (see [10]) Let 
and 
be a function defined on
. The fractional q-integral of the Riemann-Liouville type is 
and
(2.13)
Definition 2.4. (see [14-16]) The fractional q-derivative of the Riemann-Liouville type of order 
is defined by 
and
(2.14)
where 
is the smallest integer greater than or equal to
.
Next, we list some properties that are already known in the literature.
Lemma 2.5. (see [14-16]) Let 
and 
be a function defined on
, Then, the next formulas hold:
1) 
2) 
Lemma 2.6. (see [14-16]) Let 
and 
be a positive integer. Then, the following equality holds:
(2.15)
Let
, in view of Lemma 2.5 and Lemma 2.6, we see that
for some constants 
Using the boundary condition 
we have 
Differentiating both side of the above equality, one gets
(2.16)
Using the boundary condition
, we have 
similarly, we have 
From
(2.17)
and boundary value problem
, one can obtain
(2.18)
Putting all things together we finally have
(2.19)
If we define a function 
by
(2.20)
Hence, in order to solve the problem (P), it is sufficient to find positive solutions of the following integral equation
(2.21)
Some properties of the function 
needed in the sequel are now stated and proved.
Lemma 2.7. Function 
defined above satisfies the following conditions:
(2.22)
(2.23)
Proof. Let
(2.24)
and
(2.25)
It is clear that
. Now,
. For
, in view of (2.3) and Remark 2.2, we have
(2.26)
Therefore,
. Moreover,
(2.27)
i.e., 
is an increasing function of x. Obviously, 
is increasing in x, therefore, 
is an increasing function of x for fixed 
This concludes the proof of (2.22).
Suppose now that 
then
(2.28)
If
, then
(2.29)
and this finishes the proof of (2.23).
Let 
be the Banach space endowed with norm 
Define the cone 
by
It follows from the non-negativeness and continuity of 
and 
that the operator 
defined by
(2.30)
is completely continuous [18]. Moreover, for 
in view of (2.22) and (2.23), we have 
on 
and
(2.31)
that is 
Lemma 2.8. (see [19]) Let 
be a Banach space, 
a cone, and 
two bounded open balls of 
centered at the origin with
. Suppose that 
is a completely continuous operator such that either 1) 
or 2) 
holds. Then 
has a fixed point in 
3. Main Results
Let 
where 
will be defined later.
Theorem 3.1. Suppose that 
is a nonnegative continuous function on
. In addition, suppose that one of the following two conditions holds:
(H1) 
(H2) 
Then problem (P) has at least one positive solution.
Proof. Note that the operator 
is completely continuous. Now, assume that condition (H1) holds. Since
, there exists an 
such that
(3.1)
where the constant 
such that
(3.2)
Thus,
This, together with the definitions of 
and Lemma 2.7, implies that for any 
(3.3)
That is, for any 
On the other hand, from
it follows that there exists a 
such that
(3.4)
where the constant 
satisfies
(3.5)
Let
. Then we have
(3.6)
Let 
and
, then
(3.7)
Thus, the operator 
satisfies condition of Lemma 2.8. Consequently, the operator 
has at least one fixed point
, which is one positive solution of the problem (P).
Next, we suppose that condition (H2) holds. The proof is similar to that of the case in which (H1) holds and will only be sketched here. Let 
with
. Select two positive constants 
with 
and
, respectively. Then, there exist two positive constants 
and 
such that
(3.8)
(3.9)
It follows from that for
,
(3.10)
In addition, let
. If
, then
(3.11)
and
(3.12)
In view of Remark 2.1, for
, we have
(3.13)
Thus, 
for
. Consequently, the operator 
has at least one fixed point 
, which is one positive solution of the problem (P).
Example 3.2
where 
Obviously,
Thus, by the first part of Theorem 3.1, we can get that the problem above has at least one positive solution.
NOTES