Received 19 December 2015; accepted 23 January 2016; published 26 January 2016
This paper deals with the existence of solutions for the following fractional boundary value problem with q-difference
where, and may be singular at (and/or).
For problem (1.1), there have been paid attention to the existences of solutions. Rui  investigated the exi- stence of positive solutions by applying a fixed point theorem in cones. By fixed point theorem again, Li and Han  considered a similar fractional q-difference equations given as
subject to the boundary conditions. In this work, we will apply the iterative technology (    ), and as far as we know, there are few papers to establish the existence of solutions by the iterative technology for the boundary value problem with q-difference.
Motivated by the work mentioned above, with the iterative technology and properties of, explicit iterative sequences are given to approximate the solutions and the error estimations are also given.
2. Preliminaries and Some Lemmas
In this section, we introduce some definitions and lemmas.
Definition 2.1  . Let, and f be a function defined on. The fractional q-integral of the Riemann-Liouville type is defined by and
The q-integral of a function f defined in the interval is given by
and q-integral of higher order is defined by
Remark 1:,. The q-gamma function is defined by , , and satisfies, where,.
Definition 2.2  . Let,. The fractional q-derivative of the Riemann-Liouville type of order
is defined by and
where m is the smallest integer greater than or equal to. The q-derivative of a function f is defined by
and q-derivatives of higher order by
Lemma 2.1  . Suppose and is q-integrable on. Then the boundary value problem
has the unique solution
Lemma 2.2  . Function G defined as (2.2). Then G satisfies the following properties:
(1), and for all.
(2) for all.
Lemma 2.3. Function G defined as (2.2). Then
Proof. Note that (2.2) and, it follows that for all. This, with Lemma 2.2, implies that
3. Main Result
First, for the existence results of problem (1.1), we need the following assumptions.
(A1) is continuous.
(A2) For, f is non-decreasing respect to x and for any, there exists a constant such that
Then, we let the Banach space, and
Clearly P is a normal cone and Q is a subset of P in the Banach space E.
In what follows, we define the operator
where are given by (2.1) and (2.2).
Now, we are in the position to give the main results of this work.
Theorem 3.1. Suppose (A1), (A2) hold. Then problem (1.1) has at least one positive solution in Q if
Proof. We shall prove the existence of solution in three steps.
Step 1. The operator T defined in (3.2) is.
For any, there exists a positive constant such that
Then from (A2): is non-decreasing respect to x and (3.1), we can imply that for
is implied by the equivalent form to (3.1): if,
From (3.4) and Lemma 2.3, we can have
This implies T is.
Step 2. There exist iterative sequences, satisfying
Since for, there exists a constant such that
For defined in (3.5), there exist constants satisfying
Then it follows that
In fact, from (3.6)-(3.8) , we have
Then, by (3.9)-(3.11), (A2) and induction, the iterative sequences, satisfy
Step 3. There exists such that
Note that. By induction it is easy to obtain
Thus, for we have
This yields that there exists such that
Moreover, from (3.12) and
Letting in (3.8), is a fixed point of T. That is, is a positive solution of problem (1.1).
Theorem 3.2. Suppose the conditions hold in Theorem 3.1. Then for any initial, there exists a se- quence such that uniformly on as, where is the positive solu- tion of problem (1.1). And the error estimation for the sequence is
where k is a constant with and determined by.
Proof. Let be given. Then there exists a constant such that
For defined in (3.14), choose constants such that
Then define as (3.7), (3.8), and we can have converges uniformly to the positive solution of problem (1.1) on as.
For the error estimation (3.13), it can be obtained by letting in (3.12).
Example 3.3. Consider the function
satisfies (A2) and is singular at. Let,. Then
By Theorem 3.1, the following problem
has at least one positive solution.
The author is grateful to the referees for their valuable comments and suggestions.
Project supported by Program for Scientific research innovation team in Colleges and universities of Shandong Province, the Doctoral Program Foundation of Education Ministry of China (20133705110003), the Natural Science Foundation of Shandong Province of China (ZR2014AM007), the Natural Science Foundation of China (11571197).
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