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In this paper, we apply the iterative technology to establish the existence of solutions for a fractional boundary value problem with
*q*-difference. Explicit iterative sequences are given to approxinate the solutions and the error estimations are also given.

This paper deals with the existence of solutions for the following fractional boundary value problem with q-difference

where

Fractional differential equations have been of great interest recently because of their intensive applications in economics, financial mathematics and other applied science (see [

For problem (1.1), there have been paid attention to the existences of solutions. Rui [

subject to the boundary conditions

Motivated by the work mentioned above, with the iterative technology and properties of

In this section, we introduce some definitions and lemmas.

Definition 2.1 [

The q-integral of a function f defined in the interval

and q-integral of higher order

Remark 1:

Definition 2.2 [

where m is the smallest integer greater than or equal to

and q-derivatives of higher order by

Lemma 2.1 [

has the unique solution

where

Lemma 2.2 [

(1)

(2)

Lemma 2.3. Function G defined as (2.2). Then

Proof. Note that (2.2) and

First, for the existence results of problem (1.1), we need the following assumptions.

(A_{1})

(A_{2}) For

Then, we let the Banach space

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

Now, we are in the position to give the main results of this work.

Theorem 3.1. Suppose (A_{1}), (A_{2}) hold. Then problem (1.1) has at least one positive solution

Proof. We shall prove the existence of solution in three steps.

Step 1. The operator T defined in (3.2) is

For any

Then from (A_{2}):

where

is implied by the equivalent form to (3.1): if

From (3.4) and Lemma 2.3, we can have

and

where

This implies T is

Step 2. There exist iterative sequences

Since

For

Let

Then it follows that

In fact, from (3.6)-(3.8) , we have

Then, by (3.9)-(3.11), (A_{2}) and induction, the iterative sequences

Step 3. There exists

Note that

Thus, for

This yields that there exists

Moreover, from (3.12) and

we have

Letting

Theorem 3.2. Suppose the conditions hold in Theorem 3.1. Then for any initial

where k is a constant with

Proof. Let

For

Then define

For the error estimation (3.13), it can be obtained by letting

Example 3.3. Consider the function

_{2}) and is singular at

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).

XiuliLin,ZengqinZhao,YongliangGuan, (2016) Iterative Technology in a Singular Fractional Boundary Value Problem with q -Difference. Applied Mathematics,07,91-97. doi: 10.4236/am.2016.71008