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.
(A1)
(A2) 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 (A1), (A2) 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 (A2):
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), (A2) 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
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