Abstract
In this paper, we study threepoint boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative:
where
MSC: 34A08, 34K37.
Keywords:
fractional functional differential equation; delay; threepoint boundary value problems; fixed point theorem; existence of solutions1 Introduction
Fractional calculus is a branch of mathematics, it is an emerging field in the area of the applied mathematics that deals with derivatives and integrals of arbitrary orders as well as with their applications. The origins can be traced back to the end of the seventeenth century. During the history of fractional calculus, it was reported that the pure mathematical formulations of the investigated problems started to be addressed with more applications in various fields. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. Therefore, fractional differential equations have received much attention and the theory and its application have been greatly developed; see [16].
Recently, there have been many papers focused on boundary value problems of fractional ordinary differential equations [715] and an initial value problem of fractional functional differential equations [1628]. But the results dealing with the boundary value problems of fractional functional differential equations with delay are relatively scarce [2935]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past. Thus, in many cases, we must consider fractional functional differential equations with delay in order to solve practical problems. Consequently, our aim in this paper is to study the existence of solutions for boundary value problems of fractional functional differential equations.
In 2011, Rehman [12] studied the existence and uniqueness of solutions to nonlinear threepoint boundary value problems for the following fractional differential equation:
where
For
If
Enlightened by literature [12], in this paper we study the following threepoint boundary value problem for the fractional functional differential equation:
where
and
To the best of our knowledge, no one has studied the existence of positive solutions
for problem (1.1)(1.2). The aim of this paper is to fill the gap in the relevant
literatures. In this paper, we firstly give the fractional Green function and some
properties of the Green function. Consequently, boundary value problem (1.1) and (1.2)
is reduced to an equivalent Fredholm integral equation. Then we extend the existence
results for boundary value problems of an ordinary fractional differential equation
of δorder (
2 Preliminaries
For the convenience of the reader, we give the following background material from fractional calculus theory to facilitate the analysis of boundary value problem (1.1) and (1.2). This material can be found in the recent literature; see [1,2,36].
Definition 2.1 ([1])
The fractional integral of order α (
where
Definition 2.2 ([1])
The Caputo fractional derivative of order α (
where
Obviously, the Caputo derivative for every constant function is equal to zero.
From the definition of the Caputo derivative, we can acquire the following statement.
Lemma 2.1 ([2])
Let
Lemma 2.2 ([2])
Let
for some
Next, we introduce the Green function of fractional functional differential equations boundary value problems.
Lemma 2.3Let
has a unique solution
where
Proof From equation (2.1), we know
From Lemma 2.2, we have
According to (2.1), we know that
By
Therefore,
Now, for
For
Hence, we can conclude (2.2) holds, where
The proof is completed. □
Lemma 2.4 ([36] Schauder fixed point theorem)
Let
3 Main results
In this section, we discuss the existence and uniqueness of solutions for boundary value problem (1.1) and (1.2) by the Schauder fixed point theorem and the Banach contraction principle.
For convenience, we define the Banach space
For
Thus, we have
Since
We define an operator
and
Theorem 3.1Assume the following:
(H_{1}) There exists a nonnegative function
for each
(H_{2}) There exists a nonnegative function
for each
Then boundary value problem (1.1) and (1.2) has a solution.
Proof Suppose (H_{1}) holds. Choose
and define the cone
For any
Also,
Hence,
In view of (3.1) and (3.3), we obtain
which implies that
Now, if (H_{2}) holds, we choose
and by the same process as above, we obtain
which implies that
Now, we show that T is a completely continuous operator.
Let
If
If
Hence, if
If
If
Hence, if
If
If
In any case, it implies that
For convenience, we denote
Theorem 3.2Assume that
(H_{3}) There exists a constant
then boundary value problem (1.1) and (1.2) has a unique solution.
Proof Consider the operator
By a similar method, we get
In view of the definition of
Hence,
Clearly, for each
and T is a contraction. As a consequence of the Banach contraction principle, we get that T has a fixed point which is a solution of boundary value problem (1.1) and (1.2). □
4 Example
In this section, we will present some examples to illustrate our main results.
Example 4.1
Consider boundary value problems of the following fractional functional differential equations:
where
Choose
Then, for
For
Example 4.2
Consider boundary value problems of the following fractional functional differential equations:
where
Choose
Set
Let
For each
Thus the condition (H_{3}) holds with
By
and
It implies that
Then by Theorem 3.2, boundary value problem (4.3) and (4.4) has a unique solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).
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