Anti-periodic boundary value problems of fractional differential equations with the Riemann-Liouville fractional derivative
© Chai; licensee Springer. 2013
Received: 25 July 2013
Accepted: 11 October 2013
Published: 8 November 2013
In this paper, the author puts forward a kind of anti-periodic boundary value problems of fractional equations with the Riemann-Liouville fractional derivative. More precisely, the author is concerned with the following fractional equation:
with the anti-periodic boundary value conditions
where denotes the standard Riemann-Liouville fractional derivative of order , and the nonlinear function may be singular at . By applying the contraction mapping principle and the other fixed point theorem, the author obtains the existence and uniqueness of solutions.
where denotes the standard Riemann-Liouville fractional derivative of order , and the nonlinear function may be singular at .
Differential equations with fractional order are a generalization of ordinary differential equations to non-integer order. This generalization is not a mere mathematical curiosity but rather has interesting applications in many areas of science and engineering such as electrochemistry, control, porous media, electromagnetism, etc. (see [1–4]). There has been a significant development in the study of fractional differential equations in recent years; see, for example, [5–21].
and taking into account the consistency with integer order anti-periodic boundary value problems, we consider the anti-periodic boundary value condition (1.2) in the present paper to be more natural and suitable. It is noteworthy that such a form of anti-periodic boundary value conditions (1.2) is very convenient to construct an appropriate Banach space which coordinates the feature of the solution u because of the fact that () may occur and function may be singular at . Moreover, when in (1.2), the anti-periodic boundary value conditions in (1.2) are changed into , , which are coincident with anti-periodic boundary value conditions of second-order differential equations (see ).
The rest of this paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Section 3, we put forward and prove our main results. By applying the contraction mapping principle and the other fixed point theorem, we obtain the existence and uniqueness of solutions for boundary value problems (simply denoted by BVP). Finally, we give two examples to demonstrate our main results.
In this section, we introduce some preliminary facts which are used throughout this paper.
Let ℕ be the set of positive integers, ℝ be the set of real numbers.
Definition 2.1 ()
Definition 2.2 ()
where , denotes the integer part of α.
Lemma 2.1 ()
for some , , where .
respectively. Moreover, we have the following lemma.
Lemma 2.2 is a Banach space with the norm .
To summarize, with . □
We have the following lemma.
for some .
where , are given by (2.7) and (2.8), respectively.
The proof is complete. □
We first establish the following lemma.
Lemma 2.4 .
Thus, from (2.7)-(2.8), we know that the integrals and converge. So, from (2.12) and (2.22), it follows that exists on . That is, the operator T is well defined.
In what follows, we show that .
From , we know that and . Let , . Then and , , and so and by setting , .
Hence, formula (2.25) together with (2.13) implies that exists, and formula (2.26) together with (2.13)-(2.27) implies that exists.
Summing up the above analysis, we obtain that . The proof is complete. □
We need the following lemma, which is important in establishing our main result in the next section.
Lemma 2.5 is completely continuous.
Proof We divide the proof into two parts.
Part 1. First, we show that T is a continuous operator.
So, T is a continuous operator on .
Part 2. Now we show that T is a compact operator.
and so , where D is as in (2.33). It means that T Ω is bounded.
Now, we show that the set of functions B is equicontinuous on , where .
and , .
The above inequality shows that the set is equicontinuous on .
keeping in mind that .
when , .
- (1)If and with , then(2.40)
- (2)If , then(2.43)
when , noting that because .
when and .
The above inequality (2.49) shows that the set is equicontinuous on . As a consequence of the Arzela-Ascoli theorem, we have that T Ω is a compact set in . The proof is complete. □
Finally, for the remainder of this section, we give the following lemma, which will be used to obtain our main results.
Lemma 2.6 ()
Then A has a fixed point in Ω.
3 Main results
Let us introduce some assumptions which will be used throughout this paper.
for all , , and .
for all , , and .
where T is defined as before.
We first establish the following lemma to obtain our results.
Lemma 3.1 Suppose that (H1), (H2), (H3) hold. Then the operator A maps into .
and so . Thus, according to hypothesis (H1) and (3.4)-(3.5), we know that and . That is, . Therefore, in view of Lemma 2.5, it follows that . Thus, . The proof is complete. □
The following lemma is significant to obtaining the result in this article.
Lemma 3.2 Suppose that (H1) and (H4) hold. Then the operator is completely continuous.
Proof We first show that the operator F maps into under (H4).
noting that and .
Thus, according to hypothesis (H1), formulae (3.6)-(3.7) ensure that , noting that .
Now, we prove that the operator is completely continuous.
First of all, in view of Lemma 2.5, we know that the operator because of the fact that and .
It remains to show that the operator A is completely continuous. The following proof is divided into two steps.
Step 1. We show that the operator A is compact on .
Therefore, the set is bounded in , and therefore, TB is a compact set in view of Lemma 2.5. That is, A Ω is a compact set, owing to the fact that and . Hence, A is a compact operator.
Step 2. We show that the operator A is continuous on .
The following proof is divided into two parts.
holds for , because , , , observing (3.8).
when , noting that (3.11)-(3.12), where .
holds for .
because of the fact that and .
- (i)If , then from (3.8) it follows that(3.17)
- (ii)If , then(3.18)
when , where .
holds when .
Formulae (3.15), (3.24) yield that in . That is, the operator A is continuous on . The proof is complete. □
In view of Lemma 2.3, it is easy to know that is a solution of BVP (1.1)-(1.2) if and only if is a fixed point of the operator A. Therefore, we can focus on seeking the existence of a fixed point of A in .
Let , where , is given by (H3).
We are now in a position to state the first theorem in the present paper.
Theorem 3.1 Suppose that (H1)-(H3) hold. If , then BVP (1.1)-(1.2) has a unique solution.
As , A is a contraction mapping. So, by the contraction mapping principle, A has a unique fixed point . That is, is the unique solution of BVP (1.1)-(1.2). □
We give another result in this paper as follows.
Theorem 3.2 If (H1), (H4) hold, then BVP (1.1)-(1.2) has at least one solution.
Proof First, by Lemma 3.2, we know that is completely continuous.
Let , , , , , and . Put , .
noting that (2.28), (2.29) and keeping in mind the choice of R and .
noting that (2.28), (2.29), (2.31) and the choice of R as well as .
Summing up (3.28) and (3.29), we have that . That is, because . So, the relation (3.27) holds. As a consequence of Lemma 2.6, the operator A has at least one point . That is, is a solution of BVP (1.1)-(1.2). □