Abstract
In this paper, the author is concerned with the fractional equation
with the anti-periodic boundary value conditions
where denotes the Caputo fractional derivative of order γ, the constants α, , , , satisfy the conditions , , . Different from the recent studies, the function f involves the Caputo fractional derivative and . In addition, the author put forward new anti-periodic boundary value conditions, which are more suitable than those studied in the recent literature. By applying the Banach contraction mapping principle and the Leray-Schauder degree theory, some existence results of solutions are obtained.
MSC:34A08, 34B15.
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1 Introduction
In the present paper, we are concerned with the existence of solutions for the fractional differential equation
with anti-periodic boundary value conditions
where denotes the Caputo fractional derivative of order γ, the constants α, , , , satisfy conditions , , , and f is a given continuous function.
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, etc. (see [1–5]). There has been a significant development in the study of fractional differential equations and inclusions in recent years; see the monographs of Podlubny [5], Kilbas et al. [6], Lakshmikantham et al. [7], Samko et al. [8], Diethelm [9], and the survey by Agarwal et al. [10]. For some recent contributions on fractional differential equations, see [11–30] and the references therein.
Anti-periodic boundary value problems occur in the mathematic modeling of a variety of physical processes and have recently received considerable attention. For examples and details of anti-periodic fractional boundary conditions, see [16–22]. In [16], Agarwal and Ahmad studied the solvability of the following anti-periodic boundary value problem for nonlinear fractional differential equation:
where denotes the Caputo fractional derivative of order α. The existence results are obtained by nonlinear alternative theorem.
In [17], Wang, Ahmad, Zhang investigated the following impulsive anti-periodic fractional boundary value problem:
where denotes the Caputo fractional derivative of order α. By applying some well-known fixed point principles, some existence and uniqueness results are obtained.
In [18], Ahmad, Nieto studied the following anti-periodic fractional boundary value problem:
where denotes the Caputo fractional derivative of order q. By applying some standard fixed point principles, some existence and uniqueness results are obtained.
In [19], Wang and Liu considered the following anti-periodic fractional boundary value problem:
By using Schauder’s fixed point theorem and the contraction mapping principle, some existence and uniqueness results are obtained.
By careful analysis, we have found that the anti-periodic boundary value condition () in equations (1.5) and (1.6) actually is equivalent to the boundary value condition (). It means that, in a sense, in (1.5)-(1.6), the feature of anti-periodicity partially disappears. So, in the present paper, we put forward new anti-periodic boundary value conditions (1.2) so that the anti-periodicity is expressed. In fact, when , , the anti-periodic boundary value conditions in (1.2) are changed into the boundary value conditions
which are coincident with anti-periodic boundary value conditions (1.3) and (1.4) mentioned above. So, the anti-periodic boundary value conditions in (1.2) in the present paper are more suitable than those in (1.5) and (1.6). Moreover, different from the literature mentioned above, the function f in (1.1) involves the Caputo fractional derivative and , which brings more difficulty to the study. To investigate the existence, researchers often equip a Banach space with the norm . However, if such a norm was taken in the study, the introduced conditions would get more complex. So, we take the norm by finding some implicit relations. As a result, the conditions introduced are quite simple. By applying the Banach contraction mapping principle and the Leray-Schauder degree theory, some existence results of solutions are obtained in this paper.
The organization of this paper is 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. Finally, we give two examples to demonstrate our main results.
2 Preliminaries
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 ([6])
The Riemann-Liouville fractional integral of order of a function is given by
Definition 2.2 ([6])
The Riemann-Liouville fractional derivative of order of a function is given by
where , denotes the integer part of α.
Definition 2.3 ([6])
The Caputo fractional derivative of order of a function y on is defined via the above Riemann-Liouville derivatives by
Lemma 2.1 ([6])
Let and . Then
holds on .
Lemma 2.2 ([6])
If and , then
where .
Lemma 2.3 ([13])
Let with , . If and , then
holds on .
Let . It is well known that X is a Banach space endowed with the norm , where .
For any , from , there exists a such that . So, from the fact that , , it follows that , . Thus, , and so .
In what follows, we regard X as the Banach space with the norm
We have the following lemma.
Lemma 2.4 For a given , the function u is a solution of the following anti-periodic boundary value problem:
If and only if is a solution of the integral equation
Proof Let be a solution of (2.1). Then, by Lemma 2.3, we have
for some . Furthermore,
From (2.4)-(2.5), we have
Now, we show that
In fact, since , there exists an such that for all . Then, from the fact that , we have
Thus,
and so
noting that .
Similarly, we also have
and so
noting that .
So, from (2.6)-(2.7), we have
Thus, by the boundary value condition in (2.1), combined with (2.3), (2.8)-(2.9), we have
From (2.10)-(2.12), we have
Substituting (2.13)-(2.15) into (2.3), we obtain
That is, u satisfies (2.2).
Conversely, if u is a solution of the fractional integral equation (2.2), then by finding the second derivative for both sides of (2.2), we have
Noting that , it follows from (2.16) that . Again, by Lemma 2.1 and (2.2), a direct computation shows that the solution given by (2.2) satisfies (2.1). This completes the proof. □
Now, we define the operator A as
for .
From the proof of Lemma 2.4, we know that the operator A maps into X.
Now, we establish the following lemma, which will play an important role in the forthcoming analysis.
Lemma 2.5 For any and , we have
-
(i)
, and so ;
-
(ii)
, and so ;
-
(iii)
, and so .
Proof Conclusion (i) has been proved as before. We come to show that conclusions (ii)-(iii) are true. Obviously, when , , the conclusions are true. So, we only consider the case . In fact, by Lemma 2.2, for any , we have
So,
Similarly, we have that . □
We also need the following lemmas.
Lemma 2.6 ([31])
Let X be a Banach space. Assume that Ω is an open bounded subset of X with , and is a completely continuous operator such that
Then T has a fixed point in .
Lemma 2.7 (Leray-Schauder [31])
Let X be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then T has a fixed point in X.
3 Main results
We list the following hypotheses which will be used in the sequel:
() .
() , , .
() There exist constants , , and such that
for , , and .
First, we establish the following lemma to obtain our main results.
Lemma 3.1 Assume that ()-() hold. Then the operator is completely continuous, where T is defined by
and the operator A is given by (2.17).
Proof For any , we have that . Then from the hypothesis (). Thus, from (2.17) and the proof of Lemma 2.4.
First, by a direct computation, we know that the following relations hold:
Now, we show that T is a compact operator.
Let V be an arbitrary bounded set in X. Then there exists an such that . Thus, by Lemma 2.5, it follows that , , for all and .
So, by the hypothesis (), there exists an such that
Consequently, by (3.1), (3.3) and observing that , , we have
and so
Similarly, by (3.2), (3.3), we have
Thus,
That is, TV is uniformly bounded.
Now, we show that TV is equicontinuous.
In fact, for any and with , since , , from (3.1) it follows that
and
So, inequalities (3.6)-(3.7) imply that TV is equicontinuous. By the Arzela-Ascoli theorem, T is a compact operator.
Finally, we prove that T is continuous.
Assume that is an arbitrary sequence in X with , . Then there is an such that , , , , and so
from Lemma 2.5.
On the other hand, for an arbitrary , there is a such that
for all , , , with , , , because of the uniform continuity of f on
In view of the fact that , there is an such that
when .
Thus, from (3.1)-(3.2) together with (3.8)-(3.9), by a similar deducing as (3.4)-(3.5), we have
and
Hence,
That is, T is continuous in X. □
To state our main results in this paper, we first introduce some notations for convenience.
Let , .
Set , , , where () are given in ().
We are in a position to state the first result in the present paper.
Theorem 3.1 Suppose ()-() hold. If , then BVP (1.1)-(1.2) has a unique solution.
Proof For any , by () and Lemma 2.5, we have
So, it follows from (3.1) that
Thus, observing that , we have
Similarly,
Thus,
As , T is a contraction mapping. So, by the contraction mapping principle, T has a unique fixed point u. That is, u is the unique solution of BVP (1.1)-(1.2) by Lemma 2.4. □
Our next existence result is based on Lemma 2.6.
Theorem 3.2 Assume that ()-() hold. If , then BVP (1.1)-(1.2) has at least one solution, where .
Proof By Lemma 3.1, we know that is completely continuous. Again, in view of , there exists an such that
when .
Take , where . Set . For any , we have that with . Thus, , , and . So, Lemma 2.5 ensures that
Therefore,
Thus, from (3.10), it follows that
Thus, by a similar deducing to that in (3.4) and (3.5), we have
and
Thus, , where . As , we have that .
So, by virtue of Lemma 2.6, T has at least one fixed point u. That is, u is a solution of BVP (1.1)-(1.2) by Lemma 2.4. The proof is complete. □
The last result of this section is based on the Leray-Schauder fixed point theorem, namely Lemma 2.7.
Theorem 3.3 Suppose that ()-() hold. If , then BVP (1.1)-(1.2) has at least one solution, where .
Proof As before, is completely continuous. From , we can choose a such that . Then there is an such that
holds when for .
Let . Then we always have
Set . Now, we show that V is a bounded set.
In fact, for any , from Lemma 2.5 and (3.11), it follows that
Thus, by (3.1) combined with (3.12) and observing that , we have immediately
Similarly, we have
So, from (3.13)-(3.14), we have
Now, the relation with implies
Because , the above inequality implies . That is, V is a bounded set. So, by Lemma 2.7, we have that T has at least one fixed point u. That is, u is a solution of BVP (1.1)-(1.2) by Lemma 2.4. This completes the proof. □
Example 3.1
Consider the following anti-periodic boundary value problem:
where , , , , , and . Clearly, the function satisfies . Further, , . As , all the assumptions of Theorem 3.1 are satisfied. Hence BVP (3.15) has a unique solution.
Example 3.2
Consider the following anti-periodic boundary value problem:
where , , , , , and . Clearly, the function satisfies , where . Further, , , . As , all the assumptions of Theorem 3.2 are satisfied. Hence BVP (3.16) has at least one solution.
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Acknowledgements
The author sincerely thanks the anonymous referees for their valuable suggestions and comments which have greatly helped improve this article. Supported by the Natural Science Foundation of Hubei Provincial Education Department (D20102502).
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Chai, G. Existence results for anti-periodic boundary value problems of fractional differential equations. Adv Differ Equ 2013, 53 (2013). https://doi.org/10.1186/1687-1847-2013-53
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DOI: https://doi.org/10.1186/1687-1847-2013-53