Abstract
A class of nonlinear fractional multipoint boundary value problems at resonance is considered in this article. The existence results are obtained by the method of the coincidence degree theory of Mawhin. An example is given to illustrate the results.
MSC: 34A08.
Keywords:
coincidence degree; fractional differential equation; resonance; multipoint boundary conditions1 Introduction
The subject of fractional calculus has gained considerable popularity during the past decades, due mainly to its frequent appearance in a variety of different areas such as physics, aerodynamics, polymer rheology, etc. (see [13]). Many methods have been introduced for solving fractional differential equations (FDEs for short in the remaining), such as the Laplace transform method, the iteration method, the Fourier transform method, etc. (see [4]).
Recently, there have been many works related to the existence of solutions for multipoint boundary value problems (BVPs for short in the remaining) at nonresonance of FDEs (see [511]). Motivated by the above articles and recent studies on FDEs (see [1219]), we consider the existence of solutions for a nonlinear fractional multipoint BVPs at resonance in this article.
In [16], Zhang and Bai considered the following fractional threepoint boundary value problems at resonance:
where
In 2010, Bai and Jiang studied the fractional differential equation of boundary value
problems at resonance with the case of
is referred to as a condition in [18]. We will show that the assumption like above
In this article, we will use the coincidence degree theory to study the existence of solutions for a nonlinear FDEs at resonance which is given by
with boundary conditions
where
BVPs (1.1)(1.2) being at resonance means that the associated linear homogeneous equation
We will always suppose that the following conditions hold:
The rest of this article is organized as follows: In Section 2, we give some definitions, lemmas and notations. In Section 3, we establish theorems of existence result for BVPs (1.1)(1.2). In Section 4, we give an example to illustrate our result.
2 Preliminaries
We present here some necessary basic knowledge and definitions of the fractional calculus theory, which can be found in [13].
Definition 2.1 The RiemannLiouville fractional integral of order
where
Definition 2.2 The RiemannLiouville fractional derivative of order
where
Definition 2.3 ([18])
We say that the map
(i) for each
(ii) for almost every
(iii) for each
Lemma 2.4 ([2])
Assume
Lemma 2.5 ([2])
Let
Now, we briefly recall some notations and an abstract existence result, which can
be found in [20]. Let YZ be real Banach spaces,
It follows that
Lemma 2.6 ([20])
LetLbe a Fredholm operator of index zero andNbeLcompact on
(i)
(ii)
(iii)
where
In this article, we use the Banach space
Lemma 2.7 ([16])
Given
with the norm defined by
Lemma 2.8 ([16])
and equicontinuation means that there exists a
In this article, let
where
Thus, BVP (1.1) can be written as
3 Main results
First, let us introduce the following notations for convenience, with setting
Then, let us make some assumptions which will be used throughout the article.
(H1) There exist functions
(H2) For any
(H3) For any
Theorem 3.1If conditions (C), (H1)(H3) hold, then BVPs (1.1)(1.2) have at least one solution provided that
In order to obtain our main result, we first present and prove Lemmas 3.23.8. Now,
let us define operators
Lemma 3.2If condition (C) holds andLis defined by (2.1), then
Proof By (2.1) and Lemma 2.5,
Combining with the condition (1.2), we get
Suppose
Then in view of condition (C), (1.2) and Lemma 2.4, x satisfies
On the other hand, suppose
□
Lemma 3.3If condition (C) holds, then there exist two constants
Proof From
If
It is equal to
Since the determinant of coefficients is not equal to zero, we have that
If
Similarly, we can deduce that the determinant of coefficients is not equal to zero,
so we have that
Similarly, from
Let
we shall prove that S is a finite set. If else, there exists a strict increasing sequence
Since
which is a contradiction to (3.2). Therefore, there exists two constants
Lemma 3.4If the condition (C) holds andLis defined by (2.1), thenLis a Fredholm operator of index zero. Define the linear operator
Proof For each
where
It is clear that
similarly, we can derive that
Hence, for each
Furthermore, Q is a continuous linear projector.
For each
However, the determinant of coefficients is as follows
then we have
Take any
That is,
but the determinant of coefficients is as follows
we can deduce that
Let operator
It is easy to calculate that
It is clear that
For any
It follows from Lemma 2.4 that
Then, we have
By the definition of the norm in space Y, we get
Lemma 3.5Assume
Proof In order to prove N is Lcompact, we only need to prove that
For all
and
Since
Lemma 3.6Suppose (H1)(H3) hold, then the set
Proof Taking any
Furthermore, we have that, with setting
By (3.7)(3.9) and Lemma 2.4, we have that
As before, for any
Furthermore, we have
By (H1) and the definition of N, we have
where
which yield that
Furthermore, from the previous inequalities, we know that
Since
Therefore,
Lemma 3.7Suppose (H2) and (H3) hold, then the set
Proof For any
Lemma 3.8If the first parts of (H2) and (H3) hold, then the set
Proof Taking any
By the definition of the set
If
By the first parts of (H2) and (H3), similar to the proof of Lemma 3.7, then
Therefore,
If
If
Remark 3.9 If the other parts of (H2) and (H3) hold, then the set
Now with Lemmas 3.23.8 in hands, we can begin to prove our main result  Theorem 3.1.
Proof of Theorem 3.1 Assume that Ω is a bounded open set of Y with
(i)
(ii)
Finally, we will prove that (iii) of Lemma 2.6 is satisfied. We let I as the identity operator in the Banach space Y and
so (iii) of Lemma 2.6 is satisfied.
Consequently, by Lemma 2.6, the equation
According to Theorem 3.1, we have the following corollary.
Corollary 3.10Suppose that (H1) is replaced by the following condition,
(H4) there exist functions
and the others in Theorem 3.1 are not changed, then BVPs (1.1)(1.2) have at least one solution.
4 An example
Example Consider the following boundary value problem for all
Let
Thus, we have
Taking
By Corollary 3.10, the BVP (4.1) has at least one solution in
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
NX designed all the steps of proof in this research and also wrote the article. WBL suggested many good ideas in this article. LSX helped to draft the first manuscript and gave an example to illustrate our result. All authors read and approved the final manuscript.
Acknowledgement
The authors would like to acknowledge the anonymous referee for many helpful comments and valuable suggestions on this article. This work is sponsored by Fundamental Research Funds for the Central Universities (2012LWB44).
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