Abstract
In this paper, some Banach spaces are introduced. Based on these spaces and the coincidence degree theory, a 2mpoint boundary value problem for a coupled system of impulsive fractional differential equations at resonance is considered, and the new criterion on existence is obtained. Finally, an example is also given to illustrate the availability of our main results.
MSC: 34A08, 34B10, 34B37.
Keywords:
coupled system; impulsive fractional differential equations; at resonance; coincidence degree1 Introduction
Recently, Wang et al.[1] presented a counterexample to show an error formula of solutions to the traditional boundary value problem for impulsive differential equations with fractional derivative in [25]. Meanwhile, they introduced the correct formula of solutions for an impulsive Cauchy problem with the Caputo fractional derivative. Shortly afterwards, many works on the better formula of solutions to the Cauchy problem for impulsive fractional differential equations have been reported by Li et al.[6], Wang et al.[7], Fečkan [8], etc.
Fractional differential equations have been paid much attention to in recent years due to their wide applications such as nonlinear oscillations of earthquakes, Nutting’s law, charge transport in amorphous semiconductors, fluid dynamic traffic model, nonMarkovian diffusion process with memory etc.[911]. For more details, see the monographs of Hilfer [12], Miller and Ross [13], Podlubny [14], Lakshmikantham et al.[15], Samko et al.[16], and the papers of [2,1719] and the references therein.
In recent years, many researchers paid much attention to the coupled system of fractional differential equations due to its applications in different fields [2025]. Zhang et al.[25] investigated a threepoint boundary value problem at resonance for a coupled system of nonlinear fractional differential equations given by
where
where
There are significant developments in the theory of impulses especially in the area of impulsive differential equations with fixed moments, which provided a natural description of observed evolution processes, regarding as important tools for better understanding several real word phenomena in applied sciences [1,7,2629]. In addition, motivated by the better formula of solutions cited by the work of Zhou et al.[1,7,8], the aim of this work is to discuss a boundary value problem for a coupled system of impulsive fractional differential equation. Exactly, this paper deals with the 2mpoint boundary value problem of the following coupled system of impulsive fractional differential equations at resonance:
where
The coupled system (1.1) happens to be at resonance in the sense that the associated linear homogeneous coupled system
has
The plan of this work is organized as follows. Section 2 contains some necessary notations, definitions and lemmas that will be used in the sequel. In Section 3, we establish a theorem on the existence of solutions for the coupled system (1.1) based on the coincidence degree theory due to Mawhin [30,31].
2 Background materials and preliminaries
For the convenience of the readers, we recall some notations and an abstract existence theorem [30,31].
Let Y, Z be real Banach spaces,
The main tool we used is Theorem 2.4 of [30].
Theorem 2.1LetLbe a Fredholm operator of index zero, and letNbeLcompact on
(i)
(ii)
(iii)
Then the equation
Now, we present some basic knowledge and definitions about fractional calculus theory, which can be found in the recent works [13,16,32].
Definition 2.1 The fractional integral of order
provided the righthand side is pointwise defined on
Definition 2.2 The fractional derivative of order
where
Remark 2.1 It can be directly verified that the RiemannLiouville fractional integration and
fractional differentiation operators of the power functions
Proposition 2.1[17]
Assume that
for some
Proposition 2.2[32]
If
is satisfied for a continuous functiony.
If
If
is satisfied for a continuous functiony.
If
holds for a continuous functiony.
Let
with the norm
where
Thus,
Set
thus
Define the operator
with
Let
Then the coupled system of boundary value problem (1.1) can be written as
For the sake of simplicity, we define the operators
By the same way, we define the operators
In what follows, we present the following lemmas which will be used to prove our main results.
Lemma 2.1If the following condition is satisfied:
(H_{1})
then
and
Proof It is clear that (2.7) holds. For
If
and
Proposition 2.1 together with (2.10)(2.12) gives that
Substituting the boundary condition
and substituting the boundary condition
By the same way, if we substitute the condition (2.12) into (2.14), then we can obtain that
and
Conversely, if (2.15)(2.18) hold, set
It is easy to check that the above u, v satisfy equation (2.10)(2.12). Thus, (2.8) and (2.9) hold.
Define the operator
In what follows, we will show that
As a result,
Similarly, we can see that
Now, we show that
The condition (H_{1}) guarantees that
For
Define the operator
Moreover, we define
Lemma 2.2Assume that
Proof Obviously,
Similarly,
In what follows, we will show that
If
On the other hand, for
Since
By some calculations, (2.19) and (2.20) imply that
It means that
Finally, we show that N is Lcompact on
Then we can see that
where
So, we can see that
For
The equicontinuity of
3 Main results
In this section, we present the existence results of the coupled system (1.1). To do this, we need the following hypotheses.
(H_{2}) There exist functions
where
here
(H_{3}) For
(1) if either
(2) if either
(H_{4}) For
(1)
here
(2)
here
Lemma 3.1Suppose that (H_{2})(H_{3}) hold. Then the set
is bounded inY.
Proof For
Since
and
Then for
Similarly, for
Substitute (3.11) and (3.12) into (3.8), then we have
It means that
similarly,
Substituting the above two into (3.9) and (3.12), we can see that
and
From the condition (H_{2}), (3.14) and (3.15) give that
Lemma 3.2Suppose that the condition (H_{3}) holds. Then the set
is bounded inY.
Proof For
and
From (H_{3}), there exist positive constants
which means that
which means that
The above two arguments imply that
Lemma 3.3The set
is bounded inY, where
and
Proof For
From (3.16) and (3.17), we have
the condition (H_{4}) gives that
where
which is a contradiction. As a result, there exist positive constants
Theorem 3.1Suppose that (H_{1})(H_{4}) hold. Then the problem (1.1) has at least one solution inY.
Proof Let Ω be a bounded open set of Y such that
(i)
(ii)
Then we need only to prove
(iii)
Take
According to Lemma 3.3, we know
Then, by Theorem 2.1,
4 An example
Example 4.1
Consider the following boundary value problem for coupled systems of impulsive fractional differential equations:
where
and
Due to the coupled problem (1.1), we have that
It is easy to see that
where
So,
where
Taking
If
Similarly, assume that
If
So, from the above arguments, the first part of the condition (H_{3}) is true for
Taking
If
By the same way, taking
If
So, from the above arguments, the second part of the condition (H_{3}) holds for
On the other hand, for
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CZ conceived the main idea of the study, XZ carried out the main parts of the draft. CZ gave many valuable suggestions and corrected the main theorems in the discussion. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editor and referee for their valuable comments and remarks which lead to a great improvement of the article. This research is supported by the National Natural Science Foundation of China (11071108), the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147, 20114BAB201007).
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