In this paper, some Banach spaces are introduced. Based on these spaces and the coincidence degree theory, a 2m-point 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 degree
Recently, Wang et al. presented a counterexample to show an error formula of solutions to the traditional boundary value problem for impulsive differential equations with fractional derivative in [2-5]. 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., Wang et al., Fečkan , 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, non-Markovian diffusion process with memory etc.[9-11]. For more details, see the monographs of Hilfer , Miller and Ross , Podlubny , Lakshmikantham et al., Samko et al., and the papers of [2,17-19] 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 [20-25]. Zhang et al. investigated a three-point boundary value problem at resonance for a coupled system of nonlinear fractional differential equations given by
where , , , , is the standard Riemann-Liouville fractional derivative and are continuous. And Wang et al. considered a 2m-point boundary value problem (BVP) at resonance for a coupled system as follows:
where . With the help of the coincidence degree theory, many existence results have been given in the above literatures. It is worth mentioning that the orders of derivative in the nonlinear function on the right-hand of equal signs are all fixed in the above works, but the opposite case is more difficult and complicated, then this work attempts to deal exactly with this case. What is more, this case of arbitrary order derivative included in the nonlinear functions is very important in many aspects [20,22].
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 [1,7,26-29]. 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 2m-point boundary value problem of the following coupled system of impulsive fractional differential equations at resonance:
where , , and , , , . satisfy Carathéodory conditions, . , , here , , and denote the right and left limits of at , respectively, and the fractional derivative is understood in the Riemann-Liouville sense. k, m, , , , ( ) are fixed constant satisfying and .
The coupled system (1.1) happens to be at resonance in the sense that the associated linear homogeneous coupled system
has , , as a nontrivial solution. To solve this interesting and important problem and to overcome the difficulties caused by the impulses, we will construct some Banach spaces, then we shall obtain the new solvability results for the coupled system (1.1) with the help of a coincidence degree continuation theorem. The main contributions of this work are Lemma 2.1 and Lemma 3.1 in Section 3 since the calculations are disposed well.
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
Let Y, Z be real Banach spaces, be a Fredholm map of index zero and , be continuous projectors such that , and , . It follows that is invertible. We denote the inverse of the map by . If Ω is an open bounded subset of Y such that , the map will be called L-compact on if is bounded and is compact.
The main tool we used is Theorem 2.4 of .
Theorem 2.1LetLbe a Fredholm operator of index zero, and letNbeL-compact on . Assume that the following conditions are satisfied:
(i) for every ;
(ii) for every ;
(iii) , where is a projection as above with .
Then the equation has at least one solution in .
Definition 2.1 The fractional integral of order of a function is defined by
provided the right-hand side is pointwise defined on .
Definition 2.2 The fractional derivative of order of a function is defined by
where , provided the right-hand side is pointwise defined on .
Remark 2.1 It can be directly verified that the Riemann-Liouville fractional integration and fractional differentiation operators of the power functions yield power functions of the same form. For , , we have
Assume that with a fractional derivative of order that belongs to . Then
for some , , whereNis the smallest integer grater than or equal toα.
If , , then the equation
is satisfied for a continuous functiony.
If , and , the fractional derivatives and exist, then
If , then the equation
is satisfied for a continuous functiony.
If , then the relation
holds for a continuous functiony.
Let with the norm and
with the norm . Denote
where , with the norm
Thus, is a Banach space with the norm defined by .
Set equipped with the norm
thus is a Banach space with the norm defined by .
Define the operator , , , where
Let be defined as , where
Then the coupled system of boundary value problem (1.1) can be written as
For the sake of simplicity, we define the operators for as follows:
By the same way, we define the operators for as follows:
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:
(H1) , , where
then is a Fredholm operator of index zero. Moreover, , where
and , here
Proof It is clear that (2.7) holds. For , we have , i.e., , , then , , so . Similarly, it is not difficult to see that . Next, we will show that (2.8) and (2.9) hold.
If , , then there exist and such that
Proposition 2.1 together with (2.10)-(2.12) gives that
Substituting the boundary condition into (2.13), one has
and substituting the boundary condition into (2.13), one has
By the same way, if we substitute the condition (2.12) into (2.14), then we can obtain that
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 , with , , here
In what follows, we will show that and are linear projectors. By some direct computations, we have
As a result,
Similarly, we can see that . Then for , we have . It means that the operator is a projector.
Now, we show that . Obviously, . On the other hand, for , then implies that
The condition (H1) guarantees that , , then . Hence, .
For , let . Then , , it means that . Moreover, gives that . Thus, . Then , L is a Fredholm map of index zero. □
Define the operator with , here , are defined as follows:
Moreover, we define as , where , is defined as follows:
Lemma 2.2Assume that is an open bounded subset with , thenNisL-compact on .
Proof Obviously, . By a direct computation, we have that
Similarly, . This gives that , that is to say, the operator P is a linear projector. It is easy to check from that . Moreover, we can see that . Thus, .
In what follows, we will show that defined above is the inverse of .
If , then , , which gives that
On the other hand, for , we have
Since and , then
By some calculations, (2.19) and (2.20) imply that
It means that . Analogously, . Thus, . So, is the inverse of .
Finally, we show that N is L-compact on . Denote , , where
Then we can see that
So, we can see that is bounded and is uniformly bounded.
For , we have
The equicontinuity of , together with (2.21) and (2.22) gives that as , which yields that is equicontinuous. By the Ascoli-Arzela theorem, we can see that is compact. By the same way, is bounded and is compact. Since and , then QN is bounded and is compact. This means that N is L-compact on . □
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.
(H2) There exist functions , , such that
where , ( ) satisfy
(H3) For , there exist constants ( ), ( ) such that
(1) if either or for , then either or ;
(2) if either or , , then either or .
(H4) For , there exist constants ( ) such that if either or , either or , then either (1) or (2) holds, where
here , are positive constants;
here , are negative constants.
Lemma 3.1Suppose that (H2)-(H3) hold. Then the set
is bounded inY.
Proof For , by and , we have
Since , , then , . Then we can see, from the condition (H3), that there exist constants such that , for and , for . So, we can see from (3.1) and (3.2) that
Then for and , we have
Similarly, for , we have that
Substitute (3.11) and (3.12) into (3.8), then we have
It means that
Substituting the above two into (3.9) and (3.12), we can see that
From the condition (H2), (3.14) and (3.15) give that and are bounded, then and are also bounded. Thus, by the definition of the norm on Y, and are bounded. That is, is bounded in Y. □
Lemma 3.2Suppose that the condition (H3) holds. Then the set
is bounded inY.
Proof For , we have that , where , . Since , so we have
From (H3), there exist positive constants , , , , such that for ,
which means that . And for ,
which means that . So, we can see that for ,
The above two arguments imply that is bounded. In the same way, is bounded. Thus, is bounded in Y. □
Lemma 3.3The set
is bounded inY, where is the linear isomorphism given by
Proof For , set , , then implies that
From (3.16) and (3.17), we have
the condition (H4) gives that
which is a contradiction. As a result, there exist positive constants , such that , . Similarly, from (3.18)-(3.19) and the second part of (1) or (2) of (H4), there exist two positive constants , such that , . It follows that , are bounded, that is, is bounded in Y. □
Theorem 3.1Suppose that (H1)-(H4) hold. Then the problem (1.1) has at least one solution inY.
Proof Let Ω be a bounded open set of Y such that . It follows from Lemma 2.2 that N is L-compact on . By means of above Lemmas 3.1-3.3, one obtains that
(i) for every ;
(ii) for every .
Then we need only to prove
According to Lemma 3.3, we know for all . Thus, the homotopy invariance property of degree theory gives that
Then, by Theorem 2.1, has at least one solution in , i.e., the problem (1.1) has at least one solution in Y, which completes the proof. □
4 An example
Consider the following boundary value problem for coupled systems of impulsive fractional differential equations:
Due to the coupled problem (1.1), we have that , , , , , , , , , , , . , ; , ; , ; , . Obviously, and . By direct calculation, we obtain that
It is easy to see that
So, , , , . And
where , . Thus, the condition (H2) holds.
Taking , for any , assume that holds for any . Thus either or for any . If , , then
If , , then
Similarly, assume that holds for any . Thus either or for any . If , , then
If , , then
So, from the above arguments, the first part of the condition (H3) is true for , .
Taking , assume that holds for any . Then either or for . If for , then
If for , then
By the same way, taking , assume that holds for any . Then either or for . If for , then
If for , then
So, from the above arguments, the second part of the condition (H3) holds for , .
On the other hand, for , taking , assume that , , then for , for . And for , for . Then we can see, from the above arguments, that , , , , where , . Thus,
where , . So, the condition (H4) holds. Hence, from Theorem 3.1, the coupled problem (4.1) has at least one solution in .
The authors declare that they have no competing interests.
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.
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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