Solvability for a coupled system of fractional differential equations with impulses at resonance
© Zhang et al.; licensee Springer. 2013
Received: 9 November 2012
Accepted: 18 March 2013
Published: 8 April 2013
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.
Keywordscoupled 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.
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].
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 .
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 .
for every ;
for every ;
, where is a projection as above with .
Then the equation has at least one solution in .
provided the right-hand side is pointwise defined on .
where , provided the right-hand side is pointwise defined on .
Proposition 2.1 
for some , , where N is the smallest integer grater than or equal to α.
Proposition 2.2 
is satisfied for a continuous function y.
is satisfied for a continuous function y.
holds for a continuous function y.
Thus, is a Banach space with the norm defined by .
thus is a Banach space with the norm defined by .
In what follows, we present the following lemmas which will be used to prove our main results.
Lemma 2.1 If the following condition is satisfied:
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.
It is easy to check that the above u, v satisfy equation (2.10)-(2.12). Thus, (2.8) and (2.9) hold.
Similarly, we can see that . Then for , we have . It means that the operator is a projector.
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. □
Lemma 2.2 Assume that is an open bounded subset with , then N is L-compact on .
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 .
It means that . Analogously, . Thus, . So, is the inverse of .
So, we can see that is bounded and is uniformly bounded.
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.
if either or for , then either or ;
if either or , , then either or .
here , are negative constants.
is bounded in Y.
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. □
is bounded in Y.
The above two arguments imply that is bounded. In the same way, is bounded. Thus, is bounded in Y. □
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 3.1 Suppose that (H1)-(H4) hold. Then the problem (1.1) has at least one solution in Y.
for every ;
for every .
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
where , . Thus, the condition (H2) holds.
So, from the above arguments, the first part of the condition (H3) is true for , .
So, from the above arguments, the second part of the condition (H3) holds for , .
where , . So, the condition (H4) holds. Hence, from Theorem 3.1, the coupled problem (4.1) has at least one solution in .
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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