Abstract
In this paper, by using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations at resonance. A new result on the existence of solutions for a fractional boundary value problem is obtained.
MSC:34B15.
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1 Introduction
In recent years, the fractional differential equations have received more and more attention. The fractional derivative has been occurring in many physical applications such as a non-Markovian diffusion process with memory [1], charge transport in amorphous semiconductors [2], propagations of mechanical waves in viscoelastic media [3], etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order (see [4–9]).
Recently, boundary value problems for fractional differential equations have been studied in many papers (see [10–25]). Moreover, the existence of solutions to a coupled systems of fractional differential equations have been studied by many authors (see [26–33]). But the existence of solutions for a coupled system of fractional differential equations at resonance are seldom considered. Motivated by all the works above, in this paper, we consider the following boundary value problem (BVP for short) for a coupled system of fractional differential equations given by
where , are the standard Caputo fractional derivatives, , and is continuous.
The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, we establish a theorem on the existence of solutions for BVP (1.1) under nonlinear growth restriction of f and g, basing on the coincidence degree theory due to Mawhin (see [34]). Finally, in Section 4, an example is given to illustrate the main result.
2 Preliminaries
In this section, we will introduce some notations, definitions and preliminary facts which are used throughout this paper.
Let X and Y be real Banach spaces, and let be a Fredholm operator with index zero, and , be projectors such that
It follows that
is invertible. We denote the inverse by .
If Ω is an open bounded subset of X, and , the map will be called L-compact on if is bounded and is compact, where I is an identity operator.
Lemma 2.1 [27]
Let be a Fredholm operator of index zero and L-compact on . Assume that the following conditions are satisfied:
-
(1)
for every ;
-
(2)
for every ;
-
(3)
, where is a projection such that .
Then the equation has at least one solution in .
Definition 2.1 The Riemann-Liouville fractional integral operator of order of a function x is given by
provided that the right-hand side integral is pointwise defined on .
Definition 2.2 The Riemann-Liouville fractional derivative of order of a function x is given by
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on .
Definition 2.3 The Caputo fractional derivative of order of a function x is given by
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on .
Lemma 2.2 [35]
Assume that with a Caputo fractional derivative of order that belongs to . Then
where , , here n is the smallest integer greater than or equal to α.
Lemma 2.3 [35]
Assume that and . Then
In this paper, we denote with the norm and with the norm , where . Then we denote with the norm and with the norm . Obviously, both and are Banach spaces.
Define the operator by
where
Define the operator by
where
Define the operator by
where
Let be the Nemytski operator
where
and
Then BVP (1.1) is equivalent to the operator equation
3 Main result
In this section, a theorem on the existence of solutions for BVP (1.1) will be given.
Theorem 3.1 Let be continuous. Assume that
() there exist nonnegative functions () with
such that for all ,
and
where , , ();
() there exists a constant such that for , , either
or
() there exists a constant such that for every satisfying either
or
Then BVP (1.1) has at least one solution.
Now, we begin with some lemmas below.
Lemma 3.1 Let L be defined by (2.1), then
Proof By Lemma 2.2, has the solution
Combining it with the boundary value conditions of BVP (1.1), one has
For , there exists such that . By Lemma 2.2, we have
Then, we have
By the conditions of BVP (1.1), we can get that x satisfies
On the other hand, suppose and satisfies . Let , then . By Lemma 2.3, we have so that . Then we have
Similarly, we can get
Then, the proof is complete. □
Lemma 3.2 Let L be defined by (2.1), then L is a Fredholm operator of index zero, and the linear continuous projector operators and can be defined as
Furthermore, the operator can be written by
Proof Obviously, and . It follows from that . By simple calculation, we can get that . Then we get
For , we have
By the definition of , we can get
Similar proof can show that . Thus, we have .
Let , where . It follows from and that . Then, we have
Thus
This means that L is a Fredholm operator of index zero.
Now, we will prove that is the inverse of . By Lemma 2.3, for , we have
Moreover, for , we have and
which, together with , yields that
Combining (3.3) with (3.4), we know that . The proof is complete. □
Lemma 3.3 Assume is an open bounded subset such that , then N is L-compact on .
Proof By the continuity of f and g, we can get that and are bounded. So, in view of the Arzelá-Ascoli theorem, we need only prove that is equicontinuous.
From the continuity of f and g, there exists a constant , , such that
Furthermore, for , , we have
By
and
Similar proof can show that
Since , , and are uniformly continuous on , we can get that is equicontinuous.
Thus, we get that is compact. The proof is complete. □
Lemma 3.4 Suppose (), () hold, then the set
is bounded.
Proof Take , then . By (3.2), we have
Then, by the integral mean value theorem, there exist constants such that and . So, from (), we get and .
From , we get , then
By and , we have
and
Then we get
and
Take , we get
Together with , () and (3.6), we have
So, we have
Similarly, we can get
Together with (3.7) and (3.8), we have
Thus, from , we obtain that
and
Together with (3.5) and (3.6), we get
So is bounded. The proof is complete. □
Lemma 3.5 Suppose () holds, then the set
is bounded.
Proof For , we have , . Then from , we get
which, together with (), implies . Thus, we have
Hence, is bounded. The proof is complete. □
Lemma 3.6 Suppose the first part of () holds, then the set
is bounded.
Proof For , we have , and
If , then because of the first part of (). If , then . For , we can obtain . Otherwise, if or , in view of the first part of (), one has
or
which contradicts (3.9) or (3.10). Therefore, is bounded. The proof is complete. □
Remark 3.1 If the second part of () holds, then the set
is bounded.
Proof of Theorem 3.1 Set . It follows from Lemma 3.2 and 3.3 that L is a Fredholm operator of index zero and N is L-compact on . By Lemma 3.4 and 3.5, we get that the following two conditions are satisfied:
-
(1)
for every ;
-
(2)
for every .
Take
According to Lemma 3.6 (or Remark 3.1), we know that for . Therefore,
So, the condition (3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that has at least one solution in . Therefore, BVP (1.1) has at least one solution. The proof is complete. □
4 Example
Example 4.1 Consider the following BVP:
Choose , , , , , .
By simple calculation, we can get that (), () and the first part of () hold.
By Theorem 3.1, we obtain that BVP (4.1) has at least one solution.
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Acknowledgements
The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the Fundamental Research Funds for the Central Universities (2010LKSX09) and the National Natural Science Foundation of China (11271364).
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Hu, Z., Liu, W. & Chen, T. Existence of solutions for a coupled system of fractional differential equations at resonance. Bound Value Probl 2012, 98 (2012). https://doi.org/10.1186/1687-2770-2012-98
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DOI: https://doi.org/10.1186/1687-2770-2012-98