Abstract
In this paper we study the existence of solutions of nonlinear fractional differential equations at resonance. By using the coincidence degree theory, some results on the existence of solutions are obtained.
MSC: 34A08, 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 (BVPs for short) for fractional differential equations have been studied in many papers (see [10]–[33]).
In [10], by means of a fixed point theorem on a cone, Agarwal et al. considered two-point boundary value problem at nonresonance given by
where , are real numbers, and is the Riemann-Liouville fractional derivative.
Zhao et al.[18] studied the following two-point BVP of fractional differential equations:
where denotes the Riemann-Liouville fractional differential operator of order α, . By using the lower and upper solution method and fixed point theorem, they obtained some new existence results.
Liang and Zhang [19] studied the following nonlinear fractional boundary value problem:
where is a real number, is the Riemann-Liouville fractional differential operator of order α. By means of fixed point theorems, they obtained results on the existence of positive solutions for BVPs of fractional differential equations.
In [20], Bai considered the boundary value problem of the fractional order differential equation
where is a real number, is the Riemann-Liouville fractional differential operator of order α.
Motivated by the above works, in this paper, we consider the following BVP of fractional equation at resonance
where denotes the Caputo fractional differential operator of order α, . 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 existence of solutions for BVP (1.1) under nonlinear growth restriction of f, 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 introduce 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 identity operator.
Lemma 2.1
([34])
If Ω is an open bounded set, letbe a Fredholm operator of index zero andL-compact on. Assume that the following conditions are satisfied:
-
(1)
for every ;
-
(2)
for every ;
-
(3)
, where is a projection such that .
Then the equationhas 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 side integral is pointwise defined on .
Definition 2.2
The Caputo fractional derivative of order of a function x with absolutely continuous on is given by
where .
Lemma 2.2
([35])
Letand. If, then
In this paper, we denote with the norm and with the norm , where . Obviously, both X and Y are Banach spaces.
Define the operator by
where
Let be the operator
Then BVP (1.1) is equivalent to the operator equation
3 Main result
In this section, a theorem on existence of solutions for BVP (1.1) will be given.
Theorem 3.1
Letbe continuous. Assume that
(H1): there exist nonnegative functionswithsuch that
where, , , , ;
(H2): there exists a constantsuch that for allwitheither
or
Then BVP (1.1) has at least one solution in X.
Now, we begin with some lemmas below.
Lemma 3.1
Let L be defined by (2.1), then
Proof
By Lemma 2.2, has solution
Combining with the boundary value condition of BVP (1.1), one sees that (3.1) holds.
For , there exists such that . By Lemma 2.2, we have
Then we have
By the conditions of BVP (1.1), we see that y satisfies
Thus we get (3.2). On the other hand, suppose and satisfies . Let , then and . So . 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 operatorsandcan be defined as
Furthermore, the operatorcan be written by
Proof
Obviously, and . It follows from that . By a simple calculation, we get . Then we get
For , we have
Let , where , . It follows from and that . Then we have
Thus
This means that L is a Fredholm operator of index zero.
From the definitions of P, , it is easy to see that the generalized inverse of L is . In fact, for , we have
Moreover, for , we get . By Lemma 2.2, we obtain
which together with yields
Combining (3.3) with (3.4), we know that . The proof is complete. □
Lemma 3.3
Assumeis an open bounded subset such that, then N is L-compact on.
Proof
By the continuity of f, we can see that and are bounded. So, in view of the Arzelà-Ascoli theorem, we need only prove that is equicontinuous.
From the continuity of f, there exists constant such that , , . Furthermore, denote and for , , we have
and
Since , , , and are uniformly continuous on , we see that , , and are equicontinuous. Thus, we find that is compact. The proof is completed. □
Lemma 3.4
Suppose (H1), (H2) hold, then the set
is bounded.
Proof
Take , then . By (3.2), we have
Then, by the integral mean value theorem, there exists a constant such that . Then from (H2), we have .
From , we get , , and . Therefore
and
That is
By and , we have
Then we get
Take , we get
Together with , (H1), and (3.5), we have
Then we have
Thus, from , we obtain
Thus, together with (3.5), we get
Therefore,
So is bounded. The proof is complete. □
Lemma 3.5
Suppose (H2) holds, then the set
is bounded.
Proof
For , we have and . Then we get
which together with (H2) implies . Thus, we have
Hence, is bounded. The proof is complete. □
Lemma 3.6
Suppose the first part of (H2) holds, then the set
is bounded.
Proof
For , we have and
If , then because of the first part of (H2). If , we can also obtain . Otherwise, if , in view of the first part of (H2), one has
which contradicts (3.6).
Therefore, is bounded. The proof is complete. □
Remark 3.1
Suppose the second part of (H2) hold, then the set
is bounded.
Proof of Theorem 3.1
Set . It follows from Lemmas 3.2 and 3.3 that L is a Fredholm operator of index zero and N is L-compact on . By Lemmas 3.4 and 3.5, we see 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 find that has at least one solution in . Therefore, BVP (1.1) has at least one solution. The proof is complete. □
4 An example
Example 4.1
Consider the following BVP:
Here
Choose , , , , , . We get , , , , and
Then all conditions of Theorem 3.1 hold, so 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 (2013QNA33).
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Hu, Z., Liu, W. & Liu, J. Boundary value problems for fractional differential equations. Bound Value Probl 2014, 176 (2014). https://doi.org/10.1186/s13661-014-0176-5
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DOI: https://doi.org/10.1186/s13661-014-0176-5