Abstract
In this paper, a functional boundary value problem of fractional differential equations is studied. Based on Mawhin’s coincidence degree theory, some existence theorems are obtained in the case of nonresonance and the cases of and at resonance.
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1 Introduction
The subject of fractional calculus has gained considerable popularity and importance because of its frequent appearance in various fields such as physics, chemistry, and engineering. In consequence, the subject of fractional differential equations has attracted much attention. Many methods have been introduced to solve fractional differential equations, such as the popular Laplace transform method, the iteration method, the Fourier transform method and the operational method. For details, see [1–3] and the references therein. Recently, there have been some papers dealing with the basic theory for initial value problems of nonlinear fractional differential equations; for example, see [4, 5]. Also, there are some articles which deal with the existence and multiplicity of solutions for nonlinear boundary value problems of fractional order differential equations using techniques of topological degree theory. We refer the reader to [6–16] for some recent results at nonresonance and to [17–26] at resonance.
In [18], by making use of the coincidence degree theory of Mawhin, Zhang and Bai discussed the existence results for the following nonlinear nonlocal problem at resonance under the case :
Recently, Jiang [26] studied the existence of a solution for the following fractional differential equation at resonance under the case :
Being directly inspired by [18, 20, 26], we intend in this paper to study the following functional boundary value problems (FBVP) of fractional order differential equation:
where , and are the standard Riemann-Liouville differentiation and integration, and ; are continuous linear functionals.
In this paper, we shall give some sufficient conditions to construct the existence theorems for FBVP (1.1), (1.2) at nonresonance and resonance (both cases of and ), respectively. To the best of our knowledge, the method of Mawhin’s theorem has not been developed for fractional order differential equation with functional boundary value problems at resonance. So, it is interesting and important to discuss the existence of a solution for FBVP (1.1), (1.2). Many difficulties occur when we deal with them. For example, the construction of the generalized inverse of L. So, we need to introduce some new tools and methods to investigate the existence of a solution for FBVP (1.1), (1.2).
The rest of this paper is organized as follows. In Section 2, we give some notations and lemmas. In Section 3, we establish the existence results of a solution for functional boundary value problem (1.1), (1.2).
2 Preliminaries and lemmas
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature. The readers who are unfamiliar with this area can consult, for example, [1, 2, 4] for details.
The Riemann-Liouville fractional integral of order of a function is given by
provided that the right-hand side is pointwise defined on . Here is the Gamma function given by .
The Riemann-Liouville fractional derivative of order of a continuous function is given by
where , provided that the right-hand side is pointwise defined on .
We use the classical spaces with the norm , with the norm . We also use the space defined by
and the Banach space ()
with the norm .
Lemma 2.1 [2]
Let , . Assume that with a fractional integration of order that belongs to . Then the equality
holds almost everywhere on .
Remark 2.1 If u satisfies and , then . In fact, with Lemma 2.1, one has
Combine with , there is . So,
In the following lemma, we use the unified notation both for fractional integrals and fractional derivatives assuming that for .
Lemma 2.2 [2]
Assume , then:
-
(i)
Let . If and exist, then
-
(ii)
If , , then
is satisfied at any point on for and ;
-
(iii)
Let . Then holds on ;
-
(iv)
Note that for , , we have
Lemma 2.3 [18]
is a sequentially compact set if and only if F is uniformly bounded and equicontinuous. Here ‘F is uniformly bounded and equicontinuous’ means that there exists such that for every ,
and that , , for all , , , , there hold
respectively.
Next, consider the following conditions:
(A1) .
(A2) , , .
(A3) , , .
(A4) , , .
(A5) , , .
We shall prove that: If (A1) holds, then . It is the so-called nonresonance case. If (A2) holds, then . If (A3) or (A4) holds, then . If (A5) holds, then .
In the nonresonance case, FBVP (1.1), (1.2) can be transformed into an operator equation.
Lemma 2.4 Assume that (A1) holds. Then functional boundary value problem (1.1) and (1.2) has a solution if and only if the operator , defined by
has a fixed point, where .
Proof If u is a solution to , by Lemma 2.2, we get
and
Considering the linearity of (), we have
So, u is a solution to FBVP (1.1), (1.2).
If u is a solution to (1.1), by Lemma 2.1, we can reduce (1.1) to an equivalent integral equation
By , there is , and
Applying and to (2.2) and (2.3), respectively, we obtain
Thus,
Substituting (2.4) and (2.5) into (2.1), we obtain
The proof is complete. □
The following definitions and lemmas are a preparation for the existence of solutions to (1.1), (1.2) at resonance.
Definition 2.3 Let Y, Z be real Banach spaces, let be a linear operator. L is said to be a Fredholm operator of index zero provided that:
-
(i)
ImL is a closed subset of Z,
-
(ii)
.
Let Y, Z be real Banach spaces and be a Fredholm operator of index zero. , are continuous projectors such that , , and . It follows that is invertible. We denote the inverse of the mapping by (generalized inverse operator of L). If Ω is an open bounded subset of Y such that , the mapping will be called L-compact on , if is bounded and is compact.
We need the following known result for the sequel (Theorem 2.4 [27]).
Theorem 2.1 Let L be a Fredholm operator of index zero, and let N be L-compact on . Assume that the following conditions are satisfied:
-
(i)
for every .
-
(ii)
for every .
-
(iii)
, where is a projector as above with .
Then the equation has at least one solution in .
Let , . Let the linear operator with
be defined by . Let the nonlinear operator be defined by
Then (1.1), (1.2) can be written as
Now, we give KerL, ImL and some necessary operators at and , respectively.
Lemma 2.5 Let L be the linear operator defined as above. If (A2) holds, then
and
Proof Let . Clearly, and . Considering (A2), and . So,
If , then . Considering and (A2), we can obtain that . It yields and .
We now show that
If , then there exists such that . Hence,
for some . It yields
Therefore
On the other hand, suppose satisfies
Let
Obviously, and . Considering (A2) and the linearity of (), we have
and
It yields
The proof is complete. □
Lemma 2.6 If , then:
-
(i)
L is a Fredholm operator of index zero and .
-
(ii)
The linear operator can be defined by
-
(iii)
, where and is the norm of a continuous linear functional .
-
(iv)
The linear operator is completely continuous.
Proof Firstly, we construct the mapping defined by
Noting that
we get is a well-defined projector.
Now, it is obvious that . Noting that Q is a linear projector, we have . Hence, and 1. This means L is a Fredholm mapping of index zero. Taking as
then the generalized inverse of L can be rewritten
In fact, for , we have
and
which implies that is well defined on ImL. Moreover, for , we have
and for , we know
means that . So,
That is, . Since
then
It follows that
Finally, we prove that is completely continuous. Let be a bounded set. From the above discussion, we only need to prove that is equicontinuous on . For , with , we have
and
Therefore, is equicontinuous. Thus, the operator is completely continuous. The proof is complete. □
Similar to Lemmas 2.5 and 2.6, we can obtain the following lemma.
Lemma 2.7 If (A3) holds, then and
Furthermore, if also holds, then L is a Fredholm operator of index zero and . Here, the projectors , can be defined as follows:
The generalized inverse operator of can be defined by
Also,
where .
Lemma 2.8 If (A4) holds, then and
Furthermore, if also holds, then L is a Fredholm operator of index zero and . Here, the projectors , can be defined as follows:
The generalized inverse operator of can be defined by
Also,
where and is the norm of the continuous linear functional .
Lemma 2.9 If (A5) holds, then
and
Proof Let . Clearly, and . Considering (A5), and . So,
If , then . Considering and (A5), we can obtain that
We now show that
If , then there exists such that . Hence,
for some . It yields
and
Therefore,
On the other hand, suppose satisfies
Let
Obviously, and . Considering (A5) and the linearity of (), we have
and
It yields
The proof is complete. □
Lemma 2.10 If , then L is a Fredholm operator of index zero and . Furthermore, the linear operator can be defined by
Also,
Proof Firstly, we construct the mapping defined by
Let
and
We have
Noting that
and
we have, for each , that
So, is a well-defined projector.
Now we will show that . If , from , we have and . Considering the definitions of and , we have
Since
so , which yields . On the other hand, if , from and the definition of Q, it is obvious that , thus . Hence, .
For , from , , , we have . And for any , from , there exist constants such that . From , we obtain
In view of
therefore (2.8) has a unique solution , which implies and . Since , thus L is a Fredholm map of index zero. Let be defined by
Then the generalized inverse of L can be rewritten
In fact, for , we have
and
which implies that is well defined on ImL. Moreover, for , we have
and for , we know
means that . So,
That is, . Since
then
It follows that
The proof is complete. □
3 Main results
From Lemma 2.4, we can obtain the existence theorem for FBVP (1.1), (1.2).
Theorem 3.1 Assume that (A1) and the following conditions hold:
Then FBVP (1.1), (1.2) has a unique solution in provided that
Proof We shall prove that has a unique solution in . For each , considering the linearity of (), we have
Then
and
So,
The above inequality implies that T is a contraction. By using Banach’s contraction principle, has a unique solution in . From Lemma 2.4, FBVP (1.1), (1.2) has a unique solution in . The proof is complete. □
From Lemmas 2.5-2.8 and Theorem 2.1, we can obtain the existence theorem for FBVP (1.1), (1.2) in the case of .
Theorem 3.2 Let be a continuous function. Assume that , (A2) and the following conditions (H1)-(H3) hold:
(H1) There exist functions such that for all , ,
(H2) There exists a constant such that for , if for all , then .
(H3) There exists a constant such that either for each ,
or for each ,
Then FBVP (1.1), (1.2) has at least one solution in provided
where is the same as in Lemma 2.6.
Proof Set
Then, for , since , so , , hence
Thus, from (H2), there exists such that
Now,
and so
Again, for , , then and . Thus, from Lemma 2.6, we have
From (3.1), (3.2), we have
By this and (H1), we have
and
Therefore, is bounded. Let
For , there is , and , thus
From (H2), we get , thus is bounded.
Next, according to the condition (H3), for any , if , then either
or else
If (3.3) holds, set
here Q is given by (2.6) and is the linear isomorphism given by , , . For ,
If , then . Otherwise, if , in view of (3.3), one has
which contradicts . Thus, is bounded.
If (3.4) holds, then define the set
here J is as above. Similar to the above argument, we can show that is bounded too.
In the following, we shall prove that all the conditions of Theorem 2.1 are satisfied. Let Ω be a bounded open subset of Y such that . By Lemma 2.6 and standard arguments, we can prove that is compact, thus N is L-compact on . Then, by the above argument, we have
-
(i)
, for every ,
-
(ii)
for .
Finally, we will prove that (iii) of Theorem 2.1 is satisfied. Let . According to the above argument, we know
Thus, by the homotopy property of degree, we have
Then, by Theorem 2.1, has at least one solution in , so that FBVP (1.1), (1.2) has a solution in . The proof is complete. □
Theorem 3.3 Let be a continuous function. Assume that , (A3), (H1) and the following conditions (H4), (H5) hold:
(H4) There exists a constant such that for , if for all , then .
(H5) There exists a constant such that either for each ,
or for each ,
Then FBVP (1.1), (1.2) has at least one solution in provided
where is the same as in Lemma 2.7.
Theorem 3.4 Let be a continuous function. Assume that , (A4), (H1) and the following conditions (H6), (H7) hold:
(H6) There exists a constant such that for , if for all , then .
(H7) There exists a constant such that either for each ,
or for each ,
Then FBVP (1.1), (1.2) has at least one solution in provided
where is the same as in Lemma 2.8.
The proofs of Theorem 3.3 and Theorem 3.4 are similar to that of Theorem 3.2. So, we omit them.
The above Theorem 3.2, Theorem 3.3 and Theorem 3.4 are the existence of solutions to FBVP (1.1), (1.2) in the case of . By making use of Theorem 2.1, Lemma 2.9 and Lemma 2.10, we obtain the existence of solutions for FBVP (1.1), (1.2) in the case of .
Theorem 3.5 Let be a continuous function. Assume that , (A5), (H1) and the following conditions (H8), (H9) hold:
(H8) There exists a constant such that for , if for all , then
(H9) There exists a constant such that for satisfying , either
or
Then FBVP (1.1), (1.2) has at least one solution in provided
Proof Set
Then, for , since , so , , hence
and
Thus, from (H8), there exists such that
Now,
and so
Again, for , , then and . Thus, from Lemma 2.10, we have
From (3.7), (3.8) and (3.9), we have
By this and (H1), we have
and
Therefore, is bounded. Let
For , there is , and , thus
and
From (H8), we get . Then, for , we have
thus is bounded.
Next, for any , define a linear isomorphism by
If (3.5) holds, set
where Q is given by (2.7). For , from , we obtain
and
By , it yields
If , then . Otherwise, if , considering the above equalities and (3.5), we have
which contradicts . If (3.6) holds, then we take
and, again, obtain a contradiction. Thus, in either case,
for all , that is, is bounded.
In the following, we shall prove that all the conditions of Theorem 2.1 are satisfied. Let Ω be a bounded open subset of Y such that . By Lemma 2.2 and standard arguments, we can prove that is compact, thus N is L-compact on . Then, by the above argument, we have
-
(i)
for every ,
-
(ii)
for .
Finally, we will prove that (iii) of Theorem 2.1 is satisfied. Let . According to the above argument, we know
Thus, by the homotopy property of degree, we have
Then, by Theorem 2.1, has at least one solution in , so that FBVP (1.1), (1.2) has a solution in . The proof is complete. □
References
Podlubny I Mathematics in Science and Engineering. In Fractional Differential Equations. Academic Press, New York; 1999.
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Babakhani A, Gejji VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278: 434-442. 10.1016/S0022-247X(02)00716-3
Lakshmikantham V, Leela S, Devi JV: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
Lakshmikantham V, Leela S: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal. TMA 2009, 71: 2886-2889. 10.1016/j.na.2009.01.169
Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. TMA 2010, 72: 916-924. 10.1016/j.na.2009.07.033
Bai Z, Lü H: Positive solutions of boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052
Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 2009, 22: 64-69. 10.1016/j.aml.2008.03.001
Jiang D, Yuan C: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. TMA 2010, 72(2):710-719. 10.1016/j.na.2009.07.012
Zhang S: The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 2000, 252: 804-812. 10.1006/jmaa.2000.7123
Zhang S: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278: 136-148. 10.1016/S0022-247X(02)00583-8
Feng M, Zhang X, Ge W: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 720702
Zhou Y: Existence and uniqueness of solutions for a system of fractional differential equations. Fract. Calc. Appl. Anal. 2009, 12: 195-204.
Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010, 59: 1363-1375. 10.1016/j.camwa.2009.06.029
Salem HAH: On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order. Math. Comput. Model. 2008, 48: 1178-1190. 10.1016/j.mcm.2007.12.015
Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576
Cui Y: Solvability of second-order boundary-value problems at resonance involving integral conditions. Electron. J. Differ. Equ. 2012., 2012: Article ID 45
Zhang Y, Bai Z: Existence of solutions for nonlinear fractional three-point boundary value problems at resonance. J. Appl. Math. Comput. 2011, 36: 417-440. 10.1007/s12190-010-0411-x
Bai Z, Zhang Y: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 2011, 218: 1719-1725. 10.1016/j.amc.2011.06.051
Zhao Z, Liang J: Existence of solutions to functional boundary-value problem of second-order nonlinear differential equation. J. Math. Anal. Appl. 2011, 373: 614-634. 10.1016/j.jmaa.2010.08.011
Zhang X, Feng M, Ge W: Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 2009, 353: 311-319. 10.1016/j.jmaa.2008.11.082
Wang G, Liu W, Zhu S, Zheng T: Existence results for a coupled system of nonlinear fractional 2 m -point boundary value problems at resonance. Adv. Differ. Equ. 2011., 2011: Article ID 44
Webb JRL: Remarks on non-local boundary value problems at resonance. Appl. Math. Comput. 2010, 216: 497-500. 10.1016/j.amc.2010.01.056
Webb JRL, Zima M: Multiple positive solutions of resonance and nonresonance non-local fourth-order boundary value problems. Glasg. Math. J. 2012, 54: 225-240. 10.1017/S0017089511000590
Hu Z, Liu W: Solvability for fractional order boundary value problem at resonance. Bound. Value Probl. 2011., 2011: Article ID 20
Jiang W: The existence of solutions for boundary value problems of fractional differential equations at resonance. Nonlinear Anal. 2011, 74: 1987-1994. 10.1016/j.na.2010.11.005
Mawhin J: Topological degree and boundary value problems for nonlinear differential equations. Lecture Notes in Mathematics 1537. In Topological Methods for Ordinary Differential Equations. Edited by: Fitzpertrick PM, Martelli M, Mawhin J, Nussbaum R. Springer, Berlin; 1991.
Acknowledgements
The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179, 11071141), NSF (BS2010SF023, BS2012SF022) of Shandong Province.
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Zou, Y., Cui, Y. Existence results for a functional boundary value problem of fractional differential equations. Adv Differ Equ 2013, 233 (2013). https://doi.org/10.1186/1687-1847-2013-233
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DOI: https://doi.org/10.1186/1687-1847-2013-233