Solvability for a coupled system of fractional differential equations with integral boundary conditions at resonance
© Jiang; licensee Springer. 2013
Received: 5 June 2013
Accepted: 10 September 2013
Published: 19 November 2013
By constructing suitable operators, we investigate the existence of solutions for a coupled system of fractional differential equations with integral boundary conditions at resonance. Our analysis relies on the coincidence degree theory due to Mawhin. An example is given to illustrate our main result.
MSC:34A08, 70K30, 34B10.
Fractional differential equations arise in a variety of different areas such as rheology, fluid flows, electrical networks, viscoelasticity, chemical physics, electron-analytical chemistry, biology, control theory etc. (see [1, 2]). Recently, more and more authors have paid their close attention to them (see [3–24]). The existence of solutions for differential equations at resonance has been studied by many authors (see [19–23, 25–29] and references cited therein). In papers [19–22], the authors investigated the fractional differential equations with multi-point boundary conditions at resonance. In paper , the authors discussed a coupled system of fractional differential equations with two-point boundary condition at resonance. In paper , the authors showed the existence of solutions for higher-order fractional differential inclusions with multi-strip fractional integral boundary conditions. In paper , the authors studied solvability of integer-order differential equations with integral boundary conditions at resonance, which was the generalization of two, three, multi-point and nonlocal boundary value problems.
where , is the standard Riemann-Liouville fractional derivative, . To the best of our knowledge, this is the first paper to study the boundary value problems of a coupled system of fractional differential equations with integral boundary conditions at resonance with .
In this paper, we will always suppose that the following conditions hold.
() , , , , .
where , , , , , , , , , , , , , .
For convenience, we introduce some notations and a theorem. For more details, see .
is invertible. We denote the inverse by .
Assume that Ω is an open bounded subset of X, . The map will be called L-compact on if is bounded and is compact.
Theorem 2.1 
for every ;
for every ;
, where is a projection such that .
Then the equation has at least one solution in .
provided the right-hand side is pointwise defined on .
provided the right-hand side is pointwise defined on , where .
where n is the smallest integer greater than or equal to α.
where n is the smallest integer greater than or equal to α, , .
, . Then problem (1.1) is .
By Lemma 2.3 in , we get that X is a Banach space.
Definition 2.3 is a solution of problem (1.1) if it satisfies (1.1), i.e., .
3 Main result
It is clear that , , , .
Obviously, , .
This means that L is a Fredholm operator of index zero.
It follows from that . This, together with , means that . So, . Therefore, . The proof is completed. □
Lemma 3.2 Suppose that (), () and () hold. If is an open bounded subset and , then N is L-compact on .
i.e., is bounded. Now we will prove that is compact.
where is an identical mapping.
the uniform continuity of and on , the absolute continuity of integral of on , , and the Ascoli-Arzela theorem that is compact. The proof is completed. □
In order to obtain our main results, we present the following conditions.
where , , are the same as in ().
is bounded in X.
where , .
By (), (3.4), (3.6) and (3.7), we can get that is bounded in X. The proof is completed. □
is bounded in X.
Proof For , we have , and = = = = 0. By (), we get that , , , . These imply that is bounded in X. □
This is a contradiction, too. Thus, , . By the same methods, we can obtain that , . This means that is bounded in X. □
Theorem 3.1 Suppose that ()-() hold. Then problem (1.1) has at least one solution in X.
for every ,
for every .
We need only to prove
By Theorem 2.1, we can get that has at least one solution in , i.e., (1.1) has at least one solution in X. The proof is completed. □
So, () holds. Set , , , . By a simple calculation, we can obtain that condition () is satisfied.
By Theorem 3.1, problem (4.1) has at least one solution.
This work is supported by the National Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108). The author is grateful to anonymous referees for their constructive comments and suggestions which led to improvement of the original manuscript.
- Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.MATHGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.MATHGoogle Scholar
- Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Anal. 2008, 69: 2677-2682. 10.1016/j.na.2007.08.042MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Andrade Bd, Siracusa G: On fractional integro-differential equations with state-dependent delay. Comput. Math. Appl. 2011, 62: 1143-1149. 10.1016/j.camwa.2011.02.033MATHMathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Vatsala AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 2008, 21: 828-834. 10.1016/j.aml.2007.09.006MATHMathSciNetView ArticleGoogle Scholar
- Kou C, Zhou H, Yan Y: Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal. 2011, 74: 5975-5986. 10.1016/j.na.2011.05.074MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Nonlinear Anal. 2010, 59: 1095-1100.MATHMathSciNetGoogle Scholar
- Zhou Y, Jiao F, Li J: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. 2009, 71: 3249-3256. 10.1016/j.na.2009.01.202MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72: 2859-2862. 10.1016/j.na.2009.11.029MATHMathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Leela S: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal. 2009, 71: 2886-2889. 10.1016/j.na.2009.01.169MATHMathSciNetView ArticleGoogle Scholar
- Wang Y, Liu L, Wu Y: Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Anal. 2011, 74: 6434-6441. 10.1016/j.na.2011.06.026MATHMathSciNetView ArticleGoogle Scholar
- Staněk S: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 2011, 62: 1379-1388. 10.1016/j.camwa.2011.04.048MATHMathSciNetView ArticleGoogle Scholar
- Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006., 2006: Article ID 36Google Scholar
- Jafari H, Daftardar-Gejji V: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007MATHMathSciNetView ArticleGoogle Scholar
- 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. 2010, 72: 710-719. 10.1016/j.na.2009.07.012MATHMathSciNetView ArticleGoogle Scholar
- Su X: Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. Nonlinear Anal. 2011, 74: 2844-2852. 10.1016/j.na.2011.01.006MATHMathSciNetView ArticleGoogle Scholar
- Rehman M, Khan RA, Asif NA: Three point boundary value problems for nonlinear fractional differential equations. Acta Math. Sci. B 2011, 31: 1337-1346. 10.1016/S0252-9602(11)60320-2MATHMathSciNetView ArticleGoogle Scholar
- Liang S, Zhang J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 2009, 71: 5545-5550. 10.1016/j.na.2009.04.045MATHMathSciNetView ArticleGoogle Scholar
- 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, 44: 1-17.MathSciNetView ArticleGoogle Scholar
- Jiang W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. TMA 2011, 74: 1987-1994. 10.1016/j.na.2010.11.005MATHView ArticleGoogle Scholar
- Jiang W: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Anal., Real World Appl. 2012, 13: 2285-2292. 10.1016/j.nonrwa.2012.01.023MATHMathSciNetView ArticleGoogle Scholar
- Bai Z: Solvability for a class of fractional m -point boundary value problem at resonance. Comput. Math. Appl. 2011, 62: 1292-1302. 10.1016/j.camwa.2011.03.003MATHMathSciNetView ArticleGoogle Scholar
- Hu Z, Liu W, Chen T: Existence of solutions for a coupled system of fractional differential equations at resonance. Bound. Value Probl. 2012, 98: 1-13.Google Scholar
- Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013, 20: 1-19.MathSciNetView ArticleGoogle Scholar
- Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 2158-2171. 10.1016/j.na.2007.01.038MATHMathSciNetView ArticleGoogle Scholar
- Zhang X, Feng M, Ge W: Existence results of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 2009, 353: 311-319. 10.1016/j.jmaa.2008.11.082MATHMathSciNetView ArticleGoogle Scholar
- Feng W, Webb JRL: Solvability of m -point boundary value problems with nonlinear growth. J. Math. Anal. Appl. 1997, 212: 467-480. 10.1006/jmaa.1997.5520MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Existence results of an m -point boundary value problem at resonance. J. Math. Anal. Appl. 2004, 294: 147-157. 10.1016/j.jmaa.2004.02.005MATHMathSciNetView ArticleGoogle Scholar
- Du Z, Lin X, Ge W: Some higher-order multi-point boundary value problem at resonance. J. Comput. Appl. Math. 2005, 177: 55-65. 10.1016/j.cam.2004.08.003MATHMathSciNetView ArticleGoogle Scholar
- Mawhin J NSFCBMS Regional Conference Series in Mathematics. In Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.