- Open Access
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 equation
- integral boundary conditions
- Fredholm operator
- coincidence degree theory
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., .
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.
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