Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance
© Qu and Liu; licensee Springer 2013
Received: 11 January 2013
Accepted: 26 April 2013
Published: 16 May 2013
We consider the fractional differential equation
satisfying the boundary conditions
where is the Riemann-Liouville fractional order derivative. The parameters in the multi-point boundary conditions are such that the corresponding differential operator is a Fredholm map of index zero. As a result, the minimal and maximal nonnegative solutions for the problem are obtained by using a fixed point theorem of increasing operators.
Keywordsfractional order coincidence degree at resonance
where , , , , , , . We assume that is continuous. A boundary value problem at resonance for ordinary or fractional differential equations has been studied by several authors, including the most recent works [1–7] and the references therein. In the most papers mentioned above, the coincidence degree theory was applied to establish existence theorems. But in , Wang obtained the minimal and maximal nonnegative solutions for a second-order m-point boundary value problem at resonance by using a new fixed point theorem of increasing operators, and in this paper we use this method of Wang to establish the existence theorem of equations (1.1) and (1.2).
For the convenience of the reader, we briefly recall some notations.
Let X, Z be real Banach spaces, be a Fredholm map of index zero and , be continuous projectors such that , and , . It follows that is invertible. We denote the inverse of the map by . Since , there exists an isomorphism . Let Ω be an open bounded subset of X. The map will be called L-compact on if and are compact. We take , then is a linear bijection with bounded inverse and . We know from  that is a cone in Z.
Theorem 1.1 
and is uniquely determined.
We also need the following definition and theorem.
Definition 1.1 
Theorem 1.2 
Let be a Fredholm operator of index zero, K be a normal cone in a Banach space X, , , and be L-compact and continuous. Suppose that the following conditions are satisfied:
(C1) and are coupled lower and upper solutions of the equation ;
(C2) is an increasing operator.
In this section, we present some necessary basic knowledge and definitions about fractional calculus theory.
Here is the gamma function.
where means the integral part of q.
Lemma 2.1 
hold a.e. on .
Lemma 2.2 (see )
where () are some constants.
Lemma 2.3 (see Corollary 2.1 in )
Let and , the equation is valid if and only if , where () are arbitrary constants.
Let with the norm , then X and Z are Banach spaces.
Let . It follows from Theorem 1.1.1 in  that K is a normal cone.
then BVPs (1.1) and (1.2) can be written as , .
- (ii)If , then there exists a function such that , by Lemma 2.2, we have
Then , hence .
On the other hand, if , let , then , and , which implies that , thus . In general . Clearly, is closed in Z and , thus L is a Fredholm operator of index zero. This completes the proof. □
Lemma 2.5 Let Ω be any open bounded subset of , then and are compact, which implies that N is L-compact on for any open bounded set .
which implies that is bounded.
It is concluded that N is L-compact on . This completes the proof. □
3 Main result
In this section, we establish the existence of the nonnegative solution to equations (1.1) and (1.2).
Theorem 3.1 Suppose
where and , then problems (1.1) and (1.2) have a minimal solution and a maximal solution in , respectively.
so condition (C1) in Theorem 1.1 holds.
where . Finally, by Theorem 3.1, equation (4.1) with BCs (4.2) has a minimal solution and a maximal solution in .
The authors would like to thank the referees for their many constructive comments and suggestions to improve the paper.
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