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# Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance

Haidong Qu1* and Xuan Liu2

Author Affiliations

1 Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

2 Department of Basic Education, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

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Boundary Value Problems 2013, 2013:127  doi:10.1186/1687-2770-2013-127

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/127

 Received: 11 January 2013 Accepted: 26 April 2013 Published: 16 May 2013

© 2013 Qu and Liu; licensee Springer

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.

### Abstract

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.

MSC: 26A33, 34A08.

##### Keywords:
fractional order; coincidence degree; at resonance

### 1 Introduction

Let us consider the fractional differential equation

(1.1)

with the boundary conditions (BCs)

(1.2)

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 [8], 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 [9] that is a cone in Z.

Theorem 1.1[9]

, where

andis uniquely determined.

From the above theorem, the author [9] obtained that the assertions

(i) and

(ii) are equivalent.

We also need the following definition and theorem.

Definition 1.1[8]

Let K be a normal cone in a Banach space X, , and are said to be coupled lower and upper solutions of the equation if

Theorem 1.2[8]

Letbe a Fredholm operator of index zero, Kbe a normal cone in a Banach spaceX, , , andbeL-compact and continuous. Suppose that the following conditions are satisfied:

(C1) andare coupled lower and upper solutions of the equation;

(C2) is an increasing operator.

Then the equationhas a minimal solutionand a maximal solutionin. Moreover,

where

and.

### 2 Preliminaries

In this section, we present some necessary basic knowledge and definitions about fractional calculus theory.

Definition 2.1 (see Equation 2.1.1 in [10])

The R-L fractional integral of order () is defined by

Here is the gamma function.

Definition 2.2 (see Equation 2.1.5 in [10])

The R-L fractional derivative of order () is defined by

where means the integral part of q.

Lemma 2.1[11]

If, , then, for, the relations

and

hold a.e. on.

Lemma 2.2 (see [11])

Let, , , then we have the equality

where () are some constants.

Lemma 2.3 (see Corollary 2.1 in [10])

Letand, the equationis 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 [12] that K is a normal cone.

Let .

We define the operators by

(2.1)

and by

then BVPs (1.1) and (1.2) can be written as , .

Lemma 2.4If the operatorLis defined in (2.1), then

(i) ,

(ii) .

Proof (i) It can be seen from Lemma 2.3 and BCs (1.2) that .

(ii) If , then there exists a function such that , by Lemma 2.2, we have

It follows from BCs (1.2) and the equation that

and noting the definition of , we have

Thus,

which is

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. □

In what follows, some property operators are defined. We define continuous projectors by

and by

where

is the beta function defined by

By calculating, we easily obtain , , and , . We also define by

and by

thus

and

Lemma 2.5LetΩbe any open bounded subset of, thenandare compact, which implies thatNisL-compact onfor any open bounded set.

Proof For a positive integer n, let , , . It is easy to see that is compact. Now, we prove that is compact. For , we have

which implies that is bounded.

Moreover, for each , let and , then

Thus

such that

for

and each

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 3.1Suppose

(H1) There existsuch thatand

(H2) For any, satisfying

whereand, then problems (1.1) and (1.2) have a minimal solutionand a maximal solutionin, respectively.

Proof By condition (H1), we know that

so condition (C1) in Theorem 1.1 holds.

In addition, for each ,

Thus , that is, by virtue of the equivalence. From condition (H2), we have that is a monotone increasing operator. Then, in accordance with Lemma 2.5 and Theorem 1.2, we obtain a minimal solution and a maximal solution in for problems (1.1) and (1.2). Thus we can define iterative sequences and by

and

Then from Theorem 1.2 we get and converge uniformly to and , respectively. Moreover,

□

### 4 Example

We consider the following problem:

(4.1)

subject to BCs

(4.2)

We can choose

then

Let , then for any , we have

where . Finally, by Theorem 3.1, equation (4.1) with BCs (4.2) has a minimal solution and a maximal solution in .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the referees for their many constructive comments and suggestions to improve the paper.

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