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
Let us consider the fractional differential equation
with the boundary conditions (BCs)
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.
and is uniquely determined.
From the above theorem, the author  obtained that the assertions
(ii) are equivalent.
We also need the following definition and theorem.
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
Let be a Fredholm operator of index zero, Kbe a normal cone in a Banach spaceX, , , and beL-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.
Then the equation has a minimal solution and a maximal solution in . Moreover,
In this section, we present some necessary basic knowledge and definitions about fractional calculus theory.
Definition 2.1 (see Equation 2.1.1 in )
The R-L fractional integral of order ( ) is defined by
Here is the gamma function.
Definition 2.2 (see Equation 2.1.5 in )
The R-L fractional derivative of order ( ) is defined by
where means the integral part of q.
If , , then, for , the relations
hold a.e. on .
Lemma 2.2 (see )
Let , , , then we have the equality
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.
We define the operators by
then BVPs (1.1) and (1.2) can be written as , .
Lemma 2.4If the operatorLis defined in (2.1), then
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
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
is the beta function defined by
By calculating, we easily obtain , , and , . We also define by
Lemma 2.5LetΩbe any open bounded subset of , then and are compact, which implies thatNisL-compact on for 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
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).
(H1) There exist such that and
(H2) For any , satisfying
where and , then problems (1.1) and (1.2) have a minimal solution and a maximal solution in , 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
Then from Theorem 1.2 we get and converge uniformly to and , respectively. Moreover,
We consider the following problem:
subject to BCs
We can choose
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 .
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
The authors would like to thank the referees for their many constructive comments and suggestions to improve the paper.
Infantea, G, Zima, M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal.. 69, 2458–2465 (2008). Publisher Full Text
Kosmatov, N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal.. 68, 2158–2171 (2008). Publisher Full Text
Yang, L, Shen, CF: On the existence of positive solution for a kind of multi-point boundary value problem at resonance. Nonlinear Anal.. 72, 4211–4220 (2010). Publisher Full Text
Wang, F, Cui, YJ, Zhang, F: Existence of nonnegative solutions for second order m-point boundary value problems at resonance. Appl. Math. Comput.. 217, 4849–4855 (2011). Publisher Full Text