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.
From the above theorem, the author  obtained that the assertions
We also need the following definition and theorem.
In this section, we present some necessary basic knowledge and definitions about fractional calculus theory.
Definition 2.1 (see Equation 2.1.1 in )
Definition 2.2 (see Equation 2.1.5 in )
Lemma 2.2 (see )
Lemma 2.3 (see Corollary 2.1 in )
Let . It follows from Theorem 1.1.1 in  that K is a normal cone.
Lemma 2.4If the operatorLis defined in (2.1), then
3 Main result
In this section, we establish the existence of the nonnegative solution to equations (1.1) and (1.2).
Proof By condition (H1), we know that
so condition (C1) in Theorem 1.1 holds.
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
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