In this section, we give the definition of a fractional derivative and some lemmas, which will be used later.
Definition 2.1 
Let for a function . The fractional integral of order α of y is defined by
provided the integral exists. The Caputo derivative of a function is given by
provided the right-hand side is pointwise defined on , where n is an integer, with .
Γ denotes the gamma function, that is,
From Definition 2.1, we can obtain the following lemma.
Lemma 2.1 Let . If we assume , the fractional differential equation
has a unique solution
Throughout this paper, we always suppose the following condition holds.
(H0) The parameters in the boundary value problem (1) satisfy the following conditions:
Lemma 2.2 Suppose that (H0) holds and . Then the boundary value problem
has a unique solution
That is, every solution of (2) is also a solution of (3) and vice versa.
Proof The definition of the Caputo derivative implies that , and from (2) we have
By using the property of the fractional derivatives and integrals, we can get
From the boundary condition , and , we can obtain that
Substituting (5) and (6) into (4), we can obtain that
The proof is completed. □
Let , , then is a Banach space. Set
then P is a cone on E.
Define the operator by
It is clear that is the solution of the boundary value problem (1) if and only if is the fixed point of the operator T.
Lemma 2.3 Suppose that (H0) holds and the function . Then is completely continuous.
Proof By the definition of the operator T, it is easy to see for any . And using the property of the fractional integrals and derivatives, we can get that
Then Tx is nonnegative, monotone increasing and convex on . They imply that
Thus, , so .
It is easy to prove that T is continuous and compact if the conditions of the lemma hold.
The proof is complete. □
For convenience of the readers, we provide some background material from the theory of cones in Banach spaces and the Avery-Peterson fixed point theorem.
Definition 2.2 Let E be a Banach space and let be a cone. A continuous map γ is called a concave (resp. convex) functional on P if and only if (resp. ) for all and .
Let β and ρ be nonnegative continuous convex functionals on a cone P, let ω be a nonnegative continuous concave functional on a cone P, and let ψ be a nonnegative continuous functional on a cone P. Then, for positive real numbers a, b, c and d, we define the following convex sets:
Lemma 2.4 (Avery-Peterson fixed point theorem )
Let P be a cone in a real Banach space E. Let β and ρ be nonnegative continuous convex functionals on P, let ω be a nonnegative continuous concave functional on P, and let ψ be a nonnegative continuous functional on P satisfying for , such that for some positive numbers M and d, and for all .
Suppose that is completely continuous and there exist positive numbers a, b and c with such that
(A1) , and for all ;
(A2) for all with ;
(A3) and for with .
Then T has at least three fixed points such that , ; ; , with ; and .