For the convenience of the reader, we present here the necessary definitions and properties from fractional calculus theory, which are used throughout this paper.

Definition 2.1.

The Riemann-Liouville fractional integral of order of a function is given by

provided that the right-hand side is pointwise defined on .

Definition 2.2.

The Caputo fractional derivative of order of a continuous function is given by

where , , provided that the right-hand side is pointwise defined on .

Lemma 2.3 (see [28]).

Let , , , then

Lemma 2.4 (see [28]).

The relation

is valid when , , .

Lemma 2.5 (see [9]).

Let , ; is a continuous function and . If is continuous function on , then the function

is continuous on , where

Lemma 2.6.

Let , ; is a continuous function and . If is continuous function on , then the boundary value problems (1.2) are equivalent to the Volterra integral equations

Proof.

From Lemma 2.5, the Volterra integral equation (2.7) is well defined. If satisfies the boundary value problems (1.2), then applying to both sides of (1.2) and using Lemma 2.3**,** one has

where , . Since is continuous in , there exists a constant , such that , for . Hence

where denotes the beta function. Thus, as . In the similar way, we can prove that as .

By Lemma 2.4 we have

From the boundary conditions , one has

Therefore, it follows from (2.8) that

Namely, (2.7) follows.

Conversely, suppose that satisfies (2.7), then we have

From Lemmas 2.3 and 2.4 and Definition 2.2, one has

as well as

Thus, from (2.12), (2.14), and (2.15), it is follows that

Namely, (1.2) holds. The proof is therefore completed.

Remark 2.7.

For , since , we can obtain

Hence, it is follow from (2.6) that , for and .

Let is the Banach space endowed with the infinity norm, is a nonempty closed subset of defined as . The positive solution which we consider in this paper is a function such that .

According to Lemma 2.6, (1.2) is equivalent to the fractional integral equation (2.7). The integral equation (2.7) is also equivalent to fixed-point equation , , where operator is defined as

then we have the following lemma.

Lemma 2.8 (see [9]).

Let , , is a continuous function and . If is continuous function on , then the operator is completely continuous.

Let , , is a continuous function, , and is continuous function on . Take , and . For any , , we define the upper-control function , and lower-control function , it is obvious that are monotonous non-decreasing on and .

Definition 2.9.

Let , , and satisfy, respectively

then the function are called a pair of order upper and lower solutions for (1.2).