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]).

Lemma 2.4 (see [28]).

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
.

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

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).