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).