For convenience, we denote
and
. In the remainder of this paper, the following notations will be used:
(1)
if at least one of
approaches
;
(2)
if
for
;
(3)
if
for
and at least one of them is strict.
Let
. Then
is a Banach space, where
is defined as usual by the sup norm.
Our main result is the following theorem.
Theorem 3.1.
Assume that (A1)–(A3) hold. Then the BVP (1.5)–(1.7) has a unique positive solution
for any
, where
. Furthermore, such a solution
satisfies the following properties:
(P1) 
(P2)
is strictly increasing in
, that is,
(P3)
is continuous in
, that is, for any given 
Proof.
Let
. Then
is a normal solid cone in
with
For any
, if we define an operator
as follows:
then it is not difficult to verify that
is a positive solution of the BVP (1.5)–(1.7) if and only if
is a fixed point of
.
Now, we will prove that
has a unique fixed point by using Lemma 2.3.
First, in view of Lemma 2.5, we know that 
Next, we claim that
is a
-concave operator.
In fact, for any
and
it follows from (3.3) and (A3) that
which shows that
is
-concave.
Finally, we assert that
is an increasing operator.
Suppose that
and
By (3.3) and (A2), we have
which indicates that
is increasing.
Therefore, it follows from Lemma 2.3 that
has a unique fixed point
which is the unique positive solution of the BVP (1.5)–(1.7). The first part of the theorem is proved.
In the rest of the proof, we will prove that such a positive solution
satisfies properties (P1), (P2), and (P3).
First,
which together with
for
implies (P1).
Next, we show (P2). Assume that
Let
Then
for
We assert that
Suppose on the contrary that
Since
is a
-concave increasing operator and for given
,
is strictly increasing in
, we have
which contradicts the definition of
Thus, we get
for
And so,
which indicates that
is strictly increasing in
.
Finally, we prove (P3). For any given
we first suppose that
with
From (P2), we know that
Let
Then
and
for
If we define
then
and
which together with the definition of
implies that
So,
Therefore,
In view of (3.10) and (3.16), we obtain that
which together with the fact that
as
shows that
Similarly, we can also prove that
Hence, (P3) holds.