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).

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

Then

and

for

If we define

then

and

which together with the definition of

implies that

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.