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.