Lemma 2.1.

The BVP (1.2) is equivalent to the integral equation

is called the Green's function for the corresponding linear BVP, here

is the Green's function for the BVP:

Proof.

Let

be a solution of the BVP:

Then, it is easy to know that

Now, if

is a solution of the BVP (1.2), then it can be expressed by

which together with the boundary conditions in (1.2) and (2.6) implies that

On the other hand, if
satisfies (2.1), then it is easy to verify that
is a solution of the BVP (1.2).

Lemma 2.2.

For any

one has

Proof.

Since it is obvious from the expression of

that

we know that (2.9) is fulfilled.

Our main result is the following theorem.

Theorem 2.3.

Assume that (H1)–(H3) are satisfied. Then, the BVP (1.2) has at least one positive solution

. Furthermore, there exist

such that

Proof.

Define an operator

:

Then it is obvious that fixed points of the operator
in
are positive solutions of the BVP (1.2).

First, in view of (H2), it is easy to know that
is increasing.

Next, we may assert that

, which implies that for any

, there exist positive constants

and

such that

In fact, for any

, there exist

such that

which together with (H2), (H3), and Remark 1.1 implies that

By Lemma 2.2 and (2.16), for any

, we have

then it follows from (2.17) and (2.18) that

which shows that
.

Now, for any fixed

, we denote

Then, it is easy to know from (2.20), (2.21), (2.22), (2.23), (2.24), (H3), and Remark 1.1 that

Moreover, if we let

, then it follows from (2.22), (2.23), (2.24), and (H3) by induction that

which together with (2.25) implies that for any positive integers

and

,

Therefore, there exists a

such that

and

converge uniformly to

on

and

Since

is increasing, in view of (2.28), we have

which shows that

is a positive solution of the BVP (1.2). Furthermore, since

, there exist

such that