In this section, we discuss the BVP (1.9).

Assume that
,
.

with
.

Suppose that

is a solution of the BVP (3.1). Then it can be expressed as

Lemma 3.1 (see Guo et al. [11]).

Assume that

is a Banach space and

is a cone in

. Let

. Furthermore, assume that

is a completely continuous operator and

for

. Thus, one has the following conclusions:

- (1)

- (2)

Assume that

. Then (3.1) may be rewritten as

Let

be a solution of (3.3). Then by (3.2) and

, it can be expressed as

Let

be a solution of BVP (3.1) and

. Then for

we have

and

Then
is a Banach space endowed with norm
and
is a cone in
.

For

, we have by (H

) and the definition of

,

For every

,

, by the definition of

and (3.5), if

, we have

If

, we have, by (3.4),

hence by the definition of

, we obtain for

Lemma 3.2.

For every
, there is
, such that

Proof.

For

, and

, by the definitions of

and

, we have

Obviously, there is a
, such that (3.11) holds.

Define an operator

by

Then we may transform our existence problem of positive solutions of BVP (3.1) into a fixed point problem of operator (3.13).

Lemma 3.3.

Consider that
.

Proof.

If

and

,

and

, respectively. Thus, (H

) yields

It follows from the definition of

that

which implies that
.

Lemma 3.4.

Suppose that (
) holds. Then
is completely continuous.

We assume that

(H
)

(H
)
.

We have the following main results.

Theorem 3.5.

Assume that (
)–(
) hold. Then BVP (3.1) has at least one positive solution if the following conditions are satisfied:

(H
) there exists a
such that, for
, if
, then
;

(H
) there exists a
such that, for
, if
, then

or

(H
)
;

(H
) there exists a
such that, for
, if
, then
;

(H
) there exists an
, such that, for
, if
, then

Proof.

Assume that (

) and (

) hold. For every

, we have

, thus

which implies by Lemma 3.1 that

For every

, by (3.8)–(3.10) and Lemma 3.2, we have, for

,

. Then by (3.13) and (

), we have

which implies by Lemma 3.1 that

So by (3.18) and (3.20), there exists one positive fixed point
of operator
with
.

Assume that (

)–(

) hold, for every

and

,

, by (

), we have

Thus we have from Lemma 3.1 that

For every

, by (3.8)–(3.10), we have

,

Thus we have from Lemma 3.1 that

So by (3.22) and (3.24), there exists one positive fixed point
of operator
with
.

Consequently,
or
is a positive solution of BVP (3.1).

Theorem 3.6.

Assume that (
)–(
) hold. Then BVP (3.1) has at least one positive solution if (
) and (
) or (
) and (
) hold.

Theorem 3.7.

Assume that (
)–(
) hold. Then BVP (3.1) has at least two positive solutions if (H
), (H
), and (H
) or (H
), (H
), and (H
) hold.

Theorem 3.8.

Assume that (H
)–(H
) hold. Then BVP (3.1) has at least three positive solutions if (H
)–(H
) hold.

Assume that
,
, and

(
)

Define

as follows:

which satisfies

H
.

Obviously,
exists.

Assume that

is a solution of (1.9). Let

By (1.9), (3.26), (3.27), (H

), (H

), and the definition of

, we have

and, for

,

Then by (3.27), (H

), (H

), and the definition of

, we have

for

Thus, the BVP (1.9) can be changed into the following BVP:

with
and
.

Similar to the above proof, we can show that (1.9) has at least one positive solution. Consequently, (1.9) has at least one positive solution.

Example 3.9.

Consider the following BVP:

where

By calculation, we can see that (H
)–(H
) hold, then by Theorem 3.8, the BVP (3.33) has at least three positive solutions.