Theorem 4.1.

Assume that (H1) and (H2) hold and that

. Then the BVP (1.1) has at least two positive solutions for

Proof.

Let

. Then it follows from (4.1) and Lemma 3.3 that

In view of Theorem 2.1, we have

Now, combined with the definition of

, we may choose

such that

for

and

uniformly, where

satisfies

In view of (4.1), (4.4), (4.5), and Lemma 3.2, we have

It follows from Theorem 2.1 that

By (4.3) and (4.7), we get

This shows that
has a fixed point in
, which is a positive solution of the BVP (1.1).

Now, by the definition of

, there exits an

such that

for

and

, where

is chosen so that

In view of (4.1), (4.9), and Lemma 3.2, we have

It follows from Theorem 2.1 that

By (4.3) and (4.12), we get

This shows that
has a fixed point in
, which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.1, we have the following results.

Theorem 4.2.

Suppose that (H1) and (H2) hold and

Then,

(i)equation (1.1) has at least one positive solution if
,

(ii)equation (1.1) has at least one positive solution if
,

(iii)equation (1.1) has at least two positive solutions if
.

Theorem 4.3.

Assume that (H1) and (H2) hold. If

, then the BVP (1.1) has at least two positive solutions for

Proof.

Let

. Then it follows from (4.15) and Lemma 3.3 that

In view of Theorem 2.1, we have

Since

, we may choose

such that

for

and

, where

satisfies

So,

In view of (4.15), (4.18), and Lemma 3.1, we have

It follows from Theorem 2.1 that

By (4.17) and (4.20), we get

This shows that
has a fixed point in
, which is a positive solution of the BVP (1.1).

Now, by the definition of

, there exists an

such that

for

and

, where

satisfies

Combined with (4.22) and Lemma 3.1, we have

It follows from Theorem 2.1 that

By (4.17) and (4.25), we get

This shows that
has a fixed point in
, which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.3, we have the following results.

Theorem 4.4.

Suppose that (H1) and (H2) hold and that

Then,

(i)equation (1.1) has at least one positive solution if
,

(ii)equation (1.1) has at least one positive solution if

,

- (iii)
equation (1.1) has at least two positive solutions if
.

Theorem 4.5.

Suppose that (H1) and (H2) hold. If

, then the BVP (1.1) has at least one positive solution for

Proof.

We only deal with the case that
,
. The other three cases can be discussed similarly.

Let

satisfy (4.28) and let

be chosen such that

From the definition of

, we know that there exists a constant

such that

for

and

. So,

This combines with (4.29) and Lemma 3.2, we have

It follows from Theorem 2.1 that

On the other hand, from the definition of

, there exists an

such that

for

and

. Let

. Then for

,

,

. So,

Combined with (4.29) and Lemma 3.1, we have

It follows from Theorem 2.1 that

By (4.32) and (4.35), we get

which implies that the BVP (1.1) has at least one positive solution in
.

Remark 4.6.

By making some minor modifications to the proof of Theorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:

(i)
,
and
.

(ii)
,
and
.

(iii)
,
and
.

(iv)
,
and
.

Remark 4.7.

From Conditions (ii) and (iv) of Remark 4.6, we know that the conclusion in Theorem 4.5 holds for

in these two cases. By

and

, there exist two positive constants

such that, for

,

This is the condition of Theorem

of [

13]. By

and

, there exist two positive constants

such that for

,

This is the condition of Theorem
of [13]. So, our conclusions extend and improve the results of [13].