where
denotes
or

In this section, we apply Lemma 1.1 to establish the existence of positive solutions for BVP (1.3).

Theorem 3.1.

Assume that
and
hold. In addition, letting
and
satisfy the following conditions:

and
;

or
,

BVP (1.3) has at least one positive solution.

Proof.

Considering

, there exists

such that

where

satisfy

Now, for

, we prove that

In fact, if there exists

such that

. Noticing (3.2), then we have

Therefore,
, which is a contraction. Hence, (3.2) holds.

Next, turning to

. Case (

).

. There exists

such that

where

. Choose

In fact, if there exists

such that

, then

This and (3.9) imply that

In fact, if
, then
, for
. Since
. Hence,
, which contracts
So, (3.15) holds. Therefore,
, this is also a contraction. Hence, (3.10) holds.

Case (

).

. There exists

such that

where

. If we define

, then

. Choose

We prove that (3.10) holds.

In fact, if there exists

such that

, then

This and (3.17) imply that

From (3.20), we obtain that

From the definition of

, we can find that

Similar to the proof in case (
), we can show that
. Then, from (3.23), we have
, which is a contraction. Hence, (3.10) holds.

Applying (i) of Lemma 1.1 to (3.2) and (3.10) yields that
has a fixed point
. Thus, it follows that BVP (1.3) has at least one positive solution, and the theorem is proved.

Theorem 3.2.

Assume that
and
hold. In addition, letting
and
satisfy the following conditions:

and
;

or
,

BVP (1.3) has at least one positive solution.

Proof.

Considering
, there exists
such that
, for
, where
satisfy
.

Similar to the proof of (3.2), we can show that

Next, turning to

. Under condition

, similar to the proof of (3.10), we can also show that

Applying (i) of Lemma 1.1 to (3.24) and (3.25) yields that
has a fixed point
. Thus, it follows that BVP (1.3) has one positive solution, and the theorem is proved.

Theorem 3.3.

Assume that
and
hold. In addition, letting
and
satisfy the following condition:

there is a

such that

and

implies

where
satisfy
, BVP (1.3) has at least two positive solutions
and
with
.

Proof.

We choose

with

. If

holds, similar to the proof of (3.2), we can prove that

If

holds, similar to the proof of (3.24), we have

In fact, if there exists

with

then by (2.23), we have

and it follows from

that

that is,
, which is a contraction. Hence, (3.29) holds.

Applying Lemma 1.1 to (3.27), (3.28), and (3.29) yields that
has two fixed points
with
. Thus it follows that BVP (1.3) has two positive solutions
with
. The proof is complete.

Our last results corresponds to the case when problem (1.3) has no positive solution. Write

Theorem 3.4.

Assume
, and
, then problem (1.3) has no positive solution.

Proof.

Assume to the contrary that problem (1.3) has a positive solution, that is,

has a fixed point

. Then

for

, and

which is a contradiction, and this completes the proof.

To illustrate how our main results can be used in practice we present an example.

Example 3.5.

Consider the following boundary value problem:

Conclusion.

BVP (3.34) has at least one positive solution.

Proof.

BVP (3.34) can be regarded as a BVP of the form (1.3), where

It is not difficult to see that conditions

and

hold. In addition,

Then, conditions
and
of Theorem 3.1 hold. Hence, by Theorem 3.1, the conclusion follows, and the proof is complete.