In this section, we present our main results with respect to BVP (1.4).
For the sake of convenience, we define
number of zeros in the set
, and
number of
in the set 
Clearly,
or 2 and there are six possible cases:
-
(i)
-
(ii)
-
(iii)
-
(iv)
-
(v)
-
(vi)
Theorem 3.1.
BVP (1.4) has at least one positive solution in the case
and 
Proof.
First, we consider the case
and
Since
then there exists
such that
for
where
satisfies
If
with
then
It follows that if
then
for 
Since
then there exists
such that
for
where
is chosen such that
Set
and 
If
with
then
So that
In other words, if
then
Thus by
of Lemma 2.3, it follows that
has a fixed point in
with
.
Now we consider the case
and
Since
there exists
such that
for
, where
is such that
If
with
then we have
Thus, we let
so that
for 
Next consider
By definition, there exists
such that
for
, where
satisfies
Suppose
is bounded, then
for all
pick
If
with
then
Now suppose
is unbounded. From condition
it is easy to know that there exists
such that
for
If
with
then by using (3.8) we have
Consequently, in either case we take
so that for
we have
Thus by (ii) of Lemma 2.3, it follows that
has a fixed point
in
with 
The proof is complete.
Theorem 3.2.
Suppose
, and the following conditions hold,
-
(C3):
there exists constant
such that
for
where
-
(C4):
there exists constant
such that
for
where
furthermore,
Then BVP (1.4) has at least one positive solution
such that
lies between
and 
Proof.
Without loss of generality, we may assume that 
Let
for any
In view of
we have
which yields
Now set
for
we have
Hence by condition
we can get
So if we take
then
Consequently, in view of
(3.16), and (3.19), it follows from Lemma 2.3 that
has a fixed point
in
Moreover, it is a positive solution of (1.4) and 
The proof is complete.
For the case
or
we have the following results.
Theorem 3.3.
Suppose that
and
hold. Then BVP (1.4) has at least one positive solution.
Proof.
It is easy to see that under the assumptions, the conditions
and
in Theorem 3.2 are satisfied. So the proof is easy and we omit it here.
Theorem 3.4.
Suppose that
and
hold. Then BVP (1.4) has at least one positive solution.
Proof.
Since
for
there exists a sufficiently small
such that
Thus, if
, then we have
by the similar method, one can get if
then
So, if we choose
then for
we have
which yields condition
in Theorem 3.2.
Next, by
for
there exists a sufficiently large
such that
where we consider two cases.
Case 1.
Suppose that
is bounded, say
In this case, take sufficiently large
such that
then from (3.24), we know
for
which yields condition
in Theorem 3.2.
Case 2.
Suppose that
is unbounded. it is easy to know that there is
such that
Since
then from (3.23) and (3.25), we get
Thus, the condition
of Theorem 3.2 is satisfied.
Hence, from Theorem 3.2, BVP (1.4) has at least one positive solution.
The proof is complete.
From Theorems 3.3 and 3.4, we have the following two results.
Corollary 3.5.
Suppose that
and the condition
in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.
Corollary 3.6.
Suppose that
and the condition
in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.
Theorem 3.7.
Suppose that
and
hold. Then BVP (1.4) has at least one positive solution.
Proof.
In view of
similar to the first part of Theorem 3.1, we have
Since
for
there exists a sufficiently small
such that
Similar to the proof of Theorem 3.2, we obtain
The result is obtained, and the proof is complete.
Theorem 3.8.
Suppose that
and
hold. Then BVP (1.4) has at least one positive solution.
Proof.
Since
similar to the second part of Theorem 3.1, we have
for 
By
similar to the second part of proof of Theorem 3.4, we have
where
Thus BVP (1.4) has at least one positive solution.
The proof is complete.
From Theorems 3.7 and 3.8, we can get the following corollaries.
Corollary 3.9.
Suppose that
and the condition
in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.
Corollary 3.10.
Suppose that
and the condition
in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.
Theorem 3.11.
Suppose that
and the condition
of Theorem 3.2 hold. Then BVP (1.4) has at least two positive solutions
such that 
Proof.
By using the method of proving Theorems 3.1 and 3.2, we can deduce the conclusion easily, so we omit it here.
Theorem 3.12.
Suppose that
and the condition
of Theorem 3.2 hold. Then BVP (1.4) has at least two positive solutions
such that 
Proof.
Combining the proofs of Theorems 3.1 and 3.2, the conclusion is easy to see, and we omit it here.