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.