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
So,
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
Let . Then for , , . So,
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
Let . Then for , , . So,
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].