For convenience, we introduce the following notations:
We also define as [50] = number of zeros in the set and = number of infinities in the set . Sun and Li [51] pointed out that , and there are six possible cases: (i) and ; (ii) and ; (iii) and ; (iv) and ; (v) and ; and (vi) and . By using Krasnosel’skii’s fixed point theorem in a cone, some results are obtained for the existence of at least one or two positive solutions of problem (1.1) for on J under the above six possible cases.
3.1 For the case on J under and
In this subsection, we discuss the existence of a single positive solution for problem (1.1) for on J under and .
For convenience, we introduce the following notations:
Theorem 3.1 Assume that (H1)-(H4) hold. If and , then problem (1.1) has at least one positive solution.
Proof First, we consider the case and . Since , then there exists such that
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for any and , (2.9) and (2.12) imply
which implies
(3.1)
Next turning to , there exists satisfying such that
Since on J, it follows from on that
Let . Then, for , we have
Hence, for , it follows from (2.9) and (2.12) that
which implies
(3.2)
Thus by (i) of Lemma 2.6, it follows that T has a fixed point y in with
Lemma 2.5 implies that problem (1.1) has at least one positive solution x with
This gives the proof of Theorem 3.1. □
Remark 3.1 For and , there is another case and . However, at the moment, we give no information on the existence of a positive solution for problem (1.1) if we change and into and in Theorem 3.1.
3.2 For the case on J under and
In this subsection, we discuss the existence for the positive solutions of problem (1.1) under and . For convenience, we introduce the following notations:
and
Now, we shall state and prove the following main result.
Theorem 3.2 Suppose that (H1)-(H4) hold and on J. In addition, let the following two conditions hold:
(H5) There exist and such that ;
(H6) There exist and such that for , ; furthermore, .
Then problem (1.1) has at least one positive solution.
Proof Without loss of generality, we may assume that . Considering , we have for , .
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for any and , (2.9) and (2.12) imply
which implies
(3.3)
On the other hand, from (H6), when is fixed, there exists such that
for and . Since on J, it follows from on J that
Let . Then, for , we have
Hence, for , it follows from (2.9) and (2.12) that
which implies
(3.4)
Thus by (i) of Lemma 2.6, it follows that T has a fixed point y in with
Thus, it follows from Lemma 2.5 that problem (1.1) has at least one positive solution x with
This finishes the proof of Theorem 3.2. □
We remark that condition (H5) in Theorem 3.2 can be replaced by the following condition:
,
which is a special case of (H5).
Corollary 3.1 Suppose that (H1)-(H4), , (H6) hold and on J. Then problem (1.1) has at least one positive solution.
Proof We show that implies (H5). Suppose that holds. Then there exists a positive number such that
Hence, we obtain
Therefore, (H5) holds. Hence, by Theorem 3.2, problem (1.1) has at least one positive solution. □
Theorem 3.3 Suppose that (H1)-(H5) hold and on J. In addition, let the following condition hold:
(H7) .
Then problem (1.1) has at least one positive solution.
Proof The proof is similar to those of (3.2) and (3.3), respectively. □
Corollary 3.2 Suppose that (H1)-(H4), , (H7) hold and on J. Then problem (1.1) has at least one positive solution.
3.3 For the case on J under and or and
In this subsection, we discuss the existence for the positive solutions of problem (1.1) for the case on J under and or and .
Theorem 3.4 Suppose that (H1)-(H4) hold, on J and and . Then problem (1.1) has at least one positive solution.
Proof The proof is similar to that of Theorem 3.2. □
Theorem 3.5 Suppose that (H1)-(H4) hold, on J and and . Then problem (1.1) has at least one positive solution.
Proof Consider , then there exists such that for , .
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for , it follows from (2.9) and (2.12) that
which implies
(3.5)
Next, turn to . In fact, we can show that implies (H5).
Let . Then there exists such that for . Let
Then we have
This implies that . Hence, implies that (H5).
Similarly to the proof of (3.3), we have
(3.6)
where .
Thus by (ii) of Lemma 2.6, it follows that T has a fixed point y in with
This finishes the proof of Theorem 3.5. □
From Theorems 3.4 and 3.5, we have the following result.
Corollary 3.3 Suppose that and condition (H6) in Theorem 3.2 hold. Then problem (1.1) has at least one positive solution.
Theorem 3.6 Suppose that (H1)-(H4), on J, and . Then problem (1.1) has at least one positive solution.
Proof The proof is similar to that of Theorem 3.2. □
Theorem 3.7 Suppose (H1)-(H4), on J, and . Then problem (1.1) has at least one positive solution.
Proof The proof is similar to that of Theorem 3.2. □
From Theorems 3.6 and 3.7, the following corollaries are easily obtained.
Corollary 3.4 Suppose that and condition (H5) in Theorem 3.2 hold. Then problem (1.1) has at least one positive solution.
Corollary 3.5 Suppose that and condition (H5) in Theorem 3.2 hold. Then problem (1.1) has at least one positive solution.
3.4 For the case on J under and or and
In this subsection we study the existence of multiple positive solutions for problem (1.1) for the case on J under and or and .
Combining the proofs of Theorems 3.1 and 3.2, the following theorem is easily proved.
Theorem 3.8 Suppose that (H1)-(H4), on J, and and condition (H5) of Theorem 3.2 hold. Then problem (1.1) has at least two positive solutions.
Corollary 3.6 Suppose that (H1)-(H4), on J, and and condition of Corollary 3.1 hold. Then problem (1.1) has at least two positive solutions.
Remark 3.2 Noticing Remark 3.1, at the moment we give no information on the existence of a positive solution for problem (1.1) under and .