For convenience we introduce the following notation:
and
where is a constant.
Proof of Theorem 2.1 Part (i). Noticing that , () for all t and , we can define
where , and
Let
Then, for and , , we have
which implies that
(4.1)
If , , then there exist , , and such that
where satisfies
(4.2)
satisfies
(4.3)
Let . Thus, when we have
and then we get
(4.4)
(4.5)
where
and
It follows from (4.4) and (4.5) that
(4.6)
Applying (b) of Lemma 3.4 to (4.1) and (4.6) shows that has a fixed point with . Hence, since for we have , , it follows that (2.2) holds. This gives the proof of part (i).
Part (ii). Noticing that , () for all t and , we can define
where , and
Let
Then, for and , , we have
which implies that
(4.7)
If , , then there exist , , and such that
where and satisfy (4.2) and (4.3), respectively.
Similar to the proof of (4.6), we can prove that
(4.8)
Applying (a) of Lemma 3.4 to (4.7) and (4.8) shows that has a fixed point with . Hence, since for we have for , it follows that (2.3) holds. This gives the proof of part (ii).
Consider part (iii). Choose two numbers and satisfying (2.1). By part (i) and part (ii), there exist and such that
(4.9)
Since , from the proof of part (i) and part (ii), it follows that
(4.10)
and
(4.11)
Applying Lemma 3.4 to (4.9)-(4.11) shows that has two fixed points and such that and . These are the desired distinct positive solutions of problem (1.1) for and satisfying (2.4). Then the result of part (iii) follows. □
Proof of Theorem 2.2 Part (i). Noticing that , () for all t and , we can define
where , and
Let
Then, for and , , we have
(4.12)
Similar to the proof of (4.5), we can prove
(4.13)
It follows from (4.12) and (4.13) that
(4.14)
If , , then there exist , , and such that
where satisfies
(4.15)
satisfies
Let . Thus, when we have
and then we get
This yields
(4.17)
Applying (b) of Lemma 3.4 to (4.14) and (4.17) shows that has a fixed point with . Hence, since for we have , , it follows that (2.2) holds. This gives the proof of part (i).
Part (ii). Noticing that , () for all t and , we can define
where , and
Let
Then, for and , , we have
(4.18)
Similar to the proof of (4.5), we can prove
(4.19)
It follows from (4.18) and (4.19) that
(4.20)
If , , then there exist , , and such that
where and satisfy (4.15) and (4.16), respectively.
Therefore, for , we obtain
This yields
(4.21)
Applying (a) of Lemma 3.4 to (4.20) and (4.21) shows that has a fixed point with . Hence, since for we have , , it follows that (2.3) holds. This gives the proof of part (ii).
Consider part (iii). Choose two numbers and satisfying (2.1). By part (i) and part (ii), there exist and such that
(4.22)
Since , from the proof of part (i) and part (ii), it follows that
(4.23)
and
(4.24)
Applying Lemma 3.4 to (4.22)-(4.24) shows that has two fixed points and such that and . These are the desired distinct positive solutions of problem (1.1) for and satisfying (2.5). Then the proof of part (iii) is complete. □
Remark 4.1 Comparing with Feng [31], the main features of this paper are as follows.
-
(i)
Two parameters and are considered.
-
(ii)
, not only for .
-
(iii)
It follows from the proof of Theorem 2.1 that the conditions of Corollary 3.2 in [31] are not the optimal conditions, which guarantee the existence of at least one positive solution for problem (1.1). In fact, if , or , , we can prove that problem (1.1) has at least one positive solution, respectively.