Define the operator by
(3.1)
where are defined by (2.1). It is clear that u is a solution of (1.1) if and only if u is a fixed point of T.
Theorem 3.1 Assume (H1)-(H3) hold. Then BVP (1.1) has at least one symmetric positive solution.
Proof
Claim 3.1 is completely continuous and nondecreasing.
In fact, for , it is obvious that , for and . (2.3), (2.9) and a change of variables imply
(3.2)
For any , from (2.4), (2.6), (2.8), and (H3), we have
(3.3)
(3.4)
for , where satisfies
Thus, it follows from (3.3) and (3.4) that , and so . Next by a standard method and the Ascoli-Arzela theorem one can prove that is completely continuous, we omit it here. From (H2), it is easy to see that T is nondecreasing in u. Hence, Claim 3.1 holds.
Claim 3.2
Let
be fixed number satisfying
(3.5)
where λ is defined in (H2) in which , and assume
(3.6)
(3.7)
Then
(3.8)
and there exists
such that
(3.9)
In fact, since . So, from (3.5) and noting that , . From (3.6), we have and .
On the other hand, from (2.4) and (2.6), we have
Since and T is nondecreasing, by induction, (3.8) holds.
Let , then . It follows from
that, for any natural number n,
Thus, for all natural numbers n and p, we have
which implies that there exists such that (3.9) holds, and Claim 3.2 holds.
Letting in (3.7), we obtain , which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem. □
Theorem 3.2 Assume (H1), (H2′) and (H3) hold. Then BVP (1.1) has at least one symmetric positive solution.
Proof
Claim 3.3 is completely continuous and nonincreasing.
The proof of Claim 3.3 is similar to the proof of Claim 3.1, so this is omitted.
Claim 3.4 Let be fixed number, be sufficiently large constant satisfying
(3.10)
where λ is defined in (H2′) in which , and assume
(3.11)
Then
(3.12)
and there exists
such that
(3.13)
In fact, since and . So from (3.11),
From (2.5), (3.10), (3.14), and noting that T is nonincreasing in u, we have
(3.15)
(3.16)
Therefore,
(3.17)
(3.18)
From (3.15), (3.17), (3.18), and noting that is nondecreasing, by induction, (3.12) holds.
On the other hand, from (2.5) and (2.7), for ,
(3.19)
Then from (3.16) and (3.19), we have
and thus
Therefore, for all natural numbers n and p, we have
(3.20)
(3.21)
From (3.20) and (3.21), there exists such that (3.13) holds, and Claim 3.4 holds.
Letting in (3.11), we obtain , which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem. □
Remark 3.1 [3, 5] only considered that f is nondecreasing or nonincreasing in u, and , in (1.1), so our results extend the work in the literature.
Example 3.1 Consider the BVP
(3.22)
where for , , , .
It is easy to see that function satisfies (H1) and (H3). If , there exists constant λ with such that , so (H2) is also satisfied. Therefore, from Theorem 3.1, (3.22) has at least one symmetric positive solution.