In this section, we will apply the Avery-Peterson fixed point theorem to establish the existence of at least three concave symmetric positive solutions of BVP (1.1).
Let α, γ, θ, ψ be maps on P with α a nonnegative continuous concave functional; γ, θ nonnegative continuous convex functionals, and ψ a nonnegative continuous functional. Then for positive numbers a, b, c, d we define the following subsets of P:
Now we state the Avery-Peterson fixed point theorem.
Theorem 3.1 ([2, 3, 9, 13])
Let P be cone in Banach space E. Let γ and θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P leading to
and
Suppose is completely continuous and there exist positive numbers a, b, c with such that
Then
T
has at least three fixed points
such that
Lemma 3.1 Assume that (H1) is satisfied and let . Then the operator T is completely continuous.
Proof For any , from the expression of Tu, we know
Clearly, Tu is concave. From the definition of Tu, Lemma 2.2, and Lemma 2.3 we see that Tu is nonnegative on . We now show that Tu is symmetric about . From Lemma 2.3 and (H1), for , we have
therefore, .
The continuity of T with respect to is clear. We now show that T is compact. Suppose that is a bounded set. Then there exists r such that
For any , we have
So, we have from (2.7)
and
These equations imply that the operator T is uniformly bounded. Now we show that Tu is equi-continuous.
We separate these three conditions:
Case (i). ;
Case (ii). ;
Case (iii). .
We solely need to deal with Case (i) since the proofs of the other two are analogous. For , we have
In addition
So, we see that Tu is equi-continuous. By applying the Arzela-Ascoli theorem, we can guarantee that is relatively compact, which means T is compact. Then we find that T is completely continuous. This completes the proof. □
For convenience, we denote
Theorem 3.2 Suppose (H1) holds and let . Moreover, there exist nonnegative numbers such that
then BVP (1.1) has at least three concave symmetric positive solutions , , such that
Proof BVP (1.1) has a solution if and only if u solves the operator equation . So we need to verify that operator T satisfies the Avery-Peterson fixed point theorem, which will prove the existence of at least three fixed points of T.
Complete continuity of T is clear from Lemma 3.1. Define the nonnegative functionals α, θ, γ, and ψ by
Then in the cone P, θ and γ are convex as α is concave. It is well known that for all and . Moreover, from Lemma 2.5, . Now, we will prove the main theorem in four steps.
Step 1. We will show that .
If , then . Lemma 2.5 yields , then the condition (B1) implies that . On the other hand, for any , there is , then is concave, symmetric, and positive on and , and we have
Then . Therefore .
Step 2. To check if condition (S1) of Theorem 3.1 is satisfied, we choose . Clearly,
Thus, and .
If , then we have , . From condition (B2), we have , and it follows that
This shows that condition (S1) of Theorem 3.1 is satisfied.
Step 3. We will show that condition (S2) of Theorem 3.1 is satisfied. Take with . Then from Lemma 2.6, we have
so condition (S2) holds.
Step 4. We will show that condition (S3) of Theorem 3.1 is also satisfied. Obviously, , and we have . Assume that with . Then, from condition (B3), we have
It proves that condition (S3) holds. All conditions of Theorem 3.1 are satisfied and we assert that BVP (1.1) has at least three concave symmetric positive solutions such that
Therefore, our proof is complete. □