Definition 2.1 Let E be a Banach space, a nonempty convex closed set is said to be a cone provided the following hypotheses are satisfied:
-
(i)
if , , then ;
-
(ii)
if and , then .
Every cone induces a partial ordering ‘⩽’ on E defined by
Definition 2.2 Let be an ordered Banach space. An operator is said to be nondecreasing (nonincreasing) provided that () for all with . If the inequality is strict, then φ is said to be strictly nondecreasing (nonincreasing).
Definition 2.3 Let , is said to be concave on if
for any and .
We consider the Banach space equipped with the norm , where . In this paper, a symmetric positive solution of (1.1) means a function which is symmetric and positive on and satisfies equation (1.1) as well as the boundary conditions (1.2).
In this paper, we always suppose that the following assumptions hold:
(H1) , for , and for all ;
(H2) is nondecreasing for each , is nondecreasing for ;
(H3) is nonnegative and , where .
Denote
It is easy to see that P is a cone in E.
For any , suppose that u is a solution of the following BVP:
Then we can easily get the solution:
(2.1)
where
and
During the process of getting the above solution, we can also know
(2.2)
for .
Lemma 2.1 If (H3) is satisfied, the following results are true:
-
1.
, for ; for .
-
2.
, for .
For any , is defined
(2.3)
Lemma 2.2 If (H3) is satisfied, is completely continuous, i.e., T is continuous and compact. Moreover, T is nondecreasing provided that (H2) holds.
Proof For any , from the definition of Ty, we know
Obviously, Ty is concave. From the expression of Ty, combining with Lemma 2.1, we know that Ty is nonnegative on . We now prove that Ty is symmetric about .
For , then, and
Similarly, we have
So, . The continuity of T is obvious. We now prove that T is compact. Let be a bounded set. Then there exists R such that
For any , we have
Therefore, from (2.3), we have
So, is uniformly bounded. Now we prove Ty is equi-continuous. For , we have
Moreover,
And the similar results can be obtained for and .
The Arzelà-Ascoli theorem guarantees that T Ω is relatively compact, which means T is compact.
Finally, we show that Ty is nondecreasing about y.
For any () with . By the properties of a cone, we have for . Then is concave and symmetric about . Therefore,
Hence, for , by (H2) and the definition of Ty, we have
Furthermore, we have
In order to prove is concave, we need to prove is nonincreasing. Let , then
A similar result can be obtained for . And it is easy to see that is symmetric about . So, and thus T is nondecreasing. □