Theorem 3.1 Let a cone P be normal and condition () be satisfied. If () and () or () and () are satisfied, then BVP (1.1) has at least one positive solution.
Proof Set
It is clear that is a cone of the Banach space and . For any , by (2.1), we can obtain , then
We first assume that () and () are satisfied. Let
In the following, we prove that W is bounded.
For any , we have , that is, , . And so , set , by ()
(3.1)
For , let , then is a bounded linear operator. From (3.1), one deduces that
Since is the first eigenvalue of T, by (), the first eigenvalue of , . Therefore, by [14], the inverse operator exists and
It follows from that . So, we know that , and W is bounded.
Taking , we have
(3.2)
Next, we are going to verify that for any ,
If this is false, then there exists such that . This together with () yields
For , let , then the above inequality can be written in the form
(3.4)
It is easy to see that
In fact, implies for , and consequently, in contradiction to . Now, notice that is a -positive operator with , then by Lemma 2.2, we have for some . This together with and (3.4) implies that
which is a contradiction to . So, (3.3) holds.
By Lemma 2.4, A is a strict set contraction on . Observing (3.2) and (3.3) and using Theorem 2.1, we see that A has a fixed point on .
Next, in the case that () and () are satisfied, by the method as in establishing (3.3), we can assert from () that for any ,
(3.5)
Let
It is clear that is a completely continuous linear -operator with and in which . In addition, the spectral radius and is the positive eigenfunction of corresponding to its first eigenvalue .
Let
where . It is clear that is a completely continuous linear -operator with and . Thus, the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
Take () satisfying and (). For , , we have
By [12], we have . Let , by Gelfand’s formula, we have . Let as .
In the following, we prove that .
Let be the positive eigenfunction of corresponding to , i.e.,
(3.6)
satisfying . Without loss of generality, by standard argument, we may suppose by the Arzela-Ascoli theorem and that as . Thus, and by (3.6), we have
that is, . This together with Lemma 2.2 guarantees that .
By the above argument, it is easy to see that there exists a such that
Choose
(3.7)
Now, we assert that
If this is not true, then there exists with such that , then . Moreover, by the definition of , we know
Thus, , , which implies by (3.7) we have
and
So, by (), we get
It is easy to see that
In fact, implies for , and consequently, in contradiction to . Now, notice that is a -positive operator with . Then by Lemma 2.2, we have for some , where is the positive eigenfunction of corresponding to . This together with implies that
which is a contradiction to . So, (3.8) holds.
By Lemma 2.4, A is a strict set contraction on . Observing (3.5) and (3.8) and using Theorem 2.1, we see that A has a fixed point on . This together with Lemma 2.3 implies that BVP (1.1) has at least one positive solution. □
Theorem 3.2 Let a cone P be normal. Suppose that conditions (), (), () and () are satisfied. Then BVP (1.1) has at least two positive solutions.
Proof We can take the same as in Theorem 3.1. As in the proof of Theorem 3.1, we can also obtain that . And we choose , with such that
On the other hand, it is easy to see that
(3.11)
In fact, if there exists with such that , then observing and , we get
and so
(3.12)
where, by virtue of (),
(3.13)
It follows from (3.12) and (3.13) that
a contradiction. Thus (3.11) is true.
By Lemma 2.4, A is a strict set contraction on , and also on . Observing (3.9), (3.10), (3.11) and applying, respectively, Theorem 2.1 to A, and , we assert that there exist and such that and and, by Lemma 2.3 and (3.11), , are positive solutions of BVP (1.1). □
Theorem 3.3 Let a cone P be normal. Suppose that conditions (), () and () and () are satisfied. Then BVP (1.1) has at least two positive solutions.
Proof We can take the same as in Theorem 3.1. As in the proof of Theorem 3.1, we can also obtain that . And we choose , with such that
On the other hand, it is easy to see that
In fact, if there exists with such that , then
Observing and , we get
which is a contradiction. Hence, (3.16) holds.
By Lemma 2.4, A is a strict set contraction on and also on . Observing (3.14), (3.15), (3.16) and applying, respectively, Theorem 2.1 to A, and , we assert that there exist and such that and and, by Lemma 2.3 and (3.16), , are positive solutions of BVP (1.1). □