We make the following assumptions throughout this paper:
(H0) A and B are functions of bounded variation satisfying for and ;
(H1) are decreasing in second and third variables and such that
(H2) for all , there exist constants such that, for any ,
Remark 3.1 The conditions (H1)-(H2) imply that f, g have a powder singularity at , and some typical functions are , , with and , , .
Theorem 3.1 Suppose (H0)-(H2) hold. Then the system (1.1) has at least a positive solution , which satisfies
where
In particular, if , then is positive solution of the system (1.1).
Proof Define a cone
(3.1)
then P is nonempty since . Now let us denote an operator T by
(3.2)
where
We claim that T is well defined and .
In fact, for any , we have
So from Lemma 2.1 and (H1)-(H2), one gets
(3.3)
and
(3.4)
On the other hand, by Lemma 2.1 and (H1)-(H2), we also have
(3.5)
and
(3.6)
Thus it follows from (3.3)-(3.6) that T is well defined and . Moreover, by Lemma 2.2, we have
(3.7)
Now take
(3.8)
(3.9)
since , , we have
(3.10)
Let
(3.11)
then by (3.8)-(3.11) and (H2), we have
(3.12)
(3.13)
Consequently, it follows from (3.7) and (3.10)-(3.13) that
(3.14)
(3.15)
and
(3.16)
(3.17)
(3.18)
It follows from (3.12) and (3.14)-(3.18) that , are lower and upper solutions of the system (1.1), and .
Define the functions , , and the operator in E by
(3.19)
(3.20)
and where
It follows from the assumption that and are continuous. Consider the following boundary value problem:
(3.21)
Obviously, a fixed point of the operator is a solution of the BVP (3.21).
For all , by (3.19)-(3.20), we have
So , which implies that is uniformly bounded. In addition, it follows from the continuity of , and the uniform continuity of , , and (H1) that is continuous.
Let be bounded, by standard discuss and the Arzela-Ascoli theorem, we easily know is equicontinuous. Thus is completely continuous, and by using Schauder fixed point theorem, has at least a fixed point such that .
Now we prove
(3.22)
We firstly prove . Otherwise, suppose . According to the definition of , , we have
(3.23)
On the other hand, as is an upper solution of (1.1), we have
(3.24)
Let , , (3.23)-(3.24) imply that
On the other hand, since is an upper solution of the BVP (1.1) and is a fixed point of , we know
It follows from Lemma 2.3 that
i.e., on , which contradicts . Thus we have on . In the same way, on . Consequently, (3.22) is satisfied; then is a positive solution of the problem (1.1).
It follows from and (3.22) that
The proof is completed. □