Let , , for . Let , and . and are Banach spaces with the norms and . Let . A function is called a solution of PBVP (1.1) if it satisfies (1.1).
Definition 2.1 We say that the functions are lower and upper solutions of PBVP (1.1), respectively, if there exist , , , , , , , such that
Now we are in the position to establish some new comparison principles which play an important role in the monotone iterative technique.
where, , , , are constants and, , , and they satisfy
Proof Suppose, to the contrary, that for some . We divide the proof into two cases:
Case (i). There exists a such that and for all .
From (2.1), we have for . Since , then is nondecreasing in and so . However, by (2.1) , then , which implies for all . Thus, , a contradiction.
Case (ii). There exists such that , .
Let , then there exists , for some , such that or . Without loss of generality, we only consider . For the case the proof is similar. It follows that
If for all , then , . Hence, is strictly increasing on J, which contradicts . Then there exists a such that .
Let , . By mean value theorem, we have
Summing up the above inequalities, we obtain
Let , . If by using the method to get (2.3), then we have
If , then the above method together with (2.1), (2.3) implies that
Let for some . We first assume that , then . By the mean value theorem, we have
Summing up, we get
which contradicts (2.2).
For the case , the proof is similar, and thus we omit it. This completes the proof. □
where, , , , are constants and, , , and they satisfy (2.2). Thenfor all.
Proof Let , , and define
Note that , for . If we prove that , then and the proof is complete. Since , then we get
Hence, . Indeed, for ,
Meanwhile, for , ,
Then by Lemma 2.1, we get for all , which implies that , . □
Consider the linear PBVP
where constants , , , , , , are constants and , , , .
Lemma 2.3is a solution of (2.4) if and only ifis a solution of the following impulsive integral equation:
This proof is similar to the proof of Lemma 2.1 in , and we omit it.
Lemma 2.4 Let, , , , are constants and, , . If
then (2.4) has a unique solution x in E.
Proof For any , we define an operator F by
where , are given by Lemma 2.3. Then and
By computing directly, we have
On the other hand, for , we have
By the Banach fixed-point theorem, F has a unique fixed point , and by Lemma 2.3, is also the unique solution of (2.4). This completes the proof. □