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
where
and
where
Now we are in the position to establish some new comparison principles which play an important role in the monotone iterative technique.
Lemma 2.1
Assume that
satisfies
(2.1)
where, , , , are constants and, , , and they satisfy
(2.2)
Then, .
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
(2.3)
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
Thus,
Let for some . We first assume that , then . By the mean value theorem, we have
Summing up, we get
Hence,
which contradicts (2.2).
For the case , the proof is similar, and thus we omit it. This completes the proof. □
Lemma 2.2
Assume that
satisfies
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 ,
and
Meanwhile, for , ,
Then by Lemma 2.1, we get for all , which implies that , . □
Consider the linear PBVP
(2.4)
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:
(2.5)
where
This proof is similar to the proof of Lemma 2.1 in [36], 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
and
On the other hand, for , we have
Similarly,
Thus,
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. □