Theorem 3.1.
Let be an ordered Banach space, whose positive cone is normal, and , . Assume that the IPBVP(1.1) has coupled lower and upper quasisolutions and with . Suppose that the following conditions are satisfied:

(H1)
There exist constants and such that
for any and , .

(H2)
The impulsive function satisfies
for any and ,

(H3)
There exist a constant such that
for all and increasing or decreasing monotonic sequences and

(H4)
The sequences and are convergent, where , ,
Then the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and , respectively.
Proof.
By the definition of and Lemma 2.5, is continuous, and the coupled quasisolutions of the IPBVP(1.1) is equivalent to the coupled fixed point of operator Combining this with the assumptions and , we know is a mixed monotone operator (about the mixed monotone operator, please see [6, 7]).
Next, we show , . Let by (2.9), and , By Lemma 2.5
namely, . Similarly, it can be show that . So,
Now, we define two sequences and in by the iterative scheme
Then from the mixed monotonicity of , it follows that
We prove that and are uniformly convergent in
For convenience, let , , , and . Since, and by (2.11) and the boundedness of and we easy see that and is equicontinuous in every interval so, is equicontinuous in every interval where , , From and it follows that and for Let , by Lemma 2.3, . Going from to interval by interval we show that in
For from (2.11), using Lemma 2.1 and assumption and we have
Hence by the Belman inequality, in In particular, , this means that and are precompact in Thus and are precompact in and ,
Now, for by (2.11) and the above argument for we have
Again by Belman inequality, in from which we obtain that , and ,
Continuing such a process interval by intervai up to we can prove that in every ,
For any if we modify the value of , at via , , then and it is equicontinuous. Since , is precompact in for every By the ArzelaAscoli theorem, is precompact in Hence, has a convergent subsequence in Combining this with the monotonicity (3.6), we easily prove that itself is convergent in In particular, is uniformly convergent over the whole of Hence, is uniformly convergent in Set
Letting in (3.5) and (3.6), we see that and , By the mixed monotonicity of it is easy to see that and are the minimal and maximal coupled fixed points of in and therefore, they are the minimal and maximal coupled quasisolutions of the IPBVP(1.1) in respectively.
In Theorem 3.1, if is weakly sequentially complete, condition and hold automatically. In fact, by Theorem in [12], any monotonic and orderbounded sequence is precompact. By the monotonicity (3.6) and the same method in proof of Theorem 3.1, we can easily see that and are convergent on In particular, and are convergent. So, condition holds. Let and be increasing or decreasing sequences obeying condition then by condition , is a monotonic and orderbounded sequence, so Hence, condition holds. From Theorem 3.1, we obtain the following corollary.
Corollary 3.2.
Let be an ordered and weakly sequentially complete Banach space, whose positive cone is normal, and , If the IPBVP(1.1) has coupled lower and upper quasisolutions and with and conditions and are satisfied, then the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and respectively.
If we replace the assumption by the following assumption:

(H5)
There exist positive constants and such that
for any and ,
We have the following result.
Theorem 3.3.
Let be an ordered Banach space, whose positive cone is normal, and , If the IPBVP(1.1) has coupled lower and upper quasisolutions and with and conditions , , and hold, then the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and respectively.
Proof.
For let be a increasing sequence and be a decreasing sequence. For with by and
By this and the normality of cone we have
From this inequality and the definition of the measure noncompactness, it follows that
where If is a increasing sequence and is a decreasing sequence, the above inequality is also valid. Hence holds.
Therefore, by Theorem 3.1, the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and , respectively.
Now, we discuss the existence of the solution to the IPBVP(1.1) between the minimal and maximal coupled quasisolutions and If we replace the assumptions and by the following assumptions:
The impulsive function satisfies
for any and , and there exist , such that
for any countable sets and in
There exist a constant such that
for any where and are countable sets in
We have the following existence result.
Theorem 3.4.
Let be an ordered Banach space, whose positive cone is normal, and , If the IPBVP(1.1) has coupled lower and upper quasisolutions and with such that assumptions , , and hold, then the IPBVP(1.1) has minimal and maximal coupled quasisolutions and between and and at least has one solution between and
Proof.
We can easily see that , Hence, by the Theorem 3.1, the IPBVP(1.1) has minimal and maximal coupled quasisolutions and between and Next, we prove the existence of the solution of the equation between and Let clearly, is continuous and the solution of the IPBVP(1.1) is equivalent to the fixed point of operator Since is bounded and equicontinuous for any by Lemma 2.2, there exist a countable set such that
By assumptions and and Lemma 2.1,
Since is equicontinuous, by Lemma 2.3, . Combing (3.17) and .
We have
Hence, the operator is condensing, by the Lemma 2.4, has fixed point in
Lastly, since , by the mixed monotonity of
Similarly, in general, letting we get Therefore, the IPBVP(1.1) at least has one solution between and
Remark 3.5.
If and then Theorems 3.1, 3.3 and 3.4 are generalizations of the main results of [5] in Banach spaces.
Remark 3.6.
If and then Theorems 3.1, 3.3 and 3.4 are generalizations of the Theory of [4] in Banach spaces.