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)

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 Arzela-Ascoli 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 order-bounded 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 order-bounded 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
.

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.