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 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
.
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.