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Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces
Boundary Value Problems volume 2009, Article number: 978605 (2009)
Abstract
The cone theory together with Mönch fixed point theorem and a monotone iterative technique is used to investigate the positive solutions for some boundary problems for systems of nonlinear second-order differential equations with multipoint boundary value conditions on infinite intervals in Banach spaces. The conditions for the existence of positive solutions are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is also derived.
1. Introduction
In recent years, the theory of ordinary differential equations in Banach space has become a new important branch of investigation (see, e.g., [1–4] and references therein). By employing a fixed point theorem due to Sadovskii, Liu [5] investigated the existence of solutions for the following second-order two-point boundary value problems (BVP for short) on infinite intervals in a Banach space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ1_HTML.gif)
where On the other hand, the multipoint boundary value problems arising naturally from applied mathematics and physics have been studied so extensively in scalar case that there are many excellent results about the existence of positive solutions (see, i.e., [6–12] and references therein). However, to the best of our knowledge, only a few authors [5, 13, 14] have studied multipoint boundary value problems in Banach spaces and results for systems of second-order differential equation are rarely seen. Motivated by above papers, we consider the following singular
-point boundary value problem on an infinite interval in a Banach space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ2_HTML.gif)
where and
with
In this paper, nonlinear terms
and
may be singular at
, and/or
, where
denotes the zero element of Banach space
. By singularity, we mean that
as
or
Very recently, by using Shauder fixed point theorem, Guo [15] obtained the existence of positive solutions for a class of th-order nonlinear impulsive singular integro-differential equations in a Banach space. Motivated by Guo's work, in this paper, we will use the cone theory and the Mönch fixed point theorem combined with a monotone iterative technique to investigate the positive solutions of BVP (1.2). The main features of the present paper are as follows. Firstly, compared with [5], the problem we discussed here is systems of multipoint boundary value problem and nonlinear term permits singularity not only at
but also at
. Secondly, compared with [15], the relative compact conditions we used are weaker. Furthermore, an iterative sequence for the solution under some normal type conditions is established which makes it very important and convenient in applications.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is worked out to illustrate the main results.
2. Preliminaries and Several Lemmas
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ3_HTML.gif)
Evidently, . It is easy to see that
is a Banach space with norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ4_HTML.gif)
and is also a Banach space with norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ6_HTML.gif)
Let with norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ7_HTML.gif)
Then is also a Banach space. The basic space using in this paper is
.
Let be a normal cone in
with normal constant
which defines a partial ordering in
by
. If
and
, we write
. Let
. So,
if and only if
. For details on cone theory, see [4].
In what follows, we always assume that . Let
. Obviously,
for any
. When
, we write
, that is,
. Let
and
. It is clear,
are cones in
and
, respectively. A map
is called a positive solution of BVP (1.2) if
and
satisfies (1.2).
Let denote the Kuratowski measure of noncompactness in
and
, respectively. For details on the definition and properties of the measure of noncompactness, the reader is referred to [1–4]. Let
be all Lebesgue measurable functions from
to
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ8_HTML.gif)
Let us list some conditions for convenience.
(H1) for any
and there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ9_HTML.gif)
uniformly for , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ10_HTML.gif)
(H2) For any and countable bounded set
, there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ11_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ12_HTML.gif)
where .
(H3) imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ13_HTML.gif)
In what follows, we write and
. Evidently,
, and
are closed convex sets in
and
, respectively.
We will reduce BVP (1.2) to a system of integral equations in . To this end, we first consider operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ14_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ16_HTML.gif)
Lemma 2.1.
If condition is satisfied, then operator
defined by (2.12) is a continuous operator from
into
.
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ18_HTML.gif)
By virtue of condition , there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ19_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ20_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ21_HTML.gif)
Let , we have, by (2.19)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ22_HTML.gif)
which together with condition implies the convergence of the infinite integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ23_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ24_HTML.gif)
which together with (2.13) and implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ25_HTML.gif)
Therefore, by (2.15) and (2.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ26_HTML.gif)
Differentiating (2.13), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ27_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ28_HTML.gif)
It follows from (2.24) and (2.25) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ29_HTML.gif)
So, . On the other hand, it can be easily seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ30_HTML.gif)
So, . In the same way, we can easily get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ31_HTML.gif)
where Thus,
maps
into
and we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ32_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ33_HTML.gif)
Finally, we show that is continuous. Let
. Then
is a bounded subset of
. Thus, there exists
such that
for
and
. Similar to (2.24) and (2.26), it is easy to have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ34_HTML.gif)
It is clear,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ35_HTML.gif)
and by (2.20),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ36_HTML.gif)
It follows from (2.33) and (2.34) and the dominated convergence theorem that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ37_HTML.gif)
It follows from (2.32) and (2.35) that as
. By the same method, we have
as
. Therefore, the continuity of
is proved.
Lemma 2.2.
If condition is satisfied, then
is a solution of BVP (1.2) if and only if
is a fixed point of operator
.
Proof.
Suppose that is a solution of BVP (1.2). For
integrating (1.2) from
to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ38_HTML.gif)
Integrating (2.36) from 0 to , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ40_HTML.gif)
Thus, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ41_HTML.gif)
which together with the boundary value conditions imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ43_HTML.gif)
Substituting (2.40) and (2.41) into (2.37) and (2.38), respectively, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ44_HTML.gif)
It follows from Lemma 2.1 that the integral and the integral
are convergent. Thus,
is a fixed point of operator
.
Conversely, if is fixed point of operator
, then direct differentiation gives the proof.
Lemma 2.3.
Let be satisfied,
is a bounded set. Then
and
are equicontinuous on any finite subinterval of
and for any
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ45_HTML.gif)
uniformly with respect to as
Proof.
We only give the proof for operator , the proof for operator
can be given in a similar way. By (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ46_HTML.gif)
For we obtain by (2.44)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ47_HTML.gif)
Then, it is easy to see by (2.45) and that
is equicontinuous on any finite subinterval of
.
Since is bounded, there exists
such that for any
. By (2.25), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ48_HTML.gif)
It follows from (2.46) and and the absolute continuity of Lebesgue integral that
is equicontinuous on any finite subinterval of
.
In the following, we are in position to show that for any there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ49_HTML.gif)
uniformly with respect to as
Combining with (2.45), we need only to show that for any there exists sufficiently large
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ50_HTML.gif)
for all as
The rest part of the proof is very similar to Lemma
in [5], we omit the details.
Lemma 2.4.
Let be a bounded set in
. Assume that
holds. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ51_HTML.gif)
Proof.
The proof is similar to that of Lemma in [5], we omit it.
Mönch Fixed-Point Theorem. Let be a closed convex set of
and
Assume that the continuous operator
has the following property:
countable,
is relatively compact. Then
has a fixed point in
.
Lemma 2.6.
If is satisfied, then for
imply that
Proof.
It is easy to see that this lemma follows from (2.13), (2.25), and condition . The proof is obvious.
Lemma 2.7 (see [16]).
Let and
are bounded sets in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ52_HTML.gif)
where and
denote the Kuratowski measure of noncompactness in
and
, respectively.
Lemma 2.8 (see [16]).
Let be normal (fully regular) in
,
then
is normal (fully regular) in
.
3. Main Results
Theorem 3.1.
If conditions and
are satisfied, then BVP (1.2) has a positive solution
satisfying
for
Proof.
By Lemma 2.1, operator defined by (2.13) is a continuous operator from
into
, and, by Lemma 2.2, we need only to show that
has a fixed point
in
. Choose
and let
. Obviously,
is a bounded closed convex set in space
. It is easy to see that
is not empty since
. It follows from (2.27) and (3.6) that
implies
, that is,
maps
into
. Let
satisfying
for some
. Then
We have, by (2.13) and (2.25),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ53_HTML.gif)
By Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ54_HTML.gif)
where , and
.
By (2.21), we know that the infinite integral is convergent uniformly for
So, for any
we can choose a sufficiently large
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ55_HTML.gif)
Then, by [1, Theorem ], (2.44), (3.1), (3.3),
, and Lemma 2.7, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ56_HTML.gif)
It follows from (3.2) and (3.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ57_HTML.gif)
In the same way, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ58_HTML.gif)
On the other hand, . Then, (3.5), (3.6),
, and Lemma 2.7 imply
that is,
is relatively compact in
Hence, the Mönch fixed point theorem guarantees that
has a fixed point
in
. Thus, Theorem 3.1 is proved.
Theorem 3.2.
Let cone be normal and conditions
be satisfied. Then BVP (1.2) has a positive solution
which is minimal in the sense that
for any positive solution
of BVP (1.2). Moreover,
and there exists a monotone iterative sequence
such that
as
uniformly on
and
as
for any
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ59_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ60_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ61_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ62_HTML.gif)
Proof.
From (3.7), one can see that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ63_HTML.gif)
By (3.7) and (3.11), we have that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ64_HTML.gif)
which imply that . Similarly, we have
. Thus,
. It follows from (2.13) and (3.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ65_HTML.gif)
By Lemma 2.1, we get and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ66_HTML.gif)
By Lemma 2.6 and (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ67_HTML.gif)
It follows from (3.14), by induction, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ68_HTML.gif)
Let Then,
is a bounded closed convex set in space
and operator
maps
into
. Clearly,
is not empty since
Let
Obviously,
and
Similar to above proof of Theorem 3.1, we can obtain
that is,
is relatively compact in
So, there exists an
and a subsequence
such that
converges to
uniformly on
Since that
is normal and
is nondecreasing, it is easy to see that the entire sequence
converges to
uniformly on
Since
and
are closed convex sets in space
, we have
It is clear,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ69_HTML.gif)
By and (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ70_HTML.gif)
Noticing (3.17) and (3.18) and taking limit as in (3.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ71_HTML.gif)
In the same way, taking limit as in (3.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ72_HTML.gif)
which together with (3.19) and Lemma 2.2 implies that and
is a positive solution of BVP (1.2). Differentiating (3.9) twice, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ73_HTML.gif)
Hence, by (3.17), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ74_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ75_HTML.gif)
Let be any positive solution of BVP (1.2). By Lemma 2.2, we have
and
for
It is clear that
for any
So, by Lemma 2.6, we have
for any
Assume that
for
Then, it follows from Lemma 2.6 that
for
that is,
for
Hence, by induction, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ76_HTML.gif)
Now, taking limits in (3.24), we get for
and the theorem is proved.
Theorem 3.3.
Let cone be fully regular and conditions
and
be satisfied. Then the conclusion of Theorem 3.2 holds.
Proof.
The proof is almost the same as that of Theorem 3.2. The only difference is that, instead of using condition , the conclusion
is implied directly by (3.15) and (3.16), the full regularity of
and Lemma 2.4.
4. An Example
Consider the infinite system of scalar singular second order three-point boundary value problems:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ77_HTML.gif)
Proposition 4.1.
Infinite system (4.1) has a minimal positive solution satisfying
for
Proof.
Let with the norm
. Obviously,
is a real Banach space. Choose
. It is easy to verify that
is a normal cone in
with normal constants 1. Now we consider infinite system (4.1), which can be regarded as a BVP of form (1.2) in
with
. In this situation,
in which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ78_HTML.gif)
Let . Then
for
. It is clear,
for any
. Notice that
for
, by (4.2), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ79_HTML.gif)
which imply is satisfied for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ80_HTML.gif)
Let ,
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ81_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ82_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ83_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ84_HTML.gif)
Let be given, and
be any sequence in
, where
. By (4.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ85_HTML.gif)
So, is bounded and by the diagonal method together with the method of constructing subsequence, we can choose a subsequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ86_HTML.gif)
which implies by virtue of (4.9)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ87_HTML.gif)
Hence It is easy to see from (4.9)–(4.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ88_HTML.gif)
Thus, we have proved that is relatively compact in
For any , we have by (4.6)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ89_HTML.gif)
where is between
and
. By (4.13), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ90_HTML.gif)
In the same way, we can prove that is relatively compact in
, and we can also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ91_HTML.gif)
Thus, by (4.14) and (4.15), it is easy to see that holds for
. Thus, our conclusion follows from Theorem 3.1. This completes the proof.
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The project is supported financially by the National Natural Science Foundation of China (10671167) and the Natural Science Foundation of Liaocheng University (31805).
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Zhang, X. Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces. Bound Value Probl 2009, 978605 (2009). https://doi.org/10.1155/2009/978605
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DOI: https://doi.org/10.1155/2009/978605