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Existence of Solutions for
-point Boundary Value Problems on a Half-Line
Advances in Difference Equations volume 2009, Article number: 609143 (2009)
Abstract
By using the Leray-Schauder continuation theorem, we establish the existence of solutions for -point boundary value problems on a half-line
, where
and
are given.
1. Introduction
Multipoint boundary value problems (BVPs) for second-order differential equations in a finite interval have been studied extensively and many results for the existence of solutions, positive solutions, multiple solutions are obtained by use of the Leray-Schauder continuation theorem, Guo-Krasnosel'skii fixed point theorem, and so on; for details see [1–4] and the references therein.
In the last several years, boundary value problems in an infinite interval have been arisen in many applications and received much attention; see [5, 6]. Due to the fact that an infinite interval is noncompact, the discussion about BVPs on the half-line is more complicated, see [5–14] and the references therein. Recently, in [15], Lian and Ge studied the following three-point boundary value problem:

where and
are given. In this paper, we will study the following
-point boundary value problems:

where have the same signal, and
are given. We first present the Green function for second-order multipoint BVPs on the half-line and then give the existence results for (1.2) using the properties of this Green function and the Leray-Schauder continuation theorem.
We use the space exists,
exists
with the norm
, where
is supremum norm on the half-line, and
is absolutely integrable on
with the norm
.
We set

and we suppose are the same signal in this paper and we always assume
2. Preliminary Results
In this section, we present some definitions and lemmas, which will be needed in the proof of the main results.
Definition 2.1 (see [15]).
It holds that  is called an S-Carathéodory function if and only if
-
(i)
for each
is measurable on
-
(ii)
for almost every
is continuous on
,
-
(iii)
for each
, there exists
with
on
such that
implies
, for a.e.
.
Lemma 2.2.
Suppose if for any
with
, then the BVP,

has a unique solution. Moreover, this unique solution can be expressed in the form

where is defined by

here note
Proof.
Integrate the differential equation from to
, noticing that
, then from
to
and one has

Since , from (2.4), it holds that

For , the unique solution of (2.1) can be stated by

If the unique solution of (2.1) can be stated by

If the unique solution of (2.1) can be stated by

We note then

Therefore, the unique solution of (2.1) is which completes the proof.
Remark of Lemma 2.2 . Obviously satisfies the properties of a Green function, so we call
the Green function of the corresponding homogeneous multipoint BVP of (2.1) on the half-line.
Lemma 2.3.
For all , it holds that

Proof.
For each ,
is nondecreasing in
. Immediately, we have

Further, we have

Therefore, we get the result.
Lemma 2.4.
For the Green function , it holds that

Lemma 2.5.
For the function it is satisfied that

and have the same signal,
, then there exists
satisfying

where .
Proof.
Let are positive, and note
, then for every
, we have
so
that is,
Because
is continuous on the interval
, there exists
satisfying
, where
.
Theorem 2.6 (see [5]).
Let . Then
is relatively compact in
if the following conditions hold:
-
(a)
is uniformly bounded in
;
-
(b)
the functions from
are equicontinuous on any compact interval of
;
-
(c)
the functions from
are equiconvergent, that is, for any given
, there exists a
such that
, for any
.
3. Main Results
Consider the space and define the operator
by

The main result of this paper is following.
Theorem 3.1.
Let be an S-Carathéodory function. Suppose further that there exists functions
with
such that

for almost every and all
. Then (1.2) has at least one solution provided:

Lemma 3.2.
Let be an S-Carathéodory function. Then, for each
is completely continuous in
.
Proof.
First we show is well defined. Let
; then there exists
such that
. For each
, it holds that

Further, is continuous in
so the Lebesgue dominated convergence theorem implies that


where . Thus,
.
Obviously, . Notice that

so we can get .
We claim that is completely continuous in
, that is, for each
,
is continuous in
and maps a bounded subset of
into a relatively compact set.
Let as
in
. Next we prove that for each
,
as
in
. Because
is a S-Carathéodory function and

where is a real number such that
, we have

Also, we can get


Similarly, we have

For any positive number , when
, we have

Combining (3.9)(3.13), we can see that
is continuous. Let
be a bounded subset; it is easy to prove that
is uniformly bounded. In the same way, we can prove (3.5),(3.6), and (3.12), we can also show that
is equicontinuous and equiconvergent. Thus, by Theorem 2.6,
is completely continuous. The proof is completed.
Proof of Theorem 3.1.
In view of Lemma 2.2, it is clear that is a solution of the BVP (1.2) if and only if
is a fixed point of
Clearly,
for each
If for each
the fixed points
in
belong to a closed ball of
independent of
then the Leray-Schauder continuation theorem completes the proof. We have known
is completely continuous by Lemma 3.2. Next we show that the fixed point of
has a priori bound
independently of
. Assume
and set

According to Lemma 2.5, we know that for any , there exists
satisfying
. Hence, there are three cases as follow.
Case 1 ().
For any holds and, therefore, there exists a
such that
. Then, we have

and so it holds that

therefore,

At the same time, we have

and so

Set , which is independent of
.
Case 2 ().
For any , we have

which implies that for all
. In the same way as for Case 1, we can get

Set , which is independent of
and is what we need.
Case 3 ().
For , we have

and so for all
.
Similarly, we obtain

Set and which is we need. So (1.2) has at least one solution.
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Acknowledgment
The Natural Science Foundation of Hebei Province (A2009000664) and the Foundation of Hebei University of Science and Technology (XL200759) are acknowledged.
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Yu, C., Guo, Y. & Ji, Y. Existence of Solutions for -point Boundary Value Problems on a Half-Line.
Adv Differ Equ 2009, 609143 (2009). https://doi.org/10.1155/2009/609143
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DOI: https://doi.org/10.1155/2009/609143