Existence of Solutions for -point Boundary Value Problems on a Half-Line

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:

(1.1)

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

(1.2)

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

(1.3)

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,

(2.1)

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

(2.2)

where is defined by

(2.3)

here note

Proof.

Integrate the differential equation from to , noticing that , then from to and one has

(2.4)

Since , from (2.4), it holds that

(2.5)

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

(2.6)

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

(2.7)

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

(2.8)

We note then

(2.9)

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

(2.10)

Proof.

For each , is nondecreasing in . Immediately, we have

(2.11)

Further, we have

(2.12)

Therefore, we get the result.

Lemma 2.4.

For the Green function , it holds that

(2.13)

Lemma 2.5.

For the function it is satisfied that

(2.14)

and have the same signal, , then there exists satisfying

(2.15)

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

(3.1)

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

(3.2)

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

(3.3)

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

(3.4)

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

(3.5)

(3.6)

where . Thus, .

Obviously, . Notice that

(3.7)

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

(3.8)

where is a real number such that , we have

(3.9)

Also, we can get

(3.10)

(3.11)

Similarly, we have

(3.12)

For any positive number , when , we have

(3.13)

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

(3.14)

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

(3.15)

and so it holds that

(3.16)

therefore,

(3.17)

At the same time, we have

(3.18)

and so

(3.19)

Set , which is independent of .

Case 2 ().

For any , we have

(3.20)

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

(3.21)

Set , which is independent of and is what we need.

Case 3 ().

For , we have

(3.22)

and so for all .

Similarly, we obtain

(3.23)

Set and which is we need. So (1.2) has at least one solution.