- Research Article
- Open Access

# Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces

- Xingqiu Zhang
^{1}Email author

**Received:**27 April 2009**Accepted:**12 June 2009**Published:**14 July 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.

## Keywords

- Banach Space
- Fixed Point Theorem
- Infinite System
- Iterative Sequence
- Infinite Interval

## 1. Introduction

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

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 us list some conditions for convenience.

where .

In what follows, we write and . Evidently, , and are closed convex sets in and , respectively.

Lemma 2.1.

If condition is satisfied, then operator defined by (2.12) is a continuous operator from into .

Proof.

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.

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.

uniformly with respect to as

Proof.

Then, it is easy to see by (2.45) and that is equicontinuous on any finite subinterval of .

It follows from (2.46) and and the absolute continuity of Lebesgue integral that is equicontinuous on any finite subinterval of .

uniformly with respect to as

for all as The rest part of the proof is very similar to Lemma in [5], we omit the details.

Lemma 2.4.

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]).

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.

where , and .

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.

Proof.

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

Proposition 4.1.

Infinite system (4.1) has a minimal positive solution satisfying for

Proof.

Thus, we have proved that is relatively compact in

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.

## Declarations

### Acknowledgment

The project is supported financially by the National Natural Science Foundation of China (10671167) and the Natural Science Foundation of Liaocheng University (31805).

## Authors’ Affiliations

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