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# Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems

*Boundary Value Problems*
**volume 2009**, Article number: 273063 (2009)

## Abstract

Existence of positive solutions for a coupled system of nonlinear three-point boundary value problems of the type , , , , , , , is established. The nonlinearities , are continuous and may be singular at , and/or , while the parameters , satisfy . An example is also included to show the applicability of our result.

## 1. Introduction

Multipoint boundary value problems (BVPs) arise in different areas of applied mathematics and physics. For example, the vibration of a guy wire composed of parts with a uniform cross-section and different densities in different parts can be modeled as a Multipoint boundary value problem [1]. Many problems in the theory of elastic stability can also be modeled as Multipoint boundary value problem [2].

The study of Multipoint boundary value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev, [3, 4], and extended to nonlocal linear elliptic boundary value problems by Bitsadze et al. [5, 6]. Existence theory for nonlinear three-point boundary value problems was initiated by Gupta [7]. Since then the study of nonlinear three-point BVPs has attracted much attention of many researchers, see [8–11] and references therein for boundary value problems with ordinary differential equations and also [12] for boundary value problems on time scales. Recently, the study of singular BVPs has attracted the attention of many authors, see for example, [13–18] and the recent monograph by Agarwal et al. [19].

The study of system of BVPs has also fascinated many authors. System of BVPs with continuous nonlinearity can be seen in [20–22] and the case of singular nonlinearity can be seen in [8, 21, 23–26]. Wei [25], developed the upper and lower solutions method for the existence of positive solutions of the following coupled system of BVPs:

where , and may be singular at , , and/or .

By using fixed point theorem in cone, Yuan et al. [26] studied the following coupled system of nonlinear singular boundary value problem:

are allowed to be superlinear and are singular at and/or . Similarly, results are studied in [8, 21, 23].

In this paper, we generalize the results studied in [25, 26] to the following more general singular system for three-point nonlocal BVPs:

where , , . We allow and to be singular at , , and also and/or . We study the sufficient conditions for existence of positive solution for the singular system (1.3) under weaker hypothesis on and as compared to the previously studied results. We do not require the system (1.3) to have lower and upper solutions. Moreover, the cone we consider is more general than the cones considered in [20, 21, 26].

By singularity, we mean the functions and are allowed to be unbounded at , , , and/or . To the best of our knowledge, existence of positive solutions for a system (1.3) with singularity with respect to dependent variable(s) has not been studied previously. Moreover, our conditions and results are different from those studied in [21, 24–26]. Throughout this paper, we assume that are continuous and may be singular at , , , and/or . We also assume that the following conditions hold:

(*A*_{1}) and satisfy

(*A*_{2})There exist real constants such that , , and for all , ,

(*A*_{3})There exist real constants such that , , and for all , ,

for example, a function that satisfies the assumptions and is

where , , ; such that

The main result of this paper is as follows.

Theorem 1.1.

Assume that hold. Then the system (1.3) has at least one positive solution.

## 2. Preliminaries

For each , we write . Let . Clearly, is a Banach space and is a cone. Similarly, for each , we write . Clearly, is a Banach space and is a cone in . For any real constant , define . By a positive solution of (1.3), we mean a vector such that satisfies (1.3) and , on . The proofs of our main result (Theorem 1.1) is based on the Guo's fixed-point theorem.

Lemma 2.1 (Guo's Fixed-Point Theorem [27]).

Let be a cone of a real Banach space , , be bounded open subsets of and . Suppose that is completely continuous such that one of the following condition hold:

(i) for and for ;

(ii) for and for .

Then, has a fixed point in .

The following result can be easily verified.

Result.

Let such that . Let , and concave on . Then, for all .

Choose such that . For fixed and , the linear three-point BVP

has a unique solution

where is the Green's function and is given by

We note that as , where

is the Green's function corresponding the boundary value problem

whose integral representation is given by

Lemma 2.2 (see [9]).

Let . If and , then then unique solution of the problem (2.5) satisfies

where .

We need the following properties of the Green's function in the sequel.

Lemma 2.3 (see [11]).

The function can be written as

where

Following the idea in [10], we calculate upper bound for the Green's function in the following lemma.

Lemma 2.4.

The function satisfies

where

Proof.

For , we discuss various cases.

Case 1.

, ; using (2.3), we obtain

If , the maximum occurs at , hence

and if , the maximum occurs at , hence

Case 2.

, ; using (2.3), we have

Case 3.

, ; using (2.3), we have

Case 4.

, ; using (2.3), we have

For , the maximum occurs at , hence

For , the maximum occurs at , so

Now, we consider the nonlinear nonsingular system of BVPs

We write (2.19) as an equivalent system of integral equations

By a solution of the system (2.19), we mean a solution of the corresponding system of integral equations (2.20). Define a retraction by and an operator by

where operators are defined by

Clearly, if is a fixed point of , then is a solution of the system (2.19).

Lemma 2.5.

Assume that holds. Then is completely continuous.

Proof.

Clearly, for any , . We show that the operator is uniformly bounded. Let be fixed and consider

Choose a constant such that , , . Then, for every , using (2.22), Lemma 2.4, and , we have

which implies that

that is, is uniformly bounded. Similarly, using (2.22), Lemma 2.4, and , we can show that is also uniformly bounded. Thus, is uniformly bounded. Now we show that is equicontinuous. Define

Let such that . Since is uniformly continuous on , for any , there exist such that implies

For , using (2.22)–(2.27), we have

Hence,

which implies that is equicontinuous. Similarly, using (2.22)–(2.27), we can show that is also equicontinuous. Thus, is equicontinuous. By Arzelà-Ascoli theorem, is relatively compact. Hence, is a compact operator.

Now we show that is continuous. Let such that Then by using (2.22) and Lemma 2.4, we have

Consequently,

By Lebesgue dominated convergence theorem, it follows that

Similarly, by using (2.22) and Lemma 2.4, we have

From (2.32) and (2.33), it follows that

that is, is continuous. Hence, is completely continuous.

## 3. Main Results

Proof of Theorem 1.1.

Let . Choose a constant such that

Choose a constant such that , , , . For any , using (2.22), (3.1), , and , we have

Since,

it follows that

Similarly, using (2.22), (3.1), , and , we have

From (3.4), and (3.5), it follows that

Choose a real constant such that

Choose a constant such that , , , . For any , using (2.22), (3.7), , and , we have

We used the fact that

Thus,

Similarly, using (2.22), (3.7), and , we have,

From (3.10) and (3.11), it follows that

Hence by Lemma 2.1, has a fixed point , that is,

Moreover, by (3.4), (3.5), (3.10) and (3.11), we have

Since is a solution of the system (2.19), hence and are concave on . Moreover, and . For , using result (2.2) and (3.14), we have

which implies that is uniformly bounded on . Now we show that is equicontinuous on . Choose and and consider the integral equation

Using Lemma 2.3, we have

Differentiating with respect to *t*, we obtain

which implies that

In view of and (3.15), we have

which implies that

Similarly, consider the integral equation

using and (3.15), we can show that

In view of (3.21) and (3.23), is equicontinuous on . Hence by Arzelà-Ascoli theorem, the sequence has a subsequence converging uniformly on to . Let us consider the integral equation

Letting , we have

Differentiating twice with respect to *t*, we have

Letting , we have

Similarly, consider the integral equation

we can show that

Now, we show that also satisfies the boundary conditions. Since,

Similarly, we can show that

Equations (3.27)–(3.31) imply that is a solution of the system (1.3). Moreover, is positive. In fact, by (3.27) is concave and by Lemma 2.2

implies that for all . Similarly, for all . The proof of Theorem 1.1 is complete.

Example.

Let

where the real constants satisfy , with and the real constants satisfy , with . Clearly, and satisfy the assumptions . Hence, by Theorem 1.1, the system (1.3) has a positive solution.

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## Acknowledgment

Research of R. A. Khan is supported by HEC, Pakistan, Project 2- 3(50)/PDFP/HEC/2008/1.

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### Keywords

- Integral Equation
- Couple System
- Real Constant
- Singular System
- Order Ordinary Differential Equation