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

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## Abstract

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

## Keywords

Integral Equation Couple System Real Constant Singular System Order Ordinary Differential Equation## 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 Open image in new window 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, 9, 10, 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, 14, 15, 16, 17, 18] and the recent monograph by Agarwal et al. [19].

where Open image in new window , and may be singular at Open image in new window , Open image in new window , Open image in new window and/or Open image in new window .

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

where Open image in new window , Open image in new window , Open image in new window . We allow Open image in new window and Open image in new window to be singular at Open image in new window , Open image in new window , and also Open image in new window and/or Open image in new window . We study the sufficient conditions for existence of positive solution for the singular system (1.3) under weaker hypothesis on Open image in new window and Open image in new window 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 Open image in new window and Open image in new window are allowed to be unbounded at Open image in new window , Open image in new window , Open image in new window , and/or Open image in new window . 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, 25, 26]. Throughout this paper, we assume that Open image in new window are continuous and may be singular at Open image in new window , Open image in new window , Open image in new window , and/or Open image in new window . We also assume that the following conditions hold:

*A*

_{2})There exist real constants Open image in new window such that Open image in new window , Open image in new window , Open image in new window and for all Open image in new window , Open image in new window ,

*A*

_{3})There exist real constants Open image in new window such that Open image in new window , Open image in new window , Open image in new window and for all Open image in new window , Open image in new window ,

The main result of this paper is as follows.

Theorem 1.1.

Assume that Open image in new window hold. Then the system (1.3) has at least one positive solution.

## 2. Preliminaries

For each Open image in new window , we write Open image in new window . Let Open image in new window . Clearly, Open image in new window is a Banach space and Open image in new window is a cone. Similarly, for each Open image in new window , we write Open image in new window . Clearly, Open image in new window is a Banach space and Open image in new window is a cone in Open image in new window . For any real constant Open image in new window , define Open image in new window . By a positive solution of (1.3), we mean a vector Open image in new window such that Open image in new window satisfies (1.3) and Open image in new window , Open image in new window on Open image in new window . 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 Open image in new window be a cone of a real Banach space Open image in new window , Open image in new window , Open image in new window be bounded open subsets of Open image in new window and Open image in new window . Suppose that Open image in new window is completely continuous such that one of the following condition hold:

(i) Open image in new window for Open image in new window and Open image in new window for Open image in new window ;

(ii) Open image in new window for Open image in new window and Open image in new window for Open image in new window .

Then, Open image in new window has a fixed point in Open image in new window .

The following result can be easily verified.

Result.

Let Open image in new window such that Open image in new window . Let Open image in new window , Open image in new window and concave on Open image in new window . Then, Open image in new window for all Open image in new window .

Choose Open image in new window such that Open image in new window . For fixed Open image in new window and Open image in new window , the linear three-point BVP

Lemma 2.2 (see [9]).

where Open image in new window .

We need the following properties of the Green's function Open image in new window in the sequel.

Lemma 2.3 (see [11]).

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

Lemma 2.4.

where Open image in new window

Proof.

For Open image in new window , we discuss various cases.

Case 1.

Case 2.

Case 3.

Case 4.

For Open image in new window , the maximum occurs at Open image in new window , hence

For Open image in new window , the maximum occurs at Open image in new window , so

Now, we consider the nonlinear nonsingular system of BVPs

Clearly, if Open image in new window is a fixed point of Open image in new window , then Open image in new window is a solution of the system (2.19).

Lemma 2.5.

Assume that Open image in new window holds. Then Open image in new window is completely continuous.

Proof.

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

that is, Open image in new window is continuous. Hence, Open image in new window is completely continuous.

## 3. Main Results

Proof of Theorem 1.1.

*t*, we obtain

*t*, we have

implies that Open image in new window for all Open image in new window . Similarly, Open image in new window for all Open image in new window . The proof of Theorem 1.1 is complete.

Example.

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

## Notes

### Acknowledgment

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

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