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Boundary Value Problems

, 2009:273063 | Cite as

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

  • NaseerAhmad Asif
  • RahmatAli Khan
Open Access
Research Article
Part of the following topical collections:
  1. Singular Boundary Value Problems for Ordinary Differential Equations

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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].

The study of system of BVPs has also fascinated many authors. System of BVPs with continuous nonlinearity can be seen in [20, 21, 22] and the case of singular nonlinearity can be seen in [8, 21, 23, 24, 25, 26]. Wei [25], developed the upper and lower solutions method for the existence of positive solutions of the following coupled system of BVPs:

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 .

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

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].

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

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:

for example, a function that satisfies the assumptions Open image in new window and Open image in new window is

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

has a unique solution
where Open image in new window is the Green's function and is given by
is the Green's function corresponding the boundary value problem
whose integral representation is given by

Lemma 2.2 (see [9]).

Let Open image in new window . If Open image in new window and Open image in new window , then then unique solution Open image in new window of the problem (2.5) satisfies

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]).

The function Open image in new window can be written as

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.

and if Open image in new window , the maximum occurs at Open image in new window , hence

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

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 Open image in new window by Open image in new window and an operator Open image in new window by
where operators Open image in new window are defined by

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.

Clearly, for any Open image in new window , Open image in new window . We show that the operator Open image in new window is uniformly bounded. Let Open image in new window be fixed and consider
which implies that
that is, Open image in new window is uniformly bounded. Similarly, using (2.22), Lemma 2.4, Open image in new window and Open image in new window , we can show that Open image in new window is also uniformly bounded. Thus, Open image in new window is uniformly bounded. Now we show that Open image in new window is equicontinuous. Define
For Open image in new window , using (2.22)–(2.27), we have

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.

Now we show that Open image in new window is continuous. Let Open image in new window such that Open image in new window 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, Open image in new window is continuous. Hence, Open image in new window is completely continuous.

3. Main Results

Proof of Theorem 1.1.

it follows that
Similarly, using (2.22), (3.1), Open image in new window , and Open image in new window , we have
From (3.4), and (3.5), it follows that
Choose a real constant Open image in new window such that
We used the fact that
Similarly, using (2.22), (3.7), Open image in new window and Open image in new window , we have,
From (3.10) and (3.11), it follows that
Hence by Lemma 2.1, Open image in new window has a fixed point Open image in new window , that is,
Moreover, by (3.4), (3.5), (3.10) and (3.11), we have
Since Open image in new window is a solution of the system (2.19), hence Open image in new window and Open image in new window are concave on Open image in new window . Moreover, Open image in new window and Open image in new window . For Open image in new window , using result (2.2) and (3.14), we have
which implies that Open image in new window is uniformly bounded on Open image in new window . Now we show that Open image in new window is equicontinuous on Open image in new window . Choose Open image in new window and Open image in new window and consider the integral equation
Using Lemma 2.3, we have
Differentiating with respect to t, we obtain
which implies that
In view of Open image in new window and (3.15), we have
which implies that
Similarly, consider the integral equation
using Open image in new window and (3.15), we can show that
In view of (3.21) and (3.23), Open image in new window is equicontinuous on Open image in new window . Hence by Arzelà-Ascoli theorem, the sequence Open image in new window has a subsequence Open image in new window converging uniformly on Open image in new window to Open image in new window . Let us consider the integral equation
Differentiating twice with respect to t, we have
Similarly, consider the integral equation
we can show that
Now, we show that Open image in new window also satisfies the boundary conditions. Since,
Similarly, we can show that
Equations (3.27)–(3.31) imply that Open image in new window is a solution of the system (1.3). Moreover, Open image in new window is positive. In fact, by (3.27) Open image in new window is concave and by Lemma 2.2

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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Copyright information

© N. A. Asif and R. A. Khan. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Centre for Advanced Mathematics and Physics, Campus of College of Electrical and Mechanical EngineeringNational University of Sciences and TechnologyRawalpindiPakistan

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