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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 312864, 16 pages
http://dx.doi.org/10.1155/2010/312864
Research Article

Positive Solutions of a Singular Positone and Semipositone Boundary Value Problems for Fourth-Order Difference Equations

1School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2School of Mathematics and Computer, Harbin University, Harbin 150086, China

Received 13 June 2010; Revised 9 September 2010; Accepted 13 October 2010

Academic Editor: Manuel De la Sen

Copyright © 2010 Chengjun Yuan. 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.

Abstract

This paper studies the boundary value problems for the fourth-order nonlinear singular difference equations , , , . We show the existence of positive solutions for positone and semipositone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone.

1. Introduction

In this paper, we consider the following boundary value problems of difference equations: Here and . We will let denote the discrete integer set , and denotes the set of continuous function on (discrete topology) with norm .

Due to the wide applications in many fields such as computer science, economics, neural network, ecology, and cybernetics, the theory of nonlinear difference equations has been widely studied since the 70's of last century. Recently, many literatures on the boundary value of difference equations have appeared. We refer the reader to [113] and the references therein, which include work on Agarwal, Elaydi, Eloe, Erber, O'Regan, Henderson, Merdivenci, Yu, and Ma et al., concerning the existence of positive solutions and the corresponding eigenvalue problems. Recently, the existence of positive solutions of fourth-order discrete boundary value problems has been studied by several authors; for example, see [1416] and the references therein.

On the other hand, fourth-order boundary value problems of ordinary value problems have important application in various branches of pure and applied science. They arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory [1719]. For example, the deformations of an elastic beam can be described by the boundary value problems of the fourth-order ordinary differential equations. There have been extensive studies on fourth-order boundary value problems with diverse boundary conditions via many methods, for example, [2026] and the references therein. We also find that the differential equations on time scales is due to its unification of the theory of differential and difference equations, see [2730] and the references therein.

In this paper, the boundary value problem (1.1) can be viewed as the discrete analogue of the following boundary value problems for ordinary differential equation: Equation (1.2) describes an elastic beam in an equilibrium state whose both ends are simply supported. However, very little is known about the existence of solutions of the discrete boundary value problems (1.1). This motivates us to study (1.1).

In this paper, we discuss separately the cases when is positone and when is semipositone; the nonlinear term is singularity at , and we will prove our two existence results for the problem (1.1) by using Krasnosel'skii fixed point theorem. This paper is organized as follows. In Section 2, starting with some preliminary lemmas, we state the Krasnosel'skii fixed point theorem. In Section 3, we give the sufficient conditions which state the existence of multiple positive solutions to the positone boundary value problem (1.1). In Section 4, we give the sufficient conditions which state the existence of at least one positive solution to the semipositone boundary value problem (1.1).

2. Preliminaries

In this section, we state the preliminary information that we need to prove the main results. From [28, Definition ], we have the following lemma.

Lemma 2.1. is a solution of (1.1) if only and if where

Lemma 2.2. Green's function defined by (2.2) has the following properties: where

Proof. For , we have
On the other hand,
Then, for , we have
For , we have
On the other hand,
Then, for , we have also

We note that is a solution of (1.1) if and only if

For our constructions, we will consider the Banach space equipped with the standard norm , . We define a cone by

The following theorems will play a major role in our next analysis.

Theorem 2.3 (see [1]). Let be a Banach space, and let be a cone in . Let be open subsets of with , and let be a completely continuous operator such that either (1), , , , or (2), , , . Then, has a fixed point in .

3. Singular Positone Problems

Theorem 3.1. Assume that the following conditions are satisfied: (H1)(H2) on with continuous and nonincreasing on , continuous on , and nondecreasing on , with for all , (H3) there exists such that for ; (H4)there exists such that for . Then, for each , there exists a positive number such that the positone problem (1.1) has at least two positive solutions and with for .

Proof. Now, we define the integral operator by where
It is easy to check that . In fact, for each , we have by Lemma 2.2 that This implies . On the other hand, we have Thus, we have . In addition, standard argument shows that is completely continuous.
For any given , we fix it, and take . Choose
For , from () and (3.4), we have Thus, Further, choose a constant satisfying that where .
By (H3), there is a constant such that Let and . For , we have that It follows that
Then, for , we have
Therefore, by the first part of the Fixed Point Theorem 2.3, has a fixed point with .
Finally, choose a constant satisfying that
By , there is a constant and such that Let and . For , we have It follows that
Then, for , we have
Therefore, by the first part of the Fixed Point Theorem 2.3, has a fixed point with . It follows from (3.6) that .
Then, for each , there exists a positive number such that the positone problem (1.1) has at least two positive solutions with for .

From the proof of Theorem 3.1, we have the following result.

Theorem 3.2. Assume that (H1)–(H3) are satisfied. Then, for each , there exists a positive number such that the positone problem (1.1) has at least one positive solution with for .

Theorem 3.3. Assume that (H1), (H2), and (H4) are satisfied. Then, for each , there exists a positive number such that the positone problem (1.1) has at least one positive solution with for .

Example 3.4. Consider the boundary value problem where are constants. Then, for each , there exists a positive number such that the problem (3.17) has at least two positive solutions for .
In fact, it is clear that
Letting , , and , we have with continuous and nonincreasing on , continuous on , and nondecreasing on ; with for ,
Then, by Theorem 3.1, for each given , we choose such that the problem (3.17) has at least two positive solutions for .

4. Singular Semipositone Problems

Before we prove our next main result, we first state a result.

Lemma 4.1. The boundary value problem has a solution with where .

In fact, from Lemma 2.1, (4.1) has solution According to Lemma 2.2, we have

Theorem 4.2. Assume that the following conditions are satisfied: (B1) is continuous and there exists a function with for ; (B2) on with continuous and nonincreasing on , continuous on , and nondecreasing on (B3) with for all (B4) there exists such that for . Then, for each , there exists a positive number such that the semipositone problem (1.1) has at least one positive solution for .

Proof. To show that (1.1) has a nonnegative solution, we will look at the boundary value problem where and is as in Lemma 4.1.
We will show, using Theorem 2.3, that there exists a solution to (4.4) with for . If this is true, then is a nonnegative solution (positive on ) of (1.1), since
Next, let be defined by In addition, standard argument shows that and is completely continuous.
For any given , fix it. We choose where .
Now, let We show that To see this, let . Then, and for Now, for , the Lemma 4.1 implies so for , we have This yields so (4.9) is satisfied.
Further, choose a constant satisfying that where .
By , there is a constant such that Let and .
Next, we show that To see this, let . We have It follows that, for , we have
Then, we have This yields so (4.14) holds.
Therefore, by the first part of the Fixed Point Theorem 2.3, has a fixed point with , since Namely, is a positive solution of the semipositone problem (1.1).
Then, for each , there exists a positive number such that the semipositone problem (1.1) has at least one positive solution for .

Example 4.3. Consider the boundary value problem where are constants. Then, for each , there exists a positive number such that the problem (4.19) has at least one positive solution for .
To see this, we will apply Theorem 4.2 (here will be chosen later). From we let
It is clear that , , and , hold.
Then, the (B1)–(B4) of Theorem 4.2 hold. Now, we have
For each , we can choose Thus, all the conditions of Theorem 4.2 are satisfied, so the existence of positive solution is guaranteed for .

Acknowledgments

This work was supported by Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032) and NNSF of China (no. 10971021).

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