Discrete Dynamics in Nature and Society

Volume 2010, Article ID 312864, 16 pages

http://dx.doi.org/10.1155/2010/312864

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

^{1}School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China^{2}School 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 [1–13] 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 [14–16] 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 [17–19]. 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, [20–26] 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 [27–30] 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).

#### References

- R. P. Agarwal,
*Difference Equations and Inequalities: Theory, Methods, and Application*, vol. 155 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1992. View at Zentralblatt MATH - S. Elaydi,
*An Introduction to Difference Equations*, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005. View at Zentralblatt MATH - R. P. Agarwal and J. Henderson, “Positive solutions and nonlinear eigenvalue problems for third-order difference equations,”
*Computers & Mathematics with Applications*, vol. 36, no. 10–12, pp. 347–355, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal and D. O'Regan, “Multiple solutions for higher-order difference equations,”
*Computers & Mathematics with Applications*, vol. 37, no. 9, pp. 39–48, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal and D. ORegan, “Discrete conjugate boundaryvalue problems,”
*Applied Mathematics Letters*, vol. 37, pp. 34–39, 1999. View at Google Scholar - R. I. Avery, C. J. Chyan, and J. Henderson, “Twin solutions of boundary value problems for ordinary differential equations and finite difference equations,”
*Computers & Mathematics with Applications*, vol. 42, no. 3–5, pp. 695–704, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. W. Eloe, “A generalization of concavity for finite differences,”
*Computers & Mathematics with Applications*, vol. 36, no. 10–12, pp. 109–113, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. H. Erbe, H. Xia, and J. S. Yu, “Global stability of a linear nonautonomous delay difference equation,”
*Journal of Difference Equations and Applications*, vol. 1, no. 2, pp. 151–161, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. C. Hao, “Nonnegative solutions for semilinear third-order difference equation boundary value problems,”
*Acta Mathematica Scientia. Series A*, vol. 21, no. 2, pp. 225–229, 2001 (Chinese). View at Google Scholar · View at Zentralblatt MATH - J. Henderson and P. J. Y. Wong, “Positive solutions for a system of nonpositive difference equations,”
*Aequationes Mathematicae*, vol. 62, no. 3, pp. 249–261, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. - H. Matsunaga, T. Hara, and S. Sakata, “Global attractivity for a nonlinear difference equation with variable delay,”
*Computers & Mathematics with Applications*, vol. 41, no. 5-6, pp. 543–551, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Yuan, D. Jiang, D. O'Regan, and R. P. Agarwal, “Existence and uniqueness of positive solutions of boundary value problems for coupled systems of singular second-order three-point non-linear differential and difference equations,”
*Applicable Analysis*, vol. 87, no. 8, pp. 921–932, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. He and J. Yu, “On the existence of positive solutions of fourth-order difference equations,”
*Applied Mathematics and Computation*, vol. 161, no. 1, pp. 139–148, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. He and Y. Su, “On discrete fourth-order boundary value problems with three parameters,”
*Journal of Computational and Applied Mathematics*, vol. 233, no. 10, pp. 2506–2520, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Yuan, D. Jiang, and D. O'Regan, “Existence and uniqueness of positive solutions for fourth-order nonlinear singular continuous and discrete boundary value problems,”
*Applied Mathematics and Computation*, vol. 203, no. 1, pp. 194–201, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. M. Momani,
*Some problems in non-Newtonian fluid mechanics*, Ph.D. thesis, Walse University, Walse, UK, 1991. - T. F. Ma and J. da Silva, “Iterative solutions for a beam equation with nonlinear boundary conditions of third order,”
*Applied Mathematics and Computation*, vol. 159, no. 1, pp. 11–18, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. M. Chawla and C. P. Katti, “Finite difference methods for two-point boundary value problems involving high order differential equations,”
*BIT*, vol. 19, no. 1, pp. 27–33, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Pei and S. K. Chang, “Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem,”
*Mathematical and Computer Modelling*, vol. 51, no. 9-10, pp. 1260–1267, 2010. View at Publisher · View at Google Scholar - D. Ma and X. Yang, “Upper and lower solution method for fourth-order four-point boundary value problems,”
*Journal of Computational and Applied Mathematics*, vol. 223, no. 2, pp. 543–551, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Feng, D. Ji, and W. Ge, “Existence and uniqueness of solutions for a fourth-order boundary value problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 10, pp. 3561–3566, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Ramadan, I. F. Lashien, and W. K. Zahra, “Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 4, pp. 1105–1114, 2009. View at Publisher · View at Google Scholar - Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 2, pp. 357–368, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. R. Graef, C. Qian, and B. Yang, “A three point boundary value problem for nonlinear fourth order differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 287, no. 1, pp. 217–233, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Zhang, M. Feng, and W. Ge, “Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 10, pp. 3310–3321, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. M. Atici and G. Sh. Guseinov, “On Green's functions and positive solutions for boundary value problems on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 75–99, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Bohner and H. Luo, “Singular second-order multipoint dynamic boundary value problems with mixed derivatives,”
*Advances in Difference Equations*, vol. 2006, Article ID 54989, 15 pages, 2006. View at Google Scholar · View at Zentralblatt MATH - M. Feng, X. Zhang, and W. Ge, “Positive solutions for a class of boundary value problems on time scales,”
*Computers & Mathematics with Applications*, vol. 54, no. 4, pp. 467–475, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M.-Q. Feng, X.-G. Li, and W.-G. Ge, “Triple positive solutions of fourth-order four-point boundary value problems of $p$-Laplacian dynamic equations on time scales,”
*Advances in Difference Equations*, vol. 2008, Article ID 496078, 9 pages, 2008. View at Google Scholar