International Scholarly Research Notices
Volume 2014 (2014), Article ID 531978, 6 pages
http://dx.doi.org/10.1155/2014/531978

Research Article

Existence and Uniqueness of Positive Solution for Discrete Multipoint Boundary Value Problems

Huili Ma and Huifang Ma

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 15 May 2014; Revised 9 August 2014; Accepted 10 August 2014; Published 29 October 2014

Academic Editor: Evangelos J. Sapountzakis

Copyright © 2014 Huili Ma and Huifang Ma. 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.

It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: , subject to boundary conditions either or , where . The proof of the main result is based upon a fixed point theorem of a sum operator. It is expected in this paper not only to establish existence and uniqueness of positive solution, but also to show a way to construct a series to approximate it by iteration.

1. Introduction

Let be an integer; , where are positive integers. Difference equation appears as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and so it arises in many physical problems, as nonlinear elasticity theory or mechanics, and engineering topics. In recent years, the study of positive solutions for discrete boundary value problems has attracted considerable attention, but most research dealt with two-point boundary value problem; see [1–4] and the references therein. For multipoint boundary value problem, there appeared a small sample of work related to the existence of positive solution; we refer the reader to [5–7]. However all of them do not address the problem with the uniqueness of positive solution.

In this paper, we consider the existence uniqueness and positive solutions for difference equation

subject to boundary conditions

or

where , , and .

Let denote the class of real valued functions on with norm . Observe that is a Banach space. Set , to be the normal cone in with the normality constant 1. For , the notation means that there exist and such that . Clearly, is an equivalence relation. Given , we denote by the set .

Remark 1. As suggested by the notation, by equipping with the discrete topology, every is continuous.

Definition 2 (see [8]). Let or and let be a real number with . An operator is said to be -concave if it satisfies

Definition 3 (see [8]). An operator is said to be homogeneous if it satisfies

An operator is said to be subhomogeneous if it satisfies

The main tool of this paper is the following fixed point theorem.

Theorem 4 (see [9]). Let be a normal cone in a real Banach space , an increasing -concave operator, and an increasing subhomogeneous operator. Assume that

 (i)there is such that and ; (ii)there exists a constant such that, for any , .

Then the operator equation has a unique solution . Moreover, constructing successively the sequence , , for any initial value , one has as .

Remark 5. When is a null operator, Theorem 4 also holds.

2. Positive and Uniqueness of Solutions to BVP (1)-(2)

In this section, we will apply Theorem 4 to study the existence and uniqueness of positive solution for (1)-(2).

Lemma 6 (see [6]). If , , and are real numbers with and , for any defined in , the nonlocal boundary value problem

has a unique solution

where

Lemma 7. For , the Green function in Lemma 6 has the following properties:

 (i), , ; (ii), , where

 (iii)for , , , where

Proof. Since (i) and (iii) are obvious, here we just prove (ii). For , , notice that ,

For ,

For ,

For ,

That is, for any and , .

Theorem 8. Assume that

 (A1) are continuous and increasing with respect to the second variable, ; (A2) for , , and there exists a constant such that for , , ; (A3)there exists a constant such that , , .

The problem (1)-(2) has a unique positive solution , where , . Moreover, for any initial value , constructing successively the sequence

we have as , where is given as (9).

Proof. Define two operators and by

It is easy to see that is a solution of (1)-(2) if and only if . From (A2) and Lemma 7, we know that and . In the sequel we check that satisfy all assumptions of Theorem 4.
Firstly, we prove that are two increasing operators. In fact, by (A1) and Lemma 6, for with , we know that , , and obtain

Similarly, .
Next we show that is a -concave operator and is a subhomogeneous operator. In fact, for any and , from (A2), we know that

That is, is a -concave operator. At the same time, for any and , from (A2), we get

So is subhomogeneous.
Now we show that and . From (A3) and Lemma 7,

where and . Hence we have , ; that is, . We can similarly prove that . Thus condition (i) of Theorem 4 is satisfied.
In the following we show that condition (ii) of Theorem 4 holds. From (A3),

Then we get . By applying Theorem 4, it can be obtained that the operator equation has a unique solution . Moreover, constructing successively the sequence , for any initial value , we have as . That is, problem (1)-(2) has a unique positive solution . In addition, for any initial value , constructing successively the sequence

we have as .

The following result can be obtained by Remark 5 and Theorem 4.

Corollary 9. Assume that

 (A1)′ is continuous and increasing with respect to the second variable, ; (A2)′there exists a constant such that for , , .

Then problem

has a unique positive solution , where . Moreover, for any initial value , constructing successively the sequence

we have as , where is given as (9).

Remark 10. In a similar way, we can get the corresponding results for the difference equation (1) subject to boundary conditions

which are symmetric to the boundary condition (2).

Remark 11. The results can also be generalized to discrete -point boundary value problems:

Example 12. Consider the following nonlinear discrete problem:

If we set

then are continuous and increasing with respect to the second variable, , and for , , ,

In addition,

Thus, all conditions of Theorem 8 are satisfied and so problem (28) has a unique positive solution in .

3. Positive and Uniqueness of Solutions to BVP (1) and (3)

Lemma 13 (see [6]). If , the nonlocal boundary value problem

has a unique solution

where

Lemma 14. For , , the Green function in Lemma 13 has the following property:

 (i), , ; (ii), , , where

Proof. We omit it since it is obvious.

Theorem 15. Assume that (A1), (A2), and (A3) are satisfied; then the problem (1)–(3) has a unique positive solution , where , . Moreover, for any initial value , constructing successively the sequence

we have as , where is given as (34).

Proof. It is similar to the proof of Theorem 8.

The following corollary can be obtained by Remark 5 and Theorem 4.

Corollary 16. Assume that (A1)′ and (A2)′ are satisfied; then the problem

has a unique positive solution , where , . Moreover, for any initial value , constructing successively the sequence

we have as , where is given as (34).

Remark 17. In a similar way, we can get the corresponding results for the difference equation (1) subject to boundary conditions

which are symmetric to the boundary condition (3).

Remark 18. The results can also be generalized to discrete -point boundary value problems:

Example 19. Consider the following nonlinear discrete problem:

If we set

then are continuous and increasing with respect to the second variable, , and for , , ,

In addition,

Thus, all conditions of Theorem 15 are satisfied and so problem (41) has a unique positive solution in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61363058 and 11061030), the Scientific Research Fund for Colleges and Universities of Gansu Province (2013B-007, 2013A-016), Natural Science Foundation of Gansu Province (145RJZA232, 145RJYA259), and Promotion Funds for Young Teachers in Northwest Normal University (NWNU-LKQN-12-14).

References

F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic boundary value problems,” Computers and Mathematics with Applications, vol. 45, no. 6–9, pp. 1417–1427, 2003.

R. Ma and H. Ma, “Unbounded perturbations of nonlinear discrete periodic problem at resonance,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 7, pp. 2602–2613, 2009.

I. Rachůnková and C. C. Tisdell, “Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions,” Nonlinear Analysis, Theory, Methods and Applications, vol. 67, no. 4, pp. 1236–1245, 2007.

J. Sun and W. Li, “Existence of positive solutions of boundary value problem for a discrete difference system,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 857–870, 2004.

R. Ma and Y. N. Raffoul, “Positive solutions of three-point nonlinear discrete second order boundary value problem,” Journal of Difference Equations and Applications, vol. 10, no. 2, pp. 129–138, 2004.

W. Cheung, J. Ren, and P. J. Y. Wong, “Multiple positive solutions for discrete nonlocal boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 900–915, 2007.

G. Zhang and R. Medina, “Three-point boundary value problems for difference equations,” Computers & Mathematics with Applications, vol. 48, no. 12, pp. 1791–1799, 2004.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988.

C. Zhai and D. R. Anderson, “A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 388–400, 2011.