# A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems

- Ruyun Ma
^{1}Email author, - Youji Xu
^{1}and - Chenghua Gao
^{1}

**2009**:671625

**DOI: **10.1155/2009/671625

© Ruyun Ma et al. 2009

**Received: **12 February 2009

**Accepted: **20 August 2009

**Published: **7 October 2009

## Abstract

Let be an integer with , , . We consider boundary value problems of nonlinear second-order difference equations of the form , , , where , and, for , and , , . We investigate the global structure of positive solutions by using the Rabinowitz's global bifurcation theorem.

## 1. Introduction

Here is a positive parameter, and are continuous. Denote and .

There are many literature dealing with similar difference equations subject to various boundary value conditions. We refer to Agarwal and Henderson [1], Agarwal and O'Regan [2], Agarwal and Wong [3], Rachunkova and Tisdell [4], Rodriguez [5], Cheng and Yen [6], Zhang and Feng [7], R. Ma and H. Ma [8], Ma [9], and the references therein. These results were usually obtained by analytic techniques, various fixed point theorems, and global bifurcation techniques. For example, in [8], the authors investigated the global structure of sign-changing solutions of some discrete boundary value problems in the case that . However, relatively little result is known about the global structure of solutions in the case that , and no global results were found in the available literature when . The likely reason is that the Rabinowitz's global bifurcation theorem [10] cannot be used directly in this case.

- (A1)
;

- (A2)
is continuous and for ;

- (A3)
, where ;

- (A4)
, where .

Let denote the Banach space defined by

- (i)
is the set of eigenvalues of (1.7);

- (ii)
for ;

- (iii)
for , is one-dimensional subspace of ;

- (iv)
for each , if , then has exactly simple generalized zeros in .

Let denote the closure of set of positive solutions of (1.1) in .

Let be a subset of . A component of is meant a maximal connected subset of , that is, a connected subset of which is not contained in any other connected subset of .

The main results of this paper are the following theorem.

Theorem 1.2.

for some . Moreover, there exists such that (1.1) has at least two positive solutions for .

## 2. Some Preliminaries

In this section, we give some notations and preliminary results which will be used in the proof of our main results.

Definition 2.1 (see [12]).

*superior limit*of is defined by

Definition 2.2 (see [12]).

A *component* of a set
is meant a maximal connected subset of
.

Lemma 2.3 ([12, Whyburn]).

Suppose that is a compact metric space, and are nonintersecting closed subsets of , and no component of interests both and . Then there exist two disjoint compact subsets and , such that , , .

Using the above Whyburn's lemma, Ma and An [13] proved the following lemma.

Lemma 2.4 ([13, Lemma ]).

- (i)
there exist , , and , such that ;

- (ii)
, where ;

- (iii)for every , is a relatively compact set of , where(2.2)

Then there exists an unbounded component in and .

Then the operator satisfies .

Using the standard arguments, we may prove the following lemma.

Lemma 2.5.

Assume that (A1)–(A2) hold. Then and is completely continuous.

Lemma 2.6.

Proof.

## 3. Proof of the Main Results

Similarly we may extend to be an odd function for each .

as a bifurcation problem from the trivial solution .

Equation (3.8) can be converted to the equivalent equation

Further we note that for near in .

The results of Rabinowitz [10] for (3.8) can be stated as follows. For each integer , , there exists a continuum of solutions of (3.8) joining to infinity in . Moreover,

Lemma 3.1.

Let (A1)–(A4) hold. Then, for each fixed , joins to in .

Proof.

We divide the proof into two steps.

Step 1.

We show that .

Step 2.

We show that .

(where ), which yields that is bounded. However, this contradicts (3.19).

Therefore, joins to in .

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Proof of Theorem 1.2.

Notice that satisfies all conditions in Lemma 2.4, and consequently, contains a component which is unbounded. However, we do not know whether joins with or not. To answer this question, we have to use the following truncation method.

- (1)
;

- (2)
joins with infinity in .

We claim that satisfies all of the conditions of Lemma 2.4.

that is, condition (ii) in Lemma 2.4 holds. Condition (iii) in Lemma 2.4 can be deduced directly from the Arzelà -Ascoli theorem and the definition of . Therefore, the superior limit of contains a component joining with infinity in .

- (1)
;

- (2)
joins with infinity in ,

and the superior limit of contains a component joining with infinity in .

From Lemma 2.4, it follows that is closed in , and furthermore, is compact in .

Let

If for some , , then Theorem 1.2 holds.

Assume on the contrary that for all .

where and are the boundary and closure of in , respectively.

where and are the boundary and closure of in , respectively.

However, this contradicts (3.50).

Therefore, there exists such that which is unbounded in both and .

Finally, we show that joins with . This will be done by the following three steps.

Step 1.

We show that .

This is impossible by (A3) and the assumption .

Step 2.

We show that .

where . By (A2), it follows that . Obviously, (3.65) implies that is bounded. This is a contradiction.

Step 3.

We show that .

where . By (A2), it follows that . Obviously, (3.68) implies that is bounded. This is a contradiction.

To sum up, there exits a component which joins and .

## Declarations

### Acknowledgments

This work was supported by the NSFC (no. 10671158), the NSF of Gansu Province (no. 3ZS051-A25-016), NWNU-KJCXGC-03-17, the Spring-Sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006 ).

## Authors’ Affiliations

## References

- Agarwal RP, Henderson J:
**Positive solutions and nonlinear eigenvalue problems for third-order difference equations.***Computers & Mathematics with Applications*1998,**36**(10–12):347-355.MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D:
**Boundary value problems for discrete equations.***Applied Mathematics Letters*1997,**10**(4):83-89. 10.1016/S0893-9659(97)00064-5MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Wong F-H:
**Existence of positive solutions for nonpositive difference equations.***Mathematical and Computer Modelling*1997,**26**(7):77-85. 10.1016/S0895-7177(97)00186-6MathSciNetView ArticleMATHGoogle Scholar - Rachunkova I, Tisdell CC:
**Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(4):1236-1245. 10.1016/j.na.2006.07.010MathSciNetView ArticleMATHGoogle Scholar - Rodriguez J:
**Nonlinear discrete Sturm-Liouville problems.***Journal of Mathematical Analysis and Applications*2005,**308**(1):380-391. 10.1016/j.jmaa.2005.01.032MathSciNetView ArticleMATHGoogle Scholar - Cheng SS, Yen H-T:
**On a discrete nonlinear boundary value problem.***Linear Algebra and Its Applications*2000,**313**(1–3):193-201.MathSciNetView ArticleMATHGoogle Scholar - Zhang G, Feng W:
**On the number of positive solutions of a nonlinear algebraic system.***Linear Algebra and Its Applications*2007,**422**(2-3):404-421. 10.1016/j.laa.2006.10.026MathSciNetView ArticleMATHGoogle Scholar - Ma R, Ma H:
**Existence of sign-changing periodic solutions of second order difference equations.***Applied Mathematics and Computation*2008,**203**(2):463-470. 10.1016/j.amc.2008.05.125MathSciNetView ArticleMATHGoogle Scholar - Ma R:
**Nonlinear discrete Sturm-Liouville problems at resonance.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(11):3050-3057. 10.1016/j.na.2006.09.058MathSciNetView ArticleMATHGoogle Scholar - Rabinowitz PH:
**Some global results for nonlinear eigenvalue problems.***Journal of Functional Analysis*1971,**7:**487-513. 10.1016/0022-1236(71)90030-9MathSciNetView ArticleMATHGoogle Scholar - Jirari A:
**Second-order Sturm-Liouville difference equations and orthogonal polynomials.***Memoirs of the American Mathematical Society*1995,**113**(542):138.MathSciNetView ArticleGoogle Scholar - Whyburn GT:
*Topological Analysis*. Princeton University Press, Princeton, NJ, USA; 1958:119.MATHGoogle Scholar - Ma R, An Y:
**Global structure of positive solutions for superlinear seconde order****-point boundary value problems.***Topological Methods in Nonlinear Analysis*. In press

## Copyright

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.