- Research Article
- Open Access

# 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

https://doi.org/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.

## Keywords

- Banach Space
- Open Covering
- Global Structure
- Connected Subset
- Global Bifurcation

## 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)

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

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