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

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## Abstract

Let Open image in new window be an integer with Open image in new window , Open image in new window , Open image in new window . We consider boundary value problems of nonlinear second-order difference equations of the form Open image in new window , Open image in new window , Open image in new window , where Open image in new window , Open image in new window and, Open image in new window for Open image in new window , and Open image in new window , Open image in new window , Open image in new window . 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 Open image in new window is a positive parameter, Open image in new window and Open image in new window are continuous. Denote Open image in new window and Open image in new window .

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 Open image in new window . However, relatively little result is known about the global structure of solutions in the case that Open image in new window , and no global results were found in the available literature when Open image in new window . The likely reason is that the Rabinowitz's global bifurcation theorem [10] cannot be used directly in this case.

- (A1)
- (A2)
Open image in new window is continuous and Open image in new window for Open image in new window ;

- (A3)
- (A4)

Let Open image in new window denote the Banach space defined by

- (i)
Open image in new window is the set of eigenvalues of (1.7);

- (ii)
- (iii)
for Open image in new window , Open image in new window is one-dimensional subspace of Open image in new window ;

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

Let Open image in new window denote the closure of set of positive solutions of (1.1) in Open image in new window .

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

The main results of this paper are the following theorem.

Theorem 1.2.

for some Open image in new window . Moreover, there exists Open image in new window such that (1.1) has at least two positive solutions for Open image in new window .

## 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*Open image in new window of Open image in new window is defined by

Definition 2.2 (see [12]).

A *component* of a set Open image in new window is meant a maximal connected subset of Open image in new window .

Lemma 2.3 ([12, Whyburn]).

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

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

Lemma 2.4 ([13, Lemma Open image in new window ]).

- (i)
there exist Open image in new window , Open image in new window , and Open image in new window , such that Open image in new window ;

- (ii)
- (iii)for every Open image in new window , Open image in new window is a relatively compact set of Open image in new window , where(2.2)

Then there exists an unbounded component Open image in new window in Open image in new window and Open image in new window .

Then the operator Open image in new window satisfies Open image in new window .

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

Lemma 2.5.

Assume that (A1)–(A2) hold. Then Open image in new window and Open image in new window is completely continuous.

Lemma 2.6.

Proof.

## 3. Proof of the Main Results

Similarly we may extend Open image in new window to be an odd function Open image in new window for each Open image in new window .

as a bifurcation problem from the trivial solution Open image in new window .

Equation (3.8) can be converted to the equivalent equation

Further we note that Open image in new window for Open image in new window near Open image in new window in Open image in new window .

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

Lemma 3.1.

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

Proof.

We divide the proof into two steps.

Step 1.

We show that Open image in new window .

Step 2.

We show that Open image in new window .

(where Open image in new window ), which yields that Open image in new window is bounded. However, this contradicts (3.19).

Therefore, Open image in new window joins Open image in new window to Open image in new window in Open image in new window .

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Proof of Theorem 1.2.

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

- (1)
- (2)
Open image in new window joins Open image in new window with infinity in Open image in new window .

We claim that Open image in new window 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 Open image in new window . Therefore, the superior limit of Open image in new window contains a component Open image in new window joining Open image in new window with infinity in Open image in new window .

- (1)
- (2)
Open image in new window joins Open image in new window with infinity in Open image in new window ,

and the superior limit of Open image in new window contains a component Open image in new window joining Open image in new window with infinity in Open image in new window .

From Lemma 2.4, it follows that Open image in new window is closed in Open image in new window , and furthermore, Open image in new window is compact in Open image in new window .

Let

If for some Open image in new window , Open image in new window , then Theorem 1.2 holds.

Assume on the contrary that Open image in new window for all Open image in new window .

where Open image in new window and Open image in new window are the boundary and closure of Open image in new window in Open image in new window , respectively.

where Open image in new window and Open image in new window are the boundary and closure of Open image in new window in Open image in new window , respectively.

However, this contradicts (3.50).

Therefore, there exists Open image in new window such that Open image in new window which is unbounded in both Open image in new window and Open image in new window .

Finally, we show that Open image in new window joins Open image in new window with Open image in new window . This will be done by the following three steps.

Step 1.

We show that Open image in new window .

This is impossible by (A3) and the assumption Open image in new window .

Step 2.

We show that Open image in new window .

where Open image in new window . By (A2), it follows that Open image in new window . Obviously, (3.65) implies that Open image in new window is bounded. This is a contradiction.

Step 3.

We show that Open image in new window .

where Open image in new window . By (A2), it follows that Open image in new window . Obviously, (3.68) implies that Open image in new window is bounded. This is a contradiction.

To sum up, there exits a component Open image in new window which joins Open image in new window and Open image in new window .

## Notes

### 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 Open image in new window ).

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