- Research Article
- Open Access

# Oscillatory Criteria for the Two-Dimensional Difference Systems

- Jin-Fa Cheng
^{1}and - Yu-Ming Chu
^{2}Email author

**2010**:209309

https://doi.org/10.1155/2010/209309

© Jin-Fa Cheng and Yu-Ming Chu. 2010

**Received:**5 April 2010**Accepted:**17 June 2010**Published:**8 July 2010

## Abstract

We establish some necessary and sufficient conditions for oscillation of the solutions of the following two-dimensional difference system: , , where and are strongly superlinear or sublinear functions.

## Keywords

- Banach Space
- Positive Integer
- Physical Application
- Difference Equation
- Difference System

## 1. Introduction

where , , and are strongly superlinear or sublinear functions.

Now we pose some conditions on functions and :

(H1): and for ;

(H2): and are continuous real-valued functions, and nondecreasing with respect to ;

(H3): it is

for each .

Definition 1.1.

Suppose that are real-valued functions. and are the quotients of positive odd numbers.

( ) and are said to be strongly superlinear if there exist constants and with , such that and are nondecreasing with respect to for each fixed .

( ) and are said to be strongly sublinear if there exist constants and with , such that and are nonincreasing with respect to for each fixed .

The solutions of (1.1) are said to be nonoscillatory if or is eventually positive or negative. Otherwise the solutions are called oscillatory.

For (1.4), the following well-known Theorem A was established by Hooker and Patula [1, 2].

Theorem A.

For (1.4), the following statements are true.

then the following theorem is proved in [3].

Theorem B.

If (1.9) holds, then the following statements are true.

The problem of oscillation of second-order nonlinear difference equations has attracted the attention of many mathematicians because of its physical applications [2, 4]. For some results regarding the growth of solutions of some equations related to the above mentioned see book [5], as well as the following papers [6–8]. It is an interesting problem to extend oscillation criteria for second-order nonlinear difference equations to the case of nonlinear two-dimensional difference systems since such systems include, in particular, the second-order nonlinear, half-linear, and quasilinear difference equations that are the special cases of the nonlinear two-dimensional difference systems [5, 9, 10].

The main purpose of this paper is to establish some necessary and sufficient conditions for oscillation of the nonlinear two-dimensional difference systems.

## 2. Main Results

In order to establish our main results, we need the following lemma.

Lemma 2.1.

Proof.

Without loss of generality, we assume that is eventually positive; that is, for . From (1.1), we clearly see that , then we know that either or eventually holds.

This contradiction completes the proof of the lemma.

Theorem 2.2.

for every , where .

Proof.

*Sufficiency*. If (1.1) has a nonoscillatory solution , then without loss of generality, we assume that is eventually positive. Then, by Lemma 2.1, for sufficiently large,

which leads to a contradiction.

Then by (1.2) and the continuity of function , we have that and which leads to a contradiction and the proof of Theorem 2.2 is completed.

Example 2.3.

Here , , and and are strongly sublinear. It is easy to verify that the conditions of Theorem 2.2 are satisfied and hence all solutions are oscillatory. In fact, we clearly see that the sequence is such a solution for the difference system.

Example 2.4.

Here , , and and are strong sublinear. We clearly see that the conditions of Theorem 2.2 are not satisfied and hence there exists a nonoscillatory solution. In fact, the sequence is such a solution.

Theorem 2.5.

for every .

Proof.

*Sufficiency*. If (1.1) has a nonoscillatory solution , then without loss of generality, we may assume that is eventually positive. Then by Lemma 2.1, we have for sufficiently large,

which leads to a contradiction.

Since , now from the continuity of and together with the well-known Lebesgue's dominated convergence theorem (see [11, page 263]), we know that for .

for any . From Schauder's fixed-point theorem (see [11]), we know that there exists such that .

Therefore, and which leads to a contradiction. The proof of Theorem 2.5 is completed.

Example 2.6.

Here , , and and are strongly suplinear. We clearly see that the conditions of Theorem 2.5 are satisfied and hence all solutions are oscillatory. In fact, the sequence is such a solution.

Example 2.7.

Here , , and and are strong sublinear. However, it is easy to see that the conditions of Theorem 2.5 are not satisfied and hence there exists a nonoscillatory solution. In fact, the sequence is such a solution.

If we set , then the difference system (1.1) is reduced to (1.5). From Theorems 2.2 and 2.5, we get the following results for (1.5).

Corollary 2.8.

where .

Corollary 2.9.

Remark 2.10.

It is easy to see that Theorems A and B are the special cases of our Corollaries 2.8 and 2.9, respectively.

## Declarations

### Acknowledgment

The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions.

## Authors’ Affiliations

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