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# Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients

*Advances in Difference Equations*
**volume 2010**, Article number: 606149 (2010)

## Abstract

This paper is concerned with a class of nonlinear delay partial difference equations with positive and negative coefficients, which also contains forcing terms. By making use of frequency measures, some new oscillatory criteria are established.

## 1. Introduction

Partial difference equations are difference equations that involve functions with two or more independent integer variables. Such equations arise from considerations of random walk problems, molecular structure problems, and numerical difference approximation problems. Recently, there have been a large number of papers devoted to partial difference equations, and the problem of oscillatory of solutions and frequent oscillatory solutions for partial difference equations is receiving much attention.

In [1], authors considered oscillatory behavior of the partial difference equations with positive and negative coefficients of the form

but they have not discussed frequent oscillations of this equation.

In [2], authors considered oscillatory behavior for nonlinear partial difference equations with positive and negative coefficients of the form

In [3], authors considered frequent oscillation in the nonlinear partial difference equation

In [4], authors considered oscillations of the partial difference equations with several nonlinear terms of the form,

and in [5] authors considered frequent oscillations of these equations.

In [6], authors considered unsaturated solutions for partial difference equations with forcing terms

Let be the set of integers, and .

In this paper, we will consider the equation of the following form:

where ,, , and

; are nonnegative integers;

, , are real double sequences.

The usual concepts of oscillation or stability of steady state solutions do not catch all their fine details, and it is necessary to use the concept of frequency measures introduced in [7] to provide better descriptions. In this paper, by employing frequency measures, some new oscillatory criteria of (1.6) are established.

Let

In addition to and , we also assume

;

, ;

For the sake of convenience, will be denoted by in the sequel. Given a double sequence , the partial differences and will be denoted by and respectively.

To the best of our knowledge, nothing is known regarding the qualitative behaviour of the solutions of (1.6), because these equations contain positive and negative coefficients, and also contain forcing terms.

Our plan is as follows. In the next section, we will recall some of the basic results related to frequency measures. Then we obtain several criteria for all solutions of (1.6) to be frequently oscillatory and unsaturated. In the final section, we give one example to illustrate our results.

## 2. Preliminary

The union, intersection, and difference of two sets and will be denoted by and respectively. The number of elements of a set will be denoted by Let be a subset of then

are the translations of Let , and be integers satisfying and The union will be denoted by Clearly,

for and

For any we set

If

exists, then the superior limit, denoted by will be called the upper frequency measure of Similarly, if

exists, then the inferior limit, denoted by will be called the lower frequency measure of If then the common limit is denoted by and is called the frequency measure of

Clearly, and for any subset of furthermore, if is finite, then

The following results are concerned with the frequency measures and their proofs are similar to those in [8].

Lemma 2.1.

Let and be subsets of Then Furthermore, if and are disjoint, then

so that

Lemma 2.2.

Let be a subset of and let and be integers such that and Then

Lemma 2.3.

Let be subsets of Then

Lemma 2.4.

Let and be subsets of If then the intersection is infinite.

For any real double sequence defined on a subset of the level set is denoted by The notations and are similarly defined. Let be a real double sequence. If then is said to be frequently positive, and if , then is said to be frequently negative.

is said to be frequently oscillatory if it is neither frequently positive nor frequently negative. If then is said to have unsaturated upper positive part, and if then is said to have unsaturated lower positive part. is said to have unsaturated positive part if .

The concepts of frequently oscillatory and unsaturated double sequences were introduced in [5–11]. It was also observed that if a double sequence is frequently oscillatory or has unsaturated positive part, then it is oscillatory; that is, is not positive for all large and nor negative for all large and Thus if we can show that every solution of (1.6) is frequently oscillatory or has unsaturated positive part, then every solution of (1.6) is oscillatory.

## 3. Frequently Oscillatory Solutions

Lemma 3.1.

Suppose there exist and such that

for Let be a solution of (1.6). If , for then

and if , for then

Proof.

If , for it follows from (1.6) and that

Hence for

Similarly, we also have , for

Theorem 3.2.

Suppose that

where and . Then every nontrivial solution of (1.6) is frequently oscillatory.

Proof.

Suppose to the contrary that is a frequently positive solution of (1.6). Then By Lemmas 2.1–2.3, we have

Therefore, by Lemma 2.4, the intersection

is infinite. This implies that there exist and such that

hold for In view of (3.9) and Lemma 3.1, we may see that and for , and hence so by (3.9) and , we have that

which is a contradiction.

In a similar manner, if is a frequently negative solution of (1.6) such that then we may show that

is infinite. Again we may arrive at a contradiction as above. The proof is complete.

Theorem 3.3.

Suppose that

where , and . Then every nontrivial solution of (1.6) is frequently oscillatory.

Proof.

Suppose to the contrary that is frequently positive solution of (1.6). Then . By Lemmas 2.1–2.3, we know

Therefore, by Lemma 2.4, we know that

is infinite. This implies that there exist and such that (3.8) and

hold for . By similar discussions as in the proof of Theorem 3.2, we may arrive at a contradiction against (3.8).

In case is a frequently negative solution of (1.6), then . In an analogous manner, we may see that

is infinite. This can lead to a contradiction again. The proof is complete.

## 4. Unsaturated Solutions

The methods used in the above proofs can be modified to obtain the following results for unsaturated solutions.

Theorem 4.1.

Suppose there exists constant such that

where , and . Then every nontrivial solution of (1.6) has unsaturated upper positive part.

Proof.

Let be a nontrivial solution of (1.6). We assert that Otherwise, then or In the former case, applying arguments similar to the proof of Theorem 3.2, we may then arrive at the fact that

is infinite and a subsequent contradiction.

In the latter case, we have By Lemmas 2.1–2.3, we have

Therefore, by Lemma 2.4, we know that the set

is infinite. Then by discussions similar to these in the proof of Theorem 3.2 again, we may arrive at a contradiction. The proof is complete.

Combining Theorems 3.3 and 4.1, we have the following Theorem 4.2 and the proof of this theorem is omitted.

Theorem 4.2.

Suppose there exists constant such that

where , and . Then every nontrivial solution of (1.6) has unsaturated upper positive part.

Theorem 4.3.

Suppose there exists constant such that

where , and . Then every nontrivial solution of (1.6) has an unsaturated upper positive part.

Proof.

We claim that . First, we prove that . Otherwise, if , by Lemmas 2.1–2.3, we have

Hence, by Lemma 2.4, we see that

is infinite. Then there exist and such that (3.8) and

hold for . Applying similar discussions as in the proof of Theorem 3.2, we can get a contradiction. Next, we prove that . Otherwise, . Analogously, we see that

is infinite. Then, we can also lead to a contradiction. The proof is complete.

We remark that every nontrivial solution of (1.6) has an unsaturated lower positive part under the same conditions as Theorems 4.1, 4.2, or 4.3. So we can obtain that every nontrivial solution of (1.6) has an unsaturated positive part.

## 5. Examples

We give one example to illustrate our previous results.

Example 5.1.

Consider the partial difference equation

where

and .

It is clear that , , , , and .

Moreover,

Then according to Theorems 3.2 or 3.3, we know that every nontrivial solution of (5.1) is frequently oscillatory. If , we see that all conditions in Theorems 4.1, 4.2, or 4.3 are satisfied. Thus, every nontrivial solution of (5.1) has an unsaturated upper positive part.

## References

- 1.
Liu ST, Zhang BG:

**Oscillatory behavior of delay partial difference equations with positive and negative coefficients.***Computers & Mathematics with Applications*2002,**43**(8-9):951-964. 10.1016/S0898-1221(02)80005-0 - 2.
Liu ST, Liu YQ, Deng FQ:

**Oscillation for nonlinear delay partial difference equations with positive and negative coefficients.***Computers & Mathematics with Applications*2002,**43**(10-11):1219-1230. 10.1016/S0898-1221(02)00093-7 - 3.
Yang J, Zhang YJ, Cheng SS:

**Frequent oscillation in a nonlinear partial difference equation.***Central European Journal of Mathematics*2007,**5**(3):607-618. 10.2478/s11533-007-0017-1 - 4.
Zhang BG, Xing Q:

**Oscillation of certain partial difference equations.***Journal of Mathematical Analysis and Applications*2007,**329**(1):567-580. 10.1016/j.jmaa.2006.07.002 - 5.
Yang J, Zhang YJ:

**Frequent oscillatory solutions of a nonlinear partial difference equation.***Journal of Computational and Applied Mathematics*2009,**224**(2):492-499. 10.1016/j.cam.2008.05.035 - 6.
Zhu ZQ, Cheng SS:

**Unsaturated solutions for partial difference equations with forcing terms.***Central European Journal of Mathematics*2006,**4**(4):656-668. 10.2478/s11533-006-0030-9 - 7.
Tian CJ, Xie S-L, Cheng SS:

**Measures for oscillatory sequences.***Computers & Mathematics with Applications*1998,**36**(10–12):149-161. 10.1016/S0898-1221(98)80017-5 - 8.
Cheng SS:

*Partial Difference Equations, Advances in Discrete Mathematics and Applications*.*Volume 3*. Taylor & Francis, London, UK; 2003:xii+267. - 9.
Zhu ZQ, Cheng SS:

**Frequently oscillatory solutions of neutral difference equations.***Southeast Asian Bulletin of Mathematics*2005,**29**(3):627-634. - 10.
Zhu ZQ, Cheng SS:

**Frequently oscillatory solutions for multi-level partial difference equations.***International Mathematical Forum*2006,**1**(29–32):1497-1509. - 11.
Zhang B, Zhou Y:

*Qualitative Analysis of Delay Partial Difference Equations, Contemporary Mathematics and Its Applications*.*Volume 4*. Hindawi Publishing Corporation, New York, NY, USA; 2007:viii+374.

## Acknowledgments

This project was supported by the NNSF of P.R.China (60604004) and by Natural Science Foundation of Hebei province (no. 07M005).

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Xu, L., Yang, J. Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients.
*Adv Differ Equ* **2010, **606149 (2010). https://doi.org/10.1155/2010/606149

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### Keywords

- Frequent Oscillation
- Steady State Solution
- Nontrivial Solution
- Frequency Measure
- Negative Coefficient