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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

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

(1.5)

Let be the set of integers, and .

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

(1.6)

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

(1.7)

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

(2.1)

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

(2.2)

for and

For any we set

(2.3)

If

(2.4)

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

(2.5)

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

(2.6)

so that

(2.7)

Lemma 2.2.

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

(2.8)

Lemma 2.3.

Let be subsets of Then

(2.9)

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

(3.1)

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

(3.2)

and if , for then

(3.3)

Proof.

If , for it follows from (1.6) and that

(3.4)

Hence for

Similarly, we also have , for

Theorem 3.2.

Suppose that

(3.5)

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

(3.6)

Therefore, by Lemma 2.4, the intersection

(3.7)

is infinite. This implies that there exist and such that

(3.8)
(3.9)

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

(3.10)

which is a contradiction.

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

(3.11)

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

Theorem 3.3.

Suppose that

(3.12)

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

(3.13)

Therefore, by Lemma 2.4, we know that

(3.14)

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

(3.15)

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

(3.16)

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

(4.1)

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

(4.2)

is infinite and a subsequent contradiction.

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

(4.3)

Therefore, by Lemma 2.4, we know that the set

(4.4)

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

(4.5)

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

Theorem 4.3.

Suppose there exists constant such that

(4.6)

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

(4.7)

Hence, by Lemma 2.4, we see that

(4.8)

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

(4.9)

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

(4.10)

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

(5.1)

where

(5.2)

and .

It is clear that , , , , and .

Moreover,

(5.3)

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.