#### Abstract

New sufficient conditions for the oscillation of all solutions of difference equations with several deviating arguments and variable coefficients are presented. Examples illustrating the results are also given.

#### 1. Introduction

In this paper we study the oscillation of all solutions of difference equation with several variable retarded arguments of the form and the (dual) difference equation with several variable advanced arguments of the form where , , , are sequences of positive real numbers and , , are sequences of integers such that and , , are sequences of integers such that Here, denotes the forward difference operator and denotes the backward difference operator .

Strong interest in is motivated by the fact that it represents a discrete analogue of the differential equation (see [1–3] and the references cited therein) where, for every , is a continuous real-valued function in the interval and is a continuous real-valued function on such that while represents a discrete analogue of the advanced differential equation (see [1, 2] and the references cited therein) where, for every , is a continuous real-valued function in the interval and is a continuous real-valued function on such that

By a* solution* of , we mean a sequence of real numbers which satisfies for all . Here
It is clear that, for each choice of real numbers , , , , there exists a unique solution of which satisfies the initial conditions , , , .

By a solution of , we mean a sequence of real numbers which satisfies for all .

A solution (or ) of (or ) is called* oscillatory, *if the terms of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be* nonoscillatory*.

In the last few decades, the oscillatory behavior of the solutions of difference and differential equations with several deviating arguments and variable coefficients has been studied. See, for example, [1–14] and the references cited therein.

In 2006, Berezansky and Braverman [5] proved that if where , for all , then all solutions of oscillate.

Recently, Chatzarakis et al. [7–9] established the following theorems.

Theorem 1 (see [9]). *Assume that the sequences , , are increasing, (1) (2) holds, and
**
where , for all , , for all , or*

*then all solutions of [] oscillate.*

Theorem 2 (see [7, 8]). *Assume that the sequences , , are increasing and (1) (2) holds. Set
**
If , and
**
or
**
then all solutions of [] oscillate.*

The authors study further and and derive new sufficient oscillation conditions. These conditions are the improved and generalized discrete analogues of the oscillation conditions for the corresponding differential equations, which were studied in 1982 by Ladas and Stavroulakis [2]. Examples illustrating the results are also given.

#### 2. Oscillation Criteria

##### 2.1. Retarded Difference Equations

We present new sufficient conditions for the oscillation of all solutions of .

Theorem 3. *Assume that , , are increasing sequences of integers such that (1) holds and , , are sequences of positive real numbers and define , , by (11). If , , and
**
then all solutions of oscillate.*

*Proof. *Assume, for the sake of contradiction, that is a nonoscillatory solution of . Then it is either eventually positive or eventually negative. As is also a solution of , we may restrict ourselves only to the case where for all large . Let be an integer such that for all . Then, there exists such that
In view of this, becomes
which means that the sequence is eventually decreasing.

Next choose a natural number such that
Set
It is obvious that
Now we will show that for . Indeed, assume that for some , . For this , by , we have

At this point, we will establish the following claim.*Claim **1* (cf. [8])*.* For each , there exists an integer for each such that , and
where is an arbitrary real number with .

To prove this claim, let us consider an arbitrary real number with . Then by (11) we can choose an integer such that
Assume, first, that and choose . Then . Moreover, we have
and, by (23),
So, (21) and (22) are fulfilled. Next, we suppose that . It is not difficult to see that (23) guarantees that . In particular, it holds
Thus, as , there always exists an integer so that
and (21) holds. We assert that . Otherwise, . We also have . Hence, in view of (27), we get
On the other hand, (23) gives
We have arrived at a contradiction, which shows our assertion that . Furthermore, by using (23) (for the integer ) as well as (27), we obtain
and consequently (22) holds true. Our claim has been proved.

Now, summing up (20) from to , we find
or
Summing up (20) from to , we find
or
Combining (32) and (34), we obtain
or
which means that is bounded. This contradicts our assumption that . Therefore for every .

Dividing both sides of by , for , we obtain
or
Summing up (38) from to for , we find
But
or
Combining (39) and (41), we obtain
or
Taking limit inferiors on both sides of the above inequalities (43), we obtain
and by adding we find
Set
Clearly
Since
for
the function has a maximum at the critical point
since the quadratic form
Since , the maximum of at the critical point should be nonnegative. Thus,
that is,
Hence
or
which contradicts (14).

The proof of the theorem is complete.

Theorem 4. *Assume that , , are increasing sequences of integers such that (1) holds and , , are sequences of positive real numbers and define , , by (11). If , , and
**
then all solutions of oscillate.*

*Proof. *Assume, for the sake of contradiction, that is a nonoscillatory solution of . Then it is either eventually positive or eventually negative. As is also a solution of , we may restrict ourselves only to the case where for all large . Let be an integer such that for all . Then, there exists such that
In view of this, becomes
which means that the sequence is eventually decreasing.

Taking into account the fact that for (see proof of Theorem 3), by using (44) and the fact that
we obtain
Adding these inequalities we have
or
which contradicts (56).

The proof of the theorem is complete.

##### 2.2. Advanced Difference Equations

Similar oscillation theorems for the (dual) advanced difference equation can be derived easily. The proofs of these theorems are omitted, since they follow a similar procedure as in Section 2.1.

Theorem 5. *Assume that , , are increasing sequences of integers such that (2) holds and , , are sequences of positive real numbers and define , , by (11). If , , and
**
then all solutions of oscillate.*

Theorem 6. *Assume that , , are increasing sequences of integers such that (2) holds and , , are sequences of positive real numbers and define , , by (11). If , , and
**
then all solutions of oscillate.*

##### 2.3. Special Cases

In the case where , , are positive real constants and are constant retarded arguments of the form , [ are constant advanced arguments of the form ], , , equation [] takes the form For this equation, as a consequence of Theorems 3 [5] and 4 [6], we have the following corollary.

Corollary 7. *Assume that
**
or
**
Then all solutions of oscillate.*

*Remark 8. *A research question that arises is whether Theorems 3–6 are valid, even in the case where the coefficients oscillate (see [15, 16]). Then our results would be comparable to those in [15, 16]. This is a question that we currently study and expect to have some results soon.

#### 3. Examples

The following two examples illustrate that the conditions for oscillations (65) and (66) are independent. They are chosen in such a way that only one of them is satisfied.

*Example 1. *Consider the retarded difference equation
Here , , , , and
It is easy to see that
That is, condition (65) of Corollary 7 is satisfied and therefore all solutions of equation (67) oscillate.

However,
That is, condition (66) of Corollary 7 is not satisfied.

Observe that
Thus
Also,
and therefore none of the conditions (9), (12), (13), (8), and (10) are satisfied.

*Example 2. *Consider the advanced difference equation
Here , , , and
It is easy to see that
That is, condition (66) of Corollary 7 is satisfied and therefore all solutions of (74) oscillate.

However,
That is, condition (65) of Corollary 7 is not satisfied.

Observe that
Thus
Also,
and therefore none of the conditions (9), (12), and (13) are satisfied.

At this point, we give an example with general retarded arguments illustrating the main result of Theorem 3. Similarly, one can construct examples to illustrate Theorems 4–6.

*Example 3. *Consider the delay difference equation
with .

Here and denote the integer parts of and . Observe that the sequences and are increasing, , , and
Observe that, for a positive decreasing function , the following inequality holds:
Based on the above inequality, we will show that
for any , , and any real number . Indeed,
It is easy to see that
From the above, it follows that
Therefore
Hence
That is, condition (14) of Theorem 3 is satisfied and therefore all solutions of (81) oscillate.

Observe, however, that
Thus
Also,
and therefore none of the conditions (8), (9), (12), and (13) are satisfied.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The second author was supported by the Serbian Ministry of Science, Technology and Development for Scientific Research Grant no. III44006.