Consider a nonlinear neutral type two-dimensional delay difference system of the form
Subject to the following conditions:
, and are nonnegative real sequences such that
, is a positive real sequence.
, f,g : are continous non-decreasing with , , for and , where k is a constant.
, k and l are nonnegative integers.
Let . By a solution of the system (1.1), we mean a real sequence which is defined for all and satisfies (1.1) for all .
Let W be the set of all solutions of the system (1.1) which exists for and satisfies
A real sequence defined on is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise.
A solution is said to be oscillatory if both components are oscillatory and it will be called nonoscillatory otherwise.
Some oscillation results for difference system (1.1) when for and have been presented in  , In particular when for all . The difference system (1.1) reduces to the second order nonlinear neutral difference equation
If , in Equation (1.2), we have a second order linear equation
For oscillation criteria regarding Equations (1.1)-(1.3), we refer to  -  and the references cited therein. In Section 2, we present some basic lemmas. In Section 3, we establish oscillation criteria for oscillation of all solutions of the system (1.1). Examples are given in Section 4 to illustrate our theorems.
2. Some Basic Lemmas
Denote For any we define by
We begin with the following lemma.
2.1. Let hold and let be a solution of system (1.1) with either eventually positive or eventually negative for . Then is nonoscillatory and and are monotone for .
Proof. Let and let be nonoscillatory on . Then from the second equation of system (1.1), we have for all and ,and are not identically zero for infinitely many values of n. Thus is monotone for . Hence is either eventually positive or eventually negative for . Then, is nonoscillatory. Further from the first equation of the system (1.1). We have eventually. Hence is monotone and nonoscillatory for all . The proof is similar when is eventually negative.
Lemma 2.2. In addition to conditions assume that for all . Let be a nonoscillatory solution of the inequality
for sufficiently large n. If for for all . Then, is bounded.
Proof. Without loss of generality we may assume that be an eventually positive solution of the inequality (2.1), the proof for the case eventually negative is similar. From (2.1) we have
and , we have from (2.2), for all . Hence is bounded.
Next, we state a lemma whose proof can be found in  .
Lemma 2.3. Assume that is a non negative real sequence and not identically zero for infinitely many values of n and l is a positive integer. If
Then the difference inequality
cannot have an eventually positive solution and
cannot have an eventually negative solution.
3. Oscillation Theorems for the System (1.1)
Theorem 3.1. Assume that is bounded and there exists an integer j such that . If
Then every solution is a nonoscillatory solution of system (1.1), with bounded. Without loss of generality we may assume that is eventually positive and bounded for all . From the second equation of (1.1), we obtain for sufficiently large . In view of Lemma 2.1, we have two cases for sufficiently large
1) for ;
2) for .
Case (1). Because is negative and nonincreasing there is constant L > 0. Such that
Since and are bounded. defined by (2.1) is bounded. Summing the first equation of (1.1) from to and then using (3.3), we obtain
From (3.3), we see that which contradicts the fact that is bounded. Case (1) cannot occur.
Case (2). Let for where is sufficiently large. Because is nondecreasing there is a positive constant M, such that
From (2.1), we have , and by hypothesis, we obtain
summing the second equation of (1.1) from n to i, using (3.5) and then letting , we obtain
From condition (3.1), we have
we claim that the condition (3.1) implies
Otherwise, if , we can choose an integer . So large that which contradicts (3.6).
Using a summation by parts formula, we have
From (3.3), (3.4) and (3.6) and the second equation of (1.1), we have
combining (3.6) with (3.8), we obtain
The last inequality together with (3.4) and the monotonocity of implies
and , which contradicts (1.1). This case cannot occur. The proof is complete.
Theorem 3.2. Assume that , then there exists an integer j such that and the conditions (3.1) and (3.2) are satisfied. Then all solutions of (1.1) are oscillatory.
Proof . Let be a nonoscillatory solution of (1.1). Without loss of generality we may assume that is positive for n . As in the proof of above theorem we have two cases.
Case (1). Analogus to the proof of case (1) of above theorem, we can show that . By Lemma 2.2, is bounded and hence is bounded which is a contradiction. Hence case (1) cannot occur.
Case (2). The proof of case (2) is similar to that of the above theorem and hence the details are omitted. The proof is now complete.
Theorem 3.3. Assume that and
Then all solutions of (1.1) are oscillatory.
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality we may assume that is positive for . As in the proof of above theorem we have two cases.
Case 1. From (2.1), we have
where is sufficiently large. Then the following equality
Combining the last inequality with the second equations of (1.1) and (3.17), we have
Let and using the monotonocity of , from the last inequality, we obtain
which contradicts the condition (3.14).
Case 2. The proof for this case is similar to that of Theorem (3.1). Here we use condition (3.16) instead of condition (2.1). The proof is complete.
Example 4.1. Consider the difference system
The conditions (3.1) and (3.2) are
All conditions of Theorem 3.2 are satisfied and so all solutions of the system (4.1) are oscillatory.
Example 4.2. Consider the difference systems
where c is a positive constant. The conditions (3.1) and (3.2) are
For , all conditions of Theorem 3.2 are satisfied and so all solutions of
the system (4.2) are oscillatory.
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 Thandapani, E. and Mohankumar, P. (2007) Oscillation of Difference Systems of the Neutral type. Computers and Mathematics with Applications, 54, 556-566.
 Thandapani, E., Sundaram, P., Graef, J.R. and Spikes, P.W. (1995) Asymptotic Properties of Solutions of Nonlinear Second Order Neutral Delay Difference Equations. Dynam. Systems Appl., 4, 125-136.
 Zhang, Z., Chen, J. and Zhang, C. (2001) Oscillation of Solutions of Second Order Nonlinear Difference Equations with Nonlinear Neutral Term. Computers and Mathematics with Applications, 41, 1571-1584.