Oscillation and Asymptotic Behaviour of Solutions of Nonlinear Two-Dimensional Neutral Delay Difference Systems

This paper deals with the some oscillation criteria for the two-dimensional neutral delay difference system of the form ( ) ( ) ( ) 1 0 , 1, , , , 2 3 n n n k n n n n n l x p x b y y a x n n − − + ∆ + = ∆ = − ∈ =   Examples illustrating the results are inserted.

1 2), we have a second order linear equation .
For oscillation criteria regarding Equations (1.1)-(1.3),we refer to [2]- [12] 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.We begin with the following lemma.

Some Basic Lemmas
2.1.Let ( ) ( ) Next, we state a lemma whose proof can be found in [1].
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.

Oscillation Theorems for the System (1.1)
Theorem 3.1.Assume that { } n p is bounded and there exists an integer j such that 2 l j k > + + .If and Then every solution } ( ) { , . In view of Lemma 2.1, we have two cases for sufficiently large ( ) Case (1).Because { } n y is negative and nonincreasing there is constant L > 0. ,

Such that
summing the second equation of (1.1) from n to i, using (3.5) and then letting i → ∞ , we obtain Using a summation by parts formula, we have ( ) ( ) , .
The last inequality together with (3.4) and the monotonocity of { } x y W ∈ be a nonoscillatory solution of (1.1).Without loss of generality we may assume that { } n x is positive for n ( ) . As in the proof of above theorem we have two cases.
Case (1).Analogus to the proof of case ( 1 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.( ) Then all solutions of (1.1) are oscillatory.
Combining the last inequality with the second equations of (1.1) and (3.17 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.

Examples
Example 4.1.Consider the difference system

..
Consider a nonlinear neutral type two-dimensional delay difference system of the form By a solution of the system (1.1), we mean a real sequence { } , n n x y which is defined for all 0 n n θ ≥ − and satisfies (1.1) for all ( ) How to cite this paper: Thangavelu, K. and Saraswathi, G. (2017) Oscillation and Asymptotic Behaviour of Solutions of Nonlinear Two-Dimensional Neutral Delay Di-Let W be the set of all solutions be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise.A solution X W ∈ is said to be oscillatory if both components are oscillatory and it will be called nonoscillatory otherwise.Some oscillation results for difference system (1 for sufficiently large n.If for n k − for all ( )0 n n ∈  .Then, { } n x is bounded.Proof.Without loss of generality we may assume that { } n x be an eventually positive solution of the inequality (2.1), the proof for the case { } n x eventually negative is similar.From (2.1) we have

From ( 3
.3), we see that lim n n z →∞ = −∞ which contradicts the fact that { }

1
there exists an integer j such that l j k > + and the conditions (3.1) and (3.2) are satisfied.Then all solutions of ( ) of above theorem, we can show that lim a contradiction.Hence case (1) cannot occur.
nonoscillatory solution of (1.1).Without loss of generality we may assume that { } k = − + + and using the monotonocity of { } n z , from the last inequality, we obtain a positive constant.The conditions (3.1) and (3.2) are all conditions of Theorem 3.2 are satisfied and so all solutions of the system (4.2) are oscillatory.