Oscillatory and Asymptotic Behaviour of Solutions of Two Nonlinear Dimensional Difference Systems

This paper deals with the some oscillation criteria for the two dimensional difference system of the form: 0 , 1, 2,3, n n n n n n x b y y a x n N α β ∆ = ∆ = − ∈ =  . Examples illustrating the results are inserted.


Introduction
Consider a nonlinear two dimensional difference system of the form In the last few decades there has been an increasing interest in obtaining necessary and sufficient conditions for the oscillation and nonoscillation of two dimensional difference equation.See for example [1]- [10] [11] and the references cited therein.
Further it will be assumed that { } n b is non-negative for all 0 n n ≥ , ( ) The oscillation criteria for system (1.1), when studied in [12].Therefore in this paper we consider the other case that is and investigated the oscillatory behaviour of solutions of the system (1.1).Hence the results obtained in this paper complement to that of in [12].
We may introduce the function n A defined by ( ) Throughout this paper condition (1.2) is tacitly assumed; n A always denotes the function defined by (1.3).
In Section 2, we establish necessary and sufficient conditions for the system (1.1) to have solutions which behave asymptotically like nonzero constants or linear functions and in Section 3, we present criteria for the oscillation of all solutions of the system (1.1).Examples are inserted to illustrate some of the results in Section 4.

Existence of Bounded/Unbounded Solutions
In this section first we obtain necessary and sufficient conditions for the system x y .such that ( ) as n → ∞ , where Proof.We may assume without loss of generality that and let ( ) be large enough such that ( ) and ( ) Let B be the space of all real sequences { }, n y y n N = ≥ with the topology of pointwise convergence.We now define X to be the set of sequences x B , , .
where ( ) sup : and define Y to be the set of sequences y B ∈ .

Such that
, .
and ( ) Clearly X Y × is a bounded, closed and convex subset of B B × .
First we show that T maps X Y we have and so, using (2.6) and (2.7), we see that ( ) .
Finally, in order to apply Schauder-Tychonoff fixed point theorem, we need to show that ( ) In view of recent result of cheng and patula [8] it suffices to show that ( ) , , , , are uniformly cauchy and so ( ) Therefore by Schauder-Tychonoff fixed point theorem, there is an element ( ) T x y x y = . From (2.12), (2.13) and (2.14) as n → ∞ .The proof is left to the reader.
Before stating and proving our next results, we give a lemma which is concerned with the nonoscillatory solution of (1.1).
x y be a solution of (1.1) for ( ) and , where θ is a nonnegative constant.
This lemma has been proved by Graef and Thandapani [3] and is very useful in the following theorems.In our next theorem, we establish a necessary condition for the system (1.1) to have nonoscillatory solution satisfying condition (2.17).
Proof.Let ( ) x y be a nonoscillatory solution of the system (1.1) for ( ) . Since n b is not identically zero for ( ) , .
, from the first equation of system (1.1), we obtain and hence ( ) ( )  .
From the second inequality of (2.21) and the following inequality ( ) ( ) ( ) where "d" being the constant, we see that as n → ∞ , from the first equation of system (1.1), we obtain for n N ≥ ( ) ( ) ( ) which in view of boundedness of n x , implies that The inequalities (2.24) and (2.25) clearly imply (2.20).This completes the proof.
we conclude this section with the following theorem which gives a necessary condition for the system (1.1) to have a nonoscillatory solution of the form Using the inequality ( ) Because of condition (3.1), the last inequality implies ( ) Next from the second inequality (2.21), we have ( ) ( )( ) ( ) , .
Again using the argument as in the proof of Theorem 2.5, we obtain ( ) for all n N ≥ .So by condition on (3.1), we have .
1 n + to j, we obtain ∈ , α and β are ratio of odd positive integers.By a solution of Equation (1.1), we mean a real sequence { }
of the boundedness of n x implies that