ON THE OSCILLATION OF SECOND ORDER NONLINEAR NEUTRAL DELAY DIFFERENCE EQUATIONS

In this paper sufficient conditions are obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form


Introduction
Recently, a good deal of work has been published on the oscillation theory of difference equations.Most of the works in first and higher order neutral delay difference equations are concerned with the study of the behaviour of the solution which oscillates or tends to zero (see [3], [4], [5], [7]).But very few papers are available on oscillatory higher order nonlinear delay difference equations.EJQTDE, 2008 No. 11, p. 1 In [7], authors Parhi and Tripathy has considered a class of nonlinear neutral delay difference equations of higher order of the form where m ≥ 2. They have obtained the results which hold good when G is sub linear only.However, the behaviour of solutions of (E) under the superlinear nature of G is still in progress.In fact, various ranges of p(n) are restricting for all solutions as oscillatory.
In this paper, author has studied the second order nonlinear neutral delay difference equation of the form where ∆ is the forward difference operator defined by ∆ y(n) = y(n + 1) − y(n), p, q are real valued functions defined on ) is nondecreasing and x G(x) > 0 for x = 0 and m > 0, k ≥ 0 are integers.Here, an attempt is made to establish sufficient conditions under which every solution of Eq.( 1) oscillates.
The motivation of present work has come under two directions.Firstly, due to the work in [6] and second is due to the work in [7], where G is almost sublinear.It is interesting to observe that unlike differential equation, Eq. ( 1) is converting immediately into a first order difference inequality and hence study of both are interrelated hypothetically.In this regard the work in [8] provides a good input for the completion of the present work.
By a solution of Eq. ( 1) we mean a real valued function y(n) defined on N(−r) = {−r, −r + 1, • • •} which satisfies (1) for n ≥ 0, where r = max{k, m}.If are given, then (1) admits a unique solution satisfying the initial condition (2).A solution y(n) of ( 1) is said to be oscillatory, if for every integer N > 0, there exists an n ≥ N such that y(n)y(n + 1) ≤ 0 : otherwise, it is called nonoscillatory.
The following two results are useful for our discussion in the next sections.
Theorem 1.1 [2].If q(n) ≥ 0 for n ≥ 0 and (H) Suppose there are a function g(u) ∈ C(R, R + ) and a number > 0 such that

Corollary 1.3
If all the conditions of Theorem 1.2 are satisfied, then

Sublinear Oscillation
This section deals with the sufficient conditions for the oscillation of all solutions of Eq.( 1) when G is sublinear.The following conditions are needed for our use in the sequel.

Proof
Suppose for contrary that y(n) is a nonoscillatory solution of (1).Then there exists n 1 > 0 such that y(n) > 0 or < 0 for n ≥ n 1 .Let the former hold.Setting we have from (1) that is, Using (H 2 ), (H 3 ) and (H 5 ), the last inequality implies that is, Since lim n→∞ z(n) exists, then the above inequality implies that a contradiction to (H 2 ).If ∆z(n) > 0 for n ≥ n 2 , then z(n) is nondecreasing and hence there exists a constant α > 0 such that z(n) > α, n ≥ n * .Application of Eq.( 4) gives Thus, Suppose the later holds.Then setting x(n) = −y(n) > 0, for n ≥ n 1 and using (H 4 ), Eq.( 1) can be written as Following the above procedure to Eq.( 7), similar contradictions can be obtained.Hence the proof of the theorem is complete.

Remark
The prototype of G satisfying (H 3 ), (H 4 ) and (H 5 ) is Clearly, the above equation satisfies all the conditions of Theorem 2.1 and hence it is oscillatory.In particular, y(n) = (−1) 3n is such an oscillatory solution.

Proof
Proceeding as in Theorem 2.1, we get the inequality (5), where ∆z(n) < 0 and z(n Following the similar steps of Theorem 2.1, we have a contradiction to (H 1 ).If z(n) < 0, for n ≥ n 2 , then y(n) < y(n − m), that is, y(n) is bounded.Consequently, z(n) is bounded, a contradiction to the fact that ∆z(n) < 0 and z(n) < 0 for n ≥ n 2 .Hence ∆z(n) > 0 for n ≥ n 2 .If z(n) > 0, then using the similar argument as in Theorem 2.1, the contradiction is EJQTDE, 2008 No. 11, p. 6 obtained to (H 1 ).Suppose that z(n) < 0, for n ≥ n 2 .Then y(n − k) > (1/b)z(n + m − k).Thus Eq.( 1) becomes due to increasing z(n).Consequently, the last inequality can be made as The case y(n) < 0 for n ≥ n 1 is similar.This completes the proof of the theorem.

Superlinear Oscillation
This section deals with the oscillation of all solutions of Equation ( 1) such that G is superlinear.

Proof
Proceeding as in Theorem 2.1, we get the inequality (6).Using the fact that both z(n) and ∆z(n) are non-increasing, inequality (6) can be written as q(s) = ∞ hold.Then every solution of (1) oscillates.