SUFFICIENT CONDITIONS FOR OSCILLATION OF SECOND ORDER NEUTRAL ADVANCED DIFFERENCE EQUATIONS

The aim of this paper is to study the oscillation of the second order neutral advanced difference equations

(H 1 ) σ(n) > n + 1 and {σ(n)} n≥0 is nondecreasing; Let {x(n)} be a real sequence.We will also define a companion or associated sequence {z(n)} of it by where {p(n)} and {τ (n)} have been defined above.
Let n 0 be a fixed nonnegative integer.By a solution of (1), we mean a nontrivial real sequence {x(n)} which is defined for n ≥ min {n 0 , τ (n 0 )} and satisfies the equation (1) for n ≥ n 0 .A solution {x(n)} of ( 1) is said to be oscillatory if for every positive integer N > 0, there exists an n ≥ N such that x(n)x(n + 1) ≤ 0, otherwise {x(n)} is said to be nonoscillatory.Equation ( 1) is said to be oscillatory if all its solutions are oscillatory.
Recently, there has been a lot of interest in studying the oscillatory behavior of difference equations.See, for example [1][2][3][4][5][6]10] and references cited there in.They have mainly concerned with the oscillation and nonoscillatory of solutions of (1).The advanced equations have wide use.For example, the population of the future limit to population growth can be described through (1) with p(n) ≡ 0. Sternal et al. [9] showed that −1 ≤ p 1 ≤ p n ≤ 0 together with ∞ n=0 1 rn = ∞ and ∞ n=0 q n = ∞ guarantee the oscillation of unbounded solutions of the neutral equation For the same equation R.N. Rath et al. [8] established oscillation criteria .This results has been improved and generalized by other authors.We mention Tripathy [11] who studied oscillation of Zhang et al. [12] established oscillation criteria for the equation (1) with In this paper, we discuss the oscillation of solutions of (1).We obtain some better sufficient conditions for (1) to be oscillatory.They are delicate criteria.Our technique permits to relax restrictions usually imposed on the coefficients of (1).So that our results are of high generality and can be easily extended also to the nonlinear neutral difference equations.Obtained results are easily applicable and are illustrated on a suitable example.
In the sequel for convenience when we write a functional inequality without specifying its domain of validity, we assume that it holds for all sufficiently large positive integer n.

Some Lemmas
First we state a lemma which is due to [5].
A slight modification in the proof of the Theorem 3 in [7] leads to the following lemma about the advanced difference inequality.
For our further references, let us denote and where n ≥ n 1 , n 1 is a large enough.

Main Results
Theorem 5. Let τ (n) ≥ n and 0 ≤ p(n) ≤ p 0 < ∞.Assume that at least one of the first order advanced difference inequalities has no positive solution.Then (1) is oscillatory.
Proof.Assume that {x(n)} is an eventually positive solution of (1).Then from (2), we have where we used to the hypothesis (H 4 ).On the other hand, it follows from (1) that ∆(r(n)∆z(n)) + q(n)x(σ(n)) = 0; (10) and more over, taking (H 3 ) into account, we have Combining ( 10) and ( 11), we are lead to which in view of ( 9) and (4) provides Summing the previous inequality from n to ∞, we get On the other hand, since {r(n)∆z(n)} is decreasing and τ (n) ≥ n, it follows from (13) that Using that {z(σ(n))} is increasing, an summation from n 1 to n − 1, yields, That is, Let us denote the right hand side of (15) by w(n).Then w(n) > 0 and using z(n) ≥ w(n), one can see that Thus {w(n)} is a positive solution of ( 7).This contradicts our assumptions and thus the absence of the eventually positive solutions of ( 7) implies the oscillatory of (1).Now, we shall show that the absence of the eventually positive solutions of (8) also yields the oscillation of ( 1).An summation of (14) from n 1 to n − 1, provides That is, Let us denote the right hand side of ( 16) by w(n).Then w(n) > 0 and using thatz(n) ≥ w(n); one can see that {w(n)} is an eventually positive solution of (8).This is a contradiction and the proof is complete now.Theorem 6.Let τ (n) ≥ n and 0 ≤ p(n) ≤ p 0 < ∞.Assume that at least one of the following conditions holds.Then ( 1) is oscillatory.
Proof.Lemma 4 guarantees that ( 7) and ( 8) have no eventually positive solutions provided that ( 17) and ( 18) hold respectively.The assertion now follows from Theorem 5.
For our incoming references, let us denote Theorem 7. Assume that τ (n) = n − τ and σ(n) = n + σ where τ and σ are positive integers and 0 ≤ p(n) ≤ p 0 < ∞.Assume further that at least one of the first order advanced difference inequalities has no eventually positive solution.Then (1) is oscillatory.
On the other hand, since {r(n)∆z(n)} is decreasing, then it follows from (22) that Multiplying by 1 r(n−τ ) and then summing from n 1 to n − 1, we get Let us denote the right hand side of (24) by w(n).Noting that z(n − τ ) ≥ w(n).One can see that {w(n)} is an eventually positive solution of (20).This contradicts our assumptions and thus the absence of the positive solutions of (20) implies the oscillation of (1).Now we shall show that the absence of the positive solutions of (21) implies the oscillation of ( 1).An summation of (23) from n 1 to n − 1, gives That is, Let us denote the right hand side of (25) by w(n).Then w(n) > 0 and using that z(n − τ ) ≥ w(n), one can see that {w(n)} is an eventually positive solution of (21).This is a contradiction and the proof is complete now.
Proof.Lemma 4 guarantees that (20) and ( 21) have no eventually positive solutions provided that (26) and ( 27) hold respectively.The assertion now follows from Theorem 7.
Theorem 9. Assume that τ (n) = n − τ , σ(n) = n + σ, where τ and σ are positive integers and −1 < p ≤ p(n) ≤ 0. Assume further that at least one of the first order advanced difference inequalities (20) and ( 21) has no eventually positive solution.Then every solution of (1) is either oscillatory or tends to zero as n → ∞.
Proofs of Theorem 9 and 10.Without loss of generality, we may assume that {x(n)} is an eventually positive solution of (1) such that lim sup n→∞ x(n) > 0.Then, by Lemma 3, we have z(n) > 0, r(n)∆z(n) > 0 and ∆(r(n)∆z(n)) > 0, eventually.Theorem 9 and Theorem 10 can be proved by applying the procedures that are used in the Theorem 7 and 8 respectively.
Remark 12.All our conclusions can be very easily extend to nonlinear neutral difference equations of the form ∆(r(n)∆ [x(n) + p(n)x(τ (n))] + q(n)f (x(σ(n)) = 0. (29) Adding the additional condition the reader can verify that our results here hold also for (29), provided that we replace in the assumption of our achievements the sequences {q(n)} by {λq(n)}.

Lemma 4 .
Consider the advanced difference inequality