Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients

In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. In Theorem 5 and Theorem 8, We have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates or | | → ∞ as → ∞. Examples are given to illustrate the obtained results.


Introduction
In this paper the oscillation for bounded and all solutions of second order neutral delay difference equation with positive and negative coefficients: will be studied, where Δ is the forward difference operator, , are nonnegative infinite sequences of real numbers and , , are infinite sequences of real numbers. ∈ ( , ) is function ( ) > 0 . The purpose of this research is to obtain new sufficient conditions for the oscillation of all solutions of equation (1). The following assumptions are used: (H 4 ) ( ) ≤ 3 .

Main Result
The next results provide sufficient conditions for the oscillation of all bounded solutions of Eq. (1). For a simplicity set Let the sequence 7 be defined as The following theorem based on Theorem 7.6.1, [3] Then every bounded solution of equation (1) oscillates.
Case 2: 7 < 0, ∆7 > 0, ∆ 7 ≤ 0 . By taking the summation of both sides of (6) from to + D, 0 < D < 9 − E, we get From (3) we get Since B is arbitrary, it follows that Substituting (8) in (7) to obtain By theorem 1-ii and in virtue of (5), it follows that the last inequality cannot have eventually negative solution, which is a contradiction.
If k = ∞, From (3) we get which implies that lim →' = ∞ , otherwise if is bounded it follows from the last inequality 7 < + B , which is a contradiction.
If 0 < k < ∞, then there exists G > 0 such that 7 ≥ G > 0, for ≥ . If lim →' < ∞, then from (3) we get By taking the summation of both sides of (6) from to − 1, it follows that In virtue of (9) the last inequality implies that lim →' 7 = ∞. Leads to a contradiction.
Case 2: 7 < 0, ∆7 > 0, ∆ 7 ≤ 0. In this case 7 is bounded, we claim that is bounded, otherwise there exists a subsequence + $of +such that lim $→' $ = ∞, lim $→' J = ∞ and J = maxn : ≤ ≤ $ p , from (3) we get From (3) we get Since B is arbitrary, it follows that for sufficiently large : By theorem 1-ii and in virtue of (10), it follows that the last inequality cannot have eventually negative solution, which is a contradiction.
In the next theorem we will use the sequence ; already defined in (4).
Then every bounded solution of equation (1) oscillates.
If −∞ < k < 0 , then there exists G < 0 such that ; ≤ G < 0, for ≥ . If lim →' < ∞. From (4) we get By taking summation to both sides of (13) from to − 1, it follows that In virtue of (12) the last inequality implies that lim →' ; = −∞. Leads to a contradiction. By Theorem 1-i and in virtue of (12), it follows that the last inequality cannot have eventually positive solution, which is a contradiction.

Conclusion
1. In this paper we used two series 7 and ; , and obtained necessary and sufficient conditions for every solution of neutral difference equation of second order with positive and negative coefficients, to be oscillates or tends to infinity as → ∞.