Some New Oscillation Theorems for Second Order Difference Equations with Mixed Neutral Terms

Some new sufficient conditions are established for the oscillation of all solutions of the second order neutral difference equation of the form ∆ (rn∆ (xn + axn−τ + bxn+σ)) + pnx α n−k + qnx β n+m = 0, n ≥ n0, where ∑∞ n=n0 1/rn <∞. The results obtained here are new and further improve and complement some known results in the literature. Examples are provided to illustrate the main results.


Introduction
This paper deals with the oscillatory behavior of solutions of second order neutral difference equation of the form where n0 is a positive integer, ∆ is the forward difference operator defined by ∆xn = xn+1 − xn, and α and β are ratios of odd positive integers.Further, we assume the following conditions without further mention: (C1) {rn} is a sequence of positive real numbers for all n ≥ n0 with ∞ n=n 0 1/rn < ∞; (C2) a and b are nonnegative real constants; (C3) {pn} and {qn} are nonnegative real sequences, and not identically zero for many values of n; (C4) τ, σ, k and m are nonnegative integers.
By a solution of equation (1.1), we mean a real sequence {xn} defined for all n ≥ n0 − θ1 where θ1 = max {τ, k} and satisfies equation (1.1) for all n ≥ n0.Let θ2 = max {σ, m} .Clearly, if the initial conditions xn = φn for all n ∈ [n0 − θ1, n0 + θ2 − 1] are given, then equation (1.1) has a unique solution satisfying the initial conditions.A solution of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, and nonoscillatory otherwise.
However when b = 0 and either pn = 0 or qn = 0, there are some results available in the literature dealing with the oscillatory behavior of equation (1.1) with ∞ n=n 0 1/rn < ∞, see for example [1], [2], [3] and [6], and the references cited therein.In particular in [17], the authors considered equation (1.1) under the condition (C1), and obtained criteria which imply that all solutions of equation (1.1) are either oscillatory or tend to zero as n → ∞.
Equation (1.1) may be considered as a discrete analogue of the continuous equation where z(t) = x(t) + ax(t − τ ) + bx(t + σ).The oscillatory and asymptotic behavior of solutions of equation (1.2) is studied in [19], [20] and [6] under various conditions on the known functions.However we have found some results on the oscillatory behavior of solutions of equation (1.2) for the particular case b = 0, q(t) = 0 and ∞ t 0 1/r(t)dt < ∞, see for example [21], and the references contained therein.
Therefore the natural question arises whether all solutions of equation (1.1) are oscillatory if the hypothesis (C1) is satisfied.Our purpose in this paper is to give answer to this question in a affirmative way by obtaining some new sufficient conditions which ensure that all solutions of equation (1.1) are oscillatory.In Section 2, we present some preliminary lemmas and in Section 3, we obtain some new oscillation criteria for equation (1.1).Section 4 contains some examples to illustrate the main results.Thus, the results obtained in this paper are new and further improve and complement to those established in [2], [3], [6] and [17].

Some Preliminary Lemmas
In this section we present some lemmas which will be needed to prove the main results.For the sake of convenience, we define the following notations: We begin with the following lemma.
Proof.The proof can be found in [5].
Proof.Since rn∆zn is nonincreasing, we have Dividing the last inequality by rs and summing it from n to l, we obtain Letting l → ∞, we have The proof is now completed.

Oscillation Theorems
In this section, we obtain some new sufficient conditions for the oscillation of all solutions of equation (1.1).We begin with the following theorem.
has no eventually positive nonincreasing solutions, and the difference inequality has no eventually positive nondecreasing solutions, then every solution of equation (1.1) is oscillatory.
Proof.Let {xn} be a nonoscillatory solution of equation (1.1).Without loss of generality, we may assume that x n−θ 1 > 0 for all n ≥ N1 ≥ n0, where N1 is chosen so that both the cases of Lemma 2.1 hold for all n ≥ N1.From the equation (1.1), we have Since β ≤ α, a ≤ 1 and b ≤ 1, we have Using the Lemma 2.2, we obtain Next we consider the two cases of Lemma 2.1.Case (I).Assume that case (I) of Lemma 2.1 holds for all n ≥ N1.From (3.4) and Lemma 2.3, we obtain where yn = rn∆zn and yn > 0 is nonincreasing for all n ≥ N1.Now we define, wn = yn + a β yn−τ + b β 2 β−1 yn+σ, then wn > 0, n ≥ N1, and Using the last inequality in (3.5), we obtain that inequality (3.1) has an eventually positive nonincreasing solution, a contradiction.Case (II).Assume that case (II) of Lemma 2.1 holds for all n ≥ N1.From (3.4) and Lemma 2.4, we have where yn = −rn∆zn and yn > 0 is nondecreasing for all n ≥ N1.Now we define Using the last inequality in (3.6), we obtain that the inequality (3.2) has an eventually positive nondecreasing solution, a contradiction.This completes the proof. and where M = (1 + a + b) are satisfied, then every solution of equation (1.1) is oscillatory.
In the following we derive some new oscillation criteria for the equation (1.1) where qn = 0 for all n ≥ n0.In this case the equation (1.1) takes the form > 0 for all n ≥ n0, and there exists a positive nondecreasing real sequence {ρn} such that for any constant M1 > 0 If there exists a positive real sequence {δn} such that and for any constant M2 > 0 then every solution of equation (3.9) is oscillatory.
Proof.Let {xn} be a nonoscillatory solution of equation (1.1).Without loss of generality, we may assume that x n−θ 1 > 0 for all n ≥ N ≥ n0, where N is chosen so that both the cases of Lemma 2.1 hold for all n ≥ N. From the equation (3.9), we have where Then wn > 0 and from (3.15), we have ) α where we have used (2.1).From the monotonicity of {rn∆zn} and 0 < α ≤ 1, we have from the last inequality where M1 = r N −k ∆z N −k .Summing the inequality (3.16) from N to n − 1, we obtain Letting n → ∞ in (3.17), we obtain a contradiction to (3.10).Case (II).Assume that case (II) of Lemma 2.1 holds for all n ≥ N. Define Then vn < 0 for n ≥ N. From Lemma 2.4, we have By virtue of (3.18) and (3.19), we have where . From (3.18) and (3.22), we have where we have used {zn} is positive and decreasing.From (3.23), we have for some constant M2 = z N +1−k .Multiplying (3.24) by An+1 and then summing it from N to n − 1, we have Using the summation by parts formula in the first term of (3.25), and then rearranging, we obtain Using completing the square in the last term of the above inequality, we obtain .
Letting n → ∞ in the last inequality and using (3.20), we obtain a contradiction to (3.12).The proof is now completed.
> 0 for all n ≥ n0, and there exists a positive increasing sequence {ρn} such that for any constant M > 0 such that If there exists a positive real sequence {δn} such that (3.11) holds and for any constant D > 0 then every solution of equation (3.9) is oscillatory.
Proof.Proceeding as in the proof of Theorem 2.2, we see that Lemma Since {zn} is increasing and {rn∆zn} is nonincrasing, we have from (3.31) where M = z N −k .Summing the last inequality from N to n − 1, and using completing the square we have Letting n → ∞ in the last inequality, we obtain a contradiction to (
13))Case (I).Assume that case (I) of Lemma 2.1 holds for all n ≥ N. From Lemma 2.3, we have {zn/Rn} is nonincreasing for all n ≥ N. By the definition of zn, we have