Oscillatory Behavior of Higher Order Nonlinear Diﬀerence Equations

. The authors present some new oscillation criteria for higher order nonlinear diﬀerence equations with nonnegative real coeﬃcients of the form . Both of the cases n even and n odd are considered. They give examples to illustrate their results.


Introduction
In this paper, we study the oscillatory behavior of all solutions of the higher order nonlinear difference equation (ii) {a(t)} and {q(t)} are sequences of positive real numbers for t ≥ t 0 ; (iii) m > 1.
We let , v ≥ u ≥ t 0 , and assume that lim In recent years there has been much research activity concerning the oscillation and asymptotic behavior of solutions of various classes of difference equations and we mention [1,2,3,4,5,6,7] and the references cited therein as examples of some recent contributions in this area. There have been numerous studies on second order difference equations due to their use in the natural sciences and as well as for theoretical interests. Recent results on the oscillatory and asymptotic behavior of solutions of second order difference equations can be found, for example, in [8,9,10,11,12,13,17,18,19]. However, it appears that there are no known results regarding the oscillation of solutions of higher order difference equations of the form of Equation (1.1). In view of this, our aim in this paper is to present some new sufficient conditions that ensure that all solutions of Equation (1.1) are oscillatory.

Main results
We begin with some useful lemmas. Then there exists an integer p, 0 ≤ p ≤ n, with p + n even if ∆ n x(t) ≥ 0 and p + n odd for ∆ n x(t) ≤ 0, such that: Let {x(t)} be defined for t ≥ t 0 with x(t) > 0 and ∆ n x(t) ≤ 0 and not identically zero for t ≥ t 0 . Then there exists t 1 ≥ t 0 such that where p is defined as in Lemma 1. Furthermore, if {x(t)} is increasing, then The following lemma is an extension of the discrete analogue of known results in [15] and [16,Corollary 1]; it can also be found in [2,Lemma 6.2.2] and [14,Corollary 7.4.1]. The proof is immediate. Lemma 3. Let {q(t)} be a sequence of positive real numbers, m and p be positive numbers, and f : R → R be a continuous nondecreasing function with xf (x) > 0 for x = 0. If the first order delay inequality has an eventually positive solution, then so does the delay equation We are now ready for our first oscillation result; it is for the case where n is even. Theorem 1. Let n be even and assume that there is a number k such that m > (n − 2)k + 1. If the first order equations and Hence, a(t) ∆ n−1 x(t) α is nonincreasing and eventually of one sign.
That is, there exists t 2 ≥ t 1 such that First, we consider Case (I). By Lemma 1, we have ∆x(t) > 0 for t ≥ t 2 . From Lemma 2, we see that and so It follows from Lemma 3 that the corresponding Equation (2.1) also has a positive solution, which is a contradiction. Next, we consider Case (II). Since n is even, we distinguish the following two cases (recall that (2.3) holds): If (III) holds, then by Lemma 2, we see that The remainder of the proof in this case is similar to that of Case (I) and is omitted.

Using this inequality in (2.3), we have
Summing from t 2 to t − 1 yields and hence we obtain where W (t) = ∆ n−2 x(t) > 0. The rest of the proof is similar to that of Case (I) and is left to the reader. This completes the proof of the theorem.
To obtain some consequences of the above theorem, we let The following corollaries are immediate consequences of known results.
Corollary 1. Let n be even and k be a number such that m > (n − 2)k + 1. If the first order delay equation is oscillatory, then Equation (1.1) is oscillatory.
Corollary 2. Let n be even and k be a number such that m > (n − 2)k + 1. If then Equation (1.1) is oscillatory.
We now turn our attention to the case where n is odd.
Theorem 2. Let n be odd and k be a number such that m > (n − 1)k + 1. If the first order Equations (2.1)-(2.2), and are oscillatory, then Equation (1.1) is oscillatory.
If Case (I) holds, either we have (A) ∆x(t) > 0, or (B) ∆x(t) < 0 for t ≥ t 2 . By Lemma 2 and (A), we obtain (2.4) and (2.5) which leads to a contradiction. If (B) holds, we see that (2.7) holds, and as in the proof of Theorem 1, we have The rest of the proof is similar to that of the proof of Case I in Theorem 1 and hence is omitted.
Next, we consider Case (II). By Lemma 2 and as in the proof of Theorem 1, we obtain (2.6). The rest of the proof is similar to that of Case (III) in Theorem 1 and so we omit the details. This proves the theorem. Now let The following corollaries are analogous to those for the case where n is even.   it is easy to obtain the following results.
Theorem 3. Let n be even, condition (2.9) hold, and there exists a number k with m > (n − 2)k + 1. If the first order Equation (2.1) is oscillatory, then Equation (1.1) is oscillatory.
As an example to illustrate our results, consider the difference equation It is easy to see that if n is even and m > (n − 2)k + 1 (n is odd and m > (n − 1)k + 1) the conditions of Corollary 2.2 (respectively Corollary 2.4) are satisfied and hence we conclude that Equation (2.10) is oscillatory. We conclude this paper with a suggestion for future research, namely, to study the oscillatory behavior of solutions of Equation (1.1) in case β > α.