A Comparison Theorem for Oscillation of the Even-Order Nonlinear Neutral Difference Equation

Abstract

A comparison theorem on oscillation behavior is firstly established for a class of even-order nonlinear neutral delay difference equations. By using the obtained comparison theorem, two oscillation criteria are derived for the class of even-order nonlinear neutral delay difference equations. Two examples are given to show the effectiveness of the obtained results.

1. Introduction

Recently there have been a lot of research papers in connection with the oscillation of solutions of difference equations with or without neutral terms. The literature on the oscillation of neutral delay difference equations is growing very fast, and it can be widely applied to the reality. In fact, neutral delay difference equations arise in modelling of the networks containing lossless transmission lines (as in high speed computers where the lossless transmission lines are used to interconnect switching circuits). For recent contributions regarding the theoretical part and providing systematic treatment of oscillation of solutions of neutral type difference equations, the readers can refer to the recent monographs by Agarwal [1], Györi and Ladas [2].

The oscillation behavior of the even-order nonlinear neutral differential equation has been established by Zhang et al. [3]. In this paper, the discrete analogue of the above equation is considered. We consider the even-order nonlinear neutral difference equation, where is an even and ; let denote the set of all natural numbers; ; is a nonnegative integer; denotes the forward difference operator defined by , .

Throughout this paper, the following conditions are assumed to hold: is a sequence of nonnegative real number, , and is a sequence of nonnegative real number with being not eventually identically equal to zero; is a continuous odd function, and for all .

Before deriving the main results, the following definitions are given.

Definition 1. By a solution of (2), one means a real sequence defined for which satisfies (2) for .
In this paper, we restrict our attention to nontrivial solutions of (2).

Definition 2. A nontrivial solution of (2) is said to be oscillatory if the terms of the sequence are neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory.

Definition 3. An equation is said to be oscillatory if all its solutions are oscillatory.

In 2004, Stavroulakis [4] studied the oscillatory behavior of all solutions of first-order delay difference equation, and established one new oscillation criterion. Thandapani et al. [5] studied the oscillatory behavior of all solutions of second-order neutral delay difference equation, and established a number of new oscillation criteria. In 2000, Zhou et al. [6] studied the oscillatory behavior of all solutions of even-order neutral delay difference equation, and established three new oscillation criteria under certain conditions. The studies on oscillatory behavior of all solutions of even-order delay difference equations, we recommend referring to [7–10]. On the basis of the above work, we studied the oscillatory behavior of all solutions of (2). Firstly, a comparison theorem on oscillation behavior is established for (2). The comparison theorem changes the discriminant criteria of the oscillation of (2) into the oscillation’s discriminant criteria in the first-order nonneutral delay difference equations. Then, by using the above comparison theorem, we obtain some oscillation criteria for (2) and improve the well-known results of Ladas et al. [11], Erbe and Zhang [12], and Stavroulakis [4]. In particular, the results are new when , .

The paper is organized as follows. In Section 2, a comparison theorem on oscillation behavior is firstly established for a class of even-order nonlinear neutral delay difference equations. Then the comparison theorem changes the discriminant of the oscillation in the even-order nonlinear neutral delay difference equation into the oscillation’s discriminant in the first-order nonneutral delay difference equations. In Section 3, some oscillation criteria are obtained for the class of even-order nonlinear neutral delay difference equation by using the above comparison theorem. In Section 4, two examples are given.

2. Comparison Theorem

To obtain the comparison theorem in this section, we need the following lemmas which can be founded in [1]; see also Chen [7] and Thandapani and Arul [8].

Lemma 4. Let be a sequence of real numbers for . Let and be of constant sign, where is not identically zero for . If then(i)there is a natural number such that the sequences are of constant sign for ;(ii)there exists a number with such that

Lemma 5. Observe that under the hypotheses of Lemma 4, if is increasing for , then there exists a natural number such that, for all , where .

Theorem 6. Assume that conditions and hold. Let for all . If there exists a constant , such that the first-order difference equation is oscillatory, then (2) is oscillatory.

Proof. Suppose that (2) has a nonoscillatory solution . Without the loss of generality, we assume that is an eventually positive solution of (2); then there is a natural number such that , , , and for all . Let Then, from () and (), there exists a natural number such that By Lemma 4, there exist an integer and an integer , where is an odd integer. For all , we can get Thus from (12), and for . By Lemma 5, there exists an integer . For all , we derive From (10), Consequently, we have Noting that , for all , we obtain By , , and , we obtain Therefore, we have Now, by using (13), we have that for , Thus, we get where . Let ; then for large enough , we get where . Therefore, inequality (21) has an eventually positive solution. By Lemma  5 in [9], (9) has an eventually positive solution which contradicts that (9) is oscillatory. This completes the proof.

3. Applications of the Comparison Theorem

The following lemma is well known (see, e.g., [2, 11, 12] and the references therein).

Lemma 7. Let be a sequence of eventually nonnegative real number and ; if either or then the first-order difference equation is oscillatory.

Thus, from Theorem 6 and Lemma 7, we can obtain the following results.

Theorem 8. Assume that conditions and hold. Let for all . For , if either or then (2) is oscillatory.

Proof. From (25) and (26), we can obtain or where . By Lemma 7, we know (9) is oscillatory. Then similar to the proof of Theorem 6, the results follow immediately. This completes the proof.

According to Theorem 8, we obtain Corollary 9.

Corollary 9. Assume that conditions and hold. Let for all . For , when , , if either or then the second-order difference equation is oscillatory.

The following lemma is given in [4, Theorem 2.6].

Lemma 10. Let be a sequence of nonnegative real numbers and a positive integer. Assume that if either or then (24) is oscillatory.

Thus, from Theorem 6 and Lemma 10, we can obtain the following results.

Theorem 11. Assume that conditions and hold. Let for all and let be a positive integer. Assume that if either or then (2) is oscillatory.

Proof. From (36) and (37), we can obtain or where . By Lemma 10, we know (9) is oscillatory. Then similar to the proof of Theorem 6, the results follow immediately. This completes the proof.

According to Theorem 11, we can obtain the following corollary.

Corollary 12. Assume that conditions and hold; let for all and let be a positive integer. For , , assume that if either or then (31) is oscillatory.

4. Examples

Example 1. Considering the equation where , is an even, and is a positive integer, then we have where is a positive sequence. Then Thus Therefore, by Theorem 8, (43) is oscillatory.

Example 2. Considering the equation where , is an even, and is a positive integer, then we have where is a positive sequence. Denote ; then Thus Therefore, by Theorem 11, (47) is oscillatory.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author sincerely thanks the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This work was supported by a Grant from the Natural Science Foundation of Shandong Province of China (no. ZR2013AM003) and the Development Program in Science and Technology of Shandong Province of China (no. 2010GWZ20401).