Abstract and Applied Analysis
Volume 2018 (2018), Article ID 2590158, 6 pages
https://doi.org/10.1155/2018/2590158

Research Article

Some Oscillation Results for Even Order Delay Difference Equations with a Sublinear Neutral Term

Govindasamy Ayyappan and Gunasekaran Nithyakala

Department of Mathematics, Periyar University College of Arts and Science, Pappireddipatti 636 905, Tamilnadu, India

Correspondence should be addressed to Govindasamy Ayyappan; ayyapmath@gmail.com

Received 5 September 2018; Accepted 18 October 2018; Published 8 November 2018

Academic Editor: Agacik Zafer

Copyright © 2018 Govindasamy Ayyappan and Gunasekaran Nithyakala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, some new results are obtained for the even order neutral delay difference equation , where is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results.

1. Introduction

The aim of this paper is to investigate the oscillatory behavior of even order nonlinear neutral difference equation

where is a positive integer, subject to the following conditions:

is an even integer, and and are ratio of odd positive integers with and ; is a positive increasing sequence of real number for all ; and are positive real sequences for all with ; and are positive integers.

Let . Under a solution of (1), we mean a real sequence defined for and satisfying (1) for all . As usual a solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; else it is nonoscillatory.

In the past few years, there is a great interest in studying the oscillatory and asymptotic behavior of solutions of higher order neutral type difference equations, since such type of equations naturally arises in the applications including problems in population dynamics or in cobweb models in economics and so on. The problem of finding sufficient conditions which ensure that all solutions of the neutral type difference equations are oscillatory has been investigated by many authors; see, for example, [1–12] and the references cited therein. In all the results the neutral term is linear and few results are available when the neutral term is nonlinear; see [13–21].

In [20], the authors considered (1) with and and established sufficient conditions for the oscillation of all solutions. In view of these facts, in this paper our purpose is to obtain sufficient conditions for the oscillation of solution of (1) when

or

Thus the results presented here extend and generalize some of the results in [13, 14, 16, 18, 19, 21], complement the results in [20], and correct some of the results in [8].

2. Some Preliminary Lemmas

In this section, we provide some lemmas which will be useful in proving our main results. We begin with the following lemma that can be found in [22, Theorem 41, page 39].

Lemma 1. If and , then .

Lemma 2 (Discrete Kneser’s Theorem). Let be a sequence of real number and with being of constant sign eventually and not identically zero eventually. Then there exists an integer , with odd for and even for and such that

and

for all .

Lemma 3. Let be defined for and with for all and not identically zero. Then there exists a large such that

where is defined in Lemma 2. Further, if is increasing, then

The proof of the last two lemmas can be found in [1].

Next we define the sequence by

Lemma 4. Assume condition (2) holds. Let be a positive solution of (1). Then there is an integer such that

for all .

Proof. The proof is similar to that of Lemma 3 of [8], and hence the details are omitted.

3. Oscillation Theorems

In this section, we present some sufficient conditions for the oscillation of all solutions of (1). To simplify our notation, for any positive real sequence which is decreasing to zero, we set

and

Theorem 5. Let condition (2) hold. Assume that there is a positive decreasing real sequence tending to zero such that is positive for all . If

then every solution of (1) is oscillatory.

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may assume that is a positive solution of (1). Then there exists an integer such that and for all . From Lemma 4, we have and for all .
From the definition of , we have

where we have used Lemma 1. Since is positive and increasing and is positive and decreasing to zero, there is an integer such that

Using (14) in (13), one obtains

and substituting this in (1) yields

Now summing the last inequality from to , we obtain

for all . That is

Since and eventually, there exists a positive constant such that for all . Using this and the positivity of in (18) and letting , we obtain

which is a contradiction to (12). This completes the proof.

Remark 6. In the above theorem, we did not impose any condition on and hence our result is more general than some of the existing results in the literature.

In the following, we present other oscillation criteria using Lemma 3.

Theorem 7. Let condition (2) hold. Assume that there is a positive decreasing real sequence tending to zero such that is positive for all . If

(i)

 for ,(ii)

 for ,(iii)there exists a such that

 for ,

then every solution of (1) is oscillatory.

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may assume that there is an integer such that and for all . Now proceeding as in the proof of the previous theorem, we obtain (16). That is,

Since , and using Lemma 3, we have from (23) that

Set . Then and the last inequality becomes

where . Now, using Lemma 1.1 of [20], we see that the equation

has an eventually positive solution.
(i) If (20) holds, then by Theorem 7.6.1 of [23], (26) with has no positive solution, which is a contradiction.
(ii) If (21) holds, then by Theorem 1 of [24], (26) with has no positive solution, which is a contradiction.
(iii) If (22) holds, then by Theorem 2 of [24], (26) with has no positive solution which is a contradiction. This completes the proof of the theorem.

Theorem 8. Assume that (3) and hold. Assume that there is a positive decreasing real sequence tending to zero such that is positive for all . If (20) holds and

where , then every solution of (1) either is oscillatory or tends to zero as .

Proof. Assume that (1) has a nonoscillatory solution which is eventually positive such that . From the definition of , we have for all . By virtue of (1) and Lemma 2 there are two possibilities, either

or

for all
Case (i). Suppose conditions (28) hold for all ; then the proof for this case is similar to that of Case (i) of Theorem 7 and hence the details are omitted.
Case (ii). Assume now that conditions (29) hold for all . Since is decreasing, then we have

Dividing the last inequality by and summing the resulting inequality from to , we obtain

Letting , we obtain

Define

and then , and using (16), we have

where we have used as positive and decreasing. Now using in the above inequality, it follows that

Now from Lemma 3, we obtain

Since and , we have

Combining the inequalities (35) and (37), we have

where Completing the square in the above inequality, we have

or

By summing the last inequality from to , we obtain

Taking lim sup as , in the above inequality we obtain a contradiction with (27). This completes the proof.

Theorem 9. Assume that (3) and hold. Further assume that there is a positive decreasing real sequence tending to zero such that is positive for all . If (21) holds and

for some constant , then every solution of (1) either is oscillatory or tends to zero as .

Proof. Assume that is an eventually positive solution of (1) such that . Proceeding as in the proof of Theorem 8, we see that satisfies two possible cases (28) and (29) for all .
Case (i). Suppose conditions (28) hold for all ; then the proof for this case is similar to that of Case (ii) of Theorem 7 and hence the details are omitted.
Case (ii). Assume now that conditions (29) hold for all ; proceeding as in Case (ii) of Theorem 8 we have

Now using (36) and (37) in (43), we obtain

Since is positive and decreasing and , there is a constant such that for all . Using this in (44) and then summing the resulting inequality from to , we obtain

Taking lim sup as , in the above inequality, we obtain a contradiction with (42). This completes the proof.

Theorem 10. Assume that (3) and hold. Further assume that there is a positive decreasing and sequence tending to zero such that is positive for all . If (22) holds and

for some constant , then every solution of (1) either is oscillatory or tends to zero as .

Proof. Let us assume that is an eventually positive solution of (1) such that . Proceeding as in the proof of Theorem 8, we see that satisfies two possible cases (28) and (29) for all .
Case (i). If conditions (28) hold for all , then the proof is similar to that of Case (iii) of Theorem 7 and hence the details are omitted.
Case (ii). Assume now that conditions (29) hold for all . Proceeding as in Case (ii) of Theorem 9, we have

Now from (32), one can see that is nondecreasing and hence there is a constant such that for all . Using this in (47) and since , we have

Summing the last inequality from to , we obtain

Taking lim sup as in the above inequality, we get a contradiction with (46). This completes the proof.

4. Examples

In this section, we present two examples to illustrate the importance of the main results.

Example 1. Consider the neutral difference equation

where is an even integer. Here , and . By taking , we see that for all . Now it is easy to see that the hypotheses are satisfied. Also condition (12) holds and therefore, by Theorem 5, every solution of (50) is oscillatory.

Example 2. Consider the neutral difference equation

where is an even integer. Here . By taking , we see that for all . Now condition (21) becomes

since . Also a simple calculation shows that and, using this, condition (42) becomes

Thus all conditions of Theorem 9 are satisfied and hence every solution of (51) either is oscillatory or tends to zero as .

5. Conclusion

The results obtained in this paper extend and complement some of the results reported in the literature. Further, Theorem 8, where , corrects the conclusion of Theorem 4 established in [8]. The results reported in the papers [3, 4, 6–12, 17, 20] cannot be applicable to (50) and (51) to yield this conclusion since these equations have sublinear neutral terms. It would be interesting to improve Theorems 8, 9, and 10 so that all solutions are oscillatory instead of either being oscillatory or tending to zero.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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