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

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 equationwhere 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) whenorThus 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 thatandfor all .

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

Theorem 5. Let condition (2) hold. Assume that there is a positive decreasing real sequence tending to zero such that is positive for all . Ifthen 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 havewhere we have used Lemma 1. Since is positive and increasing and is positive and decreasing to zero, there is an integer such thatUsing (14) in (13), one obtainsand substituting this in (1) yieldsNow summing the last inequality from to , we obtainfor all . That isSince and eventually, there exists a positive constant such that for all . Using this and the positivity of in (18) and letting , we obtainwhich 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) thatSet . Then and the last inequality becomeswhere . Now, using Lemma 1.1 of [20], we see that the equationhas 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 andwhere , 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, eitherorfor 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 haveDividing the last inequality by and summing the resulting inequality from to , we obtainLetting , we obtainDefineand then , and using (16), we havewhere we have used as positive and decreasing. Now using in the above inequality, it follows thatNow from Lemma 3, we obtainSince and , we haveCombining the inequalities (35) and (37), we havewhere Completing the square in the above inequality, we haveorBy summing the last inequality from to , we obtainTaking 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 andfor 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 haveNow using (36) and (37) in (43), we obtainSince 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 obtainTaking 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 andfor 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 haveNow 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 haveSumming the last inequality from to , we obtainTaking 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 equationwhere 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 equationwhere is an even integer. Here . By taking , we see that for all . Now condition (21) becomessince . Also a simple calculation shows that and, using this, condition (42) becomesThus 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.