- Research
- Open Access
- Published:

# Asymptotic behavior of third order delay difference equations with a negative middle term

*Advances in Difference Equations*
**volume 2021**, Article number: 248 (2021)

## Abstract

In this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term

are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration.

## Introduction

In this paper, we are concerned with the asymptotic behavior of solutions of third order delay difference equations with a negative middle term of the form

where \(n_{0}\) is a nonnegative integer and *α* is a quotient of odd positive integers. Throughout this paper, we assume without further mention that: \(\{a_{n}\}\) is a positive real sequence, \(\{p_{n}\}\) is a nonnegative real sequence, and \(\{q_{n}\}\) is a positive real sequence for all \(n\geq n_{0}\), *l* is a positive integer, *h* is a continuous, nondecreasing real-valued function such that \(\eta h(\eta )>0\) for \(\eta \neq 0\), and \(h(\eta \xi )\geq h(\eta )h(\xi )\) for \(\eta \xi >0\).

By a solution of equation (1.1) we mean a nontrivial real sequence \(\{w_{n}\}\) that is defined for all \(n\geq n_{0}-l\) and satisfies equation (1.1) for all \(n\geq n_{0}\). A nontrivial solution \(\{w_{n}\}\) of equation (1.1) is said to be nonoscillatory if it is either eventually positive or eventually negative, and oscillatory otherwise. A difference equation is called nonoscillatory (oscillatory) if all its solutions are nonoscillatory (oscillatory). Following the terms used in [6], we define

Using these notations, one can write equation (1.1) as

We introduce the following class of nonoscillatory (without loss of generality we say positive) solutions which give the sign structure of possible nonoscillatory solutions to equation (1.1):

for all \(n\geq n_{1}\geq n_{0}\). In Lemma 2.3, we will prove that if

then the set *W* of all positive solutions of equation (1.1) has the following decomposition:

According to the well-known results in [1, 2], the oscillation criteria are often accomplished by introducing the concepts having property \((A)\) and/or \((B)\). Equation (1.1) is said to have property \((B)\) if \(W=W_{3}\).

In recent years the asymptotic behavior of nonoscillatory solutions and the oscillatory behavior of solutions to different classes of third order difference equations have been the interest of many researchers, see for example [3–8, 10, 12–19] and the references cited therein. Recently, in the papers [6, 16], the authors used the comparison method and the summation averaging technique to establish some sufficient conditions for oscillation of all solutions of the third order delay difference equation

where \(\{p_{n}\}\) and \(\{q_{n}\}\) are positive real sequences, and the auxiliary equation of second order

is nonoscillatory. In [11] the authors used the oscillation of a third order difference equation of the form

to obtain oscillation conditions for solutions of second order neutral delay difference equations.

Following this trend, in this paper, we study the asymptotic behavior of solutions of equation (1.1). Our approach depends on the application of a technique imposing one restrictive condition on the coefficients of the corresponding auxiliary equation. We show that any nonoscillatory solution \(\{w_{n}\}\) of equation (1.1) satisfies \(w_{n}\Delta w_{n}>0\). Further, we obtain new sufficient conditions for all solutions of equation (1.1) to have property \((B)\). Two examples are provided to illustrate the main results.

## Main results

For the sake of simplicity, we define the following:

for \(s\geq n\geq n_{1}\), where \(n_{1}\geq n_{0}\). Throughout we assume that \(R_{1}(n,n_{1})\rightarrow \infty \) as \(n\rightarrow \infty \). To make sense of the definitions \(P_{n}\) and \(\bar{P_{n}}\), we also assume that

In the sequel, and without loss of generality, we can deal only with the positive solutions of equation (1.1), since the proof for the opposite case is similar. From our technique, which will be described later, we will see that the properties of solutions to equation (1.1) are closely related to nonoscillatory solutions of an auxiliary second order difference equation

First, we prove the following lemmas which will be used in the proofs of the main results.

### Lemma 2.1

*Let* \(\{z_{n}\}\) *be a positive solution of* (2.1) *for all* \(n\geq n_{0}\). *Then* (1.1) *can be written in the form*

*for all* \(n\geq n_{0}\).

### Proof

It is easy to see that

where we have used (2.1). Using the above equality in equation (1.1) and rearranging, we obtain equation (2.2). This completes the proof. □

We recall that equation (2.1) (see Theorem 6.3.4 of [1]) always has a couple of nonoscillatory solutions \(\{z_{n}\}\) such that either

or

for all \(n\geq n_{0}\).

To find the structure of positive nonoscillatory solutions of equation (1.1), the following property of a nonoscillatory solution \(\{z_{n}\}\) satisfying (2.4) plays a crucial role.

### Lemma 2.2

*If* (1.2) *holds*, *then* (2.1) *has a positive solution* \(\{z_{n}\}\) *satisfying*

### Proof

Let \(\{z_{n}\}\) be a positive solution of equation (2.1) such that (2.4) holds for all \(n\geq n_{1}\geq n_{0}\). It is clear from the fact that \(\Delta z_{n}<0\), there is a constant \(M>0\) such that \(z_{n}\leq M \). Hence

On the other hand, since

then \(a_{n}\Delta z_{n}\) is increasing and there exists a constant \(c\leq 0\) such that \(\lim_{n\rightarrow \infty }a_{n}\Delta z_{n}=c\). We claim that \(c=0\), if not, then

a contradiction. Hence \(c=0\). Summing (2.1) from *n* to ∞, we have

or

Then from (2.6) we obtain

which yields

Now summing (2.7) from \(n_{1}\) to \(n-1\) and then combining with (1.2) implies that the second summation in (2.5) is divergent. This completes the proof. □

### Lemma 2.3

*Assume that condition* (1.2) *holds*. *If* \(\{w_{n}\}\) *is a positive solution of* (1.1) *for all* \(n\geq n_{0}\), *then there is an integer* \(n_{1}\) *such that either* \(w_{n}\in W_{1}\) *or* \(w_{n}\in W_{3}\) *for all* \(n\geq n_{1}\geq n_{0}\).

### Proof

Assume that \(\{w_{n}\}\) is a positive solution of equation (1.1) for all \(n\geq n_{0}\). By Lemma 2.1, we may write (1.1) in an equivalent form (2.2). From Lemma 2.2, there is a positive sequence \(\{z_{n}\}\) of (2.1) which satisfies (2.5), and so we see that

Then, by discrete Kneser’s theorem [1], we have

or

for all \(n\geq n_{1}\geq n_{0}\). Note that in both cases we have \(\Delta w_{n}>0\), and by virtue of (1.1) we see that \(L_{3}w_{n}>0\). The rest sign properties of \(L_{i}w_{n}\), \(i=1,2\), immediately follow from discrete Kneser’s theorem. The proof is now complete. □

Next, we state and prove some useful estimates which will play an important role in the proofs of our main results.

### Lemma 2.4

*Let* \(w_{n}\in W_{1}\) *be a positive solution of* (1.1) *for all* \(n\geq n_{1}\geq n_{0}\). *Then*

*and there is an integer* \(n_{2}>n_{1}\) *such that*

### Proof

Let \(w_{n}\in W_{1}\) be a positive solution of equation (1.1) for \(n\geq n_{1}\). From the monotonicity of \(L_{1}w_{n}\), we have

Therefore

and so \(w_{n}/(n-n_{1})\) is nonincreasing. Next, summing (1.1) from *n* to ∞, we obtain

Again summing, we obtain

This completes the proof. □

### Lemma 2.5

*Let* \(w_{n}\in W_{3}\) *be a positive solution of* (1.1) *for all* \(n\geq n_{1}\geq n_{0}\). *If*

*and there is an integer* \(n_{2}>n_{1}\) *such that*

### Proof

Let \(w_{n}\in W_{3}\) be a positive solution of (1.1) for all \(n\geq n_{1}\). Since \(L_{2}w_{n}\) is increasing, there is a constant \(M>0\) such that \(L_{2}w_{n}\geq M\) for all \(n\geq n_{1}\). Clearly,

We claim that condition (2.11) implies \(\lim_{n\rightarrow \infty }L_{2}w_{n}=\infty \). Using the above estimates into (1.1), we obtain

By summing (2.13) from \(n_{1}\) to ∞, we see that the claim holds. Therefore, for any \(n\geq n_{2}\geq n_{1}\), we have

which yields

and hence \(L_{1}w_{n}/R_{1}(n,n_{2})\) is nondecreasing for all \(n\geq n_{2}\). Again for any \(n\geq n_{3}\geq n_{2}\), we have

It follows from discrete L’Hospital rule [1] that

and so we have

Then

Thus \(w_{n}/B(n,n_{1})\) is nondecreasing for all \(n\geq n_{3}\). The proof is complete. □

We conclude this section with the following remark.

### Remark 2.6

It is easy to see that from Lemma 2.5, if (1.1) has property \((B)\), then any positive solution of (1.1) satisfies

which gives us information about the rate of convergence of possible positive solutions.

In the following, we present some sufficient conditions which ensure that equation (1.1) has property \((B)\).

### Theorem 2.7

*Let condition* (1.2) *hold for all* \(n\geq n_{1}\). *If the first order delay difference equation*

*is oscillatory*, *then* (1.1) *has property* (*B*).

### Proof

Let \(\{w_{n}\}\) be a positive solution of (1.1) for \(n\geq n_{0}\). From Lemma 2.3 there exists an integer \(n_{1}\geq n_{0}\) such that either \(w_{n}\in W_{1}\) or \(w_{n}\in W_{3}\) for all \(n\geq n_{1}\). If \(w_{n}\in W_{1}\), then by equation (1.1) and (2.9), we have

Summing (2.15) from *n* to ∞, we find

Using (2.10) in the above inequality, we obtain

Letting \(x_{n}=L_{1}w_{n}\), we see that the difference inequality

has a positive solution. By Lemma 2.7 of [19], we see that equation (2.14) also has a positive solution, which is a contradiction. Therefore \(w_{n}\in W_{3}\), which implies that equation (1.1) has property \((B)\). This completes the proof. □

Applying some known criteria for oscillation of first order delay difference equation (2.14), one can easily obtain criteria for equation (1.1) to have property \((B)\). The following one is given in [9].

### Corollary 2.8

*Assume that* \(h(u)=u^{\alpha }\) *and condition* (1.2) *hold*. *If*

*then* (1.1) *has property* \((B)\).

Finally, we present another result for equation (1.1) to have property \((B)\) which is applicable even to the ordinary equation.

### Theorem 2.9

*Let condition* (1.2) *hold for all* \(n\geq n_{1}\). *Assume that*

*and the function* *h* *satisfies*

*If*

*then* (1.1) *has property* \((B)\).

### Proof

Let \(\{w_{n}\}\) be a positive solution of (1.1) for all \(n\geq n_{0}\). Then from Lemma 2.3 there exists an integer \(n_{1}\geq n_{0}\) such that either \(w_{n}\in W_{1}\) or \(w_{n}\in W_{3}\) for all \(n\geq n_{1}\). If \(w_{n}\in W_{1}\), then as in the proof of Theorem 2.6 we obtain (2.16), and by summing it from *n* to ∞, we find that

Now, summing (2.21) from \(n_{1}\) to \(n-1\), one can easily see that

Using the monotonicity property (2.8) in (2.22), we have

Applying hypothesis \((H_{3})\) assumed on the function *h* and then dividing both sides of the last inequality by \(h^{1/\alpha }(w_{n})\), we see that

It follows from (2.18) that \(\lim_{n\rightarrow \infty }w_{n}=\infty \). Taking the limit supremum on both sides of (2.23), we are led to a contradiction with (2.20). Thus \(w_{n}\in W_{3}\), which means that equation (1.1) has property \((B)\). This completes the proof. □

We conclude this section with the following remark.

### Remark 2.10

If all conditions of Theorem 2.7 (Theorem 2.9) are satisfied, then one can conclude that all bounded solutions of equation (1.1) are oscillatory.

## Applications

In the following, we present two examples to illustrate the main results.

### Example 3.1

Consider the third order delay difference equation

Here

A simple calculation shows that

and

Hence all conditions of Corollary 2.8 are satisfied, and therefore (3.1) has property \((B)\).

### Example 3.2

Consider the third order delay difference equation

Here

A simple calculation shows that

Since

and

we see that all conditions of Theorem 2.9 are satisfied, and hence (3.2) has property \((B)\).

## Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

## References

- 1.
Agarwal, R.P.: Difference Equations and Inequalities, Theory, Methods and Applications, 2nd edn. Dekker, New York (2000)

- 2.
Agarwal, R.P., Bohner, M., Grace, S.R., O’Regan, D.: Discrete Oscillation Theory. Hindawi Publ. Corp., New York (2005)

- 3.
Agarwal, R.P., Grace, S.R.: Oscillation of certain third order difference equations. Comput. Math. Appl.

**42**, 379–384 (2001) - 4.
Agarwal, R.P., Grace, S.R., O’Regan, D.: On the oscillation of certain third order difference equations. Adv. Differ. Equ.

**2005**(3), 345–367 (2005) - 5.
Aktas, M.F., Tiryaki, A., Zafer, A.: Oscillation of third order nonlinear delay difference equations. Turk. J. Math.

**36**, 422–436 (2012) - 6.
Bohner, M., Dharuman, C., Srinivasan, R., Thandapani, E.: Oscillation criteria for third-order nonlinear functional difference equations with damping. Appl. Math. Inf. Sci.

**11**(3), 669–676 (2017) - 7.
Grace, S.R., Agarwal, R.P., Graef, J.R.: Oscillation criteria for certain third order nonlinear difference equations. Appl. Anal. Discrete Math.

**3**, 27–38 (2009) - 8.
Graef, J.R., Thandapani, E.: Oscillatory and asymptotic behavior of solutions of third order delay difference equations. Funkc. Ekvacioj

**42**, 355–369 (1999) - 9.
Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon, Oxford (1991)

- 10.
Parhi, N., Panda, A.: Oscillatory and nonoscillatory behavior of solutions of difference equations of the third order. Math. Bohem.

**133**, 99–112 (2008) - 11.
Pinelas, S., Saker, S.H., Alrohet, M.A.: Oscillation criteria of second order neutral difference equations via third order difference equations. Int. J. Difference Equ.

**12**, 131–143 (2017) - 12.
Saker, S.H., Alzabut, J.O.: Oscillatory behavior of third order nonlinear difference equations with delayed argument. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.

**17**, 707–723 (2010) - 13.
Saker, S.H., Alzabut, J.O., Mukheimer, A.A.: On the oscillatory behavior for a certain class of third order nonlinear delay difference equations. Electron. J. Qual. Theory Differ. Equ.

**2010**, 67 (2010) - 14.
Smith, B.: Oscillation and nonoscillation theorems for third order quasi adjoint difference equations. Port. Math.

**45**, 229–243 (1988) - 15.
Smith, B., Taylor, W.E. Jr.: Nonlinear third order difference equation: oscillatory and asymptotic behavior. Tamkang J. Math.

**19**, 91–95 (1988) - 16.
Srinivasan, R., Dharuman, C., Greaf, J.R., Thandapani, E.: Oscillatory and property (B) for the third order delay difference equations with damping term. Preprint

- 17.
Thandapani, E., Mahalingam, K.: Oscillatory properties of third order neutral delay difference equations. Demonstr. Math.

**35**, 325–337 (2002) - 18.
Thandapani, E., Pandian, S., Balasubtamanian, R.K.: Oscillatory behavior of solutions of third order quasilinear delay difference equations. Stud. Univ. Žilina Math. Ser.

**19**, 65–78 (2005) - 19.
Thandapani, E., Selvarangam, S.: Oscillation theorems of second order quasilinear neutral difference equations. J. Math. Comput. Sci.

**2**, 866–879 (2012)

## Acknowledgements

J. Alzabut would like to thank Prince Sultan University for supporting this work.

## Funding

Not applicable.

## Author information

### Affiliations

### Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Saker, S.H., Selvarangam, S., Geetha, S. *et al.* Asymptotic behavior of third order delay difference equations with a negative middle term.
*Adv Differ Equ* **2021, **248 (2021). https://doi.org/10.1186/s13662-021-03407-8

Received:

Accepted:

Published:

### MSC

- 39A10

### Keywords

- Third order delay difference equations
- Comparison method
- Oscillation
- Nonoscillation