## Advances in Difference Equations

Impact Factor 0.335

Open Access

# Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

Advances in Difference Equations20102010:693867

DOI: 10.1155/2010/693867

Received: 5 January 2010

Accepted: 31 March 2010

Published: 7 April 2010

## Abstract

A linear second-order discrete-delayed equation with a positive coefficient is considered for . This equation is known to have a positive solution if fulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality for , all solutions of the equation considered are oscillating for .

## 1. Introduction

The existence of a positive solution of difference equations is often encountered when analysing mathematical models describing various processes. This is a motivation for an intensive study of the conditions for the existence of positive solutions of discrete or continuous equations. Such analysis is related to an investigation of the case of all solutions being oscillating (for relevant investigation in both directions, we refer, e.g., to [115] and to the references therein). In this paper, sharp conditions are derived for all the solutions being oscillating for a class of linear second-order delayed-discrete equations.

We consider the delayed second-order linear discrete equation
(1.1)

where , is fixed, , and . A solution of (1.1) is positive (negative) on if ( ) for every . A solution of (1.1) is oscillating on if it is not positive or negative on for arbitrary .

Definition 1.1.

Let us define the expression , , by , where and , , and instead of , , we will only write and .

In [2] a delayed linear difference equation of higher order is considered and the following result related to (1.1) on the existence of a positive solution is proved.

Theorem 1.2.

Let be sufficiently large and . If the function satisfies
(1.2)

for every , then there exist a positive integer and a solution , of (1.1) such that holds for every .

Our goal is to answer the open question whether all solutions of (1.1) are oscillating if inequality (1.2) is replaced by the opposite inequality
(1.3)

assuming and is sufficiently large. Below we prove that if (1.3) holds and , then all solutions of (1.1) are oscillatory. The proof of our main result will use a consequence of one of Domshlak's results [8, Corollary , page 69].

Lemma 1.3.

Let and be fixed natural numbers such that . Let be a given sequence of positive numbers and a positive number such that there exists a number satisfying
(1.4)
Then, if and for
(1.5)
holds, then any solution of the equation
(1.6)

has at least one change of sign on .

Moreover, we will use an auxiliary result giving the asymptotic decomposition of the iterative logarithm [7]. The symbols " " and " " used below stand for the Landau order symbols.

Lemma 1.4.

For fixed and fixed integer , the asymptotic representation
(1.7)

holds for .

## 2. Main Result

In this part, we give sufficient conditions for all solutions of (1.1) to be oscillatory as .

Theorem 2.1.

Let be sufficiently large, , and . Assuming that the function satisfies inequality (1.3) for every , all solutions of (1.1) are oscillating as .

Proof.

We set
(2.1)
and consider the asymptotic decomposition of when is sufficiently large. Applying Lemma 1.4 (for , , and ), we get
(2.2)
Finally, we obtain
(2.3)
Similarly, applying Lemma 1.4 (for , , and ), we get
(2.4)
Using the previous decompositions, we have
(2.5)
Recalling the asymptotical decomposition of when : , we get (since )
(2.6)
as . Due to (2.3) and (2.4) we have and as . Then it is easy to see that, for the right-hand side of the inequality (1.5), we have
(2.7)
where
(2.8)
Moreover, for , we will get an asymptotical decomposition as . We represent in the form
(2.9)
As the asymptotical decompositions for
(2.10)
have been derived above (see (2.3)–(2.5)), after some computation, we obtain
(2.11)
Thus we have
(2.12)
Finalizing our decompositions, we see that
(2.13)
It is easy to see that inequality (1.5) becomes
(2.14)
and will be valid if (see (1.3))
(2.15)
or
(2.16)

for . If where is sufficiently large, then (2.16) holds for sufficiently small with fixed because . Consequently, (2.14) is satisfied and the assumption (1.5) of Lemma 1.3 holds for . Let in Lemma 1.3 be fixed and let be so large that inequalities (1.4) hold. This is always possible since the series is divergent. Then Lemma 1.3 holds and any solution of (1.1) has at least one change of sign on . Obviously, inequalities (1.4) can be satisfied for another couple of , say with and sufficiently large, and by Lemma 1.3 any solution of (1.1) has at least one change of sign on . Continuing this process, we get a sequence of intervals with such that any solution of (1.1) has at least one change of sign on . This fact concludes the proof.

## Declarations

### Acknowledgments

The first author was supported by Grants 201/07/0145 and 201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government MSM 0021630529. The second author was supported by Grants 201/07/0145 and 201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government MSM 00216 30519. The third author was supported by the Council of Czech Government MSM 00216 30503 and MSM 00216 30529.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology
(2)
Brno University of Technology

## References

1. Agarwal RP, Zafer A: Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities. Advances in Difference Equations 2009, 2009:-20.Google Scholar
2. Baštinec J, Diblík J, Šmarda Z: Existence of positive solutions of discrete linear equations with a single delay. Journal of Difference Equations and Applications 2010,16(5):1165-1177.
3. Berezansky L, Braverman E: On existence of positive solutions for linear difference equations with several delays. Advances in Dynamical Systems and Applications 2006,1(1):29-47.
4. Berezansky L, Braverman E: Oscillation of a logistic difference equation with several delays. Advances in Difference Equations 2006, 2006:-12.Google Scholar
5. Bohner M, Karpuz B, Öcalan Ö: Iterated oscillation criteria for delay dynamic equations of first order. Advances in Difference Equations 2008, 2008:-12.Google Scholar
6. Chatzarakis GE, Koplatadze R, Stavroulakis IP: Oscillation criteria of first order linear difference equations with delay argument. Nonlinear Analysis: Theory, Methods & Applications 2008,68(4):994-1005. 10.1016/j.na.2006.11.055
7. Diblík J, Koksch N:Positive solutions of the equation in the critical case. Journal of Mathematical Analysis and Applications 2000,250(2):635-659. 10.1006/jmaa.2000.7008
8. Domshlak Y: Oscillation properties of discrete difference inequalities and equations: the new approach. In Functional-Differential Equations, Functional Differential Equations Israel Sem.. Volume 1. Coll. Judea Samaria, Ariel, Israel; 1993:60-82.Google Scholar
9. Domshlak Y, Stavroulakis IP: Oscillations of first-order delay differential equations in a critical state. Applicable Analysis 1996,61(3-4):359-371. 10.1080/00036819608840464
10. Dorociaková B, Olach R: Existence of positive solutions of delay differential equations. Tatra Mountains Mathematical Publications 2009, 43: 63-70.
11. Györi I, Ladas G: Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs. The Clarendon Press, New York, NY, USA; 1991:xii+368.
12. Hanuštiaková L, Olach R: Nonoscillatory bounded solutions of neutral differential systems. Nonlinear Analysis: Theory, Methods and Applications 2008,68(7):1816-1824. 10.1016/j.na.2007.01.014
13. Kikina LK, Stavroulakis IP: A survey on the oscillation of solutions of first order delay difference equations. Cubo 2005,7(2):223-236.
14. Medina R, Pituk M: Nonoscillatory solutions of a second-order difference equation of Poincaré type. Applied Mathematics Letters 2009,22(5):679-683. 10.1016/j.aml.2008.04.015
15. Stavroulakis IP: Oscillation criteria for first order delay difference equations. Mediterranean Journal of Mathematics 2004,1(2):231-240. 10.1007/s00009-004-0013-7

## Copyright

© J. Baštinec et al. 2010

This article is published under license to BioMed Central Ltd. 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.