# On a Max-Type Difference Equation

- Ali Gelisken
^{1}Email author, - Cengiz Cinar
^{1}and - Ibrahim Yalcinkaya
^{1}

**2010**:584890

**DOI: **10.1155/2010/584890

© Ali Gelisken et al. 2010

**Received: **8 December 2009

**Accepted: **23 April 2010

**Published: **30 May 2010

## Abstract

We prove that every positive solution of the max-type difference equation , converges to where are positive integers, , and .

## 1. Introduction

Recently, the study of max-type difference equations attracted a considerable attention. Although max-type difference equations are relatively simple in form, it is unfortunately extremely difficult to understand thoroughly the behavior of their solutions; see, for example, [1–20] and the relevant references cited therein. The max operator arises naturally in certain models in automatic control theory (see [13, 14]). Furthermore, difference equation appear naturally as a discrete analogue and as a numerical solution of differential and delay differential equations having applications and various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology.

converges to or eventually periodic with period 4, where and

converges to or eventually periodic with period 2, where and

converges to where , and

where are positive integers, , , and initial conditions are positive real numbers.

In this paper, we investigate the asymptotic behavior of the positive solutions of (1.4). We prove that every positive solution of (1.4) converges to Clearly, we can assume that without loss of generality.

## 2. Main Results

### 2.1. The Case

In this section, we consider the asymptotic behavior of the positive solutions of (1.4) in the case

where and the initial conditions are real numbers. Since we have

We need the following two lemmas in order to prove the main result of this section.

Lemma 2.1.

Proof.

- (i)
- (ii)
- (iii)
- (iv)

From the above statements, we have for all Therefore, the proof is complete.

Lemma 2.2.

Proof.

- (i)
- (ii)
- (iii)
- (iv)

From the above statements, we have for all Therefore, the proof is complete.

Theorem 2.3.

Let be a solution of (1.4) where Then converges to

Proof.

Assume that is a solution of (2.2). If it is proved that converges to zero as , then converges to

From (2.9), it is clear that converges to zero as

Then, the rest of proof is similar to the case and will be omitted. Therefore, the proof is complete.

### 2.2. The Case

In this section, we consider the asymptotic behavior of the positive solutions of (1.4) in the case

where initial conditions are real numbers.

We need the following lemma in order to prove the main result of this section.

Lemma 2.4.

Proof.

- (i)
- (ii)
- (iii)
- (iv)

From the above statements, we have for all Therefore, the proof is complete.

Theorem 2.5.

Let be a solution of (1.4) where Then converges to

Proof.

Let be a solution of (2.12). To prove the desired result, it suffices to prove that converges to zero.

From (2.15), it is clear that converges to zero as Therefore, the proof is complete.

## Declarations

### Acknowledgment

The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.

## Authors’ Affiliations

## References

- Abu-Saris RM, Allan FM:
**Periodic and nonperiodic solutions of the difference equation****max****.**In*Advances in Difference Equations (Veszprém, 1995)*. Gordon and Breach, Amsterdam, The Netherlands; 1997:9-17.Google Scholar - Amleh AM, Hoag J, Ladas G:
**A difference equation with eventually periodic solutions.***Computers & Mathematics with Applications*1998,**36**(10–12):401-404. 10.1016/S0898-1221(98)80040-0MathSciNetView ArticleMATHGoogle Scholar - Berenhaut KS, Foley JD, Stević S:
**Boundedness character of positive solutions of a max difference equation.***Journal of Difference Equations and Applications*2006,**12**(12):1193-1199. 10.1080/10236190600949766MathSciNetView ArticleMATHGoogle Scholar - Briden WJ, Grove EA, Ladas G, Kent CM:
**Eventually periodic solutions of**.*Communications on Applied Nonlinear Analysis*1999,**6**(4):31-43.MathSciNetMATHGoogle Scholar - Briden WJ, Grove EA, Ladas G, McGrath LC:
**On the nonautonomous equation**. In*New Developments in Difference Equations and Applications (Taipei, 1997)*. Gordon and Breach, Amsterdam, The Netherlands; 1999:49-73.Google Scholar - Çinar C, Stević S, Yalçinkaya I:
**On positive solutions of a reciprocal difference equation with minimum.***Journal of Applied Mathematics & Computing*2005,**17**(1-2):307-314. 10.1007/BF02936057MathSciNetView ArticleMATHGoogle Scholar - Gelişken A, Çinar C, Karataş R:
**A note on the periodicity of the Lyness max equation.***Advances in Difference Equations*2008,**2008:**-5.Google Scholar - Gelişken A, Çinar C, Yalçinkaya I:
**On the periodicity of a difference equation with maximum.***Discrete Dynamics in Nature and Society*2008,**2008:**-11.Google Scholar - Gelişken A, Çinar C:
**On the global attractivity of a max-type difference equation.***Discrete Dynamics in Nature and Society*2009,**2009:**-5.Google Scholar - Grove EA, Kent C, Ladas G, Radin MA:
**On the****with a period 3 parameter.**In*Fields Institute Communications*.*Volume 29*. American Mathematical Society, Providence, RI, USA; 2001:161-180.Google Scholar - Ladas G:
**On the recursive sequence**.*Journal of Difference Equations and Applications*1996,**2**(3):339-341. 10.1080/10236199608808067MathSciNetView ArticleGoogle Scholar - Mishev DP, Patula WT, Voulov HD:
**A reciprocal difference equation with maximum.***Computers & Mathematics with Applications*2002,**43**(8-9):1021-1026. 10.1016/S0898-1221(02)80010-4MathSciNetView ArticleMATHGoogle Scholar - Myškis AD:
**Some problems in the theory of differential equations with deviating argument.***Uspekhi Matematicheskikh Nauk*1977,**32**(2(194)):173-202.Google Scholar - Popov EP:
*Automatic Regulation and Control*. Nauka, Moscow, Russia; 1966.Google Scholar - Szalkai I:
**On the periodicity of the sequence**.*Journal of Difference Equations and Applications*1999,**5**(1):25-29. 10.1080/10236199908808168MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**On the recursive sequence**.*Applied Mathematics Letters*2008,**21**(8):791-796. 10.1016/j.aml.2007.08.008MathSciNetView ArticleMATHGoogle Scholar - Sun F:
**On the asymptotic behavior of a difference equation with maximum.***Discrete Dynamics in Nature and Society*2008,**2008:**-6.Google Scholar - Voulov HD:
**On the periodic character of some difference equations.***Journal of Difference Equations and Applications*2002,**8**(9):799-810. 10.1080/1023619021000000780MathSciNetView ArticleMATHGoogle Scholar - Yalçinkaya I, Iričanin BD, Çinar C:
**On a max-type difference equation.***Discrete Dynamics in Nature and Society*2007,**2007**(1):-10. - Yang X, Liao X, Li C:
**On a difference equation with maximum.***Applied Mathematics and Computation*2006,**181**(1):1-5. 10.1016/j.amc.2006.01.005MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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