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# On a Max-Type Difference Equation

*Advances in Difference Equations*
**volume 2010**, Article number: 584890 (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.

In [20], Yang et al. proved that every positive solution of the difference equation

converges to or eventually periodic with period 4, where and

In [9], We proved that every positive solution of the difference equation

converges to or eventually periodic with period 2, where and

In [17], Sun proved that every positive solution of the difference equation

converges to where , and

The following difference equation is more general than (1.3):

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

It is easy to see that by the change

Equation (1.4) is transformed into the difference equation

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.

Let be a solution of (2.2). If , then

Proof.

Clearly, (2.2) implies the following difference equation:

From (2.4), we get the following statements.

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

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

Lemma 2.2.

Let be a solution of (2.2). If , then

Proof.

Assume that . Then (2.2) implies the following difference equation:

From (2.6), we get the following statements.

- (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 Lemma 2.1, we have that

Let Immediately, we have that the following inequality

From (2.8) and by induction, we get

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

Now, we assume that From Lemma 2.2, we have that

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

It is easy to see that by the change

Equation (1.4) is transformed into the difference equation:

where initial conditions are real numbers.

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

Lemma 2.4.

Let be a solution of (2.12). Then

Proof.

From (2.12), we get the following statements.

- (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 Lemma 2.4, we have that

From (2.14) and by induction, we get

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

## 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.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)80040-0Berenhaut 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. 10.1080/10236190600949766Briden WJ, Grove EA, Ladas G, Kent CM:

**Eventually periodic solutions of**.*Communications on Applied Nonlinear Analysis*1999,**6**(4):31-43.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.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/BF02936057Gelişken A, Çinar C, Karataş R:

**A note on the periodicity of the Lyness max equation.***Advances in Difference Equations*2008,**2008:**-5.Gelişken A, Çinar C, Yalçinkaya I:

**On the periodicity of a difference equation with maximum.***Discrete Dynamics in Nature and Society*2008,**2008:**-11.Gelişken A, Çinar C:

**On the global attractivity of a max-type difference equation.***Discrete Dynamics in Nature and Society*2009,**2009:**-5.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.Ladas G:

**On the recursive sequence**.*Journal of Difference Equations and Applications*1996,**2**(3):339-341. 10.1080/10236199608808067Mishev 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)80010-4Myškis AD:

**Some problems in the theory of differential equations with deviating argument.***Uspekhi Matematicheskikh Nauk*1977,**32**(2(194)):173-202.Popov EP:

*Automatic Regulation and Control*. Nauka, Moscow, Russia; 1966.Szalkai I:

**On the periodicity of the sequence**.*Journal of Difference Equations and Applications*1999,**5**(1):25-29. 10.1080/10236199908808168Stević S:

**On the recursive sequence**.*Applied Mathematics Letters*2008,**21**(8):791-796. 10.1016/j.aml.2007.08.008Sun F:

**On the asymptotic behavior of a difference equation with maximum.***Discrete Dynamics in Nature and Society*2008,**2008:**-6.Voulov HD:

**On the periodic character of some difference equations.***Journal of Difference Equations and Applications*2002,**8**(9):799-810. 10.1080/1023619021000000780Yalçinkaya 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.2006.01.005

## Acknowledgment

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

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Gelisken, A., Cinar, C. & Yalcinkaya, I. On a Max-Type Difference Equation.
*Adv Differ Equ* **2010**, 584890 (2010). https://doi.org/10.1155/2010/584890

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DOI: https://doi.org/10.1155/2010/584890