Research Article

# On a Max-Type Difference Equation

Ali Gelisken*, Cengiz Cinar and Ibrahim Yalcinkaya

Author Affiliations

Mathematics Department, Ahmet Kelesoglu Education Faculty, Selcuk University, Meram Yeni Yol, 42090 Konya, Turkey

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Advances in Difference Equations 2010, 2010:584890  doi:10.1155/2010/584890

 Received: 8 December 2009 Revisions received: 20 April 2010 Accepted: 23 April 2010 Published: 30 May 2010

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.

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, [120] 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

(11)

converges to or eventually periodic with period 4, where and

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

(12)

converges to or eventually periodic with period 2, where and

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

(13)

converges to where , and

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

(14)

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

(21)

Equation (1.4) is transformed into the difference equation

(22)

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

(23)

Proof.

Clearly, (2.2) implies the following difference equation:

(24)

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

(25)

Proof.

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

(26)

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

(27)

Let Immediately, we have that the following inequality

(28)

From (2.8) and by induction, we get

(29)

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

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

(210)

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

(211)

Equation (1.4) is transformed into the difference equation:

(212)

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

(213)

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

(214)

From (2.14) and by induction, we get

(215)

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

### Acknowledgment

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

### References

1. Abu-Saris, RM, Allan, FM: Periodic and nonperiodic solutions of the difference equation max. Advances in Difference Equations (Veszprém, 1995), pp. 9–17. Gordon and Breach, Amsterdam, The Netherlands (1997)

2. Amleh, AM, Hoag, J, Ladas, G: A difference equation with eventually periodic solutions. Computers & Mathematics with Applications. 36(10–12), 401–404 (1998). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

3. Berenhaut, KS, Foley, JD, Stević, S: Boundedness character of positive solutions of a max difference equation. Journal of Difference Equations and Applications. 12(12), 1193–1199 (2006). Publisher Full Text

4. Briden, WJ, Grove, EA, Ladas, G, Kent, CM: Eventually periodic solutions of . Communications on Applied Nonlinear Analysis. 6(4), 31–43 (1999)

5. Briden, WJ, Grove, EA, Ladas, G, McGrath, LC: On the nonautonomous equation . New Developments in Difference Equations and Applications (Taipei, 1997), pp. 49–73. Gordon and Breach, Amsterdam, The Netherlands (1999)

6. Çinar, C, Stević, S, Yalçinkaya, I: On positive solutions of a reciprocal difference equation with minimum. Journal of Applied Mathematics & Computing. 17(1-2), 307–314 (2005). PubMed Abstract | Publisher Full Text

7. Gelişken, A, Çinar, C, Karataş, R: A note on the periodicity of the Lyness max equation. Advances in Difference Equations. 2008, (2008)

8. Gelişken, A, Çinar, C, Yalçinkaya, I: On the periodicity of a difference equation with maximum. Discrete Dynamics in Nature and Society. 2008, (2008)

9. Gelişken, A, Çinar, C: On the global attractivity of a max-type difference equation. Discrete Dynamics in Nature and Society. 2009, (2009)

10. Grove, EA, Kent, C, Ladas, G, Radin, MA: On the with a period 3 parameter. Fields Institute Communications, pp. 161–180. American Mathematical Society, Providence, RI, USA (2001)

11. Ladas, G: On the recursive sequence . Journal of Difference Equations and Applications. 2(3), 339–341 (1996). Publisher Full Text

12. Mishev, DP, Patula, WT, Voulov, HD: A reciprocal difference equation with maximum. Computers & Mathematics with Applications. 43(8-9), 1021–1026 (2002). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

13. Myškis, AD: Some problems in the theory of differential equations with deviating argument. Uspekhi Matematicheskikh Nauk. 32(2(194)), 173–202 (1977)

14. Popov, EP: Automatic Regulation and Control, Nauka, Moscow, Russia (1966)

15. Szalkai, I: On the periodicity of the sequence . Journal of Difference Equations and Applications. 5(1), 25–29 (1999). Publisher Full Text

16. Stević, S: On the recursive sequence . Applied Mathematics Letters. 21(8), 791–796 (2008). Publisher Full Text

17. Sun, F: On the asymptotic behavior of a difference equation with maximum. Discrete Dynamics in Nature and Society. 2008, (2008)

18. Voulov, HD: On the periodic character of some difference equations. Journal of Difference Equations and Applications. 8(9), 799–810 (2002). Publisher Full Text

19. Yalçinkaya, I, Iričanin, BD, Çinar, C: On a max-type difference equation. Discrete Dynamics in Nature and Society. 2007(1), (2007)

20. Yang, X, Liao, X, Li, C: On a difference equation with maximum. Applied Mathematics and Computation. 181(1), 1–5 (2006). Publisher Full Text