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Theory and Modern Applications

On a Max-Type Difference Equation

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, [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.

  1. (i)
  2. (ii)
  3. (iii)
  4. (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.

  1. (i)
  2. (ii)
  3. (iii)
  4. (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.

  1. (i)
  2. (ii)
  3. (iii)
  4. (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.

References

  1. 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 

  2. 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)80040-0

    Article  MathSciNet  MATH  Google Scholar 

  3. 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. 10.1080/10236190600949766

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. 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 

  6. 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/BF02936057

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. 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 

  9. 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 

  10. 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 

  11. Ladas G:On the recursive sequence . Journal of Difference Equations and Applications 1996,2(3):339-341. 10.1080/10236199608808067

    Article  MathSciNet  Google Scholar 

  12. 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)80010-4

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Stević S:On the recursive sequence . Applied Mathematics Letters 2008,21(8):791-796. 10.1016/j.aml.2007.08.008

    Article  MathSciNet  MATH  Google Scholar 

  17. Sun F: On the asymptotic behavior of a difference equation with maximum. Discrete Dynamics in Nature and Society 2008, 2008:-6.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Yalçinkaya I, Iričanin BD, Çinar C: On a max-type difference equation. Discrete Dynamics in Nature and Society 2007,2007(1):-10.

  20. 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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgment

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

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Correspondence to Ali Gelisken.

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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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