Discrete Dynamics in Nature and Society

Volume 2010, Article ID 675413, 13 pages

http://dx.doi.org/10.1155/2010/675413

## On the Max-Type Difference Equation

^{1}Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11120, Serbia^{2}Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 29 October 2009; Revised 24 December 2009; Accepted 25 January 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Bratislav D. Iričanin and E. M. Elsayed. 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.

#### Abstract

We show that every well-defined solution of the fourth-order difference equation , , where parameter , is eventually periodic with period four.

#### 1. Introduction

The study of max-type difference equations attracted recently a considerable attention, see, for example, [1–27], and the references listed therein. This type of difference equations stems from, for example, certain models in automatic control theory (see [28]). In the beginning of the study of these equations experts have been focused on the investigation of the behavior of some particular cases of the following general difference equation of order :

where , , are real sequences (mostly constant or periodic ones) and where the initial values are different from zero (see, e.g., [2, 3, 6, 7, 9–12, 22–25] and the references cited therein).

The study of max-type equations of the following general form

where , are natural numbers such that , , and , are sequences of real numbers, was proposed by Stević in numerous talks, for example, in [13, 14]. For some results in this direction see [1, 4, 15–17, 19–21, 26, 27]. For some nonlinear difference equations related to (1.2) see, for example, [7, 15, 17, 18, 29–38].

*Definition 1.1. *A sequence is said to be *eventually periodic with period * if there is an index such that for all . Specially, if , then the sequence is *periodic with period *.

Motivated by some ideas due to Stević (e.g., the main lemmas there, Lemmas and are suggested by him), the authors of [26] considered the following second-order max-type difference equation:

Equation(1.3) is not difficult for handling since, by the change , it is transformed into one of the following first-order difference equations

Using these equations, it is easy to see that for the case every solution of (1.3) is eventually periodic with period two.

Recently, in the paper [5] it was showed that every solution of the third-order max-type difference equation

where the initial conditions are arbitrary nonzero real numbers and is eventually periodic with period three. The fact that all solutions of (1.5) are periodic is not a surprising fact (for an explanation see [4]).

For some recent papers on difference equations all the solutions of which are periodic see, for example, [7, 39–45] and the references cited therein.

Here we show that every well-defined solution of the following fourth-order max-type difference equation

where the parameter is eventually periodic with period four.

*Remark 1.2. *Note that if , then (1.6) becomes , from which it follows that every solution is periodic with period four. Hence, in the sequel we will consider the case

In the sequel we will frequently use the following simple lemma, given without a proof (for related results see [4, 5]).

Lemma 1.3. *Assume that is a solution of (1.6) and there is such that
**
Then this solution is eventually periodic with period four.*

#### 2. Main Results

In this subsection we give a specific form of the solutions of the difference equation (1.6) when the parameter and in each case we can deduce that every solution of this equation is periodic with period four.

Depending on the positivity of four initial values of (1.6), there are the following 16 cases to be considered:

First, we prove another auxiliary result.

Lemma 2.1. *Assume that the parameter . Then every solution of (1.6) is eventually positive if initial values satisfy one of conditions (i), (iii)–(xvi).*

*Proof. *If or , then
From this, (1.6), and by induction it follows that for every .

If , then

From this, (1.6), and by induction it follows that for every .

If , then

Similar to the previous case, by induction it follows that for every .

Now, we can formulate and prove our main results.

Theorem 2.2. *Assume that the parameter . Then every solution of (1.6) with positive initial values is eventually periodic with period four. *

*Proof. *From (1.6), we see that
We consider the following two cases. In this case , and we see that
Now, there exists two subcases. which occurs when . We have
If , then and
Hence, , , , and , which implies that is an eventually (from ) periodic solution with period four. In this case we see that the solution has the following form:
If , then and
Hence, , , , and , which implies that is an eventually (from ) periodic solution with period four (in this case minimal period is two). This solution takes the form
which occurs when , and
If , then and
If , then and
Hence, , , , and , which implies that is an eventually (from ) periodic solution with period four. It can be written in the form
If , then and
Hence, , , , and , which implies that is an eventually (from ) periodic solution with period four. Moreover, it can be written as follows:
If , then and
As above, the solution is eventually (from ) periodic with period four and it has the form
In this case , and we see that
There again exist two subcases. which occurs when . So,
If , then and
Then we see that the solution is
and is an eventually (from ) periodic solution with period four. If , then and
So, the solution takes the following form which is an eventually (from ) periodic solution with period four:
which occurs when . So,
If , then and
If , then .

Therefore is a periodic solution with period four and the solution takes the form

If , then and
Therefore, again is an eventually (from ) periodic solution with period four and the solution takes the form
If , then and
If , then and
Thus, the solution is in the following form which is eventually (from ) periodic with period four:
If , then and
Thus, the solution is of the following form
Therefore is eventually (from ) periodic with period four. The proof is completed.

From Lemma 1.3 and Theorem 2.2 we obtain the following result.

Theorem 2.3. *Assume the parameter and that initial values of (1.6) satisfy one of conditions (i), (iii)–(xvi) in Lemma 2.1. Then every such solution of (1.6) is eventually periodic with period four.*

*Proof. *If initial values of (1.6) satisfy one of conditions (i), (iii)–(xvi) in Lemma 2.1, then by the same lemma it follows that the corresponding solution is eventually positive. This means that there is such that for every In particular, we have that . Since equation (1.6) is autonomous if is a solution of (1.6), then is also a solution of (1.6) but such that Hence, the problem is reduced to the case when all the initial values are positive. Applying Theorem 2.2, the result follows.

In the next theorem we study those solutions of (1.6) such that We would like to thank Professor Stević for giving us the elegant proof below which drastically reduced our original proof.

Theorem 2.4. *Assume that the parameter and all the initial values are negative Then every solution of (1.6) is eventually periodic with period four.*

*Proof. *Since , , , and , by induction we have for each . By the change , (1.6) becomes
where for every Hence by Theorem 2.2 the result follows.

Since all the cases are reduced to the case when all initial values are positive, it is of interest to investigate this case in more details. The next theorem describes all eventually constant solutions of (1.6). Our idea stems from [46] (see also [47–50]).

Theorem 2.5. * Assume that the parameter . Then all positive solutions to (1.6) which are eventually equal to the positive equilibrium have the following form:
*

*Proof. *First note that by the change the problem is reduced to the case . Assume that for and Then , , which implies and . On the other hand, we have so that , which is a contradiction. Hence , that is, . Since , we get (case above). Since and , it follows that , as claimed.

#### 3. Conclusions and Future Works

We finish this article with some comments which can motivate further works.

*Case *

The case when the parameter is negative is not treated in this article. By similar calculations as above, we have managed to show that all the solutions of (1.6) are eventually periodic with period four in many subcases. However, there are too many subcases and calculations without some new ideas, so we decided not to present any result when We conjecture that every solution of (1.6), when , is eventually periodic with the same period four. What is more interesting is to find a reasonably short proof of the conjecture without using tiresome calculations similar to those in Theorem 2.2.

*Periodicity*

We also want to mention that in [4] we proved that every solution of the equation
where , is periodic. It is of some interest to find the minimal period of the equation, as well as to get a general result concerning this problem.

#### Acknowledgments

The authors would like to express their sincere thanks to the anonymous referees for numerous comments which considerably improved this article. The research of the first author was partly supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, Project no. 144013.

#### References

- K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max difference equation,”
*Journal of Difference Equations and Applications*, vol. 12, no. 12, pp. 1193–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Çinar, S. Stević, and I. Yalçinkaya, “On positive solutions of a reciprocal difference equation with minimum,”
*Journal of Applied Mathematics & Computing*, vol. 17, no. 1-2, pp. 307–314, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the periodic nature of some max-type difference equations,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 14, pp. 2227–2239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elsayed, B. Iričanin, and S. Stević, “On the max-type equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{{A}_{n}/{x}_{n},{x}_{n-1}\}$,” to appear in
*Ars Combinatoria*. - E. M. Elsayed and S. Stević, “On the max-type equation ${x}_{n+1}=\{A/{x}_{n},{x}_{n-2}\}$,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 3-4, pp. 910–922, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Feuer, “On the eventual periodicity of ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{1/{x}_{n},{A}_{n}/{x}_{n-1}\}$ with a period-four parameter,”
*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 467–486, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. A. Grove and G. Ladas,
*Periodicities in Nonlinear Ddifference Equations*, vol. 4 of*Advances in Discrete Mathematics and Applications*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. View at MathSciNet - B. D. Iričanin,
*The qualitative analysis of some classes of nonlinear difference equations*, Ph.D. thesis, Prirodno-matematički fakultet, Univerzitet u Novom Sadu, Novi Sad, Serbia, 2009. - C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n-1}\}$ with periodic parameters,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series B*, vol. 2003, supplement, pp. 11–15, 2003. View at Google Scholar · View at MathSciNet - D. P. Mishev, W. T. Patula, and H. D. Voulov, “A reciprocal difference equation with maximum,”
*Computers & Mathematics with Applications*, vol. 43, no. 8-9, pp. 1021–1026, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. P. Mishev, W. T. Patula, and H. D. Voulov, “Periodic coefficients in a reciprocal difference equation with maximum,”
*PanAmerican Mathematical Journal*, vol. 13, no. 3, pp. 43–57, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. T. Patula and H. D. Voulov, “On a max type recurrence relation with periodic coefficients,”
*Journal of Difference Equations and Applications*, vol. 10, no. 3, pp. 329–338, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Some open problems and conjectures on difference equations,” http://www.mi.sanu.ac.rs/colloquiums/mathcoll_programs/mathcoll.apr2004.htm.
- S. Stević, “Boundedness character of a max-type difference equation,” in
*Book of Abstracts, Conference in Honour of Allan Peterson*, p. 28, Novacella, Italy, July-August 2007. - S. Stević, “On the recursive sequence ${x}_{n+1}=A+({x}_{n}^{p}/{x}_{n-1}^{r})$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 40963, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{c,{x}_{n}^{p}/{x}_{n-1}^{p}\}$,”
*Applied Mathematics Letters*, vol. 21, no. 8, pp. 791–796, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness character of a class of difference equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 2, pp. 839–848, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness character of two classes of third-order difference equations,”
*Journal of Difference Equations and Applications*, vol. 15, no. 11-12, pp. 1193–1209, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Stević, “Global stability of a difference equation with maximum,”
*Applied Mathematics and Computation*, vol. 210, no. 2, pp. 525–529, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On a generalized max-type difference equation from automatic control theory,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1841–1849, 2010. View at Publisher · View at Google Scholar · View at Scopus - F. Sun, “On the asymptotic behavior of a difference equation with maximum,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 243291, 6 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Szalkai, “On the periodicity of the sequence ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{{A}_{0}/{x}_{n},{A}_{1}/{x}_{n-1},\dots ,{A}_{k}/{x}_{n-k}\}$,”
*Journal of Difference Equations and Applications*, vol. 5, no. 1, pp. 25–29, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. D. Voulov, “Periodic solutions to a difference equation with maximum,”
*Proceedings of the American Mathematical Society*, vol. 131, no. 7, pp. 2155–2160, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. D. Voulov, “On the periodic nature of the solutions of the reciprocal difference equation with maximum,”
*Journal of Mathematical Analysis and Applications*, vol. 296, no. 1, pp. 32–43, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. D. Voulov, “On a difference equation with periodic coefficients,”
*Journal of Difference Equations and Applications*, vol. 13, no. 5, pp. 443–452, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, B. D. Iričanin, and C. Çinar, “On a max-type difference equation,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 47264, 10 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yang, X. Liao, and C. Li, “On a difference equation wtih maximum,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 1–5, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - E. P. Popov,
*Automatic Regulation and Control*, Nauka, Moscow, Russia, 1966. - K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation ${x}_{n}=A+{({x}_{n-2}/{x}_{n-1})}^{p}$,”
*Journal of Difference Equations and Applications*, vol. 12, no. 9, pp. 909–918, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Gutnik and S. Stević, “On the behaviour of the solutions of a second-order difference equation,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 27562, 14 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - B. Iričanin and S. Stević, “On a class of third-order nonlinear difference equations,”
*Applied Mathematics and Computation*, vol. 213, no. 2, pp. 479–483, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha}_{n}+({x}_{n-1}/{x}_{n})$. II,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 10, no. 6, pp. 911–916, 2003. View at Google Scholar · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=(A/{\prod}_{i=0}^{k}{x}_{n-i})+1/({\prod}_{j=k+2}^{2(k+1)}{x}_{n-j})$,”
*Taiwanese Journal of Mathematics*, vol. 7, no. 2, pp. 249–259, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “A note on periodic character of a difference equation,”
*Journal of Difference Equations and Applications*, vol. 10, no. 10, pp. 929–932, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=\alpha +({x}_{n-1}^{p}/{x}_{n}^{p})$,”
*Journal of Applied Mathematics & Computing*, vol. 18, no. 1-2, pp. 229–234, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the difference equation ${x}_{n+1}=\alpha +({x}_{n-1}/{x}_{n})$,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1159–1171, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness character of a fourth order nonlinear difference equation,”
*Chaos, Solitons & Fractals*, vol. 40, no. 5, pp. 2364–2369, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević, “On a class of higher-order difference equations,”
*Chaos, Solitons & Fractals*, vol. 42, no. 1, pp. 138–145, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - F. Balibrea and A. Linero, “On global periodicity of ${x}_{n+2}=f({x}_{n+1},{x}_{n})$,” in
*Difference Equations, Special Functions and Orthogonal Polynomials. (Munich, July 25–30, 2005)*, pp. 41–50, World Scientific, Hackensack, NJ, USA, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Balibrea, A. Linero Bas, G. S. López, and S. Stević, “Global periodicity of ${x}_{n+k+1}={f}_{k}({x}_{n+k})\cdots {f}_{1}({x}_{n+1})$,”
*Journal of Difference Equations and Applications*, vol. 13, no. 10, pp. 901–910, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Berg and S. Stević, “Periodicity of some classes of holomorphic difference equations,”
*Journal of Difference Equations and Applications*, vol. 12, no. 8, pp. 827–835, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,”
*Canadian Journal of Mathematics*, vol. 26, pp. 1356–1371, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Rubió-Massegú and V. Mañosa, “Normal forms for rational difference equations with applications to the global periodicity problem,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 2, pp. 896–918, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On global periodicity of a class of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 23503, 10 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević and K. S. Berenhaut, “The behaviour of the positive solutions of the difference equation ${x}_{n}=f({x}_{n-2})/g({x}_{n-1})$,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 53243, 9 pages, 2008. View at Google Scholar - S. Stevich, “Nontrivial solutions of higher-order rational difference equations,”
*Matematicheskie Zametki*, vol. 84, no. 5, pp. 772–780, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - B. Iričanin and S. Stević, “Eventually constant solutions of a rational difference equation,”
*Applied Mathematics and Computation*, vol. 215, no. 2, pp. 854–856, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Global stability and asymptotics of some classes of rational difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 316, no. 1, pp. 60–68, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotics of some classes of higher-order difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 56813, 20 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević, “Existence of nontrivial solutions of a rational difference equation,”
*Applied Mathematics Letters*, vol. 20, no. 1, pp. 28–31, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet