Abstract and Applied Analysis

Volume 2012 (2012), Article ID 258718, 5 pages

doi:10.1155/2012/258718

Research Article

On a Periodic System of Difference Equations

Stevo Stević,¹ Josef Diblík,2,3 Bratislav Iričanin,⁴ and Zdeněk Šmarda³

¹Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia

²Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic

³Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic

⁴Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia

Received 9 April 2012; Accepted 26 April 2012

Academic Editor: Norio Yoshida

Copyright © 2012 Stevo Stević et al. 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.

Here we give a very short and elegant proof of an extension of two recent results on global periodicity of two systems of difference equations in a work of Elsayed et al. 2012.

1. Introduction

For a solution , , , of the system of difference equations

is said that it is eventually periodic with period , if there is an such that

If , then for the solution is said that it is periodic with period . If all well-defined solutions of system (1.1) are eventually periodic with period , it is said that the system is -periodic.

For some periodic difference equations and systems see, for example, [1–11] and the references therein. Periodic systems are essentially subclasses of the systems which can be solved explicitly, an area that has again attracted attention recently (see, e.g., [12–21]). For some classical results in the topic see, for example, [22].

In the recent paper [23] are formulated two results on global periodicity of particular systems of difference equations. Namely, they gave some tedious straightforward calculations which suggest that all the solutions with nonzero real initial values of the following systems of difference equations

are periodic with periods and , respectively, without giving complete proofs of these statements.

2. Main Result

Here we prove a result which naturally extends the results on systems (1.3) and (1.4), and give a very short and elegant proof of it.

Theorem 2.1. Assume that , , functions , , satisfy the following condition

for every .

Consider the following system of difference equations

Then every solution of system (2.2) whose initial values are arbitrary nonzero real numbers is eventually periodic with period .

Proof. Since for every and , functions , , map the set to and

map the set to , from (2.2) with , we see that for every .

Assume we have proved that for and every . Then from this, (2.2) and since functions , , map the set to , and map the set to , we get for every . So, by induction it follows that for every and . Hence, every solution of system (2.2) whose initial values are arbitrary nonzero real numbers is well-defined.

Multiplying all the equations in (2.2), then using condition (2.1) and the above proven fact that , for every , we get

from which it follows that

that is, the sequence is eventually periodic with period . In particular, the sequences are eventually periodic with period , for every .

This implies that the sequences

are eventually periodic with period , which along with (2.2) implies that the sequences are eventually periodic with period , for , from which the theorem follows.

Now we present the main results in paper [23] as two corollaries.

Corollary 2.2. Assume that , , is an arbitrary solution of system (1.3) whose initial values are arbitrary nonzero real numbers. Then the solution is eventually periodic with period .

Proof. Note that for system (1.3), , , and , so that

Hence, all the conditions of Theorem 2.1 are fulfilled from which the corollary follows.

Corollary 2.3. Assume that , , is an arbitrary solution of system (1.4) whose initial values are arbitrary nonzero real numbers. Then the solution is eventually periodic with period .

Proof . For system (1.4), we have that , ,

so that

Hence, all the conditions of Theorem 2.1 are fulfilled from which the corollary follows.

The next corollary, which corresponds to case in Theorem 2.1, is obtained similarly, so we omit the proof.

Corollary 2.4. Assume that , , is an arbitrary solution of the following system

, whose initial values are arbitrary nonzero real numbers. Then the solution is eventually periodic with period .

Acknowledgments

The second author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government Grant MSM 00 216 30519. The fourth author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.

References

B. D. Iričanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 13, no. 3-4, pp. 499–507, 2006.

C. M. Kent, M. Kustesky, A. Q. Nguyen, and B. V. Nguyen, “Eventually periodic solutions of max when the parameters are two cycles,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 10, no. 1-3, pp. 33–49, 2003.

W. Kosmala, “A period 5 difference equation,” International Journal of Nonlinear Analysis and Applications, vol. 2, no. 1, pp. 82–84, 2011.

R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,” Canadian Journal of Mathematics, vol. 26, pp. 1356–1371, 1974.

R. C. Lyness, “Notes 1581,” Mathematical Gazette, vol. 26, p. 62, 1942.

R. C. Lyness, “Notes 1847,” Mathematical Gazette, vol. 29, p. 231, 1945.

G. Papaschinopoulos and C. J. Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations,” Communications on Applied Nonlinear Analysis, vol. 5, no. 2, pp. 47–59, 1998.

G. Papaschinopoulos and C. J. Schinas, “Invariants for systems of two nonlinear difference equations,” Differential Equations and Dynamical Systems, vol. 7, no. 2, pp. 181–196, 1999.

G. Papaschinopoulos and C. J. Schinas, “Invariants and oscillation for systems of two nonlinear difference equations,” Nonlinear Analysis. Series A, vol. 46, no. 7, pp. 967–978, 2001.

S. Stević, “Periodicity of max difference equations,” Utilitas Mathematica, vol. 83, pp. 69–71, 2010.

S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562–9566, 2011.

L. Berg and S. Stević, “On some systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1713–1718, 2011.

B. Iričanin and S. Stević, “On some rational difference equations,” Ars Combinatoria, vol. 92, pp. 67–72, 2009.

S. Stević, “More on a rational recurrence relation,” Applied Mathematics E-Notes, vol. 4, pp. 80–85, 2004.

S. Stević, “A short proof of the Cushing-Henson conjecture,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 37264, 5 pages, 2006.

S. Stević, “On a nonlinear generalized max-type difference equation,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 317–328, 2011.

S. Stević, “On a system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372–3378, 2011.

S. Stević, “On the difference equation ,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4507–4513, 2011.

S. Stević, “On a third-order system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7649–7654, 2012.

S. Stević, “On some solvable systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5010–5018, 2012.

S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, “On a third-order system of difference equations with variable coefficients,” Abstract and Applied Analysis, vol. 2012, Article ID 508523, 22 pages, 2012.

H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.

E. M. Elsayed, M. M. El-Dessoky, and A. M. Alotaibi, “On the solutions of a general system of difference equations,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 892571, 12 pages, 2012.