Copyright © 2009 Dağistan Simsek 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

We study the behavior of the solutions of the following system of difference equations xn+1=max⁡{A/xn,yn/xn}, yn+1=max⁡{A/yn,xn/yn} where the constant A and the initial conditions are positive real numbers.

1. Introduction

Recently, there has been a great interest in studying the periodic nature of nonlinear difference equations. Although difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the periodic behavior of their solutions. The periodic nature of nonlinear difference equations of the max type has been investigated by many authors. See, for example [1–24].

In this paper we study the behavior of the solutions of the following system of difference equations: (1.1) where the constant and the initial conditions are positive real numbers.

2. Main Result

Definition 2.1. Fibonacci sequence is and for .

Definition 2.2. The symbol symbolizes the greatest integer function.

Definition 2.3. The sequence of

Definition 2.4. The sequence of (2.1)

Theorem 2.5. Let be the solution of the system of difference equations (1.1) for and

If , then (2.2) , and if (2.3)

Proof. Let then (2.4) then (2.5) then (2.6)

Theorem 2.6. Let be the solution of the system of difference equations (1.1) for .

and if (2.7)

If , then (2.8)

Proof. Similarly we can obtain the proof as the proof of Theorem 2.5.

Theorem 2.7. Let be the solution of the system of difference equations (1.1) for and (2.9)

Proof. (a) We obtain that (2.10)

Similarly we can obtain the proof of (b) as the proof of (a).

Theorem 2.8. Let be the solution of the system of difference equations (1.1) for and (2.11)

Proof. Similarly we can obtain the proof as the proof of Theorem 2.7.

Theorem 2.9. Let be the solution of the system of difference equations (1.1) for

If , then (2.12)

If , then (2.13)

Proof. Let (2.14), then (2.15)

Theorem 2.10. Let be the solution of the system of difference equations (1.1) for (2.16)

Proof. (a) We obtain that (2.17)

Similarly we can obtain the proof of (b) as the proof of (a).

Lemma 2.11. Let be the initial condition of (1.1) for ; there is at least an such that every for .

Proof. We consider that hence and that proofs the existing of defined in hypothesis.

Theorem 2.12. Let be the solution of the system of difference equations (1.1) for , and is the number, defined by Lemma 2.11.

(2.18)

(2.19) and when , the solutions will be different for every different constant

Proof. Let then (2.20)(2.21)

Lemma 2.13. Let be the initial condition of (1.1) for ; there is at least an such that every for .

Proof. Similarly we can obtain the proof as the proof of Lemma 2.11.

Theorem 2.14. Let be the solution of the system of difference equations (1.1) for , and is the number, defined by Lemma 2.13.

(2.22)

(2.23) and when , the solutions will be different for every different constant

Proof. Similarly we can obtain the proof of be as the proof of Theorem 2.12.

Lemma 2.15. Let be the initial condition of (1.1) for ; there is at least an such that every for .

Proof. We consider that hence and that proofs the existing of defined in hypothesis.

Theorem 2.16. Let be the solution of the system of difference equations (1.1) for , and is the number, defined by Lemma 2.15.

and if (2.24)

, and if (2.25) and when , the solutions will be different for every different constant

Proof. Let then (2.26) then (2.27) then (2.28)

Lemma 2.17. Let be the initial condition of (1.1) for ; there is at least an such that every for .

Proof. Similarly we can obtain the proof as the proof of Lemma 2.15.

Theorem 2.18. Let be the solution of the system of difference equations (1.1) for , and is the number, defined by Lemma 2.17.

, and if (2.29)

, and if (2.30) and when , the solutions will be different for every different constant

Proof. Similarly we can obtain the proof as the proof of Theorem 2.16, which completes the proofs of theorems.

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