On the positive solutions of the difference equation system xn+1=1yn, yn+1=ynxn−1yn−1

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Abstract

We study the positive solutions of the difference equation systemxn+1=1yn,yn+1=ynxn−1yn−1,n=0,1,2,…,where x−1, x0 and y0, y−1 are positive real numbers.

Introduction

Our aim in this paper is to investigate the solutions of the difference equation systemxn+1=1yn,yn+1=ynxn−1yn−1,n=0,1,2,…,wherex−1,x0andy0,y−1arepositiverealnumbers.

Some papers related to this subject are following:

Schinas [3] has studied some invariants for difference equations and systems of difference equations of rational form. Clark and Kulenovic [1] has investigated the global stability properties and asymptotic behavior of solutions of the recursive sequencexn+1=xna+cyn,yn+1=ynb+dxn.Grove et al. [2] has studied existence and behavior of solutions of the rational systemxn+1=axn+byn,yn+1=cxn+dyn.

Similar to the references above, in this paper, we define Eq. (1.1) with condition (1.2) and investigate the solutions of this difference equation system.

Section snippets

Main results

Theorem 2.1

Suppose that x−1, x0 and y0, y−1 are positive real numbers. Also, assume that (1.2) holds. Let {xn,yn} be a solution of equation system (1.1). Then all solutions of equation system (1.1) are periodic with period four.

Proof

From our assumption (1.2), we have all solutions of equation system (1.1) which are positive. Thus, by (1.1) we have the following equal:xn+1=1yn,yn+1=ynxn−1yn−1,xn+2=yn−1xn−1yn,yn+2=1xn−1yn−1xn,xn+3=xn−1yn−1xn,yn+3=1xn,xn+4=xn,yn+4=yn.Therefore, the proof was completed. 

Theorem 2.2

Suppose

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