# On the Solutions of Systems of Difference Equations

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## Abstract

We show that every solution of the following system of difference equations Open image in new window , Open image in new window as well as of the system Open image in new window , Open image in new window is periodic with period 2 Open image in new window if Open image in new window ( Open image in new window 2), and with period Open image in new window if Open image in new window ( Open image in new window 2) where the initial values are nonzero real numbers for Open image in new window .

### Keywords

Differential Equation Qualitative Analysis Partial Differential Equation Ordinary Differential Equation Functional Analysis## 1. Introduction

Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on [1]. So, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations and systems of difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and the references cited therein).

where Open image in new window are positive constants and the initial values Open image in new window are positive.

where the initial values Open image in new window Open image in new window are positive numbers and Open image in new window is a positive integer.

where the initial values Open image in new window are positive real numbers.

where Open image in new window is a nonnegative integer, Open image in new window is a positive integer, and the initial values Open image in new window are positive real numbers.

where Open image in new window fixed.

where Open image in new window (for Open image in new window are positive constants, Open image in new window is an integer, and the initial values Open image in new window (for Open image in new window are positive real numbers.

which can be considered as a natural generalizations of (1.8).

In order to prove main results of the paper we need an auxiliary result which is contained in the following simple lemma from number theory. Let Open image in new window denote the greatest common divisor of the integers Open image in new window and Open image in new window

Lemma 1.1.

Proof.

Suppose the contrary, then we have Open image in new window for some Open image in new window

Since Open image in new window it follows that Open image in new window is a divisor of Open image in new window On the other hand, since Open image in new window we have Open image in new window which is a contradiction.

Remark 1.2.

From Lemma 1.1 we see that the rests Open image in new window for Open image in new window of the numbers Open image in new window for Open image in new window obtained by dividing the numbers Open image in new window by Open image in new window , are mutually different, they are contained in the set Open image in new window , make a permutation of the ordered set Open image in new window , and finally Open image in new window is the first number of the form Open image in new window such that Open image in new window

## 2. The Main Results

In this section, we formulate and prove the main results in this paper.

Theorem 2.1.

Consider (1.9) where Open image in new window Then the following statements are true:

(a)if Open image in new window , then every solution of (1.9) is periodic with period 2*k*,

(b)if Open image in new window , then every solution of (1.9) is periodic with period k.

Proof.

First note that the system is cyclic. Hence it is enough to prove that the sequence Open image in new window satisfies conditions (a) and (b) in the corresponding cases.

From this and since by Lemma 1.1 the numbers Open image in new window are pairwise different, the result follows in this case.

which yields the result.

Remark 2.2.

In order to make the proof of Theorem 2.1 clear to the reader, we explain what happens in the cases Open image in new window and Open image in new window .

that is, the sequence Open image in new window is periodic with period 6.

Remark 2.3.

Similarly to Theorem 2.1, using Lemma 1.1 with Open image in new window for Open image in new window the following theorem can be proved.

Theorem 2.4.

Consider (1.10) where Open image in new window Then the following statements are true:

(a)if Open image in new window , then every solution of (1.10) is periodic with period 2k,

(b)if Open image in new window , then every solution of (1.10) is periodic with period k.

Proof.

First note that the system is cyclic. Hence, it is enough to prove that the sequence Open image in new window satisfies conditions (a) and (b) in the corresponding cases.

From this and since by Lemma 1.1 the numbers Open image in new window are pairwise different, the result follows in this case.

which yields the result.

Corollary 2.5.

then all solutions of (1.9) are positive.

Proof.

for Open image in new window and Open image in new window .

for Open image in new window and Open image in new window .

From (2.16) and (2.17), all solutions of (1.9) are positive.

Corollary 2.6.

then Open image in new window are positive, Open image in new window are negative for all Open image in new window

Proof.

From (2.16), (2.17), and (2.18), the proof is clear.

Corollary 2.7.

then Open image in new window are negative, Open image in new window are positive for all Open image in new window

Proof.

From (2.16), (2.17), and (2.19), the proof is clear.

Corollary 2.8.

Let Open image in new window be solutions of (1.9) with the initial values Open image in new window , then the following statements are true (for all Open image in new window and Open image in new window

(i)if Open image in new window then Open image in new window and Open image in new window ,

(ii)if Open image in new window then Open image in new window and Open image in new window

(iii)if Open image in new window then Open image in new window and Open image in new window

(iv)if Open image in new window then Open image in new window and Open image in new window ,

(v)if Open image in new window then Open image in new window and Open image in new window ,

(vi)if Open image in new window then Open image in new window and Open image in new window .

Proof.

From (2.16) and (2.17), the proof is clear.

Corollary 2.9.

then all solutions of (1.10) are positive.

Proof.

for Open image in new window and Open image in new window .

for Open image in new window and Open image in new window .

From (2.21) and (2.22), all solutions of (1.10) are positive.

Corollary 2.10.

then Open image in new window are positive, Open image in new window are negative for all Open image in new window

Proof.

From (2.21), (2.22) and (2.23), the proof is clear.

Corollary 2.11.

then Open image in new window are negative, Open image in new window are positive for all Open image in new window

Proof.

From (2.21), (2.22) and (2.24), the proof is clear.

Corollary 2.12.

Let Open image in new window be solutions of (1.10) with the initial values Open image in new window , then following statements are true (for all Open image in new window and Open image in new window

(i)if Open image in new window then Open image in new window and Open image in new window ,

(ii)if Open image in new window then Open image in new window and Open image in new window

(iii)if Open image in new window then Open image in new window and Open image in new window

(iv)if Open image in new window then Open image in new window and Open image in new window

(v)if Open image in new window then Open image in new window and Open image in new window

(vi)if Open image in new window then Open image in new window and Open image in new window

Proof.

From (2.21), (2.22), and (2.24), the proof is clear.

Example 2.13.

Let Open image in new window . Then the solutions of (1.9), with the initial values Open image in new window and Open image in new window in its invertal of periodicity can be represented by Table 1.

Table 1

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

| | | | | | |||||||

| | | | | | |||||||

| | | | | |

## Notes

### Acknowledgment

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

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