On the Solutions of Systems of Difference Equations

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 [1–11] and the references cited therein).

Papaschinopoulos and Schinas [9, 10] studied the behavior of the positive solutions of the system of two Lyness difference equations

(1.1)

where are positive constants and the initial values are positive.

In [2] Camouzis and Papaschinopoulos studied the behavior of the positive solutions of the system of two difference equations

(1.2)

where the initial values are positive numbers and is a positive integer.

Moreover, Çinar [3] investigated the periodic nature of the positive solutions of the system of difference equations

(1.3)

where the initial values are positive real numbers.

Also, Özban [7] investigated the periodic nature of the solutions of the system of rational difference equations

(1.4)

where is a nonnegative integer, is a positive integer, and the initial values are positive real numbers.

In [12] Irćianin and Stević studied the positive solution of the following two systems of di¤erence equations

(1.5)

where fixed.

In [11] Papaschinopoulos et al. studied the system of difference equations

(1.7)

where (for are positive constants, is an integer, and the initial values (for are positive real numbers.

It is well known that all well-defined solutions of the difference equation

(1.8)

are periodic with period two. Motivated by (1.8), we investigate the periodic character of the following two systems of difference equations:

(1.9)

(1.10)

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 denote the greatest common divisor of the integers and

Lemma 1.1.

Let and then the numbers (or ) for satisfy the following property:

(1.11)

Proof.

Suppose the contrary, then we have for some

Since it follows that is a divisor of On the other hand, since we have which is a contradiction.

Remark 1.2.

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

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

Theorem 2.1.

Consider (1.9) where Then the following statements are true:

(a)if , then every solution of (1.9) is periodic with period 2k,

(b)if , 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 satisfies conditions (a) and (b) in the corresponding cases.

Further, note that for every system (1.9) is equivalent to a system of difference equations of the same form, where

(2.1)

On the other hand, we have

(2.2)

(a)Let for be the rests mentioned in Remark 1.2. Then from (2.2) and Lemma 1.1 we obtain that

(2.3)

Using (2.1) for sufficently large we obtain that (2.3) is equivalent to (here we use the condition )

(2.4)

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

(b)Let for some By (2.1) we have

(2.5)

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 and .

For , system (1.9) is equivalent to the system

(2.6)

where we consider that

(2.7)

From this and (2.2), we have

(2.8)

Using again (2.2), we get which means that the sequence is periodic with period equal to 2. If , system (1.9) is equivalent to system (2.6) where we consider that , and Using this and (2.2) subsequently, it follows that

(2.9)

that is, the sequence is periodic with period 6.

Remark 2.3.

The fact that every solution of (1.8) is periodic with period two can be considered as the case in Theorem 2.1, that is, we can take that

(2.10)

Similarly to Theorem 2.1, using Lemma 1.1 with for the following theorem can be proved.

Theorem 2.4.

Consider (1.10) where Then the following statements are true:

(a)if , then every solution of (1.10) is periodic with period 2k,

(b)if , 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 satisfies conditions (a) and (b) in the corresponding cases.

Indeed, similarly to (2.2), we have

(2.11)

(a)Let for be the rests mentioned in Remark 1.2. Then from (2.11) and Lemma 1.1 we obtain that

(2.12)

Using (2.1) for sufficiently large we obtain that (2.12) is equivalent to (here we use the condition )

(2.13)

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

(b)Let for some By (2.1) we have

(2.14)

which yields the result.

Corollary 2.5.

Let be solutions of (1.9) with the initial values Assume that

(2.15)

then all solutions of (1.9) are positive.

Proof.

We consider solutions of (1.9) with the initial values satisfying (2.15). If , then from (1.9) and (2.15), we have

(2.16)

for and .

If , then from (1.9) and (2.15), we have

(2.17)

for and .

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

Corollary 2.6.

Let be solutions of (1.9) with the initial values . Assume that

(2.18)

then are positive, are negative for all

Proof.

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

Corollary 2.7.

Let be solutions of (1.9) with the initial values . Assume that

(2.19)

then are negative, are positive for all

Proof.

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

Corollary 2.8.

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

(i)if then and ,

(ii)if then and

(iii)if then and

(iv)if then and ,

(v)if then and ,

(vi)if then and .

Proof.

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

Corollary 2.9.

Let be solutions of (1.10) with the initial values . Assume that

(2.20)

then all solutions of (1.10) are positive.

Proof.

We consider solutions of (1.10) with the initial values satisfying (2.20). If , then from (1.10) and (2.20), we have

(2.21)

for and .

If , then from (1.10) and (2.20), we have

(2.22)

for and .

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

Corollary 2.10.

Let be solutions of (1.10) with the initial values . Assume that

(2.23)

then are positive, are negative for all

Proof.

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

Corollary 2.11.

Let be solutions of (1.10) with the initial values . Assume that

(2.24)

then are negative, are positive for all

Proof.

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

Corollary 2.12.

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

(i)if then and ,

(ii)if then and

(iii)if then and

(iv)if then and

(v)if then and

(vi)if then and

Proof.

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

Example 2.13.

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