on the periodicity of some classes of systems of nonlinear difference equations

Research ArticleOn the Periodicity of Some Classes of Systems of NonlinearDifference Equations

Stevo SteviT,1,2 Mohammed A. Alghamdi,2 Dalal A. Maturi,3 and Naseer Shahzad2

1 Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia3 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 42641, Jeddah 21551, Saudi Arabia

Correspondence should be addressed to Stevo Stevic; [email protected]

Received 25 September 2013; Revised 3 December 2013; Accepted 4 December 2013; Published 23 January 2014

Academic Editor: Josef Diblik

Copyright © 2014 Stevo Stevic 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.

Some classes of systems of difference equations whose all well-defined solutions are periodic are presented in this note.

1. Introduction

There has been a great recent interest in studying differenceequations and systems of difference equations which do notstem fromdifferential ones (see, e.g., [1–19] and the referencestherein). For some results on concrete systems of nonlineardifference equations, see, for example, [1, 3–5, 9–12, 18, 19].Some classical results in the topic can be found, for example,in book [20].

Solution (𝑥(1)𝑛 , . . . , 𝑥(𝑙)𝑛 )𝑛≥−𝑘, of the system of difference

equations

𝑥(1)

𝑛 = 𝑓1 (𝑥(1)

𝑛−1, . . . , 𝑥(1)

𝑛−𝑘1

, . . . , 𝑥(𝑙)

𝑛−1, . . . , 𝑥(𝑙)

𝑛−𝑘𝑙

) ,

𝑥(2)

𝑛 = 𝑓2 (𝑥(1)

𝑛−1, . . . , 𝑥(1)

𝑛−𝑘1

, . . . , 𝑥(𝑙)

𝑛−1, . . . , 𝑥(𝑙)

𝑛−𝑘𝑙

) ,

...

𝑥(𝑙)

𝑛 = 𝑓𝑙 (𝑥(1)

𝑛−1, . . . , 𝑥(1)

𝑛−𝑘1

, . . . , 𝑥(𝑙)

𝑛−1, . . . , 𝑥(𝑙)

𝑛−𝑘𝑙

) ,

(1)

where 𝑛 ∈ N0 and 𝑘 = max{𝑘1, . . . , 𝑘𝑙}, is called eventuallyperiodic with period 𝑝, if there is an 𝑛1 ≥ −𝑘 such that

𝑥(𝑗)

𝑛+𝑝 = 𝑥(𝑗)

𝑛 , (2)

for every 𝑗 = 1, 𝑙, and 𝑛 ≥ 𝑛1. It is periodic with period𝑝 if 𝑛1 = −𝑘. Period 𝑝 is prime if there is no 𝑝 ∈ N,

𝑝 < 𝑝, which is a period. If all well-defined solutions of anequation or a system of difference equations are eventuallyperiodic with the same period, then such an equation orsystem is called periodic. For some results on the periodicity,asymptotic periodicity and periodic equations or systems ofdifference equations see, for example, [1–10, 12–14, 16–19] andthe related references therein.

In recent paper [19], the authors formulated four resultswhich claim that the following systems of difference equa-tions are periodic with period ten:

𝑥𝑛+1 =𝑦𝑛

𝑥𝑛−1 (1 + 𝑦𝑛), 𝑦𝑛+1 =

𝑥𝑛

𝑦𝑛−1 (1 + 𝑥𝑛), 𝑛 ∈ N0;

(3)


𝑥𝑛−1 (−1 + 𝑦𝑛), 𝑦𝑛+1 =

𝑥𝑛

𝑦𝑛−1 (−1 + 𝑥𝑛), 𝑛 ∈ N0;

(4)


𝑥𝑛−1 (1 + 𝑦𝑛), 𝑦𝑛+1 =

𝑥𝑛

𝑦𝑛−1 (−1 + 𝑥𝑛), 𝑛 ∈ N0;

(5)


𝑥𝑛−1 (−1 + 𝑦𝑛), 𝑦𝑛+1 =

𝑥𝑛

𝑦𝑛−1 (1 + 𝑥𝑛), 𝑛 ∈ N0.

(6)First, we show that all the results in [19] follow from

knownones in the literature and also present some extensions

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 982378, 6 pageshttp://dx.doi.org/10.1155/2014/982378

2 Abstract and Applied Analysis

of these results in the spirit of systems (3)–(6). To do this,we will use a system of difference equations related to thefollowing, so called, Lyness difference equation:

𝑥𝑛+1 =1 + 𝑥𝑛

𝑥𝑛−1, 𝑛 ∈ N0. (7)

It is easy to see that every well-defined solution of (7) isperiodic with period five. The equation arises in frieze pat-terns (for the original sources, see [21–23]).

Studying max-type equations and systems of differenceequations is another topic of a recent interest (see, e.g, [2, 3, 5–7, 10, 11, 15–19]).

Some special cases of the following max-type differenceequation:

𝑥𝑛 = max{𝐴𝑛

𝑥𝑛−𝑠, 𝑥𝑛−𝑘} , 𝑛 ∈ N0, (8)

where 𝑠, 𝑘 ∈ N, and (𝐴𝑛)𝑛∈N0

⊂ R, have been studied, forexample, in [2, 16]. Positive solutions of (8) are periodic inmany cases. However, if (𝐴𝑛)𝑛∈N

0

is not a positive sequence, itwas shown in [2] that (8) can have unbounded solutions.

In [5], it was shown that all solutions of the followingmax-type system of difference equations:

𝑥𝑛+1 = max{𝐴𝑛

𝑦𝑛, 𝑥𝑛−1} , 𝑐𝑦𝑛+1 = max{

𝐵𝑛

𝑥𝑛, 𝑦𝑛−1} ,

𝑛 ∈ N0,

(9)

where 𝑥0, 𝑥−1, 𝑦0, 𝑦−1 ∈ (0, +∞) and (𝐴𝑛)𝑛∈N0

, (𝐵𝑛)𝑛∈N0

are positive two-periodic sequences, are eventually periodicwith, not necessarily prime, period two. This was done bydirect calculation.

By using direct calculation, it can be easily shown thatpositive solutions of the following max-type system of differ-ence equations:

𝑥𝑛+1 = max{𝐴𝑛

𝑥𝑛, 𝑦𝑛−1} , 𝑦𝑛+1 = max{

𝐵𝑛

𝑦𝑛, 𝑥𝑛−1} ,

𝑛 ∈ N0,

(10)

where (𝐴𝑛)𝑛∈N0

and (𝐵𝑛)𝑛∈N0

are positive two-periodicsequences, are also periodic.

Here, we give a noncalculatory explanation of the factby proving that positive solutions of the following max-typesystem of difference equations:


𝑥𝑛−𝑠, 𝑦𝑛−𝑘} , 𝑦𝑛 = max{

𝐵𝑛

𝑦𝑛−𝑠, 𝑥𝑛−𝑘} ,

𝑛 ∈ N0,

(11)



are positive periodicsequences of a certain period, are also periodic. We alsopresent another extension of the result.

2. Some Extensions of Systems (3)–(6)In this section, we present some periodic systems of differ-ence equations in the spirit of systems (3)–(6).

Theorem 1. Consider the following system of difference equa-tions

𝑥(1)

𝑛+1 = 𝑓−1

1 (1 + 𝑓2 (𝑥

(2)𝑛 )

𝑓3 (𝑥(3)𝑛−1)

) ,

...

𝑥(𝑘−1)

𝑛+1 = 𝑓−1

𝑘−1(1 + 𝑓𝑘 (𝑥

(𝑘)𝑛 )

𝑓1 (𝑥(1)𝑛−1)

) ,

𝑥(𝑘)

𝑛+1 = 𝑓−1

𝑘 (1 + 𝑓1 (𝑥

(1)𝑛 )

𝑓2 (𝑥(2)𝑛−1)

) , 𝑛 ∈ N0,

(12)

where 𝑘 ∈ N\ {1}, and functions 𝑓𝑗, 𝑗 = 1, 𝑘, are continuous ontheir domains; map the setR\{0} onto itself and, for each fixed𝑗 ∈ {1, . . . , 𝑘}, 𝑓𝑗 is simultaneously increasing or decreasing onthe intervals (−∞, 0) and (0, +∞).

Then the following statements hold.

(a) If 𝑘 ≡ 0 (mod 5), then every well-defined solution ofsystem (12) is periodic with period 5𝑘.

(b) If 𝑘 ≡ 0 (mod5), then every well-defined solution ofsystem (12) is periodic with period 𝑘.

Proof. From the conditions of the theorem, it follows that foreach 𝑗 ∈ {1, . . . , 𝑘}, there is 𝑓−1𝑗 which continuously map theset R \ {0} onto itself. Using the change of variables

𝑦(𝑗)

𝑛 = 𝑓𝑗 (𝑥(𝑗)

𝑛 ) , 𝑗 = 1, 𝑘, (13)

system (12) is easily transformed into the next one

𝑦(1)

𝑛+1 =1 + 𝑦(2)𝑛

𝑦(1)𝑛−1

, 𝑦(2)

𝑛+1 =1 + 𝑦(1)𝑛

𝑦(2)𝑛−1

, (14)

for 𝑘 = 2,

𝑦(1)

𝑛+1 =1 + 𝑦(2)𝑛

𝑦(3)𝑛−1

, 𝑦(2)

𝑛+1 =1 + 𝑦(3)𝑛

𝑦(1)𝑛−1

, 𝑦(3)

𝑛+1 =1 + 𝑦(1)𝑛

𝑦(2)𝑛−1

,

(15)

for 𝑘 = 3, and

𝑦(1)

𝑛+1 =1 + 𝑦(2)𝑛

𝑦(3)𝑛−1

, 𝑦(2)

𝑛+1 =1 + 𝑦(3)𝑛

𝑦(4)𝑛−1

, . . . , 𝑦(𝑘)

𝑛+1 =1 + 𝑦(1)𝑛

𝑦(2)𝑛−1

,

(16)

for 𝑘 ≥ 4. In [4], it was proved that, if 𝑘 ≡ 0 (mod 5), thenevery well-defined solution of systems (14)–(16) is periodic

Abstract and Applied Analysis 3

with period 5𝑘, and, if 𝑘 ≡ 0 (mod5), then every well-defined solution of systems (14)–(16) is periodic with period𝑘. Using this along with the fact

𝑥(𝑗)

𝑛 = 𝑓−1

𝑗 (𝑦(𝑗)

𝑛 ) , 𝑗 = 1, 𝑘, (17)

following from (13), the results in (a) and (b) follow.

The following theorem is proved in a similar way. There-fore, the proof will be omitted.

Theorem 2. Consider the following system of difference equa-tions

𝑥(1)

𝑛+1 = 𝑓−1

1 (1 + 𝑓𝑘 (𝑥

(𝑘)𝑛 )

𝑓𝑘−1 (𝑥(𝑘−1)𝑛−1 )

) ,

𝑥(2)

𝑛+1 = 𝑓−1

2 (1 + 𝑓1 (𝑥

(1)𝑛 )

𝑓𝑘 (𝑥(𝑘)𝑛−1)

) ,

...

𝑥(𝑘)

𝑛+1 = 𝑓−1

𝑘 (1 + 𝑓𝑘−1 (𝑥

(𝑘−1)𝑛 )

𝑓𝑘−2 (𝑥(𝑘−2)𝑛−1 )

) , 𝑛 ∈ N0,

(18)

where 𝑘 ∈ N\ {1}, and functions 𝑓𝑗, 𝑗 = 1, 𝑘, are continuous ontheir domains; map the setR\{0} onto itself and, for each fixed𝑗 ∈ {1, . . . , 𝑘}, 𝑓𝑗 is simultaneously increasing or decreasing onthe intervals (−∞, 0) and (0, +∞).

Then the following statements hold.

(a) If 𝑘 ≡ 0 (mod 5), then every well-defined solution ofsystem (18) is periodic with period 5𝑘.

(b) If 𝑘 ≡ 0 (mod5), then every well-defined solution ofsystem (18) is periodic with period 𝑘.

Now, we show that all the results on the periodicity of thesolutions of systems (3)–(6) in [19] follow from Theorems 1and 2.

Corollary 3. Systems of difference equations (3)–(6) are allperiodic with period ten.

Proof. For the systems of difference equations (3)–(6), we usethe following changes of variables, respectively:

𝑥𝑛 =1

𝑢𝑛, 𝑦𝑛 =

1

V𝑛, 𝑛 ∈ N0;

𝑥𝑛 = −1

𝑢𝑛, 𝑦𝑛 = −

1

V𝑛, 𝑛 ∈ N0;

𝑥𝑛 = −1

𝑢𝑛, 𝑦𝑛 =

1

V𝑛, 𝑛 ∈ N0;

𝑥𝑛 =1

𝑢𝑛, 𝑦𝑛 = −

1

V𝑛, 𝑛 ∈ N0.

(19)

By using them, systems (3)–(6) are transformed into system(14). By applying Theorem 1(a), ten-periodicity of all well-defined solutions of system (14) follows, from which ten-periodicity of all well-defined solutions of systems (3)–(6)follows.

The following four examples are natural extensions ofsystems (3)–(6).

Example 4. If we use the following functions:

𝑓1 (𝑥) =1

𝑥𝑙, 𝑓2 (𝑥) =

1

𝑥𝑚, (20)

where 𝑙 and𝑚 are odd integers, we see that all the conditionsinTheorem 1 are applied with 𝑘 = 2, so by using the theoremwe obtain that the system of difference equations

𝑥𝑛+1 =1

𝑥𝑛−1(

𝑦𝑚𝑛

1 + 𝑦𝑚𝑛)

1/𝑙

,

𝑦𝑛+1 =1

𝑦𝑛−1(

𝑥𝑙𝑛

1 + 𝑥𝑙𝑛)

1/𝑚

,

(21)

𝑛 ∈ N0, is also periodic with period ten.

Example 5. For

𝑓1 (𝑥) = −1

𝑥𝑙, 𝑓2 (𝑥) = −

1

𝑥𝑚, (22)

where 𝑙 and 𝑚 are also odd integers, all the conditions ofTheorem 1 are again satisfied with 𝑘 = 2. Using the theorem,it follows that the system

𝑥𝑛+1 =1

𝑥𝑛−1(

𝑦𝑚𝑛

−1 + 𝑦𝑚𝑛)

1/𝑙

,

𝑦𝑛+1 =1

𝑦𝑛−1(

𝑥𝑙𝑛

−1 + 𝑥𝑙𝑛)

1/𝑚

,

(23)

𝑛 ∈ N0, is ten-periodic.

Example 6. For

𝑓1 (𝑥) =1

𝑥𝑙, 𝑓2 (𝑥) = −

1

𝑥𝑚, (24)

where 𝑙 and 𝑚 are odd integers, all the conditions ofTheorem 1 are also satisfiedwith 𝑘 = 2. Applying the theorem,it follows that the system

𝑥𝑛+1 =1

𝑥𝑛−1(

𝑦𝑚𝑛

−1 + 𝑦𝑚𝑛)

1/𝑙

,

𝑦𝑛+1 =1

𝑦𝑛−1(

𝑥𝑙𝑛

1 + 𝑥𝑙𝑛)

1/𝑚

,

(25)


Example 7. Finally, for

𝑓1 (𝑥) = −1

𝑥𝑙, 𝑓2 (𝑥) =

1

𝑥𝑚, (26)


where 𝑙 and𝑚 are odd integers and applying Theorem 1 with𝑘 = 2, we get that the system

𝑥𝑛+1 =1

𝑥𝑛−1(

𝑦𝑚𝑛

1 + 𝑦𝑚𝑛)

1/𝑙

,

𝑦𝑛+1 =1

𝑦𝑛−1(

𝑥𝑙𝑛

−1 + 𝑥𝑙𝑛)

1/𝑚

,

(27)


Themain results in [4] can be relatively easily extended toa very general situation, which have been noticed by Iricaninand Stevic soon after publishing [4], and later also proved byseveral other authors. Namely, the following result holds (see,e.g., [1]).

Theorem 8. Assume that the following difference equation

𝑥𝑛 = 𝑓 (𝑥𝑛−1, . . . , 𝑥𝑛−𝑘) , 𝑛 ∈ N0, (28)

is periodic with period 𝑝.Then the following system of difference equations

𝑥(𝑖)

𝑛 = 𝑓(𝑥(𝜎(𝑖))

𝑛−1 , 𝑥(𝜎[2](𝑖))

𝑛−2 , . . . , 𝑥(𝜎[𝑘](𝑖))

𝑛−𝑘) ,

𝑖 = 1, 𝑙, 𝑛 ∈ N0,

(29)

where 𝜎(𝑖) = 𝑖 + 1, for 1 ≤ 𝑖 ≤ 𝑙 − 1, 𝜎(𝑙) = 1 and 𝜎[𝑗](𝑖) =𝜎(𝜎[𝑗−1](𝑖)), 𝑗 = 1, 𝑘, and 𝜎[0](𝑖) = 𝑖, 𝑖 = 1, 𝑙, is periodic withperiod lcm(𝑝, 𝑙) (the least common multiple of numbers 𝑝and 𝑙).

Theorem 8 can be used in constructing numerous peri-odic cyclic systems of difference equations based on scalarperiodic difference equations, which, with some changes ofvariables, give some other periodic cyclic systems of differ-ence equations.

3. Periodicity of Positive Solutionsof System (11)

In this section, we study positive solutions of system (11). Bygcd(𝑠, 𝑘), we denote the greatest common divisor of naturalnumbers 𝑠 and 𝑘.

Theorem 9. Consider system (11). Assume that 𝑠, 𝑘 ∈ N,and (𝐴𝑛)𝑛∈N

0

and (𝐵𝑛)𝑛∈N0

are positive 𝑘 gcd(𝑠, 𝑘)-periodicsequences.Then every positive solution of system (11) is periodicwith, not necessarily prime, period

𝑝 = 2𝑘 gcd(𝑠, 𝑘) . (30)

Proof. Let 𝑟 = gcd(𝑠, 𝑘).Thenwehave that 𝑠 = 𝑟𝑠1 and 𝑘 = 𝑟𝑘1for some 𝑠1, 𝑘1 ∈ N such that

gcd (𝑠1, 𝑘1) = 1. (31)

Since every 𝑛 ∈ N0 can be written as 𝑛 = 𝑚𝑟 + 𝑖, for some𝑚 ∈ N0 and 𝑖 = 0, 𝑟 − 1, system (11) becomes

𝑥𝑚𝑟+𝑖 = max{𝐴𝑚𝑟+𝑖

𝑥𝑟(𝑚−𝑠1)+𝑖

, 𝑦𝑟(𝑚−𝑘1)+𝑖} ,

𝑦𝑚𝑟+𝑖 = max{𝐵𝑚𝑟+𝑖

𝑦𝑟(𝑚−𝑠1)+𝑖

, 𝑥𝑟(𝑚−𝑘1)+𝑖} ,

(32)

for every𝑚 ∈ N0 and 𝑖 = 0, 𝑟 − 1.Using the next change of variables

𝑥(𝑖)

𝑡 = 𝑥𝑡𝑟+𝑖, 𝑦(𝑖)

𝑡 = 𝑦𝑡𝑟+𝑖, (33)

where 𝑡 ≥ −max{𝑠1, 𝑘1}, 𝑖 = 0, 𝑟 − 1, in (32), we have that(𝑥(𝑖)𝑡 )𝑡≥−max{𝑠

1,𝑘1}, (𝑦(𝑖)𝑡 )𝑡≥−max{𝑠

1,𝑘1}, 𝑖 = 0, 𝑟 − 1, are 𝑟 indepen-

dent solutions of the next systems

𝑥𝑡 = max{𝐴 𝑡𝑟+𝑖

𝑥𝑡−𝑠1

, 𝑦𝑡−𝑘1

} , 𝑦𝑡 = max{𝐵𝑡𝑟+𝑖

𝑦𝑡−𝑠1

, 𝑥𝑡−𝑘1

} ,

(34)

which are systems of the form in (11) with 𝑠1 and 𝑘1 instead of𝑠 and 𝑘, and where the sequences (𝐴 𝑡𝑟+𝑖)𝑡∈N

0

and (𝐵𝑡𝑟+𝑖)𝑡∈N0

,𝑖 = 1, 𝑟, are 𝑘-periodic.

Hence, it is enough to prove the theoremwhen gcd(𝑠, 𝑘) =1 and the sequences (𝐴𝑛)𝑛∈N

0

, and (𝐵𝑛)𝑛∈N0

are positive 𝑘-periodic.

Now note that from the equations in (11), we have that

𝑥𝑛 ≥ 𝑦𝑛−𝑘, 𝑦𝑛 ≥ 𝑥𝑛−𝑘, for 𝑛 ∈ N0. (35)

Further, by using the equations in (11), we also get


𝑥𝑛−𝑠, 𝑦𝑛−𝑘} = max{

𝐴𝑛

𝑥𝑛−𝑠,𝐵𝑛−𝑘

𝑦𝑛−𝑘−𝑠, 𝑥𝑛−2𝑘} ,

𝑦𝑛 = max{𝐵𝑛

𝑦𝑛−𝑠, 𝑥𝑛−𝑘} = max{

𝐵𝑛

𝑦𝑛−𝑠,𝐴𝑛−𝑘

𝑥𝑛−𝑘−𝑠, 𝑦𝑛−2𝑘} ,

(36)

for 𝑛 ≥ 𝑘.Using relations (36), we get



𝑦𝑛−𝑘−𝑠, 𝑥𝑛−2𝑘}

= max{𝐴𝑛


𝑦𝑛−𝑘−𝑠,𝐴𝑛−2𝑘

𝑥𝑛−2𝑘−𝑠,𝐵𝑛−3𝑘

𝑦𝑛−3𝑘−𝑠, 𝑥𝑛−4𝑘} ,

𝑦𝑛 = max{𝐵𝑛


𝑥𝑛−𝑘−𝑠, 𝑦𝑛−2𝑘}

= max{𝐵𝑛


𝑥𝑛−𝑘−𝑠,𝐵𝑛−2𝑘

𝑦𝑛−2𝑘−𝑠,𝐴𝑛−3𝑘

𝑥𝑛−3𝑘−𝑠, 𝑦𝑛−4𝑘} ,

(37)

for 𝑛 ≥ 3𝑘.Now, note that, from the inequalities in (35), we have that

𝑥𝑛 ≥ 𝑥𝑛−2𝑘, 𝑦𝑛 ≥ 𝑦𝑛−2𝑘, for 𝑛 ≥ 𝑘. (38)

Abstract and Applied Analysis 5

Using (38) and 𝑘-periodicity of the sequences𝐴𝑛 and 𝐵𝑛,we obtain

𝐴𝑛

𝑥𝑛−𝑠=𝐴𝑛−2𝑘

𝑥𝑛−𝑠≤𝐴𝑛−2𝑘

𝑥𝑛−2𝑘−𝑠,

𝐵𝑛−𝑘

𝑦𝑛−𝑘−𝑠=𝐵𝑛−3𝑘

𝑦𝑛−𝑘−𝑠≤𝐵𝑛−3𝑘

𝑦𝑛−3𝑘−𝑠,

𝐵𝑛

𝑦𝑛−𝑠=𝐵𝑛−2𝑘

𝑦𝑛−𝑠≤𝐵𝑛−2𝑘

𝑦𝑛−2𝑘−𝑠,

𝐴𝑛−𝑘

𝑥𝑛−𝑘−𝑠=𝐴𝑛−3𝑘

𝑥𝑛−𝑘−𝑠≤𝐴𝑛−3𝑘

𝑥𝑛−3𝑘−𝑠.

(39)

Employing (39) into (37), we get

𝑥𝑛 = max{𝐴𝑛−2𝑘

𝑥𝑛−2𝑘−𝑠,𝐵𝑛−3𝑘

𝑦𝑛−3𝑘−𝑠, 𝑥𝑛−4𝑘} = 𝑥𝑛−2𝑘,

𝑦𝑛 = max{𝐵𝑛−2𝑘

𝑦𝑛−2𝑘−𝑠,𝐴𝑛−3𝑘

𝑥𝑛−3𝑘−𝑠, 𝑦𝑛−4𝑘} = 𝑦𝑛−2𝑘,

(40)

from which it follows that in this case the solutions of system(11) are 2𝑘-periodic. From all the above, the theorem follows.

By a slight modification of the proof of Theorem 9, thenext result can be proved. We omit the proof.

Theorem 10. Consider the following system of differenceequations


𝑥𝑛−𝑠, 𝑦𝑛−𝑘} , 𝑦𝑛 = max{

𝐵𝑛

𝑦𝑛−𝑠, 𝑧𝑛−𝑘} ,

𝑧𝑛 = max{𝐶𝑛

𝑧𝑛−𝑠, 𝑥𝑛−𝑘} , 𝑛 ∈ N0,

(41)



, and (𝐶𝑛)𝑛∈N0

are pos-itive 𝑘 gcd(𝑠, 𝑘)-periodic sequences. Then, every positive solu-tion of system (41) is periodic with, not necessarily prime,period

𝑝 = 3𝑘 gcd(𝑠, 𝑘) . (42)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was funded by the Deanship of Scientific Research(DSR), King Abdulaziz University, under Grant no. (11-130/1433 HiCi). The authors, therefore, acknowledge techni-cal and financial support of KAU.

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