The form of solutions and periodic nature for some rational difference equations systems

Research Article

The form of solutions and periodic nature for some rational difference equations systems

M. M. El-Dessoky

^a,b,∗

, E. M. Elsayed

^a,b

, E. O. Alzahrani

^b

aKing Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia.

bDepartment of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

Communicated by N. Shahzad

Abstract

In this paper, we investigate the expressions of solutions and the periodic nature of the following systems of rational difference equations with order four

x

_n+1

=

_±1±y ^yn−3

nzn−1xn−2yn−3

, y

_n+1

=

_±1±z ^zn−3

nxn−1yn−2zn−3

, z

_n+1

=

_±1±x ^xn−3

nyn−1zn−2xn−3

,

with initial conditions x

−3

, x

−2

, x

−1

, x

0

, y

−3

, y

−2

, y

−1

, y

0

, z

−3

, z

−2

, z

−1

and z

0

which are arbitrary real numbers. c 2016 All rights reserved.

Keywords: Difference equations, recursive sequences, stability, periodic solution, system of difference equations.

2010 MSC: 39A10.

1. Introduction

The goal of this paper is to obtain the form of the solutions and the periodicity character of some systems of rational difference equations

x

n+1

=

_±1±y ^yn−3

nzn−1xn−2yn−3

, y

n+1

=

_±1±z ^zn−3

nxn−1yn−2zn−3

, z

n+1

=

_±1±x ^xn−3

nyn−1zn−2xn−3

, with initial conditions, which are arbitrary real numbers.

∗

Corresponding author

Email addresses:

[email protected]

(M. M. El-Dessoky),

[email protected]

(E. M. Elsayed),

[email protected]

(E. O. Alzahrani)

Received 2016-08-04

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. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solution, see [1–4, 6–8, 11, 13, 15, 19, 20, 22, 24, 25, 27, 31, 32] and the references cited therein.

Recently, a great effort has been made in studying the qualitative analysis of rational difference equations and rational difference equations system, see [4–31, 33, 34].

Din et al. [8] investigated the qualitative behavior of the following competitive system of rational difference equations

x

n+1

=

^α1_a^+β1xn−1

1+b1yn

, y

n+1

=

^α2_a^+β2yn−1

2+b2xn

.

Elsayed et al. [15] studied the form of the solutions and the periodicity of the following rational systems of rational difference equations

x

n+1

=

_1−x^xn−5

n−5yn−2

, y

n+1

=

_±1±y^yn−5

n−5xn−2

.

Grove et al. [19] studied the existence and behavior of solution of the rational system x

_n+1

=

_x^a

n

+

_y^b

n

, y

_n+1

=

_x^c

n

+

_y^d

n

. The behavior of positive solutions of the following system

x

_n+1

=

_1+x^xn−1

n−1yn

, y

_n+1

=

_1+y^yn−1

n−1xn

, were studied by Kurbanli et al. in [24].

Ozban [25] has investigated the positive solution of the system of rational difference equations ¨ x

_n+1

=

_y^a

n−3

, y

_n+1

=

_x^byn−3

n−qyn−q

.

Also, Touafek et al. [27] studied the periodicity and gave the form of the solutions of the following systems

x

_n+1

=

_x ^yn

n−1(±1±y_n)

, y

_n+1

=

_y ^xn

n−1(±1±x_n)

.

Elabbasy et al. [11] obtained the solution of particular cases of the following general system of difference equations

x

n+1

=

_a ^a1+a2yn

3zn+a4xn−1zn

, y

n+1

=

_b ^{b1zn−1+b2zn}

3xnyn+b4xnyn−1

, z

n+1

=

_c ^{c1zn−1+c2zn}

3xn−1yn−1+c4xn−1yn+c5xnyn

. 2. On the systems: x

_n+1

=

^yn−3

1±ynz_n−1x_n−2y_n−3

, y

_n+1

=

^zn−3

1±znx_n−1y_n−2z_n−3

, z

_n+1

=

^xn−3

1±xny_n−1z_n−2x_n−3

In this section, we study the solutions of the system of three difference equations in the following form x

n+1

=

_1+y ^yn−3

nzn−1xn−2yn−3

, y

n+1

=

_1+z ^zn−3

nxn−1yn−2zn−3

, z

_n+1

=

_1+x ^xn−3

nyn−1zn−2xn−3

, n = 0, 1, ..., (2.1) with nonzero real initial conditions x

−3

, x

−2

, x

−1

, x

₀

, y

−3

, y

−2

, y

−1

, y

₀

, z

−3

, z

−2

, z

−1

, z

₀

.

Theorem 2.1. Suppose that {x

_n

, y

n

, z

n

} are solutions of system (2.1), then for n = 0, 1, 2, ..., we see that x

12n−3

= a

n−1

Q

i=0

(1+12iadgl)(1+(12i+4)adgl)(1+(12i+8)adgl) (1+(12i+1)adgl)(1+(12i+5)adgl)(1+(12i+9)adgl)

, x

12n−2

= b

n−1

Q

i=0

(1+(12i+1)behm)(1+(12i+5)behm)(1+(12i+9)behm) (1+(12i+2)behm)(1+(12i+6)behm)(1+(12i+10)behm)

,

x

12n−1

= c

n−1

Q

i=0

(1+(12i+2)cf ko)(1+(12i+6)cf ko)(1+(12i+10)cf ko) (1+(12i+3)cf ko)(1+(12i+7)cf ko)(1+(12i+11)cf ko)

, x

12n

= d

n−1

Q

i=0

(1+(12i+3)adgl)(1+(12i+7)adgl)(1+(12i+11)adgl) (1+(12i+4)adgl)(1+(12i+8)adgl)(1+(12i+12)adgl)

, x

_12n+1

=

_(1+behm)^e

n−1

Q

i=0

(1+(12i+4)behm)(1+(12i+8)behm)(1+(12i+12)behm) (1+(12i+5)behm)(1+(12i+9)behm)(1+(12i+13)behm)

, x

_12n+2

=

^f_{(1+2cf ko)}^{(1+cf ko)}

n−1

Q

i=0

(1+(12i+5)cf ko)(1+(12i+9)cf ko)(1+(12i+13)cf ko) (1+(12i+6)cf ko)(1+(12i+10)cf ko)(1+(12i+14)cf ko)

, x

12n+3

=

^g_(1+3adgl)^(1+2adgl)

n−1

Q

i=0

(1+(12i+6)adgl)(1+(12i+10)adgl)(1+(12i+14)adgl) (1+(12i+7)adgl)(1+(12i+11)adgl)(1+(12i+15)adgl)

, x

12n+4

=

^h_(1+4behm)^(1+3behm)

n−1

Q

i=0

(1+(12i+7)behm)(1+(12i+11)behm)(1+(12i+15)behm) (1+(12i+8)behm)(1+(12i+12)behm)(1+(12i+16)behm)

, x

_12n+5

=

(1+cf ko)(1+5cf ko)^{k(1+4cf ko)}

n−1

Q

i=0

(1+(12i+8)cf ko)(1+(12i+12)cf ko)(1+(12i+16)cf ko) (1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

, x

_12n+6

=

^l (1+adgl)(1+5adgl)

(1+2adgl)(1+6adgl) n−1

Q

i=0

(1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl) (1+(12i+10)adgl)(1+(12i+14)adgl)(1+(12i+18)adgl)

, x

_12n+7

=

^m (1+2behm)(1+6behm)

(1+3behm)(1+7behm) n−1

Q

i=0

(1+(12i+10)behm)(1+(12i+14)behm)(1+(12i+18)behm) (1+(12i+11)behm)(1+(12i+15)behm)(1+(12i+19)behm)

, x

12n+8

=

^o (1+3cf ko)(1+7cf ko)

(1+4cf ko)(1+8cf ko) n−1

Q

i=0

(1+(12i+11)cf ko)(1+(12i+15)cf ko)(1+(12i+19)cf ko) (1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)

, y

12n−3

= e

n−1

Q

i=0

(1+(12i)behm)(1+(12i+4)behm)(1+(12i+8)behm) (1+(12i+1)behm)(1+(12i+5)behm)(1+(12i+9)behm)

, y

12n−2

= f

n−1

Q

i=0

(1+(12i+1)cf ko)(1+(12i+5)cf ko)(1+(12i+9)cf ko) (1+(12i+2)cf ko)(1+(12i+6)cf ko)(1+(12i+10)cf ko)

, y

12n−1

= g

n−1

Q

i=0

(1+(12i+2)adgl)(1+(12i+6)adgl)(1+(12i+10)adgl) (1+(12i+3)adgl)(1+(12i+7)adgl)(1+(12i+11)adgl)

, y

12n

= h

n−1

Q

i=0

(1+(12i+3)behm)(1+(12i+7)behm)(1+(12i+11)behm) (1+(12i+4)behm)(1+(12i+8)behm)(1+(12i+12)behm)

, y

12n+1

=

_{(1+cf ko)}^k

n−1

Q

i=0

(1+(12i+4)cf ko)(1+(12i+8)cf ko)(1+(12i+12)cf ko) (1+(12i+5)cf ko)(1+(12i+9)cf ko)(1+(12i+13)cf ko)

, y

_12n+2

=

^l_(1+2adgl)^(1+adgl)

n−1

Q

i=0

(1+(12i+5)adgl)(1+(12i+9)adgl)(1+(12i+13)adgl) (1+(12i+6)adgl)(1+(12i+10)adgl)(1+(12i+14)adgl)

, y

_12n+3

=

^m_(1+3behm)^(1+2behm)

n−1

Q

i=0

(1+(12i+6)behm)(1+(12i+10)behm)(1+(12i+14)behm) (1+(12i+7)behm)(1+(12i+11)behm)(1+(12i+15)behm)

, y

_12n+4

=

^o_{(1+4cf ko)}^{(1+3cf ko)}

n−1

Q

i=0

(1+(12i+7)cf ko)(1+(12i+11)cf ko)(1+(12i+15)cf ko) (1+(12i+8)cf ko)(1+(12i+12)cf ko)(1+(12i+16)cf ko)

, y

12n+5

=

(1+adgl)(1+5adgl)^{a (1+4adgl)}

n−1

Q

i=0

(1+(12i+8)adgl)(1+(12i+12)adgl)(1+(12i+16)adgl) (1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

, y

12n+6

=

^b (1+behm)(1+5behm)

(1+2behm)(1+6behm) n−1

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm) (1+(12i+10)behm)(1+(12i+14)behm)(1+(12i+18)behm)

, y

_12n+7

=

^c(1+2cf ko)(1+6cf ko)

(1+3cf ko)(1+7cf ko) n−1

Q

i=0

(1+(12i+10)cf ko)(1+(12i+14)cf ko)(1+(12i+18)cf ko) (1+(12i+11)cf ko)(1+(12i+15)cf ko)(1+(12i+19)cf ko)

,

y

12n+8

=

^d(1+3adgl)(1+7adgl) (1+4adgl)(1+8adgl)

n−1

Q

i=0

(1+(12i+11)adgl)(1+(12i+15)adgl)(1+(12i+19)adgl) (1+(12i+12)adgl)(1+(12i+16)adgl)(1+(12i+20)adgl)

, and

z

12n−3

= k

n−1

Q

i=0

(1+(12i)cf ko)(1+(12i+4)cf ko)(1+(12i+8)cf ko) (1+(12i+1)cf ko)(1+(12i+5)cf ko)(1+(12i+9)cf ko)

, z

12n−2

= l

n−1

Q

i=0

(1+(12i+1)adgl)(1+(12i+5)adgl)(1+(12i+9)adgl) (1+(12i+2)adgl)(1+(12i+6)adgl)(1+(12i+10)adgl)

, z

12n−1

= m

n−1

Q

i=0

(1+(12i+2)behm)(1+(12i+6)behm)(1+(12i+10)behm) (1+(12i+3)behm)(1+(12i+7)behm)(1+(12i+11)behm)

, z

12n

= o

n−1

Q

i=0

(1+(12i+3)cf ko)(1+(12i+7)cf ko)(1+(12i+11)cf ko) (1+(12i+4)cf ko)(1+(12i+8)cf ko)(1+(12i+12)cf ko)

, z

_12n+1

=

_(1+adgl)^a

n−1

Q

i=0

(1+(12i+4)adgl)(1+(12i+8)adgl)(1+(12i+12)adgl) (1+(12i+5)adgl)(1+(12i+9)adgl)(1+(12i+13)adgl)

, z

_12n+2

=

^b_(1+2behm)^(1+behm)

n−1

Q

i=0

(1+(12i+5)behm)(1+(12i+9)behm)(1+(12i+13)behm) (1+(12i+6)behm)(1+(12i+10)behm)(1+(12i+14)behm)

, z

12n+3

=

^c_{(1+3cf ko)}^{(1+2cf ko)}

n−1

Q

i=0

(1+(12i+6)cf ko)(1+(12i+10)cf ko)(1+(12i+14)cf ko) (1+(12i+7)cf ko)(1+(12i+11)cf ko)(1+(12i+15)cf ko)

, z

12n+4

=

^d_(1+4adgl)^(1+3adgl)

n−1

Q

i=0

(1+(12i+7)adgl)(1+(12i+11)adgl)(1+(12i+15)adgl) (1+(12i+8)adgl)(1+(12i+12)adgl)(1+(12i+16)adgl)

, z

12n+5

=

(1+behm)(1+5behm)^{e (1+4behm)}

n−1

Q

i=0

(1+(12i+8)behm)(1+(12i+12)behm)(1+(12i+16)behm) (1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

, z

_12n+6

=

^f (1+cf ko)(1+5cf ko)

(1+2cf ko)(1+6cf ko) n−1

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko) (1+(12i+10)cf ko)(1+(12i+14)cf ko)(1+(12i+18)cf ko)

, z

_12n+7

=

^g (1+2adgl)(1+6adgl)

(1+3adgl)(1+7adgl) n−1

Q

i=0

(1+(12i+10)adgl)(1+(12i+14)adgl)(1+(12i+18)adgl) (1+(12i+11)adgl)(1+(12i+15)adgl)(1+(12i+19)adgl)

, z

12n+8

=

^h (1+3behm)(1+7behm)

(1+4behm)(1+8behm) n−1

Q

i=0

(1+(12i+11)behm)(1+(12i+15)behm)(1+(12i+19)behm) (1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)

,

where x

−3

= a, x

−2

= b, x

−1

= c, x

₀

= d, y

−3

= e, y

−2

= f, y

−1

= g, y

₀

= h, z

−3

= k, z

−2

= l, z

−1

= m, z

₀

= o and

−1

Q

i=0

A

_i

= 1.

Proof. For n = 0, the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. Then we have

x

12n−7

=

(1+cf ko)(1+5cf ko)^{k(1+4cf ko)} n−2

Q

i=0

(1+(12i+8)cf ko)(1+(12i+12)cf ko)(1+(12i+16)cf ko) (1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

, x

12n−6

=

^l(1+adgl)(1+5adgl)

(1+2adgl)(1+6adgl) n−2

Q

i=0

(1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl) (1+(12i+10)adgl)(1+(12i+14)adgl)(1+(12i+18)adgl)

, x

12n−5

=

^m (1+2behm)(1+6behm)

(1+3behm)(1+7behm) n−2

Q

i=0

(1+(12i+10)behm)(1+(12i+14)behm)(1+(12i+18)behm) (1+(12i+11)behm)(1+(12i+15)behm)(1+(12i+19)behm)

, x

12n−4

=

^o (1+3cf ko)(1+7cf ko)

(1+4cf ko)(1+8cf ko) n−2

Q

i=0

(1+(12i+11)cf ko)(1+(12i+15)cf ko)(1+(12i+19)cf ko) (1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)

, y

12n−7

=

(1+adgl)(1+5adgl)^a(1+4adgl)

n−2

Q

i=0

(1+(12i+8)adgl)(1+(12i+12)adgl)(1+(12i+16)adgl) (1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

,

y

12n−6

=

^b (1+behm)(1+5behm) (1+2behm)(1+6behm)

n−2

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm) (1+(12i+10)behm)(1+(12i+14)behm)(1+(12i+18)behm)

, y

12n−5

=

^c(1+2cf ko)(1+6cf ko)

(1+3cf ko)(1+7cf ko) n−2

Q

i=0

(1+(12i+10)cf ko)(1+(12i+14)cf ko)(1+(12i+18)cf ko) (1+(12i+11)cf ko)(1+(12i+15)cf ko)(1+(12i+19)cf ko)

, y

12n−4

=

^d(1+3adgl)(1+7adgl)

(1+4adgl)(1+8adgl) n−2

Q

i=0

(1+(12i+11)adgl)(1+(12i+15)adgl)(1+(12i+19)adgl) (1+(12i+12)adgl)(1+(12i+16)adgl)(1+(12i+20)adgl)

, z

12n−7

=

(1+behm)(1+5behm)^{e (1+4behm)}

n−2

Q

i=0

(1+(12i+8)behm)(1+(12i+12)behm)(1+(12i+16)behm) (1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

, z

12n−6

=

^f (1+cf ko)(1+5cf ko)

(1+2cf ko)(1+6cf ko) n−2

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko) (1+(12i+10)cf ko)(1+(12i+14)cf ko)(1+(12i+18)cf ko)

, z

12n−5

=

^g (1+2adgl)(1+6adgl)

(1+3adgl)(1+7adgl) n−2

Q

i=0

(1+(12i+10)adgl)(1+(12i+14)adgl)(1+(12i+18)adgl) (1+(12i+11)adgl)(1+(12i+15)adgl)(1+(12i+19)adgl)

, z

12n−4

=

^h (1+3behm)(1+7behm)

(1+4behm)(1+8behm) n−2

Q

i=0

(1+(12i+11)behm)(1+(12i+15)behm)(1+(12i+19)behm) (1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)

. Now, it follows from Eq. (2.1) that

x

12n−3

= y

12n−7

1 + y

12n−7

z

12n−5

x

12n−6

y

12n−4

=

a (1+4adgl) (1+adgl)(1+5adgl)

n−2

Q

i=0

(1+(12i+8)adgl)(1+(12i+12)adgl)(1+(12i+16)adgl) (1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

!





































1 +

(1+adgl)(1+5adgl)^{a (1+4adgl)} n−2

Q

i=0

(1+(12i+8)adgl)(1+(12i+12)adgl)(1+(12i+16)adgl) (1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

g (1+2adgl)(1+6adgl) (1+3adgl)(1+7adgl)

n−2

Q

i=0

(1+(12i+10)adgl)(1+(12i+14)adgl)(1+(12i+18)adgl) (1+(12i+11)adgl)(1+(12i+15)adgl)(1+(12i+19)adgl)

l(1+adgl)(1+5adgl) (1+2adgl)(1+6adgl)

n−2

Q

i=0

(1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl) (1+(12i+10)adgl)(1+(12i+14)adgl)(1+(12i+18)adgl)

d(1+3adgl)(1+7adgl) (1+4adgl)(1+8adgl)

n−2

Q

i=0

(1+(12i+11)adgl)(1+(12i+15)adgl)(1+(12i+19)adgl) (1+(12i+12)adgl)(1+(12i+16)adgl)(1+(12i+20)adgl)





































=

a(1+4adgl)(1+8adgl)

n−2

Q

i=0

(1+(12i+8)adgl)(1+(12i+12)adgl)(1+(12i+16)adgl)

!

(1+adgl)(1+5adgl)

n−2

Q

i=0

(1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

!

×

n−2

Q

i=0

(1+(12i+20)adgl)

!

(1+8adgl)

n−2

Q

i=0

(1+(12i+20)adgl)

! +adgl

n−2

Q

i=0

(1+(12i+8)adgl)

!

=

a(1+4adgl)(1+8adgl)

n−2

Q

i=0

(1+(12i+12)adgl)(1+(12i+16)adgl)(1+(12i+20)adgl)

!

(1+adgl)(1+5adgl)

n−2

Q

i=0

(1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

!

1

(1+(12n−4)adgl)+adgl

=

a(1+4adgl)(1+8adgl)

n−2

Q

i=0

(1+(12i+12)adgl)(1+(12i+16)adgl)(1+(12i+20)adgl)

!!

(1+adgl)(1+5adgl)(1+(12n−3)adgl)

n−2

Q

i=0

(1+(12i+9)adgl)(1+(12i+13)adgl)(1+(12i+17)adgl)

!!

= a

n−1

Q

i=0

(1+12iadgl)(1+(12i+4)adgl)(1+(12i+8)adgl) (1+(12i+1)adgl)(1+(12i+5)adgl)(1+(12i+9)adgl)

.

Also, we see that

y

4n−2

= z

12n−7

1 + z

12n−7

x

12n−5

y

12n−6

z

12n−4

,

y

4n−2

=

e (1+4behm) (1+behm)(1+5behm)

n−2

Q

i=0

(1+(12i+8)behm)(1+(12i+12)behm)(1+(12i+16)behm) (1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

!

1+





































e(1+4behm) (1+behm)(1+5behm)

n−2

Q

i=0

(1+(12i+8)behm)(1+(12i+12)behm)(1+(12i+16)behm) (1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

m(1+2behm)(1+6behm) (1+3behm)(1+7behm)

n−2

Q

i=0

(1+(12i+10)behm)(1+(12i+14)behm)(1+(12i+18)behm) (1+(12i+11)behm)(1+(12i+15)behm)(1+(12i+19)behm)

b(1+behm)(1+5behm) (1+2behm)(1+6behm)

n−2

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm) (1+(12i+10)behm)(1+(12i+14)behm)(1+(12i+18)behm)

h (1+3behm)(1+7behm) (1+4behm)(1+8behm)

n−2

Q

i=0

(1+(12i+11)behm)(1+(12i+15)behm)(1+(12i+19)behm) (1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)





































=

(e(1+4behm)(1+8behm))

n−2

Q

i=0

(1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)

!

(1+behm)(1+5behm)

n−2

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

!

×

n−2

Q

i=0

(1+(12i+8)behm)

!

((1+8behm))

n−2

Q

i=0

(1+(12i+20)behm)

! +behm

n−2

Q

i=0

(1+(12i+8)behm)

!

=

(e(1+4behm)(1+8behm))

n−2

Q

i=0

(1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)

!

(1+behm)(1+5behm)

n−2

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

!

×

n−2

Q

i=0

(1+(12i+8)behm)

!

n−1

Q

i=0

(1+(12i+8)behm)

! +behm

n−2

Q

i=0

(1+(12i+8)behm)

!

=

(e(1+4behm)(1+8behm))

n−2

Q

i=0

(1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)

!

(1+behm)(1+5behm)

n−2

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

!

×

n−2

Q

i=0

(1+(12i+8)behm)

!

n−2

Q

i=0

(1+(12i+8)behm)

!

((1+(12n−4)behm)+behm)

=

(e(1+4behm)(1+8behm))

n−2

Q

i=0

(1+(12i+12)behm)(1+(12i+16)behm)(1+(12i+20)behm)

!

(1+behm)(1+5behm)(1+(12n−3)behm)

n−2

Q

i=0

(1+(12i+9)behm)(1+(12i+13)behm)(1+(12i+17)behm)

!

= e

n−1

Q

i=0

(1+(12i)behm)(1+(12i+4)behm)(1+(12i+8)behm) (1+(12i+1)behm)(1+(12i+5)behm)(1+(12i+9)behm)

, and

z

12n−3

=

_1+x ^x12n−7

12n−7y12n−5z12n−6x12n−4

=

k(1+4cf ko) (1+cf ko)(1+5cf ko)

n−2

Q

i=0

(1+(12i+8)cf ko)(1+(12i+12)cf ko)(1+(12i+16)cf ko) (1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

!

1+





































k(1+4cf ko) (1+cf ko)(1+5cf ko)

n−2

Q

i=0

(1+(12i+8)cf ko)(1+(12i+12)cf ko)(1+(12i+16)cf ko) (1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

c(1+2cf ko)(1+6cf ko) (1+3cf ko)(1+7cf ko)

n−2

Q

i=0

(1+(12i+10)cf ko)(1+(12i+14)cf ko)(1+(12i+18)cf ko) (1+(12i+11)cf ko)(1+(12i+15)cf ko)(1+(12i+19)cf ko)

f (1+cf ko)(1+5cf ko) (1+2cf ko)(1+6cf ko)

n−2

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko) (1+(12i+10)cf ko)(1+(12i+14)cf ko)(1+(12i+18)cf ko)

o(1+3cf ko)(1+7cf ko) (1+4cf ko)(1+8cf ko)

n−2

Q

i=0

(1+(12i+11)cf ko)(1+(12i+15)cf ko)(1+(12i+19)cf ko) (1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)





































=

(k(1+4cf ko)(1+8cf ko))

n−2

Q

i=0

(1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)

!

(1+cf ko)(1+5cf ko)

n−2

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

!

×

n−2

Q

i=0

(1+(12i+8)cf ko)

!

n−1

Q

i=0

(1+(12i+8)cf ko)

! +(cf ko)

n−2

Q

i=0

(1+(12i+8)cf ko)

!

=

(k(1+4cf ko)(1+8cf ko))

n−2

Q

i=0

(1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)

!

(1+cf ko)(1+5cf ko)

n−2

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

!

×

n−2

Q

i=0

(1+(12i+8)cf ko)

!

(1+(12n−4)cf ko)

n−2

Q

i=0

(1+(12i+8)cf ko)

! +(cf ko)

n−2

Q

i=0

(1+(12i+8)cf ko)

!

=

(k(1+4cf ko)(1+8cf ko))

n−2

Q

i=0

(1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)

!

(1+cf ko)(1+5cf ko)

n−2

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

!

×

n−2

Q

i=0

(1+(12i+8)cf ko)

!

n−2

Q

i=0

(1+(12i+8)cf ko)

!

(1+(12n−4)cf ko+cf ko)

=

(k(1+4cf ko)(1+8cf ko))

n−2

Q

i=0

(1+(12i+12)cf ko)(1+(12i+16)cf ko)(1+(12i+20)cf ko)

!

(1+cf ko)(1+5cf ko)(1+(12n−3)cf ko)

n−2

Q

i=0

(1+(12i+9)cf ko)(1+(12i+13)cf ko)(1+(12i+17)cf ko)

!

,

z

12n−3

=

k

n−1

Q

i=0

(1+(12i)cf ko)(1+(12i+4)cf ko)(1+(12i+8)cf ko)

n−1

Q

i=0

(1+(12i+1)cf ko)(1+(12i+5)cf ko)(1+(12i+9)cf ko)

. Also, we can prove the other relations. This completes the proof.

Lemma 2.2. Let {x

_n

, y

n

, z

n

} be positive solutions of system (2.1), then {x

_n

}, {y

_n

} and {z

_n

} are bounded and converges to zero.

Proof. It follows from Eq. (2.1) that

x

n+1

= y

n−3

1 + y

n

z

n−1

x

n−2

y

n−3

< y

n−3

, y

n+1

= z

n−3

1 + z

n

x

n−1

y

n−2

z

n−3

< z

n−3

,

z

n+1

= x

n−3

1 + x

_n

y

n−1

z

n−2

x

n−3

< x

n−3

. Thus

x

n+5

< y

n+1

, y

n+5

< z

n+1

, z

n+5

< x

n+1

⇒ x

n+5

< z

n−3

, y

n+5

< x

n−3

, z

n+5

< y

n−3

⇒ x

n+9

< y

n+5

< x

n−3

, y

n+9

< z

n+5

< y

n−3

, z

n+9

< x

n+5

< z

n−3

.

Then the subsequences {x

_12n+i

}

^∞_n=0

, i = −3. − 2, −1, 0, 1, 2, ..., 8, are decreasing and bounded from above by M = max{x

₋₃

, x

−2

, x

−1

, x

0

, ..., x

8

}. Also, the subsequences {y

_12n+i

}

^∞_n=0

and {z

_12n+i

}

^∞_n=0

, i =

−3. − 2, −1, 0, 1, 2, ..., 8, are decreasing and bounded from above by L = max{y

₋₃

, y

−2

, ..., y

₈

} and N = max{z

−3

, z

−2

, ..., z

₈

}, respectively. This completes the proof.

Lemma 2.3. If x

i

, y

i

, z

i

, i = −3, −2, −1, 0, are arbitrary real numbers and let {x

_n

, y

n

, z

n

} be solutions of system (2.1), then the following statements are true.

(i) If x

−3

= a = 0, then we have x

12n−3

= y

_12n+5

= z

_12n+1

= 0, x

_12n

= y

_12n+8

= z

_12n+4

= d, x

_12n+6

= y

12n+2

= z

12n−2

= l and x

12n+3

= y

12n−1

= z

12n+7

= g.

(ii) If x

−2

= b = 0, then we have x

12n−2

= y

_12n+6

= z

_12n+2

= 0, x

_12n+1

= y

12n−3

= z

_12n+5

= e, x

12n+4

= y

12n

= z

12n+8

= h and x

12n+7

= y

12n+3

= z

12n−1

= m.

(iii) If x

−1

= c = 0, then we have x

12n−1

= y

_12n+7

= z

_12n+3

= 0, x

_12n+2

= y

12n−2

= z

_12n+6

= f, x

12n+5

= y

12n+1

= z

12n−3

= k and x

12n+8

= y

12n+4

= z

12n

= o.

(iv) If x

0

= d = 0, then we have x

12n

= y

12n+8

= z

12n+4

= 0, x

12n−3

= y

12n+5

= z

12n+1

= a, x

12n+3

= y

12n−1

= z

12n+7

= g and x

12n+6

= y

12n+2

= z

12n−2

= l.

(v) If y

−3

= e = 0, then we have x

12n+1

= y

12n−3

= z

12n+5

= 0, x

12n+7

= y

12n+3

= z

12n−1

= m, x

12n−2

= y

12n+6

= z

12n+2

= b and x

12n+4

= y

12n

= z

12n+8

= h.

(vi) If y

−2

= f = 0, then we have x

12n+2

= y

12n−2

= z

12n+6

= 0, x

12n+5

= y

12n+1

= z

12n−3

= k, x

_12n+8

= y

_12n+4

= z

_12n

= o and x

_12n+7

= y

_12n+3

= z

12n−1

= m.

(vii) If y

−1

= g = 0, then we have x

12n+3

= y

12n−1

= z

12n+7

= 0, x

12n+6

= y

12n+2

= z

12n−2

= l, x

12n−3

= y

_12n+5

= z

_12n+1

= a and x

_12n

= y

_12n+8

= z

_12n+4

= d.

(viii) If y

0

= h = 0, then we have x

12n+4

= y

12n

= z

12n+8

= 0, x

12n+7

= y

12n+3

= z

12n−1

= m, x

12n−2

= y

_12n+6

= z

_12n+2

= b and x

_12n+1

= y

12n−3

= z

_12n+5

= e.

(ix) If z

−3

= k = 0, then we have x

_12n+5

= y

_12n+1

= z

12n−3

= 0, x

_12n+2

= y

12n−2

= z

_12n+6

= f, x

_12n+8

= y

_12n+4

= z

_12n

= o and x

12n−1

= y

_12n+7

= z

_12n+3

= c.

(x) If z

−2

= l = 0, then we have x

_12n+6

= y

_12n+2

= z

12n−2

= 0, x

_12n+3

= y

12n−1

= z

_12n+7

= g, x

12n−3

= y

_12n+5

= z

_12n+1

= a and x

_12n

= y

_12n+8

= z

_12n+4

= d.

(xi) If z

−1

= m = 0, then we have x

_12n+7

= y

_12n+3

= z

12n−1

= 0, x

_12n+1

= y

12n−3

= z

_12n+5

= e, x

12n+4

= y

12n

= z

12n+8

= h and x

12n−2

= y

12n+6

= z

12n+2

= b.

(xii) If z

₀

= o = 0, then we have x

_12n+8

= y

_12n+4

= z

_12n

= 0 and x

_12n+2

= y

12n−2

= z

_12n+6

= f, x

12n+5

= y

12n+1

= z

12n−3

= k, x

12n−1

= y

12n+7

= z

12n+3

= c.

Proof. The proof follows from the form of the solutions of system (2.1).

Theorem 2.4. The solutions of the system x

_n+1

=

_1−y ^yn−3

n−3zn−1xn−2yn

, y

_n+1

=

_1−z ^zn−3

n−3xn−1yn−2zn

, z

_n+1

=

_1−x ^xn−3

n−3yn−1zn−2xn

, (2.2) are given by the following equations

x

12n−3

= (−1)

ⁿ

a

n−1

Q

i=0

(−1+12iadgl)(−1+(12i+4)adgl)(−1+(12i+8)adgl) (−1+(12i+1)adgl)(−1+(12i+5)adgl)(−1+(12i+9)adgl)

, x

12n−2

= b

n−1

Q

i=0

(−1+(12i+1)behm)(−1+(12i+5)behm)(−1+(12i+9)behm) (−1+(12i+2)behm)(−1+(12i+6)behm)(−1+(12i+10)behm)

, x

12n−1

= c

n−1

Q

i=0

(−1+(12i+2)cf ko)(−1+(12i+6)cf ko)(−1+(12i+10)cf ko) (−1+(12i+3)cf ko)(−1+(12i+7)cf ko)(−1+(12i+11)cf ko)

, x

12n

= d

n−1

Q

i=0

(−1+(12i+3)adgl)(−1+(12i+7)adgl)(−1+(12i+11)adgl) (−1+(12i+4)adgl)(−1+(12i+8)adgl)(−1+(12i+12)adgl)

, x

12n+1

= −

_(−1+behm)^e

n−1

Q

i=0

(−1+(12i+4)behm)(−1+(12i+8)behm)(−1+(12i+12)behm) (−1+(12i+5)behm)(−1+(12i+9)behm)(−1+(12i+13)behm)

, x

_12n+2

=

^f(−1+2cf ko)^{(−1+cf ko)}

n−1

Q

i=0

(−1+(12i+5)cf ko)(−1+(12i+9)cf ko)(−1+(12i+13)cf ko) (−1+(12i+6)cf ko)(−1+(12i+10)cf ko)(−1+(12i+14)cf ko)

, x

_12n+3

=

^g_(−1+3adgl)^(−1+2adgl)

n−1

Q

i=0

(−1+(12i+6)adgl)(−1+(12i+10)adgl)(−1+(12i+14)adgl) (−1+(12i+7)adgl)(−1+(12i+11)adgl)(−1+(12i+15)adgl)

, x

_12n+4

=

^h_(−1+4behm)^(−1+3behm)

n−1

Q

i=0

(−1+(12i+7)behm)(−1+(12i+11)behm)(−1+(12i+15)behm) (−1+(12i+8)behm)(−1+(12i+12)behm)(−1+(12i+16)behm)

, x

12n+5

= −

^k (−1+4cf ko)

(−1+cf ko)(−1+5cf ko) n−1

Q

i=0

(−1+(12i+8)cf ko)(−1+(12i+12)cf ko)(−1+(12i+16)cf ko) (−1+(12i+9)cf ko)(−1+(12i+13)cf ko)(−1+(12i+17)cf ko)

, x

12n+6

=

^l (−1+adgl)(−1+5adgl)

(−1+2adgl)(−1+6adgl) n−1

Q

i=0

(−1+(12i+9)adgl)(−1+(12i+13)adgl)(−1+(12i+17)adgl) (−1+(12i+10)adgl)(−1+(12i+14)adgl)(−1+(12i+18)adgl)

, x

_12n+7

=

^m (−1+2behm)(−1+6behm)

(−1+3behm)(−1+7behm) n−1

Q

i=0

(−1+(12i+10)behm)(−1+(12i+14)behm)(−1+(12i+18)behm) (−1+(12i+11)behm)(−1+(12i+15)behm)(−1+(12i+19)behm)

, x

_12n+8

=

^o (−1+3cf ko)(−1+7cf ko)

(−1+4cf ko)(−1+8cf ko) n−1

Q

i=0

(−1+(12i+11)cf ko)(−1+(12i+15)cf ko)(−1+(12i+19)cf ko) (−1+(12i+12)cf ko)(−1+(12i+16)cf ko)(−1+(12i+20)cf ko)

, y

12n−3

= (−1)

ⁿ

e

n−1

Q

i=0

(−1+(12i)behm)(−1+(12i+4)behm)(−1+(12i+8)behm) (−1+(12i+1)behm)(−1+(12i+5)behm)(−1+(12i+9)behm)

, y

12n−2

= f

n−1

Q

i=0

(−1+(12i+1)cf ko)(−1+(12i+5)cf ko)(−1+(12i+9)cf ko) (−1+(12i+2)cf ko)(−1+(12i+6)cf ko)(−1+(12i+10)cf ko)

, y

12n−1

= g

n−1

Q

i=0

(−1+(12i+2)adgl)(1+(12i+6)adgl)(−1+(12i+10)adgl) (−1+(12i+3)adgl)(−1+(12i+7)adgl)(−1+(12i+11)adgl)

, y

_12n

= h

n−1

Q

i=0

(−1+(12i+3)behm)(−1+(12i+7)behm)(−1+(12i+11)behm) (−1+(12i+4)behm)(−1+(12i+8)behm)(−1+(12i+12)behm)

, y

_12n+1

= −

_{(−1+cf ko)}^k

n−1

Q

i=0

(−1+(12i+4)cf ko)(−1+(12i+8)cf ko)(−1+(12i+12)cf ko) (−1+(12i+5)cf ko)(−1+(12i+9)cf ko)(−1+(12i+13)cf ko)

, y

12n+2

=

^l_(−1+2adgl)^(−1+adgl)

n−1

Q

i=0

(−1+(12i+5)adgl)(−1+(12i+9)adgl)(−1+(12i+13)adgl) (−1+(12i+6)adgl)(−1+(12i+10)adgl)(−1+(12i+14)adgl)

, y

12n+3

=

^m_(−1+3behm)^(−1+2behm)

n−1

Q

i=0

(−1+(12i+6)behm)(−1+(12i+10)behm)(−1+(12i+14)behm) (−1+(12i+7)behm)(−1+(12i+11)behm)(−1+(12i+15)behm)

, y

_12n+4

=

^o (−1+3cf ko)

(−1+4cf ko) n−1

Q

i=0

(−1+(12i+7)cf ko)(−1+(12i+11)cf ko)(−1+(12i+15)cf ko) (−1+(12i+8)cf ko)(−1+(12i+12)cf ko)(−1+(12i+16)cf ko)

,

y

12n+5

= −

(−1+adgl)(−1+5adgl)^{a (−1+4adgl)} n−1

Q

i=0

(−1+(12i+8)adgl)(−1+(12i+12)adgl)(−1+(12i+16)adgl) (−1+(12i+9)adgl)(−1+(12i+13)adgl)(−1+(12i+17)adgl)

, y

12n+6

=

^b (−1+behm)(−1+5behm)

(−1+2behm)(−1+6behm) n−1

Q

i=0

(−1+(12i+9)behm)(−1+(12i+13)behm)(−1+(12i+17)behm) (−1+(12i+10)behm)(−1+(12i+14)behm)(−1+(12i+18)behm)

, y

_12n+7

=

^c (−1+2cf ko)(−1+6cf ko)

(−1+3cf ko)(−1+7cf ko) n−1

Q

i=0

(−1+(12i+10)cf ko)(−1+(12i+14)cf ko)(−1+(12i+18)cf ko) (−1+(12i+11)cf ko)(−1+(12i+15)cf ko)(−1+(12i+19)cf ko)

, y

_12n+8

=

^d (−1+3adgl)(−1+7adgl)

(−1+4adgl)(−1+8adgl) n−1

Q

i=0

(−1+(12i+11)adgl)(−1+(12i+15)adgl)(−1+(12i+19)adgl) (−1+(12i+12)adgl)(−1+(12i+16)adgl)(−1+(12i+20)adgl)

, z

12n−3

= (−1)

ⁿ

k

n−1

Q

i=0

(−1+(12i)cf ko)(−1+(12i+4)cf ko)(−1+(12i+8)cf ko) (−1+(12i+1)cf ko)(−1+(12i+5)cf ko)(−1+(12i+9)cf ko)

, z

12n−2

= l

n−1

Q

i=0

(−1+(12i+1)adgl)(−1+(12i+5)adgl)(−1+(12i+9)adgl) (−1+(12i+2)adgl)(−1+(12i+6)adgl)(−1+(12i+10)adgl)

, z

12n−1

= m

n−1

Q

i=0

(−1+(12i+2)behm)(−1+(12i+6)behm)(−1+(12i+10)behm) (−1+(12i+3)behm)(−1+(12i+7)behm)(−1+(12i+11)behm)

, z

_12n

= o

n−1

Q

i=0

(−1+(12i+3)cf ko)(−1+(12i+7)cf ko)(−1+(12i+11)cf ko) (−1+(12i+4)cf ko)(−1+(12i+8)cf ko)(−1+(12i+12)cf ko)

, z

_12n+1

= −

_(−1+adgl)^a

n−1

Q

i=0

(−1+(12i+4)adgl)(−1+(12i+8)adgl)(−1+(12i+12)adgl) (−1+(12i+5)adgl)(−1+(12i+9)adgl)(−1+(12i+13)adgl)

, z

12n+2

=

^b_(−1+2behm)^(−1+behm)

n−1

Q

i=0

(−1+(12i+5)behm)(−1+(12i+9)behm)(−1+(12i+13)behm) (−1+(12i+6)behm)(−1+(12i+10)behm)(−1+(12i+14)behm)

, z

12n+3

=

^c (−1+2cf ko)

(−1+3cf ko) n−1

Q

i=0

(−1+(12i+6)cf ko)(−1+(12i+10)cf ko)(−1+(12i+14)cf ko) (−1+(12i+7)cf ko)(−1+(12i+11)cf ko)(−1+(12i+15)cf ko)

, z

_12n+4

=

^d_(−1+4adgl)^(−1+3adgl)

n−1

Q

i=0

(−1+(12i+7)adgl)(−1+(12i+11)adgl)(−1+(12i+15)adgl) (−1+(12i+8)adgl)(−1+(12i+12)adgl)(−1+(12i+16)adgl)

, z

_12n+5

= −

(−1+behm)(−1+5behm)^{e (−1+4behm)}

n−1

Q

i=0

(−1+(12i+8)behm)(−1+(12i+12)behm)(−1+(12i+16)behm) (−1+(12i+9)behm)(−1+(12i+13)behm)(−1+(12i+17)behm)

, z

12n+6

=

^f (−1+cf ko)(−1+5cf ko)

(−1+2cf ko)(−1+6cf ko) n−1

Q

i=0

(−1+(12i+9)cf ko)(−1+(12i+13)cf ko)(−1+(12i+17)cf ko) (−1+(12i+10)cf ko)(−1+(12i+14)cf ko)(−1+(12i+18)cf ko)

, z

12n+7

=

^g (−1+2adgl)(−1+6adgl)

(−1+3adgl)(−1+7adgl) n−1

Q

i=0

(−1+(12i+10)adgl)(−1+(12i+14)adgl)(−1+(12i+18)adgl) (−1+(12i+11)adgl)(−1+(12i+15)adgl)(−1+(12i+19)adgl)

, z

_12n+8

=

^h (−1+3behm)(−1+7behm)

(−1+4behm)(−1+8behm) n−1

Q

i=0

(−1+(12i+11)behm)(−1+(12i+15)behm)(−1+(12i+19)behm) (−1+(12i+12)behm)(−1+(12i+16)behm)(−1+(12i+20)behm)

, where

−1

Q

i=0

A

i

= 1.

3. On the systems: x

n+1

=

_±1+y ^yn−3

nzn−1xn−2yn−3

, y

n+1

=

_±1−z ^zn−3

nxn−1yn−2zn−3

, z

n+1

=

_±1−x ^xn−3

nyn−1zn−2xn−3

In this section, we study the solutions of the system of three difference equations in the following form x

n+1

=

_1+y ^yn−3

nzn−1xn−2yn−3

, y

n+1

=

_1−z ^zn−3

nxn−1yn−2zn−3

, z

n+1

=

_−1−x ^xn−3

nyn−1zn−2xn−3

, n = 0, 1, ..., (3.1) with nonzero real initial conditions.

Theorem 3.1. Suppose that {x

_n

, y

n

, z

n

} are solutions of system (3.1), we see that x

12n−3

=

_(−1+adgl)^a(1+2adgl)n ⁿ

(1+adgl)²ⁿ

, x

12n−2

=

^{(−1)nb (1+behm)}_(1+2behm)^{2n (1+3behm)}2n ⁿ

,

x

12n−1

=

_{(−1+cf ko)}^{c(1+2cf ko)}n (1+cf ko)^{n 2n}

, x

12n

=

^{d (−1+adgl)}_(1+2adgl)^{n (1+adgl)}n ²ⁿ

, x

_12n+1

=

_(1+behm)^(−1)ne2n+1^(1+2behm)(1+3behm)^{2n n}

, x

_12n+2

= −

^{f (−1+cf ko)}_{(1+2cf ko)}^{n+1 (1+cf ko)}n ²ⁿ

, x

_12n+3

=

_(−1+adgl)^{g (1+2adgl)}n (1+adgl)^n+12n+1

, x

_12n+4

=

^{(−1)nh (1+behm)}_(1+2behm)²ⁿ2n+1^(1+3behm)n+1

, x

_12n+5

= −

^{k (1+2cf ko)n+1}

(−1+cf ko)ⁿ⁺¹ (1+cf ko)²ⁿ⁺¹

, x

_12n+6

= −

^{l (−1+adgl)n+1 (1+adgl)2n+1}

(1+2adgl)ⁿ⁺¹

, x

_12n+7

=

^{(−1)nm (1+2behm)2n+1}

(1+behm)²ⁿ⁺¹ (1+3behm)ⁿ⁺¹

, x

_12n+8

= −

^{o (−1+cf ko)n+1 (1+cf ko)2n+1}

(1+2cf ko)ⁿ⁺¹

, y

12n−3

=

_(1+behm)^{(−1)n e}2n^(1+2behm)(1+3behm)²ⁿⁿ

, y

12n−2

=

^{f (−1+cf ko)}_{(1+2cf ko)}^{n(1+cf ko)}n ²ⁿ

, y

12n−1

=

_(−1+adgl)^{g (1+2adgl)}n(1+adgl)^{n 2n}

, y

_12n

=

^{(−1)nh (1+behm)}_(1+2behm)^2n(1+3behm)2n ⁿ

, y

12n+1

= −

k(1+2cf ko)ⁿ

(−1+cf ko)ⁿ⁺¹(1+cf ko)²ⁿ

, y

12n+2

=

^{l (1+adgl)}_(1+2adgl)^{2n+1(−1+adgl)}n+1 ⁿ

, y

12n+3

=

_(1+behm)^(−1)nm2n^(1+2behm)(1+3behm)^2n+1n+1

, y

12n+4

=

^{o (−1+cf ko)}_{(1+2cf ko)}^{n(1+cf ko)}n+1 ²ⁿ⁺¹

,

y

12n+5

=

_(−1+adgl)^a(1+2adgl)n+1(1+adgl)^{n 2n+1}

, y

12n+6

=

^{(−1)n+1b (1+behm)}_(1+2behm)²ⁿ⁺²2n+1^(1+3behm)n

, y

12n+7

=

_{(−1+cf ko)}^{c (1+2cf ko)}n+1(1+cf ko)^{n 2n+1}

, y

12n+8

= −

^{d (−1+adgl)}_(1+2adgl)^n(1+adgl)_n+1 ²ⁿ⁺²

, z

12n−3

=

_{(−1+cf ko)}^{k (1+2cf ko)}n(1+cf ko)^{n 2n}

, z

12n−2

=

^{l (−1+adgl)}_(1+2adgl)^n(1+adgl)n ²ⁿ

, z

12n−1

=

_(1+behm)^{(−1)n m}2n^(1+2behm)(1+3behm)²ⁿⁿ

, z

_12n

=

^{o (−1+cf ko)}_{(1+2cf ko)}^{n(1+cf ko)}n ²ⁿ

,

z

_12n+1

= −

_(−1+adgl)^{a (1+2adgl)}n (1+adgl)^{n 2n+1}

, z

_12n+2

=

^{(−1)n+1b (1+behm)}_(1+2behm)²ⁿ⁺¹2n+1^(1+3behm)n

, z

_12n+3

= −

_{(−1+cf ko)}^{c (1+2cf ko)}_{n(1+cf ko)}ⁿ _2n+1

, z

_12n+4

= −

^{d (1+adgl)}_(1+2adgl)^{2n+1(−1+adgl)}_n ⁿ

, z

_12n+5

=

_(1+behm)^(−1)n+1e2n+2^(1+2behm)(1+3behm)²ⁿ⁺¹ⁿ

, z

_12n+6

=

^{f (−1+cf ko)}_{(1+2cf ko)}^{n+1(1+cf ko)}n ²ⁿ⁺¹

,

z

_12n+7

= −

_(−1+adgl)^{g (1+2adgl)}n(1+adgl)^n+12n+2

, z

_12n+8

=

^{(−1)n+1h (1+behm)}_(1+2behm)²ⁿ⁺¹2n+2^(1+3behm)n+1

, n = 0, 1, 2, ... .

Proof. For n = 0, the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. We have

x

12n−7

= −

_{(−1+cf ko)}^{k(1+2cf ko)}n (1+cf ko)^{n 2n−1}

, x

12n−6

= −

^l(−1+adgl)_(1+2adgl)^{n (1+adgl)}_n ²ⁿ⁻¹

, x

12n−5

=

^{(−1)n−1m (1+2behm)}

2n−1

(1+behm)²ⁿ⁻¹ (1+3behm)ⁿ

, x

12n−4

= −

^{o(1+cf ko)}_{(1+2cf ko)}^{n (1+cf ko)}_n ²ⁿ⁻¹

, y

12n−7

=

_(−1+adgl)^a(1+2adgl)n(1+adgl)^n−12n−1

, y

12n−6

=

^{(−1)nb (1+behm)}_(1+2behm)^2n(1+3behm)2n−1 ⁿ⁻¹

, y

12n−5

=

_{(−1+cf ko)}^{c(1+2cf ko)}n(1+cf ko)^n−12n−1

, y

12n−4

= −

^d(−1+adgl)_(1+2adgl)^{n−1(1+adgl)}_n ²ⁿ

, z

12n−7

=

_(1+behm)^{(−1)ne(1+2behm)}2n(1+3behm)^2n−1n−1

, z

12n−6

=

^{f (−1+cf ko)}_{(1+2cf ko)}^{n(1+cf ko)}n−1 ²ⁿ⁻¹

, z

12n−5

= −

_(−1+adgl)^{g (1+2adgl)}n−1(1+adgl)^{n 2n}

, z

12n−4

=

^{(−1)nh (1+behm)}_(1+2behm)²ⁿ⁻¹2n^(1+3behm)n

. Now, it follows from Eq. (3.1) that

x

_12n

=

_1+y ^y12n−4

12n−4x12n−3z12n−2y12n−1

=

_{−d(−1+adgl)}n−1(1+adgl)²ⁿ (1+2adgl)ⁿ



1 +





_{−d(−1+adgl)}n−1(1+adgl)²ⁿ (1+2adgl)ⁿ

a(1+2adgl)ⁿ (−1+adgl)ⁿ (1+adgl)²ⁿ

l (−1+adgl)ⁿ(1+adgl)²ⁿ

(1+2adgl)ⁿ

g (1+2adgl)ⁿ (−1+adgl)ⁿ(1+adgl)²ⁿ









=

−d(−1+adgl)ⁿ⁻¹(1+adgl)²ⁿ (1+2adgl)ⁿ

1+

−adgl (−1+adgl)

=

−d(−1+adgl)ⁿ⁻¹(1+adgl)²ⁿ (1+2adgl)ⁿ

−1+adgl−adgl

−1+adgl

=

^d(−1+adgl)_(1+2adgl)^n(1+adgl)n ²ⁿ

, y

12n+1

=

_1−z ^z12n−3

12n−3y12n−2x12n−1z12n

=

k(1+2cf ko)ⁿ (−1+cf ko)ⁿ(1+cf ko)²ⁿ







 1−









_{k(1+2cf ko)}n

(−1+cf ko)ⁿ(1+cf ko)²ⁿ

f (−1+cf ko)ⁿ(1+cf ko)²ⁿ (1+2cf ko)ⁿ

_{c(1+2cf ko)}n

(−1+cf ko)ⁿ (1+cf ko)²ⁿ

o(−1+cf ko)ⁿ(1+cf ko)²ⁿ (1+2cf ko)ⁿ

















=

(−1)ⁿk(1+2cf ko)ⁿ (1−cf ko)ⁿ(1+cf ko)²ⁿ

(1−cf ko)

=

_{(1−cf ko)}^(−1)nkn+1^{(1+2cf ko)}(1+cf ko)ⁿ²ⁿ

, z

_12n+1

=

_−1−x ^x12n−3

12n−3z12n−2y12n−1x12n

=

a (1+2adgl)ⁿ (−1+adgl)ⁿ (1+adgl)²ⁿ









−1−









_a(1+2adgl)n

(−1+adgl)ⁿ (1+adgl)²ⁿ

l (−1+adgl)ⁿ(1+adgl)²ⁿ (1+2adgl)ⁿ

g (1+2adgl)ⁿ

(−1+adgl)ⁿ(1+adgl)²ⁿ

d(−1+adgl)ⁿ (1+adgl)²ⁿ (1+2adgl)ⁿ

















= −

a(1+2adgl)ⁿ (−1+adgl)ⁿ (1+adgl)²ⁿ

(1+adgl)

= −

_(−1+adgl)^{a (1+2adgl)}n (1+adgl)^{n 2n+1}

. Also, we can see that

x

_12n+1

=

_1+y ^y12n−3

12n−3x12n−2z12n−1y12n

=

(−1)ⁿ e(1+2behm)²ⁿ (1+behm)²ⁿ(1+3behm)ⁿ







 1+









₍₋₁₎n e(1+2behm)²ⁿ (1+behm)²ⁿ(1+3behm)ⁿ

(−1)ⁿb (1+behm)²ⁿ (1+3behm)ⁿ (1+2behm)²ⁿ

₍₋₁₎n m(1+2bhm)²ⁿ

(1+bhm)²ⁿ(1+3bhm)ⁿ

(−1)ⁿh (1+behm)²ⁿ(1+3behm)ⁿ (1+2behm)²ⁿ

















=

(−1)ⁿ e(1+2behm)²ⁿ (1+behm)²ⁿ(1+3behm)ⁿ

(1+behm)

=

_(1+behm)^{(−1)n e}2n+1^(1+2behm)(1+3behm)^{2n n}

, y

12n+5

=

_1−z ^z12n+1

12n+4x12n+3y12n+2z12n+1

=

− a(1+2adgl)ⁿ (−1+adgl)ⁿ (1+adgl)²ⁿ⁺¹







 1−









−

^d(1+adgl)_(1+2adgl)^{2n+1(−1+adgl)}_n ⁿ _(−1+adgl)^{g (1+2adgl)}n (1+adgl)^n+12n+1

l (1+adgl)²ⁿ⁺¹(−1+adgl)ⁿ

(1+2adgl)ⁿ⁺¹

−

_(−1+adgl)^a(1+2adgl)n (1+adgl)^{n 2n+1}

















=

− a(1+2adgl)ⁿ (−1+adgl)ⁿ (1+adgl)²ⁿ⁺¹

(1−adgl)

=

_(−1+adgl)^a(1+2adgl)n+1 (1+adgl)^{n 2n+1}

=

_(−1+adgl)^a(1+2adgl)2n(1+adgl)^{n 2n+1}

, z

_12n+6

=

_−1−x ^x12n+2

12n+5y12n+4z12n+3x12n+2

=

−f (−1+cf ko)ⁿ⁺¹ (1+cf ko)²ⁿ (1+2cf ko)ⁿ

!

−1−









−

^{k(1+2cf ko)n+1}

(−1+cf ko)ⁿ⁺¹ (1+cf ko)²ⁿ⁺¹

o (−1+cf ko)ⁿ(1+cf ko)²ⁿ⁺¹ (1+2cf ko)ⁿ⁺¹

−

_{(−1+cf ko)}^{c(1+2cf ko)}_{n(1+cf ko)}ⁿ _2n+1

−

^{f (−1+cf ko)}_{(1+2cf ko)}^{n+1 (1+cf ko)}n ²ⁿ









=

−f (−1+cf ko)ⁿ⁺¹ (1+cf ko)²ⁿ (1+2cf ko)ⁿ

!

−1+

cf ko 1+cf ko

=

−f (−1+cf ko)ⁿ⁺¹ (1+cf ko)²ⁿ (1+2cf ko)ⁿ

!

−1−cf ko+cf ko 1+cf ko

=

^{f (−1+cf ko)}_{(1+2cf ko)}^{n+1(1+cf ko)}n ²ⁿ⁺¹

.

Also, we can prove the other relations. This completes the proof.

Theorem 3.2. Assume that {x

_n

, y

n

, z

n

} are solutions of the system x

n+1

=

_−1+y ^yn−3

n−3zn−1xn−2yn

, y

n+1

=

_1−z ^zn−3

n−3xn−1yn−2zn

, z

n+1

=

_1−x ^xn−3

n−3yn−1zn−2xn

, then for n = 0, 1, 2, ..., we see that

x

12n−3

=

_(−1+adgl)^{a (1−2adgl)}2n(−1+3adgl)^{2n n}

, x

12n−2

=

^{b (−1+behm)}_(1+2behm)^{n (1+behm)}n ²ⁿ

, x

12n−1

=

^c (−1+2cf ko)ⁿ

(−1+cf ko)²ⁿ (1+cf ko)ⁿ

, x

12n

=

^d(−1+adgl)_(1−2adgl)^{2n (−1+3adgl)}2n ⁿ

, x

12n+1

=

_(−1+behm)^{e (1+2behm)}_n+1 ⁿ

(1+behm)²ⁿ

, x

12n+2

= −

^{f (−1+cf ko)2n+1 (1+cf ko)n}

(−1+2cf ko)ⁿ⁺¹

, x

12n+3

= −

^{g (−1+2adgl)2n+1}

(−1+3adgl)ⁿ⁺¹ (−1+adgl)²ⁿ

, x

12n+4

= −

^{h (1+behm)}_(1+2behm)^{2n+1 (−1+behm)}_n+1 ⁿ

, x

12n+5

=

^k(−1+2cf ko)ⁿ

(−1+cf ko)ⁿ⁺¹ (1+cf ko)²ⁿ⁺¹

, x

12n+6

=

^{l (−1+adgl)}_(−1+2adgl)^{2n+2 (−1+3adgl)}2n+1 ⁿ

, x

12n+7

=

_(−1+behm)^{m (1+2behm)}_n+1 ⁿ

(1+behm)²ⁿ⁺¹

, x

12n+8

=

^{o (−1+cf ko)}(−1+2cf ko)^{2n+2(1+cf ko)n+1 n}

, y

12n−3

=

_(−1+behm)^{e (1+2behm)}n(1+behm)^{n 2n}

, y

12n−2

=

^{f (−1+cf ko)}(−1+2cf ko)^{2n(1+cf ko)n n}

, y

12n−1

=

_(−1+adgl)^{g (1−2adgl)}2n(−1+3adgl)^{2n n}

, y

12n

=

^{h (−1+behm)}_(1+2behm)^n(1+behm)n ²ⁿ

, y

_12n+1

= −

k(−1+2cf ko)ⁿ

(−1+cf ko)²ⁿ⁺¹(1+cf ko)ⁿ

, y

_12n+2

=

^{l (−1+adgl)}_(−1+2adgl)^{2n+1(−1+3adgl)}2n+1 ⁿ

, y

_12n+3

=

_(−1+bem)^{m (1+2bem)}n(1+bem)ⁿ²ⁿ⁺¹

, y

_12n+4

= −

^{o (−1+cf ko)}(−1+2cf ko)^{2n+1(1+cf ko)n n}

, y

_12n+5

= −

_(−1+adgl)^{a(−1+2adgl)}_{2n+2(−1+3adgl)}²ⁿ⁺¹ _n

, y

_12n+6

= −

^{b (−1+behm)}_(1+2behm)^n+1(1+behm)_n ²ⁿ⁺¹

, y

_12n+7

= −

^c (−1+2cf ko)ⁿ⁺¹

(−1+cf ko)²ⁿ(1+cf ko)ⁿ

, y

_12n+8

=

^d(−1+adgl)_(1−2adgl)^{n+1(−1+3adgl)}2n+2 ²ⁿ⁺¹

, z

12n−3

=

^k (−1+2cf ko)ⁿ

(−1+cf ko)²ⁿ(1+cf ko)ⁿ

, z

12n−2

=

^{l (−1+adgl)}_(1−2adgl)^{2n(−1+3adgl)}2n ⁿ

, z

12n−1

=

_(−1+behm)^{m (1+2behm)}n(1+behm)^{n 2n}

, z

12n

=

^{o (−1+cf ko)}(−1+2cf ko)^{2n(1+cf ko)n n}

, z

12n+1

= −

^{a(1−2adgl)2n}

(−1+adgl)²ⁿ⁺¹ (−1+3adgl)ⁿ

, z

12n+2

= −

^{b (−1+behm)}_(1+2behm)^n+1(1+behm)n ²ⁿ

, z

12n+3

=

^c (−1+2cf ko)ⁿ⁺¹

(−1+cf ko)²ⁿ⁺¹(1+cf ko)ⁿ

, z

12n+4

=

^{d(−1+3adgl)}_(−1+2adgl)^{n+1(−1+adgl)}2n+1 ²ⁿ

, z

12n+5

=

_(−1+behm)^{e (1+2behm)}n+1(1+behm)^{n+1 2n+1}

, z

12n+6

= −

^{f (−1+cf ko)}(−1+2cf ko)^{2n+1(1+cf ko)n+1 n+1}

, z

12n+7

=

_(−1+adgl)^{g (−1+2adgl)}2n+1(−1+3adgl)^{2n+1 n+1}

, z

12n+8

=

^{h (−1+behm)}_(1+2behm)^n+1(1+behm)n+1 ²ⁿ⁺¹

. Theorem 3.3. Let {x

_n

, y

n

, z

n

} be solutions of the system

x

n+1

=

_−1+y ^yn−3

n−3zn−1xn−2yn

, y

n+1

=

_−1−z ^zn−3

n−3xn−1yn−2zn

, z

n+1

=

_−1−x ^xn−3

n−3yn−1zn−2xn

, then we get

x

12n−3

=

^{a (−1+2adgl)}

n

(−1+adgl)²ⁿ (1+adgl)ⁿ

, x

12n−2

=

^{b (−1+behm)}

2n (1+behm)ⁿ (−1+2behm)ⁿ

,