On the solutions of a three-dimensional system of difference equations

(1)

On the solutions of a three-dimensional system of difference equations

Yasin Yazlık^1,*, Durhasan T. Tollu², Necati Taşkara³

1Nevsehir Haci Bektas Veli University, Faculty of Science and Art, Department of Mathematics, Nevsehir, Turkey

2Necmettin Erbakan University, Science Faculty, Department of Mathematics-Computer Sciences, Konya, Turkey.

3Selcuk University, Science Faculty, Department of Mathematics, Konya, Turkey.

*Corresponding Author: E-mail: [email protected]

Abstract

In this paper, we obtain the explicit solutions of a three-dimensional system of difference equations with multiplicative terms, extending some results in literature.

Also, by using explicit forms of the solutions, we study the asymptotic behaviour of well-defined solutions of the system.

Keywords: Asymptotic behaviour; difference equations; explicit form solution;

forbidden set; system of difference equations.

AMS Classification: 39A10, 39A11.

1. Introduction

Difference equations and their systems have been argued in the literature for several decades (Kulenović & Nurkanović, 2005; Papaschinopoulos & Schinas, 1998;

Diamandescu, 2009; Papaschinopoulos & Stefanidou, 2010; Elabbasy et al., 2011;

Taskara et al., 2011; Tollu et al., 2013; Yazlik, 2014 and references therein). The dominant trend in the theory of difference equations is actually to obtain the solutions of difference equation systems in the meaning of explicit or closed form. The solution forms are both an interesting and an elegant approach to study the existence and asymptotic properties of solutions of these systems (Yalcinkaya et al., 2008; Yazlik et al., 2014). Sedaghat (2009) determined the global behaviours of all solutions of the rational difference equations

(1) Stević (2004) gave a theoretical explanation for the formula of solutions of the

(2)

Later, the author showed that the following two-dimensional system of difference equations

(3)

can be solved (Stević, 2011). Stević (2012) studied the three-dimensional system of difference equations

(4)

and showed that the system in (4) can be solved as the two-dimensional system in (3) (see also Stević et al., 2012). Then, El-Metwally (2013) obtined the solutions form for the following systems of rational difference equations:

(5)

Stević et al. (2014) solved in closed form the system of difference equations

(6)

by generalizing systems in (5), and so considerably extended the results of El- Metwally’s paper.

They used formulas in the investigation of the asymptotic behaviour of the well- defined solutions when the sequences and are all constant and k=2l in (6). They presented the domain of undefinable solutions of the system.

Remark 1. While system (3) is an extension of the first equation in (1), the system in (4) is a three-dimensional extension of the system in (3). Similarly, the system in (6) is an extension of both the second equation in (1) and the system in (5).

Another extension of the second equation in (1) is the following three-dimensional system of difference equations:

(7)

(3)

where the parameters a_i, b_iand the initial values (i = 0,1,2) are real numbers.

Note that the system in (7) can be written in the form

(8)

Therefore, the system in (7) reduces to first-order linear equations and so is solvable in explicit form. Using this approach, in this paper we get explicit solutions of the system in (7) and determine the forbidden set of the initial values (i = 0,1,2) and also study asymptotic behavior of the solutions using their explicit forms.

2. Explicit solutions of the system

In this section we show that system (7) is solvable in explicit form. Here eight possible cases rise according to parameters a_iand b_i:

Case 1: and

In this case, we obtain the system

from which it follows that

(9)

From (9), we have

(10)

(11)

(4)

(12)

where and q₂ . By decomposing (10), (11) and (12), we get the following non autonomous equations

(13)

(14)

and

(15)

where and .

Equations (13), (14) and (15) easily can be solved as the following

(16)

where and .

(5)

Case 2: and In this case, the system becomes

(17)

From (17), the general solution follows as

(18)

where and .

Case 3: and

In this case, the system is

(19)

The solution can be obtained from (19) as

(6)

(20)

where and .

Case 4: and

In this case, we get the following system

(21)

From (21), the solution of system (7) takes the form

(22)

where and .

(7)

and

In this case, the system is expressed as

(23)

From (23), we obtain the solution of system (7) as follows

(24)

where and .

Case 6: and

The case yields the following system

(25)

The solution can be obtained from (25) as

(8)

(26)

where and .

Case 7:

b

₀

= = = b

₁

b

₂

1

In this case, the system is

(27)

From (27), the solution of system (7) takes the form

(28)

where and .

(9)

Case 8: and

In this case, the solution of each equation in system (8) is as follows

(29)

From (29), we get the solution of the system (7) as

(30)

where and

.

Corollary 2. Assume that ^aⁱ^{+ =}^bⁱ ¹

(

ⁱ⁼^{0,1, 2}

)

^and

Then every solution of system (7) is six-periodic.

The next theorem establishes the forbidden set of the initial values

(

ⁱ⁼^{0,1, 2}

)

^{of (7).}

Theorem 3. . Then the forbidden

set

F

of (7) is given by

^ ` ^ `

^ `

5 5

i i

i n

n n

x y

z y

x

z x

y z

T T

T T J

T J T J

o o

o f o

f o f o

½

® ¾

¯ ¿

½ ½

® ¾ ® ¾

¯ ¿ ¯ ¿

* **

F

where

^ `

LI

i in

i

a b

i b i

n

i i

b i

a n b

J

z

°

®

°¯

(10)

the solution can not be determined for the case

x y z

₋_i _{− −}_i _i

= 0

^,

ⁱ ^∈ { } ^0,1

. Thus we can incorporate the case

x y z

₋_i _{− −}_i _i

= 0

^,

ⁱ ^∈ { } ^0,1

, into the forbidden set. If

x y z

₋_i _{− −}_i _i

≠ 0

^,

{ } ^0,1

i ∈

then we define new variables ^{( )}⁰ ⁿ²

,

n

y

n x

u =

⁻

u

^{( )}n¹

=

^zⁿy⁻_n² ^and

u

n^{( )}²

=

^xⁿz_n⁻² ^{. In this}

case, (8) can be written in the form of the linear first order difference equations ( )ⁱ¹ ( )ⁱ

^, { ^{0,1, 2 ,} } ^{N ,}

⁰

n i n i

u

₊

= b u + a i ∈ n ∈

(31) which is independent of each other. Now, we indicate that the solutions of system (7) are not defined if and only if

0 _n 0 _n 2

0 or

1 _n 1 _n 2

0 or

2 _n 2 _n 2

0. a x + b y

₋

= a y + b z

₋

= a z + b x

₋

=

That is, the terms

x

_n,

y

_nand

z

_ncannot be calculated for some

n ∈ N

, after finite number of terms are calculated. So we can establish our proof on the fact that the solutions of the system are not well-defined in the cases

x y z

_{n n n}

= 0

^{for some}

n ∈ N

^{. Let}

{ }

( ) , 0,1, 2 .

i i i

f u = b u a i + ∈

Then we can write equation (31) as the following

( )ⁱ¹

⁽

( )ⁱ

^), { ^{0,1, 2 ,} } ^{N ,}

⁰

n i n

u

₊

= f u i ∈ n ∈

which have the solutions

( )ⁱ ⁿ

⁽

( )⁰ⁱ

^), { ^{0,1, 2 ,} } ^{N .}

⁰

n i

u = f u i ∈ n ∈

Suppose that

( )

{ }

0

(

0ⁱ

) 0, 0,1, 2 ,

0

N ,

0 n

f

i

u = i ∈ n ∈

which implies

0

(0)

0

.

n

f

i⁻

= u

(32) The inverses of the functions

f

_i can be calculated as follows

{ }

1

( )

ⁱ

, 0,1, 2 .

i

f v v a i b

−

= − ∈

Now note that difference equations associated with inverse functions

f

_i⁻¹are

( ) ( )

{ }

1

, 0,1, 2 , N .

0 i

i n i

n

i

v a

v i n

+

= b − ∈ ∈

(33)

(11)

( )

( ) _{ _}

0

(0) 1 , 0,1, 2 , N,

1

n

i i

i n

i

a b

u f i n

b

−

= = ∈ ∈

−

for

b

_i

≠ 1

^{. If}

b

_i

= 1

, then from (32) and (33), we have

( )

( ) { }

0ⁱ

1 , 0,1, 2 , N .

0

u = − a n + i ∈ n ∈

Consequently, we have the eight possible cases in the theorem. So the proof is complete.

From the above theorem, it can be said that if initial values

x

₋_i,

y

₋_i,

z

₋_i

∉

^,

{ ⁱ ⁼ ^{0,1, 2} }

, then every solution of system (7) is well-defined.

Theorem 4. Assume that

b

_i

≠ 1

^,

( ⁱ ⁼ ^{0,1, 2} )

^{, and}

( x y z

_n

, ,

_n _{n n}

)

_≥−2 is a well-defined solution of system (7). Then the following statements hold.

a) If

b

₀

> 1

^and

q

0

≠ 0

^or

b

1

> 1

^and

q

1

≠ 0

^or

b

2

> 1

^and

q

2

≠ 0,

^then

6_{n j}

0 x

₋

→

^,

y

6_{n j}₋

→ 0

^and

z

6_{n j}₋

→ 0

^as

n → ∞

^,

b) If

b

₀

< 1

^,

b

1

< 1

^,

b

2

< 1

^and

( )( )( )

¹â⁰^b⁰ ¹â¹^b¹ ¹â^b²²

^< ^1,

^then

^x

⁶^{n j}⁻

^{→ ∞} ^, ^y

⁶^{n j}⁻

^{→ ∞}

and

z

_{6n j}₋

→ ∞

^as

n → ∞

^,

c) If

b

₀

< 1

^,

b

1

< 1

^,

b

2

< 1

^and

( )( )( )

¹â⁰^b⁰ ¹â¹^b¹ ¹â²^b²

^> ^1,

^then

^x

⁶^{n j}⁻

^→ ^0, ^y

⁶^{n j}⁻

^→ ⁰

and

z

₆_{n j}₋

→ 0

^as

n → ∞

^,

d) If

b

₀

< 1, b

1

< 1, b

2

< 1

^and

( )

¹^a⁰^b⁰

( )( )

¹^a¹^b¹ ¹^a²^b²

⁼ ^1,

^then

(

⁶_{n j}

)

N₀

, (

⁶_{n j}

)

N₀

n n

x

y

and

( z

⁶n j n

)

N₀

are convergent,

e) If

b

₀

< 1, b

1

< 1, b

2

< 1

^and ^then

( x

¹²_{n j n}

)

N₀

,

( x

¹²_n₊⁶ _{j n}

)

N₀

,

( y

¹²_{n j n}

)

N₀

,

( y

¹²_n₊⁶ _{j n}

)

N₀

,

( z

¹²n j n

)

N₀

and

( z

¹²n₊⁶ j n

)

N₀

are convergent,

where ²

0 0

0 1

ao

y

x b

q =

⁻

−

₋ ^, 1 ₀² 1¹₁

z a

y b

q =

⁻

−

₋ ^and 2 ₀² 1 ²₂

x a

z b

q =

⁻

−

₋ ^,

j ∈ − − − { 3, 2, 1,0,1, 2 }

^,

N

0

n ∈

^.

(12)

(1) 6 6 0 6 4 1 6 2 2

0 0 1 1 2 2

0 1 2

(1) 6 6 1 6 4 2 6 2 0

1 1 2 2 0 0

1 2 0

(1) 6 6 2 6 4 0

2 2 0 0

2

: ,

1 1 1

: ,

1 1 1

: 1 1

m j m j m j

m

m j m j m j

m

m j m j

m

a a a

X q b q b q b

b b b

a a a

Y q b q b q b

b b b

a

Z q b a q b

b b

+ − + − + −

+ − + −

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ ⎝ + − ⎟⎜ ⎠ ⎝ + − ⎟⎜ ⎠⎝ + − ⎟ ⎠

⎛ ⎞

⎛ ⎞⎛ ⎞

= ⎜ ⎝ + − ⎟⎜ ⎠⎝ + − ⎟ ⎠⎝ ⎜ + − ⎟ ⎠

⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠ + −

6 2 1

1 1

0 1

1 .

m j

a

q b b

⎛ ⎞⎛

+ −

+ ⎞

⎜ ⎟⎜ ⎝ − ⎟ ⎠

⎝ ⎠

After that the result follows from the asumptions in (a). Thus, we obtain

6 6 6

lim

_{n j}

lim

_{n j}

lim

_{n j}

0.

n

x

₋ n

y

₋ n

z

₋

→∞

=

→∞

=

→∞

=

As a similar approximation, by the facts in (b) and using formula (30), we have

6 6 6

lim

_{n j}

lim

_{n j}

lim

_{n j}

.

n

x

₋ n

y

₋ n

z

₋

→∞

=

→∞

=

→∞

= ∞

For the proof of (c): By reconsidering the assumptions in the beginig of the proof, a simple calculation

( )(

^{0 1 2}

)( )

(1) (1) (1)

0 1 2

lim lim lim ,

1 1 1

m m m

n n n

a a a

X Y Z

b b b

→∞

=

→∞

=

→∞

=

− − −

gives the result. For the proofs (d) and (e): In fact it will be given only the proof of (d) since (e) can be obtained with the same manner. Again reconsidering the assumptions, it is seen that

( ) ( ) ( )

(34)

( )( ) ( )( )

( )

(1) 6 6 0 6 4 1 6 2 2

0 0 1 1 2 2

0 1 2

0 0 6 6 1 1 6 4 2 2 6 2

0 1 2

0 1 0 1 6 6 6 4 0 2 0 2 6 6 6 2

0 1 0 2

1

1 1 1

1

1 1 1 1

1 1

m j m j m j

m

m j m j m j

m j m j m j m j

a a a

X q b q b q b

b b b

b q b q b q

b b b

a a a

b b q q b b q q

b b b b

a a a a

b

+ − + − + −

+ − + − + − + −

⎛ ⎞⎛ ⎞⎛ ⎞

=⎜⎝ + − ⎟⎜⎠⎝ + − ⎟⎜⎠⎝ + − ⎟⎠

− − −

= + + +

− − − −

+ +

+ −

(

²

)

^{1 2} ⁶ ⁴ ⁶ ² ⁶ ⁶ ⁶

1 2 0 1 2

1 2

( ),

m j m j m m m

b q q

b b O b b b

a a

+ − + −

− +

(13)

( ) ( ) ( )

(1) 6 6 1 6 4 2 6 2 0

1 1 2 2 0 0

1 2 0

1 1 6 6 2 2 6 4 0 0 6 2

1 2 0

1 1 1

1

m j m j m j

m

m j m j m j

a

a a

Y q b q b q b

b b b

b q b q b q

b b b

a a a

+ − + − + −

⎛ ⎞

⎛ ⎞⎛ ⎞

= ⎜ ⎝ + − ⎟⎜ ⎠⎝ + − ⎟ ⎠⎝ ⎜ + − ⎟ ⎠

− − −

= + + +

(35)

( )( ) ( )( )

( )

1 2 1 2 6 6 6 4 0 1 0 1 6 2 6 6

1 2 0 1

1 2 1 0

0

1 1 1 1

1 1

m j m j m j m j

b b q q b b q q

b b b b

a a a a

b

+ − + − + − + −

− − − −

+ +

+ − (

²

)

^{0 2} ⁶ ² ⁶ ⁴ ⁶ ⁶ ⁶

0 2 0 1 2

0 2

( ),

m j m j m m m

b q q

b b O b b b

a a

+ − + −

− +

and

( ) ( ) ( )

(36)

( )( ) ( )( )

( )

(1) 6 6 2 6 4 0 6 2 1

2 2 0 0 1 1

2 0 1

2 2 6 6 0 0 6 4 1 1 6 2

2 0 1

0 2 0 2 6 6 6 4 0 1 0 1 6 4 6 2

2 0 0 1

0 2 0 1

1

1 1 1

1

1 1 1 1

1 1

m j m j m j

m

m j m j m j

m j m j m j m j

a a a

Z q b q b q b

b b b

b q b q b q

b b b

a a a

b b q q b b q q

b b b b

a a a a

b

+ − + − + −

+ − + − + − + −

⎛ ⎞

⎛ ⎞ ⎛ ⎞

=⎜⎝ + − ⎟⎠⎜⎝ + − ⎟⎠⎜⎝ + − ⎟⎠

− − −

= + + +

− − − −

+ +

+ −

(

²

)

^{1 2} ⁶ ² ⁶ ⁶ ⁶ ⁶ ⁶

1 2 0 1 2

1 2

( ),

m j m j m m m

b q q

b b O b b b

a a

+ − + −

− +

for every

j ∈ − − − { 3, 2, 1,0,1, 2 }

and sufficiently large

m

. From (34), (35), (36), the assumption

b

₀

< 1, b

1

< 1, b

2

< 1,

and the proof is completed by a known result on the convergence of products.

Theorem 5. Let at least one of parameters

b

_i^,

( ⁱ ⁼ ^{0,1, 2 ,} )

be one and

( x y z

_n

, ,

_n _{n n}

)

_≥−2

be a well-defined solution of system (7). Then

x

₆_{n j}₋

→ 0, y

₆_{n j}₋

→ 0, z

₆_{n j}₋

→ 0.

Proof. Let

( ) ( ) ( )

( )

( ) ( ) ( )

(2) 2 2 2

0 1 2

0 0 0

2 2 2 2

1 2 0

0 0 0

(2) 2 2 2

2 0 1

0 0 0

: 6 6 6 4 6 2 ,

: 6 6 6 4 6 2

m

y z x

X a m j a m j a m j

x y z

z x y

Y a m j a m j a m j

y z x

x y z

Z a m j a m j a m j

z x y

− − −

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ + + − ⎟⎜ + + − ⎟⎜ + + − ⎟

⎝ ⎠⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ + + − ⎟⎜ + + − ⎟⎜ + + − ⎟

⎝ ⎠⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ + + − ⎟⎜ + + − ⎟⎜ + + −

⎝ ⎠⎝ ⎠⎝ ⎠ ⎟ ,

(14)

( ) ( )

( )

(3) 2 2 6 2 0

1 2 0 0

0 0 0

(3) 2 6 4 0 2

2 0 0 1

0 0 0

: 6 6 6 4 ,

1 : 6 6

1

m j

m

m j

m

z x a

Y a m j a m j q b

y z b

a

x z

Z a m j q b a

z b y

− − + −

− + − −

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ ⎝ + + − ⎟⎜ ⎠⎝ + + − ⎟⎜ ⎠⎝ + − ⎟ ⎠

⎛ ⎞⎛ ⎞

= ⎜ ⎝ + + − ⎟⎜ ⎠⎝ + − ⎟ ⎠ ^⎛ ⎜ + ( ⁶ ^m ^{+ −} ² ^j ) ^⎞ ⎟ ^,

⎝ ⎠

( ) ( )

( )

( ) ( )

(4) 2 6 4 1 2

0 1 1 2

0 1 0

4 6 6 1 2 2

1 1 2 0

1 0 0

(4) 2 2 6 2 1

2 0 1 1

0 0

: 6 6 6 2 ,

1 : 6 4 6 2 ,

1 : 6 6 6 4

m j

m

m j

m

m j

m

y a x

X a m j q b a m j

x b z

a x y

Y q b a m j a m j

b z x

x y a

Z a m j a m j q b

z x

− + − −

+ − − −

− − + −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ + + − ⎟ ⎠ ⎜ ⎝ + − ⎟ ⎠ ⎜ ⎝ + + − ⎟ ⎠

⎛ ⎞⎛ ⎞

⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠⎝ ⎜ + + − ⎟⎜ ⎠⎝ + + − ⎟ ⎠

⎛ ⎞⎛ ⎞

= ⎜ + + − ⎟⎜ + + − ⎟ +

⎝ ⎠⎝ ⎠

¹

,

1 b

⎛ ⎞

⎜ − ⎟

⎝ ⎠

( ) ( )

( )

( ) ( )

( )

(5) 2 2 6 2 2

0 1 2 2

0 0 2

5 2 6 4 2 2

1 2 2 0

0 2 0

(5) 6 6 2 2 2

2 2 0 1

2 0 0

: 6 6 6 4 ,

1 : 6 6 6 2 ,

1 : 6 4 6

1

m j

m

m j

m

m j

m

y z a

X a m j a m j q b

x y b

z a y

Y a m j q b a m j

y b x

a y z

Z q b a m j a m

b x y

− − + −

− + − −

+ − − −

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ ⎝ + + − ⎟⎜ ⎠⎝ + + − ⎟⎜ ⎠ ⎝ + − ⎟ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ + + − ⎟ ⎠ ⎜ ⎝ + − ⎟ ⎠ ⎜ ⎝ + + − ⎟ ⎠

⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠⎝ ⎜ + + − ⎟ ⎠ ^⎛ ⎜ + ( ^{+ −} ² ^j ) ^⎞ ⎟ ^,

⎝ ⎠

( )

(6) 2 6 4 1 6 2 2

0 1 1 2 2

0 1 2

6 6 6 1 6 4 2 2

1 1 2 2 0

1 2 0

(6) 6 6 2 2

2 2 0 1 1

2 0

: 6 6 ,

1 1

: 6 2 ,

1 1

: 6 4

1

m j m j

m

m j m j

m

m j

m

y a a

X a m j q b q b

x b b

a a y

Y q b q b a m j

b b x

a y

Z q b a m j q b

b x

+ − + −

−

+ − + − −

+ − −

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ ⎝ + + − ⎟⎜ ⎠ ⎝ + − ⎟⎜ ⎠⎝ + − ⎟ ⎠

⎛ ⎞

⎛ ⎞⎛ ⎞

= ⎜ ⎝ + − ⎟⎜ ⎠⎝ + − ⎟ ⎠⎝ ⎜ + + − ⎟ ⎠

⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠⎝ ⎜ + + − ⎟ ⎠

6 2 1

1

1 ,

m j

a

b

⎛

+ −

⎞

⎜ + − ⎟

⎝ ⎠

( )

(7) 6 6 0 2 6 2 2

0 0 1 2 2

0 0 2

7 2 6 4 2 6 2 0

1 2 2 0 0

0 2 0

: 6 4 ,

1 1

: 6 6 ,

1 1

m j m j

m

m j m j

m

a z a

X q b a m j q b

b y b

a

z a

Y a m j q b q b

y b b

+ − − + −

+ − + −

−

⎛ ⎞⎛ ⎞⎛ ⎞

= ⎜ ⎝ + − ⎟⎜ ⎠⎝ + + − ⎟⎜ ⎠ ⎝ + − ⎟ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ + + − ⎟ ⎠ ⎜ ⎝ + − ⎟ ⎠ ⎜ ⎝ + − ⎟ ⎠

(15)

(7) 6 6 2 6 4 0

2 2 0 0

2 0

: 1 1

m j m j

m

a

a z

Z q b q b

b b

+ −

⎛

+ −

⎞

⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠⎝ ⎜ + − ⎟ ⎠

² ¹

( )

0

6 2 .

a m j

y

⎛

−

⎞

+ + −

⎜ ⎟

⎝ ⎠

and

( )

(8) 6 6 0 6 4 1 2

0 0 1 1 2

0 1 0

8 6 6 1 2 6 2 0

1 1 2 0 0

1 0 0

(8) 2 6 4 0

2 0 0 1 1

0 0

: 6 2 ,

1 1

: 6 4 ,

1 1

: 6 6

1

m j m j

m

m j m j

m

m j

m

a a x

X q b q b a m j

b b z

a

a x

Y q b a m j q b

b z b

x a

Z a m j q b q b

z b

+ − + − −

+ − − + −

− + −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠ ⎜ ⎝ + − ⎟ ⎠ ⎜ ⎝ + + − ⎟ ⎠

⎛ ⎞⎛ ⎞

⎛ ⎞

= ⎜ ⎝ + − ⎟ ⎠⎝ ⎜ + + − ⎟⎜ ⎠⎝ + − ⎟ ⎠

⎛ ⎞⎛ ⎞

= ⎜ ⎝ + + − ⎟⎜ ⎠⎝ + − ⎟ ⎠

6 2 1

1

1 .

m j

a

b

⎛

+ −

⎞

⎜ + − ⎟

⎝ ⎠

Since

lim

_n_→∞

X

_m^{( )}^k

= lim

_n_→∞

Y

_m^{( )}^k

= lim

_n_→∞

Z

_m^{( )}^k

= ∞ ,

^{and by} (14), (18), (20), (22), (24), (26) and (28), this completes the proof.

3. Conclusion

In this paper, we investigate an extension of the second equation in (1), that is, the system given in (7). After that we reduce this system to the first-order linear equations and then we obtain explicit solutions of the related system. Additionally, we determine the forbidden set of the initial values

x

₋_i,

y

₋_i,

z

₋_i

( ⁱ ⁼ ^{0,1, 2} )

^{and also}

study asymptotic behavior of the solutions using their explicit forms. Thus, we extend some recent results in the literature.

In the future studies on systems of difference equations, we except that the following topics will bring new insight:

The system in (7) can be extended to higher-dimensional systems;

1)

The system in (7) can be extended to higher-order systems;

2)

and finally, one can investigate behaviors and solubility of these extended 3)

systems.

References

Diamandescu, A. (2009) A note on existence of Psi-bounded solutions for linear difference equations on Z. Kuwait Journal of Science & Engineering, 36(2A):35-49.

Elabbasy, E.M., El-Metwally, H. & Elsayed, E.M. (2011) Global behavior of the solutions of some difference equations. Advances in Difference Equations, 2011:28.

El-Metwally, H. (2013). Solutions form of some rational systems of difference equations. Discrete

(16)

Papaschinopoulos, G. & Schinas, C.J. (1998) On the behaviour of the solutions of a system of two nonlinear difference equations. Communications Applied Nonlinear Analysis, 5(2):47-59.

Papaschinopoulos, G. & Stefanidou, G. (2010) Asymptotic behavior of the solutions of a class of rational difference equations. International Journal of Difference Equations, 5(2):233-249.

Sedaghat, H. (2009) Global behaviours of rational difference equations of orders two and three with quadratic terms. Journal of Difference Equations and Applications, 15(3):215-224.

Stević, S. (2004) More on a rational recurrence relation. Applied Mathematics, E-Notes, 4:80-84.

Stević, S. (2011) On a system of difference equations. Applied Mathematics and Computation, 218:3372- 3378.

Stević, S. (2012) On a third-order system of difference equations. Applied Mathematics and Computation, 218:7649-7654.

Stević, S., Diblik, J., Iričanin, B.I. & Šmarda, Z. (2012) On a third-order system of difference equations with variable coefficients. Abstract Applied Analysis, Article ID 508523: 22 pages.

Stević S., Diblik J., Iričanin B.I. & Šmarda, Z. (2014) On a solvables system of rational difference equations. Journal of Difference Equations and Applications, 20(5-6):811-825.

Taskara, N., Uslu, K. & Tollu, D.T. (2011) The periodicity and solutions of the rational difference equation with periodic coefficients, Computers & Mathematics with Applications, 62:1807-1813.

Tollu, D.T., Yazlik, Y. & Taskara, N. (2013) On the solutions of two special types of Riccati difference equation via Fibonacci numbers. Advances in Difference Equations, 2013:174.

Yalcinkaya, I., Cinar, C. & Simsek D. (2008) Global asymptotic stability of a system of difference equations. Applicable Analysis, 87(6):677-687.

Yang, X., Su, W., Chen, B., Megson, G.M. & Evans, D.J. (2005) On the recursive sequence

1 2

.

n n

ax bx n c dx x

x

⁻ ⁻

− −

+

=

+ Applied Mathematics and Computation, 162:1485-1497.

Yazlik, Y., Elsayed, E.M. & Taskara, N. (2014) On the behaviour of the solutions of difference equation systems. Journal of Computational Analysis and Applications, 16(5):932-941.

Yazlik, Y. (2014) On the solutions and behavior of rational difference equations. Journal of Computational Analysis and Applications, 17(3):584-594.

Submitted : 05/05/2014 Revised : 05/05/2015 Accepted : 28/05/2015

(17)

œUF _« w ö WO d ôœUF ÂUEM ‰uK ‰u

«dJ w U$

³

¨u u u —u s Ë—œ

²

¨p “U 5 U

^*¨1

UO d ≠ dON H ≠ UO{U d « r ≠ »«œü«Ë ÂuKF « WOK ≠ wKO ”U JO w U dON H WF U

¹

ÆUO d ≠ WO u ≠ V U(« ÂuK ≠ UO{U d « r ≠ ÂuKF « WOK ≠ s b « r$ ÊUJ —√ WF U

²

ÆUO d ≠WO u ≠ UO{U d « r ≠ÂuKF « WOK ≠‚u K WF U

³

[email protected]

∫n RLK w Ëd J ù« b d «

^*

Wö

Ë ÆW d{ œËb t œUF _« w ö WO d ôœUF ÂUEM W d ‰uK vK Y « «c w qB W dB « ‰UJ _« Â«b U ÂuI UL Æ «—uAM« w …d u « ZzU M « iF lO u UM uK ÂuI

Æn dF « WM ÂUEM « ‰uK( »—UI*« „uK « W «—b ‰uK K

On the solutions of a three-dimensional system of difference equations