On the behavior of the solutions of the system of rational difference equations

(1)

Full Length Research Paper

On the behavior of the solutions of the system of

rational difference equations

1 1 1 1 1 1 1 1

,

1

n n n n n n n n n n n n

x

y

x z

x

y

z

y x

x y

y

       





Abdullah Selcuk Kurbanli

1

*, Ibrahim Yalcinkaya

1

and Ali Gelisken

2 1

Department of Mathematics, Faculty of Education, Necmettin Erbakan University, Konya, Turkey.

2

Department of Mathematics, Faculty of Science, Karamanoglu Mehmetbey University, Karaman, Turkey.

Accepted 14 January, 2013

There has been a great interest in studying difference equations and systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, geometry, economics, probability theory, genetics, physics etc. In this paper, we investigate the solutions of the system of

rational difference equations 1 1 1 1 1 1

1 1

,

1

n n n n n n n n n n n n

x

y

x z

x

y

z

y x

x y

y

       





where the initial values

1

,

0

,

1

,

0

x

_

x z

_

z

are real numbers and the initial values

y

_₁

,

y

₀ are non-zero real numbers such that

x y

0 1 and

y x

0 1 are not equal to 1. We give general solutions of the system. Also, we obtain necessary conditions for every solution of the system to be limited or unlimited.

Key words: Difference equation, system of difference equations, solutions.

INTRODUCTION

In this study, we investigate the behavior of the solutions of the difference equation system

1 1 1 1 1 1 1 1 , , 1 1 n n n n n n n n n n n n x y x z x y z y x x y y              (1)

Where

x

_₁

, ,

x

₀

y

_₁

, ,

y

₀

z

_₁

,

z

₀ are real numbers such that

y x

₀ _₁



1,

x y

₀ _₁



1,

y

_₁



0

and

y

₀



0

.

Similar nonlinear systems of rational difference equations were investigated, for examples, Kurbanlı et al. (2011a) studied the behavior of the positive solutions of the system:

*Corresponding author. E-mail: akurbanli@yahoo.com.

1 1 1 1 1 1

,

1

n n n n n n n n

x

y

x

y

y x

x y

     







Cinar (2004) studied the solutions of the system:

1 1 1 1 1 , n n n n n n y x y y x y       

Kurbanli (2011b) studied the behavior of the solutions of the system: 1 1 1 1 1 1 1 1 1 , , 1 1 1 n n n n n n n n n n n n x y z x y z y x x y y z                

(2)

investigated the global asymptotic stability of the system: 1

,

1 n n n n n n

x

y

x

y

a

cy

b

dx





_





_



Kulenović and Nurkanović (2005) studied the global asymptotic behavior of the solutions of the system:

1

,

1

,

1 n n n n n n n n n

a

x

c

y

e

z

x

y

z

b

y

d

z

f

x

  









Zhang et al. (2006) investigated the behavior of the positive solutions of the system of difference equations:

1 1 1

1 ,

n n n n p n r n s

y

x

A

y

A

y

x

y

     

 



Zhang et al. (2007) studied the boundedness, the persistence and global asymptotic stability of the positive solutions of the system:

1

,

1 n m n m n n n n

y

x

A

y

A

x

y

  

 



 



Yalcinkaya (2008) studied the global asmptotic stability of the system: 1 1 1 1 1 1

,

n n n n n n n n n n

t z

a

z t

a

z

t

z

t

     









MAIN RESULTS Theorem 1 Let

y

₀



a y

,

_₁



b x

,

₀



c x

,

_₁



d z

,

₀



e

, z

_₁



f

be real numbers such that

ad



1, cb 1,



a



0

,

b



0

and let

( ,

x y z

_n _n

,

_n

)

be a solution of the system of Equation 1. Then all solutions of of Equation 1 are:

























,

1 ,

1

2 2 1 n n n

cb

c

ad

d

x

even n odd n  

(2)

























,

1 ,

1

2 2 1 n n n

ad

a

cb

b

y

even n odd n   (3) 1 0 1 0 1 0 ( 1) , ( 1) ( 1) , ( 1) n i n i n i n i i n i n n i n i n c f cb n odd a ad z d e cb n even b ad        _  _             _  _  _ 

(4) Proof

From Equation 1, we have

1 1 0 1 1 1 x d x y x ad       _, 1 1 0 1 1 1 y b y x y cb       _, 0 1 1 0 , x z cf z y a   



cb 1



c 1 c 1 cb b c 1 x y x x 0 1 01 2        ,



ad 1



a 1 a 1 ad d a 1 y x y y 0 1 0 2        ,









1 0 2 1 1 1 1 1 d e de cb x z _ad z b y b ad cb        ,









2 1 2 1 3

1 ad

d

1

1 ad

d

1 ad

a

1 ad

d

1 x

y

x













,









2 1 2 1 3 1 cb b 1 1 cb b 1 cb c 1 cb b 1 y x y y          ,













2 2 1 3 2 2 1 ₁ 1 1 cf c cb _{c f cb} x z _a z y a ad a ad  _      

So Equations 2, 3 and 4 are true for

n



1, 2,3.

Assume that Equations 2, 3 and 4 are true for

n



4,5,...,

k

. Then

(3)





k 3 k 2 2 k 2 3 k 2 1 k 2 1 ad d 1 x y x x         ,





k 2 k 2 1 k 2 2 k 2 k 2 c cb 1 1 x y x x        _,





k 3 k 2 2 k 2 3 k 2 1 k 2 1 cb b 1 y x y y         ,





k 2 k 2 1 k 2 2 k 2 k 2 aad 1 1 y x y y        and









1 0 1 0 2 2 2 3 2 1 2 2 1 1 k i i k i i k k k k k _k c f cb x z z y a ad               ,









1 1 2 1 2 2 2 2 1 1 1 k i i k i i k k k k k k d e cb x z z y b ad            

Now, we must show that Equations 2, 3 and 4 are true for 1

n k . From Equation 1, we have

















2 1 2 1 1 2 2 1 1 1 ₁ ₁ 1 1 k k k k k k k k d ad x d x d y x _{a ad} ad ad          _ _   ,

















2 1 2 1 1 2 2 1 1 1 ₁ ₁ 1 1 k k k k k k k k b cb y b y b x y _{c cb} cb cb          _ _   and                 1 0 1 1 0 0 0 1 0 0 1 1 2 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 k i i k k k i k i i i i i k k k i i i i k k k k k k k k k k k k c f cb c cb a ad c f cb c f cb x z z y a ad a ad a ad                                      Also, we have

















_

_

k 1 k k 1 k k 1 k 2 1 k 2 k 2 2 k 2 ccb 1 1 c 1 cb b 1 cb c 1 1 cb c 1 cb b 1 cb c x y x x                          _ _k1 k k 1 k k k 2 1 k 2 k 2 2 k 2 aad 1 1 a 1 ad d 1 ad a 1 1 ad a 1 ad d 1 ad a 1 y x y y                  and                 1 1 (1) 1 1 1 1 (1) 1 1 1 1 1 2 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 k i i k k k i k i i i i i k k k i i i i k k k k k k k k k k k k d e cb d ad b ad d e cb d e cb x z z b y b ad b ad cb                         _ _              

Therefore, the proof is completed by induction.

Corollary 1

Let ( ,x y z_n _n, _n) be a solution of the system of Equation 1 and let a b c d, , , be real numbers such that

1,

0 ad



cb



a



and

b



0

, then the following results hold:

(1) If 0a b c d e f, , , , , 1, then

lim

₂_n₁

lim

₂_n₁ n

x





n

y



 

and lim ₂_n lim ₂_n 0

nx ny  , (2) If 0a b c d e f, , , , , 1,

c



a

and

cb



ad

or 0a b c d e f, , , , , 1,

c



a

and

b



d

, then 2 1 lim _n 0 nz   , (3) If 0a b c d e f, , , , , 1,

c



a

and

cb



ad

or 0a b c d e f, , , , , 1,

c



a

and

d



b

, then 2 1 lim _n nz    , (4) If 0a b c d e f, , , , , 1,

c



a

and

b



d

,

then 2 1 lim _n nz   f , (5) If 0a b c d e f, , , , , 1,

d



b

and

cb



ad

or 0a b c d e f, , , , , 1,

d



b

and

cb



ad

or 0a b c d e f, , , , , 1,

f



1

,

b



d

and

c



a

,

then 2 lim _n 0 nz  , (6) If 0a b c d e f, , , , , 1,

d



b

and

cb



ad

or 0a b c d e f, , , , , 1,

d



b

and

ad



cb

, then 2 lim _n nz   , (7) If 0a b c d e f, , , , , 1,

d



b

and

c



a

,

then 2 lim _n . nz e Proof

(1) From 0a b c d, , , 1, we have  1 ad 1 0 and

1 cb 1 0

    . Hence, we obtain

2 1

, 1

lim lim lim

, ( 1) ( 1) n n n n n n n odd d x d n even ad ad         _  _  _ _  , 2 1 , 1

lim lim lim

, ( 1) ( 1) n n n n n n n odd b y b n even cb cb          _{ }     _ and 2

lim lim ( 1)n lim( 1)n 0

n

(4)

2

lim lim ( 1)n lim( 1)n 0

n

ny nc ad anad  .

(2) From 0a b c d, , , 1,

c



a

and

cb



ad

, we have 1 1 1 cb ad  _  . Hence, we obtain 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim 0.

1 ( 1) n n i i n i i n i n n n n _i n n c f cb c cb z f a ad a ad            _         _{  } _        

Similarly, from 0a b c d, , , 1,

c



a

and

b



d

, we

have 1 1 1 cb ad  _  . Hence, we obtain 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim 0.

1 ( 1) n n i i n i i n i n n n n _i n n c f cb c cb z f a ad a ad            _         _{  } _        

(3) From 0a b c d, , , 1,

c



a

and

cb



ad

, we have 1 1 1 cb ad    . Hence, we obtain 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim .

1 ( 1) n n i i n i i n i n n n n _i n n c f cb c cb z f a ad a ad            _         _{  } _         

c



a

and

d



b

, we

have 1 1 1 cb ad  _  . Hence, we obtain 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim .

1 ( 1) n n i i n i i n i n n n n _i n n c f cb c cb z f a ad a ad            _         _{  } _         

(4) From 0a b c d, , , 1,

c



a

and

b d



, we have 1 1 1 cb ad  _  . Hence, we obtain 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim .

1 ( 1) n n i i n i i n i n n n n _i n n c f cb c cb z f f a ad a ad            _         _{  } _        

(5) From 0a b c d, , , 1,

d



b

and

bc



da

, we have 1 1 1 cb ad  _  . Hence, we obtain 1 1 1 2 ( 1) 1

lim lim lim 0.

1 ( 1) n n i i n i i n i n n n n _i n n d e cb d cb z e b ad b ad        _         _{  } _        

d



b

and

bc



da

, we

have 1 ₁ 1 cb ad  _  . Hence, we obtain 1 1 1 2 ( 1) 1

lim lim lim 0.

d



b

and

c



a

, we

have 1 1 1 cb ad  _  . Hence, we obtain 1 1 1 2 ( 1) 1

lim lim lim 0.

(6) From 0a b c d, , , 1,

d



b

and

bc da



, we have 1 1 1 cb ad  _  . Hence, we obtain 1 1 1 2 ( 1) 1

lim lim lim .

1 ( 1) n n i i n i i n i n n n n _i n n d e cb d cb z e b ad b ad        _         _{  } _         

d



b

and

bc da



, we

have 1 1 1 cb ad  _  . Hence, we obtain 1 1 1 2 ( 1) 1

lim lim lim .

1 ( 1) n n i i n i i n i n n n n _i n n d e cb d cb z e b ad b ad        _         _{  } _         

(7) From 0a,b,c,d1,

d



b

and

c a



, we have 1 1 1 cb ad  _  . Hence, we obtain 1 1 1 2 ( 1) 1

lim lim lim .

1 ( 1) n n i i n i i n i n n n n _i n n d e cb d cb z e e b ad b ad        _         _{  } _         Corollary 2

(5)

and let a b c d, , ,  



1,



and

a

  

d

c

b

.

If

1, 1 (1, )

ad cb   , then

2 1 2 1 2 1

lim n lim n lim n 0

nx  ny  nz   , and 2 2 lim _n lim _n nx ny  . Proof From a b c d, , ,  



1,



,

a

  

d

c

b

and 1, 1 (1, ) ad cb   , we have 1 1 1 cb ad    . Hence, we have





lim 1n n cb   and lim



1



n n ad  . Also, we have









2 1 1

lim lim .lim 0

1 1 n n n n n n d x d ad ad     _   _  ,









2 1 1

lim lim .lim 0

1 1 n n n n n n b y b cb cb     _   _  and 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim 0.

1 ( 1) n n i i n i i n i n n n n n i n c f cb c cb z f a ad a ad            _         _{  } _         Similarly, we have









2

lim _n lim 1n lim 1n

nx nc cb cn cb   and









2

lim _n lim 1n lim 1n

ny na ad an ad  

.

Corollary 3

Let ( ,x y z_n _n, _n)be a solution of the system of Equation 1 and let a b c d, , ,  



1,



and

a



c b

,



d

. If

1, 1 (0, 1)

ad cb  , then

2 1 2 1 2 1

lim

_n

lim

_n

lim

_n

n

x





n

y





n

z



 

, and 2 2

lim

_n

lim

_n

0

n

x



n

y



. Proof From a b c d, , ,  



1,



,

a



c b

,



d

and 1, 1 (0,1) ad cb  , we have

1

1 cb

ad



_



. Hence, we have lim



1



n 0 n cb  and

lim



1 

n

0

n

ad





. Also, we have









2 1 1

lim lim .lim

1 1 n n n n n n d x d ad ad     _   _  ,









2 1 1

lim lim .lim

1 1 n n n n n n b y b cb cb     _   _   and 1 1 0 0 1 0 2 1 ( 1) 1

lim lim lim

1 ( 1) n n i i n i i n i n n n n _i n n c f cb c cb z f a ad a ad            _         _{  } _          Similarly, we have





2 lim _n lim 1n .0 0 nx nc cb c  and





2 lim n lim 1n .0 0 ny na ad a  . REFERENCES

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