Dynamics and Behavior of a Second Order Rational

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Dynamics and Behavior of a Second Order Rational Difference equation E. M. Elsayed1;2 , M. M. El-Dessoky1;2 , and Asim Asiri1 1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected], [email protected], [email protected] ABSTRACT In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the difference equation xn+1 = axn +

b + cxn d + exn

1

;

n = 0; 1; :::;

1

where the parameters a; b; c; d and e are positive real numbers and the initial conditions x 1 and x0 are positive real numbers.

Keywords: stability, periodic solutions, boundedness, difference equations. Mathematics Subject Classi cation: 39A10 —————————————————

1

Introduction

Difference equations have been used to describe evolution phenomena since most measurements of time-evolving variables are discrete. More signi cantly, difference equations are used in the study of discretization methods for differential equations. The theory of difference equations has some results that have been acquired approximately as natural discrete analogues of corresponding results of differential equations [35]. The study of rational difference equations of order greater than one is quite ambitious and worthwhile since some paradigms for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results of rational difference equations. However, there have not been any useful general methods to study the global behavior of rational difference equations of order greater than one so far. Therefore, the study of rational difference equations of order greater than one deserves further consideration. Many research have been done to study the global attractivity, boundedness character, periodicity and the solution form of nonlinear difference equations. For example, Agarwal et al. [2]

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looked at the global stability, periodicity character and found the solution form of some special cases of the difference equation xn+1 = a +

dxn l xn b cxn

k

:

s

The form of the solutions of the difference equation xn 1 a xn xn

xn+1 =

; 1

was obtained by Aloqeili [4]. The dynamics, the global stability, periodicity character and the solution of special case of the recursive sequence bxn cxn dxn

xn+1 = axn

; 1

was investigated by Elabbasy et al in [8]. Elabbasy et al. [9] studied the behavior of the difference equation, especially global stability, boundedness, periodicity character and gave the solution of some special cases of the difference equation xn+1 =

xn Qk

k

i=0 xn

+

: i

Karatas et al. [31] researched the behavior of the solutions of the difference equation xn+1 =

axn (2k+2) Q a + 2k+2 i=0 xn

: i

In [36] Simsek et al. acquired the solution of the difference equation xn+1 =

xn 3 1 + xn

: 1

The dynamics of the difference equation xn+1 =

+

xn m ; xkn

was studied by Yalç nkaya et al. in [44]. Zayed et al. [46], [47] looked at the behavior of the following rational recursive sequences xn+1 = axn

cxn

bxn dxn

;

xn+1 = Axn + Bxn

k

k

+

pxn + xn k : q + xn k

Other related results on rational difference equations and systems can be found in refs. [1-45]. This paper aims to study the global stability character and the periodicity of solutions of the difference equation xn+1 = axn +

b + cxn d + exn

1

;

(1)

1

where the parameters a; b; c; d and e are positive real numbers and the initial conditions x

1

and x0 are positive real numbers.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

2

Some Basic Properties and De nitions

In this section, we state some basic de nitions and theorems that we need in this paper. Suppose that I is an interval of real numbers and let F :I

I ! I;

be a continuously differentiable function. Then for every set of initial conditions x

1 ; x0

difference equation xn+1 = F (xn ; xn has a unique solution fxn g1 n=

1 );

2 I; the (2)

n = 0; 1; :::;

1.

De nition 2.1. (Equilibrium Point) A point x 2 I is called an equilibrium point of Eq.(2) if x = F (x; x). That is, xn = x for n

0; is a solution of Eq.(2), or equivalently, x is a xed point of F .

De nition 2.2. (Periodicity) A sequence fxn g1 n=

1

is said to be periodic with period p if xn+p = xn for all n

1:

De nition 2.3. (Stability) (i) The equilibrium point x of Eq.(2) is locally stable if for every > 0; there exists that for all x

1 ; x0

2 I with

jx

1

xj + jx0

> 0 such

xj < ;

we have jxn

xj
0; such that for all x jx

1

xj + jx0

1;

x0 2 I with

xj < ;

we have lim xn = x:

n!1

(iii) The equilibrium point x of Eq.(2) is global attractor if for all x

1 ; x0

2 I; we have

lim xn = x:

n!1

(iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.(2). (v) The equilibrium point x of Eq.(2) is unstable if x is not locally stable. The linearized equation associated with Eq.(2) about the equilibrium pointx is the linear difference equations yn+1 = pyn + qyn 796

1:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

where p=

@F (x; x); @xn

q=

@F (x; x): @xn 1

Theorem A [34]: (Linearized Stability) (a) If both roots of the quadratic equation 2

p

(3)

q = 0:

lie in the open unit disk j j < 1, then the equilibrium x of Eq.(2) is locally asymptotically stable. (b) If at least one of the roots of Eq.(3) has absolute value greater than one, then the equilibrium

x of Eq.(2) is unstable. (c) A necessary and suf cient condition for both roots of Eq.(3) to lie in the open unit disk j j < 1, is

jpj < 1

(4)

q < 2:

In this case the locally asymptotically stable equilibrium x is also called a sink. Now, consider the following equation xn+1 = g(xn ; xn

(5)

1 ):

The following two theorems will be useful for the proof of our results in this paper. Theorem B [34]: Suppose that [ ; ] is an interval of real numbers and assume that g : [ ; ]2 ! [ ; ]; is a continuous function satisfying the following properties: (a) g(x; y) is non-decreasing in each of its arguments; (b) The equation g(x; x) = x; has a unique positive solution. Then Eq.(5) has a unique equilibrium point x 2 [ ; ] and every

solution of Eq.(5) converges to x:

Theorem C [34]: Suppose that [ ; ] is an interval of real numbers and let g : [ ; ]2 ! [ ; ]; be a continuous function that satis es the following properties : (a) g(x; y) is non-decreasing in x in [ ; ] for each y 2 [ ; ]; and is non-increasing in y 2 [ ; ]

for each x in [ ; ];

(b) If (m; M ) 2 [ ; ]

[ ; ] is a solution of the system M = g(M; m)

and

m = g(m; M );

then m = M: Then Eq.(5) has a unique equilibrium point x 2 [ ; ] and every solution of Eq.(5) converges to

x:

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3

Local Stability of the Equilibrium Point of Eq.(1)

In this section, we study the local stability character of the equilibrium point of Eq.(1). Eq.(1) has equilibrium point and is given by x = ax +

b + cx ; d + ex

or a)x2 + (d

e(1

da

c)x

b = 0:

Then the only positive equilibrium point of Eq.(1) is given by p (c d+da)+ (c d+da)2 +4be(1 x= 2e(1 a)

a)

:

Theorem 3.1. The equilibrium x of Eq. (1) is locally asymptotically stable if and only if (d + ex)2 >

jcd (1

bej ; a)

a < 1:

Proof: Let f : (0; 1)2 ! (0; 1) be a continuous function de ned by f (u; v) = au +

b + cv : d + ev

(6)

Therefore, @f (u; v) @f (u; v) (cd be) = a; = . @u @v (d + ev)2 So, we can write @f (x; x) @f (x; x) (cd be) = a = p; = = q: @u @v (d + ex)2 Then the linearized equation of Eq.(1) about x is yn+1

pyn

(7)

qyn = 0:

1

It follows by Theorem A that, Eq.(1) is asymptotically stable if and only if jpj < 1

q < 2:

Thus, jaj +

(cd be) < 1; (d + ex)2

and so (cd be) (d + ex)2 jcd or

< (1

a);

bej < (d + ex)2 (1 jcd (1

bej < (d + ex)2 ; a)

a < 1; a);

a < 1;

a < 1:

The proof is complete.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

4

Existence of Bounded and Unbounded Solutions of Eq.(1)

Here we look at the boundedness nature of solutions of Eq.(1). Theorem 4.1. Every solution of Eq.(1) is bounded if a < 1: Proof: Let fxn g1 n=

1

be a solution of Eq.(1). It follows from Eq.(1) that

xn+1 = axn + Then xn+1

axn +

b + cxn d + exn b cxn + d exn

1

= axn +

1

1

= axn +

1

b d + exn

+ 1

b c + d e

cxn 1 d + exn

for all

n

: 1

1.

By using a comparison, the right hand side can be written as follows yn+1 = ayn +

b c + : d e

So, we can write yn = an y0 + constant; and this equation is locally asymptotically stable because a < 1; and converges to the equilibbe + cd . rium point y = de(1 a) Therefore be + cd . lim supxn de(1 a) n!1 Hence, the solution is bounded. Theorem 4.2. Every solution of Eq.(1) is unbounded if a > 1: Proof: Let fxn g1 n=

1

be a solution of Eq.(1). Then from Eq.(1) we see that xn+1 = axn +

b + cxn d + exn

1

> axn

for all

n

1.

1

The right hand side can be written as follows yn+1 = ayn

) yn = an y0 ;

and this equation is unstable because a > 1; and lim yn = 1: Then by using ratio test

fxn g1 n=

5

1

n!1

is unbounded from above.

Existence of Periodic Solutions

In this section we investigate the existence of periodic solutions of Eq.(1). The following theorem states the necessary and suf cient conditions that this equation has periodic solutions of prime period two. Theorem 5.1. Eq.(1) has positive prime period two solutions if and only if (i) [c

ad

d]2 (1 + a) + 4 [be + (c

799

ad

d)ad] > 0:

(8)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Proof: Firstly, suppose that there exists a prime period two solution :::; p; q; p; q; :::; of Eq.(1). We will show that Condition (i) holds. From Eq.(1), we get p = aq +

b + cp ; d + ep

q = ap +

b + cq : d + eq

and

Therefore, dp + ep2 = adq + aepq + b + cp;

(9)

dq + eq 2 = adp + aepq + b + cq:

(10)

and Subtracting (10) from (9) gives q) + e(p2

d(p Since p 6= q; it follows that

q2) =

ad(p

c

p+q =

ad e

q) + c(p d

q).

(11)

:

Again; adding (9) and (10) yields d(p + q) + e(p2 + q 2 ) = ad(p + q) + 2aepq + 2b + c(p + q); e(p2 + q 2 ) = (ad

d + c)(p + q) + 2aepq + 2b:

(12)

By using (11); (12) and the relation p2 + q 2 = (p + q)2

2pq for all p; q 2 R;

we obtain e((p + q)2

2pq) = (ad

2(1 + a)epq =

d + c)(p + q) + 2aepq + 2b

2ad(p + q)

2b:

ad d)ad (1 + a)e2

be

Then, pq =

(c

(13)

:

Now it is obvious from Eq.(11) and Eq.(13) that p and q are the two distinct roots of the quadratic equation t2 et2

c

ad e

d

t

(c

ad

d)t

[c

ad

d]2 +

be + (c ad d)ad (1 + a)e2 be + (c ad d)ad (1 + a)e

= 0; = 0;

(14)

and so 4[be+(c ad d)ad] (1+a)

800

> 0;

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

or [c

d]2 (1 + a) + 4 [be + (c

ad

ad

d)ad] > 0:

Therefore inequality (i) holds. Conversely, suppose that inequality (i) is true. We will prove that Eq.(1) has a prime period two solution. Suppose that p= and q= where

q = [c

d]2 +

ad

c

ad d + ; 2e

c

ad d 2e

;

4[be+(c ad d)ad] : (1+a)

We see from the inequality (i) that [c

d]2 (1 + a) + 4 [be + (c

ad

ad

d)ad] > 0;

which equivalents to [c

d]2 +

ad

4[be+(c ad d)ad] (1+a)

> 0:

Therefore p and q are distinct real numbers. Set x

1

= p and x0 = q:

We would like to show that x1 = x

1

= p and x2 = x0 = q:

It follows from Eq.(1) that b + cp x1 = aq + =a d + ep

c ad d 2e

c ad d+ 2e c ad d+ d+e 2e b+c

+

:

Dividing the denominator and numerator by 2(d + ae) we get x1 = a

c ad d 2e

+

2eb+c(c ad d+ ) 2ed+e(c ad d+ ) :

Multiplying the denominator and numerator of the right side by 2ed + e (c

ad

d

) and by

computation we obtain x1 = p: Similarly as before, it is easy to show that x2 = q: Then by induction we get x2n = q

and

x2n+1 = p

for all

n

1.

Thus Eq.(1) has the prime period two solution :::;p;q;p;q;:::; where p and q are the distinct roots of the quadratic equation (14) and the proof is complete.

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6

Global Attractivity of the Equilibrium Point of Eq.(1)

In this section, the global asymptotic stability of Eq.(1) is studied. Lemma 6.1. For any values of the quotient

b d

and ec , the function f (u; v) de ned by Eq.(6) has

the monotonicity behavior in its two arguments. Proof: The proof follows by some computations and it will be omitted. Theorem 6.2. The equilibrium point x is a global attractor of Eq.(1) if one of the following statements holds be and c > d(1

(1) cd

(2) cd Proof: Suppose that

and

(15)

a); a < 1:

be and a < 1.

(16)

are real numbers and assume that g : [ ; ]2

function de ned by

! [ ; ] is a

b + cv : d + ev

g(u; v) = au + Then @g(u;v) @u

@g(u;v) @v

= a;

=

(cd be) . (d+ev)2

Now, two cases must be considered : Case (1): Suppose that (15) is true, then we can easily see that the function g(u; v) increasing in u; v: Let x be a solution of the equation x = g(x; x): Then from Eq.(1), we can write x = ax +

b+cx d+ex ;

or x(1

a) =

b+cx d+ex ;

then the equation a)x2 + fd(1

e(1

a)

has a unique positive solution when c > d(1 x=

(c d(1 a))+

cgx

b = 0;

a); a < 1 which is p

(c d(1 a))2 +4be(1 a) ; 2e(1 a)

By using Theorem B, it follows that x is a global attractor of Eq.(1) and then the proof is complete. Case (2): Suppose that (16) is true, let

and

be real numbers and assume that g : [ ; ]2 ! b + cv [ ; ] be a function de ned by g(u; v) = au + , then we can easily see that the function d + ev g(u; v) increasing in u and decreasing in v: Let (m; M ) be a solution of the system M = g(M; m) and m = g(m; M ). Then from Eq.(1), we see that M = aM +

b + cm ; d + em

m = am +

b + cM ; d + eM

M (1

b + cm ; d + em

m(1

b + cM ; d + eM

or a) =

802

a) =

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then d(1

a)M + e(1

a)M m = b + cm; d(1

a)m + e(1

a)mM = b + cM:

Subtracting we obtain (M

m)fd(1

a)(M + m) + cg = 0;

under the condition a < 1; we see that M = m: It follows by Theorem C that x is a global attractor of Eq.(1). This completes the proof of the theorem.

7

Numerical examples

To con rm the results of this paper, we consider numerical examples which represent different types of solutions to Eq. (1). Example 1. We assume that x

1

= 7; x0 = 11; a = 0:1; b = 2; c = 5; d = 3; e = 7. See Fig.

1. p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) ) 12

10

x(n)

8

6

4

2

0 0

5

10

15

n

Figure 1. Example 2. See Fig. 2, since x

1

= 13; x0 = 5; a = 0:8; b = 7; c = 2; d = 0:4; e = 2. p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) )

13 12 11

x(n)

10 9 8 7 6 5 0

5

10

15

20

25

30

n

Figure 2.

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Example 3. We consider x

1

= 2; x0 = 5; a = 1:2; b = 8; c = 5; d = 4; e = 1. See Fig. 3. p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) )

700 600 500

x(n)

400 300 200 100 0 0

2

4

6

8

10

12

14

16

18

20

n

Figure 3.

Example 0 4. Fig. 4. shows thersolutions when a = 2; b = 1 1; c = 11; d = 3; e = 4; x x0 = q: @Since p; q =

c ad d

4[be+(c ad d)ad] [c ad d] + (1+a) 2e

1

= p;

2

:A

p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) ) 1 0 .8 0 .6

x(n)

0 .4 0 .2 0 - 0 .2 - 0 .4 0

2

4

6

8

10

12

14

16

18

20

n

Figure 4.

Acknowledgements This article was funded by the Deanship of Scienti c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and nancial support.

References [1] R. P. Agarwal, Difference Equations and Inequalities, 1st edition, Marcel Dekker, New York, 1992, 2nd edition, 2000.

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[2] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Adv. Stud. Contem. Math., 17 (2) (2008), 181201. [3] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contem. Math., 20 (4), (2010), 525–545. [4] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp., 176(2), (2006), 768-774. [5] N. Battaloglu, C. Cinar and I. Yalç nkaya, The dynamics of the difference equation, ARS Combinatoria, 97 (2010), 281-288. [6] C. Cinar, On the positive solutions of the difference equation xn+1 =

xn 1 1+xn xn

1

;

Appl. Math. Comp., 150 (2004), 21-24. [7] C. Cinar, On the difference equation xn+1 =

xn 1 1+xn xn

1

; Appl. Math. Comp., 158

(2004), 813-816. [8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equation xn+1 = axn

bxn cxn dxn

1

; Adv. Differ. Equ., Volume 2006 (2006), Article ID 82579,1–

10. [9] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equations xn+1 =

+

x Qkn

; J. Conc. Appl. Math., 5(2) (2007), 101-113.

k

i=0

xn

i

[10] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Some properties and expressions of solutions for a class of nonlinear difference equation, Utilitas Mathematica, 87 (2012), 93-110. [11] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of difference equation, Adv. Differ. Equ., 2011, 2011: 28. [12] E. M. Elabbasy and E. M. Elsayed, Global attractivity and periodic nature of a difference equation, World Appl. Sci. J., 12 (1) (2011), 39–47. [13] H. El-Metwally and E. M. Elsayed, Solution and behavior of a third rational difference equation, Utilitas Mathematica, 88 (2012), 27–42. [14] H. El-Metwally and E. M. Elsayed, Form of solutions and periodicity for systems of difference equations, J. Comp. Anal. Appl., 15 (5) (2013), 852-857. [15] E. M. Elsayed, Qualitative behavior of difference equation of order two, Math. Comp. Mod., 50 (2009), 1130–1141. [16] E. M. Elsayed, Solution of a recursive sequence of order ten, General Mathematics, 19 (1) (2011), 145–162. [17] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comp. Anal. Appl., 15 (1) (2013), 73-81. [18] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Math. J., 50(2010), 483-497. [19] E. M. Elsayed, Solutions of rational difference system of order two, Math. Comp. Mod., 55 (2012), 378–384.

805

ELSAYED ET AL 794-807

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

[20] E. M. Elsayed, Qualitative behavior of difference equation of order three, Acta Sci. Math. (Szeged), 75 (1-2), 113–129. [21] E. M. Elsayed, On the Global attractivity and the solution of recursive sequence, Studia Sci. Math. Hung., 47 (3) (2010), 401-418. [22] E. M. Elsayed, Solution and behavior of a rational difference equations, Acta Universitatis Apulensis, 23 (2010), 233–249. [23] E. M. Elsayed, On the global attractivity and the periodic character of a recursive sequence, Opuscula Mathematica, 30 (4) (2010), 431–446. [24] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Disc. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [25] E. M. Elsayed, On the solutions of a rational system of difference equations, Fasciculi Mathematici, 45 (2010), 25–36. [26] E. M. Elsayed and M. M. El-Dessoky, Dynamics and behavior of a higher order rational recursive sequence, Adv. Differ. Equ., 2012, 2012:69. [27] E. M. Elsayed and H. A. El-Metwally, On the solutions of some nonlinear systems of difference equations, Adv. Differ. Equ., 2013, 2013:16, doi:10.1186/16871847-2013-161. [28] M. Mansour, M. M. El-Dessoky and E. M. Elsayed, On the solution of rational systems of difference equations, J. Comp. Anal. Appl. , 15 (5) (2013), 967-976. [29] A. Gelisken, C. Cinar and I. Yalcinkaya, On a max-type difference equation, Adv. Differ. Equ., Vol. 2010, Article ID 584890, 2010, 6 pages. [30] A. Geli¸sken, C. Cinar and I. Yalç nkaya, On the periodicity of a difference equation with maximum, Disc. Dyn. Nat. Soc., Vol. 2008, Article ID 820629, 11 pages, doi: 10.1155/2008/820629. [31] R. Karatas and C. Cinar, On the solutions of the difference equation xn+1 = axn (2k+2) Q2k+2 i=0 xn

a+

; Int. J. Contemp. Math. Sciences, 2 (13) (2007), 1505-1509. i

[32] R. Karatas, C. Cinar and D. Simsek, On positive solutions of the difference equation xn+1 =

xn 5 1+xn 2 xn

5

; Int. J. Contemp. Math. Sci., 1(10) (2006), 495-500.

[33] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [34] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [35] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations Numerical Methods and Applications, Monographs and textbooks in pure and applied mathematics, Marcel Dekker Inc, second edition, (2002). [36] D. Simsek, C. Cinar and I. Yalcinkaya, On the recursive sequence xn+1 = xn 3 1+xn 1 ;

Int. J. Contemp. Math. Sci., 1 (10) (2006), 475-480.

806

ELSAYED ET AL 794-807

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

[37] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comp. Mod., 55 (2012), 1987–1997. [38] C. Wang, S. Wang and X. Yan, Global asymptotic stability of 3-species mutualism models with diffusion and delay effects, Disc. Dyn. Nat. Soc., Volume 2009, Article ID 317298, 20 pages. [39] C. Wang, F. Gong, S. Wang, L. LI and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Adv. Differ. Equ., Volume 2009, Article ID 214309, 8 pages. [40] I. Yalç nkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations, Adv. Differ. Equ., Vol. 2008, Article ID 143943, 9 pages. [41] I. Yalç nkaya, C. Cinar and D. Simsek, Global asymptotic stability of a system of difference equations, Applicable Analysis, 87 (6) (2008), 689-699. [42] I. Yalç nkaya, B. D. Iricanin and C. Cinar, On a max-type difference equation, Disc. Dyn. Nat. Soc., Volume 2007, Article ID 47264, 10 pages. [43] I. Yalç nkaya, On the global asymptotic stability of a second-order system of difference equations, Disc. Dyn. Nat. Soc., Vol. 2008, Article ID 860152, 12 pages. [44] I. Yalç nkaya, On the difference equation xn+1 = Soc., Vol. 2008, Article ID 805460, 8 pages.

+

xn m , xkn

Disc. Dyn. Nat.

[45] I. Yalç nkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151-159. [46] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence bxn cxn dxn

xn+1 = axn

k

, Comm. Appl. Nonlin. Anal., 15 (2), (2008), 47-57.

[47] E. M. E. Zayed, Dynamics of the nonlinear rational difference equation xn+1 = Axn + Bxn

k

+

pxn +xn k q+xn k ;

Eur. J. Pure Appl. Math., 3 (2) (2010), 254-268.g

807

ELSAYED ET AL 794-807