2. Local Stability of 1.1

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Volume 2011, Article ID 982309,17pages doi:10.1155/2011/982309

Review Article

Solution and Attractivity for a Rational Recursive Sequence

E. M. Elsayed

^{1, 2}

1Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to E. M. Elsayed,[email protected] Received 15 February 2011; Accepted 26 March 2011

Academic Editor: Ibrahim Yalcinkaya

Copyrightq2011 E. M. Elsayed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the behavior of solution of the nonlinear diﬀerence equationx_n1 ax_n−1bxnx_n−1/cxndx_n−2,n0,1, . . . ,where the initial conditionsx₋₂,x₋₁,x0are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give specific form of the solution of four special cases of this equation.

1. Introduction

In this paper we deal with the behavior of the solution of the following diﬀerence equation:

x_n1ax_n−1 bx_nx_n−1

cxndx_n−2, n0,1, . . . , 1.1

where the initial conditionsx₋₂, x₋₁, x0 are arbitrary positive real numbers anda, b, c, dare positive constants. Also, we obtain the solution of some special cases of1.1.

Let us introduce some basic definitions and some theorems that we need in the sequel.

LetIbe some interval of real numbers and let

f:I^k1−→I, 1.2

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be a continuously diﬀerentiable function. Then for every set of initial conditionsx_−k, x_−k1, . . . , x0∈I, the diﬀerence equation

x_n1fxn, x_n−1, . . . , x_n−k, n0,1, . . . , 1.3 has a unique solution{xn}^∞_n−k1.

Definition 1.1equilibrium point. A pointx∈Iis called an equilibrium point of1.3if

xfx, x, . . . , x. 1.4

That is,x_nxforn≥0, is a solution of1.3, or equivalently,xis a fixed point off.

Definition 1.2stability. iThe equilibrium pointx of1.3is locally stable if for every >

0, there existsδ >0 such that for allx_−k, x_−k1, . . . , x₋₁, x0 ∈Iwith

|x−k−x||x−k1−x|· · ·|x0−x|< δ, 1.5

we have

|xn−x|< ∀n≥ −k. 1.6

iiThe equilibrium point x of1.3is locally asymptotically stable if x is locally stable solution of1.3and there existsγ >0, such that for allx_−k, x_−k1, . . . , x₋₁, x₀∈Iwith

|x_−k−x||x_−k1−x|· · ·|x0−x|< γ, 1.7

we have

nlim→ ∞x_n x. 1.8

iiiThe equilibrium pointxof1.3is global attractor if for allx_−k, x_−k1, . . . , x₋₁, x0∈ I, we have

nlim→ ∞xn x. 1.9

ivThe equilibrium pointx of1.3is globally asymptotically stable ifx is locally stable, andx is also a global attractor of1.3.

v The equilibrium point x of 1.3 is unstable if x is not locally stable. The linearized equation of1.3about the equilibriumx is the linear diﬀerence equation

y_n1 ^k

i0

∂fx, x, . . . , x

∂x_n−i y_n−i. 1.10

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Theorem Asee2. Assume thatp, q∈Rand k∈ {0,1,2, . . .}. Then

pq<1, 1.11 is a suﬃcient condition for the asymptotic stability of the diﬀerence equation

x_n1pxnqx_n−k0, n0,1, . . . . 1.12 Remark 1.3. Theorem A can be easily extended to a general linear equations of the form

x_nkp1x_nk−1· · ·pkxn0, n0,1, . . . , 1.13

wherep1, p2, . . . , pk∈Randk∈ {1,2, . . .}. Then1.13is asymptotically stable provided that k

i1

p_i<1. 1.14

Consider the following equation

x_n1gxn, x_n−1, x_n−2. 1.15

The following theorem will be useful for the proof of our results in this paper.

Theorem Bsee1. Leta, bbe an interval of real numbers and assume that

g :a, b³−→a, b, 1.16 is a continuous function satisfying the following properties:

agx, y, zis nondecreasing inxandyina, bfor eachz∈a, b, and is nonincreasing in z∈a, bfor eachxandyina, b;

bifm, M∈a, b×a, bis a solution of the system

MgM, M, m, mgm, m, M, 1.17

then

mM. 1.18

Then1.15has a unique equilibriumx∈a, band every solution of 1.15converges tox.

Definition 1.4periodicity. A sequence{xn}^∞_n−k is said to be periodic with periodpifx_np xn for alln≥ −k.

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Definition 1.5Fibonacci sequence. The sequence{Fm}^∞_m0{1,2,3,5,8,13, . . .}, that is,Fm F_m−1F_m−2, m≥0, F₋₂0, F₋₁1 is called Fibonacci sequence.

Recently there has been a great interest in studying the qualitative properties of rational difference equations. Some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results for rational difference equations.

However, there have not been any effective general methods to deal with the global behavior of rational difference equations of order greater than one so far. From the known work, one can see that it is extremely difficult to understand thoroughly the global behaviors of solutions of rational difference equations although they have simple formsor expressions. One can refer to3–23for examples to illustrate this. Therefore, the study of rational difference equations of order greater than one is worth further consideration.

Many researchers have investigated the behavior of the solution of diﬀerence equations, for example, Aloqeili24has obtained the solutions of the diﬀerence equation

x_n1 x_n−1

a−xnx_n−1. 1.19

Amleh et al.25studied the dynamics of the diﬀerence equation

x_n1 abx_n−1

ABx_n−2. 1.20

C¸ inar26,27got the solutions of the following diﬀerence equation

x_n1 x_n−1

±1ax_nx_n−1. 1.21

In28, Elabbasy et al. investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence

x_n1ax_n− bx_n

cxn−dx_n−1. 1.22

Elabbasy et al.29investigated the global stability, boundedness, and periodicity character and gave the solution of some special cases of the diﬀerence equation

x_n1 αx_n−k βγ_k

i0x_n−i. 1.23

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In30, Ibrahim got the form of the solution of the rational diﬀerence equation x_n1 x_nx_n−2

x_n−1abxnx_n−2. 1.24

Karatas et al.31got the solution of the diﬀerence equation x_n1 x_n−5

1x_n−2x_n−5. 1.25

Yalçinkaya and Ç inar32considered the dynamics of the difference equation x_n1 ax_n−k

bcx^p_n. 1.26

Yang33investigated the global asymptotic stability of the diﬀerence equation x_n1 x_n−1x_n−2x_n−3a

x_n−1x_n−2x_n−3a. 1.27

See also 1, 2,30, 31, 34–40. Other related results on rational diﬀerence equations can be found in32,33,41–48.

2. Local Stability of 1.1

In this section we investigate the local stability character of the solutions of1.1. Equation 1.1has a unique equilibrium point and is given by

xax bx²

cxdx, 2.1

or

x²1−acd bx², 2.2

ifcd1−a/b, then the unique equilibrium point isx0.

Letf:0,∞³ → 0,∞be a function defined by fu, v, w av buv

cudw. 2.3

Therefore it follows that fuu, v, w bdvw

cudw², fvu, v, w a bu

cudw, fwu, v, w −bduv cudw²,

2.4

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we see that

f_ux, x, x bd

cd², f_vx, x, x a b

cd, f_wx, x, x −bd

cd². 2.5

The linearized equation of1.1aboutxis

y_n1− bd cd²y_n−

a b

cd

y_n−1 bd

cd²y_n−1 0. 2.6 Theorem 2.1. Assume that

bc3d<1−acd². 2.7

Then the equilibrium point of1.1is locally asymptotically stable.

Proof. It follows from Theorem A that2.6is asymptotically stable if

bd cd²

a b cd

bd cd²

<1, 2.8

or

a b

cd 2bd

cd² <1, 2.9

and so,

bc3d

cd² <1−a. 2.10

The proof is complete.

3. Global Attractor of the Equilibrium Point of 1.1

In this section we investigate the global attractivity character of solutions of1.1.

Theorem 3.1. The equilibrium pointxof1.1is global attractor ifc1−a/b.

Proof. Letp, q be real numbers and assume that g : p, q³ → p, qis a function defined bygu, v, w avbuv/cudw, then we can easily see that the function gu, v, wis increasing inu, vand decreasing inw.Suppose thatm, M is a solution of the system

MgM, M, m, mgm, m, M. 3.1

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Then from1.1, we see that

MaM bM²

cMdm, mam bm²

cmdM, 3.2

or

M1−a bM²

cMdm, m1−a bm²

cmdM, 3.3

then

d1−aMmc1−aM²bM², d1−aMmc1−am²bm², 3.4

subtracting, we obtain

c1−a

M²−m² b

M²−m²

, c1−a/b. 3.5

Thus

Mm. 3.6

It follows from Theorem B thatxis a global attractor of1.1, and then the proof is complete.

4. Boundedness of Solutions of 1.1

In this section we study the boundedness of solutions of1.1.

Theorem 4.1. Every solution of 1.1is bounded ifab/c<1.

Proof. Let{xn}^∞_n−2be a solution of1.1. It follows from1.1that

x_n1ax_n−1 bxnx_n−1

cxndx_n−2 ≤ax_n−1bxnx_n−1 cxn

ab c

x_n−1. 4.1

Then

x_n1≤x_n−1 ∀n≥0. 4.2

Then the subsequences{x_2n−1}^∞_n0,{x2n}^∞_n0are decreasing and so are bounded from above by Mmax{x₋₂, x₋₁, x0}.

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5. Special Cases of 1.1

Our goal in this section is to find a specific form of the solutions of some special cases of1.1 whena, b, c, andd are integers and give numerical examples of each case and draw it by using MATLAB 6.5.

5.1. On the Difference Equationxn1xn−1xnxn−1/xnxn−2 In this subsection we study the following special case of1.1:

x_n1x_n−1 xnx_n−1

x_nx_n−2, 5.1

where the initial conditionsx₋₂, x₋₁, x0 are arbitrary positive real numbers.

Theorem 5.1. Let{xn}^∞_n−2be a solution of 5.1. Then forn0,1,2, . . .

x_2n−1 k

n−1 i0

F_4i3hF_4i2r F_4i2hF_4i1r

, x2nh

n−1 i0

, 5.2

where x₋₂ r, x₋₁ k, x0h, {Fm}^∞_m0{0,1,1,2,3,5,8,13, . . .}.

Proof. Forn0 the result holds. Now suppose thatn >0 and that our assumption holds for n−1, n−2. That is,

x_2n−3k

n−2 i0

, x_2n−2h

n−2 i0

,

x_2n−4h

n−3 i0

.

5.3

Now, it follows from5.1that x_2n−1x_2n−3 x_2n−2x_2n−3

x_2n−2x_2n−4 k

n−2 i0

h_n−2

i0F4i5hF4i4r/F4i4hF4i3rk_n−2

i0F4i3hF4i2r/F4i2hF_4i1r h_n−2

i0F4i5hF_4i4r/F4i4hF_4i3rh_n−3

i0F4i5hF4i4r/F4i4hF_4i3r k

n−2 i0

F4n−3hF_4n−4r/F4n−4hF_4n−5rk_n−2

i0F4i3hF_4i2r/F4i2hF_4i1r F4n−3hF_4n−4r/F4n−4hF_4n−5r 1

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k

n−2 i0

F_4n−3hF_4n−4rk_n−2

i0F_4i3hF_4i2r/F_4i2hF_4i1r F_4n−3hF_4n−4rF_4n−4hF_4n−5r

k

n−2 i0

F4n−3hF_4n−4rk_n−2

i0F4i3hF_4i2r/F4i2hF_4i1r F_4n−2hF_4n−3r

k

n−2 i0

1F_4n−3hF_4n−4r F_4n−2hF_4n−3r

k

n−2 i0

F_4n−1hF_4n−2r F_4n−2hF_4n−3r

.

5.4

Therefore

x_2n−1 k

n−1 i0

. 5.5

Also, we see from5.1that x_2nx_2n−2 x_2n−1x_2n−2

x_2n−1x_2n−3 h

n−2 i0

k_n−1

i0F4i3hF_4i2r/F4i2hF_4i1rh_n−2

i0F4i5hF_4i4r/F4i4hF_4i3r k_n−1

i0F4i3hF_4i2r/F4i2hF_4i1rk_n−2

i0F4i3hF_4i2r/F4i2hF_4i1r h

n−2 i0

F4n−1hF_4n−2r/F4n−2hF_4n−3rh_n−2

i0F4i5hF_4i4r/F4i4hF_4i3r F4n−1hF_4n−2r/F_4n−2hF_4n−3r 1

h

n−2 i0

F4n−1hF_4n−2rh_n−2

i0F4i5hF_4i4r/F4i4hF_4i3r F_4n−1hF_4n−2rF_4n−2hF_4n−3r

h

n−2 i0

F4n−1hF_4n−2rh_n−2

i0F4i5hF_4i4r/F4i4hF_4i3r F_4nhF_4n−1r

h

n−2 i0

1F_4n−1hF_4n−2r F4nhF_4n−1r

h

n−2 i0

F_4n1hF4nr F_4nhF_4n−1r

.

5.6

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×10⁴

n

x(n)

0 5 10 15 20 25 30 35 40

Plot ofx(n+1) =x(n−1) + (x(n)^∗x(n−1))/(x(n) +x(n−2))

Figure 1

Thus

x2nh

n−1 i0

. 5.7

Hence, the proof is completed.

For confirming the results of this section, we consider numerical example for x₋₂ 7,x₋₁6, x03. SeeFigure 1.

5.2. On the Difference Equationxn1xn−1xnxn−1/xn−xn−2

In this subsection we give a specific form of the solutions of the diﬀerence equation

x_n1x_n−1 xnx_n−1

xn−x_n−2, 5.8

where the initial conditionsx₋₂, x₋₁, x0 are arbitrary positive real numbers withx₋₂/x0. Theorem 5.2. Let{xn}^∞_n−2be a solution of 5.8. Then forn0,1,2, . . .

x_2n−1 k

n−1 i0

F_2i3h−F_2i1r F_2i1h−F_2i−1r

, x_2nh

n−1 i0

F_2i4h−F_2i2r F_2i2h−F_2ir

, 5.9

wherex₋₂ r, x₋₁k, x0 h,{Fm}^∞_m−1{1,0,1,1,2,3,5,8,13, . . .}.

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Proof. Forn0 the result holds. Now suppose thatn >0 and that our assumption holds for n−1, n−2. That is,

x_2n−3k

n−2 i0

, x_2n−2h

n−2 i0

,

x_2n−4h

n−3 i0

F_2i4h−F_2i2r F_2i2h−F2ir

.

5.10

Now, it follows from5.8that

x_2n−1 x_2n−3 x_2n−2x_2n−3 x_2n−2−x_2n−4 k

n−2 i0

h_n−2

i0F2i4h−F_2i2r/F2i2h−F2irk_n−2

i0F2i3h−F_2i1r/F2i1h−F_2i−1r h_n−2

i0F2i4h−F_2i2r/F2i2h−F2ir−h_n−3

i0F2i4h−F_2i2r/F2i2h−F2ir k

n−2 i0

F2nh−F_2n−2r/F2n−2h−F_2n−4rk_n−2

i0F2i3h−F_2i1r/F2i1h−F_2i−1r F2nh−F_2n−2r/F_2n−2h−F_2n−4r−1

k

n−2 i0

F2nh−F_2n−2rk_n−2

i0F2i3h−F_2i1r/F2i1h−F_2i−1r F_2n−1h−F_2n−3r

k

n−2 i0

1 F2nh−F_2n−2r F_2n−1h−F_2n−3r

k

n−2 i0

F_2n−1h−F_2n−3rF2nh−F_2n−2r F_2n−1h−F_2n−3r

k

n−2 i0

F_2n1h−F_2n−1r F_2n−1h−F_2n−3r

.

5.11

Therefore

x_2n−1 k

n−1 i0

. 5.12

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Also, we see from5.8that x2nx_2n−2 x_2n−1x_2n−2

x_2n−1−x_2n−3 h

n−2 i0

k_n−1

i0F2i3h−F_2i1r/F2i1h−F_2i−1rh_n−2

i0F2i4h−F_2i2r/F2i2h−F_2ir k_n−1

i0F2i3h−F_2i1r/F2i1h−F_2i−1r−k_n−2

i0F2i3h−F_2i1r/F2i1h−F_2i−1r h

n−2 i0

F2n1h−F_2n−1r/F2n−1h−F_2n−3rh_n−2

i0F2i4h−F_2i2r/F2i2h−F_2ir F2n1h−F_2n−1r/F2n−1h−F_2n−3r−1

h

n−2 i0

F2n1h−F_2n−1rh_n−2

i0F2i4h−F_2i2r/F2i2h−F2ir F_2n1h−F_2n−1r−F_2n−1hF_2n−3r

h

n−2 i0

F2n1h−F_2n−1rh_n−2

i0F2i4h−F_2i2r/F2i2h−F_2ir F_2nh−F_2n−2r

h

n−2 i0

1F_2n1h−F_2n−1r F2nh−F_2n−2r

h

n−2 i0

F_2n2h−F2nr F_2nh−F_2n−2r

.

5.13

Thus

x_2nh

n−1 i0

. 5.14

Hence, the proof is completed.

Assume that x₋₂ 0.7,x₋₁ 1.6, x₀13.SeeFigure 2, and forx₋₂ 9,x₋₁ 5, x₀ 2.SeeFigure 3.

The following cases can be treated similarly.

5.3. On the Difference Equationx_n1x_n−1−xnx_n−1/xnx_n−2 In this subsection we obtain the solution of the following diﬀerence equation

x_n1x_n−1− x_nx_n−1

xnx_n−2, 5.15

where the initial conditionsx₋₂, x₋₁, x0are arbitrary positive real numbers.

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5

×10⁸

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

n

x(n)

0 5 10 15 20 25 30 35 40

Plot ofx(n+1) =x(n−1) + (x(n)^∗x(n−1))/(x(n)−x(n−2))

Figure 2

n

x(n)

0 5 10 15 20 25 30 35 40

−6

−4

−2 0 2 4 6 8

×10⁷

Plot ofx(n+1) =x(n−1) + (x(n)^∗x(n−1))/(x(n)−x(n−2))

Figure 3

Theorem 5.3. Let{xn}^∞_n−2be a solution of 5.15. Then forn0,1,2, . . .

x_2n−1 k

n−1 i0

F_2ihF_2i1r F_2i1hF_2i2r

, x_2nh

n−1 i0

, 5.16

wherex₋₂r, x₋₁k, x0h, {Fm}^∞_m0{0,1,1,2,3,5,8,13, . . .}.

Figure 4shows the solution whenx₋₂ 3, x₋₁ 7, x012.

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0 2 4 6 8 10 12

n

x(n)

0 5 10 15 20 25 30 35 40

Plot ofx(n+1) =x(n−1)−(x(n)^∗x(n−1))/(x(n) +x(n−2))

Figure 4

n

x(n)

0 5 10 15 20 25 30 35 40

−25

−20

−15

−10

−5 0 5 10 15 20 25

Plot ofx(n+1) =x(n−1)−(x(n)^∗x(n−1))/(x(n)−x(n−2))

Figure 5

5.4. On the Difference Equationxn1xn−1−xnxn−1/xn−xn−2 In this subsection we give the solution of the following special case of1.1

x_n1x_n−1− xnx_n−1

x_n−x_n−2 5.17

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where the initial conditionsx₋₂, x₋₁, x0 are arbitrary real numbers. withx₋₂/x0, x₋₂, x₋₁, x0/0.

Theorem 5.4. Let{xn}^∞_n−2 be a solution of 5.17. Then every solution of 5.17is periodic with period 12. Moreover{xn}^∞_n−2takes the form

r, k, h, kr

r−h, h−r, hk

r−h,−r,−k,−h, kr

h−r,−h−r, hk

h−r, r, k, h, . . .

5.18

wherex₋₂r, x₋₁k, x0h.

Figure 5shows the solution whenx₋₂ 3, x₋₁ 7, x02.

6. Conclusion

This paper discussed global stability, boundedness, and the solutions of some special cases of 1.1. InSection 2we proved whenbc3d<1−acd²,1.1local stability. InSection 3 we showed that the unique equilibrium of1.1is globally asymptotically stable ifc1−a/b.

InSection 4we proved that the solution of1.1is bounded ifab/c <1. InSection 5we gave the form of the solution of four special cases of1.1and gave numerical examples of each case and drew them by using Matlab 6.5.

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10 H. El-Metwally, E. A. Grove, G. Ladas, R. Levins, and M. Radin, “On the diﬀerence equationxn1 αx_n−1e^−xn,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4623–4634, 2001.

11 H. El-Metwally, “Global behavior of an economic model,” Chaos, Solitons and Fractals, vol. 33, no. 3, pp. 994–1005, 2007.

12 H. El-Metwally, E. A. Grove, G. Ladas, and H. D. Voulov, “On the global attractivity and the periodic character of some diﬀerence equations,” Journal of Diﬀerence Equations and Applications, vol. 7, no. 6, pp. 837–850, 2001.

13 E. M. Elsayed, “On the solution of recursive sequence of order two,” Fasciculi Mathematici, no. 40, pp.

5–13, 2008.

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14 E. M. Elsayed, “Dynamics of a recursive sequence of higher order,” Communications on Applied Nonlinear Analysis, vol. 16, no. 2, pp. 37–50, 2009.

15 E. M. Elsayed, “Dynamics of recursive sequence of order two,” Kyungpook Mathematical Journal, vol.

50, no. 4, pp. 483–497, 2010.

16 E. M. Elsayed, “On the diﬀerence equationx_n1 x_n−5/−1x_n−2x_n−5,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 33–36, pp. 1657–1664, 2008.

17 E. M. Elsayed, “Qualitative behavior of a diﬀerence equation of order three,” Acta Scientiarum Mathematicarum, vol. 75, no. 1-2, pp. 113–129, 2009.

18 E. M. Elsayed, “Qualitative behavior of a rational recursive sequence,” Indagationes Mathematicae. New Series, vol. 19, no. 2, pp. 189–201, 2008.

19 E. Elsayed, “On the global attractivity and the solution of recursive sequence,” Studia Scientiarum Mathematicarum Hungarica, vol. 47, no. 3, pp. 401–418, 2010.

20 E. M. Elsayed, “Qualitative properties for a fourth order rational diﬀerence equation,” Acta Applicandae Mathematicae, vol. 110, no. 2, pp. 589–604, 2010.

21 E. M. Elsayed, “On the global attractivity and the periodic character of a recursive sequence,”

Opuscula Mathematica, vol. 30, no. 4, pp. 431–446, 2010.

22 E. M. Elsayed, B. Iriˇcanin, and S. Stevi´c, “On the max-type equation,” Ars Combinatoria, vol. 95, pp.

187–192, 2010.

23 A. E. Hamza and R. Khalaf-Allah, “On the recursive sequencexn1Ak

ilxn−2i−1/BC_k−1

ilxn−2i,”

Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1726–1731, 2008.

24 M. Aloqeili, “Dynamics of a rational diﬀerence equation,” Applied Mathematics and Computation, vol.

176, no. 2, pp. 768–774, 2006.

25 A. M. Amleh, V. Kirk, and G. Ladas, “On the dynamics ofx_n1 abxn−1/ABxn−2,” Mathematical Sciences Research Hot-Line, vol. 5, no. 7, pp. 1–15, 2001.

26 C. C¸ inar, “On the positive solutions of the diﬀerence equationx_n1 x_n−1/1axnx_n−1,” Applied Mathematics and Computation, vol. 158, no. 3, pp. 809–812, 2004.

27 C. C¸ inar, “On the positive solutions of the diﬀerence equationx_n1x_n−1/−1axnx_n−1,” Applied Mathematics and Computation, vol. 158, no. 3, pp. 809–812, 2004.

28 E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the diﬀerence equationx_n1axn−bxn/cxn− dx_n−1,” Advances in Diﬀerence Equations, vol. 2006, Article ID 82579, 10 pages, 2006.

29 E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the diﬀerence equationx_n1 αx_n−k/β γ_k

i0x_n−i,” Journal of Concrete and Applicable Mathematics, vol. 5, no. 2, pp. 101–113, 2007.

30 T. F. Ibrahim, “On the third order rational diﬀerence equationx_n1 xnx_n−2/xn−1abxnx_n−2,”

International Journal of Contemporary Mathematical Sciences, vol. 4, no. 25–28, pp. 1321–1334, 2009.

31 R. Karatas, C. Cinar, and D. Simsek, “On positive solutions of the diﬀerence equationxn1xn−5/1 x_n−2x_n−5,” International Journal of Contemporary Mathematical Sciences, vol. 1, no. 9–12, pp. 495–500, 2006.

32 I. Yalçinkaya and C. Ç inar, “On the dynamics of the difference equationxn1 axn−k/bcxn^p,”

Fasciculi Mathematici, no. 42, pp. 141–148, 2009.

33 X. Yang, “On the global asymptotic stability of the diﬀerence equation x_n1 xn−1x_n−2x_n−3 a/xn−1xn−2xn−3a,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 857–861, 2005.

34 M. R. S. Kulenović and Z. Nurkanović, “Global behavior of a three-dimensional linear fractional system of difference equations,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp.

673–689, 2005.

35 W.-T. Li and H.-R. Sun, “Dynamics of a rational diﬀerence equation,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 577–591, 2005.

36 A. Rafiq, “Convergence of an iterative scheme due to Agarwal et al,” Rostocker Mathematisches Kolloquium, no. 61, pp. 95–105, 2006.

37 M. Saleh and M. Aloqeili, “On the diﬀerence equationy_n1 Ayn/y_n−k withA < 0,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 359–363, 2006.

38 D. Simsek, C. Cinar, and I. Yalcinkaya, “On the recursive sequencex_n1x_n−3/1xn−1,” International Journal of Contemporary Mathematical Sciences, vol. 1, no. 9–12, pp. 475–480, 2006.

39 C. Wang and S. Wang, “Global behavior of equilibrium point for A class of fractional diﬀerence equation,” in Proceedings of 7th Asian Control Conference (ASCC ’09), pp. 288–291, Hong Kong, August 2009.

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40 C.-Y. Wang, S. Wang, and X.-P. Yan, “Global asymptotic stability of 3-species mutualism models with diﬀusion and delay eﬀects,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 317298, 20 pages, 2009.

41 C.-Y. Wang, F. Gong, S. Wang, L.-R. Li, and Q.-H. Shi, “Asymptotic behavior of equilibrium point for a class of nonlinear diﬀerence equation,” Advances in Diﬀerence Equations, vol. 2009, Article ID 214309, 8 pages, 2009.

42 I. Yalçınkaya, “Global asymptotic stability in a rational difference equation,” Selçuk Journal of Applied Mathematics, vol. 6, no. 2, pp. 59–68, 2005.

43 I. Yalc¸inkaya, “On the diﬀerence equationxn1 αxn−2/x^k_n,” Polytechnica Posnaniensis. Institutum Mathematicum. Fasciculi Mathematici, no. 42, pp. 133–139, 2009.

44 I. Yalc¸inkaya, “On the diﬀerence equationx_n1αxn−m/x^k_n,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 805460, 8 pages, 2008.

45 I. Yalc¸inkaya, “On the global asymptotic stability of a second-order system of diﬀerence equations,”

Discrete Dynamics in Nature and Society, vol. 2008, Article ID 860152, 12 pages, 2008.

46 I. Yalçinkaya, C. Ç inar, and M. Atalay, “On the solutions of systems of difference equations,” Advances in Difference Equations, vol. 2008, Article ID 143943, 9 pages, 2008.

47 E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequencex_n1 αβxnγxn−1/A βxnCx_n−1,” Communications on Applied Nonlinear Analysis, vol. 12, no. 4, pp. 15–28, 2005.

48 E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequencexn1 αxnβxn−1 γx_n−2γx_n−3/AxnBx_n−1Cx_n−2Dx_n−3,” Communications on Applied Nonlinear Analysis, vol. 12, pp. 15–28, 2005.

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2. Local Stability of 1.1

Review Article

Solution and Attractivity for a Rational Recursive Sequence

E. M. Elsayed

1. Introduction

2. Local Stability of 1.1

3. Global Attractor of the Equilibrium Point of 1.1

4. Boundedness of Solutions of 1.1

5. Special Cases of 1.1

6. Conclusion

References

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