Dynamics and behavior of a second order rational difference equation

Dynamics and Behavior of a Second Order RationalDifference equation

E. M. Elsayed1;2, M. M. El-Dessoky1;2, and Asim Asiri11King Abdulaziz University, Faculty of Science,Mathematics Department, P. O. Box 80203,

Jeddah 21589, Saudi Arabia.2Department 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 periodicityof solutions of the difference equation

xn+1 = axn +b+ cxn�1d+ exn�1

; n = 0; 1; :::;

where the parameters a; b; c; d and e are positive real numbers and the initial conditionsx�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 measure-ments of time-evolving variables are discrete. More signi�cantly, difference equations are usedin the study of discretization methods for differential equations. The theory of difference equa-tions has some results that have been acquired approximately as natural discrete analoguesof corresponding results of differential equations [35].The study of rational difference equations of order greater than one is quite ambitious andworthwhile since some paradigms for the development of the basic theory of the global behav-ior of nonlinear difference equations of order greater than one come from the results of rationaldifference equations. However, there have not been any useful general methods to study theglobal behavior of rational difference equations of order greater than one so far. Therefore, thestudy of rational difference equations of order greater than one deserves further consideration.Many research have been done to study the global attractivity, boundedness character, peri-odicity and the solution form of nonlinear difference equations. For example, Agarwal et al. [2]

794

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

ELSAYED ET AL 794-807

https://www.researchgate.net/publication/267192539_Periodicity_and_stability_of_solutions_of_higher_order_rational_difference_equation?el=1_x_8&enrichId=rgreq-10af39b3-38f0-486a-bd11-725050ae1244&enrichSource=Y292ZXJQYWdlOzI2NzAxMzQ0MDtBUzoxNjc2NzQ5OTQ3NjU4MjVAMTQxNjk4ODI0MjU0MA==

looked at the global stability, periodicity character and found the solution form of some specialcases of the difference equation

xn+1 = a+dxn�lxn�kb� cxn�s

:

The form of the solutions of the difference equation

xn+1 =xn�1

a� xnxn�1;

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

xn+1 = axn �bxn

cxn � dxn�1;

was investigated by Elabbasy et al in [8].Elabbasy et al. [9] studied the behavior of the difference equation, especially global stabil-ity, boundedness, periodicity character and gave the solution of some special cases of thedifference equation

xn+1 =�xn�k

� + Qki=0 xn�i

:

Karatas et al. [31] researched the behavior of the solutions of the difference equation

xn+1 =axn�(2k+2)

�a+Q2k+2i=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�mxkn

;

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 �bxn

cxn � dxn�k; xn+1 = Axn +Bxn�k +

pxn + xn�kq + 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 thedifference equation


; (1)

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

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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 2 I; thedifference equation

xn+1 = F (xn; xn�1); n = 0; 1; :::; (2)

has a unique solution fxng1n=�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 fxng1n=�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 � > 0 suchthat for all x�1; x0 2 I with

jx�1 � xj+ jx0 � xj < �;

we havejxn � xj < � for all n � �1:

(ii) The equilibrium point x of Eq.(2) is locally asymptotically stable if x is locally stable solutionof Eq.(2) and there exists > 0; such that for all x�1; x0 2 I with

jx�1 � xj+ jx0 � xj < ;

we havelimn!1

xn = x:

(iii) The equilibrium point x of Eq.(2) is global attractor if for all x�1; x0 2 I; we have

limn!1

xn = x:

(iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is locally stable, andx 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 differ-ence equations

yn+1 = pyn + qyn�1:

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wherep =

@F

@xn(x; x); q =

@F

@xn�1(x; x):

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

�2 � p�� q = 0: (3)

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

jpj < 1� q < 2: (4)

In this case the locally asymptotically stable equilibrium x is also called a sink.Now, consider the following equation

xn+1 = g(xn; xn�1): (5)

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 everysolution 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);

thenm =M:

Then Eq.(5) has a unique equilibrium point x 2 [�; �] and every solution of Eq.(5) converges tox:

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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;

ore(1� a)x2 + (d� da� c)x� b = 0:

Then the only positive equilibrium point of Eq.(1) is given by

x =(c�d+da)+

p(c�d+da)2+4be(1�a)2e(1�a) :

Theorem 3.1. The equilibrium x of Eq. (1) is locally asymptotically stable if and only if

(d+ ex)2 >jcd� bej(1� 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)

@u= a;

@f(u; v)

@v=(cd� be)(d+ ev)2

.

So, we can write@f(x; x)

@u= a = p;

@f(x; x)

@v=(cd� be)(d+ ex)2

= q:

Then the linearized equation of Eq.(1) about x is

yn+1 � pyn�1 � qyn = 0: (7)

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

jpj < 1� q < 2:

Thus,

jaj+�� (cd� be)(d+ ex)2

�� < 1;and so �� (cd� be)(d+ ex)2

�� < (1� a); a < 1;

jcd� bej < (d+ ex)2(1� a); a < 1;

orjcd� bej(1� a) < (d+ ex)

2; a < 1:

The proof is complete.

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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 fxng1n=�1 be a solution of Eq.(1). It follows from Eq.(1) that


= axn +b

d+ exn�1+

cxn�1d+ exn�1

:

Thenxn+1 � axn +

b

d+cxn�1exn�1

= axn +b

d+c

efor all n � 1.

By using a comparison, the right hand side can be written as follows

yn+1 = ayn +b

d+c

e:

So, we can writeyn = a

ny0 + constant;

and this equation is locally asymptotically stable because a < 1; and converges to the equilib-

rium point y =be+ cd

de(1� a) .Therefore

lim supn!1

xn �be+ cd

de(1� a) .

Hence, the solution is bounded.

Theorem 4.2. Every solution of Eq.(1) is unbounded if a > 1:

Proof: Let fxng1n=�1 be a solution of Eq.(1). Then from Eq.(1) we see that


> axn for all n � 1.

The right hand side can be written as follows

yn+1 = ayn ) yn = any0;

and this equation is unstable because a > 1; and limn!1

yn = 1: Then by using ratio testfxng1n=�1 is unbounded from above.

5 Existence of Periodic Solutions

In this section we investigate the existence of periodic solutions of Eq.(1). The following theo-rem states the necessary and suf�cient conditions that this equation has periodic solutions ofprime 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� ad� d)ad] > 0: (8)

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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;

andq = ap+

b+ cq

d+ eq:

Therefore,dp+ ep2 = adq + aepq + b+ cp; (9)

anddq + eq2 = adp+ aepq + b+ cq: (10)

Subtracting (10) from (9) gives

d(p� q) + e(p2 � q2) = �ad(p� q) + c(p� q).

Since p 6= q; it follows thatp+ q =

c� ad� de

: (11)

Again; adding (9) and (10) yields

d(p+ q) + e(p2 + q2) = ad(p+ q) + 2aepq + 2b+ c(p+ q);

e(p2 + q2) = (ad� d+ c)(p+ q) + 2aepq + 2b: (12)

By using (11); (12) and the relation

p2 + q2 = (p+ q)2 � 2pq for all p; q 2 R;

we obtain

e((p+ q)2 � 2pq) = (ad� d+ c)(p+ q) + 2aepq + 2b

2(1 + a)epq = �2ad(p+ q)� 2b:

Then,

pq =�(c� ad� d)ad� be

(1 + a)e2: (13)

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

t2 ��c� ad� d

e

�t�

�be+ (c� ad� d)ad

(1 + a)e2

�= 0;

et2 � (c� ad� d)t��be+ (c� ad� d)ad

(1 + a)e

�= 0; (14)

and so[c� ad� d]2 + 4[be+(c�ad�d)ad]

(1+a) > 0;

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or[c� ad� d]2 (1 + a) + 4 [be+ (c� 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 twosolution.Suppose that

p =c� ad� d+ �

2e;

andq =

c� ad� d� �2e

;

where � =q[c� ad� d]2 + 4[be+(c�ad�d)ad]

(1+a) :

We see from the inequality (i) that

[c� ad� d]2 (1 + a) + 4 [be+ (c� ad� d)ad] > 0;

which equivalents to[c� ad� d]2 + 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

x1 = aq +b+ cp

d+ ep= a

�c�ad�d��

2e

�+

b+c

�c�ad�d+�

2e

�d+e

�c�ad�d+�

2e

� :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 bycomputation 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.

801



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 bd andce , 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 followingstatements holds

(1) cd � be and c > d(1� a); a < 1: (15)

(2) cd � be and a < 1. (16)

Proof: Suppose that � and � are real numbers and assume that g : [�; �]2 �! [�; �] is afunction de�ned by

g(u; v) = au+b+ cv

d+ ev:

Then@g(u;v)@u = a; @g(u;v)

@v = (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) increasingin u; v:Let x be a solution of the equation x = g(x; x): Then from Eq.(1), we can write

x = ax+ b+cxd+ex ;

orx(1� a) = b+cx

d+ex ;

then the equatione(1� a)x2 + fd(1� a)� cgx� b = 0;

has a unique positive solution when c > d(1� a); a < 1 which is

x =(c�d(1�a))+

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 iscomplete.Case (2): Suppose that (16) is true, let � and � be real numbers and assume that g : [�; �]2 �![�; �] be a function de�ned by g(u; v) = au +

b+ cv

d+ ev, then we can easily see that the function

g(u; v) increasing in u and decreasing in v:Let (m;M) be a solution of the systemM = g(M;m) and m = g(m;M). Then from Eq.(1), wesee that

M = aM +b+ cm

d+ em; m = am+

b+ cM

d+ eM;

orM(1� a) = b+ cm

d+ em; m(1� a) = b+ cM

d+ eM;

802



thend(1� a)M + e(1� a)Mm = 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 thatM = m:

It follows by Theorem C that x is a global attractor of Eq.(1). This completes the proof of thetheorem.

7 Numerical examples

To con�rm the results of this paper, we consider numerical examples which represent differenttypes 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.

0 5 10 150

2

4

6

8

10

12

n

x(n)

p lo t of x( n+ 1) = ax( n) + ( b+ c x( n 1) ) /( d+ ex( n 1) )

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.

0 5 10 15 20 25 305

6

7

8

9

10

11

12

13

n

x(n)

p lo t o f x( n+ 1) = ax( n) + ( b+ c x( n 1) ) /( d+ ex( n 1) )

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.

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

n

x(n)


Figure 3.

Example 4. Fig. 4. shows the solutions when a = 2; b = 1; c = 11; d = 3; e = 4; x�1 = p;

x0 = q:

0@Since p; q = c�ad�d�r[c�ad�d]2+4[be+(c�ad�d)ad](1+a)

2e :

1A

0 2 4 6 8 10 12 14 16 18 20 0.4

0.2

0

0.2

0.4

0.6

0.8

1

n

x(n)


Figure 4.

AcknowledgementsThis article was funded by the Deanship of Scienti�c Research (DSR), King Abdulaziz

University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and �nan-cial support.

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Dynamics and behavior of a second order rational difference equation