# Qualitative Study of Solutions of Some Difference Equations

Authors: Hamdy El-Metwally; E. M. Elsayed
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We obtain in this paper the solutions of the following recursive sequencesxn+1=xnxn−3/xn−2(±1±xnxn−3), n=0,1,…, where the initial conditions are arbitrary real numbers and we study the
behaviors of the solutions and we obtained the equilibrium points of the
considered equations. Some qualitative behavior of the solutions such as the
boundedness, the global stability, and the periodicity character of the solutions
in each case have been studied.
We presented some numerical examples by giving some numerical values
for the initial values and the coefficients of each case. Some figures have been
given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the
obtained results.


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## Body

## 1. Introduction

In this paper, we obtain the solutions of the following recursive sequences:(1.1)xn+1=xnxn-3xn-2(±1±xnxn-3),n=0,1,…,
where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solutions.Recently, there has been a great interest in studying the qualitative properties of rational difference equations. For the systematical studies of rational and nonrational difference equations, one can refer to the papers [1–31] and references therein.The study of rational difference equations of order greater than one is quite challenging and rewarding because 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. Therefore, the study of rational difference equations of order greater than one is worth further consideration.Recently, Agarwal and Elsayed [4]  investigated the global stability and periodicity character and gave the solution of some special cases of the difference equation: (1.2)xn+1=a+dxn-lxn-kb-cxn-s.
Aloqeili [5] has obtained the solutions of the difference equation:(1.3)xn+1=xn-1a-xnxn-1.Çinar [8, 9] investigated the solutions of the following difference equations:(1.4)xn+1=xn-11+axnxn-1,xn+1=xn-1-1+axnxn-1.
Elabbasy et al. [10, 12] investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequences:(1.5)xn+1=axn-bxncxn-dxn-1,xn+1=αxn-kβ+γ∏i=0kxn-i.
Ibrahim [19] got the solutions of the rational difference equation:(1.6)xn+1=xnxn-2xn-1(a+bxnxn-2).
Karatas et al. [20] got the form of the solution of the difference equation:(1.7)xn+1=xn-51+xn-2xn-5.
Simsek et al. [26] obtained the solutions of the following difference equations:(1.8)xn+1=xn-31+xn-1.
Here, we recall some notations and results which will be useful in our investigation.LetI be some interval of real numbers and let(1.9)f:Ik+1→I
be a continuously differentiable function. Then, for every set of initial conditions x-k,x-k+1,…,x0∈I, the difference equation(1.10)xn+1=f(xn,xn-1,…,xn-k),n=0,1,…,
has a unique solution {xn}n=-k∞ [21].Definition 1.1 (Equilibrium Point).
A pointx¯∈I is called an equilibrium point of (1.10) if(1.11)x¯=f(x¯,x¯,…,x¯).
That is, xn=x¯, for n≥0,is a solution of (1.10), or equivalently, x¯ is a fixed point of f.Definition 1.2 (Stability).
(i) The equilibrium pointx¯of (1.10) is locally stable if, for every ϵ>0,there exists δ>0such that for all x-k,x-k+1,…,x-1,x0∈Iwith
(1.12)|x-k-x¯|+|x-k+1-x¯|+⋯+|x0-x¯|<δ,we have(1.13)|xn-x¯|<ϵ∀n≥-k.(ii)
The equilibrium pointx¯of (1.10) is locally asymptotically stable if x¯is locally stable solution of (1.10) and there exists γ>0, such that for all x-k,x-k+1,…,x-1,x0∈Iwith(1.14)|x-k-x¯|+|x-k+1-x¯|+⋯+|x0-x¯|<γ,we have(1.15)lim⁡n→∞xn=x¯.(iii)
The equilibrium pointx¯of (1.10) is global attractor if, for all x-k,x-k+1,…,x-1,x0∈I,we have(1.16)lim⁡n→∞xn=x¯.(iv)
The equilibrium pointx¯of (1.10) is globally asymptotically stable if x¯is locally stable, and x¯is also a global attractor of (1.10).(v)
The equilibrium pointx¯of (1.10) is unstable if x¯is not locally stable.The linearized equation of (1.10) about the equilibrium x¯is the linear difference equation(1.17)yn+1=∑i=0k∂f(x¯,x¯,…,x¯)∂xn-iyn-i.Theorem A (see [22]).
Assume thatpi∈R,i=1,2,…,k,andk∈{0,1,2,…}. Then,
(1.18)∑i=1k|pi|<1
is a sufficient condition for the asymptotic stability of the difference equation:
(1.19)xn+k+p1xn+k-1+⋯+pkxn=0,n=0,1,….Definition 1.3 (Periodicity).
A sequence{xn}n=-k∞is said to be periodic with period p if xn+p=xnfor all n≥-k.


## 2. On the EquationXn+1=xnxn-3/(xn-2(1+xnxn-3))

In this section, we give a specific form of the solution of the first equation in the form:(2.1)xn+1=xnxn-3xn-2(1+xnxn-3),n=0,1,…,
where the initial values are arbitrary positive real numbers.Theorem 2.1.
Let{xn}n=-3∞ be a solution of (2.1). Then, for n=0,1,…,
(2.2)x6n-3=d∏i=0n-1(1+6iad1+(6i+3)ad),x6n-2=c∏i=0n-1(1+(6i+1)ad1+(6i+4)ad),x6n-1=b∏i=0n-1(1+(6i+2)ad1+(6i+5)ad),x6n=a∏i=0n-1(1+(6i+3)ad1+(6i+6)ad),x6n+1=adc(1+ad)∏i=0n-1(1+(6i+4)ad1+(6i+7)ad),x6n+2=adb(1+2ad)∏i=0n-1(1+(6i+5)ad1+(6i+8)ad),
wherex-3=d,x-2=c,x-1=b,x0=a.Proof.
Forn=0, the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is,
(2.3)x6n-9=d∏i=0n-2(1+6iad1+(6i+3)ad),x6n-8=c∏i=0n-2(1+(6i+1)ad1+(6i+4)ad),x6n-7=b∏i=0n-2(1+(6i+2)ad1+(6i+5)ad),x6n-6=a∏i=0n-2(1+(6i+3)ad1+(6i+6)ad),x6n-5=adc(1+ad)∏i=0n-2(1+(6i+4)ad1+(6i+7)ad),x6n-4=adb(1+2ad)∏i=0n-2(1+(6i+5)ad1+(6i+8)ad).
Now, it follows from (2.1) that
(2.4)x6n-3=x6n-4x6n-7x6n-6(1+x6n-4x6n-7)=adb(1+2ad)∏i=0n-2(1+(6i+5)ad1+(6i+8)ad)b∏i=0n-2(1+(6i+2)ad1+(6i+5)ad)a∏i=0n-2(1+(6i+3)ad1+(6i+6)ad)(1+adb(1+2ad)∏i=0n-2(1+(6i+5)ad1+(6i+8)ad)b∏i=0n-2(1+(6i+2)ad1+(6i+5)ad))=(ad1+2ad)∏i=0n-2(1+(6i+5)ad1+(6i+8)ad)∏i=0n-2(1+(6i+2)ad1+(6i+5)ad)a∏i=0n-2(1+(6i+3)ad1+(6i+6)ad)(1+ad(1+2ad)∏i=0n-2(1+(6i+2)ad1+(6i+8)ad))=(ad(1+(6n+2)ad))(a∏i=0n-2(1+(6i+3)ad1+(6i+6)ad))(1+ad(1+(6n+2)ad))=d(∏i=0n-2(1+(6i+3)ad1+(6i+6)ad))((1+(6n+2)ad)+ad)=d∏i=0n-2(1+(6i+6)ad1+(6i+3)ad)1(1+(6n+3)ad).
Hence, we have
(2.5)x6n-3=d∏i=0n-1(1+6iad1+(6i+3)ad).
Similarly,
(2.6)x6n+1=x6nx6n-3x6n-2(1+x6nx6n-3)=ad∏i=0n-1(1+(6i+3)ad1+(6i+6)ad)d∏i=0n-1(1+6iad1+(6i+3)ad)c∏i=0n-1(1+(6i+1)ad1+(6i+4)ad)(1+a∏i=0n-1(1+(6i+3)ad1+(6i+6)ad)d∏i=0n-1(1+6iad1+(6i+3)ad))=ad∏i=0n-1(1+6iad1+(6i+6)ad)(c∏i=0n-1(1+(6i+1)ad1+(6i+4)ad))(1+ad∏i=0n-1(1+6iad1+(6i+6)ad))=(ad(1+6nad))(c∏i=0n-1(1+(6i+1)ad1+(6i+4)ad))(1+ad1+6nad)=ad(c∏i=0n-1(1+(6i+1)ad1+(6i+4)ad))(1+6nad+ad)=∏i=0n-1(1+(6i+4)ad1+(6i+1)ad)(adc(1+7nad)).
Hence, we have
(2.7)x6n+1=adc(1+ad)∏i=0n-1(1+(6i+4)ad1+(6i+7)ad).
Similarly, one can easily obtain the other relations. Thus, the proof is completed.Theorem 2.2.
Equation (2.1) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.Proof.
For the equilibrium points of (2.1), we can write
(2.8)x¯=x¯2x¯(1+x¯3).
Then, we have
(2.9)x¯2(1+x¯2)=x¯2,x¯2(1+x¯2-1)=0,
or
(2.10)x¯4=0.
Thus the equilibrium point of (2.1) is x¯=0.
Letf:(0,∞)3→(0,∞) be a function defined by
(2.11)f(u,v,w)=uwv(1+uw).
Therefore, it follows that
(2.12)fu(u,v,w)=wv(1+uw)2,fv(u,v,w)=-uwv2(1+uw),fw(u,v,w)=uv(1+uw)2,
we see that
(2.13)fu(x¯,x¯,x¯)=1,fv(x¯,x¯,x)=1,fw(x¯,x¯,x¯)=1.
The proof follows by using Theorem A.Numerical Examples
For confirming the results of this section, we consider numerical examples which represent different types of solutions to (2.1).
Example 2.3.
We assumex-3=11,x-2=7,x-1=13,x0=3, (see Figure 1).Example 2.4.
See Figure2, since x-3=2,x-2=9,x-1=3,x0=5.Figure 1Figure 2


## 3. On the EquationXn+1=xnxn-3/(xn-2(-1+xnxn-3))

In this section, we obtain the solution of the second equation in the form:(3.1)xn+1=xnxn-3xn-2(-1+xnxn-3),n=0,1,…,
where the initial values are arbitrary nonzero real numbers with x0x-3≠1.Theorem 3.1.
Let{xn}n=-3∞ be a solution of (3.1). Then, (3.1) has unboundedness solution and, for n=0,1,…,(3.2)x6n-3=d(-1+ad)n,x6n-2=c(-1+ad)n,x6n-1=b(-1+ad)n,x6n=a(-1+ad)n,x6n+1=adc(-1+ad)n+1,x6n+2=adb(-1+ad)n,
wherex-3=d,x-2=c,x-1=b,x0=a.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is,
(3.3)x6n-9=d(-1+ad)n-1,x6n-8=c(-1+ad)n-1,x6n-7=b(-1+ad)n-1,x6n-6=a(-1+ad)n-1,x6n-5=adc(-1+ad)n,x6n-4=adb(-1+ad)n-1.
Now, it follows from (3.1) that
(3.4)x6n-3=x6n-4x6n-7x6n-6(-1+x6n-4x6n-7)=(ad/b)(-1+ad)n-1b/(-1+ad)n-1a(-1+ad)n-1(-1+(ad/b)(-1+ad)n-1b/(-1+ad)n-1)=d(-1+ad)n-1(-1+ad)=d(-1+ad)n,x6n-2=x6n-3x6n-6x6n-5(-1+x6n-3x6n-6)=(d/(-1+ad)n)a(-1+ad)n-1ad/c(-1+ad)n(-1+(d/(-1+ad)n)a(-1+ad)n-1)=c(-1+ad)n-1(-1+(ad/(-1+ad)))(-1+ad)(-1+ad)=c(-1+ad)n,x6n-1=x6n-2x6n-5x6n-4(-1+x6n-2x6n-5)=c(-1+ad)nad/c(-1+ad)n(ad/b)(-1+ad)n-1(-1+c(-1+ad)n(ad/c(-1+ad)n))=b(-1+ad)n-1(-1+ad)=b(-1+ad)n.
Similarly, one can easily obtain the other relations. Thus, the proof is completed.Theorem 3.2.
Equation (3.1) has a periodic solution of period six iff ad=2 and will take the form:   {d,c,b,a,ad/c,ad/b,d,c,b,a,ad/c,ad/b,…}.Proof.
First suppose that there exists a prime period six solution:(3.5)d,c,b,a,adc,adb,d,c,b,a,adc,adb,…,
of (3.1), we see from the form of the solution of (3.1) that
(3.6)d=d(-1+ad)n,c=c(-1+ad)n,b=b(-1+ad)n,a=a(-1+ad)n,adc=adc(-1+ad)n+1,adb=adb(-1+ad)n,
or
(3.7)(-1+ad)n=1.
Then,
(3.8)ad=2.
Second, assume that ad=2. Then, we see from the form of the solution of (3.1) that
(3.9)x6n-3=d,x6n-2=c,x6n-1=b,x6n=a,x6n+1=adc,x6n+2=adb.
Thus, we have a periodic solution of period six and the proof is complete.Theorem 3.3.
Equation (3.1) has two equilibrium points which are 0,23 and these equilibrium points are not locally asymptotically stable.Proof.
For the equilibrium points of (3.1), we can write
(3.10)x¯=x¯2x¯(-1+x¯2).
Then, we have
(3.11)x¯2(-1+x¯2)=x¯2,
or
(3.12)x¯2(x¯2-2)=0,
Thus, the equilibrium points of (3.1) are 0,±2.
Letf:(0,∞)3→(0,∞) be a function defined by
(3.13)f(u,v,w)=uwv(-1+uw).
Therefore, it follows that
(3.14)fu(u,v,w)=-wv(-1+uw)2,fv(u,v,w)=-uwv2(-1+uw),fw(u,v,w)=-uv(-1+uw)2,
we see that
(3.15)fu(x¯,x¯,x¯)=-1,fv(x¯,x¯,x)=±1,fw(x¯,x¯,x¯)=-1.
The proof follows by using Theorem A.Numerical Examples
Here, we will represent different types of solutions of (3.1).
Example 3.4.
We considerx-3=2,x-2=9,x-1=3,x0=5, (see Figure 3).Example 3.5.
See Figure4 since x-3=7,x-2=2,x-1=8,x0=2/7.Figure 3Figure 4The following cases can be proved similarly.


## 4. On the EquationXn+1=xnxn-3/(xn-2(1-xnxn-3))

In this section, we get the solution of the third following equation:(4.1)xn+1=xnxn-3xn-2(1-xnxn-3),n=0,1,…,
where the initial values are arbitrary positive real numbers.Theorem 4.1.
Let{xn}n=-3∞ be a solution of (4.1). Then, for n=0,1,…,
(4.2)x6n-3=d∏i=0n-1(1-6iad1-(6i+3)ad),x6n-2=c∏i=0n-1(1-(6i+1)ad1-(6i+4)ad),x6n-1=b∏i=0n-1(1-(6i+2)ad1-(6i+5)ad),x6n=a∏i=0n-1(1-(6i+3)ad1-(6i+6)ad),x6n+1=adc(1-ad)∏i=0n-1(1-(6i+4)ad1-(6i+7)ad),x6n+2=adb(1-2ad)∏i=0n-1(1-(6i+5)ad1-(6i+8)ad).Theorem 4.2.
Equation (4.1) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.Example 4.3.
Assume thatx-3=1,x-2=8,x-1=3,x0=9,(see Figure 5).Figure 5Example 4.4.
See Figure6 since x-3=11,x-2=8,x-1=18,x0=9.Figure 6


## 5. On the EquationXn+1=xnxn-3/(xn-2(-1-xnxn-3))

Here, we obtain a form of the solutions of the equation(5.1)xn+1=xnxn-3xn-2(-1-xnxn-3),n=0,1,…,
where the initial values are arbitrary nonzero real numbers with x-3x0≠-1.Theorem 5.1.
Let{xn}n=-3∞ be a solution of (5.1). Then, (5.1) has unboundedness solution and, for n=0,1,…,
(5.2)x6n-3=d(-1-ad)n,x6n-2=c(-1-ad)n,x6n-1=b(-1-ad)n,x6n=a(-1-ad)n,x6n+1=adc(-1-ad)n+1,x6n+2=adb(-1-ad)n.Theorem 5.2.
Equation (5.1) has a periodic solution of period six if and only if ad=-2 and will take the form:{d,c,b,a,ad/c,ad/b,d,c,b,a,ad/c,ad/b,…}.Theorem 5.3.
Equation (5.1) has a unique equilibrium point which is 0,and this equilibrium point is not locally asymptotically stable.Example 5.4.
Considerx-3=6,x-2=8,x-1=12,x0=4,(see Figure 7).Figure 7Example 5.5.
Figure8 shows the solutions when x-3=-6,x-2=11,x-1=3,x0=2/6.Figure 8



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