Periodicity and Solution of Rational Recurrence Relation of Order Six ()
where the initial conditions 
and 
are arbitrary real numbers with 
and 
not equal to be zero. On the other hand, we will study the local stability of the solutions of Equation (1). Moreover, we give graphically the behavior of some numerical examples for this difference equation with some initial conditions.
1. Introduction
Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. Every dynamical system 
determines a difference equation and vise versa. Recently, there has been great interest in studying difference equations. One of the reasons for this is a necessity for some techniques whose can be used in investigating equations arising in mathematical models decribing real life situations in population biology, economic, probability theory, genetics, psychology, ...etc. Difference equations usually describe the evolution of certain phenomenta over the course of time. Recently there are a lot of interest in studying the global attractivity, boundedness character the periodic nature, and giving the solution of nonlinear difference equations. Recently there has been a lot of interest in studying the boundedness character and the periodic nature of nonlinear difference equations. Difference equations have been studied in various branches of mathematics for a long time. First results in qualitative theory of such systems were obtained by Poincaré and Perron in the end of nineteenth and the beginning of twentieth centuries. For some results in this area, see for example [1-13].
Although difference equations are sometimes very simple in their forms, they are extremely difficult to understand throughly the behavior of their solutions.
Many researchers have investigated the behavior of the solution of difference equations for examples.
Cinar [1,2] investigated the solutions of the following difference equations
Karatas et al. [4] gave that the solution of the difference equation
G. Ladas, M. Kulenovic et al. [12] have studied period two solutions of the difference equation
Simsek et al. [13] obtained the solution of the difference equation
Ibrahim [5] studied the third order rational difference Equation
In this paper we obtain the solution and study the periodicity of the following difference equation
, (1)
where the initial conditions 
, and 
are arbitrary real numbers with
, 
and 
not equal to be zero. On the other hand, we will study the local stability of the solutions of Equation (1). Moreover, we give graphically the behavior of some numerical examples for this difference equation with some initial conditions.
Here, we recall some notations and results which will be useful in our investigation.
Let I be some interval of real numbers and Let 
be a continuously differentiable function. Then for every set of initial conditions 
, the difference equation
(2)
has a unique solution 
[11].
Definition (1.1) A point 
is called an equilibrium point of Equation (2) if
That is, 
for
, is a solution of Equation (2), or equivalently, 
is a fixed point of F.
Definition (1.2) The difference Equation (2) is said to be persistence if there exist numbers m and M with 0 < m ≤ M < ∞ such that for any initial 
there exists a positive integer N which depends on the initial conditions such that m ≤ xn ≤ M for all n ≥ N.
Definition (1.3) (Stability)
Let I be some interval of real numbers.
1) The equilibrium point 
of Equation (2) is locally stable if for every ε > 0, there exists δ > 0 such that for 
with
we have 
for all n ≥ −k.
2) The equilibrium point 
of Equation (2) is locally asymptotically stable if 
is locally stable solution of Equation (2) and there exists γ > 0, such that for all 
with
we have 
3) The equilibrium point 
of Equation (2) is global attractor if for all
, we have 
4) The equilibrium point 
of Equation (2) is globally asymptotically stable if 
is locally stable, and 
is also a global attractor of Equation (2).
5) The equilibrium point 
of Equation (2) is unstable if 
not locally stable.
The linearized equation of Equation (2) about the equilibrium 
is the linear difference equation
Theorem (1.4) [10] Assume that 
(real numbers) and 
Then
is a sufficient condition for the asymptotic stability of the difference equation
Remark (1.5) Theorem (1.4) can be easily extended to a general linear equations of the form
where 
(real numbers) and
. Then Equation (4) is asymptotically stable provided that
.
Definition (1.6) (Periodicity)
A sequence 
is said to be periodic with period p if 
for all n ≥ −k.
2. Solution and Periodicity
In this section we give a specific form of the solutions of the difference Equation (1).
Theorem (2.1)
Let 
be a solution of Equation (1). Then Equation (1) have all solutions and the solutions are
where
.
Proof:
For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. We shall show that the result holds for n. By using our assumption for n − 1, we have the following:
Now, it follows from Equation (1) that
similarly we can derive,
Thus, the proof is completed.
Theorem (2.2)
Suppose that 
be a solution of Equation (1). Then all solutions of Equation (1) are periodic with period fourteen.
Proof:
From Equation (1), we see that
which completes the proof.
3. Stability of Solutions
In this section we study the local stability of the solutions of Equation (1).
Lemma (3.1)
Equation (1) have two equilibrium points which are 0 and 1.
Proof:
For the equilibrium points of Equation (1), we can write
Then
, i.e. 
Thus the equilibrium points of Equation (1) is are 0 and 1.
Theorem (3.2)
The equilibrium points 
and 
are unstable.
Proof:
We will prove the theorem at the equilibrium point 
and the proof at the equilibrium point 
by the same way.
Let 
be a continuous function defined by
Therefore it follows that
At the equilibrium point 
we have
, 
,
, 
,
, 
Then the linearized equation of Equation (1) about 
is
i.e.
Whose characteristic equation is
By the generalization of theorem (1.4) we have
which is impossible. This means that the equilibrium point 
is unstable. Similarly, we can see that the equilibrium point 
is unstable.
4. Numerical Examples
For confirming the results of this section, we consider numerical examples which represent different types of solutions to Equation (1).
Example 4.1
Consider x−5 = 1, x−4 = 2, x−3 = 4, x−2 = −1, x−1 = −2, and x0 = 7. See Figure 1.
Example 4.2
Consider x−5 = 1, x−4 = −1, x−3 = 2, x−2 = −6, x−1 = 5, and x0 = −4. See Figure 2.
Example 4.3
Consider x−5 = 3, x−4 = 5, x−3 = −7, x−2 = −3, x−1 = 2, and x0 = −1. See Figure 3.
Example 4.4
Consider x−5 = −4, x−4 = 3, x−3 = −2, x−2 = 9, x−1 = 17,
Figure 1. The periodicity of solutions with period 14 with unstable equilibrium points 
= 1 and 
= 0.
Figure 2. Periodicity of solutions with period 14 with unstable equilibrium points 
= 1, 
= 0.
Figure 3. Periodicity of solutions with period 14 with unstable equilibrium points 
= 1 and 
= 0.
Figure 4. The periodicity of solutions with period 14 with unstable equilibrium points 
= 1, 
= 0.
and x0 = −6. See Figure 4.
5. Acknowledgements
We want to thank the referee for his useful suggestions.