Qualitative Properties of Difference Equation of Order Six

where the initial conditions x−5, x−4, x−3, x−2, x−1, x0 are arbitrary positive real numbers and α, β, γ, δ are constants. In addition, we obtain the form of solution of some special cases. Recently, there has been great interest in studying difference equation systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economics, probability theory, genetics, psychology, and so forth. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solution (see [1–9] and the references cited therein). Recently, a great effort has been made in studying the qualitative analysis of rational difference equations and rational difference system (see [10–23]). Elabbasy et al. [8] studied the boundedness, global stability, periodicity character and gave the solution of some special cases of the difference equation.


Introduction
This paper deals with behavior of the solutions of the difference equation x n+1 = αx n−2 + βx 2 n−2 γx n−2 + δx n−5 , n = 0, 1, ..., (1.1) Elabbasy and Elsayed [9] investigated the local and global stability, boundedness, and gave the solution of some special cases of the difference equation x n+1 = ax n−l x n−k bx n−p + cx n−q .
In [13], Elsayed investigated the solution of the following non-linear difference equation Keratas et al. [24] obtained the solution of the following difference equation x n+1 = x n−5 1 + x n−2 x n−5 .
Saleh et al. [25] investigated the dynamics of the solution of difference equation Yalçınkaya [26] has studied the following difference equation For other related work on rational difference equations, see . Below, we outline some basic definitions and some theorems that we will need to establish our results.
Let I be some interval of real numbers and let F : I k+1 → I, be a continuously differentiable function. Then for every set of initial conditions x −k , x −k+1 , ..., x 0 ∈ I, the difference equation x n+1 = F(x n , x n−1 , ..., x n−k ), n = 0, 1, ..., (1.2) has a unique solution {x n } ∞ n=−k . A point x ∈ I is called an equilibrium point of Equation (1.2) if That is, x n = x for n ≥ 0, is a solution of Equation (1.2), or equivalently, x is a fixed point of f .
That is, x n = x for n ≥ 0, is a solution of Equation ( we have |x n − x| < for all n ≥ −k.
The following theorem will be useful for establishing the results in this paper. Theorem B. [39] Let [α, β] be an interval of real numbers assume that g : [α, β] 2 → [α, β], is a continuous function and consider the following equation

Local Stability of the Equilibrium Point of Equation (1.1)
In this section we study the local stability properties of the equilibrium point of Equation (1.1). The equilibrium points of Equation (1.1) are given by the relation Then the linearized equation of Equation (1.1) about x is Then the equilibrium point x = 0 of Equation (1.1) is locally asymptotically stable.

Proof: From Theorem A, it follows that Equation (2.2) is asymptotically stable if
and so which completes the proof.

Global Attractivity of the Equilibrium Point of Equation (1.1)
In this section we investigate the global attractivity character of solutions of Equation (1.1).

Theorem 2. The equilibrium point x of Equation (1.1) is a global attractor if
Proof: Let α, β are real numbers and assume that g : Then from Equation (1.1), we see that From Theorem B, it follows that x is a global attractor of Equation (1.1) and then the proof is complete.

Boundedness of Solutions of Equation (1.1)
In this section we study the boundedness of solution of Equation (1.1) Proof: Let {x n } ∞ n=−5 be a solution of Equation (1.1). It follows from Equation (1.1) that Then In order to confirm the result in this section we consider some numerical examples for Figure 1) and Figure 2).

First Equation
In this section we study the following special case of Equation (1.1) x 0 are arbitrary real numbers.

Proof:
We prove that the forms given are solutions of Equation (5.1) by using mathematical induction. First, we let n = 0, then the result holds. Second, we assume that the expressions are satisfied for n − 1, n − 2. Our objective is to show that the expressions are satisfied for n. That is; Now, it follows from Equation (5.1) that, Also, we see from Equation (5.1) that, Also, wee see from Equation (5.1) that, Hence, the proof is complete. We will confirm our result by considering some numerical examples assume x −5 = 3, Figure 3).

Second Equation
In this section we solve a more specific form of Equation (1.1) where the initial conditions x −5 , x −4 , x −3 , x −2 , x −1 , x 0 are arbitrary real numbers.

Proof:
The proof is the same as for Theorem 4 and is therefore omitted.
To confirm our result assume Figure 4).

Third Equation
In this section we deal with the following special case of Equation (1.1) where the initial conditions x −5 , x −4 , x −3 , x −2 , x −1 , x 0 are arbitrary real numbers.
Theorem 6. Let{x n } ∞ n=−5 be a solution of Equation(5.5) then for n = 0, 1, 2, .... Proof: For n = 0, the result holds. Now suppose that n > 0 and that our assumption holds for n − 2, n − 3. That is, Also, from Equation (5.3), we get, Hence, the proof is complete. We consider a numerical example of this special case, assume Figure 5).

Fourth Equation
In this section we deal with the form of solution of the following equation where the initial conditions , t, q, p, r, k, h, ... or, The proof is the same as the proof of Theorem 6 and thus will be omitted. Figure 6 shows the solution when x −5 = 4, x −4 = 7, x −3 = 5, x −2 = 14, x −1 = 19, x 0 = 11.

Conclusions
In this paper we investigated the global attractivity, boundedness and the solutions of some special cases of Equation (1.1). In Section 2 we proved when β(γ + 3δ) < (γ + δ) 2 (1 − α), Equation (1.1) has local stability. In Section 3 we showed that the unique equilibrium of Equation