On the Nonlinear Difference Equation

In this paper, we investigate some qualitative behavior of the solutions of the difference equation ∑ n k n k i n i i bx x a n c x − +

are arbitrary positive real numbers.There is a class of nonlinear difference equations, known as the rational difference equations, each of which consists of the ratio of two polynomials in the sequence terms in the same form.There has been a lot of work concerning the global asymptotics of solutions of rational difference equations [1]- [8].
Many researchers have investigated the behavior of the solution of difference equation.For example: Amleh et al. [9] has studied the global stability, boundedness and the periodic character of solutions of the equation Our aim in this paper is to extend and generalize the work in [9], [10] and [11].That is, we will investigate the global behavior of (1.1) including the asymptotical stability of equilibrium points, the existence of bounded solution, the existence of period two solution of the recursive sequence of Equation ( 1).Now we recall some well-known results, which will be useful in the investigation of (1.1) and which are given in [12].
Let I be an interval of real numbers and let 1 : , where F is a continuous function.Consider the difference equation ( ) with the initial condition 1 0 , , , .
2) An equilibrium point y of Equation (1.2) is called locally asymptotically stable if y is locally stable and there exists 0 γ > such that, if ( )  The characteristic equation associated with Equation (1.3) is , p q be an interval of real numbers and assume that is a continuous function satisfying the following properties: (a) ( ) x + is non-increasing in the first (k) terms for each p q and non-decreasing in the last term for each i x in [ ] , p q for all 1, 2, , .
, , , , , and , , , , , , C -function and let y be an equilibrium point of Equation (1.2).Then the following statements are true: 1) If all roots of Equation (1.4) lie in the open unit disk 1 λ < , then he equilibrium point y is locally asymptotically stable.2) If at least one root of Equation (1.4) has absolute value greater than one, then the equilibrium point y is unstable.
3) If all roots of Equation (1.4) have absolute value greater than one, then the equilibrium point y is a source.
is a sufficient condition for the asymptotically stable of Equation (1.5)

Local Stability of Equation (1.1)
In this section we investigate the local stability character of the solutions of Equation (1.1).Equation (1.1) has a unique nonzero equilibrium point 0 0 , .
Therefore it follows that ( ) ( ) Then the linearized equation of (1.1) about x is Then the equilibrium point of Equation (1.1) is locally stable.
Hence, the proof is completed.

Periodic Solutions
In this section we investigate the periodic character of the positive solutions of Equation (1.1).Theorem 3.1.Equation (1.1) has positive prime period-two solution only if ( )( ) Proof.Assume that there exists a prime period-two solution , , , , , p q p q of (1.1).Let 1 , .
n n x q x p + = = Since k odd − , we have .2 , , R, p q p q pq p q Assume that p and q are two distinct real roots of the quadratic equation Thus, the proof is completed.

Bounded Solution
Our aim in this section we investigate the boundedness of the positive solutions of Equation (1.1).
be a solution of Equation (1.1).We see from Equation (1.1) that On the other hand, we see that the change of variables 1 , Hence, we obtain   ( ) ) ) From (4.1) and ( 4.2) we see that for all 1.
Therefore every solution of Equation (1.1) is bounded.

Global Stability of Equation (1.1)
Our aim in this section we investigate the global asymptotic stability of Equation ( 1 , , , k f u u u is decreasing in the rest of arguments and increasing in k u .
Then from Equation (2.1), we see that ( ) ( ) It follows by Theorem (1.1) that x is a global attractor of Equation (1.1) and then the proof is complete.

Numerical Examples
For confirming the results of this section, we consider numerical examples which represent different types of solution of Equation (1.1).Example 6.1.Consider the difference equation this paper is to study with some properties of the solutions of the difference equilibrium point y of Equation (1.2) is called a global attractor if for all Figure 2, shows that Equation (1.1) which is periodic with period two.Where the initial data satisfies condition (3.1) of Theorem (3.1