DYNAMICAL BEHAVIOR AND SOLUTION OF NONLINEAR DIFFERENCE EQUATION VIA FIBONACCI SEQUENCE∗

In this paper, we study the behavior of the difference equation xn+1 = axn + bxnxn−1 cxn−1 + dxn−2 , n = 0, 1, . . . , where the initial conditions x−2, x−1, x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.


Introduction
In this paper, we deal with the behavior of the solutions of the following difference equation x n+1 = ax n + bx n x n−1 cx n−1 + dx n−2 , n = 0, 1, ..., (1.1) where the initial conditions x −2 , x −1 , x 0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we obtain the solution of some special cases of the same equation. Let us introduce some basic definitions and some theorems that we need in the sequel.
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 [34]. That is, x n = x for n ≥ 0, is a solution of Eq. (1.2), or equivalently, x is a fixed point of f .
we have |x n − x| < for all n ≥ −k.
Consider the following equation The following theorem will be useful for the proof of our results in this paper.
Theorem B ( [35]). Let [a, b] be an interval of real numbers and assume that is a continuous function satisfying the following properties : 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 et al. [4] investigated the global stability, periodicity character and gave the solution of some special cases of the difference equation Aloqeili [6] has obtained the solutions of the difference equation Cinar [12,13] deal with the solutions of the following difference equations Elabbasy et al. [17,18] investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequences Ibrahim [27] has got the solutions of the rational difference equation .
The study of these equations is quite challenging and rewarding and is still in its infancy. We believe that the nonlinear rational difference equations are of paramount importance in their own right, and furthermore we believe that these results about such equations over prototypes for the development of the basic theory of the global behavior of nonlinear rational difference equations.

Local Stability of Eq. (1.1)
In this section we investigate the local stability character of the solutions of Eq. (1.1). Eq. (1.1) has a unique equilibrium point and is given by or, Therefore it follows that we see that Then the equilibrium point of Eq. (1.1) is locally asymptotically stable.
Proof. It is follows by Theorem A that, Eq.
The proof is completed. Subtracting we obtain It follows by Theorem B that x is a global attractor of Eq. (1.1) and then the proof is completed.

Boundedness of solutions of Eq. (1.1)
In this section, we study the boundedness of solutions of Eq. (1.1).
Then x n+1 ≤ x n for all n ≥ 0.
Then the sequence {x n } ∞ n=0 is decreasing and so are bounded from above by M = max{x −2 , x −1 , x 0 }.

First Equation
In this subsection, we deal with the following special case of Eq. (1.1) where the initial conditions x −2 , x −1 , x 0 are arbitrary positive real numbers.  Proof. For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1, n − 2. That is; Now, it follows from Eq. (5.1) that Also, from Eq. (5.1), we see that Hence, the proof is completed. For confirming the results of this section, we consider numerical example for Eq. (5.1) put x −2 = 3, x −1 = 6, x 0 = 7. [See Fig. 1].

Second Equation
In this subsection, we give a specific form of the solutions of the difference equation where the initial conditions x −2 , x −1 , x 0 are arbitrary positive real numbers with Theorem 5.2. Let {x n } ∞ n=−2 be a solution of Eq. (5.2). Then for n = 0, 1, 2, ...

Third Equation
In this subsection, we obtain the solution of the following special case of Eq. (1.1) where the initial conditions x −2 , x −1 , x 0 are arbitrary positive real numbers.

Fourth Equation
In this subsection, we study the following special case of Eq. (1.1) where the initial conditions x −2 , x −1 , x 0 are arbitrary non zero real numbers.with