QUALITATIVE ANALYSIS OF A FOURTH ORDER DIFFERENCE EQUATION

In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation xn+1 = axn−1 + bxn−1 cxn−1−dxn−3 , n = 0, 1, ..., where the initial conditions x−3,x−2, x−1 and x0 are arbitrary real numbers and the values a, b, c and d are defined as positive real numbers.


Introduction
Our main objective in this paper is to obtain the qualitative behavior of the solutions of the following recursive equation: x n+1 = ax n−1 + bx n−1 cx n−1 − dx n− 3 , n = 0, 1, ..., (1.1) where the initial conditions x −3 , x −2, x −1 and x 0 are arbitrary nonzero real numbers and a, b, c,and d are positive constants. In recent years, the theory of difference equations has been studied by a large number of researchers due to the importance of this field in modeling a large number of real-life problems. Difference equations are used in modeling some natural phenomena that appear in biology, physics, economy, engineering, etc. Difference equations become apparent in the study of discretization methods for differential equations. Some results in the theory of difference equations have been obtained in the corresponding results of differential equations as more or less natural discrete analogues. Some recent studies of the dynamics of difference equations are given as follows. Agarwal and Elsayed [3] studied the periodicity character and global stability and provided a solution form for several special cases of the recursive sequence Cinar [7] investigated the solution of the difference equation Ibrahim [25] presented some relevant results of the difference equation Elsayed [16] analyzed the global stability and examined the periodic solution of the following difference equation: Elabbasy et al. [9] investigated the global stability and periodicity character and gave the solution of the special case of the difference equation Additionally, Yalçınkaya [39] addressed the difference equation Yang et al. [40] examined the global and local stability of the equilibrium points of the following recursive equation: Other results of the qualitative behavior of difference equations can be obtained in refs.

Some Basic Properties and Definitions
Here, we recall some basic definitions and some theorems that we need in the sequel. Let I be some interval of real numbers, and the function f have continuous partial derivatives on I k+1 ,where I k+1 = I × I × · · · × I (k + 1− times). Then, for initial conditions x −k , x −k+1 , ..., x 0 ∈ I, the difference equation That is, x n = x for n ≥ 0 is a solution of Eq.(2.1), or equivalently, x is a fixed point of f .
we have |x n − x| < ϵ for all n ≥ −k.
(ii) The equilibrium point x of Eq.(2.1) is locally asymptotically stable if x is a locally stable solution of Eq.(2.1) and there exists γ > 0 such that for all (iv) The equilibrium point x of Eq.(2.1) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.
The linearized equation of Eq.(2.1) about the equilibrium x is the linear difference equation Now, assume that the characteristic equation associated with Eq.(2.2) is is a sufficient condition for the asymptotic stability of the difference equation Next, we introduce a fundamental theorem to prove the global attractor of the fixed points. Suppose that g satisfies the following conditions.
(2) If m, M is a solution of the system

Local Stability of the Equilibrium Point of Eq.(1.1)
This section studies the local stability character of the equilibrium point of Eq.(1.1). Eq.(1.1) has an equilibrium point given by Let f : (0, ∞) 2 −→ (0, ∞) be a continuous function defined by Therefore, it follows that Then, we see that That is, Thus According to Theorem A, the fixed point of Eq.(1.1) is asymptotically stable. Hence, the proof is complete.

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

Thus, M = m.
It follows by Theorem B that x is a global attractor of Eq.(1.1). Therefore, the proof is complete. Case (2) If a − bdv (cu−dv) 2 < 0, let α and β be a real numbers and assume that g : Then, we can easily see that Then, from Eq. (1), we see that Thus, M = m.
It follows by Theorem B that x is a global attractor of Eq.(1.1). Hence, the proof is complete.

Existence of Periodic Solutions
In this section, we study the existence of periodic solutions of Eq.(1.1). The following theorem states the necessary and sufficient condition that this equation does not have periodic solutions of prime period two. Then, This result contradicts the fact that p ̸ = q. Hence, this completes the proof.

Numerical examples
We now present some numerical examples to confirm the theoretical work.