Global Analysis of Solutions of a New Class of Rational Difference Equation

The study suggests asymptotic behavior of the solution to a new class of difference equations: ( ) ( ) 2 1 1 0 2 2 1 , 0,1, 2, k i i i i i a b η η η η ψ ψ η αψ βψ − + + = − − + = + = + ∑  . where , , i a b α and β are positive real numbers for 0,1, , i k =  , and the initial conditions 1 0 , , , j j ψ ψ ψ − − +  are randomly positive real numbers where 2 1 j k = + . Accordingly, we consider the stability, boundedness and periodicity of the solutions of this recursive sequence. Indeed, we give some interesting counter examples in order to verify our strong results.

+ .Accordingly, we consider the stability, boundedness and periodicity

Introduction
The global asymptotic behavior of the solutions and oscillation of solution are two such qualitative properties which are very important for applications in many areas such as control theory, mathematical biology, neural networks, etc.It is impossible to use computer based (numerical) techniques to study the oscillation or the asymptotic behavior of all solutions of a given equation due to the global nature of these properties.Therefore, these properties have received the attention of several mathematicians and engineers.Currently, much attention has given to study the properties of the solutions of the recursive sequences from scientists in various disciplines.Specifically, the topics dealt with include the following: -Finding equilibrium points for the recursive sequences; -Investigating the local stability of the solutions of the recursive sequences; -Finding conditions which insure that the solutions of the recursive sequences are bounded; -Investigating the global asymptotic stability of the solutions of the recursive sequences; -Finding conditions which insure that the solutions of equation are periodic with positive prime period two or more; -Finding conditions for oscillation of solutions.
Elsayed [9] studied the periodicity, the boundedness of the positive solution of the recursive sequences Abdelrahman [1] considered analytical investigation of the solution of the recursive sequence By new method, Elsayed [10] investigated the periodic solution of the equation Moaaz [18] completed the results of [10].
In this work, we deal with some qualitative behaviour of the solutions of the recursive sequence where , , In the next, we will and to many of the basic concepts.Before anything, the concept of equilibrium point is essential in the study of the dynamics of any physical system.A point ψ in the domain of the function Φ is called an equilibrium point of the equation ( )  2) is said to be locally stable if for all 0 ε > there exists As well, ψ is said to be locally asymptotically stable if it is locally stable and there exists 0 γ > such that, if ( ) Also, ψ is said to be a global attractor if used for every ( ) On the other hand, ψ is said to be unstable if it is not locally stable.
Finally, Equation (1.2) is called permanent and bounded if there exists numbers r and R with 0 r R < < < ∞ such that for any initial conditions ( )  there exists a positive integer N which depends on these initial conditions such as r R The linearized equation of Equation (1.1) about the equilibrium point ψ is

Global Stability of Equation (1.1)
In the following theorem, we check into the global stability of the recursive sequence (1.1).
B a α β = − American Journal of Computational Mathematics Proof.We consider the function as follow: ( ) , , , .
2) and (2.3), we note that f is increasing in , we have that µ λ = .Hence, the proof of Theorem 3.1 is complete.□

Periodic Solutions
In this section, we enumerate some basic facts concerning the existence of two Proof.Assume that Equation (1.1) has a prime period-two solution , , , , , , , ρ σ ρ σ ρ σ   We shall prove that condition (4.1) holds.From Equation (1.1), we see that , , By combining (4.2) and (4.3), we obtain Now, evident is that (4.4) and (4.5) that ρ and σ are both two positive distinct roots of the quadratic equation ( ) Hence, we obtain which has the same extent as Hence, the proof is complete.□ The next numerical example is mimicry to enhance our results.
be a Solution (1.1), we can conclude from (1.1) that .
Also, from Equation (1.1), we see that  -Every solution of (1.1) is bounded and persists.

i a b α and β are positive real numbers for 0
For a stability of equilibrium O. Moaaz et al.DOI: 10.4236/ajcm.2017.74036American Journal of Computational Mathematics point, equilibrium point ψ of equation (

Figure 2 )
Figure 2) solution is bounded and persists and the proof is complete.□ Conclusion 1.In this paper, we study a asymptotic behavior of solutions of a general class of difference Equation (1.1).Our results extend and generalize to the earlier ones.Moreover, we obtain the next results: -The ve + equilibrium point ψ of Equation (1.1) is local stable if ψ is global attractor.