Investigation of the Class of the Rational Difference Equations

This paper is concerned with asymptotic behavior of the solution of a new class of rational Difference Equations. We consider the local and global stability of the solution. Moreover we investigate the new periodic character (periodic two) of solutions of these equations. Finally, we give some interesting counter examples in order to verify our strong results.


Introduction
The objective of this work is to investigate the asymptotic behavior of the solutions of the following difference equation  In recent years, there are a great interest in studying the rational difference equations.These equations describe real life situations in queuing theory, stochastic time series, combinatorial analysis, electrical network, number theory, genetics in biology, psychology, probability theory, statistical problems, economics, etc.The study of rational difference equations of high order (greater than one) is a big challenge.However, there have not been any effective general methods to deal with the global behavior of rational difference equations of order greater than one.Therefore, the study of such equations is so interesting.
There has been a lot of work concerning the global asymptotic of solutions of ra- In fact, there has been a lot of interest in studying the behavior of the nonlinear difference equation of the form For multiple delay and order, see [3] [18] [19] [20] and their references for more results of this equation.In Theorems 4.7.1-4.7.5 in [12], Kulenovic investigated the asymptotic behavior of the solutions of the equation .
Metwally et al. in [14] established a global convergence result and then apply it to show that under appropriate hypotheses every positive solution of the difference equation In [8], Elsayed studied the periodicity, the boundedness, and the global stability of the positive solution of the difference equation where the parameters , α β and γ are positive real numbers and the initial conditions 1 0 , x x − are positive real numbers.Recently, Moaaz et al. [15] investigated some qualitative behavior of the following nonlinear difference equations In this paper, in section 2, we state the sufficient condition for the asymptotic stability of Equation (1).Next, in section 3, we study the existence of periodic solutions of Equation (1).Finally, we study the boundedness nature of the solutions of Equation (1).Some numerical examples will be given to explicate our results.
During this study, we will need 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 ω in the domain of the function Φ is called an equilibrium point of the equation ( ) ].For a stability of equilibrium point, equilibrium point ω of Equation ( 4) is said to be locally stable if for all 0 ε > there exists As well, ω is said to be locally asymptotically stable if it is locally stable and there exists 0 On the other hand, ω is said to be unstable if it is not locally stable.
Finally, Equation ( 4) is called permanent and bounded if there exists numbers m and M with 0 m M < < < ∞ such that for any initial conditions ( ) .
is a sufficient condition for the asymptotic stability of Equation(1).
n n n n n y p y p y p y p y From Theorem 1.1, we have that Equation ( 1) is locally stable if Then, we find  Equation ( 1) is global attractor.Proof.We consider the following function ( ) Note that f non-decreasing for , v r and non-increasing for , u w .Let ( ) , ψ Ψ a solution of the system ( ) From (11), we have By subtracting ( 12) and ( 13), we have This completes the proof of this Theorem.

Periodic Solution of Period Two
In this section, we investigate the existence of periodic solutions of Equation (1). ) ( ) Proof.Suppose that there exists a prime period-two solution , , , , , , , of Equation ( 1).We will prove that condition ( 14) holds.We see from Equation ( 1) and so, By subtracting ( 15) and ( 16), we have By combining ( 15) and ( 16),  ( ) Now,it is clear from ( 17) and ( 18) that σ and ρ are both two positive dis- tinct roots of the quadratic equation ( ) Hence, we obtain ) ( )  .By Theorem 3.1, Equation (1) has prime period two solution (see Figure 3).

Boundedness of the Solutions
In this section, we study the characteristic task of boundedness of the positive solutions of Equation (1).Theorem 4.1.Every solution of (1) is bounded and persists.Proof.From Equation (1), we have .-Every solution of ( 1) is bounded and persists.
and the initial conditions

ω
are arbitrary positive real numbers.
and , α β and γ are positive constants.
linearized equation of Equation (1) about the equilibrium point ω is 1 0

2 ,
then the equilibrium point ω
+ = Hence, the proof is completed.E. M. Elabbasy et al.DOI: 10.4236/am.2017.8101071471 Applied Mathematics Conclusion 1.This work is concerned with studying a dynamics and behavior of solutions of a new class of difference Equation (1).Our results extend and generalize to previous studies, for example, Equation (2) (if 1 .Furthermore, we obtain the following results: -The ve + equilibrium point ω of equation (1) is local stable if ω is global attractor.-Equation (1) has a prime period-two solutions if 1