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.
Closely related global convergence results were well-gained from these articles  -  . Khuong in  studied the dynamics the recursive sequences
For further related and special cases of this difference equations see    ,    .
Elsayed  studied the periodicity, the boundedness of the positive solution of the recursive sequences
Abdelrahman  considered analytical investigation of the solution of the recursive sequence
By new method, Elsayed  investigated the periodic solution of the equation
Also, Moaaz  completed the results of  .
In this work, we deal with some qualitative behaviour of the solutions of the recursive sequence
where and are positive real numbers for , and the the initial conditions are arbitrary positive real numbers 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 in the domain of the function is called an equilibrium point of the equation
if is a fixed point of [ ]. For a stability of equilibrium point, equilibrium point of equation (2) is said to be locally stable if for all there exists such that, if for with . As well, is said to be locally asymptotically stable if it is locally stable and there exists such that, if for with , then . Also, is said to be a global attractor if used for every for , we have . On the other hand, 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 such that for any initial conditions for there exists a positive integer N which depends on these initial conditions such as for all .
The linearized equation of Equation (1.1) about the equilibrium point is
Theorem 1.1.  Assume that for . The equilibrium of (1.1) is locally asymptotically stable if
2. Local Stability of Equation (1.1)
The equilibrium point of Equation (1.1) is
Let defined by
Therefore it follows that
Theorem 2.1. Let be equilibrium of Equation (1.1). If
than is locally stable.
Proof. From (2.2) to (2.3), we obtain
for . Thus, the linearized equation of (1.1) is
It follows by Theorem 1.1 that Equation (1.1) is locally stable if
where , and hence,
Thus, we find
Hence, the proof is complete.
In order to verify and support our theoretical outcomes and discussions, in this concern, we investigate several interesting numerical examples.
Example 2.1. By Theorem 2.1, the equilibrium Equation (1.1) with , , , and , is locally stable (see Figure 1).
3. Global Stability of Equation (1.1)
In the following theorem, we check into the global stability of the recursive sequence (1.1).
Theorem 3.1. The equilibrium of Equation (1.1) is global attractor if
Figure 1. The stable solution corresponding to difference Equation (1.1).
Proof. We consider the function as follow:
From (2.2) and (2.3), we note that is increasing in and decreasing in for all . Suppose that is a solution of the system
Then, we find
Hence, we get
By (3.1) and (3.2), we obtain
Since , we have that . Hence, the proof of Theorem 3.1 is complete. □
4. Periodic Solutions
In this section, we enumerate some basic facts concerning the existence of two period solutions.
Theorem 4.1. Equation (1.1) has prime period-two solutions if
Proof. Assume that Equation (1.1) has a prime period-two solution
We shall prove that condition (4.1) holds. From Equation (1.1), we see that
Thus, we get
From (4.3) and (4.2), we have
Dividing , then we find
By combining (4.2) and (4.3), we obtain
Since , we get
Now, evident is that (4.4) and (4.5) that and 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.
Example 4.1. By Theorem 4.1, Equation (1.1) with , , , , and , has prime period two solution (see Figure 2)
Theorem 5.1. Every solution of Equation (1.1) is bounded and persists.
Proof. Let be a Solution (1.1), we can conclude from (1.1) that
Also, from Equation (1.1), we see that
Thus, the 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 equilibrium point of Equation (1.1) is local stable if . Also, if , then is global attractor.
Figure 2. Prime period two solution of Equation (1.1).
- Equation (1.1) has a prime period-two solutions if .
- Every solution of (1.1) is bounded and persists.
The author is very grateful to the reviewers for their valuable suggestions and useful comments on this paper.
 Amleh, A.M., Grove, E.A., Georgiou, A. and Ladas, G. (1999) On the Recursive Sequence ωη+1 = α + ωη-1/ωη. The Journal of Mathematical Analysis and Applications, 233, 790-798.
 Berenhaut, K.S. and Stevic, S. (2006) The Behaviour of the Positive Solutions of the Difference Equation ψη = a + (ψη-2/ψη-1)σ. Journal of Difference Equations and Applications, 12, 909-918.
 Camouzis, E., DeVault, R. and Ladas, G. (2001) On the Recursive Sequence ωη+1 = -1 + ωη-1/ωη. Journal of Difference Equations and Applications, 7, 477-482.
 Elabbasy, E.M., EL-Metwally, H.A. and Elsayed, E.M. (2006) On the Difference Equation ωη+1 = aωη - bωη/(cωη - dωη-1). Advances in Difference Equations, 2006, 1-10.
 Kulenovic, M.R.S. and Ladas, G. (2001) Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Florida.
 El-Metwally, H., Ladas, G., Grove, E.A. and Voulov, H.D. (2001) On the Global Attractivity and the Periodic Character of Some Difference Equations. Journal of Difference Equations and Applications, 7, 837-850.
 Metwally, H.E., Grove, E.A. and Ladas, G. (2000) A Global Convergence Result with Applications to Periodic Solutions. Journal of Mathematical Analysis and Applications, 245, 161-170.
 Moaaz, O. (2016) Comment on New Method to Obtain Periodic Solutions of Period Two and Three of a Rational Difference Equation. Nonlinear Dynamics, 79, 241-250.
 Ocalan, O. (2014) Dynamics of the Difference Equation ωη+1 = Pη + ωη-k/ωη with a Period-Two Coefficient. Applied Mathematics and Computation, 228, 31-37.
 Sun, T. and Xi, H. (2007) On Convergence of the Solutions of the Difference Equation ωη+1 = 1 + ψη/ψη-k. Journal of Mathematical Analysis and Applications, 325, 1491-1494.
 Zhang, L., Zhang, G. and Liu, H. (2005) Periodicity and Attractivity for a Rational Recursive Sequence. Journal of Applied Mathematics and Computing, 19, 191-201.