Qualitative Behavior of Rational Difference Equation of Big Order

Recently, there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodicity nature of nonlinear difference equations see for example, [1– 22]. The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order; recently, many researchers have investigated the behavior of the solution of difference equations. For example, in [8]. Elabbasy et al. investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence:


Introduction
Recently, there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodicity nature of nonlinear difference equations see for example, .
The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order; recently, many researchers have investigated the behavior of the solution of difference equations.For example, in [8].Elabbasy et al. investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence: Elabbasy et al. [9] investigated the global stability, boundedness, and periodicity character and gave the solution of some special cases of the difference equation Elabbasy et al. [10] investigated the global stability and periodicity character and gave the solution of some special cases of the difference equation Saleh and Aloqeili [23] investigated the difference equation Wang et al. [24] studied the global attractivity of the equilibrium point and the asymptotic behavior of the solutions of the difference equation In [25], Wang et al. investigated the asymptotic behavior of equilibrium point for a family of rational difference equation Yalc ¸inkaya [26] considered the dynamics of the difference equation Zayed and El-Moneam [27,28] studied the behavior of the following rational recursive sequences: For some related works see [29][30][31][32][33][34][35][36][37][38][39].Our goal in this paper is to investigate the global stability character and the periodicity of solutions of the recursive sequence where the parameters , , , and  are positive real numbers and the initial conditions  − ,  −+1 , . . .,  −1 and  0 are positive real numbers where  = max{, }.

Local Stability of the Equilibrium
Point of (9) This section deals with the local stability character of the equilibrium point of ( 9) Equation ( 9) has equilibrium points given by then Then the equilibrium points of (9) are given by  = 0 or  = √  +  −   when  +  > .
(2) If  +  > , then we see from (14) that Then, the linearized equation of ( 9) about  is whose characteristic equation is Then, ( 19) is asymptotically stable if all roots of (20) which is true if The proof is complete.

Boundedness of the Solutions of (9)
Here, we study the boundedness nature of the solutions of (9).
Proof.Let {  } ∞ =− be a solution of (9).It follows from (9) that By using a comparison, we can write the right-hand side as follows: and this equation is locally asymptotically stable if  +  <  and converges to the equilibrium point  = 0. Therefore, lim sup Thus, the solution is bounded.

Existence of Periodic Solutions
In this section, we study the existence of periodic solutions of (9).The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two.
Proof.We will prove the theorem when condition (1) is true, and the proof of the other cases is similar and so we will be omit it.First suppose that there exists a prime period two solution . . ., , , , , . . ., of ( 9).We will prove that Condition (1) holds.We see from ( 9) that Then, +  2 =  + .
Assume that We see from inequality (1) that Therefore,  and  are distinct real numbers. Set We wish to show that It follows from ( 9) that Similarly, we see that Then, it follows by induction that Thus, ( 9) has the prime period two solution . . ., , , , , . . ., where  and  are distinct roots of a quadratic equation, and the proof is complete.

Global Attractor of the Equilibrium
Point of (9) In this section, we investigate the global asymptotic stability of ( 9).If we take the function (, V) defined by ( 16), then we have four cases of the monotonicity behavior in its arguments (all of these cases we suppose that  +  > ).
Proof.Let {  } ∞ =− be a solution of ( 9) and again let  be a function defined by (16).
We will prove the theorem when (, V) is nondecreasing and the proof of the other cases is similar, and so we will omit it.
Suppose that (, ) is a solution of the systems  = (, ) and  = (, ).Then, from (9), we see that or Subtracting these two equations, we obtain Under the condition  > 0, we see that It follows by Theorem 2 that  is a global attractor of (9), and then the proof is complete.
Theorem 5.If the function (, V) defined by ( 16) is nondecreasing in  and nonincreasing in V, then the positive equilibrium point  = √( +  − )/ is a global attractor of (9) if  +  > .
Proof.Let {  } ∞ =− be a solution of ( 9) and again let  be a function defined by (16).
Suppose that (, ) is a solution of the systems  = (, ) and  = (, ).Then, from (9), we see that Under the condition  +  > , we see that It follows by Theorem 2 that  is a global attractor of (9), and then the proof is complete.Theorem 6.If the function (, V) defined by ( 16) is nondecreasing in V, nonincreasing in .Then the positive equilibrium point  = √( +  − )/ is a global attractor of (9) if + > .
Proof.The proof is similar to the previous Theorem and so we will be omit it.Lemma 7. When  ≥ + then the equilibrium point  = 0 of (9) is global attractor.
Proof.If  ≥  + , then the proof follows by Theorem 2.

Numerical Examples
For confirming the results of this paper, we consider numerical examples which represent different types of solutions to (9).