ABSTRACT

Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. Every dynamical system determines a difference equation and vise versa. We obtain in this paper the solution and periodicity of the following difference equation.

where the initial conditions and are arbitrary real numbers with and not equal to be zero. On the other hand, we will study the local stability of the solutions of Equation (1). Moreover, we give graphically the behavior of some numerical examples for this difference equation with some initial conditions.

1. Introduction

Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. Every dynamical system determines a difference equation and vise versa. Recently, there has been great interest in studying difference equations. One of the reasons for this is a necessity for some techniques whose can be used in investigating equations arising in mathematical models decribing real life situations in population biology, economic, probability theory, genetics, psychology, ...etc. Difference equations usually describe the evolution of certain phenomenta over the course of time. Recently there are a lot of interest in studying the global attractivity, boundedness character the periodic nature, and giving the solution of nonlinear difference equations. Recently there has been a lot of interest in studying the boundedness character and the periodic nature of nonlinear difference equations. Difference equations have been studied in various branches of mathematics for a long time. First results in qualitative theory of such systems were obtained by Poincaré and Perron in the end of nineteenth and the beginning of twentieth centuries. For some results in this area, see for example [1-13].

Although difference equations are sometimes very simple in their forms, they are extremely difficult to understand throughly the behavior of their solutions.

Many researchers have investigated the behavior of the solution of difference equations for examples.

Cinar [1,2] investigated the solutions of the following difference equations

Karatas et al. [4] gave that the solution of the difference equation

G. Ladas, M. Kulenovic et al. [12] have studied period two solutions of the difference equation

Simsek et al. [13] obtained the solution of the difference equation

Ibrahim [5] studied the third order rational difference Equation

In this paper we obtain the solution and study the periodicity of the following difference equation

, (1)

where the initial conditions , and are arbitrary real numbers with, and not equal to be zero. On the other hand, we will study the local stability of the solutions of Equation (1). Moreover, we give graphically the behavior of some numerical examples for this difference equation with some initial conditions.

Here, we recall some notations and results which will be useful in our investigation.

Let I be some interval of real numbers and Let be a continuously differentiable function. Then for every set of initial conditions , the difference equation

(2)

has a unique solution [11].

Definition (1.1) A point is called an equilibrium point of Equation (2) if

That is, for, is a solution of Equation (2), or equivalently, is a fixed point of F.

Definition (1.2) The difference Equation (2) is said to be persistence if there exist numbers m and M with 0 < m ≤ M < ∞ such that for any initial there exists a positive integer N which depends on the initial conditions such that m ≤ x_n ≤ M for all n ≥ N.

Definition (1.3) (Stability)

Let I be some interval of real numbers.

1) The equilibrium point of Equation (2) is locally stable if for every ε > 0, there exists δ > 0 such that for with

we have for all n ≥ −k.

2) The equilibrium point of Equation (2) is locally asymptotically stable if is locally stable solution of Equation (2) and there exists γ > 0, such that for all with

we have

3) The equilibrium point of Equation (2) is global attractor if for all, we have

4) The equilibrium point of Equation (2) is globally asymptotically stable if is locally stable, and is also a global attractor of Equation (2).

5) The equilibrium point of Equation (2) is unstable if not locally stable.

The linearized equation of Equation (2) about the equilibrium is the linear difference equation

Theorem (1.4) [10] Assume that (real numbers) and Then

is a sufficient condition for the asymptotic stability of the difference equation

Remark (1.5) Theorem (1.4) can be easily extended to a general linear equations of the form

where (real numbers) and. Then Equation (4) is asymptotically stable provided that

.

Definition (1.6) (Periodicity)

A sequence is said to be periodic with period p if for all n ≥ −k.

2. Solution and Periodicity

In this section we give a specific form of the solutions of the difference Equation (1).

Theorem (2.1)

Let be a solution of Equation (1). Then Equation (1) have all solutions and the solutions are

where.

Proof:

For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. We shall show that the result holds for n. By using our assumption for n − 1, we have the following:

Now, it follows from Equation (1) that

similarly we can derive,

Thus, the proof is completed.

Theorem (2.2)

Suppose that be a solution of Equation (1). Then all solutions of Equation (1) are periodic with period fourteen.

Proof:

From Equation (1), we see that

which completes the proof.

3. Stability of Solutions

In this section we study the local stability of the solutions of Equation (1).

Lemma (3.1)

Equation (1) have two equilibrium points which are 0 and 1.

Proof:

For the equilibrium points of Equation (1), we can write

Then, i.e.

Thus the equilibrium points of Equation (1) is are 0 and 1.

Theorem (3.2)

The equilibrium points and are unstable.

Proof:

We will prove the theorem at the equilibrium point and the proof at the equilibrium point by the same way.

Let be a continuous function defined by

Therefore it follows that

At the equilibrium point we have

, ,

, ,

,

Then the linearized equation of Equation (1) about is

i.e.

Whose characteristic equation is

By the generalization of theorem (1.4) we have

which is impossible. This means that the equilibrium point is unstable. Similarly, we can see that the equilibrium point is unstable.

4. Numerical Examples

For confirming the results of this section, we consider numerical examples which represent different types of solutions to Equation (1).

Example 4.1

Consider x₋₅ = 1, x₋₄ = 2, x₋₃ = 4, x₋₂ = −1, x₋₁ = −2, and x₀ = 7. See Figure 1.

Example 4.2

Consider x₋₅ = 1, x₋₄ = −1, x₋₃ = 2, x₋₂ = −6, x₋₁ = 5, and x₀ = −4. See Figure 2.

Example 4.3

Consider x₋₅ = 3, x₋₄ = 5, x₋₃ = −7, x₋₂ = −3, x₋₁ = 2, and x₀ = −1. See Figure 3.

Example 4.4

Consider x₋₅ = −4, x₋₄ = 3, x₋₃ = −2, x₋₂ = 9, x₋₁ = 17,

and x₀ = −6. See Figure 4.

5. Acknowledgements

We want to thank the referee for his useful suggestions.

REFERENCES