On the Nonlinear Difference Equation

Elmetwally M. Elabbasy^*, Abdulmuhaemn A. El-Biaty

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt.

DOI: 10.4236/jamp.2016.41014                

Abstract

In this paper, we investigate some qualitative behavior of the solutions of the difference equation where the coefficients a, b and c_i are positive real numbers, and where the initial conditions are arbitrary positive real numbers.

Keywords

Difference Equation, Stability, Periodicity, Boundedness, Global Stability

Received 17 November 2015; accepted 22 January 2016; published 26 January 2016

1. Introduction

Our aim in this paper is to study with some properties of the solutions of the difference equation

(1.1)

where the coefficients and are positive real numbers, and where the initial conditions are arbitrary positive real numbers. There is a class of nonlinear difference equations, known as the rational difference equations, each of which consists of the ratio of two polynomials in the sequence terms in the same form. There has been a lot of work concerning the global asymptotics of solutions of rational difference equations [1] -[8] .

Many researchers have investigated the behavior of the solution of difference equation. For example:

Amleh et al. [9] has studied the global stability, boundedness and the periodic character of solutions of the equation

Our aim in this paper is to extend and generalize the work in [9] , [10] and [11] . That is, we will investigate the global behavior of (1.1) including the asymptotical stability of equilibrium points, the existence of bounded solution, the existence of period two solution of the recursive sequence of Equation (1).

Now we recall some well-known results, which will be useful in the investigation of (1.1) and which are given in [12] .

Let I be an interval of real numbers and let

where F is a continuous function. Consider the difference equation

(1.2)

with the initial condition

Definition 1. (Equilibrium Point)

A point is called an equilibrium point of Equation (1.2) if

That is, for, is a solution of Equation (1.2), or equivalently, is a fixed point of f.

Definition 2. (Stability)

Let be in equilibrium point of Equation (1.2) then

1) An equilibrium point of Equation (1.2) is called locally stable if for every there exists such that, if with then for all.

2) An equilibrium point of Equation (1.2) is called locally asymptotically stable if is locally stable and there exists such that, if with

then

3) An equilibrium point of Equation (1.2) is called a global attractor if for all we have

4) An equilibrium point of Equation (1.2) is called globally asymptotically stable if is locally stable and a global attractor.

5) An equilibrium point of Equation (1.2) is called unstable if is not locally stable.

Definition 3. (Permanence)

Equation (1.2) is called permanent if there exists numbers m and M with such that for any initial conditions there exists a positive integer N which depends on the initial conditions such that

Definition 4. (Periodicity)

A sequence is said to be periodic with period p if for all. A sequence is said to be periodic with prime period p if p is the smallest positive integer having this property.

The linearized equation of Equation (1.2) about the equilibrium point is defined by the equation

(1.3)

where

The characteristic equation associated with Equation (1.3) is

(1.4)

Theorem 1.1. [13] Let be an interval of real numbers and assume that

is a continuous function satisfying the following properties:

(a) is non-increasing in the first (k) terms for each in and non-decreasing in the last term for each in for all

(b) If is a solution of the system

implies

Theorem 1.2. [12] Assume that F is a -function and let be an equilibrium point of Equation (1.2). Then the following statements are true:

1) If all roots of Equation (1.4) lie in the open unit disk, then he equilibrium point is locally asymptotically stable.

2) If at least one root of Equation (1.4) has absolute value greater than one, then the equilibrium point is unstable.

3) If all roots of Equation (1.4) have absolute value greater than one, then the equilibrium point is a source.

Theorem 1.3. [14] Assume that Then

is a sufficient condition for the asymptotically stable of Equation (1.5)

(1.5)

2. Local Stability of Equation (1.1)

In this section we investigate the local stability character of the solutions of Equation (1.1). Equation (1.1) has a unique nonzero equilibrium point

Let

Then, we get

Let be a function defined by

(2.1)

Therefore it follows that

and

Then we see that

and

Then the linearized equation of (1.1) about is

(2.2)

Theorem 2.1. Assume that

Then the equilibrium point of Equation (1.1) is locally stable.

Proof. It is follows by Theorem (1.3) that, Equation (2.2) is locally stable if

That is

This implies that

then

Thus

Hence, the proof is completed.

3. Periodic Solutions

In this section we investigate the periodic character of the positive solutions of Equation (1.1).

Theorem 3.1. Equation (1.1) has positive prime period-two solution only if

(3.1)

Proof. Assume that there exists a prime period-two solution

of (1.1). Let Since, we have Thus, from Equation (1.1), we get

and

Let

and

Then

and

Then

(3.2)

and

(3.3)

Subtracting (3.2) from (3.3) gives

Since, we have

(3.4)

Also, since p and q are positive, should be positive. Again, adding (3.2) and (3.3) yields

(3.5)

It follows by (3.4), (3.5) and the relation

that

(3.6)

Assume that p and q are two distinct real roots of the quadratic equation

and so

which is equivalent to

Thus, the proof is completed.

4. Bounded Solution

Our aim in this section we investigate the boundedness of the positive solutions of Equation (1.1).

Theorem 4.1. The solutions of Equation (1.1) are bounded.

Proof. Let be a solution of Equation (1.1). We see from Equation (1.1) that

Then

(4.1)

On the other hand, we see that the change of variables

transforms Equation (1.1) to the following form:

Hence, we obtain

Thus

and so,

It follows that

Thus we obtain

(4.2)

From (4.1) and (4.2) we see that

Therefore every solution of Equation (1.1) is bounded.

5. Global Stability of Equation (1.1)

Our aim in this section we investigate the global asymptotic stability of Equation (1.1).

Theorem 5.1. If then the equilibrium point of Equation (1.1) is global attractor.

Proof. Let be a function defined by

then we can see that the function is decreasing in the rest of arguments and increasing in.

Suppose that is a solution of the system

Then from Equation (2.1), we see that

then

Thus

It follows by Theorem (1.1) that is a global attractor of Equation (1.1) and then the proof is complete.

6. Numerical

For confirming the results of this section, we consider numerical examples which represent different types of solution of Equation (1.1).Example

Examples 6.1. Consider the difference equation

where Figure 1 shows that the equilibrium point of Equation (1.1) has locally stable, with initial data (see Table 1).

Example 6.2. Consider the difference equation

where Figure 2, shows that Equation (1.1) which is periodic with period two. Where the initial data satisfies condition (3.1) of Theorem (3.1) (see Table 2).

Figure 1. Ref. b1.

Table 1. The equilibrium point of Equation (1.1).

Figure 2. Ref. b4.

Table 2. The initial data satisfies condition (3.1) of Theorem (3.1).

Remark 6.1. Note that the special cases of Equation (1.1) have been studied in [9] when and in [10] when and in [11] when

Conflicts of Interest

The authors declare no conflicts of interest.

References

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