GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION

Abstract

In this paper, we investigate the boundedness character, the periodic character and the global behavior of positive solutions of the difference equation $$x_{n+1}=p_n+\frac{x_n}{x_{n-1}},\;n=0,1,{\cdots}$$ where $\{p_n\}$ is a two periodic sequence of nonnegative real numbers and the initial conditions $x_{-1}$, $x_0$ are arbitrary positive real numbers.

1. Introduction

Recently, there has been an increasing interest in the study of the qualitative analyses of rational difference equations. For example, see[1 − 8] and the references cited therein.

This work studies the boundedness character and the global asymptotic stability for the positive solutions of the difference equation

where {pn} is a two periodic sequence of nonnegative real numbers and the initial conditions x−1, x0 are arbitrary positive numbers.

As far as we can examine, this is the first work devote to the investigation of the type Eq.(1.1).

Now, we assume p2n = α and p2n+1 = β in Eq.(1.1). Then we have

and

The autonomous case of Eq.(1.1) is

where p > 0 and the initial conditions x−1, x0 are arbitrary positive numbers. We now consider the local asymptotic stability of the unique equilibrium of Eq.(1.4).

The linearized equation for Eq. (1.4) about the positive equilibrium is

The following theorem is given in [1].

Theorem A.Consider Eq. (1.4) and assume that x−1, x0, p ∈ (0,∞). Then the unique positive equilibrium of Eq. (1.4) is globally asymptotically stable.

 

2. Boundedness Character of Eq. (1.1)

In this section, we investigate the boundedness character of Eq. (1.1). So, we have the following result.

Theorem 2.1. Suppose that α > 1 and β > 1 with α ≠ β, then every positive solution of Eq.(1.1) is bounded.

Proof. It is clear from Eq. (1.2) and (1.3) that

Then, from (1.2) and (2.1) we obtain

and from (1.3) and (2.1) we obtain

From (2.3), (2.4) using induction we get

The result now follows.

 

3. Stability and Periodicity for Eq. (1.1)

In this section, we investigate the periodicity and stability character of positive solutions of Eq. (1.1). Now, we have the following result.

Proposition 3.1. Consider Eq. (1.1) when the case α ≠ β and assume that α, β ∈ (0,∞). Then there exist prime two periodic solutions of Eq. (1.1).

Proof. In order Eq. (1.1) to possess a periodic solution {xn} of prime period 2, we must find positive numbers x−1, x0 such that

Let x−1 = x, x0 = y, then from (3.1) we obtain the system of equations

We prove that (3.2) has a solution From the first relation of (3.2) we have

From (3.3) and the second relation of (3.2) we obtain

Now we consider the function

Then from (3.4) we get

Hence Eq. (3.4) has a solution Then if we have that the solution of Eq. (1.1) with initial values is a periodic solution of period two.

Theorem 3.2. Consider Eq. (1.1) when the case α ≠β and assume that α, β ∈ (0,∞). Suppose that

Then the two periodic solutions of Eq. (1.1) are locally asymptotically stable.

Proof. From equations (1:3), (1:4) and Proposition 3:1 there exist such that

We set x2n−1 = un, x2n= vn in equations (1.3), (1.4) and so we have

Then is the positive equilibrium of Eq. (3.8), and the linearised system of Eq. (3.8) about is the system

The characteristic equation of B is

Using Eq. (3.6), from Eq. (3.7), since we have

and we obtain

Then, from (3.10) and Theorem 1.3.7 of Kocic and Ladas in [4], all the roots of Eq. (3.9) are modulus less than 1. Therefore, from Proposition 3.1, system (3.8) is asymptotically stable. The proof is complete.

Theorem 3.3. Consider Eq. (1.1) when the case α ≠ β. Assume that α > 1, β > 1. Then every positive solution of Eq. (1.1) converges to a two-periodic solution of Eq. (1.1).

Proof. Since α > 1, β > 1, we know by Theorem 2.1 that every positive solution of Eq. (1.1) is bounded, it follows that there are finite

exist. Then it is easy to see from Eq. (1.2) and (1.3) that

and

Thus, we have

and

This implies that

and

Then, we get

and

Now, we shall prove that s = S and l = L. It is clear that if l = L, then by (3.11) it must be s = S. Similarly, if s = S, then by (3.12) it must be l = L.

Hence we assume that s < S and l < L. From (3.11) and (3.12) we have

then we obtain a contradiction. So, we get s = S and l = L

Moreover, it is obvious that since α ≠ β, then from Eq. (1.2) and Eq. (1.3)

Then it is clear that every positive solution of Eq. (1.1) converges to a twoperiodic solution of Eq. (1.1). The proof is complete.

Finally, using Proposition 3.1, Theorems 3.1 and 3.2, we have the following Theorem.

Theorem 3.4. Consider Eq. (1.1) when the case α ≠ β. Assume that α > 1, β > 1 and that (3.6) holds. Then two-period solutions of Eq. (1.1) are globally asymptotically stable.