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# Qualitative behavior of a rational difference equation

- Xiao Qian
^{1}Email author and - Shi Qi-hong
^{1}

**2011**:6

https://doi.org/10.1186/1687-1847-2011-6

© Qian and Qi-hong; licensee Springer. 2011

**Received:**10 February 2011**Accepted:**3 June 2011**Published:**3 June 2011

## Abstract

This article is concerned with the following rational difference equation *y*
_{
n+1}= (*y*
_{
n
}+ *y*
_{
n-1})/(*p* + *y*
_{
n
}
*y*
_{
n-1}) with the initial conditions; *y*
_{-1}, *y*
_{0} are arbitrary positive real numbers, and *p* is positive constant. Locally asymptotical stability and global attractivity of the equilibrium point of the equation are investigated, and non-negative solution with prime period two cannot be found. Moreover, simulation is shown to support the results.

## Keywords

- Global stability
- attractivity
- solution with prime period two
- numerical simulation

## Introduction

with initial conditions *y*
_{-1}, *y*
_{0} ∈ (0, + ∞), *p* ∈ *R*
^{+}.

## Preliminaries and notation

Let us introduce some basic definitions and some theorems that we need in what follows.

*x*

_{-k },

*x*

_{-k+1}, ...,

*x*

_{0}∈

*I*the difference equation

has a unique solution .

**Definition 2**(Stability).

- (1)The equilibrium point of Equation 2 is locally stable if for every
*ε*> 0, there exists*δ*> 0, such that for any initial data*x*_{-k },*x*_{-k+1}, ...,*x*_{0}∈*I*, withwe have , for all

*n*≥ -*k*. - (2)
- (3)
The equilibrium point of Equation 2 is a global attractor if for all

*x*_{-k },*x*_{-k+1}, ...,*x*_{0}∈*I*, 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 is not locally stable.

**Definition 3**The linearized equation of (2) about the equilibrium is the linear difference equation:

Moreover, suppose *p*
_{2} > 0, then, |*p*
_{1}| + |*p*
_{2}| < 1 is also a necessary condition for the asymptotic stability of Equation 4.

**Lemma 3**[5]. Let

*g*:[

*p, q*]

^{2}→ [

*p, q*] be a continuous function, where

*p*and

*q*are real numbers with

*p*<

*q*and consider the following equation:

*g*satisfies the following conditions:

- (1)
*g*(*x, y*) is non-decreasing in*x*∈ [*p, q*] for each fixed*y*∈ [*p, q*], and*g*(*x, y*) is non-increasing in*y*∈ [*p, q*] for each fixed*x*∈ [*p, q*]. - (2)
If (

*m, M*) is a solution of system

*M* = *g*(*M, m*) and *m* = *g*(*m, M*),

then *M* = *m*.

Then, there exists exactly one equilibrium of Equation 5, and every solution of Equation 5 converges to .

## The main results and their proofs

**Theorem 1**.

It follows by Lemma 2, Equation 7 is asymptotically stable, if *p* > 2.

Equilibrium point is unstable, it follows from Lemma 2. This completes the proof.

**Theorem 2**. Assume that
, the equilibrium point
and
of Equation 1 is a global attractor.

**Proof**. Let *p, q* be real numbers and assume that *g*:[*p, q*]^{2} → [*p, q*] be a function defined by
, then we can easily see that the function *g*(*u, v*) increasing in *u* and decreasing in *v*.

Suppose that (*m, M*) is a solution of system

*M* = *g*(*M, m*) and *m* = *g*(*m, M*).

Lemma 3 suggests that is a global attractor of Equation 1 and then, the proof is completed.

**Theorem 3**. (1) has no non-negative solution with prime period two for all *p* ∈ *R*
^{+}.

**Proof**. Assume for the sake of contradiction that there exist distinctive non-negative real numbers

*φ*and

*ψ*, such that

is a prime period-two solution of (1).

so *φ* = *ψ*, which contradicts the hypothesis *φ* ≠ *ψ*. The proof is complete.

## Numerical simulation

*x*

_{0}= 1,

*x*

_{1}= 1.2. Figure 2 shows the equilibrium point of Equation 14 is locally asymptotically stable with initial data

*x*

_{0}= 1,

*x*

_{1}= 1.2. Figure 3 shows the equilibrium point of Equation 15 is locally asymptotically stable with initial data

*x*

_{0}= 1,

*x*

_{1}= 1.2.

## Declarations

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.