Theory and Modern Applications

# Qualitative behavior of a rational difference equation

## 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.

## Introduction

Difference equations are applied in the field of biology, engineer, physics, and so on [1]. The study of properties of rational difference equations has been an area of intense interest in the recent years [6, 7]. There has been a lot of work deal with the qualitative behavior of rational difference equation. For example, Ã‡inar [2] has got the solutions of the following difference equation:

Karatas et al. [3] gave that the solution of the difference equation:

In this article, we consider the qualitative behavior of rational difference equation:

(1)

with initial conditions y -1, y 0 âˆˆ (0, + âˆž), p âˆˆ R +.

## Preliminaries and notation

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

Lemma 1. Let I be some interval of real numbers and

be a continuously differentiable function. Then, for every set of initial conditions, x -k , x -k+1, ..., x 0 âˆˆ I the difference equation

(2)

has a unique solution .

Definition 1 (Equilibrium point). A point is called an equilibrium point of Equation 2, if

Definition 2 (Stability).

1. (1)

The equilibrium point of Equation 2 is locally stable if for every Îµ > 0, there exists Î´ > 0, such that for any initial data x -k , x -k+1, ..., x 0 âˆˆ I, with

we have , for all n â‰¥ - k.

2. (2)

The equilibrium point of Equation 2 is locally asymptotically stable if is locally stable solution of Equation 2, and there exists Î³ > 0, such that for all x -k , x -k+1, ..., x 0 âˆˆ I, with

we have

3. (3)

The equilibrium point of Equation 2 is a global attractor if for all x -k , x -k+1, ..., x 0 âˆˆ I, we have .

4. (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. (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:

(3)

Lemma 2 [4]. Assume that p 1, p 2 âˆˆ R and k âˆˆ {1, 2, ...}, then

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

(4)

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 â†’ [p, q] be a continuous function, where p and q are real numbers with p < q and consider the following equation:

(5)

Suppose that g satisfies the following conditions:

1. (1)

g(x, y) is non-decreasing in x âˆˆ [p, q] for each fixed y âˆˆ [p, q], and g(x, y) is non-increasing in y âˆˆ [p, q] for each fixed x âˆˆ [p, q].

2. (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

In this section, we investigate the local stability character of the equilibrium point of Equation 1. Equation 1 has an equilibrium point

Let f:(0, âˆž)2 â†’ (0, âˆž) be a function defined by

(6)

Therefore, it follows that

Theorem 1.

1. (1)

Assume that p > 2, then the equilibrium point of Equation 1 is locally asymptotically stable.

2. (2)

Assume that 0 < p < 2, then the equilibrium point of Equation 1 is locally asymptotically stable, the equilibrium point is unstable.

Proof. (1) when ,

The linearized equation of (1) about is

(7)

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

(2) when ,

The linearized equation of (1) about is

(8)

It follows by Lemma 2, Equation 8 is asymptotically stable, if

Therefore,

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 â†’ [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).

Then, from Equation 1

Therefore,

(9)
(10)

Subtracting Equation 10 from Equation 9 gives

Since p+Mm â‰  0, it follows that

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 âˆˆ R +.

Proof. Assume for the sake of contradiction that there exist distinctive non-negative real numbers Ï† and Ïˆ, such that

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

Ï† and Ïˆ satisfy the system

(11)
(12)

Subtracting Equation 11 from Equation 12 gives

so Ï† = Ïˆ, which contradicts the hypothesis Ï† â‰  Ïˆ. The proof is complete.

## Numerical simulation

In this section, we give some numerical simulations to support our theoretical analysis. For example, we consider the equation:

(13)
(14)
(15)

We can present the numerical solutions of Equations 13-15 which are shown, respectively in Figures 1, 2 and 3. Figure 1 shows the equilibrium point of Equation 13 is locally asymptotically stable with initial data 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.

## References

1. Berezansky L, Braverman E, Liz E: Sufficient conditions for the global stability of nonautonomous higher order difference equations. J Diff Equ Appl 2005,11(9):785-798. 10.1080/10236190500141050

2. Ã‡inar C: On the positive solutions of the difference equation x n+1 = ax n-1 /1+ bx n x n-1 . Appl Math Comput 2004,158(3):809-812. 10.1016/j.amc.2003.08.140

3. Karatas R, Cinar C, Simsek D: On positive solutions of the difference equation x n+1 = x n-5 /1+ x n-2 x n-5 . Int J Contemp Math Sci 2006,1(10):495-500.

4. Li W-T, Sun H-R: Global attractivity in a rational recursive sequence. Dyn Syst Appl 2002,3(11):339-345.

5. Kulenovic MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC Press; 2001.

6. Elabbasy EM, El-Metwally H, Elsayed EM: On the difference equation x n+1 = ax n - bx n /( cx n - dx n-1 ). Adv Diff Equ 2006, 1-10.

7. Memarbashi R: Sufficient conditions for the exponential stability of nonautonomous difference equations. Appl Math Lett 2008,3(21):232-235.

## Author information

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Correspondence to Xiao Qian.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

Xiao Qian carried out the theoretical proof and drafted the manuscript. Shi Qi-hong participated in the design and coordination. All authors read and approved the final manuscript.

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Qian, X., Qi-hong, S. Qualitative behavior of a rational difference equation . Adv Differ Equ 2011, 6 (2011). https://doi.org/10.1186/1687-1847-2011-6