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Qualitative behavior of a rational difference equation

Advances in Difference Equations20112011:6

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

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

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

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, Çinar [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 (0, + ∞), p 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 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 ε > 0, there exists δ > 0, such that for any initial data x -k , x -k+1, ..., x 0 I, with

    we have , for all n ≥ - k.

     
  2. (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 x -k , x -k+1, ..., x 0 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 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 R and k {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 → [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 [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. (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, ∞)2 → (0, ∞) 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 → [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 ≠ 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 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).

φ and ψ satisfy the system
(11)
(12)
Subtracting Equation 11 from Equation 12 gives

so φ = ψ, which contradicts the hypothesis φψ. 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.
Figure 1
Figure 1

Plot of x ( n +1) = ( x ( n )+ x ( n -1))/(1.1+ x ( n )* x ( n -1)). This figure shows the solution of , where x 0 = 1, x 1 = 1.2

Figure 2
Figure 2

Plot of x ( n +1) = ( x ( n )+ x ( n -1))/(1.5+ x ( n )* x ( n -1)). This figure shows the solution of , where x 0 = 1, x 1 = 1.2

Figure 3
Figure 3

Plot of Plot of x ( n + 1) = ( x ( n ) + x ( n -1))/(5 + x ( n )* x ( n - 1)). This figure shows the solution of , where x 0 = 1, x 1 = 1.2

Declarations

Authors’ Affiliations

(1)
Department of Basic Courses, Hebei Finance University, Baoding, 071000, China

References

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Copyright

© Qian and Qi-hong; licensee Springer. 2011

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.

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