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, 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) 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) 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) 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:

(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) 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

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

• (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 φ ≠ ψ. 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

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

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

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