ON A HIGHER-ORDER RATIONAL DIFFERENCE EQUATION

Abstract

In this paper, we investigate the global behavior of the solutions of the difference equation $x_{n+1}=\frac{A+Bx_{n-2k-1}}{C+D\prod_{i=l}^{k}x_{n-2i}^{m_i}}$, n=0, 1, ..., with non-negative initial conditions, the parameters A, B are non-negative real numbers, C, D are positive real numbers, k, l are fixed non-negative integers such that l ≤ k, and m_i, i=l, k are positive integers.

1. Introduction and Preliminaries

In [3] we investigated the global behavior of the rational third-order difference equation

where the initial conditions x0, x−1, x−2 and the parameter B are non-negative real numbers, the parameters A, C, D are positive real numbers and p, q are fixed positive integers. Abo-Zeid [1] discussed the global behavior and boundedness of the solutions of the difference equation

where A, B are non-negative real numbers, C, D > 0 and l, k are non-negative integers such that l ≤ k. Inspired and motivated by these aforementioned works, our aim in this paper is to investigate the global asymptotic behavior of the difference equation

with non-negative initial conditions, the parameters A, B are non-negative real numbers, C, D are positive real numbers, k, l are fixed non-negative integers such that l ≤ k, and mi, are positive integers.

We note that if mi = 1, for all Eq.(3) is reduced to Eq.(2). Clearly, the results obtained in [1] will follow from the results we shall exhibit here.

In what follows, we present some definitions and results which will be useful in our investigation, for more details we refer to [6], [11], [14] and [15].

Let I be some interval of real numbers and let

be a continuously differentiable function. Then, for every set of initial conditions {x0, x−1, ..., x−k} ⊂ I, the difference equation

has a unique solution .

Definition 1.1. A point ∈ I is called an equilibrium point of Eq.(4) if

Definition 1.2. Let be an equilibrium point of Eq.(4).

Let denote the partial derivatives of f(u0, u1, …, uk) with respect to ui evaluated at the equilibrium of Eq.(4). Then, the equation

is called the linearized equation of Eq.(4) about the equilibrium point , and the equation

is called the characteristic equation of Eq.(4) about .

Theorem 1.3. Let be an equilibrium of Eq.(4). Then, the following statements are true

Theorem 1.4 (Rouché’s Theorem). Let D be a bounded domain with piecewise smooth boundary ∂D. Let f and g be two analytic functions on D ∪ ∂D. If |g(z)| < |f(z)| for z ∈ ∂D, then f and f + g have the same number of zeros in D, counting multiplicities.

Definition 1.5. Let be an equilibrium of Eq.(4) and assume that is a solution of the same equation.

Definition 1.6. A solution of Eq.(4) is called non-oscillatory about , or simply non-oscillatory, if there exists N ≥ −k such that either

or

Otherwise, the solution is called oscillatory about , or simply oscillatory.

From now on, we let

Remark 1.7. The change of variables reduces Eq.(3) to the difference equation

where . It suffices to study Eq.(6) instead of Eq.(3).

 

2. Main results

2.1. Case α > 0.

2.1.1. Local stability. Here, we determine the equilibrium points of Eq.(6) and discuss their local stability.

Lemma 2.1. The following statements are true.

Proof. A point is an equilibrium point of Eq.(6) if and only if is a zero of the function

If we consider the above function, we get

Theorem 2.2. Assume that is the positive equilibrium point of Eq.(6). Then, the following statements are true

Proof. The linearized equation associated with Eq.(6) about is

The characteristic equation associated with this equation is

2.1.2. Oscillation and Boundedness of Solutions.

Theorem 2.3. Let be the positive equilibrium of Eq.(6) and let be a solution of the same equation. Then

Proof. (1) Assume that the condition (a1) is satisfied. Then

Lemma 2.4. Let be a solution of Eq.(6). Then

(1)

(2)

(3)

Proof. Let be a solution of Eq.(6). Then

Corollary 2.5. Assume that β < 1. Then, every solution of Eq.(6) is bounded and persists.

Lemma 2.6. Suppose β < 1 and let be a solution of Eq.(6). If , then Λ and λ satisfy the following inequalities

Proof. Let β < 1. From Corollary 2.5 the solution is bounded. Then, for every ε ∈ (0, λ), there exists n0 ∈ ℕ such that

so,

Therefore,

Lemma 2.7. Suppose β > 2. Then, the following statements are true

Proof. (1) Let , then

Theorem 2.8. Assume that β > 2. Then, Eq.(6) has solutions which are neither bounded nor persist.

Proof. Let be a solution of Eq.(6) with initial conditions

and

Then,

and

By applying Lemma 2.7, we get y2 < y−2k.

Now consider the subsequences

where 0 ≤ j ≤ k. We will prove that

and

for all n ≥ 1. For n = 1, we have

and

By applying Lemma 2.7, we obtain y2j+2 < y−2(k+1)+2j+2. We also have

and

By applying Lemma 2.7, we get y2(k+1)(n+1)−2k+2j < y2(k+1)n−2k+2j . Now from inequality (8), we have

This implies that

and

This completes the proof. □

2.1.3. Global stability.

Theorem 2.9. Assume that β < 1. If , then, the positive equilibrium point is globally asymptotically stable.

Proof. Let be a solution of Eq.(6). As β < 1, the solution is bounded. Let . Using Lemma 2.6, we have

This implies that

Now consider the function

Hence,

and the function h(x) is increasing on , we get . In view of inequality (9), we have a contradiction.

Therefore, is a global attractor.

The global asymptotically stability of is obtained by combining the global attractivity and the local asymptotic stability of whan . □

In order to confirm our theoretical results, we consider the following numerical example.

Example 2.10. Consider the difference equation Eq.(6) with l = 2, k = 4, m2 = 2, m3 = 3, m4 = 4, α = 0.25 and β = 0.5, that is:

We have:

Clearly ≃ 0.49811911 is the unique equilibrium point of the equation Eq.(10) in

If we take y0 = 5.85, y−1 = 0.89, y−2 = 0.35, y−3 = 4.55, y−4 = 0.65, y−5 = 3.15, y−6 = 0.75, y−7 = 2.35, y−8 = 1.25, y−9 = 0.25, then, we get the solution as in figure 1. However, if we take y0 = 0.35, y−1 = 0.19, y−2 = 0.67, y−3 = 2.55, y−4 = 0.5, y−5 = 1.15, y−6 = 0.15, y−7 = 1.35, y−8 = 0.33, y−9 = 0.45, then, the solution will be as in figure 2.

FIGURE 1.Plot of the solution of Eq.(10) with the initial conditions: y0 = 5.85, y−1 = 0.89, y−2 = 0.35, y−3 = 4.55, y−4 = 0.65, y−5 = 3.15, y−6 = 0.75, y−7 = 2.35, y−8 = 1.25, y−9 = 0.25.

FIGURE 2.Plot of the solution of Eq.(10) with the initial conditions: y0 = 0.35, y−1 = 0.19, y−2 = 0.67, y−3 = 2.55, y−4 = 0.5, y−5 = 1.15, y−6 = 0.15, y−7 = 1.35, y−8 = 0.33, y−9 = 0.45.

As we can see from figure 1 and figure 2, for two different choices of the initial values we have

That is the equilibrium point ≃ 0.49811911 is globally asymptotically stable (as expected).

2.2. Case α = 0. When α = 0, Eq.(6) becomes

Clearly = 0 is always an equilibrium point of Eq.(11). When β > 1, Eq.(11) also possesses the positive equilibrium point .

Some very close equations and systems of difference equations to Eq.(11) have been studied, for example, [2], [4], [5], [10] and [17].

Following the above mentioned papers, we summarize the main results for this particular equation.

Lemma 2.11. Assume that β < 1. Then, every solution of Eq.(11) is bounded.

Proof. Let be a solution of Eq.(11). Then

(1)

(2)

(3)

The proof follows from the above inequalities. □

Theorem 2.12. Assume that β > 1. Let be a solution of the Eq.(11) and . Then if either

(b1) y−2k−1, y−2k+1, ..., y−1 < ≤ y−2k, y−2k+2, ..., y0

or

(b2) y−2k, y−2k+2, ..., y0 < ≤ y−2k−1, y−2k+1, ..., y−1

is satisfied, the solution oscillates about with semicycles of length one.

Proof. The proof is similar to that of Theorem 2.3 and will be omitted. □

Theorem 2.13. The following statements are true

Proof. (1) The linearized equation associated with Eq.(11) about the equilibrium point = 0 is

we can see that g(λ) has a real root in (−∞,−1) and a root with modulus less than one. Therefore, the point is a saddle point. □