On the higher order rational recursive sequence xn=Axn-k+Bxn-3k

doi:10.1016/j.amc.2005.04.007

Applied Mathematics and Computation

Volume 173, Issue 2, 15 February 2006, Pages 710-723

https://doi.org/10.1016/j.amc.2005.04.007 Get rights and content

Abstract

The main purpose of the current paper is to prove that every positive solution of the delay difference equation $x_{n} = \frac{A}{x_{n - k}} + \frac{B}{x_{n - 3 k}},$ where A, B ∈ (0, ∞), the initial conditions x_−3k+1, x_−3k+2, …, x₀ ∈ (0, ∞) and k ∈ {1, 2, 3, …}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion.

Section snippets

Preliminaries

Consider the recursive sequence $x_{n} = \frac{A}{x_{n - k}} + \frac{B}{x_{n - 3 k}}, n = 1, 2, \dots,$ where A, B ∈ (0, ∞), k ∈ {1, 2, 3, …}, and the initial conditions x_−3k+1, x_−3k+2, …, x₀, are arbitrary positive real numbers. We investigate the periodic character of the positive solution of Eq. (1) and we shall show that the sequence {x_n} converges eventually to a period-k solution. This confirms Conjecture 4.8.1 in [10], [11] given by Ladas and his co-workers for the above mentioned particular case.

Basically, we generalize some of the results due to

Local stability of Eq. (3)

In this section, we discuss locally asymptotically stable of Eq. (3).

Let $f (x, y) = C + \frac{x}{y},$ assume that $p ≔ p (\bar{y}, \bar{y}) = \frac{\partial f (\bar{y}, \bar{y})}{\partial x} = \frac{1}{\bar{y}} = \frac{1}{C + 1}$ and $q ≔ q (\bar{y}, \bar{y}) = \frac{\partial f (\bar{y}, \bar{y})}{\partial y} = - \frac{1}{\bar{y}} = - \frac{1}{C + 1},$ denote the partial derivates of f(x, y) evaluated at the equilibrium point $\bar{y}$ of Eq. (3). Then the equation $z_{n} = {pz}_{n - k} + {qz}_{n - 2 k} = \frac{1}{C + 1} z_{n - k} - \frac{1}{C + 1} z_{n - 2 k}$ is called the Linearized equation associated with Eq. (3) about the equilibrium point $\bar{y}$ . Its characteristic equation is $λ^{2 k} - \frac{1}{C + 1} λ^{k} + \frac{1}{C + 1} = 0 .$ By substituting θ = λ^k, it turns to the following equation: $θ^{2}$

Analysis of semi-cycle and global stability of Eq. (3)

In this section, first we investigate the semi-cycle of Eq. (3). Next we consider the global behavior of solutions of it and we prove that the equilibrium $\bar{y}$ is globally asymptotically stable for all C > 0.

Definition 3

Let ${y_{n}}_{n = - 2 k + 1}^{\infty}$ be a positive solution of Eq. (3). A positive semi-cycle of ${y_{n}}_{n = - 2 k + 1}^{\infty}$ consists of a “string” of terms {y_l, y_l+1 …, y_m}, all greater than or equal to the equilibrium $\bar{y}$ , with l ⩾ −2k + 1 and m ⩽ ∞ and such that $either l = - 2 k + 1, or l > - 2 k + 1, and y_{l - 1} < \bar{y},$ and $either m = \infty, or m < \infty, and y_{m + 1} < \bar{y} .$

Definition 4

Let ${y_{n}}_{n = -}$

The periodic behavior of the solution of Eq. (1)

Here we show that the equilibrium of the solution of Eq. (1) is periodic with period k.

From Eq. (1), we obtain $x_{2 n} = \frac{A}{x_{2 n - k}} + \frac{B}{x_{2 n - 3 k}},$ and $x_{2 n - k} = \frac{A}{x_{2 n - 2 k}} + \frac{B}{x_{2 n - 4 k}} .$ So it follows that $x_{2 n} = \frac{A}{\frac{A}{x_{2 n - 2 k}} + \frac{B}{x_{2 n - 4 k}}} + \frac{B}{\frac{A}{x_{2 n - 4 k}} + \frac{B}{x_{2 n - 6 k}}} .$ The following statements outline properties of the Eq. (1). Their proofs are based on Eq. (14).

Lemma 2

Let {x_n} be a positive solution of Eq. (1). Then the following statements hold.

(i)
For N > 0, let $m_{N} = Min {x_{2 N - 4 k}, x_{2 N - 2 k}, x_{2 N}},$ and $M_{N} = Max {x_{2 N - 4 k}, x_{2 N - 2 k}, x_{2 N}} .$ Then $m_{N} ⩽ x_{2 N + 2 lk} ⩽ M_{N} for l ⩾ 1 .$
(ii)
There exist positive

Numerical discussion

In this section, we present two test examples to support our theoretical discussion.

Example 1

Assume that Eq. (1) hold, k = 2 and A = B = 1. So the equation reduces to the following: $x_{n} = \frac{1}{x_{n - 2}} + \frac{1}{x_{n - 6}} .$ Using the change of variable $y_{n} = x_{n} x_{n - 2},$ the corresponding equation will be as follows: $y_{n} = 1 + \frac{y_{n - 2}}{y_{n - 4}} .$ In this part, to illustrate the result of this paper a numerical example is given, which was carried out using Maple 7. We assume A = B = 1 and the initial points x₋₅, x₋₄, …, x₀ in the following table.

n	x_n	n	x_n	n	x_n	n	x_n
−5	3.05	20

Conclusion

The results of this paper are generalizations of those of DeVault [2]. This solves the conjecture of Ladas and his co-workers [5], [6] for some special cases. However, the general case is still open. By a change of variable, we deduce a new difference equation which involves some interesting results in its own right. For instance, it has a unique positive equilibrium which is oscillatory and globally attractive. This was used to show that every positive solution of Eq. (1) converges to a

References (22)

R.M. Abu-Saris et al.
Global stability of x_n+1 = a + x_n/x_n−k
Applied Mathematics Letters
(2003)
M. Dehghan et al.
Bounds for solution of a six-point partial-difference scheme
Computers and Mathematics with Applications
(2004)
R. DeVault et al.
Global behavior of y_n+1 = (p + y_n−k)/(qx_n + x_n−k)
Nonlinear Analysis Theory Methods and Applications
(2001)
H.M. El-Owaidy et al.
On asymptotic behavior of the difference equation x_n+1 = a + x_n−k/x_n
Applied Mathematics and Computation
(2004)
W.-S. He et al.
Global attractivity of the difference equation x_n+1 = a + x_n−k/x_n
Applied Mathematics and Computation
(2004)
R.P. Agarwal
Difference Equations and Inequalities, Theory, Methods and Applications
(2000)
R.P. Agarwal et al.
Advanced Topics in Difference Equations
(1997)
R. DeVault et al.
On the recursive sequence x_n+1 = a + x_n−k/x_n
Journal of Difference Equations and Applications
(2003)
R. DeVault et al.
On the recursive sequence x_n+1 = A/x_n + 1/x_n−2
Proceedings of the American Mathematical Society
(1998)
V.L. Kocic et al.
Global Behavior of Nonlinear Difference Equations of Higher Order with Applications
(1993)

M.R.S. Kulenovic’ et al.

Dynamic of Second Order Rational Difference Equations

(2002)

Cited by (4)

The periodicity and solutions of the rational difference equation with periodic coefficients
2011, Computers and Mathematics with Applications
Dynamics of x<inf>n + 1</inf> = frac(x<inf>n - 2 k + 1</inf>, x<inf>n - 2 k + 1</inf> + α x<inf>n - 2 l</inf>)
2007, Applied Mathematics and Computation
We investigate the global stability of all positive solutions of the difference equation $x_{n + 1} = \frac{x_{n - 2 k + 1}}{x_{n - 2 k + 1} + α x_{n - 2 l}}$ , where α is a positive real number, k ∈ {1, 2 , …}, l ∈ {0, 1 , …} and the initial conditions are positive real numbers. When α < 1 we show the positive equilibrium $\bar{x} = \frac{1}{α + 1}$ is globally asymptotically stable. Also, when α > 1 we show that this equation possesses solutions that converge to the two-cycle …, 0, 1, 0, 1 , …
The oscillatory character of the recursive sequence x<inf>n+1</inf> = α+βx<inf>n-k+1</inf>/A+Bx<inf>n-2k+1</inf>
2006, Applied Mathematics and Computation
Global asymptotic stability of the higher order equation xn+1=axn+bxn-kA+Bxn-k
2017, Journal of Applied Mathematics and Computing

¹: Department of Mathematics and Statistics, University of Laval, Quebec, Canada.

View full text

On the higher order rational recursive sequence xn=Axn-k+Bxn-3k

Abstract

Section snippets

Preliminaries

Local stability of Eq. (3)

Analysis of semi-cycle and global stability of Eq. (3)

The periodic behavior of the solution of Eq. (1)

Numerical discussion

Conclusion

Applied Mathematics Letters

Computers and Mathematics with Applications

Nonlinear Analysis Theory Methods and Applications

Applied Mathematics and Computation

Applied Mathematics and Computation

Difference Equations and Inequalities, Theory, Methods and Applications

Advanced Topics in Difference Equations

On the recursive sequence xn+1 = a + xn−k/xn

Journal of Difference Equations and Applications

On the recursive sequence xn+1 = A/xn + 1/xn−2

Proceedings of the American Mathematical Society

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications

Dynamic of Second Order Rational Difference Equations

On the higher order rational recursive sequence $x_{n} = \frac{A}{x_{n - k}} + \frac{B}{x_{n - 3 k}}$

On the recursive sequence x_n+1 = a + x_n−k/x_n

On the recursive sequence x_n+1 = A/x_n + 1/x_n−2