On the Rational Recursive Sequence xn+1=(α-βxn)/(γ-δxn-xn-k)

Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations

For nonnegative real numbers α, β, γ , A, B, and C such that B C > 0 and α β γ > 0, the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . . . has a unique positive equilibrium. A proof is given here for the following statements: 1 For every choice of positive parameters α, β, γ , A, B, and C, all solutions to the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . ...

Dynamics of a rational difference equation

The main goal of the paper is to investigate boundedness, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of the equation xn 1 α βxn γxn−k / 1 xn−k , n ∈ N0, where the parameters α, β, γ ∈ 0,∞ , k ≥ 2 is an integer, and the initial conditions x−k, . . . , x0 ∈ 0,∞ . It is shown that the unique positive equilibrium of the equation is globally asymptotically ...

On the Rational Recursive Sequence xn + 1 = (A + sum k i = 0 alpha i xn - i)/(B + sum k i = 0 beta i xn - i)

The difference equation xn + 1 = α + xn − k ∑ k − 1 i = 0 cixn − i has solutions converging to zero

The aim of this note is to show that the following difference equation: xn+1 = α+ xn−k ∑k−1 i=0 cixn−i , n= 0,1, . . . , where k ∈ N, ci 0, i = 0, . . . , k − 1, ∑k−1 i=0 ci = 1, and α < −1, has solutions which monotonically converge to zero. This result shows the existence of such solutions which was not shown in the recently accepted paper: A.E. Hamza, On the recursive sequence xn+1 = α+ xn−1...

On the Rational Recursive Sequence

Our main objective is to study some qualitative behavior of the solutions of the difference equation xn+1 = γxn−k + (axn + bxn−k) / (cxn − dxn−k) , n = 0, 1, 2, ..., where the initial conditions x−k,..., x−1, x0 are arbitrary positive real numbers and the coefficients γ, a, b, c and d are positive constants, while k is a positive integer number.