On the recursive sequencex n +1=α-(x n /x n −1)

A short proof of the Cushing‐Henson conjecture

“…Indeed, in this case (1.2) becomes y n+1 = q n y n + 1 − q n L n , n ∈ N 0 , (1.12) where q n = 1/μ n . As above it can be obtained that 13) and that arbitrary solution (y n ) n∈N0 of the equation has the form…”

“…Recently there has been a great interest in studying nonlinear difference equations and systems, in particular those which model some real life situations in population biology and ecology, see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13], and the references therein.…”

Nontrivial closed-loop analysis for an extremely simple one-step-guess adaptive controller

“…This motivates us to investigate the related research and we find that recently there has been a lot of interest in studying the global attractivity, the bound character and the periodic nature of nonlinear difference equations. For some recent results concerning, see, for example, [13], [14], [15], [16], [17], [18], [19] and the references therein. The authors [13] investigate the global attractivity of the recursive sequences x n+1 = −αx n−1 /(β ± x n ) under specified conditions.…”

“…Among all these results, the coefficients concerned are time-invariable. Especially, our model can be viewed as a generlization to that in [18] and [19], with more complicated form where the coefficients in the equation related to the system noise are time-varing. There are only limited results on nonlinear difference equation with variable coefficients, see [20], however, the condition in that paper is very harsh that our model cannot satisfy.…”

On the Behaviour of the Solutions of a Second-Order Difference Equation

“…In [34], the authors study the boundedness, the global asymptotic stability, and the periodicity of positive and negative solutions (x n ) n∈N0 of the difference equation…”