Search results for "Rational difference equation"
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On the Second Order Rational Difference Equation $$x_{n+1}=\beta +\frac{1}{x_n x_{n-1}}$$ x n + 1 = β + 1 x n x n - 1
2016
The author investigates the local and global stability character, the periodic nature, and the boundedness of solutions of the second-order rational difference equation $$x_{n+1}=\beta +\frac{1}{x_{n}x_{n-1}}, \quad n=0,1,\ldots ,$$ with parameter \(\beta \) and with arbitrary initial conditions such that the denominator is always positive. The main goal of the paper is to confirm Conjecture 8.1 and to solve Open Problem 8.2 stated by A.M. Amleh, E. Camouzis and G. Ladas in On the Dynamics of a Rational Difference Equations I (International Journal of Difference Equations, Volume 3, Number 1, 2008, pp.1–35).
Periodic Solutions of the Second Order Quadratic Rational Difference Equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}} $$ x n + 1 = α ( 1 + x n ) x n…
2016
The aim of this article is to investigate the periodic nature of solutions of a rational difference equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}}. {(*)} $$ We explore Open Problem 3.3 given in Amleh et al. (Int J Differ Equ 3(1):1–35, 2008, [2]) that requires to determine all periodic solutions of the equation (*). We conclude that for the equation (*) there are no periodic solution with prime period 3 and 4. Period 7 is first period for which exists nonnegative parameter \(\alpha \) and nonnegative initial conditions.