Original Article
Kyungpook Mathematical Journal 2010; 50(1): 153-164
Published online March 23, 2010
Copyright © Kyungpook Mathematical Journal.
The Dynamics of Solutions to the Equation $displaystyle x_{!n+1}!!=!frac{p!+! x_{!n-k}}{q!+!x_{!n}}+frac{x_{!n-k}}{x_{!n}}$
Xiaona Xu, Yongjin Li
Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275 P. R. China.
Received: March 23, 2010; Revised: March 23, 2010; Accepted: March 23, 2010
We study the global asymptotic stability, the character of the semicycles, the periodic nature and oscillation of the positive solutions of the difference equation $$ x_{n+1}=displaystylefrac{p+ x_{n-k}}{q+x_n}+frac{x_{n-k}}{x_n}, n=0,1,2,cdots.$$ where $p, q in(0,infty), q
eq 2, k in {1,2,cdots}$ and the initial values$ x_{-k},cdots,x_0$ are arbitrary positive real numbers.
Keywords: Difference equations, Asymptotic stability, Periodicity, Semicycle, Oscillation