Asymptotic behavior of a sequence defined by iteration with applications
Volume 93 / 2002
Abstract
We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let $f:(0,\infty )^2\rightarrow (0,\infty )$ be a continuous function such that (a) $0< f(x,y)< px+(1-p)y$ for some $p\in (0,1)$ and for all $x,y\in (0,\alpha )$, where $ \alpha >0$; (b) $f(x,y)=px+(1-p)y-\sum _{s=m}^{\infty }{\cal K}_s(x,y)$ uniformly in a neighborhood of the origin, where $m>1, $ ${\cal K}_s(x,y)=\sum _{i=0}^s a_{i,s}x^{s-i}y^i$; (c) ${\cal K}_m(1,1)=\sum _{i=0}^m a_{i,m}>0$. Let $x_0,x_1\in (0,\alpha )$ and $ x_{n+1}=f(x_n,x_{n-1}),\ n\in {\mathbb N}.$ Then the sequence $(x_n)$ satisfies the following asymptotic formula: $$ x_n\sim \left ({2-p\over (m-1) \sum _{i=0}^m a_{i,m}}\right ) ^{{1}/{(m-1)}}{1\over\root{m-1}\of{n}}.$$
