ON A DIFFERENCE EQUATION MOTIVATED BY A HEAT CONDUCTION PROBLEM

Abstract

Let $\{ \tau_n \}$ be a sequence of numbers recursively defined by $$ f(\tau_n) + f(\tau_n+\tau_{n-1}) + \dots + f( \tau_n+\tau_{n-1} + \dots +\tau_1 ) =1 , $$ where $f$ is a continuous and strictly decreasing function on $(0,\infty) $ with $f(0^+) \ge 1, $ and $ f(\infty)=0 .$ Assume the convexity of $\log f$ or $\log |f'|$. It can be shown that $ \{ \tau_n \} $ is increasing. Thus $ \lim \tau_n $ exists in $ (0, \infty]$.

The difference equation above is motivated by a heat conduction problem studied in Myshkis (1997) and Chen, Chow and Hsieh (2006).