On the asymptotic behaviour of solutions to a linear functional equation

Abstract

We investigate the asymptotic behaviour at infinity of solutions of the equation $\varphi (x) = \int_S \varphi (x+M(s))\sigma(d s).$ We show among others that, under some assumptions, any positive solution of the equation which is integrable on a vicinity of infinity or vanishes at $+\infty$ tends on some sequence to zero faster than some exponential function, but it does not vanish faster than another such function.