Opuscula Math. 32, no. 3 (2012), 559-577
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.
Keywords: linear functional equations and inequalities, solutions with a constant sign, asymptotic behaviour of solutions.
Mathematics Subject Classification: 39B12, 39B22, 39B62, 26A12.