On the existence of smooth solutions of linear functional equations

Recent results on functional equations in a single variable, perspectives and open problems

“…In the real case the asymptotic behaviour of solutions of this equation is studied by M. H. Lantsman [158]. We mention here also the paper [36] by G. Belitskii and V. Tkachenko where differentiable and real analytic vector-valued solutions of (4.1) with f j 's being translations in a Euclidean space are examined and [191] by J. Morawiec and J. Walorski providing some its extension; see also [33] and [30]. Systems of such equations in a pretty general algebraic setting appear in [156] by M. Laczkovich where also some applications of them to studying operators commuting with the translation operators on function spaces are presented.…”

On Continuous and Smooth Solutions of the Schröder Equation in Normed Spaces

“…For recent results we refer the reader to [1]. In this paper we use a method inspired by [2] and [5]. The paper consists of three parts.…”