### abstract

- describes the light absorption in an interstellar medium and there is given an (incorrect) analysis of the solutions of Eq. (i). An analogous equation, but with ~ < i (i.e., with "extension" replaced by "contraction"), has been investigated by Mahler [2] in connection with a certain problem in number theory. Among other applications we mention also the theory of dynamical systems [3, 4], the theory of probability on algebraic structures, the theory of nonlinear oscillations. A complete analysis of Eq. (i) (for various values of the parameters a, b, ~) has been given by Kato and McLeod [5]. Kato has formulated the problem of the investigation of equations that contain terms with both "contractions" and "extensions" of the argument. In particular, of a special interest in physical applications is the problem of the existence of bounded (almost periodic) solutions. The purpose of this note is to give an answer to T. Kato's question for the equation l V' (t) = ~ ~ju (~/) + ~..~ (t), ( 2 ) j=-/