On some applications of convolution to linear differential equations with Levitan almost periodic coefficients

Abstract

We investigate some properties of Levitan almost periodic functions with particular emphasis on their behavior under convolution. These considerations allow us to establish the main result concerning Levitan almost periodic solutions to linear differential equations of the first order. In particular, we state a condition, which guarantees that a special linear equation possesses a Levitan almost periodic solution. We also compare the class of Levitan almost periodic functions and the class of almost periodic functions with respect to the Lebesgue measure, and simultaneously, give an answer to the open question posed by Basit and Günzler in the paper [Difference property for perturbations of vector-valued Levitan almost periodic functions and their analogs, Russ. J. Math. Phys. 12 (4) (2005), 424-438].