On Autonomous Superposition Operators in Spaces of μ–Almost Periodic Functions and Applications to Linear Differential Equations

In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly: μ -a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of μ -a.p. functions. We will indicate necessary and sufficient conditions under which the autonomous superposition operator maps the space under consideration into itself as well as conditions under which it is continuous. As a corollary from these results we indicate when the autonomous superposition operator defined on that space is a bijection. Next, we will analyse in detail the situation when the composition of μ -a.p. function with a continuous function or with a homeomorphism gives a Stepanov almost periodic function. As an application of our results we indicate a subclass of μ -a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have μ -a.p. solutions.