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

Abstract In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly: $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> -a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> -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 $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> -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 $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> -a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> -a.p. solutions.