On Stepanov almost-periodic oscillations and their discretizations

The relationship between Caratheodory almost-periodic (a.p.) solutions and their discretizations is clarified for differential equations and inclusions in Banach spaces. Our investigation was stimulated by an old result of Meisters [Proc. Am. Math. Soc. 10 (1959), pp. 113-119] about Bohr a. p. solutions which we generalize in several directions. Unlike for functions, Stepanov and Bohr a. p. sequences are shown to coincide. A particular attention is paid to purely (i.e. non-uniformly continuous) Stepanov a. p. solutions. Many ideas are explained in detail by means of examples illustrated.