On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations

Andres, Jan; Pennequin, Denis

doi:10.1090/S0002-9939-2012-11154-2

On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations
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Proc. Amer. Math. Soc. 140 (2012), 2825-2834 Request permission

Abstract:

It is shown that in uniformly convex Banach spaces, Stepanov almost-periodic functions with Stepanov almost-periodic derivatives are uniformly almost-periodic in the sense of Bohr. This in natural situations yields, jointly with the derived properties of the associated Nemytskii operators, the nonexistence of purely (i.e. nonuniformly continuous) Stepanov almost-periodic solutions of ordinary differential equations. In particular, the existence problem of such solutions, considered in a series of five papers of Z. Hu and A. B. Mingarelli, is answered in a negative way.

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