Shadowing, Hyers-Ulam stability and hyperbolicity for nonautonomous linear delay differential equations (Q6650048)

It is well known that delay differential equations form a special case of infinite dimensional systems. Recently, it has been shown that hyperbolic nonautonomous linear delay equations are Hyers-Ulam stable and shadowable, however the converse result has been proved only for autonomous and periodic equations under additional spectral conditions. It should be emphasized that for infinite dimensional systems, even in the simplest case of autonomous linear dynamics, the shadowing property alone does not imply hyperbolicity. In this paper the authors prove that the shadowing property implies hyperbolicity for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients.