  The study proposes new results on the set input-to-state stability (ISS)
subject to a small input time delay for compact, invariant sets that contains
the origin. First, using the nonlinear small-gain theory, we prove a
Razumikhin-type theorem that ensures ISS for sets in the setup of functional
differential equations (FDEs) with disturbances. Next we show how this theorem
can be used to ensure set ISS for nonlinear systems with input delays and
disturbances. In comparison to the existing research on set ISS robustness with
respect to small time delays at the input, our results are rather broad,
retaining the ISS gain and without any constraints on time delayed states. The
effectiveness of the method is illustrated through two case-studies addressing
set stability for classes of nonlinear oscillators of practical interest.
