We consider a general nonlinear input-output system governed by operat
or equations that relate the system's input, state, and output, all of
which are in extended spaces. It is assumed that the system variables
are separated. Our results give conditions under which the stability
of the nominal system is robust; i.e., it is not destroyed by any suff
iciently small admissible perturbation of the system. Theorem 1 deals
with the case when by stability we mean the incremental stability. The
orem 3 concerns the -stability; i.e., the case when the stability is
essentially the boundedness of the transmission operator. Moreover, in
Theorem 2 it is shown that, under certain conditions, the incremental
stability of the nominal system implies insensitivity. Basically, our
results show that if the operators describing the nominal system are
well behaved, and the transition from the nominal system to the pertur
bed system is not abrupt, then the nominal system stability is robust.
The applications of the results are illustrated by several examples.