  A general nonlinear regularity model for a set-valued mapping $F:X\times
R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered
using special iteration procedures, going back to Banach, Schauder, Lusternik
and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal.
Appl., 118 (1986) and employ it to obtain basic estimates for studying
regularity/openness properties. We also show that it can serve as a
substitution of the Ekeland variational principle when establishing other
regularity criteria. Then, we apply the induction theorem and the mentioned
estimates to establish criteria for both global and local versions of
regularity/openness properties for our model and demonstrate how the
definitions and criteria translate into the conventional setting of a
set-valued mapping $F:X\rightrightarrows Y$.
