An Induction Theorem and Nonlinear Regularity Models

A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for 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 paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.