An Induction Theorem and Nonlinear Regularity

A general nonlinear regularity model for a set-valued mapping F : X × R+ ⇒ 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) and employ it to obtain basic estimates for exploring 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 ⇒ Y . An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.