Normed convex processes

Abstract:

We show that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.