A proof-theoretic bound extraction theorem for monotone operators in Banach spaces

Abstract

Abstract We utilize a proof-theoretically tame approach to the dual of an abstract Banach space in systems amenable to methods from proof mining, as recently introduced by the author, to provide similar such systems with accompanying logical metatheorems on the extraction of uniform quantitative information from proofs pertaining to the theory of monotone set-valued operators on Banach spaces as introduced by Browder. With that, we finally extend proof mining methods to this important class of objects which are at the heart of many seminal results from nonlinear functional analysis, and the metatheorems presented here in particular provide the first logical basis for a range of recent applications of proof mining methods to this branch of mathematics. Further, we provide a characterization of the extensionality principle for these operators using the analytical notion of maximality, extending previous analogous results for accretive operators on Banach spaces and monotone operators on Hilbert spaces, and with that further illustrate the central importance and special position of extensionality issues in proof mining applications dealing with set-valued operators.