Lower Limits of Type (D) Monotone Operators in general Banach Spaces

TL;DR

This paper characterizes the sequential lower limit of maximal monotone operators of type (D) in general Banach spaces, extending key concepts like the variational sum and composition, and proving their representability.

Contribution

It provides a new characterization of the lower limits of type (D) operators and extends the definitions and representability of variational sums and compositions in Banach spaces.

Findings

Characterization of the sequential lower limit of type (D) operators.

Extension of the variational sum and composition to general Banach spaces.

Proof of the representability of these extended operators.

Abstract

We give, for general Banach spaces, a characterization of the sequential lower limit of maximal monotone operators of type (D) and prove its representability. As a consequence, using a recent extension of the Moreau-Yosida regularization for type (D) operators, we extend to general Banach spaces the definitions of the variational sum of monotone operators and the variational composition of monotone operators with continuous linear mappings, and we prove that both operators are representable.