Blurred maximal cyclically monotone sets and bipotentials

TL;DR

This paper establishes conditions under which blurred maximal cyclically monotone graphs can be represented by bipotentials, linking differential inclusions to bipotential functions in reflexive Banach spaces.

Contribution

It provides necessary and sufficient conditions for the existence of bipotentials for blurred maximal cyclically monotone graphs, extending the theory in Banach spaces.

Findings

Characterization of bipotential existence for blurred maximal cyclically monotone graphs

Conditions on $ abla ext{phi}$ for representing differential inclusions as bipotentials

Extension of bipotential theory to reflexive Banach spaces

Abstract

Let X be a reflexive Banach space and Y its dual. In this paper we find necessary and sufficient conditions for the existence of a bipotential for a blurred maximal cyclically monotone graph. Equivalently, we find a necessary and sufficient condition on $\phi \in \Gamma_{0}(X)$ for that the differential inclusion $y \in \bar{B}(\epsilon) + \partial \phi(x)$ can be put in the form $y \in \partial b(\cdot, y)(x)$, with $b$ a bipotential.