Blurred maximal cyclically monotone sets and bipotentials

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 $\varphi \in {\mathrm{\Gamma }}_0(X)$ for that the differential inclusion $y\in \bar{B}(ϵ)+\mathrm{\partial }\varphi (x)$ can be put in the form $y\in \mathrm{\partial }b(\cdot ,y)(x)$, with $b$ a bipotential.