Maximal Monotone Inclusions and Fitzpatrick Functions

We study maximal monotone inclusions from the perspective of (convex) gap functions. We propose a very natural gap function and will demonstrate how this function arises from the Fitzpatrick function — a convex function used effectively to represent maximal monotone operators. • This approach allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion. – We also study the special cases of a variational inequality and of a generalised variational inequality problem. – The associated notion of a scalar gap is also considered. – Corresponding local and global error bounds are developed for the maximal monotone inclusion.