# Maximal Monotone Inclusions and Fitzpatrick Functions

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## Abstract

In this paper, we study maximal monotone inclusions from the perspective of *gap functions*. We propose a very natural gap function for an arbitrary maximal monotone inclusion and will demonstrate how naturally this gap function arises from the Fitzpatrick function, which is a convex function, used to represent maximal monotone operators. This 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 generalized variational inequality problem. The associated notion of a scalar gap is also considered in some detail. Corresponding local and global error bounds are also developed for the maximal monotone inclusion.

## Keywords

Maximal monotone operator Monotone inclusions Variational inequality Fitzpatrick function Gap functions Error bounds## Mathematics Subject Classification

90C30 49J52## Notes

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