A family of enlargements of maximal monotone operators

We introduce a family of enlargements of maximal monotone operators. The Bronsted and Rockafellar epsilon -subdifferential operator can be regarded as an enlargement of the subdifferential. The family of enlargements introduced in this paper generalizes the Bronsted and Rockafellar epsilon -subdifferential (enlargement) and also generalize the enlargement of an arbitrary maximal monotone operator recently proposed by Burachik, Iusem and Svaiter. We characterize the biggest and the smallest enlargement belonging to this family and discuss some general properties of its members. A subfamily is also studied, namely the subfamily of those enlargements which are also additive. Members of this subfamily are formally closer to the epsilon -subdifferential. Existence of maximal elements is proved. In the case of the subdifferential, we prove that the epsilon -subdifferential is maximal in this subfamily.