Some Recent Results on Premonotone Operators

The notion of premonotone operator refers to a class of operators more general than monotone ones, which still enjoy a surjectivity property akin to Minty's Theorem for monotone operators. In this paper we prove that operators which are monotone outside a bounded set, are indeed premonotone. As a consequence, we show that premonotocity is preserved under rather arbitrary alterations of the operator values in a bounded subset of its domain. We also develop the basic elements of a polarity theory related to premonotonicity and we deepen the study of maximal premonotone operators. Finally, we prove the one-dimensional version of a previously stated conjecture, namely that any maximal premonotone operator contains a maximal monotone one.