Monotonicity beyond Minty and Kato on locally convex spaces

This paper is a contribution to the theory of monotone operators in topological vector spaces. On the one hand, we provide new results concerning topological and geometric properties of monotone operators satisfying mild continuity assumptions. In particular, we give fairly general conditions on the operator to become single-valued, to be closed and maximal. A fundamental tool is a generalization of the well-known Minty's Lemma that is interesting in its own right and, surprisingly, remains true for general topological vector spaces. As a consequence of Minty's, we obtain an extension of a rather remarkable theorem of Kato for multi-valued mappings defined on general locally convex spaces.