Monotone operators, subdifferentials and Asplund spaces

Abstract

A set-valued map T from a Banach space E into the subsets of its dual E* is said to be a monotone operator provided

$$ \left\langle {x* - y*,x - y} \right\rangle \geqslant 0 $$

whenever x, y ∈ E and x* ∈ T(x), y* ∈ T(y). We do not require that T(x) be nonempty. The domain (or effective domain) D(T) of T is the set of all x ∈ E such that T(x) is nonempty.