A Characterization of Maximal Monotone Operators

It is shown that a set-valued map $M:mathbb{R}^{q} rightrightarrows mathbb{R}^{q}$ is maximal monotone if and only if the following five conditions are satisfied: (i) M is monotone; (ii) M has a nearly convex domain; (iii) M is convex-valued; (iv) the recession cone of the values M(x) equals the normal cone to the closure of the domain of M at x; (v) M has a closed graph. We also show that the conditions (iii) and (v) can be replaced by Cesari’s property (Q).