Second Order Cones for Maximal Monotone Operators via Representative Functions

It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to represent differential information associated with the tangent and normal cones to the Graph T. In particular we obtain formula for the proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of proto-differentiability to the graph of T, is often associated with single valuedness of T. The second author’s research was funded by NSERC and the Canada Research Chair programme, and the first author’s by ARC grant number DP0664423. This study was commenced between August and December 2005 while the first author was visiting Dalhousie University.