FENCHEL DUALITY, FITZPATRICK FUNCTIONS AND MAXIMAL MONOTONICITY
Stephen Simons · 2004
This paper is dedicated to Simon Fitzpatrick, in recognition of his amazing insights ABSTRACT. We show in this paper how the versions of the Fenchel duality theorem due to Rockafellar and Attouch–Brezis can be applied to the Fitzpatrick function determined by a maximal monotone multifunction to obtain number of results on maximal monotonicity, including a number of sufficient conditions for the sum of maximal monotone multifunctions on a reflexive Banach space to be maximal monotone, unifying a number of the results of “Attouch–Brezis type ” that have been obtained in recent years. We also obtain generalizations of the Brezis–Crandall–Pazy result. We find various explicit formulas in terms of the Fitzpatrick function for the minimum norm of the solutions x of (S +J)x ∋ 0, where E is reflexive, S is maximal monotone on E and J is the duality map. Among the tools that we develop are a version of the Fenchel duality theorem in which we obtain an explicit formula for the minimum norm of solutions in certain cases, and a generalization of the Attouch–Brezis version of the Fenchel duality theorem to a more symmetric result for convex functions of two variables.