
Journal of Convex Analysis 22 (2015), No. 3, 687710 Copyright Heldermann Verlag 2015 Some Aspects of the Representation of cMonotone Operators by CConvex Functions Sedi Bartz Dept. of Mathematics, The Technion  Israel Institute of Technology, 32000 Haifa, Israel bartz@techunix.technion.ac.il Simeon Reich Dept. of Mathematics, The Technion  Israel Institute of Technology, 32000 Haifa, Israel sreich@techunix.technion.ac.il Several takes on generalizing the theory of representation of monotone operators by convex functions to the full generality of cconvexity have been proposed in recent years. In particular, given a monotone operator, a new family of convex antiderivatives is now associated with it, both in classical convex analysis as well as in the generality of cconvexity. In the present paper we take the generalization of the theory to cconvexity a few steps farther. In particular, we study the Cconvex separable representation in detail, construct the sequence of Fitzpatrick functions of higher orders and present its basic properties in the generality of cconvexity, and, finally, we present a new example that demonstrates why the associated family of antiderivatives is a more natural environment for the Fitzpatrick function in an even more dramatic manner than in the classical case: the Fitzpatrick function turns out to be the maximal(!) member of the Fitzpatrick family, although it is still a minimal convex antiderivative. Keywords: Abstract convexity, cconvex function, convex antiderivative, cyclic monotonicity, envelope, Fitzpatrick function, maximal monotone operator, subdifferential. MSC: 47H04, 47H05, 49N15, 52A01 [ Fulltextpdf (201 KB)] for subscribers only. 