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Journal of Convex Analysis 22 (2015), No. 3, 687--710
Copyright Heldermann Verlag 2015

Some Aspects of the Representation of c-Monotone Operators by C-Convex Functions

Sedi Bartz
Dept. of Mathematics, The Technion -- Israel Institute of Technology, 32000 Haifa, Israel

Simeon Reich
Dept. of Mathematics, The Technion -- Israel Institute of Technology, 32000 Haifa, Israel

Several takes on generalizing the theory of representation of monotone operators by convex functions to the full generality of c-convexity 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 c-convexity. In the present paper we take the generalization of the theory to c-convexity a few steps farther. In particular, we study the C-convex separable representation in detail, construct the sequence of Fitzpatrick functions of higher orders and present its basic properties in the generality of c-convexity, 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, c-convex function, convex antiderivative, cyclic monotonicity, envelope, Fitzpatrick function, maximal monotone operator, subdifferential.

MSC: 47H04, 47H05, 49N15, 52A01

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