On the Rockafellar theorem for ${\mit\Phi}^{\gamma (\cdot ,\cdot )}$-monotone multifunctions

S. Rolewicz¹

¹ Institute of Mathematics Polish Academy of Sciences P.O. Box 21, Śniadeckich 8 00-956 Warszawa, Poland

Studia Mathematica, Tome 172 (2006) no. 2, pp. 197-202

Résumé

Let $X$ be an arbitrary set, and $\gamma : X\times X \to {{\mathbb R}}$ any function. Let ${\mit \Phi }$ be a family of real-valued functions defined on $X$. Let ${\mit \Gamma }: X \to 2^{{\mit \Phi }}$ be a cyclic ${\mit \Phi }^{\gamma (\cdot ,\cdot )}$-monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function $f: X \to {{\mathbb R}}$ such that ${\mit \Gamma }$ is contained in the ${\mit \Phi }^{\gamma (\cdot ,\cdot )}$-subdifferential of $f$, ${\mit \Gamma }(x)\subset \partial _{{\mit \Phi }}^{\gamma (\cdot ,\cdot )}f |_{x}$.

Détail

DOI : 10.4064/sm172-2-6

Keywords: arbitrary set gamma times mathbb function mit phi family real valued functions defined mit gamma mit phi cyclic mit phi gamma cdot cdot monotone multifunction non empty values shown following generalization rockafellar theorem holds there function mathbb mit gamma contained mit phi gamma cdot cdot subdifferential mit gamma subset partial mit phi gamma cdot cdot