On the Rockafellar theorem for ${\mit\Phi}^{\gamma (\cdot ,\cdot )}$-monotone multifunctions
Volume 172 / 2006
Studia Mathematica 172 (2006), 197-202
MSC: 46N10, 26E15, 52A01.
DOI: 10.4064/sm172-2-6
Abstract
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 $f$, ${\mit \Gamma }(x)\subset \partial _{{\mit \Phi }}^{\gamma (\cdot ,\cdot )}f |_{x}$.
