A+ CATEGORY SCIENTIFIC UNIT

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

Volume 172 / 2006

S. Rolewicz 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 $f$, ${\mit \Gamma }(x)\subset \partial _{{\mit \Phi }}^{\gamma (\cdot ,\cdot )}f |_{x}$.

Authors

  • S. RolewiczInstitute of Mathematics
    Polish Academy of Sciences
    P.O. Box 21, Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail

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