On ${\mit\Phi}^{\gamma(\cdot,\cdot)}$-subdifferentiable and $[{\mit\Phi}+\gamma]$-subdifferentiable Functions

Volume 53 / 2005

S. Rolewicz Bulletin Polish Acad. Sci. Math. 53 (2005), 273-279 MSC: 46N10, 26E15, 52A01. DOI: 10.4064/ba53-3-4


Let $X$ be an arbitrary set. Let ${\mit\Phi}$ be a family of real-valued functions defined on $X$. Let $\gamma:X\times X\to \mathbb R$. Set $[{\mit\Phi}+\gamma]=\{ \phi(\cdot)+ \gamma(\cdot,x)\mid \phi \in {\mit\Phi},\, x \in X\}$. We give conditions guaranteeing the equivalence of ${\mit\Phi}^{\gamma(\cdot,\cdot)}$-subdifferentiability and $[{\mit\Phi}+\gamma]$-subdifferentiability.


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

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