A+ CATEGORY SCIENTIFIC UNIT

An extension of Mazur's theorem on Gateaux differentiability to the class of strongly $\alpha (\cdot )$-paraconvex functions

Volume 172 / 2006

S. Rolewicz Studia Mathematica 172 (2006), 243-248 MSC: Primary 46G05. DOI: 10.4064/sm172-3-3

Abstract

Let $(X,\| \cdot \| )$ be a separable real Banach space. Let $f$ be a real-valued strongly $\alpha (\cdot )$-paraconvex function defined on an open convex subset ${\mit \Omega } \subset X$, i.e. such that $$ f(tx+(1-t)y) \leq tf(x)+(1-t)f(y) + \mathop {\rm min}[t,(1-t)] \alpha (\| x-y \| ). $$ Then there is a dense $G_{\delta }$-set $A_{\rm G}\subset {\mit \Omega }$ such that $f$ is Gateaux differentiable at every point of $A_{\rm G} $.

Authors

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

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