## Rolewicz's Problem (Solved)

Summary: Find all even, nonnegative, and differentiable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ satisfying the inequality $f(t) - f(s) - f'(s)(t-s) \geq f(t-s), \; t,s \in {\mathbb R}$. This problem is motivated by the study of S. Rolewicz of Fréchet $\Phi$-differentiability of real-valued mappings of a metric space $(X,d)$.

Classification: Primary, functional analysis; Secondary, integral and functional equations

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**Bogdan Choczewski**

Faculty of Applied Mathematics

Department of Real and Complex Analysis

University of Mining and Metallurgy (AGH)

al. Mickiewicz 30

PL-30-059 Krakow

Poland

e-mail: smchocze@cyf-kr.edu.pl**Download Solution**[PDF]

**Michael Renardy**

Department of Mathematics

Virginia Tech

Blacksburg, VA 24061-0123

e-mail: renardym@math.vt.edu**Download Solution**[PDF]

**Roland Girgensohn**

Institute for Biomathematics and Biometry

GSF-National Research Center

Postfach 1129, 85758, Neuerberg

Germany

e-mail: girgen@janus.gsf.de