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Mat. Sb. (N.S.), 1988, Volume 136(178), Number 4(8), Pages 468–477 (Mi msb1754)  

Fixed points and differentiability of the norm

N. M. Gulevich, S. V. Konyagin, R. V. Rakhmankulov


Abstract: It is proved that in a (real) uniformly smooth Banach space $X$ a nonexpansive mapping $f\colon X\to X$ has a fixed point if
$$ \inf\{\|x-y\|:x\in f(\partial E), y\in X\setminus\operatorname{\overline{co}}E\}>0 $$
for some nonempty closed bounded (not necessarily convex) set $E\subset X$ with boundary $\partial E$ and closed convex hull $\operatorname{\overline{co}}E$.
It is also shown that a nonexpansive mapping $f\colon B\to X$, where $B$ is a closed bounded convex subset of a Hilbert space or a two-dimensional strictly convex Banach space $X$, has a fixed point if
$$ \{x+t(f(x)-x):0<t\leqslant 1\}\cap C\ne\varnothing\quadfor all\quad x\in\partial C $$
for some nonempty closed (not necessarily convex) set $C\subset B$.
Bibliography: 11 titles.

Full text: PDF file (621 kB)
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English version:
Mathematics of the USSR-Sbornik, 1989, 64:2, 461–469

Bibliographic databases:

UDC: 517.988.52
MSC: Primary 47H09, 47H10; Secondary 46B20, 46B22, 46C05
Received: 24.08.1987

Citation: N. M. Gulevich, S. V. Konyagin, R. V. Rakhmankulov, “Fixed points and differentiability of the norm”, Mat. Sb. (N.S.), 136(178):4(8) (1988), 468–477; Math. USSR-Sb., 64:2 (1989), 461–469

Citation in format AMSBIB
\Bibitem{GulKonRak88}
\by N.~M.~Gulevich, S.~V.~Konyagin, R.~V.~Rakhmankulov
\paper Fixed points and differentiability of the norm
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 136(178)
\issue 4(8)
\pages 468--477
\mathnet{http://mi.mathnet.ru/msb1754}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=965887}
\zmath{https://zbmath.org/?q=an:0678.47043|0653.47036}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 64
\issue 2
\pages 461--469
\crossref{https://doi.org/10.1070/SM1989v064n02ABEH003320}


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