On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces

Sehie Park

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 2, page 263-268
  • ISSN: 0010-2628

Abstract

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Let X be a uniformly convex Banach space, D ⊂ X , f : D → X a nonexpansive map, and K a closed bounded subset such that co ¯ K ⊂ D . If (1) f | K is weakly inward and K is star-shaped or (2) f | K satisfies the Leray-Schauder boundary condition, then f has a fixed point in co ¯ K . This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.

How to cite

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Park, Sehie. "On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces." Commentationes Mathematicae Universitatis Carolinae 37.2 (1996): 263-268. <http://eudml.org/doc/247920>.

@article{Park1996,
abstract = {Let $X$ be a uniformly convex Banach space, $D\subset X$, $f:D\rightarrow X$ a nonexpansive map, and $K$ a closed bounded subset such that $\overline\{\text\{co\}\}\,K\subset D$. If (1) $f|_K$ is weakly inward and $K$ is star-shaped or (2) $f|_K$ satisfies the Leray-Schauder boundary condition, then $f$ has a fixed point in $\overline\{\text\{co\}\}\,K$. This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.},
author = {Park, Sehie},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {uniformly convex; Banach space; Hilbert space; contraction; nonexpansive map; weakly inward map; demi-closed; Rothe condition; Leray-Schauder condition; (KR)-bounded; Opial's condition; demi-closed; Rothe condition; uniformly convex Banach space; nonexpansive map; closed bounded subset; weakly inward; star-shaped; Leray-Schauder boundary condition; fixed point},
language = {eng},
number = {2},
pages = {263-268},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces},
url = {http://eudml.org/doc/247920},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Park, Sehie
TI - On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 2
SP - 263
EP - 268
AB - Let $X$ be a uniformly convex Banach space, $D\subset X$, $f:D\rightarrow X$ a nonexpansive map, and $K$ a closed bounded subset such that $\overline{\text{co}}\,K\subset D$. If (1) $f|_K$ is weakly inward and $K$ is star-shaped or (2) $f|_K$ satisfies the Leray-Schauder boundary condition, then $f$ has a fixed point in $\overline{\text{co}}\,K$. This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.
LA - eng
KW - uniformly convex; Banach space; Hilbert space; contraction; nonexpansive map; weakly inward map; demi-closed; Rothe condition; Leray-Schauder condition; (KR)-bounded; Opial's condition; demi-closed; Rothe condition; uniformly convex Banach space; nonexpansive map; closed bounded subset; weakly inward; star-shaped; Leray-Schauder boundary condition; fixed point
UR - http://eudml.org/doc/247920
ER -

References

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