Fixed points of periodic and firmly lipschitzian mappings in Banach spaces

Krzysztof Pupka

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 4, page 573-579
  • ISSN: 0010-2628

Abstract

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W.A. Kirk in 1971 showed that if T : C → C , where C is a closed and convex subset of a Banach space, is n -periodic and uniformly k -lipschitzian mapping with k < k 0 ( n ) , then T has a fixed point. This result implies estimates of k 0 ( n ) for natural n ≥ 2 for the general class of k -lipschitzian mappings. In these cases, k 0 ( n ) are less than or equal to 2. Using very simple method we extend this and later results for a certain subclass of the family of k -lipschitzian mappings. In the paper we show that k 0 ( 3 ) > 2 in any Banach space. We also show that Fix ( T ) is a Hölder continuous retract of C .

How to cite

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Pupka, Krzysztof. "Fixed points of periodic and firmly lipschitzian mappings in Banach spaces." Commentationes Mathematicae Universitatis Carolinae 53.4 (2012): 573-579. <http://eudml.org/doc/252506>.

@article{Pupka2012,
abstract = {W.A. Kirk in 1971 showed that if $T\colon C\rightarrow C$, where $C$ is a closed and convex subset of a Banach space, is $n$-periodic and uniformly $k$-lipschitzian mapping with $k<k_0(n)$, then $T$ has a fixed point. This result implies estimates of $k_0(n)$ for natural $n\ge 2$ for the general class of $k$-lipschitzian mappings. In these cases, $k_0(n)$ are less than or equal to 2. Using very simple method we extend this and later results for a certain subclass of the family of $k$-lipschitzian mappings. In the paper we show that $k_0(3)>2$ in any Banach space. We also show that $\operatorname\{Fix\}(T)$ is a Hölder continuous retract of $C$.},
author = {Pupka, Krzysztof},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {lipschitzian mapping; firmly lipschitzian mapping; $n$-periodic mapping; fixed point; retractions; firmly Lipschitzian mapping; -periodic mapping; fixed point; Hölder continuous retraction},
language = {eng},
number = {4},
pages = {573-579},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Fixed points of periodic and firmly lipschitzian mappings in Banach spaces},
url = {http://eudml.org/doc/252506},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Pupka, Krzysztof
TI - Fixed points of periodic and firmly lipschitzian mappings in Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 4
SP - 573
EP - 579
AB - W.A. Kirk in 1971 showed that if $T\colon C\rightarrow C$, where $C$ is a closed and convex subset of a Banach space, is $n$-periodic and uniformly $k$-lipschitzian mapping with $k<k_0(n)$, then $T$ has a fixed point. This result implies estimates of $k_0(n)$ for natural $n\ge 2$ for the general class of $k$-lipschitzian mappings. In these cases, $k_0(n)$ are less than or equal to 2. Using very simple method we extend this and later results for a certain subclass of the family of $k$-lipschitzian mappings. In the paper we show that $k_0(3)>2$ in any Banach space. We also show that $\operatorname{Fix}(T)$ is a Hölder continuous retract of $C$.
LA - eng
KW - lipschitzian mapping; firmly lipschitzian mapping; $n$-periodic mapping; fixed point; retractions; firmly Lipschitzian mapping; -periodic mapping; fixed point; Hölder continuous retraction
UR - http://eudml.org/doc/252506
ER -

References

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  9. Kirk W.A., Sims B. (eds.), Handbook of Metric Fixed Point Theory, Kluwer Acad. Pub., Dordrecht-Boston-London, 2001. Zbl0970.54001MR1904271
  10. Koter M., Fixed points of lipschitzian 2 -rotative mappings, Boll. Un. Mat. Ital. C (6) 5 (1986), 321–339. Zbl0634.47053MR0897203
  11. Linhart J., Fixpunkte von Involutionen n -ter Ordnung, Österreich. Akad. Wiss. Math.-Natur., Kl. II 180 (1972), 89–93. Zbl0244.47041MR0303369
  12. Perez Garcia V., Fetter Nathansky H., Fixed points of periodic mappings in Hilbert spaces, Ann. Univ. Mariae Curie-Skłodowska Sect. A 64 (2010), no. 2, 37–48. MR2771119

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