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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  2. Goebel K., Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compositio Math. 22 (1970), 269–274. Zbl0202.12802MR0273477
  3. Goebel K., Kirk W.A., A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140. Zbl0265.47044MR0336468
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  5. Goebel K., Złotkiewicz E., Some fixed point theorems in Banach spaces, Colloquium Math. 23 (1971), 103–106. Zbl0223.47022MR0303367
  6. Górnicki J., Fixed points of involution, Math. Japonica 43 (1996), no. 1, 151–155. MR1373993
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  8. Kirk W.A., 10.1090/S0002-9939-1971-0284887-3, Proc. Amer. Math. Soc. 29 (1971), 294–298. Zbl0213.41303MR0284887DOI10.1090/S0002-9939-1971-0284887-3
  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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