| Title:
|
Fixed points of periodic and firmly lipschitzian mappings in Banach spaces (English) |
| Author:
|
Pupka, Krzysztof |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
53 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
573-579 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
W.A. Kirk in 1971 showed that if $T\colon C\to C$, 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 $n\geq 2$ 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 $C$. (English) |
| Keyword:
|
lipschitzian mapping |
| Keyword:
|
firmly lipschitzian mapping |
| Keyword:
|
$n$-periodic mapping |
| Keyword:
|
fixed point |
| Keyword:
|
retractions |
| MSC:
|
47H09 |
| MSC:
|
47H10 |
| idMR:
|
MR3016427 |
| . |
| Date available:
|
2013-03-02T13:41:54Z |
| Last updated:
|
2015-02-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143191 |
| . |
| Reference:
|
[1] Bruck R.E.: Nonexpansive projections on subsets of Banach spaces.Pacific J. Math. 48 (1973), 341–357. Zbl 0274.47030, MR 0341223, 10.2140/pjm.1973.47.341 |
| Reference:
|
[2] Goebel K.: Convexity of balls and fixed point theorems for mappings with nonexpansive square.Compositio Math. 22 (1970), 269–274. Zbl 0202.12802, MR 0273477 |
| Reference:
|
[3] Goebel K., Kirk W.A.: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant.Studia Math. 47 (1973), 135–140. Zbl 0265.47044, MR 0336468 |
| Reference:
|
[4] Goebel K., Koter M.: Regularly nonexpansive mappings.Ann. Stiint. Univ. “Al.I. Cuza” Iaşi 24 (1978), 265–269. Zbl 0402.47032, MR 0533754 |
| Reference:
|
[5] Goebel K., Złotkiewicz E.: Some fixed point theorems in Banach spaces.Colloquium Math. 23 (1971), 103–106. Zbl 0223.47022, MR 0303367 |
| Reference:
|
[6] Górnicki J.: Fixed points of involution.Math. Japonica 43 (1996), no. 1, 151–155. MR 1373993 |
| Reference:
|
[7] Górnicki J., Pupka K.: Fixed point theorems for $n$-periodic mappings in Banach spaces.Comment. Math. Univ. Carolin. 46 (2005), no. 1, 33–42. Zbl 1123.47038, MR 2175857 |
| Reference:
|
[8] Kirk W.A.: A fixed point theorem for mappings with a nonexpansive iterate.Proc. Amer. Math. Soc. 29 (1971), 294–298. Zbl 0213.41303, MR 0284887, 10.1090/S0002-9939-1971-0284887-3 |
| Reference:
|
[9] Kirk W.A., Sims B. (eds.): Handbook of Metric Fixed Point Theory.Kluwer Acad. Pub., Dordrecht-Boston-London, 2001. Zbl 0970.54001, MR 1904271 |
| Reference:
|
[10] Koter M.: Fixed points of lipschitzian $2$-rotative mappings.Boll. Un. Mat. Ital. C (6) 5 (1986), 321–339. Zbl 0634.47053, MR 0897203 |
| Reference:
|
[11] Linhart J.: Fixpunkte von Involutionen $n$-ter Ordnung.Österreich. Akad. Wiss. Math.-Natur., Kl. II 180 (1972), 89–93. Zbl 0244.47041, MR 0303369 |
| Reference:
|
[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. MR 2771119 |
| . |
