# Some cohomological aspects of the Banach fixed point principle

Mathematica Bohemica (2011)

- Volume: 136, Issue: 3, page 333-336
- ISSN: 0862-7959

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topJanoš, Ludvík. "Some cohomological aspects of the Banach fixed point principle." Mathematica Bohemica 136.3 (2011): 333-336. <http://eudml.org/doc/197048>.

@article{Janoš2011,

abstract = {Let $T\colon X\rightarrow X$ be a continuous selfmap of a compact metrizable space $X$. We prove the equivalence of the following two statements: (1) The mapping $T$ is a Banach contraction relative to some compatible metric on $X$. (2) There is a countable point separating family $\mathcal \{F\}\subset \mathcal \{C\}(X)$ of non-negative functions $f\in \mathcal \{C\}(X)$ such that for every $f\in \mathcal \{F\}$ there is $g\in \mathcal \{C\}(X)$ with $f=g-g\circ T$.},

author = {Janoš, Ludvík},

journal = {Mathematica Bohemica},

keywords = {Banach contraction; cohomology; cocycle; coboundary; separating family; core; Banach contraction; cohomology; cocycle; coboundary; separating family; core},

language = {eng},

number = {3},

pages = {333-336},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Some cohomological aspects of the Banach fixed point principle},

url = {http://eudml.org/doc/197048},

volume = {136},

year = {2011},

}

TY - JOUR

AU - Janoš, Ludvík

TI - Some cohomological aspects of the Banach fixed point principle

JO - Mathematica Bohemica

PY - 2011

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 136

IS - 3

SP - 333

EP - 336

AB - Let $T\colon X\rightarrow X$ be a continuous selfmap of a compact metrizable space $X$. We prove the equivalence of the following two statements: (1) The mapping $T$ is a Banach contraction relative to some compatible metric on $X$. (2) There is a countable point separating family $\mathcal {F}\subset \mathcal {C}(X)$ of non-negative functions $f\in \mathcal {C}(X)$ such that for every $f\in \mathcal {F}$ there is $g\in \mathcal {C}(X)$ with $f=g-g\circ T$.

LA - eng

KW - Banach contraction; cohomology; cocycle; coboundary; separating family; core; Banach contraction; cohomology; cocycle; coboundary; separating family; core

UR - http://eudml.org/doc/197048

ER -

## References

top- Bakakhanian, A., Cohomological Methods in Group Theory, Marcel Dekker, New York (1972). (1972)
- Janoš, L., The Banach contraction mapping principle and cohomology, Comment. Math. Univ. Carolin. 41 (2000), 605-610. (2000) MR1795089

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