# The Abel equation and total solvability of linear functional equations

Studia Mathematica (1998)

• Volume: 127, Issue: 1, page 81-97
• ISSN: 0039-3223

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## Abstract

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We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx) - φ(x) = γ(x). The smooth situation can also be considered in this way.

## How to cite

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Belitskii, G., and Lyubich, Yu.. "The Abel equation and total solvability of linear functional equations." Studia Mathematica 127.1 (1998): 81-97. <http://eudml.org/doc/216461>.

@article{Belitskii1998,
abstract = {We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx) - φ(x) = γ(x). The smooth situation can also be considered in this way.},
author = {Belitskii, G., Lyubich, Yu.},
journal = {Studia Mathematica},
keywords = {functional equation; Abel equation; cohomological equation; wandering set; linear functional equations; topological space; total solvability},
language = {eng},
number = {1},
pages = {81-97},
title = {The Abel equation and total solvability of linear functional equations},
url = {http://eudml.org/doc/216461},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Belitskii, G.
AU - Lyubich, Yu.
TI - The Abel equation and total solvability of linear functional equations
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 1
SP - 81
EP - 97
AB - We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx) - φ(x) = γ(x). The smooth situation can also be considered in this way.
LA - eng
KW - functional equation; Abel equation; cohomological equation; wandering set; linear functional equations; topological space; total solvability
UR - http://eudml.org/doc/216461
ER -

## References

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1. [1] N. H. Abel, Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable, in: Oeuvres complètes, Vol. II, Christiania, 1881.
2. [2] G. Belitskii and Yu. Lyubich, On the normal solvability of cohomological equations on compact topological spaces, Proc. IWOTA-95 (to appear). Zbl0889.39016
3. [3] M. Kuczma, Functional Equations in a Single Variable, Polish Sci. Publ., Warszawa, 1968.
4. [4] Z. Nitecki, Differentiable Dynamics, MIT Press, Cambridge, Mass., 1971.
5. [5] H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, N.J., 1957. Zbl0083.28204

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