# 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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topBelitskii, 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

top- [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] G. Belitskii and Yu. Lyubich, On the normal solvability of cohomological equations on compact topological spaces, Proc. IWOTA-95 (to appear). Zbl0889.39016
- [3] M. Kuczma, Functional Equations in a Single Variable, Polish Sci. Publ., Warszawa, 1968.
- [4] Z. Nitecki, Differentiable Dynamics, MIT Press, Cambridge, Mass., 1971.
- [5] H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, N.J., 1957. Zbl0083.28204

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