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.
@article{bwmeta1.element.bwnjournal-article-smv127i1p81bwm,
author = {G. Belitskii and Yu. Lyubich},
title = {The Abel equation and total solvability of linear functional equations},
journal = {Studia Mathematica},
volume = {129},
year = {1998},
pages = {81-97},
zbl = {0889.39017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv127i1p81bwm}
}
Belitskii, G.; Lyubich, Yu. The Abel equation and total solvability of linear functional equations. Studia Mathematica, Tome 129 (1998) pp. 81-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv127i1p81bwm/
[00000] [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.
[00001] [2] G. Belitskii and Yu. Lyubich, On the normal solvability of cohomological equations on compact topological spaces, Proc. IWOTA-95 (to appear). | Zbl 0889.39016
[00002] [3] M. Kuczma, Functional Equations in a Single Variable, Polish Sci. Publ., Warszawa, 1968.
[00003] [4] Z. Nitecki, Differentiable Dynamics, MIT Press, Cambridge, Mass., 1971.
[00004] [5] H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, N.J., 1957. | Zbl 0083.28204