ArticleOriginal scientific text
Title
The Abel equation and total solvability of linear functional equations
Authors 1, 2
Affiliations
- Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel
- Department of Mathematics, Technion, 32000 Haifa, Israel
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.
Keywords
functional equation, Abel equation, cohomological equation, wandering set
Bibliography
- 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.
- G. Belitskii and Yu. Lyubich, On the normal solvability of cohomological equations on compact topological spaces, Proc. IWOTA-95 (to appear).
- M. Kuczma, Functional Equations in a Single Variable, Polish Sci. Publ., Warszawa, 1968.
- Z. Nitecki, Differentiable Dynamics, MIT Press, Cambridge, Mass., 1971.
- H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, N.J., 1957.
Pages:
81-97
Main language of publication
English
Received
1996-11-04
Accepted
1997-06-17
Published
1998
Exact and natural sciences