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The Abel equation and total solvability of linear functional equations

G. Belitskii, Yu. Lyubich (1998)

Studia Mathematica

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)...

The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

Janusz Brzdęk (1996)

Annales Polonici Mathematici

Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation f ( x + f ( x ) n y ) = f ( x ) f ( y ) , then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.

The continuous solutions of a generalized Dhombres functional equation

L. Reich, Jaroslav Smítal, M. Štefánková (2004)

Mathematica Bohemica

We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J → J is a given increasing homeomorphism of an open interval J ⊂ ( 0 , ∞ ) and f ( 0 , ∞ ) → J is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under ϕ and which contains in its interior no fixed point except for 1 . They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

The converse problem for a generalized Dhombres functional equation

L. Reich, Jaroslav Smítal, M. Štefánková (2005)

Mathematica Bohemica

We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J → J is a given homeomorphism of an open interval J ⊂ ( 0 , ∞ ) and f ( 0 , ∞ ) → J is an unknown continuous function. A characterization of the class 𝒮 ( J , ϕ ) of continuous solutions f is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f ( 0 , ∞ ) → J , where J is an interval, there is an increasing homeomorphism ϕ of J such that f ∈ 𝒮 ( J , ϕ ) . We...

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