EUDML

Let M be a non-empty set endowed with a dense linear order without the smallest and greatest elements. Let (G,+) be a group which has a non-trivial uniquely divisible subgroup. There are given conditions under which every solution F: M×G → M of the translation equation is of the form $F (a, x) = f^{- 1} (f (a) + c (x))$ for a ∈ M, x ∈ G with some non-trivial additive function c: G → ℝ and a strictly increasing function f: M → ℝ such that f(M) + c(G) ⊂ f(M). In particular, a problem of J. Tabor is solved.

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.

On some iteration semigroups

Janusz Brzdęk — 1995

Archivum Mathematicum

Let $F$ be a disjoint iteration semigroup of $C^{n}$ diffeomorphisms mapping a real open interval $I \neq ⌀$ onto $I$ . It is proved that if $F$ has a dense orbit possesing a subset of the second category with the Baire property, then $F = {f_{t} f_{t} (x) = f^{- 1} (f (x) + t) for every x \in I, t \in R}$ for some $C^{n}$ diffeomorphism $f$ of $I$ onto the set of all reals $R$ . The paper generalizes some results of J.A.Baker and G.Blanton [3].