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On the increasing solutions of the translation equation

Janusz Brzdęk — 1996

Annales Polonici Mathematici

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 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 I , t 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].

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