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

100%

Brzdęk J.

Annales Polonici Mathematici

1996

tom 64

nr 3

195-205

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 $f(x + f(x)^ny) = f(x)f(y)$, then it is continuous or the set {x ∈ X : f(x) ≠ 0} is a Christensen zero set.

On the increasing solutions of the translation equation

100%

Brzdęk J.

Annales Polonici Mathematici

1996

tom 64

nr 3

207-214

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.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Brzdęk J.

2 1996

Wyniki wyszukiwania

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

On the increasing solutions of the translation equation