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Subgroups of the groupZ n and a generalization of the Gołąb-Schinzel functional equation

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Summary

Letn be a positive integer and letX be a linear space over a commutative fieldK. In the setĀ = (K\{0}) × X we define a binary operation ·:Ā × Ā → Ā by

$$(a,x) \cdot (b,y) = (ab,y + b^{n - 1} x),(a,x),(b,y) \in \bar A.$$

Then\(\bar Z_n (X,K): = (\bar A, \cdot )\) is a group (in general, non-commutative). Moreover the group\(\bar Z_2 (X,K)\) is isomorphic to the group of functionsf: X → X, f(z) = az + x forz ∈ X, f ↷ (a, x), a ∈ K\{0}, x ∈ X:

Theorem 1. A function f: X → K, f ≠ 0, satisfies the functional equation

$$f(x + f(x)^n y) = f(x)f(y)$$
((1))

iff the set {(f(x), x): x ∈ X, f(x) ≠ 0} is a subgroup of\(\bar Z_{n + 1} (X,K)\).

The general solution of equation (1) is analogous to that of the Gołą—Schinzel functional equation.

Theorem 2.Let K be either R or C and let Y be a topological linear space over the field K. A continuous function f: Y → K, f ≠ 0, is a solution of equation (1) iff

$$f(Y) \subseteq Rorn = 1$$

and the following two conditions are valid:

(i) if f(Y)\( \subseteq \) R, then there exists a continuous R-linear functional g: Y→R such that,

1° in the case where n is odd,

$$f(x) = \sqrt[n]{{g(x) + 1}},x \in Y$$

or

$$f(x) = \sqrt[n]{{\sup (g(x) + 1,0)}},x \in Y;$$

2° in the case where n is even,

$$f(x) = \sqrt[n]{{\sup (g(x) + 1,0)}},x \in Y;$$

(ii) if f(Y)\(\not \subseteq \) R and n=1, there exists a continuous C-linear functional g: Y→C such that f(x)=g(x)+1, x ∈Y

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Brzdęk, J. Subgroups of the groupZ n and a generalization of the Gołąb-Schinzel functional equation. Aeq. Math. 43, 59–71 (1992). https://doi.org/10.1007/BF01840475

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