Subgroups of the group \(Z_ n\) and a generalization of the Gołąb- Schinzel functional equation. (English) Zbl 0757.39003

Brzdȩk, Janusz

Aequationes Math. 43, No. 1, 59-71 (1992).

Let \(X\) be a linear space over a commutative field \(K\) and let \(n\) be a positive integer. Finding subgroups of some forms leads the author to the generalized Gołąb-Schinzel functional equation (1) \(f(x+f(x)^ ny)=f(x)f(y)\) for functionals \(f:X\to K\). Its general solution may be described similarly as in the case of the Gołąb-Schinzel equation (the case of \(n=1)\).
Assuming that \(K\in\{\mathbb{R},\mathbb{C}\}\) the author gives all the solutions \(f:X\to K\) of (1) such that the set (2) \(\{x\in X:f(x)\neq 0\}\) has an algebraically interior point. Then he considers the case where \(X\) is a (real or complex) topological vector space. If the set (2) is open then the solution \(f\) is necessarily continuous and its explicit form is given. Quasicontinuous solutions are also considered.

Reviewer: K.Baron (Katowice)

MSC:

39B12	Iteration theory, iterative and composite equations
39B52	Functional equations for functions with more general domains and/or ranges
20E07	Subgroup theorems; subgroup growth

Keywords:

quasicontinuous solutions; linear space; Gołąb-Schinzel functional equation; topological vector space

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