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

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
Full Text:

### References:

 [1] Aczél, J. andGołlb S.,Remark on one-parameter subsemigroups of the affine group and their homo- and isomorphisms. Aequationes Math.4 (1970), 1–10. · Zbl 0205.14802 [2] Baron, K.,On the continuous solutions of the Gołąb–Schinzel equation. Aequationes Math.38 (1989), 155–162. · Zbl 0702.39005 [3] Brillouët, N. andDhombres, J.,Equations fonctionnelles et recherche de sous groupes. Aequationes Math.31 (1986), 253–293. · Zbl 0611.39004 [4] Ilse, D., Lechmann, I. andSchulz, W.,Gruppoide und Funktionalgleichungen. Deutscher Verlag der Wissenschaften, Berlin, 1984. [5] Javor, P.,On the general solution of the functional equation f(x + yf(x)) = f(x)f(y). Aequationes Math.1 (1968), 235–238. · Zbl 0165.17102 [6] Kempisty, S.,Sur les fonctions quasicontinues. Fundamenta Math.19 (1932), 184–197. · Zbl 0005.19802 [7] Kuczma, M.,Note on microperiodic functions. Rad. Mat.5 (1989), 127–140. · Zbl 0726.26005 [8] Marcus, S.,Sur les fonctions quasicontinues au sens de S. Kempisty. Colloq. Math.8 (1961), 47–53. · Zbl 0099.04501 [9] Midura, S.,Sur la determination de certains sous-groupes du groupe L s l a l’aide d’équations fonctionnelles. [Dissertationes Math., No. 105]. IMPAN, Warsaw, 1973. · Zbl 0271.53010 [10] Wołodźko, S.,Solution générale de l’équation fonctionnelle f(x + yf(x)) = f(x)f(y). Aequationes Math.2 (1968), 12–29. · Zbl 0162.20402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.