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The cardinality of the set of discontinuous solutions of a class of functional equations. (English) Zbl 0616.39003

The paper deals with the following equation \[ (1)\quad f([f(x)]^ ky+[f(y)]^{\ell}x)=\lambda f(x)f(y) \] where k,\(\ell \in {\mathbb{N}}\) and \(\lambda\in {\mathbb{R}}\) are given and f:\({\mathbb{R}}\to {\mathbb{R}}\) is unknown. The following theorem is proved: equation (1) has \(2^{\aleph}\) discontinuous solutions if a) \(\lambda\geq 0\), b) \(\lambda <0\) and k,\(\ell \in 2{\mathbb{N}}\), c) \(0>\lambda\) is transcendental, d) \(0>\lambda\) is algebraic over \({\mathbb{Q}}\) and its minimal polynomial has a positive root. The author uses in the proof some facts concerning derivation known e.g. form J. Aczél’s book [Lectures on functional equations and their applications (1966; Zbl 0139.093)].
Reviewer: M.Sablik

MSC:

39B99 Functional equations and inequalities

Citations:

Zbl 0139.093
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References:

[1] Aczél, J.,Lectures on functional equations and their applications. Academic Press, New York-London, 1966. · Zbl 0139.09301
[2] Brillouët, N.,Equations fonctionelles et théorie des groupes. Publ. Math. Univ. Nantes, 1983.
[3] Dhombres, J.,Finding subgroups. Aequationes Math.24 (1982), 267–269.
[4] Sablik, M.,Remark. Aequationes Math.26 (1984), 274.
[5] Urban, P.,Continuous solutions of the functional equation f(xf(y) k +yf(x) l )=f(x)f(y). Demonstratio Math.16 (1983), 1019–1025. · Zbl 0543.39001
[6] Sablik, M. andUrban, P.,On the solutions of the equation f(xf(y) k +yf(x) l )=f(x)f(y). Demonstratio Math.18 (1985), 863–867. · Zbl 0599.39006
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