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Solution of generalized Jensen and quadratic functional equations. (English) Zbl 1448.39039

Anastassiou, George A. (ed.) et al., Frontiers in functional equations and analytic inequalities. Cham: Springer. 273-292 (2019).
The authors solve some extensions to abelian monoids of Jensen’s functional equation and the quadratic functional equation. They generalize some of the results obtained by R. Łukasik [Aequationes Math. 83, No. 1–2, 75–86 (2012; Zbl 1239.39019)].
Let \(S\) be an abelian monoid, \(\Phi\) a group of automorphisms of \(S\) of cardinality \(\lvert \Phi\rvert < \infty\), \(\{ a_{\lambda} \mid \lambda \in \Phi\} \subseteq S\) and \((H,+)\) be an abelian group. The authors find solutions \(f:S \to H\) of each of the following functional equations \begin{align*} \sum_{\lambda \in \Phi} f(x + \lambda y + a_{\lambda}) & = \lvert \Phi\rvert f(x) \text{ for all } x,y \in S, \\ \sum_{\lambda \in \Phi} f(x + \lambda y + a_{\lambda}) & = \lvert \Phi\rvert f(x) + \lvert \Phi\rvert f(y) \text{ for all } x,y \in S. \end{align*} For the first (resp., the second) functional equation it is assumed that \(H\) is uniquely divisible by \(\lvert \Phi\rvert !\) (resp., \((\lvert \Phi\rvert +1)!\)). The solutions are expressed in terms of \(\lvert \Phi\rvert\) (resp., \(\lvert \Phi\rvert +1\)), constant, additive and multi-additive functions on \(S\) with values in \(H\).
For the entire collection see [Zbl 1432.39001].

MSC:

39B52 Functional equations for functions with more general domains and/or ranges

Citations:

Zbl 1239.39019
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References:

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