Abstract
AbstractIn this paper we deal with the product of two or three Cauchy differences equaled to zero. We show that in the case of two Cauchy differences, the condition of absolute continuity and differentiability of the two functions involved implies that one of them must be linear, i.e., we have a trivial solution. In the case of the product of three Cauchy differences the situation changes drastically: there exists non trivial $${\mathcal {C}}^{\infty }$$
C
∞
solutions, while in the case of real analytic functions we obtain that at least one of the functions involved must be linear. Some open problems are then presented.
Funder
Università degli Studi di Milano
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,General Mathematics
Reference5 articles.
1. Fenyö, I., Forti, G.L.: On the inhomogeneous Cauchy functional equation. Istituto Matematico F. Enriques, Università degli Studi di Milano, Quaderno 46/S (1980)
2. Forti, G.L., Paganoni, L.: $$\Omega $$–additive functions on topological groups. In: Constantin Carathéodory: an International Tribute, 312–330, World Sci. Publishing, Singapore (1990)
3. Forti, G.L., Paganoni, L.: On an alternative Cauchy equation in two unknown functions. Some classes of solutions. Aequationes Math. 42, 271–295 (1991)
4. Forti, G.L., Paganoni, L.: Local solutions of an alternative Cauchy equation. Publ. Math. Debrecen 44, 51–65 (1994)
5. Kuczma, M.: Functional equations on restricted domains. Aequationes Math. 18, 1–34 (1978)
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