Abstract
More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation \({f(xy) - f(x) - f(y) \in \{ 0, 1\}}\) where \({f : S \to \mathbb{R}, S}\) is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215–226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions \({f : H \to \mathbb{R}}\) of the alternative functional equation on a semigroup \({H = \langle a, b| a^2 = a, b^2 = b \rangle }\) where the Cauchy equation is not stable.
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References
Baker J.A.: On some mathematical characters. Glasnik Mat. 25(45), 319–328 (1990)
Baron K., Kannappan Pl.: On the Cauchy difference. Aequat. Math. 46, 112–118 (1993)
Baron K., Volkmann P.: On a theorem of van der Corput. Abh. Math. Sem. Univ. Hamburg 61, 189–195 (1991)
Baron K., Volkmann P.: On the Cauchy equation modulo \({\mathbb{Z} }\). Fund. Math. 131, 143–148 (1988)
Baron K., Simon A., Volkmann P.: On functions having Cauchy differences in some prescribed sets. Aequat. Math. 52, 254–259 (1995)
Borelli Forti, C., Forti, G.L.: On an alternative functional equation in \({\mathbb{R}^n}\). In: Functional Analysis, Approximation Theory and Numerical Analysis, pp. 33–44. World Scientific, Singapore (1994)
Brzdek J.: On functions which are almost additive modulo a subgroup. Glasnik Mat. 36, 1–9 (2001)
Faĭziev V.A.: The stability of the equation f(xy)−f(x)−f(y) = 0 on groups. Acta Math. Univ. Comenianae 1, 127–135 (2000)
Forti G.L.: Problem. Aequat. Math. 73, 188 (2007)
Forti G.L.: The stability of homomorphisms and amenability with applications to functional equations. Abh. Math. Sem. Univ. Hamburg 57, 215–226 (1987)
Forti G.L.: On alternative functional equation related to the Cauchy equation. Aequat. Math. 24, 195–206 (1982)
Forti G.L.: La soluzione generale dell’equazione funzionale { cf(x + y)−af(x)−bf(y)−d} {f(x + y)−f(x)−f(y)} = 0. Matematiche (Catania) 34, 219–242 (1979)
Forti G.L., Paganoni L.: A method of solving a conditional Cauchy equation on abelian groups. Ann. Math. Pura Appl. 127, 77–99 (1981)
Ger R.: On a method of solving of conditional Cauchy equation. Univ. Beograd. Publ. Elektrochechn. Fak. Ser. Mat. Fiz. No. 544–576, 159–165 (1976)
Kuczma M.: Functional equations on restricted domains. Aequat. Math. 18, 1–34 (1978)
Paganoni L.: On an alternative Cauchy equation. Aequat. Math. 29, 214–221 (1985)
Paganoni L.: Soluzione di una equazione funzionale su dominio ristretto. Boll. Un. Mat. Ital. 17B(5), 979–993 (1980)
Paganoni L., Rätz J.: Conditional functional equations and orthogonal additivity. Aequat. Math. 50, 135–142 (1995)
Sablik M.: Functional congruence revisited. Grazer Math. Ber. 316, 181–200 (1992)
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Faĭziev, V.A., Powers, R.C. & Sahoo, P.K. An alternative Cauchy functional equation on a semigroup. Aequat. Math. 85, 131–163 (2013). https://doi.org/10.1007/s00010-012-0131-4
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DOI: https://doi.org/10.1007/s00010-012-0131-4