Abstract
In this work, we present the Cauchy functional equation in the context of connected Lie groups. We consider two generalizations of this equation with higher orders of the finite difference.
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References
Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)
Brzdȩk, J.: Remarks on hyperstability of the Cauchy functional equation. Aequ. Math. 86, 255–267 (2013)
Brzdȩk, J., El-hady, E.: On hyperstability of the Cauchy functional equation in \(n\)-Banach spaces. Mathematics 8, 1886 (2020)
Cauchy, A.-L.: Cours d’Analyse de l’École Royale Polytechnique. Debure frères, Paris (1821)
Darboux, G.: Mémoire sur les fonctions discontinues. Ann. Sci. de l’Ecole Norm. Superieure 4, 57–112 (1875)
Gilányi, A.: Hyers–Ulam stability of monomial functional equations on a general domain. Proc. Natl. Acad. Sci. U.S.A. 96, 10588–10590 (1999)
Jung, S.M., Popa, D., Rassias, M.T.: On the stability of the linear functional equation in a single variable on complete metric groups. J. Glob. Optim. 59, 165–171 (2014)
Koh, E.L.: The Cauchy functional equations in distributions. Proc. Am. Math. Soc. 106, 641–646 (1989)
Kucharski, R., Łukasik, R.: The form of multi-additive symmetric functions. Results Math. 73(150), 15 (2018)
Kuczman, M.: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jenssen’s Inequality. Basel, Birkäuser (2008)
Molla, A., Nicolay, S.: The Fréchet functional equation for Lie groups. Mediterr. J. Math. 18, 1–18 (2021)
Molla, A., Nicolay, S., Schneiders, J.-P.: On some generalizations of the Fréchet functional equations. J. Math. Anal. Appl. 466, 1400–1409 (2018)
Reem, D.: Remarks on the Cauchy functional equation and variations of it. Aequ. Math. 91, 237–264 (2017)
Stetkær, H.: Functional Equations on Groups. World Scientific, Singapore (2013)
Szekelyhidi, L.: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific Publishing Company, London (1991)
Thielman, H.P.: On generalized Cauchy functional equations. Am. Math. Monthly 56, 452–457 (1949)
Toborg, I., Volkmann, P.: On stability of the Cauchy functional equation in groupoids. Ann. Math. Sil. 31, 155–164 (2017)
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Molla, A., Nicolay, S. The monomial functional equation for connected Lie groups. Aequat. Math. 98, 625–635 (2024). https://doi.org/10.1007/s00010-023-00969-8
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DOI: https://doi.org/10.1007/s00010-023-00969-8