Abstract
Let G be an amenable metric semigroup with nonempty center, let E be a reflexive Banach space, and let ƒ: G → E be a given function. By Cƒ: G × G → E we understand the Cauchy difference of the function /, i.e.:
We prove that if the function C(f) is Lipschitz then there exists an additive function A: G → E such that f − A is Lipschitz with the same constant. Analogous result for Jensen equation is also proved.
As a corollary we obtain the stability of the Cauchy and Jensen equations in the Lipschitz norms.
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Tabor, J. Lipschitz Stability of the Cauchy and Jensen Equations. Results. Math. 32, 133–144 (1997). https://doi.org/10.1007/BF03322533
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DOI: https://doi.org/10.1007/BF03322533