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Cauchy–Rassias Stability of Cauchy–Jensen Additive Mappings in Banach Spaces

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Abstract

Let X, Y be vector spaces. It is shown that if a mapping f : X → Y satisfies

$$ f{\left( {\frac{{x + y}} {2} + z} \right)} + f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( x \right)} + 2f{\left( z \right)}, $$

(0.1)

$$ f{\left( {\frac{{x + y}} {2} + z} \right)} - f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( y \right)}, $$

(0.2) or

$$ 2f{\left( {\frac{{x + y}} {2} + z} \right)} = f{\left( x \right)} + f{\left( y \right)} + 2f{\left( z \right)} $$

(0.3) for all x, y, z ∈ X, then the mapping f : X → Y is Cauchy additive. Furthermore, we prove the Cauchy–Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.

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Authors and Affiliations

  1. Department of Mathematics, Chungnam National University, Daejeon 305–764, South Korea

    Choonkil Baak

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  1. Choonkil Baak
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Correspondence to Choonkil Baak.

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Supported by Korea Research Foundation Grant KRF-2005-070-C00009.

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Baak, C. Cauchy–Rassias Stability of Cauchy–Jensen Additive Mappings in Banach Spaces. Acta Math Sinica 22, 1789–1796 (2006). https://doi.org/10.1007/s10114-005-0697-z

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