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

作者姓名：

Choonkil BAAK

作者单位：

Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea

基金项目：

Supported by Korea Research Foundation Grant KRF-2005-070-C00009.Acknowledgments The author would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.

摘要：

Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（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.

关键词：

Cauchy附加映射 Jensen附加映射 Cauchy-Rassias稳定性同构Banach代数

收稿时间：

2005-03-18

修稿时间：

2005-03-182005-07-29

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

Choonkil BAAK.Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces[J].Acta Mathematica Sinica,2006,22(6):1789-1796.

Authors:

Choonkil Baak

Institution:

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

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

Keywords:

Cauchy additive mapping Jensen additive mapping Cauchy-Rassias stability isomorphism between Banach algebra

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