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Zbl 1135.39013
Jun, Kil-Woung; Kim, Hark-Mahn; Rassias, John Michael
Extended Hyers-Ulam stability for Cauchy-Jensen mappings. (English)
[J] J. Difference Equ. Appl. 13, No. 12, 1139-1153 (2007). ISSN 1023-6198

Suppose $X$ is a normed space, $Y$ is a Banach space and $A:X\to Y$ is a mapping. Following {\it J. M. Rassias}, and {\it M. J. Rassias } [J. Math. Anal. Appl. 281, No. 2, 516--524 (2003; Zbl 1028.39011)], the authors consider the Cauchy equation of Euler-Lagrange type $A(ax+by)+A(bx+ay)=(a+b)[A(x)+A(y)]\quad a+b\neq 0$ and the Cauchy-Jensen equation of Euler-Lagrange type $A(ax+by)+A(ax-by)=2aA(y)$, where $x, y\in X$ and $a, b\in \mathbb R$, and generalize the stability results controlled by more general mappings. They also prove that under certain conditions a bijective mapping between $C^*$-algebras is a $C^*$-algebra isomorphism.
[Maryam Amyari (Mashhad)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
46L05 General theory of C*-algebras
39B52 Functional equations for functions with more general domains

Keywords: Ulam stability problem; Jensen equation; Euler-Lagrange mappings; Jensen type mappings; Banach space; Cauchy equation; Cauchy-Jensen equation; $C^*$-algebras; isomorphism

Citations: Zbl 1028.39011

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