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Jun, Kil-Woung; Kim, Hark-Mahn; Rassias, John Michael

Extended Hyers-Ulam stability for Cauchy-Jensen mappings. (English) Zbl 1135.39013

J. Difference Equ. Appl. 13, No. 12, 1139-1153 (2007).
Suppose \(X\) is a normed space, \(Y\) is a Banach space and \(A:X\to Y\) is a mapping. Following J. M. Rassias, and 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.
Reviewer: Maryam Amyari (Mashhad)

Cited in 31 Documents

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
46L05 General theory of \(C^*\)-algebras
39B52 Functional equations for functions with more general domains and/or ranges

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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\textit{K.-W. Jun} et al., J. Difference Equ. Appl. 13, No. 12, 1139--1153 (2007; Zbl 1135.39013)
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