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A fixed point approach to the stability of a generalized Cauchy functional equation. (English) Zbl 1143.39016

Using a fixed point method, the authors prove the Hyers-Ulam-Rassias stability of a generalized Cauchy functional equation of the form f(αx+βy)=αf(x)+βf(y), where α and β are given nonzero real numbers. Indeed, one of their main theorems states:

Let A be a unital C * -algebra with unitary group U(A). Assume that X and Y are left Banach A-modules. Let φ:X 2 [0,) be a function such that lim n 2 n φ(x 2 n ,y 2 n )=0 for all x,yX and there exists a constant L<1 with 2ψ(x)Lψ(2x) for all xX, where ψ(x)=φ(x 2α,x 2β)+φ(x 2α,0)+φ(0,x 2β).

If a function f:XY satisfies f(0)=0 and

f(αx+βay)-αf(x)-βaf(y)φ(x,y)

for all x,yX and for all aU(A), then there exists a unique A-linear function T:XY such that f(x)-T(x)1 1-Lψ(x) for all xX.

The readers may also refer to the following literature for more information on this subject: S.-M. Jung [J. Math. Anal. Appl. 329, No. 2, 879–890 (2007); Fixed Point Theory Appl. 2007, Article ID 57064, 9 p. (2007; Zbl 1155.45005)].

MSC:
39B82Stability, separation, extension, and related topics
39B22Functional equations for real functions
39B52Functional equations for functions with more general domains and/or ranges