Using a fixed point method, the authors prove the Hyers-Ulam-Rassias stability of a generalized Cauchy functional equation of the form , where and are given nonzero real numbers. Indeed, one of their main theorems states:
Let be a unital -algebra with unitary group . Assume that and are left Banach -modules. Let be a function such that for all and there exists a constant with for all , where .
If a function satisfies and
for all and for all , then there exists a unique -linear function such that for all .
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)].