Fixed points and the stability of the linear functional equations in a single variable

In this paper we prove that an interesting result concerning the generalized Hyers-Ulam stability of the linear functional equation on a complete metric group, given in 2014 by S.M. Jung, D. Popa and M.T. Rassias, can be obtained using the fixed point technique. Moreover, we give a characterization of the functions that can be approximated with a given error, by the solution of the linear equation mention above.
Our results are also related to a recent result of G.H. Kim and Th.M. Rassias concerning the stability of Psi functional equation.