Stability of generalized additive Cauchy equations (Q5926131)

A functional equation NEWLINE\[NEWLINEf(ax+b)=cf(x)NEWLINE\]NEWLINE is solved in the class of functions \(f:\mathbb{R}\to \mathbb{R}\). This result is applied to investigate the Hyers-Ulam-Rassias stability of the generalized additive Cauchy equation NEWLINE\[NEWLINEf(a_1x_1+\dots+a_mx_m+x_0)=\sum_{i=1}^mb_if(a_{i1}x_1 +\dots+a_{im}x_m)NEWLINE\]NEWLINE in the same class of functions. A~partial answer to the question concerning the Hyers-Ulam-Rassias stability of the equation NEWLINE\[NEWLINEf(a_1x+a_2y+v)=b_1f(x)+b_2f(y)+wNEWLINE\]NEWLINE is also given.