On a simple linear functional equation on normed linear spaces (Q1349041)

Let \(K\) be either the set of all real or of all complex numbers, \(a, b\) fixed elements of \(K ,\) let \(E, G\) be normed linear spaces over \(K,\) and \(g: E\to G\) be a given function. The general, the continuous, and the differentiable solutions \(f: E\to G\) of the functional equation \(f(ax)-bf(x)=g(x) (x\in E)\) are offered except in some exceptional situations. (For example, if \(a\) is not a root of unity but \(|a|=|b|=1,\) then only the case \(g=0\) is considered).