On a simple linear functional equation on normed linear spaces

In the present paper, we consider the linear functional equation¶¶\( \Phi (\alpha x) - \beta \Phi(x) = F(x) \qquad (x \in E) \),(1)¶where E and G are normed linear spaces over K, K is either \( {\Bbb R} \) or \( {\Bbb C} \)\( \alpha \) and \( \beta \) are given scalars in K, \( F : E \to G \) is a given function and \( \Phi: E \to G \) is the unknown function.¶In an earlier paper, we studied the case \( E = {\Bbb R} \) or \( [0,+\infty) \) or \( (0,+\infty), G = {\Bbb R} \), and we gave there the general solution of (1) and also its continuous and differentiable solutions by using elementary direct methods. The results presented here extend the previous ones to the general case.