Further results on stability of normal regions for linear homogeneous functional equations (Q1345326)

This is a continuation of the author's paper [ibid. 36, No. 2/3, 176-187 (1988; Zbl 0664.39003)]. Regions are determined by solutions \(\psi : (0,a) \to \mathbb{R}\) to the inequality \(\psi (f(x)) \leq g(x) \psi (x)\), and stability of such a region with respect to the family of continuous solutions \(\varphi : (0,a) \to \mathbb{R}\) of the equation (1) \(\varphi (f(x)) = g(x) \varphi (x)\) is considered (the notion of set stability being adapted from that introduced for difference equations by E. Shanholt). The paper contains a necessary and sufficient condition for instability of a region and some sufficient conditions for its stability (the main results, according to the author's claim) in case the continuous solutions to (1) do not depend continuously on the initial conditions.