Hyperstability of a functional equation

Abstract

The aim of this paper is to prove that the parametric fundamental equation of information is hyperstable on its open as well as on its closed domain, assuming that the parameter is negative. As a corollary of the main result, it is also proved that the system of equation that defines the alpha-recursive information measures is stable.Comment: 9 page

On the generalized Fréchet functional equation with constant coefficients and its stability

“…Such a method has been used in, e.g., [4][5][6]10,26,30,33,34]. Moreover, the results that we provide correspond to the outcomes in [3,9,13,16,18,21,24,25,31,32] (for more details see, e.g., [8,19,22]) and complement [4,Corollary 6]. …”

“…The assumption that β 0 < 1 can be omitted in Theorem 13, if we replace (18) by the subsequent somewhat stronger condition…”

“…Let us note that the result presented below cannot be applied to Eq. (2), since its assumptions exclude the case where (18), and β 0 , β ∈ [0, 1), where β 0 and β are defined as in Theorem 13. Let f : X → Y be a function satisfying condition (20).…”

Stability of a generalization of the Fréchet functional equation

“…The results in [2] as well as our main theorem have been motivated by the notion of hyperstability of functional equations (see, e.g., [3,4,5,13,20]), introduced in connection with the issue of stability of functional equations (for more details see, e.g., [14,17]). …”

Hyperstability of general linear functional equation

“…The first well known hyperstability result appeared probably in [4] and concerned some ring homomorphisms. However, the term hyperstability was introduced much later (in the meaning applied here probably in [16]; see also [14,15] or [10]). …”