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

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general.The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form n k=1xwhere p k and q k (k = 1, . . . , n) are given nonnegative integers and the unknown functions f 1 , . . . , f n : R → R are supposed to be additive on the ring R. It is illustrated by some explicit examples too.As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.

In this paper the following implication is verified for certain basic algebraic curves: if the additive real function f approximately (i.e., with a bounded error) satisfies the derivation rule along the graph of the algebraic curve in consideration, then f can be represented as the sum of a derivation and a linear function. When, instead of the additivity of f , it is assumed that, in addition, the Cauchy difference of f is bounded, a stability theorem is obtained for such characterizations of derivations.

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let n ≥ 2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field K and let α i

We present the general and continuous solutions of Shannon's functional equation on the positive octant. We show that extensions to the positive octant yield more general, non-separable, solutions than the entropy expression. We also show that strict monotonicity of the entropy measure is not a reqiured axiom to derive the entropy solution, rather a milder condition of it being an increasing non-constant function is sufficient.