Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

A hyperstability result for the Cauchy equation

We prove a hyperstability result for the Cauchy functional equation $f(x+y)=f(x)+f(y)$, which complements some earlier stability outcomes of J.M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function $f$, mapping a normed space $E_1$ into a normed space $E_2$, and for every real numbers $r,s$ with $r+s>0$ one of the following two conditions must be valid: \begin{align*} \sup_{x,y\in E_1} \|f(x+y)-f(x)-f(y)\| \|x\|^r \|y\|^s=\infty,\\ \sup_{x,y\in E_1} \|f(x+y)-f(x)-f(y)\|\, \|x\|^r \,\|y\|^s=0. \end{align*} In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem. 10.1017/S000497271300068

On the hyperstability of a pexiderised σ-quadratic functional equation on semigroups

DOI: 10.1017/S000497271700113

On approximately additive mappings in 22-Banach spaces

DOI: 10.1017/S000497271900086