Characterizations of derivations
Volume 539 / 2019
Abstract
The main purpose of this work is to characterize derivations through functional equations.
In Chapter 3 we examine whether the equations occurring in the definition of derivations are independent in the following sense. Let $Q$ be a commutative ring and let $P$ be a subring of $Q$. Let $\lambda, \mu\in Q\setminus\{0\}$ be arbitrary, $f\colon P\rightarrow Q$ be a function and consider the equation \[ \lambda[f(x+y)-f(x)-f(y)]+ \mu[f(xy)-xf(y)-yf(x)]=0 \quad\ (x, y\in P). \] We prove that under some assumptions on the rings $P$ and $Q$, derivations can be characterized via the above equation.
Chapter 4 is devoted to the additive solvability of the system of functional equations \[ d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \quad\ (x,y\in \mathbb{R},\,k\in\{0,\ldots,n\}), \] where $\Delta_n:=\{(i,j)\in\mathbb{Z}\times\mathbb{Z}\, \vert\, 0\leq i,j\mbox{ and }i+j\leq n\}$ and $\Gamma\colon\Delta_n\to\mathbb{R}$ is a symmetric function such that $\Gamma(i,j)=1$ whenever $i\cdot j=0$. Moreover, the linear dependence and independence of the additive solutions $d_{0},d_{1}, \ldots,d_{n} \colon\mathbb{R}\to\mathbb{R}$ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\colon\mathbb{R}\to\mathbb{R}$, the iterates $d^0,d^1,\dots,d^n$ of $d$ are shown to be linearly independent, and the graph of the mapping $x\mapsto (x,d^1(x),\dots,d^n(x))$ to be dense in $\mathbb{R}^{n+1}$.
The closing chapter deals with the following problem. Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and for an additive function $f\colon \mathbb{R}\to \mathbb{R}$, the mapping \[ \varphi(x)=f(\xi(x))-\xi’(x)f(x) \] fulfills some regularity condition on its domain. Is it true that $f$ is the sum of a derivation and a linear function?
