an:05485620
Zbl 1160.39010
Tabor, Jacek; Tabor, J??zef
Stability of the Cauchy functional equation in metric groupoids
EN
Aequationes Math. 76, No. 1-2, 92-104 (2008).
00232332
2008
j
39B82 39B52 20L05
Cauchy functional equation; square-symmetric groupoid; shadowing; Hyers-Ulam stability; metric groupoid; stability of additive mappings
A set \(G\) with a binary operation \(\circ\) satisfying
\[
(x\circ y)\circ (x\circ y)=(x\circ x)\circ (y\circ y),\qquad x,y\in G
\]
is called a \textit{square-symmetric groupoid}. If \(d\) is a metric on \(G\) and, with some \(K>0\),
\[
d(a\circ b,\tilde{a}\circ\tilde{b})\leq K(d(a,\tilde{a})+d(b,\tilde{b})),\qquad a,\tilde{a},b,\tilde{b}\in G,
\]
then \(G\) is called a \(K\)-metric groupoid.
The main result concerns the stability of additive mappings
\[
a(x\circ y)=a(x)\circ a(y).
\]
More precisely, let \((G,\circ)\) be a square-symmetric groupoid and \((X,\circ)\) be a square-symmetric complete \(K\)-metric groupoid which is locally two-divisible (it means here some local invertibility of \(x\mapsto x\circ x\)). Then for a mapping \(f\colon G\to X\) which is approximately additive:
\[
d(f(x\circ y),f(x)\circ f(y))\leq \delta
\]
there exists a unique additive mapping \(a:U\to X\) such that the distance \(d(f(x),a(x))\) is bounded (by some constant depending on \(\delta)\).
In the proof, in particular, the first author's result concerning shadowing of locally invertible mappings is used.
The above result is a generalization of the celebrated Hyers-Ulam theorem and has some connections also with other results in the theory of stability of functional equations.
Jacek Chmieli??ski (Krak??w)