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
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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)