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Stability of the Cauchy functional equation in metric groupoids. (English) Zbl 1160.39010
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 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.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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