Stability of the Cauchy functional equation in metric groupoids.

*(English)*Zbl 1160.39010A 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.

\[ (x\circ y)\circ (x\circ y)=(x\circ x)\circ (y\circ y),\qquad x,y\in G \]

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.

Reviewer: Jacek Chmieliński (Kraków)

##### 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) |