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A property of a functional inclusion connected with Hyers-Ulam stability. (English) Zbl 1189.39032
The author uses some ideas of D. Popa [Math. Inequal. Appl. 7, No. 3, 419–428 (2004; Zbl 1058.39026)] and Z. Páles [Publ. Math. 58, No. 4, 651–666 (2001; Zbl 0980.39022)] to prove that if $$(X,*)$$ is a square-symmetric divisible groupoid and $$(Y,\diamond,d)$$ is a complete metric bisymmetric divisible groupoid and $$F:X\to \mathcal{P}_0(Y)$$ is a set valued map with the property $$F(x)\diamondsuit F(Y)\subseteq F(x*y)$$, then under certain conditions there exists a unique selection $$f:X\to Y$$ of $$F$$ such that $$f(x)\diamond f(y)=f(x*y)$$.

##### 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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