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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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