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On selections of generalized convex set-valued maps. (English) Zbl 1308.54017
Let \((G,\ast )\) be a bisymmetric grupoid, \(Y\) a Banach space, \(C\) a compact convex subset of \(Y\) containing \(0_Y\), and \(F:G\to 2^Y\) a set-valued map satisfying \((1-p)F(x)+pF(y)\subset F(x\ast y)+C\), where \(p\in (0,1)\). The main result of the paper gives sufficient conditions for the existence of a function \(f:G\to Y\) with the following properties: (i) \(f(x)\in \text{cl}\, F(x)+\frac{1}{p}C\), \(x\in G\). (ii) \((1-p)f(x)+pf(y)=f(x\ast y)\), \(x,y\in G\). This theorem is applied to the study of Hyers-Ulam stability of the functional equation \((1-p)f(x)+pf(y)=f(x\ast y)\). For particular operations \(\ast\) the authors obtain generalizations of some known results.

54C65 Selections in general topology
39B82 Stability, separation, extension, and related topics for functional equations
54C60 Set-valued maps in general topology
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