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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.

MSC:
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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