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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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References:
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