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The properties of functional inclusions and Hyers-Ulam stability. (English) Zbl 1271.39031
Let \(Y\) be a normed space over \(\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}\), let \(K\) be a set and let \(n(Y):=2^Y\setminus\{\emptyset\}\). Furthermore assume that \(F: K\to n(Y)\), \(\psi: Y\to Y\), \(a: K\to K\) are given functions and that \(\lambda\in(0,1)\). Under the hypotheses that \(\psi\) is \(\lambda\)-Lipschitz and that \(\sup\{\text{diam}(F(x))\mid x\in K\}\) is finite, the author among others proves:
If \(Y\) is complete and if \(\psi(F(a(x)))\subseteq F(x)\) for all \(x\in K\), then there is a unique \(f: K\to K\) such that \(f(x)\in\overline{F(x)}\) for all \(x\in K\) and such that \(\psi\circ f\circ a=f\).
If \(F(x)\subseteq\psi(F(a(x)))\) for all \(x\) then there is some \(f: K\to K\) such that \(F(x)=\{f(x)\}\) for all \(x\). Moreover, this function \(f\) satisfies \(\psi\circ f\circ a=f\).
The (general) results are applied to prove some stability results.
Reviewer’s remark: The condition that \(\psi\) is \(\lambda\)-Lipschitz is formulated in terms of a two-place function \(d\) which seems to be undefined. From the context it should become clear that \(d(x,y)=\| x-y\|\). But probably some of the results, especially those cited above, could be proved for (general) metric spaces \(Y\).

39B82 Stability, separation, extension, and related topics for functional equations
39B05 General theory of functional equations and inequalities
54C60 Set-valued maps in general topology
54C65 Selections in general topology
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI
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