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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:
(1)
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 \(\psi\circ f\circ a=f\).
(2)
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 \(Y\).

MSC:
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
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