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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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