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Selections of set-valued maps satisfying a linear inclusion in a single variable. (English) Zbl 1205.39025
Let $$S$$ be a nonempty set, $$X$$ a Banach space over $$\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$$, $$a: S\to\mathbb{K}$$, $$b: S\to [0,\infty)$$, $$\varphi: S\to S$$, $$\psi: S\to X$$ given functions and $$B\subset X$$ a given balanced, convex and bounded set.
The main result concerns a set-valued mapping $$F: S\to 2^X$$ satisfying the functional inclusion
$a(x)F(\varphi(x))\subseteq F(x)+\psi(x)+b(x)B,\qquad x\in S.$
Under some assumptions it is proved that there exists a unique solution $$f: S\to X$$ of the functional equation
$a(x)f(\varphi(x))=f(x)+\psi(x),\qquad x\in S$
such that $$f$$ is a selection of $$\text{cl}\,(F(x)+\omega(x)B)$$ with suitably defined $$\omega: S\to [0,\infty)$$.
Several corollaries are given, in particular on the Hyers-Ulam stability of a linear functional equation and the equation of homomorphism.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 54C65 Selections in general topology 39B52 Functional equations for functions with more general domains and/or ranges 39B62 Functional inequalities, including subadditivity, convexity, etc. 54C60 Set-valued maps in general topology
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