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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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