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On the stability of additive, quadratic, cubic and quartic set-valued functional equations. (English) Zbl 1330.39029
Let \(X\) be a real vector space and \(Y\) a Banach space. By \(C_{cb}(Y)\) we denote the set of all nonempty, closed, bounded and convex subsets of \(Y\). By \(A\oplus B\) we mean the closure of \(A+B\). For \(f: X\to C_{cb}(Y)\), a fixed integer \(a>1\) and \(m=1,2,3,4\), the following equation is considered and its stability is proved. \[ \begin{split} f(ax+y)\oplus f(ax-y)= \\ a^{m-2}[f(x+y)\oplus f(x-y)]\oplus 2(a^2-1)\left[ a^{m-2}f(x)\oplus\frac{(m-2)(1-(m-2)^2)}{6}f(y)\right] \end{split} \] for all \(x,y\in X\). Namely, if the Hausdorff distance between the left and right hand sides of the above equation is bounded by a suitably contractively subhomogeneous function, then there exists a unique additive, quadratic, cubic or quartic (for \(m=1,2,3,4\), respectively) mapping which is sufficiently close to \(f\).

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
54C60 Set-valued maps in general topology
39B55 Orthogonal additivity and other conditional functional equations
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