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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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