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On the generalized Hyers-Ulam-Rassias stability of higher ring derivations. (English) Zbl 1183.39023
Let $${\mathcal A}$$, $${\mathcal B}$$ be real or complex algebras. A sequence $$H=\{h_0,h_1,\dots\}$$ of additive operators from $${\mathcal A}$$ to $${\mathcal B}$$ is called a higher ring derivation if
$h_n(zw)=\sum_{i=0}^{n}h_i(z)h_{n-i}(w),\qquad z,w\in{\mathcal A}, n=0,1,\dots.$
A sequence $$F=\{f_0,f_1,\dots\}$$ of operators from $${\mathcal A}$$ to $${\mathcal B}$$ is called a higher derivation if
$f_n(x+y+zw)=f_n(x)+f_n(y)+\sum_{i=0}^{n}f_i(z)f_{n-i}(w),\qquad x,y,z,w\in{\mathcal A}, n=0,1,\dots.$
The main goal of the paper is to consider approximate higher derivations and the problem of the stability of higher ring derivations. It is shown, in particular, that if a sequence $$F=\{f_0,f_1,\dots\}$$ satisfies, with some given control mappings $$\varphi_n:{\mathcal A}^4\to[0,\infty)$$,
$\|f_n(x+y+zw)-f_n(x)-f_n(y)-\sum_{i=0}^{n}f_i(z)f_{n-i}(w)\|\leq\varphi_n(x,y,z,w)$
for all $$x,y,z,w\in {\mathcal A}$$ and $$n=0,1,\dots$$, then there exists a unique higher ring derivation $$H=\{h_0,h_1,\dots\}$$ such that $$h_n$$ is somehow close to $$f_n$$ for each $$n$$.
Several corollaries are obtained for particular control mappings $$\varphi_n$$ and under some additional assumptions upon $${\mathcal A}$$.
The results refer in particular to D. G. Bourgin [Duke Math. J. 16, 385–397 (1949; Zbl 0033.37702)], R. Badora [Math. Inequal. Appl. 9, No. 1, 167–173 (2006; Zbl 1093.39024)], T. Miura, G. Hirasawa and S.-E. Takahasi [J. Math. Anal. Appl. 319, No. 2, 552–530 (2006; Zbl 1104.39025)].
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
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