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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)].
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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