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On the stability problem for a mixed type of quartic and quadratic functional equation. (English) Zbl 1106.39027
The problem “If we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality must be close to the solutions of the given equation?” is the essence of Hyers-Ulam-Rassias stability theory; cf. {\it Th. M. Rassias} [Acta Appl. Math. 62, No. 1, 23--130 (2000; Zbl 0981.39014)]. For a mapping $f: E_1 \to E_2$ between real vector spaces, let us define $\biguplus_{x_2}f(x_1)$ to be $f(x_1+x_2)+f(x_1-x_2)$ and $\biguplus_{x_2, \dots, x_{n+1}}^nf(x_1)= \biguplus_{x_{n+1}} (\biguplus_{x_2, \dots, x_n}^{n-1}f(x_1))$ $(n \in \Bbb N)$. In the paper under review, the author determines the general solution for the mixed type functional equation $$\biguplus_{x_2, \dots, x_n}^{n-1}f(x_1)+2^{n-1}(n-2)\sum_{i=1}^nf(x_i)=2^{n-2}\sum_{1\leq i < j \leq n}\left(\biguplus_{x_j}f(x_j)\right),$$ and proves its Hyers-Ulam-Rassias stability by using the Hyers type sequences; see {\it Th. M. Rassias} [J. Math. Anal. Appl. 158, No. 1, 106--113 (1991; Zbl 0746.46038)].

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
Full Text: DOI
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