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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. 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 ${\uplus }_{{x}_{2}}f\left({x}_{1}\right)$ to be $f\left({x}_{1}+{x}_{2}\right)+f\left({x}_{1}-{x}_{2}\right)$ and ${\uplus }_{{x}_{2},\cdots ,{x}_{n+1}}^{n}f\left({x}_{1}\right)={\uplus }_{{x}_{n+1}}\left({\uplus }_{{x}_{2},\cdots ,{x}_{n}}^{n-1}f\left({x}_{1}\right)\right)$ $\left(n\in ℕ\right)$.

In the paper under review, the author determines the general solution for the mixed type functional equation

$\underset{{x}_{2},\cdots ,{x}_{n}}{\overset{n-1}{\uplus }}f\left({x}_{1}\right)+{2}^{n-1}\left(n-2\right)\sum _{i=1}^{n}f\left({x}_{i}\right)={2}^{n-2}\sum _{1\le i

and proves its Hyers-Ulam-Rassias stability by using the Hyers type sequences; see Th. M. Rassias [J. Math. Anal. Appl. 158, No. 1, 106–113 (1991; Zbl 0746.46038)].

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges