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On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces. (English) Zbl 1117.39017
A quasi-norm is a real-valued function on a vector space $X$ satisfying the following: (1) $\Vert x\Vert = 0$ if and only if $x =0$; (2) $\Vert \lambda x\Vert = \vert \lambda\vert \cdot \Vert x\Vert $ for all scalars $\lambda$ and all $x\in X$; (3) There is a constant $K \geq 1$ such that $\Vert x+y\Vert \leq K(\Vert x\Vert + \Vert y\Vert )$ for all $x, y \in X$. Then the pair $(X, \Vert \cdot \Vert )$ is said to be a quasi-normed space. A quasi-norm $\Vert \cdot \Vert $ is called a $p$-norm $(0 < p \leq 1)$ if $\Vert x+y\Vert ^p \leq \Vert x\Vert ^p + \Vert y\Vert ^p \quad (x, y \in X)$. By the Aoki-Rolewicz theorem [see {\it S. Rolewicz}, Metric linear spaces. 2nd ed. Mathematics and its applications (East European Series), 20. Dordrecht- Boston-Lancaster: D. Reidel Publishing Company, Warszawa: PWN-Polish Scientific Publishers. (1985; Zbl 0573.46001)]), each quasi-norm is equivalent to some $p$-norm. Since it is much easier to work with $p$-norms than quasi-norms, henceforth the authors restrict their attention mainly to $p$-norms. The functional equation $$ f(ax+y) + f(x+ay) = (a+1)(a-1)^2[f(x)+ f(y)] + a(a+1)f(x+y) $$ is called the Euler-Lagrange type cubic functional equation. The authors prove the stability of this equation for fixed integers $a$ with $a \neq 0, \pm 1$ in the framework of quasi-Banach spaces by using the direct method.

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46B03Isomorphic theory (including renorming) of Banach spaces
46B20Geometry and structure of normed linear spaces
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)
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