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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) $\parallel x\parallel =0$ if and only if $x=0$; (2) $\parallel \lambda x\parallel =|\lambda |·\parallel x\parallel$ for all scalars $\lambda$ and all $x\in X$; (3) There is a constant $K\ge 1$ such that $\parallel x+y\parallel \le K\left(\parallel x\parallel +\parallel y\parallel \right)$ for all $x,y\in X$. Then the pair $\left(X,\parallel ·\parallel \right)$ is said to be a quasi-normed space. A quasi-norm $\parallel ·\parallel$ is called a $p$-norm $\left(0 if ${\parallel x+y\parallel }^{p}\le {\parallel x\parallel }^{p}+{\parallel y\parallel }^{p}\phantom{\rule{1.em}{0ex}}\left(x,y\in X\right)$. By the Aoki-Rolewicz theorem [see 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

$f\left(ax+y\right)+f\left(x+ay\right)=\left(a+1\right){\left(a-1\right)}^{2}\left[f\left(x\right)+f\left(y\right)\right]+a\left(a+1\right)f\left(x+y\right)$

is called the Euler-Lagrange type cubic functional equation. The authors prove the stability of this equation for fixed integers $a$ with $a\ne 0,±1$ in the framework of quasi-Banach spaces by using the direct method.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges 46B03 Isomorphic theory (including renorming) of Banach spaces 46B20 Geometry and structure of normed linear spaces 39B62 Functional inequalities, including subadditivity, convexity, etc. (functional equations)