*(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 =\left|\lambda \right|\xb7\parallel x\parallel $ for all scalars $\lambda $ and all $x\in X$; (3) There is a constant $K\ge 1$ such that $\parallel x+y\parallel \le K(\parallel x\parallel +\parallel y\parallel )$ for all $x,y\in X$. Then the pair $(X,\parallel \xb7\parallel )$ is said to be a quasi-normed space. A quasi-norm $\parallel \xb7\parallel $ is called a $p$-norm $(0<p\le 1)$ if ${\parallel x+y\parallel}^{p}\le {\parallel x\parallel}^{p}+{\parallel y\parallel}^{p}\phantom{\rule{1.em}{0ex}}(x,y\in X)$. 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

is called the Euler-Lagrange type cubic functional equation. The authors prove the stability of this equation for fixed integers $a$ with $a\ne 0,\pm 1$ 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) |