A quasi-norm is a real-valued function on a vector space satisfying the following: (1) if and only if ; (2) for all scalars and all ; (3) There is a constant such that for all . Then the pair is said to be a quasi-normed space. A quasi-norm is called a -norm if . 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 -norm. Since it is much easier to work with -norms than quasi-norms, henceforth the authors restrict their attention mainly to -norms.
The functional equation
is called the Euler-Lagrange type cubic functional equation. The authors prove the stability of this equation for fixed integers with in the framework of quasi-Banach spaces by using the direct method.