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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) x=0 if and only if x=0; (2) λx=|λ|·x for all scalars λ and all xX; (3) There is a constant K1 such that x+yK(x+y) for all x,yX. Then the pair (X,·) is said to be a quasi-normed space. A quasi-norm · is called a p-norm (0<p1) if x+y p x p +y p (x,yX). 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(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 a0,±1 in the framework of quasi-Banach spaces by using the direct method.

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)