Improvements on the stability of Euler-Lagrange type cubic maps in quasi-Banach spaces
The main aim of this paper is to study the stability of the following linear functional equation, called the Euler-Lagrange type cubic functional equation: \[ f (ax + y) + f (x + ay) = (a + 1)(a - 1)^{2}[f (x) + f (y)] + a(a + 1)f (x + y).\] Here \(f\colon X\to Y\) denotes the unknown function and \(X\) is assumed to be a real quasi-normed space, while \(Y\) is a real quasi-Banach space and \(a\) is a fixed integer with \(a\notin \left\{-1, 0, 1 \right\}\). This problem has been formerly considered by \textit{K.-W. Jun} and \textit{H.-M. Kim} [J. Math. Anal. Appl. 332, No. 2, 1335--1350 (2007; Zbl 1117.39017)]. The main aim of this paper is to improve the above-mentioned results. Accordingly the authors answer the open question raised in the above paper, namely to investigate the stability of the above equation for the case \(r\) with \(-\log_{\vert a\vert}K_{Y}\leq r-3\leq \log_{\vert a\vert}K_{Y}\) and \(\vert a\vert >1\).
scientific article; zbMATH DE number 7491539
scientific article