Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces (Q1758008)

Let \(X\) and \(Y\) be real linear spaces. \textit{A. Najati} and \textit{G. Z. Eskandani} [J. Math. Anal. Appl. 342, No. 2, 1318--1331 (2008; Zbl 1228.39028)] have given the general solution \(f:X\to Y\) with \(f(0)=0\) of the functional equation \[ f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+2f(2x)-4f(x) \] and have proved its generalized Hyers-Ulam stability under the additional assumption that \(X\) is a quasi-normed space and \(Y\) is a \(p\)-Banach space. In this paper, the authors consider the above problems for the following more general equation \[ f(kx+y)+f(kx-y)=kf(x+y)+kf(x-y)+2f(kx)-2kf(x), \] where \(k\) is a fixed integer different from \(-1, 0\) and \(1\). The stability results are obtained by the direct method.