Generalized Hyers-Ulam-Rassias stability for a general additive functional equation in quasi-\(\beta\)-normed spaces (Q2872196)

Let \(X\) be a linear space, let \(Y\) be a \((\beta,p)\)-Banach space and let \(n\geq 2\) be an integer. For \(f: X\to Y\) and \(x_1,\dots,x_n\in X\) define NEWLINE\[NEWLINE Df(x_1,\dots,x_n):=2\sum_{j=1}^{n}f\left(\frac{x_j}{2}+\sum_{j\neq i=1}^{n}x_i\right)+\sum_{j=1}^{n}f(x_j)-2nf\left(\sum_{j=1}^{n}x_j\right). NEWLINE\]NEWLINE The stability of the equation NEWLINE\[NEWLINE Df(x_1,\dots,x_n)=0,\qquad x_1,\dots,x_n\in X, NEWLINE\]NEWLINE (solution of which are additive mappings) is proved, i.e., it is shown that if NEWLINE\[NEWLINE \|Df(x_1,\dots,x_n)\|\leq\varphi(x_1,\dots,x_n),\qquad x_1,\dots,x_n\in X, NEWLINE\]NEWLINE (with an appropriate control mapping \(\varphi\) being, in particular, contractively subadditive or expansively superadditive), then \(f\) can be approximated by a unique additive mapping. The proof is based on the Banach contraction principle.