Stability of additive mappings in generalized normed spaces (Q2878093)

Let \(X\) be a real vectoer space and \(\lozenge\) be a binary operation. A generalized norm on \(X\) is a function \(N :X \to \mathbb{R}\) that satisfies the following properties: (i) \(N(x)=0\) if and only if \(x=0\); (ii) \(N(x) \geq 0\) for each \(x\in X\); (iii) \(N(\alpha x)=|\alpha |^tN(x)\) for some \(t\in (0, \infty)\) for each \(x\in X\) and every \(\alpha \in \mathbb{R}\); (iv) \(N(x+y)\leq N(x)\lozenge N(y)\) for each \(x, y\in X\). If \((X, N, \lozenge)\) is a generalized normed space and \((X, N', \lozenge)\) a generalized Banach space, then the authors investigate the Hyers-Ulam stability of the additive function \(f(x+y)=f(x)+f(y)\).