On the Hyers-Ulam stabilities of functional equations on \(n\)-Banach spaces (Q2816983)

Let \(n\) be a positive integer, \(X\) be a real linear space with \(\dim X \geq n\) and \(\| \cdot,\dots, \cdot \| : X^n \to \mathbb{R}\) be a functional such that: NEWLINE\(\| x_1,\dots,x_n \| = 0 \Leftrightarrow x_1, \dots , x_n \) are linearly dependent; NEWLINE\(\| x_1,\dots,x_n \| = \| x_{j_1},\dots,x_{j_n}\| \) for every permutation \(( j_{1}, \dots ,j_{n} )\) of \((1, \dots, n)\); NEWLINENEWLINE\[NEWLINE \| \alpha x_1,\dots,x_n \| = |\alpha| \;\| x_1,\dots,x_n \| , NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE \| x + y,x_2,\dots,x_n \| \leq \| x, x_2,\dots,x_n \| + \| y, x_2,\dots,x_n \| , NEWLINE\]NEWLINE for all \(\alpha \in \mathbb{R}\) and for all \(x,y, x_1 , \dots, x_n \in X\). The function \(\| \cdot,\dots, \cdot \| \) is then called an \(n\)-norm in \(X\) and the pair \((X, \| \cdot,\dots, \cdot \| )\) is called a linear \(n\)-normed space (see [\textit{A. Misiak}, Math. Nachr. 140, 299--319 (1989; Zbl 0673.46012)]).NEWLINENEWLINEA sequence \(\{x_l \}\) of an \(n\)-normed linear space \(X\) is said to be \(n\)-convergent to an element \(x \in X\) if NEWLINE\[NEWLINE \lim_{l \to \infty}\| x_l - x, y_2, \dots, y_n \| = 0 NEWLINE\]NEWLINE for all \(y_2, \dots y_n \in X\).NEWLINENEWLINEA sequence \(\{x_l \}\) of an \(n\)-normed linear space \(X\) is said to be an \(n\)-Cauchy sequence if for any \(\varepsilon> 0\) there exists \(N \in \mathbb{N}\) such that NEWLINE\[NEWLINE \| x_s - x_t, y_2, \dots, y_n \| < \varepsilon NEWLINE\]NEWLINE for all \(y_2, \dots y_n \in X\) and \(s, t > N\).NEWLINENEWLINEAn \(n\)-normed linear space \(X\) is an \(n\)-Banach space if every \(n\)-Cauchy sequence is \(n\)-convergent in \(X\).NEWLINENEWLINEThe authors prove the following Hyers-Ulam-type stability theorem for the Cauchy functional equation.NEWLINENEWLINEAssume that \(X\) is a real linear space and \((Z,\| \cdot,\dots, \cdot \| )\) is an \(n\)-Banach space. Let \(\varphi: X^{n+1} \to\mathbb{R}^+\) be a function such that the series \(\sum_{i=0}^{\infty}2^{-i}\varphi(2^ix,2^iy,x_2,\dots,x_n)\) converges for all \(x, y, x_2,\dots, x_n \in X\). Suppose that \(f: X \to Z\) is a surjective mapping satisfying NEWLINE\[NEWLINE \| f(x + y) - f(x) - f(y), f(x_2),\dots, f(x_n) \| \leq \varphi(x, y, x_2,\dots,x_n) NEWLINE\]NEWLINE for all \(x, y, x_2,\dots, x_n \in X\). Then there is a unique additive mapping \(A: X \to Z\) such that NEWLINE\[NEWLINE \| f(x) - A(x), f(x_2),\dots,f(x_n) \| \leq \sum_{i=0}^{\infty}2^{-i-1}\varphi(2^ix,2^iy,x_2,\dots,x_n) NEWLINE\]NEWLINE for all \(x, x_2,\dots,x_n \in X\).NEWLINENEWLINESimilar theorems for the Jensen and quadratic functional equations are also proved.