On a composite functional equation related to the Golab-Schinzel equation (Q2812587)

Let \(X\) be a vector space over the field \(K\) of real or complex numbers.NEWLINENEWLINEA function \(f: X \to K\) is hemicontinuous at the origin provided that for every \(x \in X\) the function \(f_x: K \to K\) given by NEWLINE\[NEWLINE f_x(t) = f(tx), \, t \in K, NEWLINE\]NEWLINE is continuous at \(0\).NEWLINENEWLINEThe main result of this paper is the following superstability theorem.NEWLINENEWLINETheorem 1. Assume that \(f: X \to K\) and \(\varphi: { \underbrace{X \times X \times... \times X}_{p-1}} \to \mathbb{R}^{+} \) are two hemicontinuous functions at the origin satisfying NEWLINE\[NEWLINE |f(x_1 + \sum_{i=2}^p x_i f(x_1)^k f(x_2)^k\dots f(x_{i-1})^k) - \prod_{i=1}^p f(x_i)| \leq \varphi(x_2, x_3,\dots,x_p), NEWLINE\]NEWLINE for \(x_1, x_2,\dots,x_p \in X\), where \(k\) and \(p\) are two nonnegative integers with \(p \geq 2\).NEWLINENEWLINEThen \(f\) is either bounded or it satisfies the functional equation NEWLINE\[NEWLINE f(x_1 + \sum_{i=2}^p x_i f(x_1)^k f(x_2)^k\dots f(x_{i-1})^k) = \prod_{i=1}^p f(x_i), \, x_1, x_2,\dots,x_p \in X. NEWLINE\]NEWLINENEWLINENEWLINEThe paper also contains several corollaries of Theorem 1.