Bulletin of the
Korean Mathematical Society BKMS

Let $X$ be a linear space and $Y$ be a complete quasi $p$-norm space. We will show that for each function $f:X \to Y$, which satisfies the inequality $$||\Delta^n_x f(y) - n! f(x)|| \leq \varphi(x, y)$$ for suitable control function $\varphi$, there is a unique monomial function $M$ of degree $n$ which is a good approximation for $f$ in such a way that the continuity of $t \mapsto f(tx)$ and $t \mapsto \varphi(tx, ty)$ imply the continuity of $t \mapsto M(tx)$.

Keywords: quasi $p$-norm, monomial functional equation, fixed point alternative, Hyers--Ulam--Rassias stability

MSC numbers: Primary 39B52, 39B82; Secondary 47H10