STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION IN QUASI NORMED SPACES

Abstract. Let X be a linear space and Y be a complete quasi p-norm space. We will show that for each function f: X → Y, which satisfies the inequality ||∆nxf(y) − n!f(x)| | ≤ ϕ(x, y) for suitable control function ϕ, 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 7 → f(tx) and t 7 → ϕ(tx, ty) imply the continuity of t 7→M(tx). 1