Certain properties of difference operator and stability of Fréchet functional equation

Abstract

Let M be a commutative monoid, and let B be a Banach space. We give a new recursive method to obtain a Gǎvruţa-type stability result for the functional equation n+1 y f (x) := n+1 k=0 (-1) n+1+k n + 1 k f (x + ky) = 0 via algebraic manipulations of the forward difference operator.