Solutions of two functional equations using a result of M. Sablik

Abstract

We rephrase Lemma 2.3 from (M. Sablik, Taylor’s theorem and functional equations, Aequationes Math. 60 (2000), 258–267) in order to solve two functional equations. First, the following “Taylor-like” functional equation $$ f(x)\, = \,{\sum\limits_{k = 0}^n {g_{k} (x + t(y - x))((t(x - y))^{k} ) + (\Phi (x) - \Phi (y))((t(x - y))^{n} )} } $$ and a second one stemming from Simpson’s rule: $$ f(x) - f(y) = [h(sx + ty) + \Phi (x) + \Psi (y)](x - y) $$ .