On some equations stemming from quadrature rules

Abstract

We deal with fun tional equations of the type \[F(y) − F(x) = (y − x) \sum_{k=1}^nf_k((1 − λ_k)x + λ_ky),\] connected to quadrature rules and, in particular, we find the solutions of the following functional equation \[f(x) − f(y) = (x − y)[g(x) + h(x + 2y) + h(2x + y) + g(y)].\] We also present a solution of the Stamate type equation \[yf(x) − xf(y) = (x − y)[g(x) + h(x + 2y) + h(2x + y) + g(y)].\] All results are valid for functions acting on integral domains.