Regularity of functional equations with few variables

Abstract. Under certain conditions weak regularity property of solutions \(f\) of the functional equation \[ \begin{gathered} f(x)=h\Bigl(x,y,f\bigl(g_1(x,y)\bigr),\ldots,f\bigl(g_n(x,y)\bigr)\Bigr),\\ (x,y)\in D\subset{{\Bbb R}^r}\times{{\Bbb R}^s}, \end{gathered} \] implies that they are \({{\cal C}^\infty}\), even if \(1\le s\le r\). Corollaries which are easy to apply and examples will be given.