On global transformations of functional-differential equations of the first order

The paper describes the general form of functional-differential equations of the first order with m(m ≥ 1) delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f(t,uv,u_1 v_{\text{1}} , \ldots ,u_m v_m ) = f(x,v,v, \ldots ,v_m )g(t,x,u,u, \ldots ,u_m )u + h(t{\text{,}}x{\text{,}}u{\text{,}}u_{\text{1}} \ldots {\text{,}}u_m )v$$ \end{document} for u ≠ 0 is solved on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document} and a method of proof by J. Aczél is applied.