Continuous solutions of a system of composite functional equations

Abstract

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set emptyset ne S subset {mathbb {R}}^n, the function F : S longrightarrow {mathbb {R}} is translation invariant if F(x) = F(y) implies F(x+t)=F(y+t) for any vectors x,y,t in {mathbb {R}}^n such that x ,y , x+t ,y+t in S. In our main results we shall consider an open, connected set emptyset ne D subset {mathbb {R}}^n. We prove that if F : D longrightarrow {mathbb {R}} is a translation invariant, continuous function, then there exists a vector a = (a_1, dots , a_n) in {mathbb {R}}^n and a strictly monotone, continuous function f such that F(x1,⋯,xn)=f(a1x1+⋯+anxn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} F(x_1, \\dots , x_n) = f (a_1 x_1 + \\dots + a_n x_n) \\end{aligned}$$\\end{document}holds for all (x_1, dots , x_n) in D ,. Using this result we also show that continuous solutions F : D longrightarrow {mathbb {R}} of the system of functional equations F(x1,⋯,xj+tj,⋯,xn)=Ψj(F(x1,⋯,xj,⋯,xn),tj)(j=1,⋯,n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} F(x_1 , \\dots , x_j + t_j , \\dots , x_n) = \\Psi _j (F (x_1 , \\dots , x_j , \\dots , x_n), t_j) \\ \\ (j=1,\\dots ,n) \\end{aligned}$$\\end{document}can be represented as the composition of a strictly monotone, continuous function and a linear functional as well. Applying the latter theorem, we give a characterization of Cobb–Douglas type utility functions.