Continuous solutions of a system of composite functional equations.

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set ∅ ≠ S ⊂ R n , the function F : S ⟶ R is translation invariant if F (x) = F (y) implies F (x + t) = F (y + t) for any vectors x , y , t ∈ R n such that x , y , x + t , y + t ∈ S . In our main results we shall consider an open, connected set ∅ ≠ D ⊂ R n . We prove that if F : D ⟶ R is a translation invariant, continuous function, then there exists a vector a = (a 1 , ⋯ , a n) ∈ R n and a strictly monotone, continuous function f such that F (x 1 , ⋯ , x n) = f (a 1 x 1 + ⋯ + a n x n) holds for all (x 1 , ⋯ , x n) ∈ D . Using this result we also show that continuous solutions F : D ⟶ R of the system of functional equations F (x 1 , ⋯ , x j + t j , ⋯ , x n) = Ψ j (F (x 1 , ⋯ , x j , ⋯ , x n) , t j) (j = 1 , ⋯ , n) 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. [ABSTRACT FROM AUTHOR]