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Continuous solutions of a system of composite functional equations.
- Source :
-
Aequationes Mathematicae . Dec2022, Vol. 96 Issue 6, p1179-1205. 27p. - Publication Year :
- 2022
-
Abstract
- 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]
- Subjects :
- *UTILITY functions
*CONTINUOUS functions
*FUNCTIONAL equations
Subjects
Details
- Language :
- English
- ISSN :
- 00019054
- Volume :
- 96
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- Aequationes Mathematicae
- Publication Type :
- Academic Journal
- Accession number :
- 160372294
- Full Text :
- https://doi.org/10.1007/s00010-022-00900-7