General Pexider equations. I. Existence of injective solutions

M. A. McKiernan

Vol. 82 (1979), No. 2, 499–502

DOI: 10.2140/pjm.1979.82.499

Abstract

Given open connected Ω, Ω ⊆ Rⁿ and given T : Ω → R continuous, F : Ω → R strictly monotonic, in each variable separately. The equation is h ∘ T = F ∘ π for the unknowns h : T(Ω) → R, π : Ω →Ω with π = (f₁,⋯,f_n) a product mapping - e.g., h{T(x,y)} = F{f(x),g(y)}. If T is one-one in each variable, then any continuous solution π must be injective or constant on Ω; conversely, if an injective solution π exists then T must be one-one in each variable separately.

Mathematical Subject Classification

Primary: 39B20

Secondary: 39B50

Milestones

Received: 13 March 1978

Revised: 20 October 1978

Published: 1 June 1979