General Pexider equations. II. An application of the theory of webs

M. A. McKiernan

Vol. 82 (1979), No. 2, 503–514

DOI: 10.2140/pjm.1979.82.503

Abstract

Given open connected Ω, Ω ⊆ Rⁿ and continuous T : Ω → R, F : Ω → R both strictly monotonic in each variable separately. The equation h{T(x₁,⋯,x_n)} = F{f₁(x₁),⋯,f_n(x_n)} for the unknowns h : T(Ω) → R and π : (f₁,⋯,f_n) : Ω →Ω can be interpreted within the theory of webs (the “Gewebe” of Blaschke-Bol). The web structure is then used to prove: any continuous solution π is uniquely determined on Ω by its value at two points of Ω; if a solution π is not continuous on Ω, then π(ω) is dense in Ω for every open ω in Ω; if a solution π is continuous at one point of Ω, it is continuous on Ω.

Mathematical Subject Classification

Primary: 39B20, 39B20

Secondary: 39B50

Milestones

Received: 13 March 1978

Revised: 20 October 1978

Published: 1 June 1979