Vol. 82, No. 2, 1979

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

M. A. McKiernan

Vol. 82 (1979), No. 2, 503–514

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,,xn)} = F{f1(x1),,fn(xn)} for the unknowns h : T(Ω) R and π : (f1,,fn) : 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
Received: 13 March 1978
Revised: 20 October 1978
Published: 1 June 1979
M. A. McKiernan
Department of Mathematics
University of Waterloo
Waterloo ON N2L 3G1