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Two functional equations preserving functional forms.

János Aczél1, R Duncan Luce

  • 1University of Waterloo, Waterloo, ON, Canada N2L 3G1; and University of California, Irvine, CA, 92697-5100, USA.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 2006
PubMed
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This study solves two functional equations related to utility theory and homogeneous functions. Solutions were fully determined for both equations, with some conditions and open problems remaining for the second equation.

Area of Science:

  • Mathematics
  • Functional Equations
  • Utility Theory

Background:

  • Explores functional equations motivated by utility theory, psychophysics, and homogeneous functions.
  • Investigates homomorphisms of degree 1 homogeneous functions and their connection to quasilinear mean values.

Purpose of the Study:

  • To find solutions for two specific functional equations.
  • To determine the conditions under which these equations hold and to identify any remaining open problems.

Main Methods:

  • Analysis of the first functional equation: h(σ(y)x + [1 - σ(y)]y) = τ(y)h(x) + [1 - τ(y)]h(y) for x ≥ y ≥ 0.
  • Analysis of the second functional equation: h[y + f(x - y)] = h(y) + g[h(x) - h(y)] for x ≥ y ≥ 0.
  • Application of differentiability and analyticity assumptions to derive solutions.

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Main Results:

  • The solutions for the first functional equation are fully determined.
  • The solutions for the second functional equation are fully determined under specific differentiability conditions (f"(0) ≠ 0 or g"(0) ≠ 0).
  • Solutions for the second equation are determined under analyticity when f"(0) = g"(0) = 0 and h" is not identically zero.

Conclusions:

  • The study successfully determined solutions for both functional equations under various conditions.
  • Open problems remain regarding the second equation, particularly concerning the elimination of differentiability assumptions.