On First-Order Partial Differential Equations: An Existence Theorem and Its Applications

  • Yuhki Hosoya
Research Article
Part of the Advances in Mathematical Economics book series (MATHECON, volume 20)


In this paper, we present an equivalence theorem between the existence of a global solution of a standard first-order partial differential equation and the extendability of the solution of corresponding ordinary differential equation. Moreover, we use this result to produce existence theorems on partial differential equation, and apply this theorem to the integrability problem in consumer theory.


Partial differential equation Global solution Nikliborc’s theorem Shephard’s lemma Expenditure function Integrability 


  1. 1.
    Hartman P (1997) Ordinary differential equations. Birkhaeuser, BaselMATHGoogle Scholar
  2. 2.
    Hurwicz L, Uzawa H (1971) On the integrability of demand functions. In: Chipman JS, Hurwicz L, Richter MK, Sonnenschein HF (eds) Preferences, utility and demand, Harcourt Brace Jovanovich, Inc., New York, pp 114–148Google Scholar
  3. 3.
    Nikliborc W (1929) Sur les équations linéaires aux différentielles totales. Studia Mathematica 1:41–49MATHGoogle Scholar
  4. 4.
    Pontryagin LS (1962) Ordinary differential equations. Addison-Wesley, Reading (translated from Russian)MATHGoogle Scholar
  5. 5.
    Pontryagin LS (1968) Ordinary differential equations, 2nd edn. Kyoritsu Shuppan, Tokyo (in Japanese)MATHGoogle Scholar
  6. 6.
    Smale S, Hirsch MW (1974) Differential equations, dynamical systems, and linear algebra. Academic, New YorkMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of EconomicsKanto-Gakuin UniversityTokyoJapan

Personalised recommendations