Abstract
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.
JEL Classification Numbers: D11
MSC Codes: 35A01, 91B08, 91B16, 91B42
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- 1.
We mainly refer to Chapter 4 of [4], which is a very good textbook. However, only the 1st edition of this book has been translated into English, and this edition is rather complicated and difficult to understand. In the 2nd edition, this problem is removed, although this has been translated into Japanese [5] but not English. The essence of results in this section are also included in [1, 6], and many other textbooks.
- 2.
We abbreviate \(f((1 - t)p + tq,c(t; p,q))\) to f, c(t; p, q) to c, and so on.
- 3.
This formulation of the Nikliborc’s theorem is in [2]. Actually, Nikliborc’s theorem holds under only differentiability and local Lipschitz condition of f. However, our Theorem 1 cannot be applied under such assumptions, because the equation \(\frac{\partial ^{2}c} {\partial t\partial q_{i}} = \frac{\partial ^{2}c} {\partial q_{i}\partial t}\) used in the proof no longer holds.
- 4.
They claimed that this theorem holds even if f is not C 1 but only differentiable. However, we doubt this claim.
- 5.
Here, P is not open and thus this assumption means that f can be extended to some open set including P and this extension is of C1-class and integrable.
- 6.
Hurwicz and Uzawa [2] also assumed this condition in their results.
- 7.
References
Hartman P (1997) Ordinary differential equations. Birkhaeuser, Basel
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 114–148
Nikliborc W (1929) Sur les équations linéaires aux différentielles totales. Studia Mathematica 1:41–49
Pontryagin LS (1962) Ordinary differential equations. Addison-Wesley, Reading (translated from Russian)
Pontryagin LS (1968) Ordinary differential equations, 2nd edn. Kyoritsu Shuppan, Tokyo (in Japanese)
Smale S, Hirsch MW (1974) Differential equations, dynamical systems, and linear algebra. Academic, New York
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© 2016 Springer Science+Business Media Singapore
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Hosoya, Y. (2016). On First-Order Partial Differential Equations: An Existence Theorem and Its Applications. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 20. Advances in Mathematical Economics, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-10-0476-6_3
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DOI: https://doi.org/10.1007/978-981-10-0476-6_3
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