# American Institute of Mathematical Sciences

December 2018, 11(6): 1143-1167. doi: 10.3934/dcdss.2018065

## First-order partial differential equations and consumer theory

 1-50-1601 Miyamachi, Fuchu, Tokyo, 183-0023, Japan

Received  February 2017 Revised  June 2017 Published  June 2018

In this paper, we show that the existence of a global solution of a standard first-order partial differential equation can be reduced to the extendability of the solution of the corresponding ordinary differential equation under the differentiable and locally Lipschitz environments. By using this result, we can produce many known existence theorems for partial differential equations. Moreover, we demonstrate that such a result can be applied to the integrability problem in consumer theory. This result holds even if the differentiability condition is dropped.

Citation: Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065
##### References:
 [1] J. Dieudonné, Foundations of Modern Analysis, Hesperides press, 2006. [2] P. Hartman, Ordinary Differential Equations, Birkhäuser Boston, Mass., 1982. [3] Y. Hosoya, On first-order partial differential equations: An existence theorem and its applications, Advances in Mathematical Economics, 20 (2016), 77-87. doi: 10.1007/978-981-10-0476-6_3. [4] L. Hurwicz and H. Uzawa, On the Integrability of Demand Functions, in Preference, Utility and Demand (eds. J. S. Chipman, L. Hurwicz, M. K. Richter and H. F. Sonnenschein) Harcourt Brace Jovanovich, Inc., New York, (1971), 114–148. [5] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, Elsevier, 1979. [6] W. Nikliborc, Sur les équations linéaires aux différentielles totales, Studia Mathematica, 1 (1929), 41-49. doi: 10.4064/sm-1-1-41-49. [7] L. S. Pontryagin, Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1962. [8] S. Smale and M. W. Hirsch, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.

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##### References:
 [1] J. Dieudonné, Foundations of Modern Analysis, Hesperides press, 2006. [2] P. Hartman, Ordinary Differential Equations, Birkhäuser Boston, Mass., 1982. [3] Y. Hosoya, On first-order partial differential equations: An existence theorem and its applications, Advances in Mathematical Economics, 20 (2016), 77-87. doi: 10.1007/978-981-10-0476-6_3. [4] L. Hurwicz and H. Uzawa, On the Integrability of Demand Functions, in Preference, Utility and Demand (eds. J. S. Chipman, L. Hurwicz, M. K. Richter and H. F. Sonnenschein) Harcourt Brace Jovanovich, Inc., New York, (1971), 114–148. [5] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, Elsevier, 1979. [6] W. Nikliborc, Sur les équations linéaires aux différentielles totales, Studia Mathematica, 1 (1929), 41-49. doi: 10.4064/sm-1-1-41-49. [7] L. S. Pontryagin, Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1962. [8] S. Smale and M. W. Hirsch, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.
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