# 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.  Google Scholar [2] P. Hartman, Ordinary Differential Equations, Birkhäuser Boston, Mass., 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, Elsevier, 1979.  Google Scholar [6] W. Nikliborc, Sur les équations linéaires aux différentielles totales, Studia Mathematica, 1 (1929), 41-49.  doi: 10.4064/sm-1-1-41-49.  Google Scholar [7] L. S. Pontryagin, Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1962.  Google Scholar [8] S. Smale and M. W. Hirsch, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.  Google Scholar

show all references

##### References:
 [1] J. Dieudonné, Foundations of Modern Analysis, Hesperides press, 2006.  Google Scholar [2] P. Hartman, Ordinary Differential Equations, Birkhäuser Boston, Mass., 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, Elsevier, 1979.  Google Scholar [6] W. Nikliborc, Sur les équations linéaires aux différentielles totales, Studia Mathematica, 1 (1929), 41-49.  doi: 10.4064/sm-1-1-41-49.  Google Scholar [7] L. S. Pontryagin, Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1962.  Google Scholar [8] S. Smale and M. W. Hirsch, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.  Google Scholar
 [1] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [2] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023 [3] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [4] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [5] Zhong Tan, Jianfeng Zhou. Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1335-1350. doi: 10.3934/cpaa.2016.15.1335 [6] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020 [7] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [8] Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591 [9] Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230 [10] Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 [11] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 [12] Rui L. Fernandes, Yuxuan Zhang. Local and global integrability of Lie brackets. Journal of Geometric Mechanics, 2021, 13 (3) : 355-384. doi: 10.3934/jgm.2021024 [13] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [14] Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211 [15] Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761 [16] Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020 [17] Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4415-4427. doi: 10.3934/dcds.2019179 [18] Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9 [19] Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503 [20] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

2020 Impact Factor: 2.425