scholarly journals Consumer Optimization and a First-Order PDE with a Non-Smooth System

2021 ◽  
Vol 2 (4) ◽  
Author(s):  
Yuhki Hosoya
Keyword(s):  
Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Local Existence ◽  
Microeconomic Theory ◽  
Frobenius Theorem ◽  
First Order ◽  
Integrability Problem ◽  
First Order Pde ◽  
Necessary And Sufficient ◽  
Smooth System

AbstractWe study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

scholarly journals Oscillation and nonoscillation of a delay differential equation

1994 ◽  
Vol 49 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Chunhai Kou ◽  
Weiping Yan ◽  
Jurang Yan
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
First Order ◽  
Oscillating Coefficients ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Oscillation And Nonoscillation

In this paper, some necessary and sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formare established. Several applications of our results improve and generalise some of the known results in the literature.

Conditions for the local existence of metric in a generic affine manifold

1980 ◽  
Vol 87 (3) ◽  
pp. 527-534 ◽  
Author(s):  
Kuo-Shung Cheng ◽  
Wei-Tou Ni
Keyword(s):  
Affine Connection ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Riemann Tensor ◽  
Affine Manifold ◽  
Absolute Value ◽  
First Order ◽  
Covariant Derivatives ◽  
Explicit Formulae ◽  
Necessary And Sufficient

AbstractFor a manifold with a generic symmetric affine connection, explicit necessary and sufficient conditions for the local existence of metric compatible with the connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives. If these conditions are satisfied, the solutions for metric are unique up to a constant scale factor and the absolute value of the signature is uniquely determined. Explicit formulae for the solutions are given in terms of integrals.

scholarly journals On periodic waves of the nonlinear systems

1998 ◽  
Vol 20 (4) ◽  
pp. 11-19
Author(s):  
Le Xuan Can
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Electric Cables ◽  
Necessary And Sufficient

The paper is concerned with the solvability and approximate solution of the nonlinear partial differential equation describing the periodic wave propagation. Necessary and sufficient conditions for the existence of the periodic wave solutions are obtained. An approximate method for solving the equation is presented. As an illustrative example, the equation of periodic waves of the electric cables is considered.

scholarly journals Property of oscillation of first order impulsive neutral differential equations with positive and negative coefficients

2019 ◽  
Vol 11 (1) ◽  
Author(s):  
Hussain Ali Mohamad ◽  
Aqeel Falih Jaddoa
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Neutral Differential Equations ◽  
First Order ◽  
Positive And Negative Coefficients ◽  
Variable Delays ◽  
Impulsive Neutral Differential Equations ◽  
Necessary And Sufficient

            In this paper, necessary and sufficient conditions for oscillation are obtained, so that every solution of the linear impulsive neutral differential equation with variable delays and variable coefficients oscillates. Most of authors who study the oscillatory criteria of impulsive neutral differential equations, investigate the case of constant delays and variable coefficients. However the points of impulsive in this paper are more general. An illustrate example is given to demonstrate our claim and explain the results.

Conditions for an affine manifold with torsion to have a Riemann–Cartan structure

1981 ◽  
Vol 90 (3) ◽  
pp. 517-527 ◽  
Author(s):  
Wei-Tou Ni
Keyword(s):  
Affine Connection ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Riemann Tensor ◽  
Scale Factor ◽  
Necessary And Sufficient Conditions ◽  
Affine Manifold ◽  
First Order ◽  
Covariant Derivatives ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions for the local existence of a metric compatible with the affine connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives in a generic affine manifold with torsion. In case these conditions are satisfied, the solutions of the metric are given in terms of integrals and are unique up to a constant scale factor. Some global conditions are also obtained and discussed.

The exact difference equation of the first order

1933 ◽  
Vol 29 (3) ◽  
pp. 373-381
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Linear Equation ◽  
Sufficient Conditions ◽  
First Order ◽  
Non Linear Equation ◽  
Exact Difference ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Limiting Case

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.

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