In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization...

Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees...

In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory: Carathéodory's theorem (conformal mapping)...

{2}{3}}t\right)^{\frac {3}{2}}} . Even more general is Carathéodory's existence theorem, which proves existence (in a more general sense) under weaker conditions...

the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential...

Bôcher's theorem (complex analysis) Borel–Carathéodory theorem (complex analysis) Branching theorem (complex manifold) Carathéodory's theorem (complex...

subjects of interest. For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any...

lower than the one provided by Carathéodory's theorem can be obtained. He is credited with the authorship of the Carathéodory conjecture claiming that a closed...

12 September 2010. Hörmander 1983, p. 56. Rudin 1991, Theorem 6.25. Stein & Weiss 1971, Theorem 1.18. Rudin 1991, §II.6.31. More generally, one only needs...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

thermodynamics, Frobenius' theorem can be used to construct entropy and temperature in Carathéodory's formalism. Specifically, Carathéodory considered a thermodynamic...

equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of existence and uniqueness...

zero means that the function itself is specified. The Cauchy–Kowalevski theorem states that If all the functions F i {\displaystyle F_{i}} are analytic...

{\displaystyle 2x^{2}{\frac {d^{2}y}{dx^{2}}}-3x{\frac {dy}{dx}}+y=2\,.} The existence of a constant term is a sufficient condition for an equation to be inhomogeneous...

the case of an ordinary differential operator of order n, Carathéodory's existence theorem implies that, under very mild conditions, the kernel of L is...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

be extended to a homeomorphism of the boundaries (see Carathéodory's theorem). Carathéodory's proof used Riemann surfaces and it was simplified by Paul...

Picard–Lindelöf theorem Peano existence theorem Carathéodory existence theorem Numerical ordinary differential equations Bendixson–Dulac theorem Gradient conjecture...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in n-dimensional Euclidean space...

second-order partial differential equations, it is not as simple to guarantee existence and uniqueness as it is for ordinary differential equations. Cauchy data...

finite element methods. For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as f ( x 0 + h ) = f ( x 0 ) + f ′...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Wolsson...

C 1 ( [ a , b ] ) . {\displaystyle f\in C^{1}([a,b]).} The mean value theorem for f , {\displaystyle f,} where x ∈ [ a , b ) , {\displaystyle x\in [a...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

of the Calabi conjecture was the proof of existence for a Monge–Ampere equation. The open problem of existence (and smoothness) of solutions to the Navier–Stokes...

Solution Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem General topics...

equations. When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More...