History[edit]

Peano first published the theorem in 1886 with an incorrect proof.^[1] In 1890 he published a new correct proof using successive approximations.^[2]

Theorem[edit]

Let $D$ be an open subset of $\mathbb {R} \times \mathbb {R}$ with $f\colon D\to \mathbb {R}$ a continuous function and $y'(x)=f\left(x,y(x)\right)$ a continuous, explicit first-order differential equation defined on D, then every initial value problem $y\left(x_{0}\right)=y_{0}$ for f with $(x_{0},y_{0})\in D$ has a local solution $z\colon I\to \mathbb {R}$ where $I$ is a neighbourhood of $x_{0}$ in $\mathbb {R}$ , such that $z'(x)=f\left(x,z(x)\right)$ for all $x\in I$ .^[3]

The solution need not be unique: one and the same initial value $(x_{0},y_{0})$ may give rise to many different solutions $z$ .

Osgood, W. F. (1898). . Monatshefte für Mathematik. 9: 331–345. doi:10.1007/BF01707876. S2CID 122312261.

"Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung"

Coddington, Earl A.; Levinson, Norman (1955). . New York: McGraw-Hill.

Theory of Ordinary Differential Equations

Murray, Francis J.; Miller, Kenneth S. (1976) [1954]. Existence Theorems for Ordinary Differential Equations (Reprint ed.). New York: Krieger.

(2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

History[edit]

Theorem[edit]

"Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung"

Theory of Ordinary Differential Equations

Teschl, Gerald