Peano existence theorem

In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.

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

Let D be an open subset of R × R with

a continuous function and

a continuous, explicit first-order differential equation defined on D, then every initial value problem

for f with ${\displaystyle (x_{0},y_{0})\in D}$ has a local solution

where ${\displaystyle I}$ is a neighbourhood of ${\displaystyle x_{0}}$ in ${\displaystyle \mathbb {R} }$, such that ${\displaystyle z'(x)=f\left(x,z(x)\right)}$ for all ${\displaystyle x\in I}$.

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