Peano theorem

2020 Mathematics Subject Classification: Primary: 34A12 [MSN][ZBL]

One of the existence theorems for solutions of an ordinary differential equation (cf. Differential equation, ordinary), established by G. Peano on [Pe]. More precisely

Theorem Let $U\subset \mathbb R^n$ be an open set and $f: U\times [0,T] \to \mathbb R^n$ a continuous function. Then, for every $x_0\in U$ there is a positive $\delta$ and a solution $x: [0,\delta]\to U$ of the ordinary differential equation $\dot{x} (t) = f (x(t), t)$ satisfying the initial condition $x(0)=x_0$.

Strictly speaking, the theorem above is the $n$-dimensional generalization of the original Peano's result, which he established in the case $n=1$. The solution $x$ of the theorem is called the integral curve through $x_0$. Peano's theorem guarantees the existence of at least one solution, but the continuity hypothesis is far from guaranteeing its uniqueness. For the latter one usually assumes a Lipschitz condition on $f$, namely $|f(x_1, t)- f (x_2, t)|\leq M |x_1-x_2|$, as in the classical Cauchy-Lipschitz theorem (see also Osgood criterion for a refinement of this statement).

References

[Am]	H. Amann, "Ordinary differential equations. An introduction to nonlinear analysis." de Gruyter Studies in Mathematics, 13. Walter de Gruyter & Co., Berlin, 1990.
[Ha]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[Pe]	G. Peano, "Démonstration de l'intégrabilité des équations différentielles ordinaires" Math. Ann. , 37 (1890) pp. 182–228
[Pet]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)