Osgood criterion

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

The term refers to a classical criterion, due to Osgood, see [Os] for the uniqueness of solutions of ordinary differential equations which generalizes the Cauchy-Lipschitz (or Picard-Lindelof) theorem. More precisely, we first introduce the so-called Osgood condition:

Definition 1 A continuous nondecreasing function $\omega:[0, \infty[\to [0, \infty[$ with $\omega (0)=0$ and $\omega(\xi)>0$ for $\xi>0$ is said to satisfy Osgood's condition if \[ \int_0^1 \frac{d\xi}{\omega (\xi)} = \infty\, . \]

Theorem 2 Let $U\subset \mathbb R^n$ be open, $f:U \times [0,T] \to \mathbb R^n$ a continuous function and $\omega:[0, \infty[\to [0, \infty[$ be a function satisfying the requirements of Definition 1. Assume \[ |f(x_1, t)- f(x_2, t)|\leq \omega (|x_1-x_2|) \qquad \mbox{for all } (x_1, t), (x_2, t)\in U\, . \] Then, for any $x_0\in U$ there is a $\delta> 0$ for which there exists a unique solution $x : [0, \delta] \to U$ of the ordinary differential equation $\dot{x} (t) = f (x(t), t)$ with $x (0) = x_0$.

Remarks

• The existence of the solutions is actually valid for merely continuous functions (see Peano's theorem). Thus the most interesting point is the uniqueness.
• The classical Cauchy-Lipschitz theorem gives the same conclusion under the stronger assumption that there is a constant $M$ such that $|f (x_1, t) - f(x_2, t)|\leq M |x_1-x_2|$ (see also Lipschitz condition). Typical examples of functions $f$ satisfying Osgood's criterion, but which do not satisfy the assumptions of the Cauchy-Lipschitz theorem, are those for which the bound $|f(x_1, t)-f(x_2, t)|\leq |x_1-x_2| |\log |x_1-x_2||$ holds.
• Classical examples of functions which do not satisfy the Osgood conditions are $\mathbb R \ni \xi \mapsto |\xi|^\alpha$, for $\alpha < 1$ (see also Holder condition). In fact for such functions it is easy to see that there are infinitely many solutions of the corresponding ordinary differential equation for the initial condition $x(0)=0$.
• The requirements in Theorem 2 can be considerably weakened. For instance, the same conclution holds for maps $f$ satisfying the bound

\[ |f(x_1, t)- f(x_2, t)| \leq \phi (t) \omega (|x_1-x_2|)\, , \] where $\phi$ is a positive summable function. Consistently the continuity assumption in the time variable can also be relaxed, see Caratheodory conditions.

References

[Os]	W.F. Osgood, "Beweis der Existenz einer Lösung der Differentialgleichung $\frac{dy}{dx}=f(x,y)$ ohne Hinzunahme der Cauchy-Lipschitz'schen Bedingung." (German) Monatsh. Math. Phys. 9 (1898), no. 1, 331–345.