Would the [Picard–Lindelöf theorem](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem#Picard.E2.80.93Lindel.C3.B6f_theorem) still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be [almost Lipschitz](https://en.wikipedia.org/wiki/Modulus_of_continuity) in y?


If not, are there any moduli of uniform continuity weaker than Lipschitz continuity that it is known suffice, or results indicating that there can't be any?



Existence/Uniqueness of solutions to quasi-Lipschitz ODEs
Yes. This follows from the classical uniqueness theorem due to Osgood (the original [paper](https://doi.org/10.1007/BF01707876 "Osgood, W.F. Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x,y) ohne Hinzunahme der Cauchy-Lipschitz'schen Bedingung. Monatsh. f. Mathematik und Physik 9, 331–345 (1898). zbMATH review at https://zbmath.org/?q=an:29.0260.03") appeared in 1898).



> 
> **Osgood's Criterion.** Let $\omega(t,u)=\phi(t)\psi(u)$ where $\phi(t)\geq 0$ is continuous
> on the interval $(0,a)$ and $\psi(u)$ is continuous on $\mathbb R\_{+}$, $\psi(0)=0$, $\psi(u)>0$ for $u>0$, and
> $$\int\_{0}^{\epsilon}\phi(t)dt<\infty,\qquad \int\_{0}^{\epsilon}\frac{du}{\psi (u)}=\infty$$
> for some $\epsilon>0$. Suppose that the mapping $f:[0,a]\times B\_R(x\_0)\to \mathbb R^d$ satisfies the condition
> $$\|f(t,x\_1)-f(t,x\_2)\|\leq\omega (t,\|x\_1-x\_2\|)$$
> for any $t\in(0,a]$ and any $x\_1,x\_2\in B\_R(x\_0)$. Then the initial value problem
> $$\dot{x}=f(t,x),\qquad x(0)=x\_0$$
> has at most one solution on the interval $[0,\delta]$ with some $\delta>0$.
> 
> 
> 


Osgood's theorem allows for the mappings $f(t,x)$ which are discontinuous at $t=0$. (Actually, the condition that $\phi(t)$ is continuous on $(0,a)$ can be replaced with an assumption
of mere integrability.) Of course, the existence of a local solution is implied by the Peano theorem under the additional assumption that $f$ is continuous in $(t,x)$.


Moreover, Wintner showed that Osgood's uniqueness condition implies the convergence of successive Picard iterations to a local solution on a sufficiently small interval (A. Wintner, ["On the Convergence of Successive Approximations"](https://www.jstor.org/stable/2371736 "doi:10.2307/2371736. zbMATH review at https://zbmath.org/?q=an:0063.08288"), *Amer. Journal of Math.* Vol. 68 (1946), pp. 13-19).


