For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If so, could someone direct me to some resources on the topic?
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1$\begingroup$ Welcome to MathOverflow! Hmm, if the vector field is not Lipschitz, the solution will not be unique, in general. So it seems to me that it is not even clear what one means by "continuous dependance on the initial parameter", since there could be many solutions for each initial value. (+1 anyway, since I find the question to be an interesting thought.) $\endgroup$– Jochen GlueckCommented Apr 12, 2020 at 14:17
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2$\begingroup$ For non-Lipschitz differentil equations, there is no uniqueness theorem. So we cannot speak of "dependence" on initial conditions at all. There are some generalizations of uniqueness theorem to more general functions, but it is something very close to the Lipschitz condition. $\endgroup$– Alexandre EremenkoCommented Apr 12, 2020 at 14:27
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$\begingroup$ mathoverflow.net/questions/234183/… $\endgroup$– Alexandre EremenkoCommented Apr 12, 2020 at 14:34
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1$\begingroup$ When we lose uniqueness, maybe we can do with an "existence" formulation of continuous dependence? Something along the lines of "for every bounded open $\Omega\subset\mathbb{R}^N$ there exists a time interval $(-T,T)$ and a continuous function $f: \Omega\times(-T,T) \to \mathbb{R}^N$ such that $\frac{d}{dt} f(\alpha,t) = F(f(\alpha,t))$ with initial data $f(\alpha,0) = \alpha$"? $\endgroup$– Willie WongCommented Apr 14, 2020 at 14:43
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$\begingroup$ In regards to my previous comment: I think in at least the case of the Peano theorem (where the vf is only continuous), the usual proof by looking at the integral formulation and using compactness should go through. $\endgroup$– Willie WongCommented Apr 14, 2020 at 14:55
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The continuous dependence on initial conditions and parameters (and even on the right-hand side in the compact-open topology) is a consequence of the uniqueness. See Theorem 3.2 of Chapter II in Hartman's "Ordinary differential equations" (I call this statement the Kamke lemma). Note that without the uniqueness, the question of continuous dependence (in the classical sense) is incorrect. So when you have this correctness, you automatically have the continuous dependence.
