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The canonical example of an ODE with nonunique solutions is $y'=\sqrt{y}$ with $y(0)=0$. The solutions are $y=\frac{1}{4}t^2$ and $y=0$. We also know that just by looking at $\sqrt{y}$, the fact that it is not Lipschitz immediately suggests this ODE may have multiple solutions. I have a few questions regarding this:

1) Let $y'=f(y,t)$ with $y(0)=0$ where $f$ is not Lipchitz in $y$. Is this always enough to guarantee a nonunique solution to $y$? I suppose this is probably equivalent to asking how multiple fixed points arise when a mapping is non-contractive.

2) Supposing we are assured that $y'=f(y,t)$ does not have a unique solution, is there a systematic way of characterizing all nonunique solutions? In particular, most of the examples I've seen such as the $\sqrt{y}$ example have the solution $y=0$ and I would be interested to see drastically different solutions. More to the point, for something like $y'=\sqrt{y}$ is there a way to conclude that $y=\frac{1}{4}(t-a)^2$ and $y=0$ are the only solutions? Does this generalize to harder problems?

On the subject of what one might call particularly badly nonunique ODEs I found papers such as this one which show the existence of equations of the form $y'=f(y,t)$ with $y(t_0)=y_0$ and $f(y,t)$ is continuous in both coordinates, that have more than one solution for every $(t_0,y_0)$ on every interval $[t_0,t_0+\epsilon]$ and $[t_0-\epsilon,t_0]$ for small enough $\epsilon$.

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up vote 2 down vote accepted
  1. No. There is Osgood's theorem that states that if $$ |f(x,y_1)-f(x,y_2)|\leq \Phi(|y_1-y_2|), $$ for any $\Phi(u)$ such that $$ \int_\epsilon^c\frac{1}{\Phi(u)}du\to \infty, $$ when $\epsilon\to 0$, then the solution to IVP is unique. Take $\Phi(u)=Ku$ and you get the usual Lipchitz condition. But you can also take $\Phi(u)=Ku\log u$, which will give you the condition weaker than Lipchitz. It is possible to present an infinite sequence of $\Phi(u)$, weaker and weaker. Hence, there is no hope to give a sharp sufficient condition for the uniqueness.

  2. I do not quite follow your question, but just one remark: $C(t-a)^2$ and $0$ are not the only solutions. We can glue as many solutions from these two as we'd like. In general, this example is used because there is a simple generalization. Consider a separable ODE $$ y'=f(x)g(y). $$ Assume that $g(y_0)=0$ and we need to solve IVP $y(x_0)=y_0$. Then solution is unique if and only if the integral $$ \int_{y_0}^y\frac{1}{g(y)}dy $$ diverges. This example also shows the way how non-uniqueness appears.

The paper by Hartman is way more involved than my example. That is probably the reason that it is in a research journal and not in a textbook. I do not exactly follow the latest development in these area (existence and uniqueness theorems) but I am quite sure that I saw several recent papers on the subject. You should try look it up, if you would like to find out more.

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