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i have the following Osgood Lemma: let $f(x,y)$ a function such as $|f(x,y_1)-f(x,y_2)| \leq h(|y_2 - y_1|)$ for all $(x,y_1)$ and $(x,y_2)$ in an opena $\Omega \subset \mathbb{R}^2$. We suppose that function $h$ is continuous on each point $u \in ]0,\alpha]$, stricly positive and $\lim_{\epsilon \to 0^+} \displaystyle\int_{\epsilon}^{\alpha} \dfrac{d u}{h(u)}=+\infty$, with $\alpha > 0$. Then for all $(x_0,y_0)$ on $\Omega$, the problem $y'=f(x,y)$ admits at last one solution.

My question is: please can you give me an example to show how we can apply this theorem to prouve existence of solution of Cauchy problem $y'=f(x,y), y(x_0)=y_0$.

Thank you in advance.

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    $\begingroup$ Welcome to MSE. The problem admits at most one solution. Osgood's lemma is used to prove not the existence but the uniqueness of a solution to an initial value problem (as regards the former, see Peano existence theorem). $\endgroup$ – user539887 Feb 20 at 10:58
  • $\begingroup$ I don't understand. So what's name of this theorem? Isn't Theorem of Osgood? The lemma who's in my fist message isn't Osgood? What's the Osgood theorem? Plase $\endgroup$ – capucine Feb 20 at 11:13
  • $\begingroup$ Where did I write that the lemma in your question is not Osgood's lemma? $\endgroup$ – user539887 Feb 20 at 12:10
  • $\begingroup$ Sorry. So the Osgoos lemma say that the Cauchy problem admits at most one solution. Why you say that we use it to prouve unicity of Cauchy problem? And can you give me an example please to show how we use Osgood? $\endgroup$ – capucine Feb 20 at 16:32
  • $\begingroup$ If the problem has at most one solution (please correct that in the question), then there can be no two solutions to the same initial value problem. Thus uniqueness if a solution exists at all. $\endgroup$ – LutzL Feb 20 at 20:23

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