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Let's say that I've got the following IVP:

$\frac{dy}{dx} = f(x,y)$

$y(x_0) = y_0$

And I want conditions that guarantee existence and uniqueness of its solution.

On the one hand I've got the Picard–Lindelöf theorem. It asks that there exists a rectangle $R = [a,b] \times [c,d]$, containing $(x_0, y_0)$ as an interior point, where $f$ is continuous in $x$ and Lipschitz continuous in $y$.

On the other hand I've got a theorem, which I've encountered in many undergraduate text books, that requires $f$ and $\frac{\partial f}{\partial y}$ to be continuous in the aforementioned rectangle.

Are these two different theorems? It seems to me that the hypotheses of the first one are implied by those of the second one. But in that case, why would some authors prefer this more restrictive form of the theorem? Could it be just so that students don't need to learn the concept of Lipschitz continuity?

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  • $\begingroup$ Yes, it's just to avoid having to explain what Lipschitz means. $\endgroup$ – Hans Lundmark Oct 14 '17 at 7:35
  • $\begingroup$ @HansLundmark, if you provide some explanation/reference that proves that the hypotheses of the second theorem do imply those of the first one, I can select that as the accepted answer. $\endgroup$ – LGenzelis Oct 14 '17 at 19:47
  • $\begingroup$ OK, I added an answer. $\endgroup$ – Hans Lundmark Oct 15 '17 at 6:15
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Suppose $\frac{\partial f}{\partial y}$ is continuous. Then $|\frac{\partial f}{\partial y}|$ has a greatest value $K$ on the rectangle in question (by the extreme value theorem). The mean value theorem for derivatives says that $$ f(x,y_2)-f(x,y_1) = \frac{\partial f}{\partial y}(x,\eta) \, (y_2-y_1) $$ for some $\eta$ between $y_1$ and $y_2$, which implies $$ |f(x,y_2)-f(x,y_1)| \le K |y_2-y_1| , $$ so $f$ is Lipschitz continuous with respect to $y$ on that rectangle, with Lipschitz constant $K$.

Hence the assumptions that $f$ and $\frac{\partial f}{\partial y}$ are continuous are just (somewhat weaker) replacements for the “proper” assumptions about Lipschitz continuity, with the advantages that they are often very easy to verify, and that you don't have to explain to your readers what Lipschitz continuity means.

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The first, that requires that the partial derivative of f with respect to y exist, is an "if and only if" theorem. The second, that requires "Lindelof" in y, says "if" but not "only if". Any function that is differentiable with respect to y is "Lindelof" but the converse is not true.

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  • $\begingroup$ It requires that the partial derivative of f with respect to y is continuous, not only that it exists. And I'm confident it's just a sufficient condition, not a necessary one. $\endgroup$ – LGenzelis Oct 13 '17 at 19:25

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