"Existence and Uniqueness" Theorems for first order IVP: two or just one?

Question

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?

Answer 1

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.

Answer 2

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.