In the page 184 of the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by H. Berezis, there is a theorem due to Cauchy, Lipschitz and Picard, which states that if $E$ be a Banach space and $F:E\to E$ be a Lipshitz map, then given any $u_0\in E$, there is a unique solution $u\in C^1([0,\infty);E)$ of the problem $$\frac{du}{dt}(t)=Fu(t)~~~~~on [0,\infty);~~~~u(0)=u_0.$$
I want to know what happens if domain of $F$ is not whole of $E$, and be just a convex (and possibly closed) set of $E$? Dose the theorem hold under this new condition? What dose happen for the solution $u$?
Any hint or comment is appreciated as well as giving any reference.