3
$\begingroup$

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$.

It seems to me that something like the Picard–Lindelöf theorem should ensure that there will be a unique maximally defined solution to the differential equation $x'(t) = f(x(t)) - x(t)$ with $x(0) = p$. Is that correct? Must this solution have a limiting value which is a fixed point of $f$? And is there any way to argue for Brouwer's fixed point theorem in general along these lines?

$\endgroup$
9
  • $\begingroup$ Picard-Lindelof and its analogues only guarantee the existence of solutions locally. $\endgroup$ Aug 26, 2014 at 17:16
  • $\begingroup$ @Christian: $f(x) = 2x$ will not be a map from $B$ to $B$, unless the ball $B$ consists solely of the zero point. $\endgroup$ Aug 26, 2014 at 17:25
  • $\begingroup$ @Christian: With the correction, the differential equation becomes $x'(t) = f(x(t)) - x(t) = -2x(t)$ with $x(0) = p$, which has unique solution $x(t) = e^{-2t}p$. The limiting value of this for large $t$ will be $0$, which is a fixed point of $f$. $\endgroup$ Aug 26, 2014 at 17:28
  • $\begingroup$ @Vidit: Yes, and also the uniqueness. Thus, any solutions defined on an open interval around 0 are compatible, and there is at least one, so we can join them all together and obtain a (unique) solution defined on a maximum open interval around 0 (not necessarily the entire real line), just as claimed, no? $\endgroup$ Aug 26, 2014 at 17:32
  • 1
    $\begingroup$ @SidharRamesh: Yes, my "examples" were nonsense. In $d=2$, the issues seem roughly those addressed by Poincare-Bendixson and its variants. This seems to indicate that it will probably be tough to make this work in $d>2$. $\endgroup$ Aug 26, 2014 at 17:36

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.