228 CHAPTER 9. INTEGRATION

61. Show, using Gronwall’s inequality of Problem 59 that in the above theorem, if

∥f(t,x)− f(t,y)∥ ≤ K ∥x−y∥ ,

then there is only one solution to the initial value problem 9.16.

62. It you let Br be the closed ball {x : ∥x∥ ≤ r} let

Prx =

{ x∥x∥ r if ∥x∥> rx if ∥x∥ ≤ r

Show that Pr is continuous as a map from Rp to Rp.

63. Using the above problem, show using Problem 60 that there is a local solution to 9.16valid for t ∈ [0,T0] for some T0 ≤ T if it is only assumed that f is continuous, withno assumption that it is bounded. The last four problems contain all that is typicallyleft out in undergraduate differential equations courses which is also that which is ofmost importance.

64. Suppose f is Riemann integral on the interval [a,b] . The integrator function is justg(t) = t. Now let h(u) ≡ f (x−u) for u ∈ [x−b,x−a]. Show h is Riemann inte-grable on this new interval. Do something similar for h(u)≡ f (x+u).