Summary: Math 243A Assignment 4
(1) Let Rn be an open set. Suppose that f : Rn.
(a) Prove that if f is locally Lipschitz on ,1
then it is continuous
(b) Prove that if f is continuously differentiable on , then it is
locally Lipschitz on .
(2) Show that the function f(x) = x , > 0, from Rn to R is locally
Lipschitz if and only if 1.
(3) Let A be an n × n matrix over R, and let x0 Rn. Calculate the
Picard iterates for the initial value problem
x = Ax, x(0) = x0,
and show that they converge uniformly to the exact solution on every
closed, bounded interval of the form [-T, T].
(4) Let Rn+1 be an open set, and suppose that f : Rn is
continuous. Let I denote the interval |t - t0| < . Suppose that the
function t (t, x(t)) is continuous from I into .
Prove that x(t) is a C1 solution of the first order system
x (t) = f(t, x(t)),