Problem 1. (i) Prove that ||f|| - ||g|| ≤ ||f - g||.
(ii) Derive Newton's method for finding the zeros of a twice continuously differentiable function f(x),
xn+1 = K(xn), K(x) = x - f(x)/f'(x).
from the contraction principle by showing that if x¯ is a zero with ft(x¯) ≠ 0, then there is a corresponding closed interval C around x¯ assumptions in part (i) are satisfied.
Problem 2. Are the following functions Lipschitz continuous near 0?
If yes, find a Lipschitz constant for some interval containing 0.
(i) f (x) = 1/1 -x2
(ii) f (x) = |x|1/2
(iii) f (x) = x2 sin(1/x)
Problem 3. Consider the ODE
x. = x2, x(0) = a
In the existence theorem, solutions exist over an interval [-T0, T0] where T0 = min{T, δ/M}, For this problem, choose a radius, δ for your ball so as to maximize, T0 for given a > 0. Solve the ODE and compare your estimated interval of existence to the actual one.
Problem 4. Let f (x) = Ax where A is a constant matrix. Show that each component of the nth Picard iteration to any solution is a polynomial of degree at most n. Can you guess a formula for xn(t) and from this guess a formula for x(t)?