AMATH 331/PMATH 331 - Assignment 5
Due: April 8th, 2021
Problem 1
Complete the proof of the Global Picard Theorem in the course notes by proving that the fixed point F ∗ of the
map T is unique.
[Hint: suppose there is another fixed point Fˆ and use a similar approach as in the proof that (Fk) is Cauchy. You
might also find Lemma 10.3.3 useful.]
Problem 2
(a) Express the initial value problem:
f ′(x) = − 1
x + 1
f(x), for − 1
3
≤ x ≤ 1
2
,
f(0) = 1,
as an integral equation and define a map T : C
[− 13 , 12]→ C [− 13 , 12] such that the fixed points of T solve the
IVP.
(b) Prove that T from Part (a) is a contraction.
(c) Find the forward orbit O(f0) for the starting point f0(x) = 1. That is, find a formula defining the kth point
(function) in the sequence (fk).
(d) Find the limit of this orbit, and hence, the fixed point of T .
(e) Verify that the fixed point is a solution of the initial value problem from Part (a).
Problem 3
Consider the initial value problem:
d2y
dx2
− (1− y)
x
= 0, for 1 ≤ x ≤ 10,
y(1) = 1, y′(1) = −1.
(a) Formulate (do not solve) the associated fixed point problem.
(b) Use the Global Picard Theorem to show that there is a unique solution.
Problem 4
Given a continuous function f : [0, 1]→ R, show that its Bernstein polynomials Bnf satisfy (Bnf)2 ≤ Bn(f2).
Hint: Consider Bn((f − a)2) where a ∈ R.
Problem 5
Prove that every continuously differentiable function f : [0, 1]→ R is the uniform limit of a sequence of polynomials
(pn) such that f
′ is the uniform limit of the sequence (p′n).
1
Problem 6
Is the statement in Problem 5 true if we replace “continuously differentiable” with simply “differentiable” (on
[0, 1])? Fully justify your answer.
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