AMATH 331/PMATH 331 - Assignment 5 Due: April 8th, 2021 Problem 1[6] 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[8] (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[6] 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[7] 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[6] 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[7] Is the statement in Problem 5 true if we replace “continuously differentiable” with simply “differentiable” (on [0, 1])? Fully justify your answer. 2
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