Homework 12, Real Analysis

Due November 16

Problem 1

If

A=\begin{pmatrix} 1/12 & 5/8\\ 5/8 & 1/12\end{pmatrix},

then the map x\mapsto Ax is a contraction in \R^2.

Problem 2

Let F(x,t) = \dfrac{tx}{x^2+1}. Then, for all t\in\R,

|F(x,t) - F(y,t)| \le |t||x-y|.

Problem 3

The function f(x) = \sqrt x on [0,\infty) is uniformly continuous but not Lipschitz.

Problem 4

Consider the operator \Phi:C([-1,1])\to C([-1,1]) given by

\displaystyle \Phi(x)(t) = 1 + 2 \int_0^t s x(s) ds,

for any x(t)\in C([-1,1]). Starting from the constant function x_0(s)=1, verify explicitly that the nth iteration of x_{n+1} = \Phi(x_n) is the nth Taylor polynomial of t\mapsto e^{t^2} around t=0.

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