Homework 12, Real Analysis

Due November 16

Problem 1

If

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

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

Problem 2

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

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

Problem 3

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

Problem 4

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

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

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