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Homework 10, Real Analysis

Due date: October 13

Problem 1


$latex \displaystyle\begin{cases} x'(t) = \sqrt{x(t)}\\ x(0)=0\end{cases}$

has an infinite number of solutions.

Problem 2


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

then, for any $latex x\in\R^2$, $latex |Ax| \le \dfrac{1}{2}|x|$.

Problem 3

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 4

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

Problem 5

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 $latex t=0$.