Tarea 9, Análisis real

Due October 9

Problem 1

1. If $latex f:X\to Y$ is a Lipschitz function, then it is uniformly continuous.


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

Problem 2

Let $latex P,Q,f:[-1,1]\to\R$ continuous and $latex a,b\in\R$. Then the IVP

$latex \begin{cases} u''(x)+P(x)u'(x)+Q(x)u(x)=f(x)\\u(0)=a,\qquad u'(0)=b\end{cases}$

has a unique solution in a neighborhood of $latex x=0$.

Problem 3

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

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

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

Problem 4

For $latex T\in L(\R^l,\R^m)$, $latex ||T||_L = \sup\{ |Tx| : x\in\bar B_1(0)\}$.

Problem 5

Let $latex B_r(x_0)$ be a ball in $latex \R^l$ and $latex f\in C^1(B_r(x_0),\R^m)$ such that $latex ||Df(x)||_L\le M$ for all $latex x\in B_r(x_0)$. Then, for $latex x,y\in B_r(x_0)$,

$latex |f(x) - f(y)| \le M|x-y|$.

(Hint: Consider $latex F(t) = f(x + t(y-x))$.)