Ir al contenido principal

## Due October 9

### Problem 1

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

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

### Problem 2

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

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

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

### Problem 3

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

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

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

### Problem 4

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

### Problem 5

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

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

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