Math 641-600 Fall 2017
Current Assignment
Assignment 11 - Due Wednesday, December 6, 2017.
- Read sections 3.6, 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1. and my notes on
Examples problems for distributions.
- Do the following problems.
- Section 4.1: 4, 7
- Section 4.2: 1, 3, 4
- Section 4.3: 3
- Show that the fixed point found in the Contraction Mapping Theorem
is unique.
- Let $F:C[0,1]\to C[0,1]$ be defined by $F[u](t) :=
\int_0^1(2+st+u(s)^2)^{-1}ds$, $0\le t\le 1$. Let $\| \cdot
\|:=\|\cdot \|_{C[0,1]}$. Let $B_r:=\{u\in C[0,1]\,|\, \|u\|\le
r\}$.
- Show that $F: B_1\to B_{1/2}\subset B_1$.
- Let $D$ be an open subset of a Banach space $V$. We say that a
map $G:D\to V$ is Lipschitz Contraction on $D$ if and only
if there is a constant $0\le \alpha$<1 such that $\|G[u]-G[v]\|\le
\alpha \|u-v\|$. Show that $F$ is a Lipschitz contacvtion on $B_1$,
with Lipschitz constant $\alpha \le 1/2$.
- Show that $F$ has a fixed point in $B_1$.
- Let $L$ be in $\mathcal B (\mathcal H)$.
- Let $A$ and $B$ be in $\mathcal B (\mathcal H)$. Show that
$\|AB\|\le \|A\|\,\|B\|$. Use this to show that $\|L^k\| \le \|L\|^k$,
$k=2,3,\ldots$.
- Let $|\lambda| \|L\|<1$. In class, we showed that the truncation
error $E_n$ satisfies
\[
E_n=\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le
\frac{|\lambda|^n \|L\|^n}{1 - |\lambda| \|L\|}.
\]
Let $L$ be as in
problem 3, HW8. Use the bound on $\|L\|$ in 3(b) to estimate how
many terms of the Neumann expansion would be required to approximate
$(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.2$.
- Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
- Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
-(2y-1)x, & 0 \le x < y \le 1\\
-(2x-1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
- Let $Kf(x) := \int_0^1G(x,y)f(y)dy$. Show that $K$ is a self-adjoint
Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $K$.
- Use (b) and the spectral theory of compact operators to show the
orthonormal set of eigenfunctions for $L$ form a complete set in
$L^2[0,1]$.
Updated 11/29/2017.