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.

1. Section 4.1: 4, 7
2. Section 4.2: 1, 3, 4
3. Section 4.3: 3
4. Show that the fixed point found in the Contraction Mapping Theorem is unique.
5. 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 $B_r:=\{u\in C[0,1]\,|\, \|u\|\le r\}$.
1. Show that $F: B_1\to B_{1/2}\subset B_1$.
2. 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 $\alpha \le 1/2$.
3. Show that $F$ has a fixed point in $B_1$.

6. Let $L$ be in $\mathcal B (\mathcal H)$.
1. Let $A$ and $B$ be in $\mathcal B (\mathcal H)$. Show that $\|AB\|\le \|A\|\,\|B\|$. Use this to show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
2. 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 $|\lambda|\le 0.2$.

7. Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
1. Show that the Green's function for this problem is $G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right.$
2. 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 $K$.
3. 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.