Math 641-600 — Fall 2016

Assignment 11 - Due Friday, December 2, 2016.

• 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: 1(b) (Skip the plots.), 4, 6
  2. Section 4.2: 1, 4, 8
  3. Section 4.3: 3
  4. 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 continuous on $D$ if and only if there is a constant $0<\alpha$ such that $\|G[u]-G[v]\|\le \alpha \|u-v\|$. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha \le 1/2$.
     3. Show that $F$ has a fixed point in $B_1$.

  5. Let $L$ be in $\mathcal B (\mathcal H)$.
     1. Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
     2. Let $|\lambda| \|L\|<1$. Show that \[ \big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^n \|L\|^n}{1 - |\lambda| \|L\|}. \]
     3. Let $L$ be as in problem 6, HW8. Use the bound on $\|L\|$ in 6(b) of this problem to estimate how many terms of the Neumann expansion it would require to approximate $(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.2$.

  6. 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 $L^2[0,1]$.