Math 641-600 — Fall 2024

Assignment 12 - Due Monday, December 2, 2024.

• Read sections 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1 and my notes on and my notes on example 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. 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. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha \le 1/2$.
     3. Does $F$ have a fixed point in $B_1$? Why or why not? Be brief.

  5. 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 $Kf(x) := \int_0^1 g(x,y)f(y)dy$. Show that $G$ is a self-adjoint Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $G$.
     3. Use (b) and the spectral theory of compact operators to show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$.

  6. Suppose that $Lu= u''+\lambda u$, with Dom$(L)=\{u\in L^2[0,\infty):Lu\in L^2[0,\infty)\ \text{and}\ u(0)=0\}$, where $\lambda\in \mathbb C \setminus [0,\infty)$. In addition, choose $\text{Im}\sqrt{\lambda}>0$. Show that the Green's function for $L$ is given by \[ g(x,y, \lambda)=\begin{cases} \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} xe^{i\sqrt{\lambda} y} & 0\le x\le y<\infty\\ \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} y)e^{i\sqrt{\lambda}x} & 0\le y\le x<\infty \end{cases}\] Hint: follow the procedure used in Keener, pg. 150, to solve a similar problem.

Updated 11/24/2024.