Ricardo A. Sáenz

The proof of Schauder's fixed point theorem we saw in class was, sadly, incomplete, as the approximating functions defined on the finite dimensional convex sets were not well defined. Here is a correct proof of the theorem. Theorem (Schauder).   Let $latex V$ be a compact convex subset of the Banach space $latex X$ and $latex f:V\to V$ continuous. Then $latex f$ has a fixed point. Proof. For a given $latex n\in\Z_+$, let $latex x_1, \ldots, x_k\in V$ be such that $latex \displaystyle V\subset\bigcup_i^k B_{1/n}(x_i)$. Such $latex x_i$ exist because $latex V$ is compact. Define, for each $latex i$, the functions $latex \lambda_i:V\to\R$ by $latex \lambda_i(x) = \begin{cases}1/n-||x_i - x||& x\in B_{1/n}(x_i)\\0 & \text{otherwise.}\end{cases}$ The functions $latex \lambda_i$ are continuous and $latex \sum_i \lambda_i(x)\not=0$ for every $latex x\in V$. Thus, if we define $latex \pi_n(x) = \dfrac{\sum \lambda_i(x)x_i}{\sum\lambda_i(x)}$, $latex \pi_n$ maps $latex V$ into the ...

Due October 30 Problem 1 Let $latex A\subset X$ be connected. Then $latex \bar A$ is connected. Problem 2 If $latex A\subset X$ is connected and $latex A\subset B\subset \bar A$, then $latex B$ is connected. Problem 3 $latex X$ is connected if and only if every continuous $latex f:X\to Y$ into a discrete space $latex Y$ is constant. Problem 4 If $latex A$ is convex, then $latex \bar A$ is convex. Problem 5 Prove $latex B_r(x_0)$ is convex from the fact that $latex B_1(0)$ is convex.

Due October 16 Problem 1 If $latex X$ is discrete, then $latex (\mathcal C_X,d_H)$ is discrete. Problem 2 Let $latex A\subset X$ be a finite set of isolated points in $latex X$. Then $latex A$ is isolated in $latex \mathcal C_X$. Problem 3 Let $latex A_n$ be a decreasing sequence of nonempty compact sets in $latex X$. Then $latex \displaystyle \lim A_n = \bigcap_{n\ge 1} A_n$ in $latex \mathcal C_H$. Problem 4 Let $latex f_1, \ldots, f_N:X\to X$ be contractions in the complete metric space $latex X$, and $latex K$ the self-similar set with respect to the $latex f_i$. If $latex A\subset X$ is compact and $latex A\subset f_1(A)\cup\ldots\cup f_N(A)$, then $latex A\subset K$. Problem 5 Under the same hypothesis of the previous problem, if $latex B\subset X$ is nonempty and $latex B\supset f_1(B)\cup\ldots\cup f_N(B)$, then $latex \bar B \supset K$.

Due October 9 Problem 1 If $latex f:X\to Y$ is a Lipschitz function, then it is uniformly continuous. The function $latex x\to\sqrt x$ is uniformly continuous on $latex [0,\infty)$ but not Lipschitz. Problem 2 Let $latex P,Q,f:[-1,1]\to\R$ continuous and $latex a,b\in\R$. Then the IVP $latex \begin{cases} u''(x)+P(x)u'(x)+Q(x)u(x)=f(x)\\u(0)=a,\qquad u'(0)=b\end{cases}$ has a unique solution in a neighborhood of $latex x=0$. Problem 3 Consider the integral operator $latex \Phi:C([-1,1])\to C([-1,1])$ given by $latex \displaystyle\Phi(x)(t) = 1 + 2\int_0^t sx(s) ds,$ for $latex x\in C([-1,1])$. Starting from the constant function $latex x_0(t)=1$, explicitly calculate the iterations of $latex x_{n+1} = \Phi(x_n)$, and verify that $latex x_n$ is the $latex n$-th Taylor polynomial of $latex e^{t^2}$ around $latex t=0$. Problem 4 For $latex T\in L(\R^l,\R^m)$, $latex ||T||_L = \sup\{ |Tx| : x\in\bar B_1(0)\}$. Problem 5 Let $latex B_r(x_0)$ be a ball in $latex \R^l$ and...