Homework 9

Due Date: Friday, April 22

1. Let \(V\) be a Banach space. A function \(F\colon V\to V\) is called a contraction map if there exists an \(r\in(0,1)\) so that \[ \|F(\mathbf{v}) - F(\mathbf{w})\| \,\leq\, r\,\|\mathbf{v} - \mathbf{w}\| \] for all \(\mathbf{v},\mathbf{w}\in V\). Prove that if \(F\colon V\to V\) is a contraction map then there exists a unique point \(\mathbf{v}_0 \in V\) such that \(F(\mathbf{v}_0) = \mathbf{v}_0\).
2. Consider an initial value problem of the form \[ y' \,=\, g(y),\quad y(0) = 0 \] where \(g\colon\mathbb{R}\to\mathbb{R}\) is continuous, and suppose there exists a constant \(K>0\) so that \[ |g(a)-g(b)| \,\leq\, K|a-b| \] for all \(a,b\in\mathbb{R}\).
   1. Let \(0<\epsilon < 1/K\), and let \(C([0,\epsilon])\) denote the Banach space of all continuous functions \([0,\epsilon]\to \mathbb{R}\) under the \(L^\infty\)-norm. Prove that the function \(I\colon C([0,\epsilon])\to C([0,\epsilon])\) defined by \[ I(f)(x) \,=\, \int_0^x \!g(f(t))\,dt \] is a contraction map.
   2. Use part (a) together with question (1) to prove that the given initial value problem has a unique solution on the interval \([0,\epsilon]\).
3. Let \(f\colon \ell^1 \to \mathbb{R}\) be a continuous linear function. Prove that there exists a bounded sequence \(\{a_n\}\) of real numbers such that \[ f(\mathbf{v}) \,=\, \sum_{n=1}^\infty a_n v_n \] for every vector \(\mathbf{v} = (v_1,v_2,\ldots)\) in \(\ell^1\).