Math 461 Bard College

# 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 $$\ell^1$$.