Day 2 (September 22 & 23, 2024)

Today, we looked more carefully at Theorem 11.3.1 of the book. Specifically, the implication clause states that

Then there exist positive constant $\epsilon$ and $\delta$ such that, for every

$$f \in U = \{f \in L^\infty(\mathbb{R}^+; \mathbb{R}^n) \ | \ ||f||_{L^\infty(\mathbb{R}^+)} \leq \epsilon\},$$

there is a unique solution $x(f) \in L^\infty(\mathbb{R}^+; \mathbb{R}^n)$ of the half-line equation with $||x(f)||_{L^\infty(\mathbb{R}^+)} \leq \delta$.

In our cases, we have that $f(t) = e_2(t) + \frac{\mu}{\nu} \left(1-e^{\nu e^{-t}}\right).$ From Equation 29 of the paper, we deduce that

$$||f||_{L^\infty(\mathbb{R})} \leq \mu + e^{\mu-1} + \frac{\mu}{\nu}(e^{\nu}-1).$$

From the proof of Theorem 3.2 on page 319, we want to choose $\delta$ so that

$$||b\delta||{L^1(\mathbb{R})} \leq \frac{c}{1+||r||_{L^1(\mathbb{R})}}, \quad |c|<1.$$

We overestimate

\begin{align}||r||{L^1(\mathbb{R})} = \left|\left| \sum{j=1}^m(-1)^{j-1}k^{\ast j}\right|\right|_{L^1(\mathbb{R})}\end{align}

mpham26uchicago / cash_dispersion

Day 2 (September 22 & 23, 2024) #2