Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2004 • Question 1

Question

Math 6410 • ODE Winter 2004 • Question 1

Picard’s Existence Theorem.

Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Show that there is an \(\epsilon >0\) so that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0 \end {cases} \] has a unique solution \(y\in C^1([t_0-\epsilon ,t_0+\epsilon ],\mathbb {R}^n)\).
Suppose that the hypothesis on \(f\) was replaced by \(f(x,t)\in C^0(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\). Would all or part of the conclusions in (a.) continue to hold?

\textbf{Picard's Existence Theorem.}
\begin{enumerate}
\item[(a.)] Suppose \(f(x,t)\in C^1(\R^n\times\R,\R^n)\) and
\((x_0,t_0)\in\R^n\times\R\). Show that there is an \(\epsilon>0\) so that the initial value
problem
\[
\begin{cases}
  \dfrac{dx}{dt}=f(x,t),\\
  x(t_0)=x_0
\end{cases}
\]
has a unique solution \(y\in C^1([t_0-\epsilon,t_0+\epsilon],\R^n)\).

\item[(b.)] Suppose that the hypothesis on \(f\) was replaced by
\(f(x,t)\in C^0(\R^n\times\R,\R^n)\). Would all or part of the conclusions in (a.)
continue to hold?
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Existence and Uniqueness Theorems Poincare Maps and Periodic Solutions