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Math 6410 • ODE

Fall 2007 • Question 1

Question

Math 6410 • ODE Fall 2007 • Question 1

Let \[ f:\mathbb {R}\times \mathbb {R}^N\to \mathbb {R}^N \] be a continuous mapping which satisfies \begin{equation} \lvert f(t,x)-f(t,y)\rvert \le L(t)\lvert x-y\rvert , \quad \forall t\in \mathbb {R},\quad \forall x,y\in \mathbb {R}^N, \tag {0.1} \end{equation} where \[ L:\mathbb {R}\to [0,\infty ) \] is a continuous function and where \(\lvert \cdot \rvert \) is a norm in \(\mathbb {R}^N\). Consider the initial value problem \begin{equation} \begin {aligned} x'&=f(t,x)\\ x(t_0)&=x_0,\quad t_0\in \mathbb {R},\quad x_0\in \mathbb {R}^N. \end {aligned} \tag {0.2} \end{equation}

State the fundamental existence uniqueness theorem for solutions of (0.2).
State a general principle about the existence intervals for solutions of (0.2) and the behavior of solutions at the endpoints of these intervals.
Use the above and (0.1) (note: \(\lvert f(t,x)\rvert \le \lvert f(t,x)-f(t,0)\rvert +\lvert f(t,0)\rvert \)) to show that solutions of (0.2) are defined on all of \(\mathbb {R}\).

Let
\[
  f:\R\times\R^N\to\R^N
\]
be a continuous mapping which satisfies
\begin{equation}
  \lvert f(t,x)-f(t,y)\rvert\le L(t)\lvert x-y\rvert,
  \quad \forall t\in\R,\quad \forall x,y\in\R^N,
  \tag{0.1}
\end{equation}
where
\[
  L:\R\to[0,\infty)
\]
is a continuous function and where \(\lvert\cdot\rvert\) is a norm in \(\R^N\).
Consider the initial value problem
\begin{equation}
\begin{aligned}
  x'&=f(t,x)\\
  x(t_0)&=x_0,\quad t_0\in\R,\quad x_0\in\R^N.
\end{aligned}
\tag{0.2}
\end{equation}
\begin{enumerate}
\item State the fundamental existence uniqueness theorem for solutions of (0.2).

\item State a general principle about the existence intervals for solutions of
(0.2) and the behavior of solutions at the endpoints of these intervals.

\item Use the above and (0.1) (note:
\(\lvert f(t,x)\rvert\le\lvert f(t,x)-f(t,0)\rvert+\lvert f(t,0)\rvert\)) to show
that solutions of (0.2) are defined on all of \(\R\).
\end{enumerate}

Topic Tags

Existence and Uniqueness Theorems Fundamental Solutions Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions