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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2012 • Question 1

Question

Math 6410 • ODE Spring 2012 • Question 1

Let \(f(t,x)\) be continuous function on \(\mathbb {R}\times \mathbb {R}^n\) that satisfies \[ \norm {f(t,x_1)-f(t,x_2)}\le \norm {x_2-x_1} \] for all \(t\in \mathbb {R}\) and \(x_1,x_2\in \mathbb {R}^n\). Let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Raw LaTeX

Let \(f(t,x)\) be continuous function on \(\R\times\R^n\) that satisfies
\[
  \norm{f(t,x_1)-f(t,x_2)}\le \norm{x_2-x_1}
\]
for all \(t\in\R\) and \(x_1,x_2\in\R^n\). Let \(x_0\in\R^n\) be a point. Show that there
is and \(\epsilon>0\) and a unique function \(y\in C^1([0,\epsilon],\R^n)\) that satisfies
the initial value problem
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(x,t),\\
    x(0)=x_0.
  \end{cases}
\]

Topic Tags

Existence and Uniqueness Theorems Gronwall Inequality and Continuous Dependence Point, Continuous, and Residual Spectrum