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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2013 • Question 1

Question

Math 6410 • ODE Fall 2013 • Question 1

Let \(f(x,t)\) be continuous for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is \(M\in \mathbb {R}\) such that \[ \norm {f(x,t)-f(y,t)}\le M\norm {x-y}, \qquad \text {for all }(x,t),(y,t)\in \mathbb {R}^n\times \mathbb {R}. \] 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 \(f(x,t)\) be continuous for all \((x,t)\in\R^n\times\R\) and let
\((x_0,t_0)\in\R^n\times\R\). Suppose that there is \(M\in\R\) such that
\[
  \norm{f(x,t)-f(y,t)}\le M\norm{x-y},
  \qquad\text{for all }(x,t),(y,t)\in\R^n\times\R.
\]
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