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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2003 • Question 2

Question

Math 6410 • ODE Fall 2003 • Question 2

Picard’s Existence Theorem. Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is a continuous function \(m(t)\in C(\mathbb {R})\) such that \(\lvert f(x,t)\rvert \le m(t)(1+\lvert x\rvert )\) for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\). Show that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \] has a unique solution \(y\in C^1(\mathbb {R},\mathbb {R}^n)\) which exists for all time.

Raw LaTeX

\textbf{Picard's Existence Theorem.}
Suppose \(f(x,t)\in C^1(\R^n\times\R,\R^n)\) and \((x_0,t_0)\in\R^n\times\R\). Suppose that
there is a continuous function \(m(t)\in C(\R)\) such that
\(\lvert f(x,t)\rvert\le m(t)(1+\lvert x\rvert)\) for all
\((x,t)\in\R^n\times\R\). Show that the initial value problem
\[
\begin{cases}
  \dfrac{dx}{dt}=f(x,t),\\
  x(t_0)=x_0.
\end{cases}
\]
has a unique solution \(y\in C^1(\R,\R^n)\) which exists for all time.

Topic Tags

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