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Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Fall 2004 • Question 1

Question

Math 6410 • ODE Fall 2004 • Question 1

Contraction Mapping. Consider the initial value problem \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\) satisfies for all \(x,y\in \mathbb {R}^n\) \[ \lvert f(x)-f(y)\rvert \le (1+\lvert x\rvert +\lvert y\rvert )\lvert x-y\rvert . \] Use the contraction mapping principle to find a condition on \(\lvert x_0\rvert \) which guarantees local existence and uniqueness of solutions. How does the local time interval upon which existence is obtained depend on \(\lvert x_0\rvert \)?

Raw LaTeX

\textbf{Contraction Mapping.}
Consider the initial value problem
\[
\begin{aligned}
  \dot{x} &= f(x),\\
  x(0) &= x_0,
\end{aligned}
\]
where \(f\) satisfies for all \(x,y\in\R^n\)
\[
  \lvert f(x)-f(y)\rvert\le(1+\lvert x\rvert+\lvert y\rvert)\lvert x-y\rvert.
\]
Use the contraction mapping principle to find a condition on \(\lvert x_0\rvert\) which
guarantees local existence and uniqueness of solutions. How does the local time interval
upon which existence is obtained depend on \(\lvert x_0\rvert\)?

Topic Tags

Contraction Mapping and Nonlinear Equations Existence and Uniqueness Theorems Picard Iteration and Contraction Mapping Topological Degree and Fixed Point Methods