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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2005 • Question 1

Question

Math 6410 • ODE Winter 2005 • Question 1

Contraction Mapping Theorem. Show that the problem \[ \begin {aligned} \dot {x}(t) &= f(x(t))+\frac {1}{a}\int _0^a x(s)\,ds,\quad t\in (0,a),\\ x(0) &= 0, \end {aligned} \] admits a unique solution \(x\in C([0,a];\mathbb {R}^n)\) for \(a>0\) sufficiently small. Assume \(f:\mathbb {R}^n\to \mathbb {R}^n\) satisfies \(\lvert f(x)-f(y)\rvert \le M\lvert x-y\rvert \) for all \(x,y\in \mathbb {R}^n\).

Raw LaTeX

\textbf{Contraction Mapping Theorem.}
Show that the problem
\[
\begin{aligned}
  \dot{x}(t) &= f(x(t))+\frac{1}{a}\int_0^a x(s)\,ds,\quad t\in(0,a),\\
  x(0) &= 0,
\end{aligned}
\]
admits a unique solution \(x\in C([0,a];\R^n)\) for \(a>0\) sufficiently small. Assume
\(f:\R^n\to\R^n\) satisfies \(\lvert f(x)-f(y)\rvert\le M\lvert x-y\rvert\) for all
\(x,y\in\R^n\).

Topic Tags

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