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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 3

Prove the fixed-point theorem for contracting maps. Let \(X\) be a complete metric space and \(f:X\to X\) be a map such that there exists \(\lambda <1\) so that \[ d(f(x),f(y))\le \lambda d(x,y) \] for all \(x,y\in X\). Then \(f\) has a fixed point.

Raw LaTeX

Prove the fixed-point theorem for contracting maps. Let \(X\) be a complete
metric space and \(f:X\to X\) be a map such that there exists \(\lambda<1\) so that
\[
  d(f(x),f(y))\le\lambda d(x,y)
\]
for all \(x,y\in X\). Then \(f\) has a fixed point.

Topic Tags

Fixed Point Theorems and Borsuk-Ulam