1
$\begingroup$

It's a rather easy exercise to construct a non-complete metric space $X$ such that any continuous mapping $f:X\to X$ has a fixed point. However, I'd like to have a reference to such an example.

$\endgroup$

1 Answer 1

1
$\begingroup$

For instance, section 4 of the paper

Suzuki, Tomonari; Takahashi, Wataru. Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 8 (1996), no. 2, 371–382 (1997); MR1483635

contains such an example.

For more information see Converse to Banach's fixed point theorem? at MathOverflow, Contraction mapping in an incomplete metric space at math.SE,

and related pages pointed out at these sites.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.