Uniformly contractive fixed points in compact metric spaces
Uniformly contractive fixed points in compact metric spaces
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding="application/x-tex">f</mml:annotation></mml:semantics></mml:math></inline-formula>be a continuous self-map on a compact metric space. Equivalent conditions are given for the existence of a uniformly contractive fixed point<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis f Superscript n Baseline x right-arrow w"><mml:semantics><mml:mrow><mml:mi>w</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mi>x</mml:mi><mml:mo stretchy="false">→<!-- → --></mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">w({f^n}x \to w</mml:annotation></mml:semantics></mml:math></inline-formula>uniformly for all<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"><mml:semantics><mml:mi>x</mml:mi><mml:annotation …