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This is an example from a text to show that this mapping does not have a fixed point because it is contractive but not a contraction:

I am not sure what the difference is between contractive and contraction. Doesn't the function satisfy all the requirements of the fixed point theorem/

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2 Answers

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A contraction map is a map $f$ such that there exists a $0 \le k < 1$ such that$$|f(x) - f(y)| \le k|x - y|$$for all $x$ and $y$ in the domain.

A contractive map, also called a shrinking map, is a map $f$ for which$$|f(x) - f(y)| < |x - y|$$for all $x$ and $y$ in the domain. Not all contractive maps are contraction maps, as the example points out. Although contraction maps must have a (unique) fixed point, contractive maps may not have any fixed point.

On compact metric spaces, contractive and contraction maps are the same.

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another example, the hyperbola (upper branch)$$ 3y^2 - 4yx + x^2 = 5 $$ or$$ y=\frac{2x+\sqrt{15+x^2}}{3}$$When $x\geq 0,$ we see $\sqrt{15+x^2} > x$ so that $2x+\sqrt{15+x^2} > 3x$

The trouble occurs because the derivative gets arbitrarily close to $1$
$$ y'=\frac{x+2\sqrt{15+x^2}}{3\sqrt{15+x^2}}$$as $x$ increases without bound

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